diff --git a/Gaussians.ipynb b/Gaussians.ipynb index d20c160..5f1e9a3 100644 --- a/Gaussians.ipynb +++ b/Gaussians.ipynb @@ -1,7 +1,7 @@ { "metadata": { "name": "", - "signature": "sha256:7987517a7cb3fbbd21339999a44f481804ef1719e02dea0e02c2456b42d2d036" + "signature": "sha256:22eea384355b20ece229d2669aa6a64132be5fada2c2565b3598ab6846b58610" }, "nbformat": 3, "nbformat_minor": 0, @@ -48,7 +48,8 @@ ], "language": "python", "metadata": {}, - "outputs": [] + "outputs": [], + "prompt_number": "" }, { "cell_type": "code", @@ -63,7 +64,8 @@ ], "language": "python", "metadata": {}, - "outputs": [] + "outputs": [], + "prompt_number": "" }, { "cell_type": "markdown", @@ -107,7 +109,8 @@ ], "language": "python", "metadata": {}, - "outputs": [] + "outputs": [], + "prompt_number": "" }, { "cell_type": "markdown", @@ -135,7 +138,8 @@ ], "language": "python", "metadata": {}, - "outputs": [] + "outputs": [], + "prompt_number": "" }, { "cell_type": "markdown", diff --git a/Introduction.ipynb b/Introduction.ipynb index 75e3719..86e2612 100644 --- a/Introduction.ipynb +++ b/Introduction.ipynb @@ -1,6 +1,7 @@ { "metadata": { - "name": "" + "name": "", + "signature": "sha256:ba2750345b8a777403243fcf1a80cb6a58d7ee1074b9095134150e1679648635" }, "nbformat": 3, "nbformat_minor": 0, @@ -66,7 +67,8 @@ "input": [], "language": "python", "metadata": {}, - "outputs": [] + "outputs": [], + "prompt_number": "" } ], "metadata": {} diff --git a/Kalman Filters.ipynb b/Kalman Filters.ipynb index ecc08cd..b11dd91 100644 --- a/Kalman Filters.ipynb +++ b/Kalman Filters.ipynb @@ -1,1107 +1,888 @@ -{ - "metadata": { - "name": "" - }, - "nbformat": 3, - "nbformat_minor": 0, - "worksheets": [ - { - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "#Kalman Filters\n", - "\n", - "\n", - "Now that we understand the histogram filter and gaussians we are prepared to implement a 1D Kalman filter. We will do this exactly as we did the histogram filter - rather than going into the theory we will just develop the code step by step. \n", - "\n", - "#Tracking A Dog\n", - "\n", - "As in the histogram chapter we will be tracking a dog in a long hallway at work. However, in our latest hackathon someone created an RFID tracker that provides a reasonable accurate position for our dog. Suppose the hallway is 100m long. The sensor returns the distance of the dog from the left end of the hallway. So, 23.4 would mean the dog is 23.4 meters from the left end of the hallway.\n", - "\n", - "Naturally, the sensor is not perfect. A reading of 23.4 could correspond to a real position of 23.7, or 23.0. However, it is very unlikely to correspond to a real position of say 47.6. Testing during the hackathon confirmed this result - the sensor is accurate, and while it had errors, the errors are small.\n", - "\n", - "Implementing and/or robustly modelling an RFID system is beyond the scope of this book, so we will write a very simple model. We will start with a simulation of the dog moving from left to right at a constant speed with some random noise added. " - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "import numpy.random as random\n", - "import math\n", - "\n", - "class DogSensor(object):\n", - " \n", - " def __init__(self, x0=0, velocity=1, noise=0.0):\n", - " \"\"\" x0 - initial position\n", - " velocity - (+=right, -=left)\n", - " noise - scaling factor for noise, 0== no noise\n", - " \"\"\"\n", - " self.x = x0\n", - " self.velocity = velocity\n", - " self.noise = math.sqrt(noise)\n", - "\n", - " def sense(self):\n", - " self.x = self.x + self.velocity\n", - " return self.x + random.randn() * self.noise\n" - ], - "language": "python", - "metadata": {}, - "outputs": [], - "prompt_number": 2 - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The constructor (__init__()) initializes the DogSensor class with an initial position (x0), velocity (vel), and an noise scaling factor. The *sense()* function has the dog move by the set velocity and returns its new position, with noise added. If you look at the code for *sense()* you will see a call to *numpy.random.randn()*. This returns a number sampled from a normal distribution with a mean of 0.0. Let's look at some example output for that.\n", - "\n" - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "for i in range(20):\n", - " print (\"%.4f\" % random.randn())," - ], - "language": "python", - "metadata": {}, - "outputs": [ - { - "output_type": "stream", - "stream": "stdout", - "text": [ - "-1.9126 0.6913 -0.5946 -1.2480 0.2278 0.7453 0.0081 0.6740 0.5837 0.5784 -1.2489 0.1916 -1.1272 -0.0133 -0.3466 0.9102 0.8873 2.1069 0.8212 0.6930\n" - ] - } - ], - "prompt_number": 3 - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "You should see a sequence of numbers near 0, some negative and some positive. Most are probably between -1 and 1, but a few might lie somewhat outside that range. This is what we expect from a normal distribution - values are clustered around the mean, and there are fewer values the further you get from the mean.\n", - "\n", - "Okay, so lets look at the output of the *DogSensor* class. We will start by setting the noise to 0 to check that the class does what we think it does" - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "%matplotlib inline\n", - "\n", - "dog = DogSensor (noise=0.0)\n", - "xs = []\n", - "for i in range(10):\n", - " x = dog.sense()\n", - " xs.append(x)\n", - " print(\"%.4f\" % x),\n", - "plot(xs)\n", - "show()" - ], - "language": "python", - "metadata": {}, - "outputs": [ - { - "output_type": "stream", - "stream": "stdout", - "text": [ - "1.0000 2.0000 3.0000 4.0000 5.0000 6.0000 7.0000 8.0000 9.0000 10.0000" - ] - }, - { - "metadata": {}, - "output_type": "display_data", - "png": 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- "text": [ - "" - ] - }, - { - "output_type": "stream", - "stream": "stdout", - "text": [ - "\n" - ] - } - ], - "prompt_number": 4 - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The constructor initialized the dog at position 0 with a velocity of 1 (move 1.0 to the right). So we would expect to see an output of 1..10, and indeed that is what we see. If you thought the correct answer should have been 0..9 recall that *sense()* returns the dog's position *after* updating his position, so the first postion is 0.0 + 1, or 1.0.\n", - "\n", - "Now let's inject some noise in the signal." - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "def test_sensor(noise_scale):\n", - " dog = DogSensor(noise=noise_scale)\n", - "\n", - " xs = []\n", - " for i in range(100):\n", - " x = dog.sense()\n", - " xs.append(x)\n", - " p1, = plot(xs, c='b')\n", - " p2, = plot([0,99],[1,100], 'r--')\n", - " xlabel('time')\n", - " ylabel('pos')\n", - " ylim([0,100])\n", - " title('noise = ' + str(noise_scale))\n", - " legend([p1, p2], ['sensor', 'actual'], loc=2)\n", - " show()\n", - " \n", - "test_sensor(4.0)" - ], - "language": "python", - "metadata": {}, - "outputs": [ - { - "metadata": {}, - "output_type": "display_data", - "png": 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- "text": [ - "" - ] - } - ], - "prompt_number": 5 - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "**note**:*numpy uses a random number generator to generate the normal distribution samples. The numbers I see as I write this are unlikely to be the ones that you see. If you run the cell above multiple times, you should get a slightly different result each time. I could use numpy.random.seed(some_value) to force the results to be the same each time. This would simplify my explanations in some cases, but would ruin the interactive nature of this chapter. To get a real feel for how normal distributions and Kalman filters work you will probably want to run cells several times, observing what changes, and what stays roughly the same.*\n", - "\n", - "So the output of the sensor should be a wavering blue line drawn over a dotted red line. The dotted red line shows the actual position of the dog, and the blue line is the noise signal produced by the simulated RFID sensor. Please note that the red dotted line was manually plotted - we do not yet have a filter that recovers that information! \n", - "\n", - "If you are running this in an interactive IPython Notebook, I strongly urge you to run the script several times in a row. You can do this by putting the cursor in the cell containing the Python code and pressing Ctrl+Enter. Each time it runs you should see a different jagged blue line wavering over the top of the dotted red line.\n", - "\n", - "I also urge you to adjust the noise setting to see the result of various values. However, since you may be reading this in a read only notebook, I will show two extreme examples. The first plot shows the noise set to 100.0, and the second shows noise set to 0.5." - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "test_sensor(100.0)" - ], - "language": "python", - "metadata": {}, - "outputs": [ - { - "metadata": {}, - "output_type": "display_data", - "png": 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BKiuNZ+lERFB9gzyjVSMU8oymG2+kvNA28MorpkeSmkqPtVUoAGDdOsP2JR0Z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\neKQ/wa5XLxpWxK4nhmE6Gh3C9fThh/SEHxhINQw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- "text": [ - "" - ] - } - ], - "prompt_number": 7 - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "You may not have a full understanding of the exact *meaning* of a noise value of 100.0, but as it turns out if you multiply *randn()* with a number $n$, the result is just a normal distribution with $\\sigma = \\sqrt{n}$. So the example with noise = 100 is using the normal distribution $N(0,100)$. Recall the notation for a normal distribution is $N(\\mu,\\sigma^2)$. If the square root is confusing, recall that normal distributions use $\\sigma^2$ for the variance, and $\\sigma$ is the standard deviation, which we do not use in this book. *dog_sensor.__init__()* takes the square root of the noise setting so that the *noise * randn()* call properly computes the normal distribution. " - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "#### Math with Gaussians\n", - "\n", - "Let's say we believe that our dog is at 23m, and the variance is 5 ($N(23,5)$). We can represent that in a plot:\n" - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "import gaussian\n", - "gaussian.norm_plot(23, 5)" - ], - "language": "python", - "metadata": {}, - "outputs": [ - { - "metadata": {}, - "output_type": "display_data", - "png": 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- "text": [ - "" - ] - } - ], - "prompt_number": 8 - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This corresponds to a fairly inexact belief. While we believe that the dog is at 23, note that roughly 21 to 25 are quite likely as well. Let's assume for the moment our dog is standing still, and we query the sensor again. This time it returns 23.2 as the position. Can we use this additional information to improve our estimate of the dog's position.\n", - "\n", - "Intuition suggests 'yes'. Consider: if we read the sensor 100 times and each time it returned a value between 21 and 25, all centered around 23, we should be very confident that the dog is somewhere very near 23. Of course, a different physical interpertation is possible. Perhaps our dog was randomly wandering back and forth in a way that exactly emulated a normal distribution. But that seems extremely unlikely - I certainly have never seen a dog do that. So the only reasonable assumption is that the dog was mostly standing still at 23.0.\n", - "\n", - "Let's look at this in a plot:" - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "dog = DogSensor(23, 0, 5)\n", - "xs = range(100)\n", - "ys = []\n", - "for i in xs:\n", - " ys.append(dog.sense())\n", - " \n", - "plot(xs,ys)\n", - "show()" - ], - "language": "python", - "metadata": {}, - "outputs": [ - { - "metadata": {}, - "output_type": "display_data", - "png": 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2BaKdOmCdq3d2Uj17R4dxPtzbGxH1eNWp63WUAvac+sAA/YhpRKzunoVTtyPq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\nv1hl6sEgPU/boSqTqRuJOmD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- "text": [ - "" - ] - } - ], - "prompt_number": 9 - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Eyeballing this confirms our intuition - no dog moves like this. However, noisy sensor data certainly looks like this. So let's proceed to see how we might solve this mathematically. But how?\n", - "\n", - "\n", - "Recall the histogram code for adding a measurement to a pre-existing belief:\n", - "\n", - " def sense(pos, measure, p_hit, p_miss):\n", - " q = array(pos, dtype=float)\n", - " for i in range(len(hallway)):\n", - " if hallway[i] == measure:\n", - " q[i] = pos[i] * p_hit\n", - " else:\n", - " q[i] = pos[i] * p_miss\n", - " normalize(q)\n", - " return q\n", - " \n", - "Note that the algorithm is essentially computing:\n", - "\n", - " new_belief = old_belief * measurement * sensor_error\n", - " \n", - "The measurement term might not be obvious, but recall that measurement in this case was always 1 or 0, and so it was left out for convience. \n", - " \n", - "If we are implementing this with gaussians, we might expect it to be implemented as:\n", - "\n", - " new_gaussian = measurement * old_gaussian\n", - " \n", - "where measurement is a gaussian returned from the sensor. But does that make sense? Can we multiply gaussians? If we multiply a gaussian with a gaussing is the result another gaussian, or something else?\n", - "\n", - "Of course the answer is 'yes', or this chapter would be for naught. It is not particularly difficult to perform the algebra to derive the equation for multiplying two gaussians, but I will just present the result:\n", - "$$ N({\\mu}_1, {{\\sigma}_1}^2)*N({\\mu}_2, {{\\sigma}_2}^2) = N(\\frac{{\\sigma}_1 {\\mu}_2 + {\\sigma}_2 {\\mu}_1}{{\\sigma}_1 + {\\sigma}_2},\\frac{1}{\\frac{1}{{\\sigma}_1} + \\frac{1}{{\\sigma}_2}}) $$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Let's immediately look at some plots of this to inform our intuition about this result. First, let's look at the result of multiplying $N(23,5) $ to itself. This corresponds to getting 23.0 as the sensor value twice in a row." - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "import gaussian\n", - "def multiply(mu1, sig1, mu2, sig2):\n", - " m = (sig1*mu2 + sig2*mu1) / (sig1+sig2)\n", - " s = 1. / (1./sig1 + 1./ sig2)\n", - " return (m,s)\n", - "\n", - "\n", - "xs = np.arange(16, 30, 0.1)\n", - "\n", - "\n", - "m1,s1 = 23, 5\n", - "m, s = multiply(m1,s1,m1,s1)\n", - "\n", - "ys = [gaussian.gaussian(x,m1,s1) for x in xs]\n", - "p1, =plot (xs,ys)\n", - "\n", - "ys = [gaussian.gaussian(x,m,s) for x in xs]\n", - "p2, = plot (xs,ys)\n", - "\n", - "legend([p1,p2],['original', 'multiply'])\n", - "show()" - ], - "language": "python", - "metadata": {}, - "outputs": [ - { - "metadata": {}, - "output_type": "display_data", - "png": 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yHYsoOTW+ZYXwISE9ATYGNtB5bovJk4Fdu7jmCSI5DQ1g0yagUyfg7bcFXK+c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\nlxEIBAgICICnpyd+++03WeY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- "text": [ - "" - ] - } - ], - "prompt_number": 10 - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The result is either amazing or what you would expect, depending on your state of mind. I must admit I vacillate freely between the two! Note that the result of the multiplation is taller and narrow than the original gaussian. If we think of the gaussians as two measurement, this makes sense. If I measure twice and get the same value, I should be more confident in my answer than if I just measured once. \"Measure twice, cut once\" is a useful saying and practice due to this fact! \n", - "\n", - "Now let's multiply two gaussians (or equivelently, two measurements) that are partially separated. What do you think the result will be? Let's find out:" - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "xs = np.arange(16, 30, 0.1)\n", - "\n", - "\n", - "m1,s1 = 23, 5\n", - "m2,s2 = 25, 5\n", - "m, s = multiply(m1,s1,m2,s2)\n", - "\n", - "ys = [gaussian.gaussian(x,m1,s1) for x in xs]\n", - "p1, = plot (xs,ys)\n", - "\n", - "ys = [gaussian.gaussian(x,m2,s2) for x in xs]\n", - "p2, = plot (xs,ys)\n", - "\n", - "ys = [gaussian.gaussian(x,m,s) for x in xs]\n", - "p3, = plot(xs,ys)\n", - "legend([p1,p2,p3],['measure 1', 'measure 2', 'multiply'])\n", - "show()" - ], - "language": "python", - "metadata": {}, - "outputs": [ - { - "metadata": {}, - "output_type": "display_data", - "png": 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KDFeurMSO2dnsLO7t26KNl+RIc9B/d3+YVzXHpr6bIHnvIwkRK6/w+jW72lZX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\nDV0Lnzs+OP/VeRhW0oyyt5xqVKsGHD0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- "text": [ - "" - ] - } - ], - "prompt_number": 11 - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Another beautiful result! If I handed you a measuring tape and asked you to measure the distance from table to a wall, and you got 23m, and then a friend make the same measurement and got 25m, your best guess must be 24m. \n", - "\n", - "That is fairly counter-intuitive, so let's consider it further. Perhaps a more reasonable assumption would be that either you or your coworker just made a mistake, and the true distance is either 23 or 25, but certainly not 24. Surely that is possible. However, suppose the two measurements you reported as 24.01 and 23.99. Surely you would agree that in this case the best guess for the correct value is 24? Which interpretation we choose depends on the properties of the sensors we are using. Humans make galling mistakes, physical sensors do not. \n", - "\n", - "This topic is fairly deep, and I will explore it once we have completed our Kalman filter. For now I will merely say that the Kalman filter requires the interpretation that measurements are accurate, with gaussian noise, and that a large error caused by misreading a measuring tape is not gaussian noise. So perhaps you would be justified in thinking that a histogram filter will perform better for the human readings, and the Kalman filter will perform better with sensor readings that have gaussian noise.\n", - "\n", - "For now I ask that you trust me. The math is correct, so we have no choice but to accept it and use it. We will see how the Kalman filter deals with movements vs error very soon. 24 is the correct answer to this problem." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "#### Implementing Sensing\n", - "\n", - "Recall the histogram filter uses a numpy array to encode our belief about the position of our dog at any time. That array stored our belief that the dog was in any position in the hallway using 10 positions. This was very crude, because with a 100m hallway that corresponded to positions 10m apart. It would have been trivial to expand the number of positions to say 1,000, and that is what we would do if using it for a real problem. But the problem remains that the distribution is discrete and multimodal - it can express strong belief that the dog is in two positions at the same time.\n", - "\n", - "Therefore, we will use a single gaussian to reflect our current belief of the dog's position. Gaussians extend to infinity on both sides of the mean, so the single gaussian will cover the entire hallway. They are unimodal, and seem to reflect the behavior of real-world sensors - most errors are small and clustered around the mean. Here is the entire implementation of the sense function for a Kalman filter:" - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "def sense(mu, sigma, measurement, measurement_sigma):\n", - " return multiply(mu, sigma, measurement, measurement_sigma)" - ], - "language": "python", - "metadata": {}, - "outputs": [], - "prompt_number": 12 - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Kalman filters are supposed to be hard! But this is very short and straightforward. All we are doing is multiplying the gaussian that reflects our belief of where the dog was with the new measurement. Perhaps this would be clearer if we used more specific names:" - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "def sense_dog(dog_pos, dog_sigma, measurement, measurement_sigma):\n", - " return multiply(dog_pos, dog_sigma, measurement, measurement_sigma)" - ], - "language": "python", - "metadata": {}, - "outputs": [], - "prompt_number": 13 - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "That is less abstract, which perhaps helps with comprehension, but it is poor coding practice. We are writing a Kalman filter that works for any problem, not just tracking dogs in a hallway, so we don't use variable names with 'dog' in them. Still, the *sense_dog()* function should make what we are doing very clear. \n", - "\n", - "Let's look at an example. We will suppose that our current belief for the dog's position is $N(2,5)$. Don't worry about where that number came from. It may appear that we have a chicken and egg problem, in that how do we know the position before we sense it, but we will resolve that shortly. We will create a *DogSensor* object initialized to be at position 0.0, and with no velocity, and modest noise. This corresponds to the dog standing still at the far left side of the hallway. Note that we mistakenly believe the dog is at postion 2.0, not 0.0." - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "dog = DogSensor(velocity=0, noise=1)\n", - "\n", - "pos,s = 2, 5\n", - "for i in range(20):\n", - " pos,s = sense(pos, s, dog.sense(), 5)\n", - " print 'time:', i, 'position = ', \"%.3f\" % pos, 'variance = ', \"%.3f\" % s\n", - "\n" - ], - "language": "python", - "metadata": {}, - "outputs": [ - { - "output_type": "stream", - "stream": "stdout", - "text": [ - "time: 0 position = 0.091 variance = 2.500\n", - "time: 1 position = -0.441 variance = 1.667\n", - "time: 2 position = -0.577 variance = 1.250\n", - "time: 3 position = -0.360 variance = 1.000\n", - "time: 4 position = -0.401 variance = 0.833\n", - "time: 5 position = -0.436 variance = 0.714\n", - "time: 6 position = -0.272 variance = 0.625\n", - "time: 7 position = -0.119 variance = 0.556\n", - "time: 8 position = -0.083 variance = 0.500\n", - "time: 9 position = -0.032 variance = 0.455\n", - "time: 10 position = -0.088 variance = 0.417\n", - "time: 11 position = -0.103 variance = 0.385\n", - "time: 12 position = -0.082 variance = 0.357\n", - "time: 13 position = -0.055 variance = 0.333\n", - "time: 14 position = -0.018 variance = 0.312\n", - "time: 15 position = -0.034 variance = 0.294\n", - "time: 16 position = -0.083 variance = 0.278\n", - "time: 17 position = -0.128 variance = 0.263\n", - "time: 18 position = -0.102 variance = 0.250\n", - "time: 19 position = -0.079 variance = 0.238\n" - ] - } - ], - "prompt_number": 14 - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Because of the random numbers I do not know the exact values that you see, but the position should have converged very quickly to almost 0 despite the initial error of believing that the position was 2.0. Furthermore, the variance should have quickly converged from the intial value of 5.0 to 0.238.\n", - "\n", - "By now the fact that we converged to a position of 0.0 should not be terribly suprising. All we are doing is computing new_position = old_position * measurement, and the measurement is a normal distribution around 0, so we should get very close to 0 after 20 iterations. But the truly amazing part of this code is how the variance became 0.238 despite every measurement having a variance of 5.0. \n", - "\n", - "If we think about the physical interpretation of this is should be clear that this is what should happen. If you sent 20 people into the hall with a tape measure to physically measure the position of the dog you would be very confident in the result after 20 measurements - more confident than after 1 or 2 measurements. So it makes sense that as we make more measurements the variance gets smaller.\n", - "\n", - "Mathematically it makes sense as well. Recall the computation for the variance after the multiplication: $\\sigma^2 = \\frac{1}{\\frac{1}{{\\sigma}_1} + \\frac{1}{{\\sigma}_2}}$. We take the reciprocals of the sigma from the measurement and prior belief, add them, and take the reciprocal of the result. Think about that for a moment, and you will see that this will always result in smaller numbers as we proceed.\n", - "\n", - "\n", - "#Implementing Updates\n", - "\n", - "That is a beautiful result, but it is not yet a filter. We assumed that the dog was sitting still, an extremely dubious assumption. Certainly it is a useless one - who would need to write a filter to track nonmoving objects? The histogram used a loop of sense and update functions, and we must do the same to accomodate movement.\n", - "\n", - "How how do we perform the update function with gaussians? Recall the histogram method:\n", - "\n", - " def update(pos, move, p_correct, p_under, p_over):\n", - " n = len(pos)\n", - " result = array(pos, dtype=float)\n", - " for i in range(n):\n", - " result[i] = \\\n", - " pos[(i-move) % n] * p_correct + \\\n", - " pos[(i-move-1) % n] * p_over + \\\n", - " pos[(i-move+1) % n] * p_under \n", - " return result\n", - " \n", - " \n", - "In a nutshell, we shift the probability vector by the amount we believe the animal moved, and adjust the probability. How do we do that with gaussians?\n", - "\n", - "It turns out that we just add gaussians. Think of the case without gaussians. I think my dog is at 7.3m, and he moves 2.6m to right, where is he now? Obviously, $7.3+2.6=9.9$. He is at 9.9m. Abstractly, the algorithm is *new_pos = old_pos + dist_moved*. It does not matter if we use floating point numbers or gaussians for these values, the algorithm must be the same. \n", - "\n", - "How is addition for gaussians performed. It turns out to be very simple:\n", - "$$ N({\\mu}_1, {{\\sigma}_1}^2)+N({\\mu}_2, {{\\sigma}_2}^2) = N({\\mu}_1 + {\\mu}_2, {\\sigma}_1 + {\\sigma}_2)$$\n", - "\n", - "All we do is add the means and the variance separately! Does that make sense? Think of the physical representation of this abstract equation.\n", - "${\\mu}_1$ is the old position, and ${\\mu}_2$ is the distance moved. Surely it makes sense that our new position is ${\\mu}_1 + {\\mu}_2$. What about the variance? It is perhaps harder to form an intuition about this. However, recall that with the *update()* function for the histogram filter we always lost information - our confidence after the update was lower than our confidence before the update. Perhaps this makes sense - we don't really know where the dog is moving, so perhaps the confidence should get smaller (variance gets larger). I assure you that the equation for gaussian addition is correct, and derived by basic algebra. Therefore it is reasonable to expect that if we are using gaussians to model physical events, the results must correctly describe those events.\n", - "\n", - "I recognize the amount of hand waving in that argument. Now is a good time to either work through the algebra to convince yourself of the mathematical correctness of the algorithm, or to work through some examples and see that it behaves reasonably. This book will do the latter.\n", - "\n", - "So, here is our implementation of the update function:\n", - "\n", - "\n" - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "def update(pos, sigma, movement, movement_sigma):\n", - " return (pos + movement, sigma + movement_sigma)" - ], - "language": "python", - "metadata": {}, - "outputs": [], - "prompt_number": 15 - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "What is left? Just calling these functions. The histogram did nothing more than loop over the *sense()* and *update()* functions, so let's do the same. " - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "# assume dog is always moving 1m to the right\n", - "movement = 1\n", - "movement_error = 2\n", - "sensor_error = 10\n", - "pos = (0, 500) # gaussian N(0,50)\n", - "\n", - "dog = DogSensor(pos[0], velocity=movement, noise=sensor_error)\n", - "\n", - "zs = []\n", - "ps = []\n", - "\n", - "for i in range(10):\n", - " pos = update(pos[0], pos[1], movement, movement_error)\n", - " print 'UPDATE:', \"%.4f\" %pos[0], \", %.4f\" %pos[1]\n", - " \n", - " Z = dog.sense()\n", - " zs.append(Z)\n", - " \n", - " pos = sense(pos[0], pos[1], Z, sensor_error)\n", - " ps.append(pos[0])\n", - " \n", - " print 'SENSE:', \"%.4f\" %pos[0], \", %.4f\" %pos[1]\n", - " print\n", - " \n", - "p1, = plot(zs,c='r', linestyle='dashed')\n", - "p2, = plot(ps, c='b')\n", - "legend([p1,p2], ['measurement', 'filter'], 2)\n", - "show()" - ], - "language": "python", - "metadata": {}, - "outputs": [ - { - "output_type": "stream", - "stream": "stdout", - "text": [ - "UPDATE: 1.0000 , 502.0000\n", - "SENSE: 10.2262 , 9.8047\n", - "\n", - "UPDATE: 11.2262 , 11.8047\n", - "SENSE: 4.7554 , 5.4138\n", - "\n", - "UPDATE: 5.7554 , 7.4138\n", - "SENSE: 5.9000 , 4.2574\n", - "\n", - "UPDATE: 6.9000 , 6.2574\n", - "SENSE: 4.2941 , 3.8490\n", - "\n", - "UPDATE: 5.2941 , 5.8490\n", - "SENSE: 4.8349 , 3.6904\n", - "\n", - "UPDATE: 5.8349 , 5.6904\n", - "SENSE: 5.3833 , 3.6267\n", - "\n", - "UPDATE: 6.3833 , 5.6267\n", - "SENSE: 5.5365 , 3.6007\n", - "\n", - "UPDATE: 6.5365 , 5.6007\n", - "SENSE: 7.4788 , 3.5900\n", - "\n", - "UPDATE: 8.4788 , 5.5900\n", - "SENSE: 7.2898 , 3.5856\n", - "\n", - "UPDATE: 8.2898 , 5.5856\n", - "SENSE: 8.9818 , 3.5838\n", - "\n" - ] - }, - { - "metadata": {}, - "output_type": "display_data", - "png": 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AkBBaPl9Wsabz54HOnenfevXME585jRyJiw6BGHJoPJo1A9avp4l4ZVHmLJQSVK4MjBsH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\n7qUn5Dp1LHt8UJV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- "text": [ - "" - ] - } - ], - "prompt_number": 16 - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "There is a fair bit of arbitrary constants code above, but don't worry about it. What does require explanation are the first few lines:\n", - "\n", - " movement = 1 \n", - " movement_error = 2\n", - " \n", - "For the moment we are assuming that we have some other sensor that detects how the dog is moving. For example, there could be an inertial sensor clipped onto the dog's collar, and it reports how far the dog moved each time it is triggered. The details don't matter. The upshot is that we have a sensor, it has noise, and so we represent it with a guassian. Later we will learn what to do if we do not have a sensor for the *update()* step.\n", - "\n", - "For now let's walk through the code and output bit by bit.\n", - "\n", - " movement = 1\n", - " movement_error = 2\n", - " sensor_error = 10\n", - " pos = (0, 500) # gaussian N(0,500)\n", - " \n", - " \n", - "The first lines just set up the initial conditions for our filter. We are assuming that the dog moves steadily to the right 1m at a time. We have a relatively low error of 2 for the movement sensor, and a higher error of 10 for the RFID position sensor. Finally, we set our belief of the dog's initial position as $N(0,500)$. Why those numbers. Well, 0 is as good as any number if we don't know where the dog is. But we set the variance to 500 to denote that we have no confidence in this value at all. 100m is almost as likely as 0 with this value for the variance. \n", - "\n", - "Next we initialize the RFID simulator with\n", - " dog = DogSensor(pos[0], velocity=movement, noise=sensor_error)\n", - "\n", - "It may seem very 'convienent' to set the simulator to the same position as our guess, and it is. Do not fret. In the next example we will see the effect of a wildly inaccurate guess for the dog's initial position.\n", - "\n", - "The next code allocates an array to store the output of the measurements and filtered positions. \n", - "\n", - " zs = []\n", - " ps = []\n", - " \n", - "This is the first time that I am introducing standard nomenclature used by the Kalman filtering literature. It is traditional to call our measurement $Z$, and so I follow that convention here. As an aside, I find the nomenclature used by the literature very obscure. However, if you wish to read the literature you will have to become used to it, so I will not use a much more readable variable name such as $m$ or $measure$.\n", - " \n", - " \n", - "Now we just enter our *sense()->update()* loop.\n", - "\n", - " for i in range(10):\n", - " pos = update(pos[0], pos[1], movement, sensor_error)\n", - " print 'UPDATE:', \"%.4f\" %pos[0], \", %.4f\" %pos[1]\n", - "\n", - "Wait, why *update()* before sense? It turns out the order does not matter once, but the first call to DogSensor.sense() assumes that the dog has already moved, so we start with the update step. In practice you will order these calls based on the details of your sensor, and you will very typically do the *sense()* first.\n", - "\n", - "So we call the update function with the gaussian representing our current belief about our position, the another gaussian representing our belief as to where the dog is moving, and then print the output. Your output will differ, but when writing this I get this as output:\n", - "\n", - " UPDATE: 1.000 502.000\n", - "\n", - "What is this saying? After the update, we believe that we are at 1.0, and the variance is now 502.0. Recall we started at 500.0. The variance got worse, which is always what happens during the update step.\n", - "\n", - " Z = dog.sense()\n", - " zs.append(Z)\n", - " \n", - "Here we sense the dog's position, and store it in our array so we can plot the results later.\n", - "\n", - "Finally we call the sense function of our filter, save the result in our *ps* array, and print the updated position belief:\n", - " pos = sense(pos[0], pos[1], Z, movement_error)\n", - " ps.append(pos[0])\n", - " print 'SENSE:', \"%.4f\" %pos[0], \", %.4f\" %pos[1]\n", - " \n", - "Your result will be different, but I get\n", - "\n", - " SENSE: 1.6279 , 9.8047\n", - " \n", - "as the result. What is happening? Well, at this point the dog is really at 1.0, however the predicted position is 1.6279. What is happening is the RFID sensor has a fair amount of noise, and so we compute the position as 1.6279. That is pretty far off from 1, but this is just are first time through the loop. Intuition tells us that the results will get better as we make more measurements, so let's hope that this is true for our filter as well. Now look at the variance: 9.8047. It has dropped tremendously from 502.0. Why? Well, the RFID has a reasonably small variance of 2.0, so we trust it far more than our previous belief. At this point there is no way to know for sure that the RFID is outputting reliable data, so the variance is not 2.0, but is has gotten much better.\n", - "\n", - "Now the software just loops, calling *update()* and *sense()* in turn. Because of the random sampling I do not know exactly what numbers you are seeing, but the final position is probably between 9 and 11, and the final variance is probably around 3.5. After several runs I did see the final position nearer 7, which would have been the result of several measurements with relatively large errors.\n", - "\n", - "Now look at the plot. The noisy measurements are plotted in with a dotted red line, and the filter results are in the solid blue line. Both are quite noisy, but notice how much noisier the measurements (red line) are. This is your first Kalman filter shown to work!\n", - "\n", - "\n", - "#More Examples\n", - "\n", - "Before I go on, I want to emphasize that this code fully implements a 1D Kalman filter. If you have tried to read the literatue, you are perhaps surprised, because this looks nothing like the complex, endless pages of math in those books. To be fair, the math gets a bit more complicated in multiple dimensions, but not by much. So long as we worry about *using* the equations rather than *deriving* them we can create Kalman filters without a lot of effort. Moreover, I hope you'll agree that you have a decent intuitive grasp of what is happening. We represent our beliefs with gaussians, and our beliefs get better over time because more measurement means more data to work with. \"Measure twice, cut once!\"\n", - "\n", - "So I didn't put a lot of noise in the signal, and I also 'correctly guessed' that the dog was at position 0. How does the filter perform in real world conditions? Let's explore and find out. I will start by injecting a lot of noise in the RFID sensor. I will inject an extreme amount of noise - noise that apparently swamps the actual measurement. What does your intution tell about how the filter will perform if the noise is allowed to be anywhere from -300 or 300. In other workds, an actual position of 1.0 might be reported as 287.9, or -189.6, or any other number in that range. Think about it before you scroll down." - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "sensor_error = 30000\n", - "movement_error = 2\n", - "pos = (0,500)\n", - "\n", - "dog = DogSensor(pos[0], velocity=movement, noise=sensor_error)\n", - "\n", - "zs = []\n", - "ps = []\n", - "\n", - "for i in range(1000):\n", - " pos = update(pos[0], pos[1], movement, movement_error)\n", - " \n", - " Z = dog.sense()\n", - " zs.append(Z)\n", - " \n", - " pos = sense(pos[0], pos[1], Z, sensor_error)\n", - " ps.append(pos[0])\n", - "\n", - "\n", - "p1, = plot(zs,c='r', linestyle='dashed')\n", - "p2, = plot(ps, c='b')\n", - "legend([p1,p2], ['measurement', 'filter'], 2)\n", - "show()" - ], - "language": "python", - "metadata": {}, - "outputs": [ - { - "metadata": {}, - "output_type": "display_data", - "png": 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onXWf7OGSygU0xaBBw38RixdTdovBkK+dvyxi3z4SxvbinXcs8xCxArt3j4K9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\nu1Y8/PDDQghxW65FZ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- "text": [ - "" - ] - } - ], - "prompt_number": 17 - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In this example the noise is extreme yet the filter still outputs a nearly straight line! This is an astonishing result! What do you think might be the cause of this performance? If you are not sure, don't worry, we will discuss it latter.\n", - "\n", - "Now let's lets look at the results when we make a bad initial estimate of position. To avoid obscuring the results I'll reduce the sensor variance to 30, but set the initial position to 1000m. Can the filter recover from a 1000m initial error?" - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "sensor_error = 30\n", - "movement_error = 2\n", - "pos = (1000,500)\n", - "\n", - "dog = DogSensor(0, velocity=movement, noise=sensor_error)\n", - "\n", - "zs = []\n", - "ps = []\n", - "\n", - "for i in range(100):\n", - " pos = update(pos[0], pos[1], movement, movement_error)\n", - " \n", - " Z = dog.sense()\n", - " zs.append(Z)\n", - " \n", - " pos = sense(pos[0], pos[1], Z, sensor_error)\n", - " ps.append(pos[0])\n", - "\n", - "\n", - "p1, = plot(zs,c='r', linestyle='dashed')\n", - "p2, = plot(ps, c='b')\n", - "legend([p1,p2], ['measurement', 'filter'], 2)\n", - "show()" - ], - "language": "python", - "metadata": {}, - "outputs": [ - { - "metadata": {}, - "output_type": "display_data", - "png": 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uXVVgYKBKTk5WSil14sQJ1ahRo3zHWnlpIRzXo48qtXJlkQ+/cUOpkSOVKueS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\nT5CVlbN9/s91qFv1Ct262e7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- "text": [ - "" - ] - } - ], - "prompt_number": 18 - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Again the answer is yes! Because we are relatively sure about our belief in the sensor ($\\sigma=30$) even after the first step we have changed our belief in the first position from 1000 to somewhere around 60.0 or so. After another 5-10 measurements we have converged to the correct value! So this is how we get around the chicken and egg problem of initial guesses. In practice we would probably just assign the first measurement from the sensor as the initial value, but you can see it doesn't matter much if we wildly guess at the initial conditions - the Kalman filter still converges very quickly.\n", - "\n", - "What about the worst of both worlds, large noise and a bad initial estimate:" - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "sensor_error = 30000\n", - "movement_error = 2\n", - "pos = (1000,500)\n", - "\n", - "dog = DogSensor(0, velocity=movement, noise=sensor_error)\n", - "\n", - "zs = []\n", - "ps = []\n", - "\n", - "for i in range(1000):\n", - " pos = update (pos[0], pos[1], movement, movement_error)\n", - " \n", - " Z = dog.sense()\n", - " zs.append(Z)\n", - " \n", - " pos = sense (pos[0], pos[1], Z, sensor_error)\n", - " ps.append(pos[0])\n", - "\n", - "\n", - "p1, = plot(zs,c='r', linestyle='dashed')\n", - "p2, = plot(ps, c='b')\n", - "legend([p1,p2], ['measurement', 'filter'], 2)\n", - "show()" - ], - "language": "python", - "metadata": {}, - "outputs": [ - { - "metadata": {}, - "output_type": "display_data", - "png": 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YGODnx/8XLQKeeSbz92zdCgwZoo0HPAyio9Nvi4tzvH/lynz87Tfg++/5f7VqwNNPA1On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- "text": [ - "" - ] - } - ], - "prompt_number": 19 - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This time the filter does struggle. Notice that the previous example only computed 100 updates, whereas this example uses 1000. By my eye it takes the filter 400 or so iterations to become reasonable accurate, but maybe over 600 before the results are good. Kalman filters are good, but we cannot expect miracles. If we have extremely noisy data and extremely bad initial conditions, this is as good as it gets.\n", - "\n", - "Finally, let's make the suggest change of making our initial position guess just be the first sensor measurement." - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "sensor_error = 30000\n", - "movement_error = 2\n", - "pos = None\n", - "\n", - "dog = DogSensor(0, velocity=movement, noise=sensor_error)\n", - "\n", - "zs = []\n", - "ps = []\n", - "\n", - "for i in range(1000):\n", - " Z = dog.sense()\n", - " zs.append(Z)\n", - " if pos == None:\n", - " pos = (Z, 500)\n", - " \n", - " pos = sense (pos[0], pos[1], Z, sensor_error)\n", - " ps.append(pos[0])\n", - "\n", - " pos = update (pos[0], pos[1], movement, movement_error)\n", - "\n", - "p1, = plot(zs,c='r', linestyle='dashed')\n", - "p2, = plot(ps, c='b')\n", - "legend([p1,p2], ['measurement', 'filter'], 2)\n", - "show()" - ], - "language": "python", - "metadata": {}, - "outputs": [ - { - "metadata": {}, - "output_type": "display_data", - "png": 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6vnxZ2VmtZk2AY5l8Do2GhLmxYjh6lOby+ut0jZIlyQKRKxZ5BlRAAGUgabXk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\nHx+PmJiYo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- "text": [ - "" - ] - } - ], - "prompt_number": 20 - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This simple change significantly improves the results. On some runs it takes 200 iterations or so to settle to a good solution, but other runs it converges very rapidly. This all depends on whether the initial measurement $Z$ had a small amount or large amount of noise. \n", - "\n", - "200 iterations may seem like a lot, but the amount of noise we are injecting is truly huge. In the real world we use sensors like thermometers, laser rangefinders, GPS satellites, computer vision, and so on. None have the enormous error as shown here. A reasonable value for the variance for a cheap thermometer might be 10, for example, and our code is using 30,000 for the variance. " - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "####Explaining the Results\n", - "\n", - "So how does the Kalman filter do so well? I have glossed over one aspect of the filter as it becomes confusing to address too many points at the same time. In these example we do not have 1 sensor but 2. The first sensor is the RFID sensor that outputs the position measurement, and the second sensor measures our dog's movement using an intertial tracker. How does our filter perform if that tracker is also noisy? Let's see:\n", - "\n" - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "sensor_error = 30000\n", - "movement_sensor = 30000\n", - "pos = (0,500)\n", - "\n", - "dog = DogSensor(0, velocity=movement, noise=sensor_error)\n", - "\n", - "zs = []\n", - "ps = []\n", - "\n", - "for i in range(1000):\n", - " Z = dog.sense()\n", - " zs.append(Z)\n", - " \n", - " pos = sense(pos[0], pos[1], Z, sensor_error)\n", - " ps.append(pos[0])\n", - "\n", - " pos = update(pos[0], pos[1], movement, movement_error)\n", - "\n", - "p1, = plot(zs,c='r', linestyle='dashed')\n", - "p2, = plot(ps, c='b')\n", - "legend([p1,p2], ['measurement', 'filter'], 2)\n", - "show()" - ], - "language": "python", - "metadata": {}, - "outputs": [ - { - "metadata": {}, - "output_type": "display_data", - "png": 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f+L9RI7pHeSURtAOlGBQUCgrmzBGhl6VLBR2zNbh6JyLC8T4MFoZmM1nRjrB6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\nDkd2SMVx6BCFze64g0JF3BfBI1IdDbAvjWPw9NTvqsqLskwbVDgtld8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1MXXjjAQW3Bs8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- "text": [ - "" - ] - } - ], - "prompt_number": 21 - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This result is worse than the example where only the measurement sensor was noisy. Instead of being mostly straight, this time the filter's output is distintly jagged. But, it still mostly tracks the dog. What is happening here?\n", - "\n", - "This illustrates the effects of *multi-sensor fusion*. Suppose the dog is actually at 10.0, and we get subsequent measurement readings of -289.78 and 301.43. From that information alone it is impossible to tell if the dog is standing still during very noisy measurements, or perhaps sprinting from -289 to 301 and being accurately measured. But we have a second source of information, his velocity. Even when the velocity is also noisy, it constrains what our beliefs might be. For example, suppose with the readings of -289.78 to 301.43 we get a velocity reading of 590. That matches the difference between the two positions quite well, so this will lead us to believe the RFID sensor and the velocity sensor. Now suppose we got a velocity reading of 1.7. This doesn't match our RFID reading very well. Finally, suppose the velocity reading was -678.8. This completely contradicts the RFID reading - we may not be sure from these few values which sensor is most inaccurate, but perhaps by now you will trust that the gaussians expressing our beliefs will correctly handle these cases. It's a bit hard to talk about while working with 1D problems, so we will take this topic up in great detail in the next chapter where we develop multidimensional Kalman filters. Remark\n", - "\n", - "Besides that issue, we are modelling the noise in our sensors using gaussians which model their real world performance. We are multiplying the gaussians (probabilities) when we get a new position measurement, adding the gaussians when we get a movement update. This is algorithmically correct (this is how the histogram filter works) and mathematically correct - why wouldn't it work? \n", - "\n", - "#### Summary\n", - "This takes some time to assimulate. To truly understand this you will probably have to work through this chapter several times. I encourage you to change the various constants and observe the results. Convince yourself that gaussians are a good representation of a unimodal belief of something like the position of a dog in a hallway. Then convince yourself that multiplying gaussians truly does compute a new belief from your prior belief and the new measurement. Finally, convince yourself that if you are measuring movement, that adding the gaussians correctly updates your belief. That is all the Kalman filter does. Even now I alternate between complacency and amazement at the results. \n", - "\n", - "If you understand this, you will be able to understand multidimensional Kalman filters and the various extensions that have been make on them. If you do not fully understand this, I strongly suggest rereading this chapter until you do understand it. Try implementing the filter from scratch, just by looking at the equations and reading the text. Change the constants. Maybe try to implement a different tracking problem, like tracking stock prices. Experimentation will build your intuition and understanding of how these marvelous filters work.\n", - "\n" - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "author notes:\n", - " clean up the code - same stuff duplicated over and over - write a 'clean implemntation' at the end.\n", - " \n", - " " - ], - "language": "python", - "metadata": {}, - "outputs": [ - { - "ename": "SyntaxError", - "evalue": "invalid syntax (, line 1)", - "output_type": "pyerr", - "traceback": [ - "\u001b[1;36m File \u001b[1;32m\"\"\u001b[1;36m, line \u001b[1;32m1\u001b[0m\n\u001b[1;33m author notes:\u001b[0m\n\u001b[1;37m ^\u001b[0m\n\u001b[1;31mSyntaxError\u001b[0m\u001b[1;31m:\u001b[0m invalid syntax\n" - ] - } - ], - "prompt_number": 22 - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "sensor_error = 30\n", - "movement_error = 2\n", - "pos = (1000,500)\n", - "\n", - "zs = []\n", - "ps = []\n", - "\n", - "\n", - "for i in range(100):\n", - " pos = update(pos[0], pos[1], movement, movement_error)\n", - "\n", - " Z = math.sin(i/3.)*5.\n", - " zs.append(Z)\n", - " \n", - " pos = sense(pos[0], pos[1], Z, sensor_error)\n", - " ps.append(pos[0])\n", - "\n", - "\n", - "p1, = plot(zs,c='r', linestyle='dashed')\n", - "p2, = plot(ps, c='b')\n", - "legend([p1,p2], ['measurement', 'filter'], 2)\n", - "show()" - ], - "language": "python", - "metadata": {}, - "outputs": [] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [], - "language": "python", - "metadata": {}, - "outputs": [] - } - ], - "metadata": {} - } - ] +{ + "metadata": { + "name": "", + "signature": "sha256:0131b9ce88d9ff5da30b8c74995895cb6cb161758766f2e1e2599c4b05cebb6c" + }, + "nbformat": 3, + "nbformat_minor": 0, + "worksheets": [ + { + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "#Kalman Filters\n", + "\n", + "\n", + "Now that we understand the histogram filter and gaussians we are prepared to implement a 1D Kalman filter. We will do this exactly as we did the histogram filter - rather than going into the theory we will just develop the code step by step. \n", + "\n", + "#Tracking A Dog\n", + "\n", + "As in the histogram chapter we will be tracking a dog in a long hallway at work. However, in our latest hackathon someone created an RFID tracker that provides a reasonable accurate position for our dog. Suppose the hallway is 100m long. The sensor returns the distance of the dog from the left end of the hallway. So, 23.4 would mean the dog is 23.4 meters from the left end of the hallway.\n", + "\n", + "Naturally, the sensor is not perfect. A reading of 23.4 could correspond to a real position of 23.7, or 23.0. However, it is very unlikely to correspond to a real position of say 47.6. Testing during the hackathon confirmed this result - the sensor is accurate, and while it had errors, the errors are small.\n", + "\n", + "Implementing and/or robustly modelling an RFID system is beyond the scope of this book, so we will write a very simple model. We will start with a simulation of the dog moving from left to right at a constant speed with some random noise added. " + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import numpy.random as random\n", + "import math\n", + "\n", + "class DogSensor(object):\n", + " \n", + " def __init__(self, x0=0, velocity=1, noise=0.0):\n", + " \"\"\" x0 - initial position\n", + " velocity - (+=right, -=left)\n", + " noise - scaling factor for noise, 0== no noise\n", + " \"\"\"\n", + " self.x = x0\n", + " self.velocity = velocity\n", + " self.noise = math.sqrt(noise)\n", + "\n", + " def sense(self):\n", + " self.x = self.x + self.velocity\n", + " return self.x + random.randn() * self.noise\n" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The constructor (__init__()) initializes the DogSensor class with an initial position (x0), velocity (vel), and an noise scaling factor. The *sense()* function has the dog move by the set velocity and returns its new position, with noise added. If you look at the code for *sense()* you will see a call to *numpy.random.randn()*. This returns a number sampled from a normal distribution with a mean of 0.0. Let's look at some example output for that.\n", + "\n" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "for i in range(20):\n", + " print (\"%.4f\" % random.randn())," + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "You should see a sequence of numbers near 0, some negative and some positive. Most are probably between -1 and 1, but a few might lie somewhat outside that range. This is what we expect from a normal distribution - values are clustered around the mean, and there are fewer values the further you get from the mean.\n", + "\n", + "Okay, so lets look at the output of the *DogSensor* class. We will start by setting the noise to 0 to check that the class does what we think it does" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "%matplotlib inline\n", + "\n", + "dog = DogSensor (noise=0.0)\n", + "xs = []\n", + "for i in range(10):\n", + " x = dog.sense()\n", + " xs.append(x)\n", + " print(\"%.4f\" % x),\n", + "plot(xs)\n", + "show()" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The constructor initialized the dog at position 0 with a velocity of 1 (move 1.0 to the right). So we would expect to see an output of 1..10, and indeed that is what we see. If you thought the correct answer should have been 0..9 recall that *sense()* returns the dog's position *after* updating his position, so the first postion is 0.0 + 1, or 1.0.\n", + "\n", + "Now let's inject some noise in the signal." + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "def test_sensor(noise_scale):\n", + " dog = DogSensor(noise=noise_scale)\n", + "\n", + " xs = []\n", + " for i in range(100):\n", + " x = dog.sense()\n", + " xs.append(x)\n", + " p1, = plot(xs, c='b')\n", + " p2, = plot([0,99],[1,100], 'r--')\n", + " xlabel('time')\n", + " ylabel('pos')\n", + " ylim([0,100])\n", + " title('noise = ' + str(noise_scale))\n", + " legend([p1, p2], ['sensor', 'actual'], loc=2)\n", + " show()\n", + " \n", + "test_sensor(4.0)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**note**:*numpy uses a random number generator to generate the normal distribution samples. The numbers I see as I write this are unlikely to be the ones that you see. If you run the cell above multiple times, you should get a slightly different result each time. I could use numpy.random.seed(some_value) to force the results to be the same each time. This would simplify my explanations in some cases, but would ruin the interactive nature of this chapter. To get a real feel for how normal distributions and Kalman filters work you will probably want to run cells several times, observing what changes, and what stays roughly the same.*\n", + "\n", + "So the output of the sensor should be a wavering blue line drawn over a dotted red line. The dotted red line shows the actual position of the dog, and the blue line is the noise signal produced by the simulated RFID sensor. Please note that the red dotted line was manually plotted - we do not yet have a filter that recovers that information! \n", + "\n", + "If you are running this in an interactive IPython Notebook, I strongly urge you to run the script several times in a row. You can do this by putting the cursor in the cell containing the Python code and pressing Ctrl+Enter. Each time it runs you should see a different jagged blue line wavering over the top of the dotted red line.\n", + "\n", + "I also urge you to adjust the noise setting to see the result of various values. However, since you may be reading this in a read only notebook, I will show two extreme examples. The first plot shows the noise set to 100.0, and the second shows noise set to 0.5." + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "test_sensor(100.0)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "test_sensor(0.5)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "You may not have a full understanding of the exact *meaning* of a noise value of 100.0, but as it turns out if you multiply *randn()* with a number $n$, the result is just a normal distribution with $\\sigma = \\sqrt{n}$. So the example with noise = 100 is using the normal distribution $N(0,100)$. Recall the notation for a normal distribution is $N(\\mu,\\sigma^2)$. If the square root is confusing, recall that normal distributions use $\\sigma^2$ for the variance, and $\\sigma$ is the standard deviation, which we do not use in this book. *dog_sensor.__init__()* takes the square root of the noise setting so that the *noise * randn()* call properly computes the normal distribution. " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "#### Math with Gaussians\n", + "\n", + "Let's say we believe that our dog is at 23m, and the variance is 5 ($N(23,5)$). We can represent that in a plot:\n" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import gaussian\n", + "gaussian.norm_plot(23, 5)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This corresponds to a fairly inexact belief. While we believe that the dog is at 23, note that roughly 21 to 25 are quite likely as well. Let's assume for the moment our dog is standing still, and we query the sensor again. This time it returns 23.2 as the position. Can we use this additional information to improve our estimate of the dog's position.\n", + "\n", + "Intuition suggests 'yes'. Consider: if we read the sensor 100 times and each time it returned a value between 21 and 25, all centered around 23, we should be very confident that the dog is somewhere very near 23. Of course, a different physical interpertation is possible. Perhaps our dog was randomly wandering back and forth in a way that exactly emulated a normal distribution. But that seems extremely unlikely - I certainly have never seen a dog do that. So the only reasonable assumption is that the dog was mostly standing still at 23.0.\n", + "\n", + "Let's look at this in a plot:" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "dog = DogSensor(23, 0, 5)\n", + "xs = range(100)\n", + "ys = []\n", + "for i in xs:\n", + " ys.append(dog.sense())\n", + " \n", + "plot(xs,ys)\n", + "show()" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Eyeballing this confirms our intuition - no dog moves like this. However, noisy sensor data certainly looks like this. So let's proceed to see how we might solve this mathematically. But how?\n", + "\n", + "\n", + "Recall the histogram code for adding a measurement to a pre-existing belief:\n", + "\n", + " def sense(pos, measure, p_hit, p_miss):\n", + " q = array(pos, dtype=float)\n", + " for i in range(len(hallway)):\n", + " if hallway[i] == measure:\n", + " q[i] = pos[i] * p_hit\n", + " else:\n", + " q[i] = pos[i] * p_miss\n", + " normalize(q)\n", + " return q\n", + " \n", + "Note that the algorithm is essentially computing:\n", + "\n", + " new_belief = old_belief * measurement * sensor_error\n", + " \n", + "The measurement term might not be obvious, but recall that measurement in this case was always 1 or 0, and so it was left out for convience. \n", + " \n", + "If we are implementing this with gaussians, we might expect it to be implemented as:\n", + "\n", + " new_gaussian = measurement * old_gaussian\n", + " \n", + "where measurement is a gaussian returned from the sensor. But does that make sense? Can we multiply gaussians? If we multiply a gaussian with a gaussing is the result another gaussian, or something else?\n", + "\n", + "Of course the answer is 'yes', or this chapter would be for naught. It is not particularly difficult to perform the algebra to derive the equation for multiplying two gaussians, but I will just present the result:\n", + "$$ N({\\mu}_1, {{\\sigma}_1}^2)*N({\\mu}_2, {{\\sigma}_2}^2) = N(\\frac{{\\sigma}_1 {\\mu}_2 + {\\sigma}_2 {\\mu}_1}{{\\sigma}_1 + {\\sigma}_2},\\frac{1}{\\frac{1}{{\\sigma}_1} + \\frac{1}{{\\sigma}_2}}) $$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Let's immediately look at some plots of this to inform our intuition about this result. First, let's look at the result of multiplying $N(23,5) $ to itself. This corresponds to getting 23.0 as the sensor value twice in a row." + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import gaussian\n", + "def multiply(mu1, sig1, mu2, sig2):\n", + " m = (sig1*mu2 + sig2*mu1) / (sig1+sig2)\n", + " s = 1. / (1./sig1 + 1./ sig2)\n", + " return (m,s)\n", + "\n", + "\n", + "xs = np.arange(16, 30, 0.1)\n", + "\n", + "\n", + "m1,s1 = 23, 5\n", + "m, s = multiply(m1,s1,m1,s1)\n", + "\n", + "ys = [gaussian.gaussian(x,m1,s1) for x in xs]\n", + "p1, =plot (xs,ys)\n", + "\n", + "ys = [gaussian.gaussian(x,m,s) for x in xs]\n", + "p2, = plot (xs,ys)\n", + "\n", + "legend([p1,p2],['original', 'multiply'])\n", + "show()" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The result is either amazing or what you would expect, depending on your state of mind. I must admit I vacillate freely between the two! Note that the result of the multiplation is taller and narrow than the original gaussian. If we think of the gaussians as two measurement, this makes sense. If I measure twice and get the same value, I should be more confident in my answer than if I just measured once. \"Measure twice, cut once\" is a useful saying and practice due to this fact! \n", + "\n", + "Now let's multiply two gaussians (or equivelently, two measurements) that are partially separated. What do you think the result will be? Let's find out:" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "xs = np.arange(16, 30, 0.1)\n", + "\n", + "\n", + "m1,s1 = 23, 5\n", + "m2,s2 = 25, 5\n", + "m, s = multiply(m1,s1,m2,s2)\n", + "\n", + "ys = [gaussian.gaussian(x,m1,s1) for x in xs]\n", + "p1, = plot (xs,ys)\n", + "\n", + "ys = [gaussian.gaussian(x,m2,s2) for x in xs]\n", + "p2, = plot (xs,ys)\n", + "\n", + "ys = [gaussian.gaussian(x,m,s) for x in xs]\n", + "p3, = plot(xs,ys)\n", + "legend([p1,p2,p3],['measure 1', 'measure 2', 'multiply'])\n", + "show()" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Another beautiful result! If I handed you a measuring tape and asked you to measure the distance from table to a wall, and you got 23m, and then a friend make the same measurement and got 25m, your best guess must be 24m. \n", + "\n", + "That is fairly counter-intuitive, so let's consider it further. Perhaps a more reasonable assumption would be that either you or your coworker just made a mistake, and the true distance is either 23 or 25, but certainly not 24. Surely that is possible. However, suppose the two measurements you reported as 24.01 and 23.99. Surely you would agree that in this case the best guess for the correct value is 24? Which interpretation we choose depends on the properties of the sensors we are using. Humans make galling mistakes, physical sensors do not. \n", + "\n", + "This topic is fairly deep, and I will explore it once we have completed our Kalman filter. For now I will merely say that the Kalman filter requires the interpretation that measurements are accurate, with gaussian noise, and that a large error caused by misreading a measuring tape is not gaussian noise. So perhaps you would be justified in thinking that a histogram filter will perform better for the human readings, and the Kalman filter will perform better with sensor readings that have gaussian noise.\n", + "\n", + "For now I ask that you trust me. The math is correct, so we have no choice but to accept it and use it. We will see how the Kalman filter deals with movements vs error very soon. 24 is the correct answer to this problem." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "#### Implementing Sensing\n", + "\n", + "Recall the histogram filter uses a numpy array to encode our belief about the position of our dog at any time. That array stored our belief that the dog was in any position in the hallway using 10 positions. This was very crude, because with a 100m hallway that corresponded to positions 10m apart. It would have been trivial to expand the number of positions to say 1,000, and that is what we would do if using it for a real problem. But the problem remains that the distribution is discrete and multimodal - it can express strong belief that the dog is in two positions at the same time.\n", + "\n", + "Therefore, we will use a single gaussian to reflect our current belief of the dog's position. Gaussians extend to infinity on both sides of the mean, so the single gaussian will cover the entire hallway. They are unimodal, and seem to reflect the behavior of real-world sensors - most errors are small and clustered around the mean. Here is the entire implementation of the sense function for a Kalman filter:" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "def sense(mu, sigma, measurement, measurement_sigma):\n", + " return multiply(mu, sigma, measurement, measurement_sigma)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Kalman filters are supposed to be hard! But this is very short and straightforward. All we are doing is multiplying the gaussian that reflects our belief of where the dog was with the new measurement. Perhaps this would be clearer if we used more specific names:" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "def sense_dog(dog_pos, dog_sigma, measurement, measurement_sigma):\n", + " return multiply(dog_pos, dog_sigma, measurement, measurement_sigma)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "That is less abstract, which perhaps helps with comprehension, but it is poor coding practice. We are writing a Kalman filter that works for any problem, not just tracking dogs in a hallway, so we don't use variable names with 'dog' in them. Still, the *sense_dog()* function should make what we are doing very clear. \n", + "\n", + "Let's look at an example. We will suppose that our current belief for the dog's position is $N(2,5)$. Don't worry about where that number came from. It may appear that we have a chicken and egg problem, in that how do we know the position before we sense it, but we will resolve that shortly. We will create a *DogSensor* object initialized to be at position 0.0, and with no velocity, and modest noise. This corresponds to the dog standing still at the far left side of the hallway. Note that we mistakenly believe the dog is at postion 2.0, not 0.0." + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "dog = DogSensor(velocity=0, noise=1)\n", + "\n", + "pos,s = 2, 5\n", + "for i in range(20):\n", + " pos,s = sense(pos, s, dog.sense(), 5)\n", + " print 'time:', i, 'position = ', \"%.3f\" % pos, 'variance = ', \"%.3f\" % s\n", + "\n" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Because of the random numbers I do not know the exact values that you see, but the position should have converged very quickly to almost 0 despite the initial error of believing that the position was 2.0. Furthermore, the variance should have quickly converged from the intial value of 5.0 to 0.238.\n", + "\n", + "By now the fact that we converged to a position of 0.0 should not be terribly suprising. All we are doing is computing new_position = old_position * measurement, and the measurement is a normal distribution around 0, so we should get very close to 0 after 20 iterations. But the truly amazing part of this code is how the variance became 0.238 despite every measurement having a variance of 5.0. \n", + "\n", + "If we think about the physical interpretation of this is should be clear that this is what should happen. If you sent 20 people into the hall with a tape measure to physically measure the position of the dog you would be very confident in the result after 20 measurements - more confident than after 1 or 2 measurements. So it makes sense that as we make more measurements the variance gets smaller.\n", + "\n", + "Mathematically it makes sense as well. Recall the computation for the variance after the multiplication: $\\sigma^2 = \\frac{1}{\\frac{1}{{\\sigma}_1} + \\frac{1}{{\\sigma}_2}}$. We take the reciprocals of the sigma from the measurement and prior belief, add them, and take the reciprocal of the result. Think about that for a moment, and you will see that this will always result in smaller numbers as we proceed.\n", + "\n", + "\n", + "#Implementing Updates\n", + "\n", + "That is a beautiful result, but it is not yet a filter. We assumed that the dog was sitting still, an extremely dubious assumption. Certainly it is a useless one - who would need to write a filter to track nonmoving objects? The histogram used a loop of sense and update functions, and we must do the same to accomodate movement.\n", + "\n", + "How how do we perform the update function with gaussians? Recall the histogram method:\n", + "\n", + " def update(pos, move, p_correct, p_under, p_over):\n", + " n = len(pos)\n", + " result = array(pos, dtype=float)\n", + " for i in range(n):\n", + " result[i] = \\\n", + " pos[(i-move) % n] * p_correct + \\\n", + " pos[(i-move-1) % n] * p_over + \\\n", + " pos[(i-move+1) % n] * p_under \n", + " return result\n", + " \n", + " \n", + "In a nutshell, we shift the probability vector by the amount we believe the animal moved, and adjust the probability. How do we do that with gaussians?\n", + "\n", + "It turns out that we just add gaussians. Think of the case without gaussians. I think my dog is at 7.3m, and he moves 2.6m to right, where is he now? Obviously, $7.3+2.6=9.9$. He is at 9.9m. Abstractly, the algorithm is *new_pos = old_pos + dist_moved*. It does not matter if we use floating point numbers or gaussians for these values, the algorithm must be the same. \n", + "\n", + "How is addition for gaussians performed. It turns out to be very simple:\n", + "$$ N({\\mu}_1, {{\\sigma}_1}^2)+N({\\mu}_2, {{\\sigma}_2}^2) = N({\\mu}_1 + {\\mu}_2, {\\sigma}_1 + {\\sigma}_2)$$\n", + "\n", + "All we do is add the means and the variance separately! Does that make sense? Think of the physical representation of this abstract equation.\n", + "${\\mu}_1$ is the old position, and ${\\mu}_2$ is the distance moved. Surely it makes sense that our new position is ${\\mu}_1 + {\\mu}_2$. What about the variance? It is perhaps harder to form an intuition about this. However, recall that with the *update()* function for the histogram filter we always lost information - our confidence after the update was lower than our confidence before the update. Perhaps this makes sense - we don't really know where the dog is moving, so perhaps the confidence should get smaller (variance gets larger). I assure you that the equation for gaussian addition is correct, and derived by basic algebra. Therefore it is reasonable to expect that if we are using gaussians to model physical events, the results must correctly describe those events.\n", + "\n", + "I recognize the amount of hand waving in that argument. Now is a good time to either work through the algebra to convince yourself of the mathematical correctness of the algorithm, or to work through some examples and see that it behaves reasonably. This book will do the latter.\n", + "\n", + "So, here is our implementation of the update function:\n", + "\n", + "\n" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "def update(pos, sigma, movement, movement_sigma):\n", + " return (pos + movement, sigma + movement_sigma)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "What is left? Just calling these functions. The histogram did nothing more than loop over the *sense()* and *update()* functions, so let's do the same. " + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "# assume dog is always moving 1m to the right\n", + "movement = 1\n", + "movement_error = 2\n", + "sensor_error = 10\n", + "pos = (0, 500) # gaussian N(0,50)\n", + "\n", + "dog = DogSensor(pos[0], velocity=movement, noise=sensor_error)\n", + "\n", + "zs = []\n", + "ps = []\n", + "\n", + "for i in range(10):\n", + " pos = update(pos[0], pos[1], movement, movement_error)\n", + " print 'UPDATE:', \"%.4f\" %pos[0], \", %.4f\" %pos[1]\n", + " \n", + " Z = dog.sense()\n", + " zs.append(Z)\n", + " \n", + " pos = sense(pos[0], pos[1], Z, sensor_error)\n", + " ps.append(pos[0])\n", + " \n", + " print 'SENSE:', \"%.4f\" %pos[0], \", %.4f\" %pos[1]\n", + " print\n", + " \n", + "p1, = plot(zs,c='r', linestyle='dashed')\n", + "p2, = plot(ps, c='b')\n", + "legend([p1,p2], ['measurement', 'filter'], 2)\n", + "show()" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "There is a fair bit of arbitrary constants code above, but don't worry about it. What does require explanation are the first few lines:\n", + "\n", + " movement = 1 \n", + " movement_error = 2\n", + " \n", + "For the moment we are assuming that we have some other sensor that detects how the dog is moving. For example, there could be an inertial sensor clipped onto the dog's collar, and it reports how far the dog moved each time it is triggered. The details don't matter. The upshot is that we have a sensor, it has noise, and so we represent it with a guassian. Later we will learn what to do if we do not have a sensor for the *update()* step.\n", + "\n", + "For now let's walk through the code and output bit by bit.\n", + "\n", + " movement = 1\n", + " movement_error = 2\n", + " sensor_error = 10\n", + " pos = (0, 500) # gaussian N(0,500)\n", + " \n", + " \n", + "The first lines just set up the initial conditions for our filter. We are assuming that the dog moves steadily to the right 1m at a time. We have a relatively low error of 2 for the movement sensor, and a higher error of 10 for the RFID position sensor. Finally, we set our belief of the dog's initial position as $N(0,500)$. Why those numbers. Well, 0 is as good as any number if we don't know where the dog is. But we set the variance to 500 to denote that we have no confidence in this value at all. 100m is almost as likely as 0 with this value for the variance. \n", + "\n", + "Next we initialize the RFID simulator with\n", + " dog = DogSensor(pos[0], velocity=movement, noise=sensor_error)\n", + "\n", + "It may seem very 'convienent' to set the simulator to the same position as our guess, and it is. Do not fret. In the next example we will see the effect of a wildly inaccurate guess for the dog's initial position.\n", + "\n", + "The next code allocates an array to store the output of the measurements and filtered positions. \n", + "\n", + " zs = []\n", + " ps = []\n", + " \n", + "This is the first time that I am introducing standard nomenclature used by the Kalman filtering literature. It is traditional to call our measurement $Z$, and so I follow that convention here. As an aside, I find the nomenclature used by the literature very obscure. However, if you wish to read the literature you will have to become used to it, so I will not use a much more readable variable name such as $m$ or $measure$.\n", + " \n", + " \n", + "Now we just enter our *sense()->update()* loop.\n", + "\n", + " for i in range(10):\n", + " pos = update(pos[0], pos[1], movement, sensor_error)\n", + " print 'UPDATE:', \"%.4f\" %pos[0], \", %.4f\" %pos[1]\n", + "\n", + "Wait, why *update()* before sense? It turns out the order does not matter once, but the first call to DogSensor.sense() assumes that the dog has already moved, so we start with the update step. In practice you will order these calls based on the details of your sensor, and you will very typically do the *sense()* first.\n", + "\n", + "So we call the update function with the gaussian representing our current belief about our position, the another gaussian representing our belief as to where the dog is moving, and then print the output. Your output will differ, but when writing this I get this as output:\n", + "\n", + " UPDATE: 1.000 502.000\n", + "\n", + "What is this saying? After the update, we believe that we are at 1.0, and the variance is now 502.0. Recall we started at 500.0. The variance got worse, which is always what happens during the update step.\n", + "\n", + " Z = dog.sense()\n", + " zs.append(Z)\n", + " \n", + "Here we sense the dog's position, and store it in our array so we can plot the results later.\n", + "\n", + "Finally we call the sense function of our filter, save the result in our *ps* array, and print the updated position belief:\n", + " pos = sense(pos[0], pos[1], Z, movement_error)\n", + " ps.append(pos[0])\n", + " print 'SENSE:', \"%.4f\" %pos[0], \", %.4f\" %pos[1]\n", + " \n", + "Your result will be different, but I get\n", + "\n", + " SENSE: 1.6279 , 9.8047\n", + " \n", + "as the result. What is happening? Well, at this point the dog is really at 1.0, however the predicted position is 1.6279. What is happening is the RFID sensor has a fair amount of noise, and so we compute the position as 1.6279. That is pretty far off from 1, but this is just are first time through the loop. Intuition tells us that the results will get better as we make more measurements, so let's hope that this is true for our filter as well. Now look at the variance: 9.8047. It has dropped tremendously from 502.0. Why? Well, the RFID has a reasonably small variance of 2.0, so we trust it far more than our previous belief. At this point there is no way to know for sure that the RFID is outputting reliable data, so the variance is not 2.0, but is has gotten much better.\n", + "\n", + "Now the software just loops, calling *update()* and *sense()* in turn. Because of the random sampling I do not know exactly what numbers you are seeing, but the final position is probably between 9 and 11, and the final variance is probably around 3.5. After several runs I did see the final position nearer 7, which would have been the result of several measurements with relatively large errors.\n", + "\n", + "Now look at the plot. The noisy measurements are plotted in with a dotted red line, and the filter results are in the solid blue line. Both are quite noisy, but notice how much noisier the measurements (red line) are. This is your first Kalman filter shown to work!\n", + "\n", + "\n", + "#More Examples\n", + "\n", + "Before I go on, I want to emphasize that this code fully implements a 1D Kalman filter. If you have tried to read the literatue, you are perhaps surprised, because this looks nothing like the complex, endless pages of math in those books. To be fair, the math gets a bit more complicated in multiple dimensions, but not by much. So long as we worry about *using* the equations rather than *deriving* them we can create Kalman filters without a lot of effort. Moreover, I hope you'll agree that you have a decent intuitive grasp of what is happening. We represent our beliefs with gaussians, and our beliefs get better over time because more measurement means more data to work with. \"Measure twice, cut once!\"\n", + "\n", + "So I didn't put a lot of noise in the signal, and I also 'correctly guessed' that the dog was at position 0. How does the filter perform in real world conditions? Let's explore and find out. I will start by injecting a lot of noise in the RFID sensor. I will inject an extreme amount of noise - noise that apparently swamps the actual measurement. What does your intution tell about how the filter will perform if the noise is allowed to be anywhere from -300 or 300. In other workds, an actual position of 1.0 might be reported as 287.9, or -189.6, or any other number in that range. Think about it before you scroll down." + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "sensor_error = 30000\n", + "movement_error = 2\n", + "pos = (0,500)\n", + "\n", + "dog = DogSensor(pos[0], velocity=movement, noise=sensor_error)\n", + "\n", + "zs = []\n", + "ps = []\n", + "\n", + "for i in range(1000):\n", + " pos = update(pos[0], pos[1], movement, movement_error)\n", + " \n", + " Z = dog.sense()\n", + " zs.append(Z)\n", + " \n", + " pos = sense(pos[0], pos[1], Z, sensor_error)\n", + " ps.append(pos[0])\n", + "\n", + "\n", + "p1, = plot(zs,c='r', linestyle='dashed')\n", + "p2, = plot(ps, c='b')\n", + "legend([p1,p2], ['measurement', 'filter'], 2)\n", + "show()" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In this example the noise is extreme yet the filter still outputs a nearly straight line! This is an astonishing result! What do you think might be the cause of this performance? If you are not sure, don't worry, we will discuss it latter.\n", + "\n", + "Now let's lets look at the results when we make a bad initial estimate of position. To avoid obscuring the results I'll reduce the sensor variance to 30, but set the initial position to 1000m. Can the filter recover from a 1000m initial error?" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "sensor_error = 30\n", + "movement_error = 2\n", + "pos = (1000,500)\n", + "\n", + "dog = DogSensor(0, velocity=movement, noise=sensor_error)\n", + "\n", + "zs = []\n", + "ps = []\n", + "\n", + "for i in range(100):\n", + " pos = update(pos[0], pos[1], movement, movement_error)\n", + " \n", + " Z = dog.sense()\n", + " zs.append(Z)\n", + " \n", + " pos = sense(pos[0], pos[1], Z, sensor_error)\n", + " ps.append(pos[0])\n", + "\n", + "\n", + "p1, = plot(zs,c='r', linestyle='dashed')\n", + "p2, = plot(ps, c='b')\n", + "legend([p1,p2], ['measurement', 'filter'], 2)\n", + "show()" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Again the answer is yes! Because we are relatively sure about our belief in the sensor ($\\sigma=30$) even after the first step we have changed our belief in the first position from 1000 to somewhere around 60.0 or so. After another 5-10 measurements we have converged to the correct value! So this is how we get around the chicken and egg problem of initial guesses. In practice we would probably just assign the first measurement from the sensor as the initial value, but you can see it doesn't matter much if we wildly guess at the initial conditions - the Kalman filter still converges very quickly.\n", + "\n", + "What about the worst of both worlds, large noise and a bad initial estimate:" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "sensor_error = 30000\n", + "movement_error = 2\n", + "pos = (1000,500)\n", + "\n", + "dog = DogSensor(0, velocity=movement, noise=sensor_error)\n", + "\n", + "zs = []\n", + "ps = []\n", + "\n", + "for i in range(1000):\n", + " pos = update (pos[0], pos[1], movement, movement_error)\n", + " \n", + " Z = dog.sense()\n", + " zs.append(Z)\n", + " \n", + " pos = sense (pos[0], pos[1], Z, sensor_error)\n", + " ps.append(pos[0])\n", + "\n", + "\n", + "p1, = plot(zs,c='r', linestyle='dashed')\n", + "p2, = plot(ps, c='b')\n", + "legend([p1,p2], ['measurement', 'filter'], 2)\n", + "show()" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This time the filter does struggle. Notice that the previous example only computed 100 updates, whereas this example uses 1000. By my eye it takes the filter 400 or so iterations to become reasonable accurate, but maybe over 600 before the results are good. Kalman filters are good, but we cannot expect miracles. If we have extremely noisy data and extremely bad initial conditions, this is as good as it gets.\n", + "\n", + "Finally, let's make the suggest change of making our initial position guess just be the first sensor measurement." + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "sensor_error = 30000\n", + "movement_error = 2\n", + "pos = None\n", + "\n", + "dog = DogSensor(0, velocity=movement, noise=sensor_error)\n", + "\n", + "zs = []\n", + "ps = []\n", + "\n", + "for i in range(1000):\n", + " Z = dog.sense()\n", + " zs.append(Z)\n", + " if pos == None:\n", + " pos = (Z, 500)\n", + " \n", + " pos = sense (pos[0], pos[1], Z, sensor_error)\n", + " ps.append(pos[0])\n", + "\n", + " pos = update (pos[0], pos[1], movement, movement_error)\n", + "\n", + "p1, = plot(zs,c='r', linestyle='dashed')\n", + "p2, = plot(ps, c='b')\n", + "legend([p1,p2], ['measurement', 'filter'], 2)\n", + "show()" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This simple change significantly improves the results. On some runs it takes 200 iterations or so to settle to a good solution, but other runs it converges very rapidly. This all depends on whether the initial measurement $Z$ had a small amount or large amount of noise. \n", + "\n", + "200 iterations may seem like a lot, but the amount of noise we are injecting is truly huge. In the real world we use sensors like thermometers, laser rangefinders, GPS satellites, computer vision, and so on. None have the enormous error as shown here. A reasonable value for the variance for a cheap thermometer might be 10, for example, and our code is using 30,000 for the variance. " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "####Explaining the Results\n", + "\n", + "So how does the Kalman filter do so well? I have glossed over one aspect of the filter as it becomes confusing to address too many points at the same time. In these example we do not have 1 sensor but 2. The first sensor is the RFID sensor that outputs the position measurement, and the second sensor measures our dog's movement using an intertial tracker. How does our filter perform if that tracker is also noisy? Let's see:\n", + "\n" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "sensor_error = 30000\n", + "movement_sensor = 30000\n", + "pos = (0,500)\n", + "\n", + "dog = DogSensor(0, velocity=movement, noise=sensor_error)\n", + "\n", + "zs = []\n", + "ps = []\n", + "\n", + "for i in range(1000):\n", + " Z = dog.sense()\n", + " zs.append(Z)\n", + " \n", + " pos = sense(pos[0], pos[1], Z, sensor_error)\n", + " ps.append(pos[0])\n", + "\n", + " pos = update(pos[0], pos[1], movement, movement_error)\n", + "\n", + "p1, = plot(zs,c='r', linestyle='dashed')\n", + "p2, = plot(ps, c='b')\n", + "legend([p1,p2], ['measurement', 'filter'], 2)\n", + "show()" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This result is worse than the example where only the measurement sensor was noisy. Instead of being mostly straight, this time the filter's output is distintly jagged. But, it still mostly tracks the dog. What is happening here?\n", + "\n", + "This illustrates the effects of *multi-sensor fusion*. Suppose the dog is actually at 10.0, and we get subsequent measurement readings of -289.78 and 301.43. From that information alone it is impossible to tell if the dog is standing still during very noisy measurements, or perhaps sprinting from -289 to 301 and being accurately measured. But we have a second source of information, his velocity. Even when the velocity is also noisy, it constrains what our beliefs might be. For example, suppose with the readings of -289.78 to 301.43 we get a velocity reading of 590. That matches the difference between the two positions quite well, so this will lead us to believe the RFID sensor and the velocity sensor. Now suppose we got a velocity reading of 1.7. This doesn't match our RFID reading very well. Finally, suppose the velocity reading was -678.8. This completely contradicts the RFID reading - we may not be sure from these few values which sensor is most inaccurate, but perhaps by now you will trust that the gaussians expressing our beliefs will correctly handle these cases. It's a bit hard to talk about while working with 1D problems, so we will take this topic up in great detail in the next chapter where we develop multidimensional Kalman filters. Remark\n", + "\n", + "Besides that issue, we are modelling the noise in our sensors using gaussians which model their real world performance. We are multiplying the gaussians (probabilities) when we get a new position measurement, adding the gaussians when we get a movement update. This is algorithmically correct (this is how the histogram filter works) and mathematically correct - why wouldn't it work? \n", + "\n", + "#### Summary\n", + "This takes some time to assimulate. To truly understand this you will probably have to work through this chapter several times. I encourage you to change the various constants and observe the results. Convince yourself that gaussians are a good representation of a unimodal belief of something like the position of a dog in a hallway. Then convince yourself that multiplying gaussians truly does compute a new belief from your prior belief and the new measurement. Finally, convince yourself that if you are measuring movement, that adding the gaussians correctly updates your belief. That is all the Kalman filter does. Even now I alternate between complacency and amazement at the results. \n", + "\n", + "If you understand this, you will be able to understand multidimensional Kalman filters and the various extensions that have been make on them. If you do not fully understand this, I strongly suggest rereading this chapter until you do understand it. Try implementing the filter from scratch, just by looking at the equations and reading the text. Change the constants. Maybe try to implement a different tracking problem, like tracking stock prices. Experimentation will build your intuition and understanding of how these marvelous filters work.\n", + "\n" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "author notes:\n", + " clean up the code - same stuff duplicated over and over - write a 'clean implemntation' at the end.\n", + " \n", + " " + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "sensor_error = 30\n", + "movement_error = 2\n", + "pos = (1000,500)\n", + "\n", + "zs = []\n", + "ps = []\n", + "\n", + "\n", + "for i in range(100):\n", + " pos = update(pos[0], pos[1], movement, movement_error)\n", + "\n", + " Z = math.sin(i/3.)*5.\n", + " zs.append(Z)\n", + " \n", + " pos = sense(pos[0], pos[1], Z, sensor_error)\n", + " ps.append(pos[0])\n", + "\n", + "\n", + "p1, = plot(zs,c='r', linestyle='dashed')\n", + "p2, = plot(ps, c='b')\n", + "legend([p1,p2], ['measurement', 'filter'], 2)\n", + "show()" + ], + "language": "python", + "metadata": {}, + "outputs": [] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [], + "language": "python", + "metadata": {}, + "outputs": [] + } + ], + "metadata": {} + } + ] } \ No newline at end of file diff --git a/Multidimensional Kalman Filters.ipynb b/Multidimensional Kalman Filters.ipynb index 0ef5418..74ac03e 100644 --- a/Multidimensional Kalman Filters.ipynb +++ b/Multidimensional Kalman Filters.ipynb @@ -1,481 +1,403 @@ -{ - "metadata": { - "name": "" - }, - "nbformat": 3, - "nbformat_minor": 0, - "worksheets": [ - { - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Multidimensional Kalman Filters\n", - "\n", - "The techniques in the last chapter are very powerful, but they only work in one dimension. The gaussians represent a mean and variance that are scalars - real numbers. They provide no way to represent multidimensional data, such as the position of a dog in a field. You may retort that you could use two Kalman filters for that case, one tracks the x coordinate and the other tracks the y coordinate. That does work in some cases, but put that thought aside, because soon you will see some enormous benefits to implementing the multidimensional case.\n", - "\n", - "\n", - "## Multivariate Normal Distributions\n", - "\n", - "What might a multivariate (meaning multidimensional) normal distribution look like? Our goal is to be able to represent a normal distribution across multiple dimensions. Consider the 2 dimensional case. Let's say we believe that x = 2 and y = 7. Therefore we can see that for N dimensions, we need N means, like so:\n", - "$$ \\mu = \\begin{bmatrix}{\\mu}_1\\\\{\\mu}_2\\\\ \\vdots \\\\{\\mu}_n\\end{bmatrix} \n", - "$$\n", - "\n", - "Therefore for this example we would have\n", - "$$\n", - "\\mu = \\begin{bmatrix}2\\\\7\\end{bmatrix} \n", - "$$\n", - "\n", - "The next step is representing our variances. At first blush we might think we would also need N variances for N dimensions. We might want to say the variance for x is 10 and the variance for y is 8. While this is possible, it does not consider the more general case. For example, suppose we were tracking house prices vs total $m^2$ of the floor plan. These numbers are *correlated*. It is not an exact correlation, but in general houses in the same neighborhood are more expensive if they have a larger floor plan. We want a way to express not only what we think the variance is in the price and the $m^2$, but also the degree to which they are correlated. It turns out that we use a matrix to denote this:\n", - "\n", - "$$\n", - "\\Sigma = \\begin{pmatrix}\n", - " {\\sigma}_{1,1} & {\\sigma}_{1,2} & \\cdots & {\\sigma}_{1,n} \\\\\n", - " {\\sigma}_{2,1} &{\\sigma}_{2,2} & \\cdots & {\\sigma}_{2,n} \\\\\n", - " \\vdots & \\vdots & \\ddots & \\vdots \\\\\n", - " {\\sigma}_{n,1} & {\\sigma}_{n,2} & \\cdots & {\\sigma}_{n,n}\n", - " \\end{pmatrix}\n", - "$$\n", - "\n", - "This is called the covariance matrix, and is probably a bit confusing at the moment. Rather than explain the math in detail at the moment, we will take our usual tactic of building our intuition first with various physical models. \n", - "\n", - "So here is the full equation for the multivarate normal distribution.\n", - "\n", - "$$\\mathcal{N}(\\mu,\\,\\Sigma) = (2\\pi)^{-\\frac{n}{2}}|\\Sigma|^{-\\frac{1}{2}}\\, e^{ -\\frac{1}{2}(\\mathbf{x}-\\mu)'\\Sigma^{-1}(\\mathbf{x}-\\mu) }$$\n", - "\n", - "I urge you to not try to remember this function. We will program it once in a function and then call it when we need to compute a specific value. However, if you look at it briefly you will note that it looks quite similar to the univarate normal distribution except it uses matrices instead of scalar values. If you are reasonably well-versed in linear algebra this equation should look quite managable; if not, don't worry, the python is coming up next!\n" - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "import numpy as np\n", - "import math\n", - "def multivariate_gaussian(x, mu, cov):\n", - " n = len(x)\n", - " det = np.sqrt(np.prod(np.diag(cov)))\n", - " frac = (2*math.pi)**(-n/2.) * (1./det)\n", - " fprime = (x - mu)**2\n", - " return frac * np.exp(-0.5*np.dot(fprime, 1./np.diag(cov)))\n" - ], - "language": "python", - "metadata": {}, - "outputs": [], - "prompt_number": 1 - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Let's use it to compute a few values just to make sure we know how to call and use the function, and then move on to more interesting things.\n", - "\n", - "First, let's find the probability for our dog being at (2.5, 7.3) if we believe he is at (2,7) with a variance of 8 for x and a variance of 10 for y. This function requires us to pass everything in as numpy arrays (we will soon provide a more robust version that works with numpy matrices, numpy arrays, and/or scalars in any combinations. That code contains a lot of boilerplate which obscures the algorithm).\n", - "\n", - "So we set x to (2.5,7.3)" - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "x = np.array([2.5, 7.3])" - ], - "language": "python", - "metadata": {}, - "outputs": [], - "prompt_number": 2 - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Next, we set the mean of our belief:" - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "mu = np.array([2,7])" - ], - "language": "python", - "metadata": {}, - "outputs": [], - "prompt_number": 3 - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Finally, we have to define our covariance matrix. In the problem statement we did not mention any correlation between x and y, and we will assume there is none. This makes sense; a dog can choose to independently wander in either the x direction or y direction without affecting the other. If there is no correlation between the values you just fill in the diagonal of the covariance matrix with the variances:" - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "cov = np.array([[8.,0],[0,10.]])" - ], - "language": "python", - "metadata": {}, - "outputs": [], - "prompt_number": 4 - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Now just call the function" - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "print multivariate_gaussian(x,mu,cov)" - ], - "language": "python", - "metadata": {}, - "outputs": [ - { - "output_type": "stream", - "stream": "stdout", - "text": [ - "0.0174395374407\n" - ] - } - ], - "prompt_number": 5 - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Let's check the probability for the dog being at exactly (2,7)" - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "x = np.array([2,7])\n", - "print multivariate_gaussian(x,mu,cov)" - ], - "language": "python", - "metadata": {}, - "outputs": [ - { - "output_type": "stream", - "stream": "stdout", - "text": [ - "0.0177940635854\n" - ] - } - ], - "prompt_number": 6 - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "These numbers are not easy to interpret. Let's plot this in 3D, with the z coordinate being the probability." - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "%matplotlib inline\n", - "pylab.rcParams['figure.figsize'] = 12,6\n", - "\n", - "from mpl_toolkits.mplot3d import Axes3D\n", - "\n", - "xs, ys = arange(-8, 13, .75), arange(-8, 20, .75)\n", - "xv, yv = meshgrid (xs, ys)\n", - "\n", - "zs = np.array([multivariate_gaussian(np.array([x,y]),mu,cov) \n", - " for x,y in zip(np.ravel(xv), np.ravel(yv))])\n", - "zv = zs.reshape(xv.shape)\n", - "\n", - "ax = plt.figure().add_subplot(111, projection='3d')\n", - "ax.plot_wireframe(xv, yv, zv)\n", - "show()\n", - "pylab.rcParams['figure.figsize'] = 6,4" - ], - "language": "python", - "metadata": {}, - "outputs": [ - { - "metadata": {}, - "output_type": "display_data", - "png": 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MFoNYyyWFvu9mg3RsKcaaxMCxCOtnn32GnTt3okePHjh48CCGDRuW0fhqa2vxq1/9Coqi\nYMaMGVi4cGGr18ydOxcvv/wy3G431qxZg/79+wMApk2bhpdeegkdO3bEJ598Enn99u3bMXv27Igl\nbfny5aiurs5onOlCYpUoGKn4ULPh4SykWJVlORItttvtEATBklYFM1iprEKU+8o18R5AyWQf699P\niV9th2z7Zdn/t2e/bFvf50TXDKt0wmwABw8exMaNG/HNN99gz549eOKJJ9CjR4/In5NOOgn9+/fH\nhRdemHDbiqJg9uzZ2LRpEyorK1FdXY3x48ejT58+kdds3LgRu3btQl1dHbZt24ZZs2Zh69atAICp\nU6dizpw5mDx5ctTnLliwAPfeey9GjRqFl19+GQsWLMAbb7yRwVFKHxKrRF5J14dq7MiUDvkWq/oo\nKvOssWhwvkl1341jFwQhbYtCKtu2UgQqXuQNSK27E9WUtda5zZR0/LLGOtBmkfq2GrFvS+c+E2w2\nW6Say+jRozF69GgAwGWXXYZVq1bh6NGj2Lt3L/bs2YO9e/fi3//+d1Jidfv27ejVqxd69OgBAJg4\ncSLWr18fJVY3bNiAKVOmAABqamrQ2NiIAwcOoHPnzhg6dCj27t3b6nO7dOmCpqYmAEBjYyMqKysz\n2f2MILFK5Bz9Mj9r+2kWeWI+SH23k2x6OPMhEsyWyllnLJYlXwiS2XdjFDufXb2K8UGWjQQfn89H\niV9tEKNfVpZliKIYiaplKyu9GGiPEzMz4gn2o0ePomvXrrDZbOjbt2/Kn71//35079498nO3bt2w\nbdu2hK/Zv38/OnfuHPNzFy9ejHPPPRfz5s2DqqrYsmVLymPLFiRWiZxgFKgtLS2RZW+jl8tY7iiX\nAikfGfHFtlRerA0HrE68BB9mrRBFMa5gocSvtoFeqCTrl41lPUnVL2sVrDimfBJLrLJnUqarhsmO\nIZX3TZ8+HUuXLsWECRPwzDPPYNq0aXjttdfSHmcmkFglsorZchfHcRG/FovwGVuGiqIIQRBynmSU\nTVKNBBdy+de4bbNJQroNB1LdNnFMsNhstqQTfJKpLVuMkTcimnSSv6xcd5gsAGHiHYdMj1FlZSXq\n6+sjP9fX16Nbt25xX7Nv376Ey/rbt2/Hpk2bAAC/+MUvMGPGjLTHmCkkVomMScaHynFcJMkoV7VE\nE5EN0RRvmT/RzabQok0/9kI2HKAHV2LStRhYuewSTViyQzbsJ/n0y5JYjX/tM8tVJgwcOBB1dXXY\nu3cvunbtiqeeegrr1q2Les348eOxbNkyTJw4EVu3bkVFRQU6deoU93N79eqFt956C8OGDcPrr7+O\nU045JaNxZgKJVSItUvGhSpIEVVXh9XohiiJKS0tzEsFLRLpiMVvL/IUSq6qqIhAIAAh7JLORrJYK\nye43e117f7AlQzyLQaxl5Hhll/IVeWuv5zaf13WiawM41vO+rftlrYL+uBo5fPgwOnTokNHn2+12\nLFu2DKNGjYKiKJg+fTr69OmDlStXAgBmzpyJMWPGYOPGjejVqxc8Hg+efPLJyPsnTZqEt956Cw0N\nDejevTsWLVqEqVOnYtWqVbjxxhsRDAbhcrmwatWqjMaZCZxG010iSfQCNV49VLbELElSJPooyzLc\nbnek1Vwh8Pv90DQNbrc7qdebFb4XBCHthK9QKAS/34+ysrK03p8KZudAkiRUVFTk3Yva3NwcqSXL\nxmV20/Z6vXC5XO3CK8smP8lei9nEKGSNkTggN5E3WZYRCoXgcrmyuTuWR9M0eL1eeDweyws9s4mO\n/vpIxy/bXs+7nnjf908++QR//vOfI8KSMIciq0RCYvlQjfVQZVmOCCRj9PHo0aOFGn4EjuMi44+F\nUeSxeqjZSPjKdWQ1UQT4yJEjln9YErkn3jJyqpG3YknusQLFcFyy4Zc1Xhvs9+151STeff/w4cM4\n4YQT8jia4oTEKmFKMj5UoHUmuSAIpm03C+3XjDeGREI719vPFGNNVKu1bTXuNy33Wxe9UMl2ck+h\nv/9E5qTjl2WTHa/XC6D9JgbG2reGhgYSq0lAYpWIYOZDBWK3PWU97ZNpe2qFh5VxDIqiRPYDSNzf\nPhtk6xikk+hlBZHIjjmAdvOQaktkmvgFtL/asoX+zuUTM78su+8wC5jRXhCrI1xbuj7iXQOHDx8u\naLH9YoHEKhG5YSTyoRrFUSo97a0iVvU+VH3JpnhCO5vbzxQWyQ4Gg5HOXslGgAt5DiRJiniGWWF0\n40NK07TIftHScnESL7lHkqTId04fdUvGD0m1ZYsXY3JRoqi9MSpb7PVlgcRi9cwzz8zziIoPEqvt\nlGR9qGYeyHR62hdSKLFl/kAgEIns5aOuq5F0j4Exkm21ZX4z2DFnZbJsNlsk8suy0tmxZw8i1jub\nPazaQ03RYBCYMMGFrl013HZbECef3HaXytm5SsVioPfLAsV5HbSnyKoZmqYl9bzIxC9rFLNWqC9r\nHHus7R85cgTHH398nkdUfJBYbUek4kM1Lo9nKo6SSW7KNvr94DguUpWgtLQ0r+MwkszDyyj27HZ7\nSpFsM/IxYdBHrjku3A2LLQHGqiWovwbtdnur68zsARVr6bCYonGHDvH4+c89+O47Dp9/ruGFFzzo\n0UPFpEkyfv7zEHr3brvC1YxULQbJLCEXQ9SNSI50LShWmOzEE+wNDQ3o2LFjzrbdViCx2sZJ1Yeq\nX6rL5vI4z/MIhUIZf04iWDa/2TI/E1KFIpnjWIytT40WEeO1o7eXpEO8peVUE36s4oH75BMbfvnL\nUjQ3c3j1VR9OP13FmjUO3HOPiKeesmP5cgc6ddJw8cUyLr5Yxqmn5neilwsyjTCmeh3Eqi2b76gb\nRVbzs/+Jrg8AKU92snWvoMhq5pBYbaOwLyIrBs8Sb8x8qJIk5byjUS6jemZRSLP9sMIDgx0H43nI\nR+vTbJ8Do3820woK6Ywv1WiLWRmmfHsk16+346abRJSUqFi0KIiqqrAQnT49hAkTQrj0Uhd27uRx\n0kka3n3XhtWrHSgt1dC3r4rHHw+gpCTrQyp6shV1a2uJPVbACmI9nr0AyL1fNt4xYBN8Ij4kVtsQ\nZj5UWZbBcVzky2DmQ81HR6NciNVUo5BsDIW8eerHoC+XVajWp6liFNbJWkT05z6f/mUrRWU1DViy\nRMDatQ5ccIGM5mYV1113bLXh+++BGTNccLuBM89UMXCgjP37eXi9HAIBDv/4B4833yzB/fcHceWV\nIVjYsmw5krkOjEIlmUkNWQzaBtn0y5rdL1RVNb1G2HuJxFh3bZFICnZjlSQJgUAgKnlFX5RZVVX4\n/X40NzejpaUFHMehrKwMZWVlcDqdOV9mzpZAYa1Dm5ub0dzcDE3TUFJSgvLy8oT7YZUHSiAQQFNT\nE7xeL3ieR3l5OUpLS3Oe8JXuOWDC2uv1orGxEZIkwel0oqKiAm63O6FQtcpxN8K+I3a7HQ6HA6Io\nwul0wu12o6SkBB6PB6IoRiYQbFnZ7/fD6/XC6/XC7/cjEAhEVieYyDEeZ58PuOYaJ157zY677w7i\nnXfseOihZrBD89lnPIYP9+DUU1WsX+/H3LkSPvvMhv/93wD+3/8LoqxMw5AhCpqagFtvFdG3rwfP\nPkuxhmzArgObzQaHwwFBEOB0OuFyueDxeODxeCJ+cX0li2AwCJ/PB6/XC5/PB7/fH5k8s7yAQk+O\nrUBb2P949wp2jbBgA0sSZdYoVl82EAhErpH/+7//w0svvYRPP/00Y6tdbW0tTjvtNPTu3RtLliwx\nfc3cuXPRu3dvVFVV4cMPP4z8ftq0aejUqRP69evX6j2PPfYY+vTpg9NPPx0LFy5Me3zZgu52RYje\nh5qo3BR7wEqSFImgZqMbU6pkIlazaVcwW4bPNfobF5s4ZKsrVi5hHl9JkqBpWl7q0FqJdDySZskc\n335rx+TJZejbV8GDDwYwYYIHTz7pxY9+FP4+PP+8HfPmiVi8OIjLLgv7yseNk3HbbSL+/ncbfvc7\nAa++6kPv3hruuEPAv/5lh8ejYcYMJ9avl/GXvwTyc0DaKelaDPS1ZYGwWGmPiV9tQawmIt69QlVV\n+Hy+qDqzn3zyCb788kt8/fXX+Prrr9GpUyecdNJJUX/GjRuHzp07x92uoiiYPXs2Nm3ahMrKSlRX\nV2P8+PHo06dP5DUbN27Erl27UFdXh23btmHWrFnYunUrAGDq1KmYM2cOJk+eHPW5b7zxBjZs2ICP\nP/4YDocDhw4dyuTwZAUSq0VErHqoZm1PmX+T3RjLysoKesNIZwle74nkeT4ryUb5XIJm50Ff9ktV\n1YjQzjfJ7LsxWSqZZgPZ2G6xEU/A6JM5tm3jcc01JZg504fqagljx1bA4dBwzTUenHeeDcGgDZ9+\nasezzx5FVZUGTeN/sO0AV1wRwo03uvDb3wYilQHuuEPC+efbcfnlMq69VsLUqS68/z6PAQOsn4DV\nVkVLokmNJElQVTWS5Bkr8SuW3aSYaWvf+3ThOC4qB2Hx4sUAgF27duGhhx7CkiVLsGfPnsifLVu2\n4Nxzz00oVrdv345evXqhR48eAICJEydi/fr1UWJ1w4YNmDJlCgCgpqYGjY2NOHDgADp37oyhQ4di\n7969rT738ccfx2233RZ5TlmhwxaJVYtj5kM1e0gy/6NR2CmKAr/fX/CbXrLbN6spWlpamrVko1wL\nJ2PrU0EQojydLEpZCOLtu5n/N516urkaXzHBrvXHH3fi7rtF3H13ECUldlxzjRvHHw8sXx5AaWkI\n11xTgoYGHrIMPPSQiJ/9LIDzz/fB4wmXudmzxwGvFxg1KgBZDkdqHQ4Oq1YF8LOfufD66z506aLh\niitc+OADLzyeAu840Qr9pMZsgqqP0CdTbqlYE7+KZZy5IN4k7dChQ+jYsSO6du2Krl27YsiQISl9\n9v79+9G9e/fIz926dcO2bdsSvmb//v1xhXBdXR3efvtt3H777XA6nfif//kfDBw4MKWxZRsSqxZE\nv6yoKAoA83qoyQg7/U2v0MRagjcu86faHSudMWQTs/HHikZaSZDlqwpBe+SbbzjceaeIkSNl3Hef\nCI4D7rsvgPvvF+F2A1deWYpx4wK4914VjY0cXnrJjmeeKcH8+eUYOlRG584KPvrIgQsvlPC3v9kx\nfbo/ImZOPJHD7NkqZs4UcdNNPjzyiAu33ipg6dJguxYFViWeWEk2Qh8v8SuekC309dBWo+mpEO8Y\nNDQ0ZBS1TPbYGp85id4nyzK+//57bN26FTt27MBll12G3bt3pz3ObEBPJYuQig+VZZAns0xrJXFk\nHEusaHCxVCXIhU0hl7B9N7Mn5DK5iyX56X9uy4RCwPjxLpxwgoZDh3gMGyZj0CAFd94pQhTDHat+\n/3s/xoxpgd3uxvHHa5gyJYQpU0JobAQ2brRj7lwnLr44hGnTFMye7cENNwA8f+w+MXu2jNdec+LI\nEQ5eL4dNm2x49lkZo0cHqQRTG0GfnZ5uhnqs6yBfFgOrPHsKSTyxevjw4YxqrFZWVqK+vj7yc319\nPbp16xb3Nfv27UNlZWXcz+3WrRsuueQSAEB1dTV4nkdDQwM6dOiQ9lgzhcRqgUnFh8pEaio1LdPx\niuYKfUZ1rGXyfIwhkxuoWemmVGwKhZo8sM5lTGAXQ8vWYuX220XU1/MoLdUwerSMX/9aAs8DL77o\nwCef8Lj4YhkTJoRg1p+iogLo21dFly4a3n7bjquvluF0anjzTRvOP1+JCE5R5LFyZRDDh7sxapQC\nQMNtt1Xg3HO96NhRiYrExasVmU/xwsRTeyNX+51p4pf+/bmc2BT6uVNoEkVW+/fvn/ZnDxw4EHV1\nddi7dy+6du2Kp556CuvWrYt6zfjx47Fs2TJMnDgRW7duRUVFBTp16hT3cy+++GK8/vrrGDZsGHbu\n3AlJkgoqVAESqwUhWR+q0UcoCELK2dhWuFHob5ItLS1FV5XAmLSWjWoE+cAs2c5ut2dUuD8dzPbZ\n7AZupVWAdNmwwY61a8PexJUrAz8ISeDgQQ6ffsrjqqtCePFFB77/noPbHfszJkwIYehQBbNmOXHj\njRJWrXLg/POVqNedeKKGRYuCeOQRAUeOcJgyJYRZs1x4/nk/HI74kTgqjN/2SaaahV7IJlsEP5WG\nGVYIkhRWr4SzAAAgAElEQVSaePe0TG0Adrsdy5Ytw6hRo6AoCqZPn44+ffpg5cqVAICZM2dizJgx\n2LhxI3r16gWPx4Mnn3wy8v5JkybhrbfeQkNDA7p3745FixZh6tSpmDZtGqZNm4Z+/fpBEASsXbs2\n7TFmC04r9qdDkRDLh6r/G2jtQxUEAaIoZlSLrbGxEaWlpXmNoulrEbJoMCt/5HQ68zYOI6y2qcvl\nSvhaVr+W9bln5yKTKInP5wMAuGMplSzASk6xcbNlfhbNzuW2zWBiueSH1kuyLEOW5VbHUX+ci5Ev\nvuDw0596EAoBL77ow/Dhx6wPixYJWLZMwK5dLVi40IkePWTMmdNsei4GDnRjxYoABg5UsXChiPp6\nDps32/HXv/pw7rnR/nNNAyZNcuLzz3nMmydhzRoBv/hFCDfckFprYzPxohe12YjKBgKBSC3T9oTf\n74fD4SgqL3i8a8Es8ct4TbDrgdUeLuQ9v9DEu6/NnDkT99xzD3r37l2AkRUXxfPtKUJS8aEaywVl\nM8EonxErY21O/TI/K45cSBIdi1hJR5kWbk52++livIbMxm30jhLZQ1GAESM8cLs1XHaZHCVUNQ14\n4gkBV1wRQlkZMHeuhHHjXLj2WrSKrn75Zbhj1Vlnhd9/111BVFV50NIC/PKXLpx4ooYxY2T87Gcy\n+vdXwfPAY48FMWCAG488IuLpp30YOdKNYcMU/OQnyZ9r/T0pkT+SorJtn1QtBiwQY7QYsNeylR2r\nJH7lk3g2kEwjq+0JEqs5gC2/JvKhGiOPuSoXlGuxaia2zZb5rbDMG2tJ2uxc5CLpKNuCUZ/klYqX\nOZ/oj7miKAgEAlBVNSJu2kI9ycWLBQSDgCgCt98ebUZdu9aOo0eBu+8O/75PHxVnnqngmWdcmDkz\n+nM2bLBj7FgZPA8cOMDh+uudOOEEDS0twOmnq7jvviD+8Q8Hrr3WBa8XEeH6yCNBzJjhxL59HO69\nN/zvN97wIVsBrURZ68ksKev/vy3VEk1EW1wKT8ZiwJK9AMSsLVuoxK98Eu/8e71elJaW5nlExQnZ\nALKEPoKqKAqamppQUVHRahapX1oGEBFFuVyib2lpibSIyxbpCDy/31+QZWg9gUAAiqLA4/G0qoma\nj3NhXBJPB7MkL2YVyfW20yEYDEa69yiKAofDYVqWBzj2EEzXI1cIGhuB004rQWmphptvlqKW4H0+\noG9fD3r10rBpky/y+7ffBm66yYX33vNDf9qGDnXjt78N4uhRYM4cJ6ZODWHhQglLlzqwaJGI+voW\nsGfbzp08XnrJjpdesuM//+HBcRpEEdi504vJk52orNSweLFJFleeYQIlEAhEtYBmgsYs0actReF8\nPl9S38+2iNkSuFnil5nFoK1E6WOdf03TMHr0aGzevLno9qkQUGQ1C2iaFhE8xtkhuzGzTP5cLC0n\nIpsRTbMWnMlmlVtlGVpRFBw9ejRrHZryQTaSvPId2WbjZQ8sj8cDh8MRibgYVxqMnX7Y66xeHP13\nvwvXUbXZgBkzor2ijzwiwOUCJk+O/v3gwQrKyjRs3GjHuHHhFqt79nDYt4/DM8/Y8c9/2vGXvwRw\nzjlhf/uvfhXCQw+J+PWvRaxaFRagp5yi4pRTJNx8s4SDBzn85S923HOPiPHjnVi6NIAxYzwYOVLG\niBHRiVn5Ri867XZ7qzrQZlHZ9hqFaw+kk/il/zlW4pdVJzeJIutWG69VIbGaBWJ9SVgt1EwzyLMx\nvkxEChMSbH/SzeYvpA1Av1wOAC6XqyDL5akeA32yFADL13JlEzd9aTKPx4NgMBg3cUofXYvV6cdY\nHN0s+SffgqaujsPatXYEg8CSJUHod7G+nsPKlQ7IMocxY2TD/gI33ODFo4+WRsTqE084IMtAczOH\nzZu9qKg49nqeB666KoQ1axyYPj2EmproSV/HjhpOOkmDzQa8/bYdP/1pCUaOlDFzphPbtvnQoYM1\nF9CS8cqmWn7JasKlLdoAkiWeX9OMVLzT+ntCrEltoSc3+vuTEVZCkEgOEqtZguf5iAWAPagDgQCc\nTmfBxUU6EU0WyUuntmu8ceRTrJp1+HK73QgGgwXLTk3mGJh5gD0eT8bR31wmd+ktIXa7PSpBkPnW\nMoFFY1JJ/slHl5/f/MaJLl00NDcD48dHC9I77hAxYoSCb7/lcPzxrY/7mDFBPPBAGd5914YPPuCx\nYoWAWbMk3HefBLMhjR0r45VX7Lj2Whc2b/airOzY//3nPzx+/WsRl10WwrPPOtCxowq3W0NTE4eL\nLnJhxw5f6w8sAlKJwrEkH4rKWodsC/V43mm2PStObsw+9/DhwwWvXVpMkFjNEoFAIOLJFAQhEkm1\nQhkeJqSTwcxTm63i8fkQq2bL5XrhJMtyQZO84h0Ds45YVkuW0mPm+TWrA5zr855K8k827QX//KcN\nX37Jo76ewyOPBKIE5rvv2rBjhw1DhigYO1Y2fb/NBsycKeHqq8OC1+0G7rrLXKgCQHW1goMHOYwb\nF8K8eU6sWhUAABw9Clx5pROLFgXx4x9reOstOySJg8sFbN/uRf/+Hjz/vB2XXGI+jnyRK+ECWDsq\n294jq/kkW5ObWPeDVIl37g8fPkyVAFKAxGqW4Hk+amnc6/Vawp8JFL5cU7LjyARjA4VYy+WFtCKY\nYXbsU+mIlQrZ2HfjZCCbnt9ceJqTFTTJ+uPYvxWFx223iejVS8F334U7TTEUBVi4UMSkSSE89JAA\nSdLw8ss2DB+uwFjed/duHt9/z2H2bAmff25DvLmtKAI1NQrOP1/B4sUinn3WjksvlTFnjhODBim4\n+moZgQDQ1MSB54HaWjsGDQq//q67hIKL1XyTqXBJpo5oIqx0rykUVhHqqVgM4pVnS2ViG0+sUtmq\n1CCxmiVEUYyKXlpJFJmNxSwCmY8e8dk8JrFEdjyhV+jzwrbPjj1bNi+UnzlZsumdtdI+pmMv0DQN\nTzzhRFmZjM2bHZgwIYBQKBh5gK1ZI+DwYQ5//KMD5eUaBg5UsXSpgGuvtWH4cBljxsgYOVLFzp12\nvPCCHYMHK3jqKQd+8xsp4XjPO0/Gli02rF7tx6WXurBzJ4/du3m8+mp4md/pBM4+W4EkAVVVCubN\nE/GnP/lx8cVuvPcej4EDrTGBLjTZjMomYy2x0jWfT4opqpxohQZAwomt8Zpg7zM7DocPH8bxxx+f\n+x1rI1gzS6MIMV6IPM9bJrKqH4uiKPD7/WhqaoLP54PNZkN5eTlKS0shimJObyzZOCbsoeH1etHY\n2AhJkuB0OlFRURHxdcajkGKVLZuztrM8z6OsrAylpaU5nSQwUt13dqxbWlrQ1NQUKflVXl4Ol8uV\ntFAt9AQhXdiDx263R7qXuVwu+P1uPPpoKTp04OHxAFdffayW5PvvK5g/34UuXUK4+eYWDB0q4dpr\nW7BhQzPef78Jo0aF8I9/2FFVVYZLLz0Ow4fLqKpSUFfHY8SIxJHP4cMVvPmmHf37q7jsMhm//72A\ntWv9UfVUzztPQceOGv7xDwfuuEPCrbc60bevinnz2m8XoVRhkxjjufd4PPB4PHC5XJFyRHrPtt/v\nh9frhc/ng9/vj0zuZFmOKtHWXigmsRoPJkRZBzZBEOB0OltdE6z0ob6qhaqqkWtix44duOWWW7B0\n6VJ89tlnCAQCaGlpSWtMtbW1OO2009C7d28sWbLE9DVz585F7969UVVVhQ8//DDy+2nTpqFTp07o\n16+f6ft+//vfg+d5HDlyJK2x5QISqznCag9oVVXR3NyM5uZmqKqKkpISlJWVpSQ6skU6x0VV1YjI\nZi1TmchOR+jl69wYBZ8shwUJE3xWrL2oP9Y+nw92ux3l5eUoKSmxdPQ3X/z2tyKGDFGwbZsNDgcw\neDAHQRDxwgslGDv2RzjzTAWvvRbErl0iamrUH2wDCsrKJFxySTNWrTqMG25owUknKbDZFKxe7YCi\nAA6HkvC6/MlPVDQ1Ad98w8FmC9dVbWqKPh/Dh8v47DMenTppOP54DWeeqaJzZxUffhj21xaKtiRc\neJ6PCBfWQtrtdkeEiyiKUd+VUCgUEbJMuAQCAUiSFGlJ2tbEbFval0ToJ7b6a8LhcEQSZEVRRIcO\nHdCjRw/s3bsXW7ZswR/+8Ad07NgRHTt2RE1NDSZNmoRHHnkk4fYURcHs2bNRW1uLzz//HOvWrcMX\nX3wR9ZqNGzdi165dqKurw6pVqzBr1qzI/02dOhW1tbWmn11fX4/XXnsNJ554YmYHJcuQDSBLWC2y\nalzmB5DzZf5EsNlpsg8tY1Y8K4OUiT8yX/uuT1TTe2g5jsP333+flzEYiTeBMjvW2fItW23ilglf\nfMHj2Wft4HnAbgfOOktBYyMwb54TH3zAw+kE/va3AGw2Du+9Z8c118itkix37QL++EcPNm5sQK9e\n4eXCZ54RcfvtDtx1V3PCxJ9hwxS8/rodL77owOTJIfzhD0Ik2QoAzjhDxeHDHK67LoiVKx14+mk/\nRoxwo7xcw69/LeLppwPG3SKyiN4ry/M8ZFmG6wezcrqll6xQTzhdinHM2UJvD7DZbOjZsyfmzp0L\nALjpppuwatUq9OvXD9999x12796N3bt3J/W527dvR69evdCjRw8AwMSJE7F+/Xr06dMn8poNGzZg\nypQpAICamho0NjbiwIED6Ny5M4YOHYq9e/eafvYtt9yC3/3ud/j5z3+e/o7nABKrOaJQD+hYiUaN\njY2WiIolc1z0fs5ctBBNRTCnQi4FXzYwO/ZWatdqdVGracCtt4ro2lXDzp08jjtOw6ef8jj55BKc\nfrqKrl01TJsWRMeO4VJWe/bw6NdPbfUZt9ziwq9+FUSPHioEwYl//9uBkSNl/OUvLkyYAAwaJMdN\n/BkyRMMzzzjhdqu4+eYWVFdXYN8+DZWVLMIDDBumwOkEdu3isWcPj7/8xY+f/tSDV18Nt36lDo/5\nwXifieeLZK83Cll91DVVr2whaSuR9ExgiXpmHDlyBCeccAI4jkPnzp3RuXNnDB48OKnP3b9/P7p3\n7x75uVu3bti2bVvC1+zfvx+dO3eO+bnr169Ht27dcMYZZyQ1jnxCNoAsUcjIqqqqCAQCkWV+TdNQ\nUlKC8vJyOJ3OyI3NCkIgliBh+9DU1ISWlhZwHIeysjKUlZVl3UubbVEky3LEQxsIBCAIQpSH1izZ\nopDngiWm6a+X0tLSnBzrWBTjQ6y21oZPPuFx4ACHCRNC8Pk4+P0cFi8OoFcvFe++a8MvfhG2eXzw\ngQ1nnKG0yu7/61/taGzkMGtW2MvY2AjU1/OYPj2E7t01TJvmwuHDtrhLzBdcAOzYIWDCBAnl5Qou\nuSSAxx/nI0vMfr8fQ4YE8NZbPKZODeLxx+3o1UvFsmUBaBqwYAEVIrcqRq9sMr7IYDAIn8/XyivL\nmri0NXtBMRFPsB85ciTtBKt0q1HEe5/P58MDDzyAe+65J+b7CwmJ1RzBBEmuTrbRCxkKheImGhVa\nIJmNg4kmvZ/T7XajvLwcbrc7Z37ObBwLTdMi4vro0aM5FdfZglWrYKJaFMWkE9MywSrXXiZIEnDT\nTU7IMhAKAV9+yaOkRMPmzT7MnClD04BBgxQ8/HBYnW7fbsPZZ0dPVg8e5HDHHSIeeywAdrh37LCh\nf38Fw4Yp+PZbHhMmhDBtWng7ZnAch27deIRCQP/+gNPpxJw5Kv7v/9wAPJGawuedp+Cttxy44ooW\nbNjgwDff+DBqVCPOOCOEdescOHo02Ga9klYim9HFWL5It9uNkpKSVl5ZZi9I5JXN1XOKIqvxxZ6q\nqmnfdysrK1FfXx/5ub6+Ht26dYv7mn379qGysjLmZ3711VfYu3cvqqqqcNJJJ2Hfvn0YMGAADh48\nmNYYsw2J1SxhXILR+zOziaIo8Pl8aGxsjEp+SZRoZBXBwJJN2D74/f68J/Ckeyz0E4TGxkaEQiG4\nXC5UVFSkJK7zeS40Ldz+tLm5GUePHgWAvEdRY42rmLj3XgENDRymTw/h1FNVfPGFDXfeGUSXLhr2\n7+fw+ut2LF8ewDPP2PHVV9wPYjW6EcfChSKuvFJGVdUxEbttmw01NQpEETj3XBlnnqmC54H7749d\ncPXtt2047jgNu3aFb98nn6xh8GAF69YJETHTu7cNbjfg87kxdqyCZ54JX6MbNvigqsBNN5W0ymBn\nUdlcReVIvOSWdKOyeiHLzn8oFMr4/NP5DmN2DDL9Tg0cOBB1dXXYu3cvJEnCU089hfHjx0e9Zvz4\n8Vi7di0AYOvWraioqECnTp1ifibzzu7Zswd79uxBt27d8MEHH6Bjx44ZjTVbkFjNIdkSJfol8ubm\nZgBhwaFf5k+EFRK+mJfT5wvXhEx1H7JFqufFWIlAX+4rnYS1fIhVvTVBX94rnl8u17B9juets6KI\n3bWLw2OPCfh//y+ITZvs+PZbDnY7cMEFYTG6erUDl18eQs+eGmbPDmHRIhE7dthQXX1MrL7yig3v\nv2/DbbcFoz5bL2ovvFDBpk12rF4dwN/+5sDLL5tPfp591oGzzlLwxBMCXn3VBr8fmD07hOXLBei/\n4sOHy3jjDRuuv17C6tUCFIXHccfZcNZZCtavF+FwRNsL9LV+zaJyfr/fNCpHWJ9Uo7LMJ53M+adr\nIDaxBLuqqhm1/LXb7Vi2bBlGjRqFvn374vLLL0efPn2wcuVKrFy5EgAwZswY9OzZE7169cLMmTOx\nfPnyyPsnTZqEwYMHY+fOnejevTuefPLJVtuw2kSD0+hKyxr6jE4AaG5ujizLpQqL4rEbA6vtlm7k\nkYkspzN/tRbNGg9omgaHwxHJji0ELS0tkRt2LIzJUuz12ejUlMl1EQ82IWDtT1l9SH3E9/vvvzdt\niZprjhw5guOOOy4i1FkCoB4W6XG73XkdWyLGjXNh3z4Oq1YFcOmlbpx6qoKDB3l89JEXfj/wk594\n8MorPvTurcHnA844wwMA2LXLCwBoaQFqajx47LEAzj8/LEzDqwtBnHZaR3z8cQs6dADq6zn89Kdu\n7NrlxXvv8bjiChf++U8fevQ4dovet4/DmWd6UFqq4ehRDgMGKPj0UxsGDVLwxRc85s8PYurUsIfg\nuefsePppB556yo+LLnLhuutCuOQSGdu387jgAjceeyyAKVMS13aNlcFu7OxjlsWun5i0tLTA4/FY\n7iGYS5ioy+d9N9ske/6NiV8sOayQ9/pComkavF6v6TXf0NCAOXPm4O9//3uBRld8UDWAHJJOBE2W\n5UjJI9YfPt1OQZmOJV2MZZsEQYjsA4uqFpJ4x0KfHc+Of6Gy45NBv5zHumExIWw25kLaQYwTAOPD\njb3GSgQCwObNNqxcGcDDDwsIBjWMHClj9+7wBOC558IF+nv3Do/b7QYuvFBGba0dmgZwHHDvvSKG\nDlUiQpXx5Zd2dOmiokOH8M/du2vo1EnDBx/wqKlRccstEiZPduHVV32w28MR3EWLBBx/vIbt270Y\nOtSDRx4JoksXFW+8YceqVQ7Mm+fEY4+Fx1hTI2PzZhtCIeD660NYvtyBSy6RcfbZKlwu4O67RUye\nLCPRpR0vgz1WZx/2bxZZYu8NhUJFX4qpvZFsBQNjZyfmkWd1sc3KcbHPb8uY7R91r0odEqtZJN2K\nAKyzUTAYhKqqEEURZWVlWU0wYjPdXMGiZcbWp8ayTbkeRzIYBZtx7KIoorS0NGdJR9kQjPprRtM0\niKJYkIhpMrDz3dzcHCmnxqLs7EGnFzisu1esOpP5fLg9/LADghDuCjVzphMLFgRRV2fDkCEKNA1Y\nsULAXXdFL+3b7YDNFq4ecMIJGp5/3o6tW72tPnvHDgdqaqIF7IUXynj1VTuqqyXccEMI27fbcM01\nTtTX86io0FBTo2LsWBllZUB1tYIdO2yYPFnFhAkyxo6V0a+fBwsXBvHNNzwef1yE1wssXSrgppsk\n3H67iA8/5NG/v4px42Q8+6wdL79sw5gxSquxJQs7F/Fa1upLMLEJS6JSTG1FyLQH3ybzyhrPv36F\nRx+J1bcpBcyjsm1hMhPv3B8+fBgnnHBCnkdU3JBYzSHxRInZMrPb7c7KMrMZPM9HZrrZwhjVY3U6\n4/k4czGOVGGCWV/P1W63R3n2cr39dBO89LaKdK6ZfEVWjdc3ALjdbgiCENcGEAgE4Ha7IwLHrM5o\nPh9uf/yjgHHjQpg/X0RZmYb580Po31/AvHkS3n037BcdMSL6en7vPRvmzJFw550ibDbg/vuDkehp\n9OscGDYseuI2apSC228X8ZvfSDh0iIPNBrzyih3XXBPCokVB9OlTgieeCBf1P/tsBdu385g8Ofxe\nhyMcQd20yY4nnghg/nwJU6Y48eCDDpx7roxrrw1hxQoBK1cGcMUVIfz973b85jciRo/2JYyupgsT\nMjzPIxgMRllvzJaXzYRMInsBYV3Y+UslKq+/JsyisfrfWZl4YrWhoYEiqylCYjWLmEVWjcLMWPCe\ndWXKdUQsmyLFLKqXbCS40FUJmEiVZRmhUKggEcl0EryYNYFFJtO1huT6+JuNtaSkBI2NjQkj1ez7\nEytSA7Qumh7r4ZaN7j/vvMOjoYHD9deHcMEFbvzhDwEcOsShuZlD794q7rvPiZkzQ9CfhqNHgd27\necycGcIf/yjA7dbwy1+a+0J37HBgwYLoqOzZZyvYvZvHkiUOrFgh4KqrZLzyig+XX+5C9+4qamoU\ndOgQPn/V1eFWrXqmTJFQVVWC//6XQ9euGkaPlvHttxyuusqFp5/24eGHPTh4kMOQIeH70jff8Ni8\n2YahQ/M/gcyWvSCWmLUC7SGyGgs2sYxFMlF54/edTVqLYTIT79wfOnSIIqspQmI1hzBhYBR3giBk\nfZk/2bGkS7YiwYUQq8aIJOvrXVpaWvAbWizMjnemrWZzBTu+gUAgqnOXXpxm67wzIRtrHEb/nHHJ\nOZUozaJFInr0UHH33SI6dNBw5ZUyXnjBjkGDFOzfz+Htt8PlqvR88IEN/fqpUBTgyBGgQweYRi2/\n+45DYyOPU06JjqzyPCCKGp591oHaWj9OPTX8/w88EMTNNzuxePGx7Z1+uopvvuHR1ASUl4d/d9xx\nwOWXh/DEEw7cdZeE6moFixaJWLhQwvTpblx0UQh/+pMDt94qYfhwGe+8Y8Mdd4h4883Ce8n1pCJk\nYp3rtmwvKAYyFerZmswUKiobb/+PHDmCk08+OedjaEuQWM0i+guTfXlCoRCamppyvsyfiHRLV2W7\nHWchE71YRDIUCiEUChXsgRXvGJhFJrMZec92hN0silrI4xrv4RYvSqN/L8dxOHTIhh07bJg+PYg/\n/1nEAw+EReKWLeHM+yeecGDSpFCrtqWsFNXatQ4MGaJgyxY7Dhzg0Llz9DHfscOOAQOio7IA8OCD\nAkpKNFRXKxGhCgCXXirjxhuBAweOHVuHA6iqUvD++7ao5K1ZsySMGOHGvHkSevYMVygYM0bG11/z\nePNNG954g8Mtt0gYNUrBd9/x+PxzHu+/z2PAgNx62rNJqueane98+yQTRReJ9Eh3MpPPqGwiG4BV\n6pcWC/QtyiIswsTqW4ZCIQBARUVF3grexyIVkWJW1zVb3ZlyLVaZH5K1ElVVtVXr2UJbEWIleB09\nehRNTU2mY7YK7KZvHGtZWRmcTmfMa8MKx9xYZ9LlckXqjLpcrqgyX8uWCeA4YMMGOxwODRMmNMPv\n92PLFg79+gWxdq0D114bbLVP27eHO1ItXSpg/nwJF10k48UXW8cEtm+3obo6FPW7t96yYfVqBx5+\nOIgtW6Lf88EHNvTsqWL1agF+/7HfV1er2LEj+mHds2e4ScBf/+oAxx17zT33BNGzpwpVDVcxGDlS\nxt69HHhew6JFuW/Bmq97n9m5NqspygIHiWqKyrJMNWXToJAWCP01wEr46b/vbrc78n1n14AkSVm9\nBsizml2s8xRsA+gzmcvKylD6Q9jFCktOTCwkSvjSt29NpztTsuPINvpJQqJWooUWTsCxyDvr5BUI\nBCAIQs7bn6a77/rGCD6fDw6HI2qs2bjGC3Ve2IPNZrP9MKEUsXatCz/+sQqPh8M114RQUSEiELCj\nrs6OnTs5DBwYQufOXsODLYgdO3j8978aTj5ZxYABCi69NITnnmtdT3f79nDGP+O77zhcd50TK1YE\nMGyYgoYGDt99d+yYbt5sw4gRCgYMCEdtGawigJHZs0P4wx/CTQLYa3geWLky3Or1xhudWLlSQHm5\nhrFjZWzdasMXX7SPxwGzkrDa1WadnhwOB3iej9wXA4FA3JalsTo9tXfPqhX33fh9Z9eAvkGG2TXA\nun21tLREroF43b4SVQOgyGpqtI+7U55gnY1cLldkxhZPIOYT9qUxjoUJJiZCkm3fmsk4snVMNE2L\nRICPHj0KjuOSigAXUqwykcqiv5qmWaL9qRnpRlGLnRdftENROBw8yKGhgcf06SHYbDZ89JETZ5yh\nYO1aN268UWnV+WnPHhucTg1r1giYPbsZXq8XZ5/djLo6Drt2SZEIjc8XLuTfv3848UpRgGuvdeKq\nq0IYMUIBzwODBinYsuWYCH3nHRvOPVfBvHkSHn1UgPSDzmVC1Hg5DxqkoLxcQ21t2JawY0f4Vv/R\nRzyCwfA233+fx5EjPF55xQFJCreUbe/EisjpRQyLyOmrofh8PtOWpXo7AlEcZCsqqygKFEWJRGXZ\nSisABAIBeDyetMdYW1uL0047Db1798aSJUtMXzN37lz07t0bVVVV+PDDDyO/nzZtGjp16oR+/fpF\nvX7+/Pno06cPqqqqcMkll6CpqSnt8eUCEqtZxOihiiUQCwWbJWrasX7xTDDla9k5U4GjjwAzq0Wq\nEeBCiFV9FFWWZfA8n/MoqhnJ7LtZFLW8vDyjKKoVotnJ8tBDAkIhYOxYBQMHKujZMzzuLVtsqKzU\noKrA8OFhjyiL0tntdmza5MKPfgSUl3O48MJwQlx5uQs/+1kIf/+7K3Ltvv++ipNOkuFyha+J3/6W\nhyw5VCkAACAASURBVCRpmDfPG1lqPOeccPITAMhy2DYweLCCgQNVnHKKinXrwtHVLl00lJRo2LUr\n+pxwHDB7toRlywScdZaCjz+2oa4uXBVg1aoALrsshH//24abbw6ic2cVHTpoqK21Y/futjkByQbG\niFwyLUtZcm17tBdYNbKaCalEZYHoRNlJkybhpJNOwnnnnYcjR47gzjvvxJNPPom33noL9fX1SeeU\nKIqC2bNno7a2Fp9//jnWrVuHL774Iuo1GzduxK5du1BXV4dVq1Zh1qxZkf+bOnUqamtrW33uhRde\niM8++wwfffQRTjnlFPz2t7/N4EhlHxKrOYYJxELDxsAEk75fvBUFkxG9gGKtY9ONAOdLOBknBUDY\n+8vKTlnpRm6MoiqKEhVFzfYExkr7rue993h89RWPjh01/Oc/PGbMOLZUv3WrDV9/HS5lpR++zwfc\nd5+ARYtEfPEFj6++4jB7togNGxw4epTHL3+p4MUXxUiE5qOP3KipUcBxHLZt8+DPf3Zh5cpmcJwS\nWWo866wWvPMOh0AggB07FHTrpqCiIrzUOH9+EL//vYAfytfGtAJcfLGMPXt47N7No1s3FRMmuDF/\nvoQLL1Rw7bUhVFRoWLpUwP79PBYsCEJRchddbYvCxYh+4sJEDM/zkchsvKVlvb2ARWXj2QusTjGO\nORvoo7IAoqKyTz/9NN59913ce++98Hg8cDgcePPNN/Gb3/wGNTU1cLvdWL9+fcJtbN++Hb169UKP\nHj3gcDgwceLEVu/bsGEDpkyZAgCoqalBY2MjDhw4AAAYOnQojjvuuFafO3LkyMh9vqamBvv27cvo\nWGQbqgaQZYxCiOPCBejzWaZKjz5jW1XVyDJ/IZN22DFJNAZj+SZWkzZTj2SuxWqiZgOFbIpg3Pdc\nVx+Itd14FFLYLFsWXmK/8UYJq1YJuPDC8Lli0U2HQ8Pll4d+GCewcaMdt94qYsAABd27q5Ak4IUX\n/PjnP+34858duOEGJ6qqFNTV8Xj1VRtGjlSwbZsNY8ZIOHTIhuuvd2PlygBOPPGYSNQ0DYMGAXv3\n2tHSYse77zpwzjmhSOm7qioNnTs7sG6dissuC2HAAB7bttkxceKxqgYAaxIgYelSAU1N4dqw110X\nHnt1tQpBAGbMkLBkiYhAgIPLpWH9egf++18JXbu2T7GRK5KpXqDPXmfLxsXa5Yl91604tnxhvI/x\nPI8uXbqgrKwM5eXluOuuu6Je7/P5kjpe+/fvR/fu3SM/d+vWDdu2bUv4mv3796Nz585Jjf1Pf/oT\nJk2alNRr8wWJ1RxTiMhqLJEXCAQiM/tCkuiYZLtcViyyKYpYFDWZRgmFXBJnEwV2fbBOWGatcXOJ\n2bEv9IPtu+84/OMfdogi8NlnNkybFgI7fZ99xkMQgKuvluHxAF99xWHBAie+/prD0qXhpKgOHUrw\nP/8TQO/eGnr3DuH660Pw+YC337bh7rtFzJjhREkJcOQIhwUL/LjhhjJcfXUoquwUgB8mDcCAAQre\nf1/E1q0CrrgiBLfbDSB87BYskLBwYQl++ctG9O8vYd06AX6/P6q+JM/zuOKKEO699zj06KGic2dV\nt41wPdaDB3mMHCnj0UfDDQiefNKB228XsWZNdP1YIj2SucfoBWcmjTAKVU80FoX+PheSePf3w4cP\nm1YCYN/vRCR7XI1jSPZ9999/PwRBwBVXXJHU6/MFidUsY/YATqe+aToYu2MZRZ4kSZZYnjETa6x8\nUzAYhKIoKXXFSmf7bAyZ3FBZgkUgEEAoFILdbo8s9cX73EKJVSZSmek/k05YbZE//tEBTQPOOUfG\nxo123H+/N/J/b75pg9cLXHWVhHvvFbB6tQM33yxh1qwQBAF47rnwdTplSnS3KrcbuOgiBWVlQfz6\n1yIefDCIiy92YepUN44/XsVtt0V3sNJzzjkK3nnHhq1bbfjDH46JR47jMGKEhrIyoLbWjdGjZeze\nbQfggccTHaELBMLX2aBBAfzrXyK8Xm9EyEyYIGPcuHK88EILhg0rRa9eKjweDevX23HoEIcTTij8\nvYJIvhGGWf1goHU9UX1ENleCsj3YPpLB7Bg0NDRk1L2qsrIS9fX1kZ/r6+vRrVu3uK/Zt28fKisr\nE372mjVrsHHjRvzzn/9Me3y5gp5SOSbXkVV9TdSWlpa4GfFWSXLRVwTQl5zS+2izWS4r3hjSgR3z\n5uZmtLS0pOyfzed5MJYk0zQNdrs9Z17UWMSaoFgFSQJWrAgX5P/RjzRcdJEcaWsKAM8950BlpYZf\n/MKNr77i8c47Ptx0U1ioAsCjj4ro00dFLOv3oEEKvv+ew5df8ujbV8WePTb07Ckj3iU+eLCCTZvs\n6NRJRceOxigJsGBBEA8+KEAQgJ/8RMUHH9haZTI/91wJzjtPwdatTjQ02ODzuSEIAmw2G3r2VFBZ\nqWDvXgVut4YVK2zo3FmBx6NhzhxHVn2T7VW85Hq/E9UP1meuA8dWrVjmur56AUv6ovOdOYnKVmVS\nY3XgwIGoq6vD3r17IUkSnnrqKYwfPz7qNePHj8fatWsBAFu3bkVFRQU6deoU93Nra2vx4IMPYv36\n9XA6nWmPL1eQWM0y+YissigkEyCyLMPtdqO8vDyuyMtnlDcRrHQTq0uby3JZsUi1wLNe9OmPucvl\nslx00pjRz7zKoihaYnnQag+yF1+0IxgEQiEOtbUOuN0a/vWvcDQ1EAiXfPL7NSxbFsCaNQFUVh67\ndj75hEddHY/Ro+WYn8/zwCWXyHjqKTvq6nj86U9ebNsm4PHHW9dgZVRXK9i5k8egQeYe54suCv++\nttZmmmSlacCaNQ7Mnx9EKBT2rL7/vj1K2EycqGDDhrCgtdl4nHaahtNPl1FbK+DAATVuWaZsCRsi\nN8SrXsAy1/XVC1hUNlZhfDZ5ofOdmERiNZPIqt1ux7JlyzBq1Cj07dsXl19+Ofr06YOVK1di5cqV\nAIAxY8agZ8+e6NWrF2bOnInly5dH3j9p0iQMHjwYO3fuRPfu3fHkk08CAObMmYOWlhaMHDkS/fv3\nxw033JD2GHMB2QByDM/zWUuo0Xs5eZ5PeRk3m2NJFRZF1dsUCtl+NtltmiUgZbp0nqvIqv4Yx/Ki\nFrrGrH6cAEyTRAoRlVm8WMRxx6loaODhcmlwu4G77xbx2Wc8bDYNmgZ8/LEPZgGHhx4S0LWrirPO\nij8RnDAhhOXL3Rg7VsbYsTJOPfV7/PznHdC9e7gwvxGPJ2wjMLZqZXAcMG+ehAcfFHHDDRKeey76\ndv6vf4XrvlZXq5g0KYSXX7Zjxw5bROQC4TauDzwgYt68cDWALVvsCATC7WF/9asKPPdcuF2WmW/S\nrA97rOXm9ojVk4xStRfovbJAfHsBRVZj7/+RI0fQq1evjD5/9OjRGD16dNTvZs6cGfXzsmXLTN+7\nbt0609/X1dVlNKZcQ2I1y2Q7smrm5SwtLU2r1FSh6osygcrEHhN6rBZdIYh3LIwJatlOQMr2eciF\noM427Hvg9XohSVJUzV9N06KivaFQuAh/vsTOe+/x+PprDscfDwgCcOutEm68MZw1//zzdsyY4QTP\nh5OsBgyI/i5/9RWHN96wwW4Hzjgj/kRw924eqgpMnhwuh/XjH6v429/8uOQSFzp2VHH22dGfrarh\nqG4gTq7T+PEy7r8/3Klq+/ZwcwB2uNasceCaa8JltiZODOGxxxwoK4sWJx07aqiuViBJQF0djyFD\nFPz73zyGD5exdm24PJbdnryw0fdh1ycAsWuenXsrJAARsUmlegHz7jOvrF6ks3tSe5u8JIqsnnPO\nOXkeUfFDYjXHpONZNUbIzMofpUO+xKpRYAuCgJKSkojADgQCBS3fBJgfC9aJRC/6clGFgJFJ9MEs\nippMWa98T1jYtSD90HLJ6XSivLwcsixH9t/40DMTO7mM2i1aJILjwr5VRQkLOwD45hsOt9wigueB\nESNkTJ/uwubNXpSUHHvvo48KmDgxhL/+VUC3brGPq98P3HWXCJsN2LnThhEjwtvo31/F448HcOWV\nLrzyii/SgAAAvviCR0WFhk8+iW1stdmAX/1KwpIl4fJUV13lxM9+JqOqSsFrr9nx+9+HlW6PHhr6\n9FGxfbsNioIor+zll4fw1FMOfPstjxUr/Bg3zoO6unB71tWr7Zg5M7a9AUhO2DDbAPs3O+f69xrP\nL/vsYqWtRheTqV7AqqKw77fZ5CXWd7otEO/cNzQ0ZORZba9YJ/TSRsgksqooSpTPMJPC92bwPJ9T\nz6o+WSoYDEIURdOmA1ZI9NIneUmShKNHj6K5uTnSUpR188rFzTOTz4zlRS0pKUl6MpOPY6/v2MXq\nzbLkD6OoYVE7JrQFQTDt1a5/8Bl9dekmiHz3HYe33rKhe3cVLS0czj9fRocOQCgETJvmwllnKejU\nScOVV8o491wZCxYc8wH8978cXnzRgbPPVlFVpSDeoV+xQsCPf6yiVy8Vr78eHSO46CIFt90m4dJL\n3WhoOPYhmzfbMGyYjPfes0UaAOhpaQGWL3fggQdEfPMNj5NOUlFRoeHll+04/3wPVFXDffeJ+Pvf\n7fj+e2DyZBkcB+zcGX38x46VsWOHDVVVCg4csGHSpBC2bbOhpkbBihWZNQnQCxFWoUSfABSrfWms\nQvnkk7U+TIzabLaYLWtZkp9+0u31etHS0tImznkisdqxY8c8j6j4ochqjtEvd5pdvLGikLmoeZkL\nkcjaCbKZtCAICUtOWUGsAogkj+SylmssUvF1pRtFjbXdXMGEJIuc68uPJVvw2kgyUbt4y8+J6k9e\neaUTHAcsXhzE5MkuzJgRjnjed5+AsjINO3fa4PcDVVUKLrhAxtChHjz/vB2XXCJj2bJw/dO9e3n0\n6xd7Enj4MIdHH3Xg6qtDaGnh8PTTDvj94aQrxrRpIXzzDYeJE13YsMEHlwt45x0bLrpIxief2PDR\nR8csCAcPclixwoEnn3Rg6FAFa9f68emnNjz6qAOlpcCyZQFUV7sxZ46ExkYOf/qTA9df70TPnioC\nAeDpp+24665jnbk8HmDUKBlNTcC779pw++0S1q51oLJSxebNDnz7LYcuXbL/fU0mQqc/t8Xmk22r\nkdVkYNYeI4m+zwBSPudWtJTE2n8AaGxsxI9+9KM8j6j4ochqljHeKGMJ1ERRyFxF9PSeonQxZsaH\nQiG4XK6E1Qj04yhEVQK2PNXc3BxZlo5V5ivXJCPYsxFFTWe7qaKPogYCgci1nKgyRabjYw8pVqbJ\nLILjcDgiERxje8vDh33Yvt2GX/wiiI0bedhswPDhMl57zYann3bgqqtCKCnRIMscevTQUFICrF7t\nx7x5Ij76iMP//q8Dc+ZI+PhjPq5f9YEHBFx2mYx9+3hUVyvo10/BO++0jhPceaeE7t1VXHedE4oS\nFqvnnqtg8GAFW7bYUFfH4aabRAwc6MH333PYtMmHtWsDGDgwnEDV1MTh7bfDdVmBcCT1pptCeOEF\nP3bvbsHixUGccIKGRx8V8OGH0bf+yy8Pi+5337WhQwcNkyaF8NJLDpSVAUuW5KYFayJYdC5WD3a9\nNSpRJrssy5TJnkfSOc76iGwy51w/OTZG4o3VC/J93uMFpzRNK1hHy2KGxGoeYMvvTHzoSzblUyyx\nz0/3i2sUT+naFPLd1UtRlFYTA6fTCbvdbrmbhnEioCgKPB5P3uuiJsJon9A0Le61nO9oul7ImtWf\ndLlcuOOOCgDAkiVevPyyAxdeGMCePX7MmiXiscca8cwzPM45R8IZZ8hQ1fBD76yzVMyZE8KkSW6M\nHRtCZaWGjz+24YwzzCdfO3fyeOEFOxYuDOLjj3lUVakYOVLBpk2tkwt5Hnj88QAaGjjMni3C5QJ+\n/GMNffuqWLpUwKhRbnTsqOGDD7x4+OEgTj752PEURWDuXAlffMFj9epjiVX6/x8yRMEtt0j4/+yd\nd3gUVdvGfzOzsyWF0FEDAhqadOkovYki0lQUFRA0oIgFRRSsIMKrn+IrotgorwoIFlDpRRCkdxUE\nVBRQQAIBkmyZ9v0xzGY32fTsJkDu6/KSzU45c87szH2ecz/3I4rQt6+LzZvT76UOHTSSkgR++UUk\nNRVefdVLWhrUr68xf75MMXG88yNQOmKRmlATFetZk1tSU5i43IlxOFYGA8c8J3lBKElJpKzXsouq\nX84R94KgeLz5LjEE3ojWDyE1NdVPPgL9OSNNlvJKFANJydmzZ/2azlKlSuXbXzQSxCUwinru3DkE\nIbhYQlGUwQ1Exj4IRxQ1N+fNKwLb6Xa7sdvt/hWB4kb8s4IZwRGZP99O48YagmDn+HGRp57SGDmy\nHEOHKtSpI7F+vUxcnEGDBkqQpm7AgGSOHxeoWdPHmTMKf/8tUKNGaLLz3HMOnnjCh8MBx46J1Kql\n07mzyqpVoRVYDgd8+qmblSttXHGFQVKSwDvvyJw9K7BnTypjx/ooXz70+CUmKhgGfPutjbvuCp0U\nNWiQgqLAU0/56N/fxfr15pjZbNCvn0rZsgbbtklER0OXLir794u43bByZcHGNpIv6NxE3LMiNaE0\nkwUpjHC5kpJIE7JQk1MrKhsTE+P3lLVWLa1k2uy07wWZwGR1/ZqmXTTPyeKGEs1qmKCqql/LCaYm\nKzY2tsgfXrklK4GeroWt6QwnWQ30cs3ORaGodbOBS5eFoUUNJzJqZjO6O+QGRSX9yAqzZ9vw+eDj\njz289ZaduDhYuFBGkuCppxTefNNJr14qBw7I9OtnFoCwlvDmzpWpX1/lq68cXH+9Qo0aKl5vKl5v\nsPfkhg12fvlFYOZML7t22ahd26xwVb++zrlzAn/+KVGnTua2lSkDTZtqrFkj0bOni9tuU1mwwNSO\n1qiR9T3rckF8vEFKCkHVtzJuU6mSwbZtIjNnerj3XicffuihUyeNO+5QmDNH5scfJdq103j5ZS+t\nW0dTqpTBm2/a6drVXVjdX2TIj2Yy0JIpOw10cfrNFjWKW/TQispmpY3OjfVaVmOe1XJ/qL+fPn26\nRK+aT5SQ1TDAWm6wEkzcbnexISDZkYaMyV6BCTLhQGE90KwoqpXk5XA4iIuLyzbqW5Rk1XooWnXa\nrQhAJJb483LdVvKcx+NBECLn3xqJcXnhBQeVKhlce63BggUydetqzJwps25dGoIAs2bJzJrl5p57\nXIwfb044zb4TeO89B6+95uWxx5wsXRpF48YGMTExQS88TTN47jkXzz57Hl33sGWLi3r1zDKXoijS\noYPM6tUytWrpIV94Bw+K/gIEL7zg4+hRkY0bbdSooWR7XdHRBn//LZKUBOXKhd6mTRuNZctsvP++\nl88+8zBggJN33vFw000aLpfBsmU2nnnGR+3aBtWr6/zzj8DWrRInTwqZyr5eSghM9sou4cv6f6BJ\nfqjkH+s5W9yIWwmCkZskzsDfthVkCCyOkHHsrX0zjntSUhLlsvphliBblMgAwgCXyxWUYBJuy6i8\nIOPyd8ZkL5/Ph9PpzDFBpiCwHg4FISWB2s7k5OSgJK/cyhMiSVYzalENw/B7jhY3LaqqqkFlZSOp\nmY3ES337doHTpwUmTfJw+LDA338L7N8v8t57Hq64wmD1aonSpQ2qVDE4f14I8j5duVLC4YD27TWG\nDfPxzTey3wkgMDnkyy+jcDpF+veXiI6OZv9+F40a4Z+0dujgZdUqR8ga7cePK/z2m8gNN6gcOSJw\n+LBAy5aaP3EqK/h8cOSIiMsFc+ZkXXCja1cVUYS1ayVatdL4/HM3I0Y4WbTIRv/+Cnv3iigXOHFi\nooLDYRYomD276Ip4FAfkVidrPdssQns5JnxdKgTdIqGhStZmJy8A/GWKT506xSOPPMIbb7zBypUr\nEUWRlJSUfLVn6dKl1K5dmxo1ajB58uSQ24wcOZIaNWrQsGFDdu7c6f/7/fffT6VKlahfv37Q9qdP\nn6ZLly7UrFmTrl27kpycnK+2hRvF4w15iSGj7VRRLzkHIvBB6vF4gpK9CtPTNbftyCsC252amprv\nJK9IPUit9p49e5bU1FS/FtVK/oj0Az2rfjcMI+h+sPq1sDSzGc9blOR82DAXDgf06aPx4Ydm8tDA\ngQodO5oZ/TNnygwerLBrl5jJP3XaNDsPPeRDEOCeexSOHhWoVCl4IpqWBuPHO5g40ePfd88eicaN\nDX+Wc5cusGmTHZstuEa7pgncd18MpUoZTJt2mgceSOX55yUaN07lxx/FbL1kN26UqF1bR1Xhww9l\nsvp5NWmiAYKf0DZtqvPll6bLQcWKBroOO3aY49O3r4rXK2AY8N57+U+0Ki7Pv3AhlE7W0k/m5ExR\n2DrZosbF2Ob8IpR7gZUTYY27zWajdu3a/P333yxfvpxvv/2WihUrUqlSJVq3bs29997LxIkTczyX\npmmMGDGCpUuX8ssvvzBnzhz27dsXtM3ixYs5dOgQBw8e5P3332f48OH+7wYPHszSpUszHXfSpEl0\n6dKFAwcO0KlTJyZNmlTwjgkDSmQAEYAoiihK9st3kYBFUi2LF1mWiYqKKhKJQl7Iaiif0YK2O5wT\niNz4ohb1BMaKfGTU+Ebqfiiqaz9xAn79VSQx0Yeuw0cf2YmKgueeM6Mhx48LrFtnY9o0D9On22nU\nKJ2d7dsn8vPPIvPmmclLTqdZ2nT9ehs9e6ZbV73zjp3mzTVatDD39XrNZf26ddOPVbasQa1aKhs3\nSrRvbyZd6DqMGuXk5EmRO+9UiYuL5vHHDZo2deDxKCQlCZw4YVC+fOjqXsuWRdOpkw+bzdS3rl8v\n0aZNZkuta681MAxYvNjG+fMQGwsNG+osWuSmVy8XUVHwyScyLVp4KVfOoH17lb17RY4fF1m3zmxv\nfnApRNryg7wuM1s62eyWmS8GnWxxbls4EahzFQSB0qVL8/DDDwPw4YcfUq5cOe655x7++ecffvvt\nN37//XfOnDmT43G3bNlCQkIC1apVA6B///4sXLiQOgHC90WLFjFw4EAAWrRoQXJyMsePH+eKK66g\nTZs2HD58ONNxFy1axNq1awEYOHAg7du3L5aEtSSyGgYUt2QeywfTciOwoqiFnWmeFxSVz2hezp9X\nZBVFzaq9RXFPWG2wnBIyRtXDdT+E6u+iuP6HHzaLAIwd62PJEom0NLj9dgUrV+yTT2Ruu02hVCnY\nvVukUaN0YjZtmsyQIeayOJi2VJUrG8ybJ3P+vPm3EyfM7P0XX/T69/vlF5Frr9X9GlQLHTr4WLHC\nPLFhwNNPO/jtN4HatXWaNzfPGxMjMG6cl5deiqJ5c51du6KzXHpetUqmfXs3jRp5qVlT4YMPhJBL\nz2DQtKlGzZoaixalxyvq1NH59ts0NA2+/jp9yf+uu1RKlzbwes3oaglyh9wshed3mbmwKriFA5eK\nBCC/yO76k5KSqFChAqIoEh8fT9u2bRk0aBCPP/54jsc9duwYVapU8X+uXLkyx44dy/M2GXHixAkq\nVaoEQKVKlThx4kSObSkKlJDVMCDjjVoUmtWM1k2GYSaBWHrOotZIZrccnZVVVmFqJguLrGbUoloa\nz5y0qEXxMLcmLYBfm5wXjW8kEM6Jnc8HK1bYaNZMo3RpePllB9HRcNttZqRU183EqsGDzVWQXbsk\nP1k9dUpg4UKTrFrYs0ekSRONdu1UPv3UJHETJ9oZMEClevX0a9i9W6Jhw8y//44dvX5LqPHj7Wza\nJPH5525275YuLNWbuOsulZQUgbg4I0i3Grj0fPKkgxMnRFq1kmnVSkQUJVavdpKcbA9aera8Rhs0\n8HDFFSqffCIFEdmEBJ2ZM90XqlmZ90TXripHjojExsLKlTZOnbp8iUikEWqZOSeT/KIujHC5k9Xs\nUJAEq9z2acbxzctYFOdofYkMIAKIZGQ1cFnXspwK1HIW9azbQsY+sayyfD6fP/M8nOVPCzomGcvM\n5jWjP1L3RCBJsRweAGJiYooNQY0U3nrLrMT0yite9u0zk6pcLtMmCvAnVjVurJOUBMnJ6clVH30k\n07OnQoUK6WNmFQNo1UojMdHJjTdqfPONje3bU4POu3t36ApXDRuqnDgh8NJLdr75xsbSpW7cbgG3\nG6pVSz+PJMGECV6GDXMSHx/6nlm1ykbHjhqSZGpSn37aQY8eKvPmuXjsMV/QtoZh0LKlwPbtEvv2\nSfz5p0Dlyop/KbpNGwGbzcXAgU527kzG4RDo3Vth2zaJn3+WmD3bxhNPFL2sqbjDsroKFywim9W5\nM7oXWBZcup7uQBGqXK117PyiOLxfihLZkfVTp075o5h5RXx8PEeOHPF/PnLkCJUrV852m6NHjxIf\nH5/tcStVquSXCvzzzz9UrFgxX+0LNy6vt1WEkFVkNVw/4sDl55SUFARByLKaUHHxuwxc0gqsghQT\nE+OPSoZzhhdoL5JbhIqiWgUe8hr1DTdZza4EalEURChqKQzAu+/KxMRA8+Y6I0c6qVpVp1o1nbg4\n8/uZM9MrP+3aJdGggYYomprTDz+UeeihYIK2d69JQlu00ChTxmD4cCdPPumjTJng8+7eLQVpXy1I\nElxzjc7MmTKLFrkpX95gxw6JJk10Mt76HTtqXHedzp49IheC40FYsUKic2czQnz11WaS1C23KMyY\nkTkpShAEmjY12LPHRq9eKl99FZ2pulfjxhrnzwtMmeJE0zT69k3h5EkBu91g6lQZtzs4GcgiR1mh\nJNoWWQRG3bOq4OZwOLKt9lSQhK/Leayzu9dPnz5NhQoV8nXcpk2bcvDgQQ4fPozP52PevHn07Nkz\naJuePXsye/ZsADZt2kTp0qVzJMc9e/Zk1qxZAMyaNYtevXrlq33hRglZDRMyEsTCRijiZFk3ZWc5\nVdSVm8AkUpqm4fF48Hg8QVWQ8mI0X1DklkBl5UBQlJrfUAiUUOSmBGpRwNIhnz9/nrS0tExlL3Mi\nPfnFrl0iSUkC997r49dfRbZtMxOFLG2olVh1++2BEgCT5S1YYOO663Suuy6d9f36q8iWLRIniJdi\nTwAAIABJREFUTgicOwcdOqgcOCAydGgwoVVVU7Nar17myOqhQxL79pklWK+6yrzm7dvFIAlAICZO\n9GIYpuVUIBQF1q610bmzuZ8gQJMmOqoqEB1tZNoeoEIFg7g4g7ZtVebMSXcOsEhOnTo6aWkCr70W\nxU03lWXhwlgURUBVBc6eFfnxRzNCX9gk51JCcSXo2elkLXmBRWStoILX6w1psxaqbGlxve5IIbvr\n93q9uFyufB3XZrMxdepUunXrxnXXXcedd95JnTp1mD59OtOnTwfg5ptv5pprriEhIYHExESmTZvm\n3/+uu+6idevWHDhwgCpVqjBjxgwAxowZw4oVK6hZsyarV69mzJgx+WpfuFEiA4gQrOhqQX1LdV33\nG+Dnx6i9qCJcGQsOWA/K2NjYiLfFQnZ9EQ4HgtyeO6/IeE84nc5sJRSRvgesvtQ0jbNnz/r70orw\nB5ptWxMZn8/nX5bMynQ7Lxg/3o5hmGVGBwxwUaqUQVqaQPv2ZjQyMLEKzKX7Hj1UDAPefdfOc8+Z\nCVObNklMmSKzaZOEKMK8eTKjRjkxDJOYfvedjZtvVv1JWAcPilx5peE/rgVFgREj4hg1ysd//+tA\nVc2Sp9u3Szz0UPCyvYXrrtOpVUtnyhQ73bunV5PaskWienU9yLC/aVON7dslBg82o6sdOmQmwE2a\naHi9AoIAW7aIfveC1aslFi2SiYoySfj+/SLx8TplyxqcOycQFWXw2mtRdOqUOWEwu+pPuq6jqupF\nl9V+OcEah4IURrBg/YYDx/lyQFZktTAm4t27d6d79+5Bf0tMTAz6PHXq1JD7zpkzJ+Tfy5Yty8qV\nKwvUrkigJLIaJhSmI0BhJh1Z7YikhtYqOOD1ev0FB4pDpC/UmEQqilpQwpgxsm7dE3FxccWib602\nWn1plR22+tIi/RmjO7Is+8lsVsuUgb6UVkQ2u8SREycE1qyxUb26zv79Env2SAwbprB1q0Tz5ro/\nsWrQoPSo6K5dEo0ba6xfL+F2g9cr0KVLFImJTjp31njzTQ+tW2ssXOhm+nQ3FSoY1K6t8+yzDqpX\nj6F3bxdvvSWzaJEtpF719dedlCmjM2qUQny8zo4d4gV/U1MGkBWGDfOxdavE4cPp47tyZboEwIJJ\nVkXuuEPh++9tnDiR+X6wCO3ddyt+z9X16yWGDnXy2WcmGf77b5Grrzb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Mn69espW7Zstsc9duwYVapU8X+uXLkyx44dy9U2f//9d5b7Hjx4kHXr1tGyZUvat2/Ptm3b8nS9\nRYESshpmhJpdF+VsOFCXCBS50Xx2hRIy6n0FofBLoBbGeOi67i+IYBVxCOU8EI5z5wYZNb1Wm1wu\nV7btCycCl9TygoMHBVautKFp0KKFxocfupk5007dumbyEMDLLzsYNcpHbKxJNGvX1nG7YeJEO23b\naui6ScDatVPZscMkk+PGORg/3kuo7ti1S6RqVZ0nnnDw/vtuYmLMv7dvrxEXB19/bcMwzAispY19\n5x2Zf/4RWLEilenTZR5+2In7Qu7TnXcqHD8uUKtW1hIQtxsee8zBSy85GDnSx5kzAp07m8ceNcrL\n4sU29u4N/r2WL29Qt67G2rXpkfulS+3ccovOI49o/O9/TjQtiptvFvn+e1emqkE+n5f+/VP58EOB\n9u09LF5s+Ins778b/PijRLlyBuPGeTl5UuCpp0zT/86dNU6cENi7V6RbN5Xvv7dxzz0KEyY4uOoq\ngypVdLxegcmT7RSTeXrYUZjE7VIgspcLQo35qVOn8m1bldt7KD/OPmfOnGHTpk289tpr3HHHHflp\nXkRRQlYjjKKIrGa1dG5FZIpyNhyKsGWs3BTOEqgF1cxaxF9VVX9CktPpzFWfhpusZkzoioqKIi4u\nDqfTWaRL/db9mLHyTG5eoq+9ZsfjMf0/v/zSzYMPOomLMzhzRqR9e9PQft8+kfvvN6OW27aJNGum\n8eKLDux2M8q6dWsaL7zg49Zbo9iwQeLffwUqVzbo2jWz56muw549EjNnygwapNCiRfpvVxBg9Ggv\nr71m5+hRAcOA+HiDhQttTJ1qZ/58N9dfb7B6dRpuN3TpEsUffwh4PAIuF/z4oz3kNR44INKxYxTJ\nyQLr1qUydKhJVq+5xmxf6dJmVHfcOAcZu+uWW1QWLzYZ9/nzsHWrRIcOKtWrG7RrpzF7tky9ejoe\nD/zxhy2oalBUVBSDBomsXOmidWuD7793+InsM8/YeeCBVO69N5UdOwx69fLyzTc2HntMRtM07rlH\nYfZsmfbtNXbvlhgxwiw9O2yYkwkTTBurU6cEvvmmJE2iMFGciOzlHlnN6tpPnTqV72X2+Ph4jhw5\n4v985MgRKleunO02R48epXLlytnuW7lyZfr06QNAs2bNEEWRJEvYXkxRQlbDjKKMrAZqUUMtnRd1\nlBfS+yOjp6dhRKYEan41s+fOneP8+fN+4p+fUrjh6P+M/RgYjQ5sX1GNvc/n809CbDabP7Ju+Rym\npZnLxoEvUauE4rFj8MUXZunPfv0UDh8WWL7cxksveVBVM9nqxRftPPOMFytnYetWCVE0mDVLpnNn\nlW++cVOpksF99ymMHu1jxw6R+fNtTJjgJdTQ/fabgCwbnD8v8PTTvkzfd+2q4XDABx/ING6ss22b\nyGOPOZgzx02VKmb/xsTAxx97uOcehc6do5g1y0aDBhpz57oyHe/TT2106+YiMVFhxgwPcXFw8KBE\n2bIGy5alL6MPHqxw9KjA8uXBk7ebbzbJqq7D6tU2mjfXiI01vxs50se0aaZsoXNnNaQrQLlyBp07\nq/zxh40//5Q4c8bBDz9Es3+/nVGjYMgQWLTISffuClWrmsltgwc7ufXWs3z+uY20tDRat/axfbtB\n584K+/ebDgkul4FhwDPPhCbolxqKA3GLNJEtDtdcVAhXQYCmTZty8OBBDh8+jM/nY968efTs2TNo\nm549ezJ79mwANm3aROnSpalUqVK2+/bq1YvVq1cDcODAAXw+H+XKlctXGyOFkmluhBHuyKpFVjwe\nT44Z/bnVi4YbiqLg8/nC6umZFXJL2gIrOEmSFFSvuqAojId8JCtg5QWGYaBpmv+lJ0kSsbGx2Gw2\nf/WwjO4EqampyLLsn8RYlWdeeKEUimJGNF999Rxdu5ambl2zOlPbtiorVkicOSPQv3/68vq6dRLn\nztm44w6F//7XG9S2li1NoqlpAlWqhP4dLFliIyVF4IMP0pBDSC7N6KqPRx910Lu3yt13u5g2zUPj\nxnqm7YYNU2jcWOO226Lo2FFhwwaZI0cUqlQxSEmBUaOcbN8u8u23burWTd//xx8l2rQx/V0tTa2l\nix03zkGnTml++UJCgkHp0gbbt4ssXWrjppvS+6JJE52rr9b56isbnTtrfPKJzLBhmXWzgwYpPP20\ng7ZtVZYulZgyxcHkyR6cToiPNwsDJCfL/Pyzja1bU3jqKSdjx5alQQON5ctj6NZNZdkymYEDU3n2\n2RgmTHDSrp2XtWsdHD0qsmKFRocOweUwSxBZFNRHNpRrQXF4lxQVcoqs5rfUqs1mY+rUqXTr1g1N\n0xgyZAh16tRh+vTpACQmJnLzzTezePFiEhISiI6OZsaMGdnuC3D//fdz//33U79+fex2u5/sFmcI\nRlGH1i4DeL3pL0ld1zl79ixlypQp1HNktHEK1KNlhZSUFP+MO5KwZvEejwdFURBFMagEaiShaRrn\nz5+ndOnSIdtpRfysUq1Op7NQpQinT5+mTJky+X5hB/ajLMs4nc5c9eO5c+f8WeThQKhJk/Wocblc\n/gi1pmlBdj+CIJCWlkZ0dHRQn5w+DddcE4MoQuPGKnfd5eGpp2JYsyaJ11+Ppn17Lx99FM2TT6bS\no4dp3fT22y5eeslJ1ao627dnJpvPPGNnxgw7/fsrHD0q8vnnbgLndIoCtWtH06yZxty5nmyuFa66\nKgaHQ2fMGCUkAQxEs2ZRnDwp0LSpj+bNBW65RWXgQCctWui89pqH6Ojg7Xv0cPHQQz4eftjJmjVp\nVKtm+M/bs6eL225TGTo0/ZwvvmhqSf/3P5k1a9KoWjX9Eb9kicTEiQ4WLUqjXr0YDh1KwZUhwKvr\ncP310fTqpbB8uY34eIP589OLDaxYIfHyyw4qV9a57TaV229XeeIJB2vW2KhUSWf2bA8tWkRz8GAK\nzZpFc/PNXtaulfjlFxmXC6KiDH7++VQQ+YlkDfdww5pwZbyHLwWEIrJW8YOMpvkXo/1WfmFFnp1O\nZ6bvhg8fzvPPP0/NmjWLoGWXDkpkABFA4I80cNm7oMioRc1rAlKkl4JDlUCNjo5GkqQiTfTJ2AeB\n7XS73djtdn/CVHHQzAZKEVJSUoKKIBRVP1rQdT1IbxwoPbG+t/Splmbaivpb0WswH/6WD60ZFTB9\nVGUZhgxRee65GLp0UWnQwMGGDQ40zYHLJXDrrSb5+fpriTfeMHWqkycnoyjBS5qnTunMmGHnrrsU\nXnvNS0oKTJoUvEQ9aZIdVRWCiGAopKaaCVGGIfDgg9lve/48HDkiMnNmKps325k61U6PHi6eesrH\ntGmZiarPB9u3S9xwg0afPirz5qUzbkEwLbkmTbJzIV8SMKUAX35po2JFI4iogmmF5XabOtx69TTW\nr898P4uiWZJ13z6Bn38WmTgxmKh36qSRnCxQt67O4sU2JAmmTPHSt69ZcvXgQYHq1U03gUGDFJKT\nRSpUMIiLM7j6atO6a/ny6KDlaFmWg0pfejwe/3K02+3OJAm5GGIslyI5y05aIMuyP1ByuSV7hUuz\nWoJ0lJDVCKMwvFaz0qLmNQEpEmQ1o51TYGUkK9GnKB9WgZMHVVWD2hkTE1NsbKcgMxEsiJNDOC27\nAvXGlkm2FUFVVZXz58+TkpKCx+Pxk9FAKUhUVJS/kpMgCBw5AqtW2YiJMRAEWLpURFXh1VfT2LvX\nJEHTptl58UUfdrvMxo0unn46lsqVDa69Vuemm+SgF6imaTzxhIOoKJ2ePc+jKGlMn57M7Nk2vv3W\nTCjatElk5kwZRcGf4R8KPh/06+fC4TA9XwN9V0Nh506JevV0mjfXqFNH5dw5eOwxX5B0IXh7kWuv\nNb1i77pLYe5cOSipqkEDna5dNf6fvfMOr6JM3/9nyukJAUJPCCGFKkWQJggKKroi4ooKrP5QESui\nrmJZ3bWsuK79a0FQEVRWRRGkCIggoHRRmiCQhNBrSE9Om/L742VOTg4nECAkFu7rOhfGnJl5553J\nzP0+z/3cz6uvlhHtCy4wyMmR6dbt+HHLMoweHeT//s/OpZfqLFgQfbx/+1uQhQtt1KljkpcXab8n\npAI7d0osXqzi9wvi/NRTAXr21BkyxE2nTsJr9aabgnz9tY3nnitG0yAzU6FhQ5N77hGLl3DrrXDy\nE97D3WazldM2Wx6yFpENL9L7LRDZP7N287dS7FXdONE1LyoqolakefM5nDLOaVarAZHEwNL3nArB\nCE+rGoZRrhPS6UKWZXS94hfxmSBaNyy3233cOdd0kZd1bKuoq6Jxni2c7PzDJROWFOFMr3tVITLV\nb1WUW8QiPEUIYLPZsNvtod9ZBNV60Ftk1vrbsD6DB9dClqFuXUhL01m0yM711wdITjZ5+20bDRoY\nOJ3Qo4ePNWsUbrnFw3/+U8pdd3n417+85Oeb1KmjhObsyy9VNm2yUVIi06WLit0u06SJycSJRdx0\nUy0++yyXkSNjePDBIt55J4aYGD+BQPl0pqiSh9tvd+LzwcUXa9x3X5C773ZyzTUaFSlr1qxRaNNG\n5+qrY2jUSOOKKzRefNHBXXcFiZJBZOVKlR49xN9o584GkgQ//ijTtWuZPvCf//TTvbuH224L0qyZ\niSwLa679+yX+9z+VI0dkDh+WOHJEfA4fltiyRWbxYgVdF16uHToYdOig07GjQbt2oruXJEHr1gbf\nfqvStWv54rKbbw5ywQUeWrQQVlmWk8Krr/rp29fFl1+qxMSIyO/FFwf54Qc7L77o5557nFx4YZAZ\nM+xMnqxyyy0V23eFayOj3XuRqWhd10P3W3j6OVJW8GclkjWJs6GR/S1JC0zTrPA+BWrUfeWPgnNk\ntQZwKgQtUotqRRp+Kx6jkTjVFqg1RVbDi36AKp3XU0FF5x9J9qu6YOpM5t3ylbUq+sPnzopchktd\nwl9UFkn1+/3IshzyerV+H0lAli+X2L5dJi7OwOk02LRJxjDgkUeEhnLJEhtbtyrMnOkjO9vBkCEu\n+vTRuO8+D6YJn39u58UXXcTEmKSm6jRqZLBggY2hQwMsXWpDUWRUVbxIevaEJ58MMHhwPFdcEaRB\nAxsdOuih87IKvcR5STz0UBx5eSY9emjUqgUXXhikZUsbEyfauOee6HKAb75R2bZN4r77/NxzTyF2\nu5ukpBiGDHExY4b3OEeCFSsUhg4NHpvHsuhq165lOvjGjU3uuivAmDEO+vTRmTZNpaAAli1TqVsX\nGjQwadTI4LzzTBo0EO1mv/xSZf9+mU2bZO6+O4CiCE/amTNt/PKLjKJAw4Ym69cLovv444Fyet4G\nDUz69hVdrebOVUNktUULg7ZtDXr10nn5ZTuffKIyfLifJ55ws2qVl2efNZk9205yssGjjzq55Zbi\n07oHT0Zkw3WU4STWIhXRyOw5IntmqIiwnQx/BCJ7smj6ufvqzHGOrFYDIm/Uk1VOno0oakXjqirt\nbGUdCKKNobqqSMP1cLquh+a1sLCwxvxmI69BOImuDNk/E5yqZdeJIrwWSbWuZSSRsPSoViFYRfpf\nSZJQlLIo6C23OHE4oF07+OknlcaNDQYNCpCQYOD1Gvzwg0qfPj527/YxYkRtQEQPnU6TTp0Mvv3W\nj66b7N8PGRnw6KMuWrY0mDvXxsGDMomJMTRubJCaqpOSYnDwoEwwaJKbK7FunUqnTmZIymDBMEye\neMJORobKtGkF3HxzLHfdVYLX6+eRRwIMGVKXwYOLiYsrX2wyaZKdNWtk3nnHx403+vH7hQb32Wf9\nPP20g1desfPww4Gw48CqVQpvvlmmGb3hhiB9+rh54QV/yDd21iyVr79W+eUXMd8dOhikpJisXasw\nYkSgXBTWQlJSgA4dYrjrrgArV6pMmODj5ptFlDMvD9q1i+G++0SHrqwsmY4d3YwZE+CGG8qixrfe\nGuSBBxysXSvx6qv+EJkdPjzI9Ok2rrlG48EHncyeHcDngzVrZF54wcett7ro3l3ns89kXnrJxpgx\nJ+/kdSqw7qFoiEZkg8FglRLZP6sM4Gycd2WIbOT1rIjIns3CvYrOPRAIHPf8OB9lgzIAACAASURB\nVIfTwzk3gGqArutoWlm6q6SkJGR/FI7Tqeg/EwSDQUpLS4mLizut7SOJ1emM1zRN8vLyzqgi/mQI\nt3WSZRmHw1GuAC0/P5/Y2NgaSa0XFhaG7oNwEm1pu84WKroHIxEt1W9peCNfFlD+5RKuSzMMA7vd\njt1ur3T0ZfJkmXvvdVC7NvTvrzFjhorLBT/95KVxY6Fdvf56B23aGGzdKrSdTZoYLF+uEgxCbKxJ\np04BUlN1UlMNfv1VZd48O4oixjdwYJAXXgiye7fojDVrlsr06XbatNHYsEHsIybGpGdPjfR0g7Q0\n8VmwQGXhQhvz5pVSuzY0axbDTz+VUL++mI8773SSkKDz2GMlmKaJz2fwxBMxfP+9Ha9X5pdfcgFC\n1cPBoEy7djEYBrzyip9Bg8SzYvNmmZtucrFuXfn+4Rdf7KZpU3HORUUSAwZoDByosXOnxEcf2YmN\nFT6ye/dKbNqk8O670Z0MHn3UgabB55/b2Ly5GEtW9/bbNn78UWHyZB/Tp6s8+qiDfv00Dh6U2bpV\n5p57Atx6a5CYGOjc2UMgAJMne7ngAkGKS0uhVasYPvjAy8iRThQFBg/2kp9v5+23fTRrFkNJCaSn\n62RlKRw5UsxvIUsaKV0Jj/ADUaN30YiPtTBzu901cRo1Bq/XGyqyqmlEI7Lh/13VRLakpCRq7cC+\nfft48sknmTZtWlWd2p8WytNPP/10TQ/ij45wWw8gtPqzvCQDgUBIXK6qKh6PJ2SRdLZX6FahTmVR\n1eOVJAmfz1flRUwWUfJ6vZSWliLLMm63+7i0M4g5sIo4qhNWOj0QCISIi8fjOSVCd7qwFk8VWVdZ\nC5GSkpJyelTr+5Ev8vAHvWEYoXvEIqmnKrMIBOCyy5y43dC3ryjWSUszGDRIp08fg3feUbn3XjuC\nz5sMHqwzY0YAn08mL0+iWTOTBx/UuOQSOHBA5ZNPHHz/vQ3DkPB6RRcpm01n61aTwkLxef99J9On\nl9K2LWRkyBw6JHHFFRrXXx/E7xfEb/x4BwsX2iguFob8M2bI7N2rkJam4fOBy2XQvbvJffe5uflm\nE7/fxrBhtZAkhWHDgui6xLXXahFpzQCSBKWlgmx27+6jUSODWbNs2Gxw1VUixb5mjcyYMQ5+/lkm\nN1fm/fd9jB0boH9/nWbNTNq1M5gwwcaGDQqvv+7jvPMMHn7Yyc03B4nGm1q0MBgzxkmPHhqaJtGx\no4Guw8iRLp57zk9Cgkl6usHbb9vJzpZZtqyUiy7SmTHDxqOPOigqkmjXTmfHDhlJgj599GP3FOze\nLVFYKLF/v8yAAQFmzHCwaZOI9BoGrF2r0KyZye7dMoWFhFrJ1iTCK90thxJLZ225FViLNMvVIhAI\nlOvAFk6Karo7YHUjGAyWc/eoSYST0fBraX2scYZLl6zrGe5CEi4fOVF03YqgRv5+586dZGVlMWDA\ngOo47T80av6u+hMg8gYWxRl6uYp+p9N51lqKnmhclQ2sR7ZArcrxVnVlus/no7CwkJKSElRVJS4u\nDo/HU+GKv7p1s+GuA0BINlHTrgOWTCKyi5hlhQZlWYJo0SZd1/F6vRQVFaHrOm63m5iYmErZqEXi\niSfKqsxjYw0CAdi1SyYnB9q2dfH11wqGAQ6HiWHIzJ2rUL++iwcftKGqsGqVzNq1Ms89Z2PqVBXD\nkHjyySBHj3rZulX4qY4caRIfr7J4sYNHHhHRvssv93D33Q5Ax+UyWbFCpXZtjdtu89O7t4GuS2zc\nWMrMmaUkJJisXGnH6TRZsEDl73930rZtLS6+2I3TadC3r4tOndw0bKjz8MM+NmxQ6dbNCL0wrQWU\nx+Ph9ttNNm608dhjpQwfHsuuXTrLlkGnTqXMmhWkf38Ht93mpFevABs3FqCqJn4/5SKSsgzXXCMW\nIQ6HKEgbMEDj44+jL0iaNRO603r1zNB35s5VqV/fpEsXcX1VFe67L4DfL/H99wodOxpMnuzju+9K\nycuTGDfOQUaGzJdflv/buuWWIB9/bOPGG4McPSpxxx0iuvvBBzZuvjmIogj7rNRUgwkT7ASrVglQ\n5bCkBVaVu7V483iEBVek24RhGFGr3CvbVvj3iN+L/KEiB4pw1wLLjcR6JlbkWqBpWrmsaSRycnJO\nu3vVOZTHORlANcAqKrGikl6vF8MwQinVmqrsPlkKPvwP9WyZ4gMUFBSckExWBuESCusBVNnIRlFR\nUUgacLYQTdfrcDjwer2VSsdXNbxeL6Zp4na7oxZzVVeqPxoOH4bUVBd165p062awcKGoWtc0QcIk\nSZCo4mJwu8XP7dsbFBRAfLyB2y2xcKFCXBzExZns3y8q9zt2NEhPN1FVk1WrFD780E9JCdxwg6jo\nv+YanZtu0oiJgU8+UVi8WBRzHT4sIrGaBs2aafh8MsXFEpddFgAkWrc2eeyx4DEtusnBgzB2rI0P\nP7TTvLnOwYMKcXEmhw5JxMcbtGmjk5Kik5Ki0bKlRFqaSXKy8HU9elQiOdnk889V9uyRadDAwG6H\n++/3MnCgH1kW12HKFAdffeXkiy/yy6UzH3zQw+rVKv/v/wW5774gP/8sM3y4i/XrS4j2Z7tunczQ\noS50HebM8fLAAw5uvz3IddeVvYDz86Flyxj69dP45JPykoLDhyWuusrF9u0yw4cLz1pL09q7t5tR\no/w89JCTDRtyefjhOObMUcnMLGbYMBdr18rYbBIFBfDXvwaZPLl8h7HfKyzbJasRRjRZQbhjQUUa\n2d8biouLq9VJpbpRkbQg/HpKkkRBQQEff/wxKSkpHDlyBIBHHnnktI45f/58HnjgAXRd5/bbb+fR\nRx897jujR49m3rx5uN1uJk+ezPnnn1+pbV955RXGjBlDTk4OdevWPa3xVSfOkdVqgK7rFBQUhLSd\niqIQDAZPWytalcjLyzuuGCpa687TiY5VFqfbTckigFYnJItknerD8mx28jpZm9bKakerGl6vN2TM\nb1X1W92vwlOdFVX1h8+9LMuhVGlV3CMDB9pZtEjB6RQEVdcFIR0yRBDJFStktmwRRLJ+fWjeXBDV\nzEwZXRc60wsuMPn6az+bN0tceaWTr77y4fdLZGRIfPKJSkaGRHGxRHGxSFt36iSq2NPSTNLTTebM\nkUlMNHn4YY0uXZzs2iXRoIGJwwHdu2vExZns2CGzdKmKpkHjxgYpKRoJCTpr1tg5fFgmEAC7XeKz\nz0rp2NEgLS2GJUuK2LHDICNDYtcuG1lZKlu2KBw5ItOkiSjwOv98jfXrxX7HjfMyZIh+HNEMBESX\nqXffLaFrV+2YnMCgQ4e6vPRSAQ88EMfy5UepW1eif//ajBnj48or9aiavAEDXLjdJnXqwA8/KGzc\nWELkunHUKAeffWYjI6OYyOZ7K1cq3Hijk7g4k9hYGD/eR/v2Bu+/b+P77xWKi02uu87P0KGQlBRD\naqrB3XcHeO45B/HxJrt3S+TmSmzfXkzDhmd8+9Q4TtTNyEJliM/vicia5h+3a9fJYEmmXC4XhmFw\n6NAhxo0bR1ZWFtu3b2f37t3UrVuXtLQ00tPTSUtLo2vXrlx66aUn3W/Lli1ZuHAhCQkJdOnShU8/\n/TTUMhVg7ty5vPXWW8ydO5fVq1dz//33s2rVqpNuu2fPHkaOHMm2bdv46aeffhdkteaV0H8CWA8c\nq3pa1/WQZVJNIzx6FlntXV0dkU41DX8yAni2j38ynKxy/mweu7JjC9fJnmpVv6XTs9lsIbeCqsLs\n2TKLFomxBAKiIh7g8ceD3HijTm6uIJsxMTByZJCOHU1efFFl2zaZ1q0N6tcXUdMlSyTq1xfG8126\nGKxerdC0qcGhQxLr18vUqmXSvr2B2w2vvRYgO1siM1MmI0Ni0SKFZctE44FXX7Xh9YoUe4cOBnff\nrdGihUmTJiag06SJyq+/eikslHj7bYVJk1wYBjRsaKBpkJMjcdNNLlJTNWJiDJYuFYR4wACTNWsk\n1q5ViI2VGD3az759Eu+/bycjQ6FFC52tWxUefdTJQw9JNG2q4/FAnz5B0tJM0tIMRo708/LLLqZP\nFzZeGzbIuN0S11xjY9EinbfeqsUzz5Ry220+Jk6007dvftTiknvvhSeecLN3r8yjjwaOI6oADz4Y\n4LPPbHz66fG2XN2768THmxw6JPPAAz4GDXJx551BRowI8MwzDv75zxKmTnXwt7/5GTPGz7vv2pk5\nU6WoSJDUxESDo0clhg1zsWiR9/iD/85QmXT4yay3IouCwm3TTmTVVNNEsaaPX1MIvw4JCQmMHTsW\ngOeee45+/frRpk0bMjIyyMzMJCMjg3Xr1p2UrK5Zs4a0tDSSk5MBGDJkCDNnzixHVmfNmsXw4cMB\n6NatG/n5+Rw8eJDs7OwTbvv3v/+dF198kWuuuaaKZ+Ls4RxZrQZYfpIWqpugnAxWb3lJknA4HFXq\n51kZVGY+okkSqsrOqyotvPx+Pz6fr9JzWV33QniqH8o6zVgygEgD/2ipfovg2u12YmNjqyzdZ5qw\nZInMuHEq8+aVXU9Jgnr1THRdmOlPmaKye7eEJIlo6Pvv23A4THw+eP75AKNHiyKd5GQXs2f7GD7c\nzo4dMjabyccfi970sgx+v5AR/PKLzA03aLz3norXK1La27bJ7NghYa0l/X4hH6hd22ThQoX9+yUO\nHpQoKJBISBBjGzXKzooVCgUFcMMNOq+9FqBWLTFvW7cGufTSWsTGGiQnS6xe7eL//k9l3z5RlNS0\nqU5MjM7zz9tJS9P5739L+O9/3dSvD4MGBXjrLTt9+2osWaLSqZOOwyFM/CdOtJOZKdwAunRx0b69\nzqFDMgkJOj//DPffH6BvXw933KFxww0mTz9t4+BBD82bG8eRoT59vJimk0BApn59L15v4LiIXkqK\nRIcOOm+/bT+OrEoS3HlnkNdft2MYEj/8UMq99zqZO9dNr14aeXkS69er7N8f4KabNF56yUFBgUSj\nRmIlctllGjt22PnxR4XVq2W6daseK7vfKk6FyP5WmiH8XvSqZwMnOvejR4/SoEEDEhMTSUxM5JJL\nLqn0fvft20fTpk1DPycmJrJ69eqTfmffvn3s37+/wm1nzpxJYmIi7du3r/RYfgs4R1arCeGkJDya\nWRN/4JE6Q6u1aHW4D0TDiQhbpCThbJHpMyGMkRZelv72VHwZzxbCo9Dhvq2RLQ1PlOq32qDa7Xbc\nbneVzb3XC1OnKowbZ2PbNgm328SaCpvN5OqrDb7+WuF///Nz5ZUGV19tZ+9eBZdLkMirr9ZYuFAh\nJgZef93O2LGQmGhSUgJ3323H65Xo1Mlg8WLR4vOrr/zUqmXSt6+T4mIJj8fko49U4uJE16f8fIlg\nUERRZVmY7Q8dqhEMSuzeLVGnjsmGDTLduxu0b6+zcqVMMCgisTabSdu2MH++wvz5LlJSdJo3D5Ka\nauO664J88omDzp11Fi60M2CAzh13+Fi+XOHVV1Xi4iQGD9YoKoLJk53k5sKSJQo//ijj98PixQpP\nP11M164GaWkQGyuH9LGvvWZjzhwFVYUVK1Ti4gzuvddFdraCJEHfvm6uvDJAixY6Dz1kZ9SoAElJ\nYLeLRYJhiH+TkuDQIZMvv/Rw440mknR8RG/MGI2hQ+vw008a7duXT1UPGRLkmWccjB9v4/bbg8yY\n4WXSJBtPPeVg5UqZXr2C/PvfDtq21alXz0BVTXRdYs8emenTbTz0kJ///MfB//t/LrZsia6v/b3g\nbD7XT0ZkzzVDqH5UhqyeDs7G+8Pr9fL888/z7bffntb2NYlzZLUGYD0cqpusRiukMU1hel6T3njR\nyKqVRrdM5M8mmbbslk4F1os83Bu1so0QIo9d1YiUIUQ2lbBeXoFAINTaVFGU0AvMMkq3rGgsu6+q\nQlaWxPvvq3zyiUqXLgb/+U+AVq0MWrUS2QebzaRRI5g5UxRW3XqrA10X/p3JySadOxsUFUGtWtC6\ntSChqgoFBfDaazbee09h82YZm03oXZ1OyMuTGDDAcez8hf71uut02rY1+OILlXffFWMZNEjjxRdt\nmCYMHaqzc6dMRoZMVpaE0wn165usXSuzZo2IjGqasF369NMAsqzj8/k5fFhn924nO3fa+e47B0uX\nyni9sGyZQr16wqlgxgwnjRubjB6tccUVOikpgqhPnarwyisy2dnQpo3JsGEB3njDxvjxbiZONNm5\nU6FuXYPmzTUaN9Y5dEjYQHm9JnY7rF1byp49Mtu2KaxaJayw5s61k5cnYRiwaJHQhSuKiE7bbOax\nuZOORbhVmjatTevWQsPbtq0e+veyywzq1jV56SU3kycXliNCdrvMgAEqixY5WLzYpE8fg1tvNXG5\nDO6808W8eXbsdvB4TK66SmPiRDuqKqQexcUmrVqJzloHD0p88IGNkSN/4/YAv0FYjgXREJk9idYM\noSJZQWWeUeciq9HPPTc397TdABISEtizZ0/o5z179pCYmHjC7+zdu5fExESCwWDUbbOysti5cycd\nOnQIfb9z586sWbPmtEl1deFcgVU1wYpQWKiKCvjK4kSV8jVV4BMOyx3B7XZHrZg/29WlFuH0eDwn\n/e7JGgycKqyuTjExMae1fTiiyRAqquo/UToRyneSCieyp3uepilI2rhxKgsWKHi9EB8vSEp6usGi\nRTK7donr3KiRSWKiSSAA+/ZJqCocOiTRtatOz54mn36q4POJSGjDhiYtWpg0bWpw5IiIckoSvPFG\ngJtu0ikthTfeUBk3zkZSks7GjQqxsUKvunmzQn4+uFzQurVBs2YmS5bIxMTArbdqPPCA6NZkGPDt\ntzKvvWZj0yYZj8ekqEiiqAhq1zbJz5ew2eC884K0a2fSvDns3y+zcKEoEhsyJMibb9opLRXFX40a\nmVxyiYFhQGamkB7s3CnIYu3aolhLloVV13vv+enUyaB3bycTJwbo1ctg4UKZt95SWbFCoWFDg/x8\n4WlqmiIl73abNGhg0Ly5jmlKZGTIJCfDunUKzZsbzJrlPVaQJpGZKfH99yrff2/DbjfJy5Ow26Fl\nS434eAgGIT9fZvduGZfLpFYtUVg2a5aXiy7SkaQyIrRmjcTQobH06BHkvffyef11N5Mnu7nuOi/z\n5jnZvVvh73/38sQTAYYM8dCzp85PPynMnCmaPUybVsJVV3lwOk1+/bWU+Pjf56vJ5/OFig5/D4gk\nsqfTDEHTNILBYDm5258FVtYv2vW+4oorWLZs2Wk9NzVNo2XLlixatIgmTZrQtWvXExZYrVq1igce\neIBVq1ZValuA5s2b/24KrM6R1WqC5a9n4XQr4CuLiqySIolfaWkpkiTV6EOmtLSUQCCAaZrV0rkr\nEpUhjJGRXqty/kxhVdTHxsae9j4iZQinU9Vvpfotw+zIVKL1sUhs+EvLinhHu15+P4wfL6KoXi/c\ndZfGTTdpeDywd6+ozp85U+GDD8rmUhAgEf3r2NFgyxaZxo0N5swJ0KiRSVKSC7sdZs/2ceSIxNtv\nqyxbphATY5KTI9GokWiX6vEIe6tmzUySkw3WrFGw20GSxPeGDNF49tkggYDEl18q/Pe/Nrp101m+\nXKFBA5MDByRiY018PgmXSzQnaNasPFn+9dfDOJ0Sb75Zi3HjHNSrJyyqYmNF1DInR5Btn08QW59P\n4r//DXDRRaKoaMIElW+/VRg8WOPqq3V0HZ56yka9eiYHDoiILkDdumJftWqZlJRI1K5tUlAgcd55\nBhdeaDBxokqvXjrPPx9kxw6JzZvhm28U1q8X864o0KyZzoEDMv36BbjkkiCpqcL54JlnXLRsaeL3\nS7z8svCpHTLET/PmJllZ8rGPQmGhRIMGBrt3y9jtJh4PXHFFkFtuCdK9u7i3LrzQzc6dMp066Wga\nTJrkJSbGS48e8fToEWD2bAdDhvgYNKiUu++uw/LlOfTtG092tkJ8vFh0rF8vGiiMH//7tLLy+Xwh\nT9bfM6znRaRTQTQiC2Ud2c5GO9PfMiq63qZpcuWVV542WQWYN29eyH5qxIgRPP7440yYMAGAO++8\nE4BRo0Yxf/58PB4PkyZNolOnThVuG4mUlBTWrl17jqyeQxkiyWpRURF2u73K7ZJOtQXqqUQVqxKR\naXRJkmqs5WlFhLGyhP9sHPtkiCw4i2zRGvlSiSSTVlW/leq3TLBP9FCNjLpE6uLCyevhwwqTJzuZ\nNMmG2w25ucKnVFhDiYr41FSDpCSTAQMcIa2qpRd94okAL71kp3ZtQf66dDHYsUPi6FFR/JSaauD3\nSxQVSVxzjUZCgsGHH4rOUi++GGDsWBvNm4uU89SpKr/+Kqyu/H5B3Bo2NDl6VKJNG4PUVINvv1V5\n/fUAXbsa9OrlZOhQjU8/Venc2aB3b43CQompU1WOHhUENidHRDJr1TLp3t1g3z5RmFWvnogIP/ts\nkO3bZSZOVImJEZHc3r0NZs+WWbVKXKNgUJDQdu1MWrc2SE0VpPqOOxz88IOPpk1NLrrIQWkp7N4t\nrLBsNmjZ0kCS4OhRUU2flGSyY4eEosDw4Rq5ucJntlcvnVGjNIqL4fHH7Xz4oZ8ZMxTef9/G5ZcH\n2LVL5tdfFQoKRNFaUpJO06Y6GRkqdjssWFBCkyYSiiJjmib5+SZTpyo8/7yT3FyJjh019u9XOHJE\nOqbxNXC7TTIyRAR3/HjRDcxm8/Hxx27mznWwZYtMcrKBaYqGDgMGBKlTx+Dpp13s2ydTp45Bbq6M\nqprMmpVPly7Gb67i/WT4o5DVEyEyO2M1CrEkVZGOE2fazvS3jIrazBqGwVVXXcWyZctqaGR/LJwj\nq9UEq/OPhapMv0f6jZ5Kb/mqTENXBtHS6JIkWq7WspqTVzOCwSBerzd0/MiipLMZ6Y089skQmeoP\n98CNTPXDyav6q6q1a1kqGJ56ysmGDQqDBvkYObKEFi2Ev2dxsUJ2tkp2to2sLIXMTIWZM5VQ9yJZ\nFin5yy/Xsdlg7VqRHv/xRy+tWolmAR06uPB6RbS0fXsDhwN+/FEQRVkWKfvateGyy/RjBFaloEDi\nqqt0unY1WLBAYfp0P9nZEt9/L/PMM3aKigQBPHJEkGHThObNTXr21GnZUmhUlyyRGT3ay+23F+Hx\n2PjXv2JZv15hwwZRBNWpk7DBysqS2LtX7MPphH79dNatkxkyRFTFz5mjUquWidcLCxcK79esLOlY\n9FJi7VqZ9etlTJMQgZckYZs1alSQ//zHzgMPBLntNrHw9Xrhyy8V/vUvERHdt09ElnUdioslkpPF\nomD9epmePQ2GD9d44w0Vh0OQ3U2bJBo3NnnppQBHjkhs3w5vvWVH16VjWlRITtZQVdi9W0SmBw0K\n8N57TgYN0vjgAz+aBnPmyEyaZGPxYjUkR1AU4boQFyckCRs32ujUSRD/7t11vvxSuDNs2lRCr14e\nAgGTw4fFoiIQgKZNDdauzQ8Ve1VU8f5bKxSqiLz8kWFlxqxGCH8kD9mTobS0NOr7Njc3l3vvvZc5\nc+bU0Mj+WDhHVqsJkWS1tLQUAHe0pt2nsM8z9RutijT0yRDuPhAMBkMRZethrmkaJSUlNdYkwWp/\n6vF4ynmjno1uXdGOXZlzD4+YR+qOTyXVD1Q5+Q4EYMYMhXfeUTl0SKTMg0FBnBo2FGQpNdUgJUUn\nOTlInTo6S5aojBvnoahIQlVNDEOiRw9Red+7t8H48TaKi0XHKpdLpNSPHhXjHTZMY8QInebNDVat\nUhg92s6gQRoffaTSp49OerrJjBnCZsrjEaTR6nZVt67J5ZfrHDkC33yjcuml4phz5yrUrWuSlmbS\nqZNBnz4G06fLzJqlUquWgc1mcuiQQu3aoiDpwAGJxESTIUM09u8XZLZtW4Off1a45RaN/v01Jk1S\nmTZNJRAQpC0mRhSJ1akDqmpSWipxyy1BVBX27pVZt04QVkkCj0e4HfTurbNli8L48SopKSb5+UI+\n0aSJSfPmJoYh/FUVBe65J8iePaLb1kcfBSgtJUSEf/hBZvJklQYNzBCZbthQyBL69tW57DJR5JWW\nZrJsmcxLL6kcOCBz7bUaX36pkpBg0KqVTjAI2dkyW7aI4rc2bTTat9dITdVZsMDBgQMyBw/KoVa4\nY8YUc+WVGocP25k0yc769QoHD8rExxscOiS8bOvXN2ndWiMrS0R5ZRkKC8W1/sc/Ajz2WJkn9YlI\nUEWFQtUdzfszklXLEu9kmcJoHrLRiOxv0UO2IpSUlOByuY5b9GdkZPDGG28wefLkmhnYHwznyGo1\nwWq5auF00+8nS/+eKs4mUYzmPhAtkqfrOkVFRdSuXbvKx1CZMXq93lDqLrwoqTpwonOPdq0tTZj1\n+6pO9Z8KDh2CF16wMXWqSkqKwV13aQwZoodM5TUNdu8WutSsLJlVqyRWrFA4eFA6Lu3fuLGB1wsf\nf1xAdjbcc08dkpM1QCYnRyIuTuhQg0GJSy7ROXJEFA6Vlgoy6nJBSYmwZKpdW3z3uecC3HGHiECW\nlMAllzi58EKNefNUiosJ2VUBIRJ65IjE+efr5OVJSJLJk08W07cvzJ3r4oMPVPbsEWR65kyFG27Q\nWLtWYccOYXllmtCggUjpl5aK8VlFSwMG6EyaJEjXokUyc+cqzJihkpcHLpcJSHi9Yh/x8aLNbEqK\nIKTJyQa6DvfdZ+c//wmyapXM558rBAIS6ekG2dmiIUJBgUR2tmgtm5Rk0rWrQZMmBl6vaITw448y\nSUkmd90VZP160XyhuFjIKHJzZQ4eFHKDkhJC51Ovnsk992j07CkWHA0biqjp5s0Sffs6adrUoE8f\nnWnTVHJyhL1XSUmZBVjDhjqFhTIDBwa45ZYAw4bF0KePznffieKzjh115s9Xad9eJytLRpYhEJDw\nhXV1vfLKAP3766SlGaSlGTRuLAh9tL+Xyugrz3ZauqJI2x8ZJyoyqix+r0TWCnREjmHFihV8++23\nvPzyyzU0sj8WzpHVakIkWbWq8ysb0azqKnQLZ4MohrsPRBb8RINhGBQUVgIsTAAAIABJREFUFFAn\nso/jWUTkGIPBIHXq1Kn2h160+TcMI6SV/S2l+i389JOo6p8/X+HCC0VR0NGjEjt2iEiZiNAJDWaz\nZgYHDojI5aFDErffrjFkiEanTi78foiLE4VCqiqI0eHDUkgWYLNB3bqCaBQWyiQlaRw+LBMTY3Lg\ngIJhCEIUF2dSXCxIWosWBpmZMnXrCsJar55JcrJJgwYms2YpKIqIcNaubXLrrRq33qpRqxbs3Cnx\nwQcKb7whbKtE+1GT3FxhO+V0QnKyQXq6iExu3izjckH//hq33CJ8Ry3XgdJS4S7QurXBunUKOTnC\nOaCsWt/Sd0J2tmj72q2bwZtvBujXz8nHH/soKpLJzpbYtUsiO1tm+3YhLzAMEaW12UyaNIE2bQy+\n+Ubhqqt0iookDh0SnrAFBWWFapJESCJhmhAbC7GxYq4VRUgggkHRFCE/Xwp9T9xbYq7sdiErMAxC\n2trly2UKCsRCwu+XmDbtKJ066aiqg+efd/Pllyp2u0n79hpLl9pCEgvr2vbv7+eiizT+9z8Hui5R\nUCDTqJFJdrbMFVcE+N//BPGRZbjhhgDZ2aLQy+uVSEvTad68jMA2b27SurVJtMdYRe4X1s9VnZb+\nM5LVs63TPdH1iyYNqc5mCBW1mZ09ezZ79uzh0UcfPWvH/zPhHFmtJlipWAuV0SpG+mVGps+rAlVF\nFCN1s06ns9LFSKZpkpeXd9bJYjRvVKfTiSRJ5OXl1UhFZPj8/x5S/c88Y6OwUGLAAI3bbtPo0MEk\nPPOXmwtZWTI//ywxY4bK2rUiRS1JwvM0Odlk/36J/HxBAE1T/H+Hg1CfeEWBJ58M0KKFydNP2wkG\nhWZz2TIZn0+itLSMTNntwk4pP1/oPD0eg8ceK2Ho0CAej8yePSpjxzr56isVwxARxzp1BLndv1+Q\nZLvdpLBQwuMxAIkLLhDtWYNBMcYRI4LUqWMya5bKpk2iVasgryYZGRKHD4viMY9HtFkFiT17RJOD\nwkKJ+Hihf92+XebwYSGTyM8XhVF16wrrqe3bRQGVosB55xl4PGLOjx4VBDQQEFFj77FupE6ncBiI\nizMJBCQcDigsFCQbRGRSVSElxWDs2CBt25o0bGgyZoyNXbskBg4UTQLsdhF11jTx/aZNTc47T9h4\nTZum0KOHzrffqgwZouH3S6xaJRYl1jULBgml/B0OKbRISUoyefddUVyWlmbwyy8KnTvr+HzCb7Zd\nO53iYonevTXWrVP45RcldI84HCaaJpo2WHKA++8PMHashmma5OaaZGaKiP2aNSrffady8KCIyrrd\nJqmpOqmpRojIijGJ6HskTiWaV1kSVFFa+I+MmpQ+VEYacraaIRiGgdfrjZohnTRpEh6PhxEjRpzR\nMc5B4BxZrSZEklVLJxktohmta9PZSk2fKVGsCt0sCDH62SKrJ4tKVxdZrmhs+fn5qKpajkCfTqrf\nkjFUZar/4EH44AOV99+30bKliBYahtAt7tghon2NGpnHUtYGNhts2iSzaZPMoEEao0ZptGsnHjGF\nhfCvf9l47z01FOmLhMMhImlXX62zYIHMRRcZPPSQRuPGBh07uggGBaGpU8dk5MggHo/EV18prFsn\nSKSiEIoOWseQZYiPN0hJ0bjxxgB160Jmpso336hs26bQuXMAkFm1yhYitLIsIrO5uSJiC1axkEhz\n22yCLNavbxITI0jbgQOC/DZoYOLxmBw+DIcORScs8fHgdIp5CQSElrWkRDp2jQUZt9nE8eLiBAHN\nzxcR2nr1TCQJDh4UY2vWTLScDQQkunTR6djRoHFjk127ZGbOVDhwQBC/4mIRaT54UERD8/Ik2rY1\nURQTTYPPPvMfkxII7ezSpaL9qc0mot+KIkhy+/Y6XboEcDh0/u//PCiKmO/4eKH3bdhQLDqWL1c4\nJs0nLo6Q40FGBuzYIZObK+zD3ngjSL9+Dvx+cV8FAsJe7C9/0Zg3zxaa/xYtNNq100hJMdB1iRUr\nbGRmKowcqXH77QHq1xf36/btkJkph30Udu2ScTpNOnfWSUuzIrImaWk6zZpJUTtmnSyaVxEJKi0t\nxe12nyOrvwGcrocsVK5hi/X+i1Z78vLLL9O5c2cGDhxYtSf1J8U5slqNsETocHxEM7IIyfLyrI4W\nqHl5eafUfSlaxPdMi5FOdQyVGaMVqYxW1HW2j38yhBNoqyFCRQb+UP2p/h9/FEU2CxYonHeewTXX\n6Fx0kUFycpluEQRJ27ZN4sMPVWbOFBrIhAQRXbQKeZKShBxAlmHOHCVExkCkmJ96KsiYMXbOO8+g\nVSuTrVtFpLFnT1HQ88svCjk5ZWOrXx9atTKOSQ8kOnY02LBB5uGHA6xdq/D996LpgNtt0qSJKN7K\nyhLRR4dDGPrLshiAppURxHA4nSLiahHe0tIyeUI4LM2tdU66XnZu5SEB5X9hHdM0yxNr678jybwk\nlR0n2jHC92el/hVF7EfTyrZXFPFzvXombreIrOblSaFFQJMmJrGxYj8//SQTG0vIf/aBB0qRJJ3d\nu+1s3Kiyfr2QY1iBJbe7rBCuaVODw4dlLrpIx+WCe+7RyMqS+OUXmffeE64I1nctqeNjjwWZOVNh\n2zaZxo1NZs700b69C0kSi4Vrrw3y3XcqPp9EbKxBXp5M7dpiIZKaqh8joAYtWgiZxKxZdt5804Yk\nwY03Bmjd2iArSwqR2KwsmcOHZZKSDJKTdWrVMrnoolPTx54KCfqtFgmdKX6P0oequIYnaobwj3/8\ng2HDhtGjR49qO6c/Ms6R1WqEZe8BZdG82rVrh9LnJypCOpvIz8+vlMfp2Yz4VnYMJ8PpeqPm5eWV\na0l6thDZTczpdFJYWBhatNRkqt/vh2nTFCZMUMnNlfjb3zTq1BF2Qjt2SOzcKaJuVtvT+vVFCjsj\nQ6ZVK4Nhw3SuuUajUSNBlADy82HXLoklS2T+8Y/yxRcWeTIMkaK98EKDpUtlHA7o0UNnxw6ZgwdF\nir1WLREZfO45P1u2KHzxhUq7diKC+NNPEjt3yqFjxsYKomlJBsLsjSsFSRJRTZfLICZGRDYbNxYF\nT4YhMWuWjWefDRAISJSUiGN4vRLx8QZJSYQI/SuvqNSvb/LYYxqmKYi9zyc+U6YovPyynX//O8CF\nFxpoGvTr52TGDB9xcWWE9MABicOHhV60qKhMNxobC7/8IrF6taigT0gwME0hrygqkkLpecOwLKRE\ndFbTys+H0L9yrPWp0J4CIemGZWASTpQbNTJp107YYZ1/vsHKlTJ+v+h85XCYXHqpzsGDMlu2yBw5\nIrY3TWjc2OTqq3Xq1jX59FOVAwdEUwMrIp2TI3xv3W5Yvlw4Ctjt0KCBwd694uI6HPDuu37++lcj\nROr37xeWW9u3C2nAli0SGzeq5OdLuFwmrVppdO2qHfP3NWjZknL+sXv3mrz6qp3PP7eRkGDQoYO4\n98L1sSkpgsCmp4v7oGVLk4pqUouLi0OtrCOJ0B/Vf/SPJH2wnr2VKdazntXWOyb8Gt5xxx38+9//\nJi0t7ZTHMH/+/JCh/+233x5V9zp69GjmzZuH2+1m8uTJnH/++SfcdsyYMcyZMwe73U5qaiqTJk2q\nMQee08E5slqNCG+5qmkahYWFWF2DwvWJ1Y2TtX6N7N50NsZaUFCA2+0+bYH+mXqjVhVZjoZIrWyk\nnjcvLy+keYqmpbIWCWcr1b9vn8T776u8/baKyyWqyHv3Fi/lZs0MkpNFqhsEeZk+XWHcOJVt22Q6\ndDBISjIoLJTYs0dm715Bqpo0ES1THQ6Tfftkfv21LIJps8E112hs3iy6I6WmGjzxRIAHHnBw9KhI\nsTdsaFJSIiKaqkrIxiq8SrxilI9i2u2CqHXpEiQ9Hdq2NY9Fe02aNhVRxHXrZO69187q1WUHCH9B\nWc0P3n3XwebNCi+/XBC1k1f4C6tHDyevvRage/coegfgm29k7rjDwYQJfmrXhgcftLNyZaVOEIA7\n7rDTqZNB584G11/vYOVKH40bl3+caxoUFIjIqfjA5MkqP/4oc/SoRMuWBkVFguB6veDziQhrePRb\nzCF4PEIuUFIiRZVw1KljLTxM7rsvSMeOJnXrmlxxhZMOHXSWLVNC29ntQv7QooVJUZGQTjRsaLJw\nocJf/6rTrZvOM88IvbLVDMFSUTmd4mcrld+ihfgXTL75RhT+XXutzogRQZxOoSvevl061l5WJiND\nRN6bNNHRNIkDBxQ6dQoyYkSAyy7TqV27jIiE62N//FFl0aIyfWxMjHmMxJZFdFNTdRo1KqFu3eML\nbirrP/pbq3avDP5IZPVEiCzWCwaDoWtnGAYfffQRX331FSkpKezfv5/bbruNjh07kpqaWmmbSl3X\nadmyJQsXLiQhIYEuXbqcsM3q6tWruf/++1m1atUJt/3222/p168fsizz2GOPAfDCCy+clXk6GzhH\nVqsRVsTPivqZponH46nxHtJFRUUhHaeF041Qni4KCwtDUeXKIlKOcCY2Xicj7KeDyEh0RVX9paWl\nIQ/ecAIEZZ3PLBlD1ckkRPRq/HiVJUsUhgzR6N9fx+sVEdRdu0S0ctcu8d9Op9BY5uWJlp89expc\ndplOUpJJ48biExsrSM6ePfDuuzY+/1wUKeXmQjBYlk5NTRUdlyzioihl0T4rpV6ZaKgkCeKiaSLa\nl5AgumNddJFO+/ZBGjTw4XLpvP56HH6/wtixWoX7euMNlexsiddei5LrD8PNN9u54gqdoUOD5chG\neFtaSZLIz1fo0iWerKwCHI6K08A//ihzww0OevQQc/nCCyc+vgXhuuBixQofiYkmzz1n48cfZb76\nyh81bW3BNKFzZydvvx1g/HiV5s1Nnn46GPZ7k4KCAAcOaGzdqvLTT05WrLCxfr1CvXqCWJaWSqGI\nK4iorGmKRYHTKYivJTuw3i6KIjSrBw7ING8uFkBff63gcJj062ewaJFCSorB5s1Cs2pJI0A0ibD0\nr1a09513ArRubbJxo8TXXyusWaNQVCS+Hx9v0qqVILHp6eKeSE83j2mR4YcfZF57TWXNGuFm0ayZ\nxsGDQt+alaXg8RikpmqhQi1ZhqVLHWzcqHD77Rp33CH0sfv3H6+PzcqS2b1bweUyueAC7RiBFcdP\nTdVJSqqcPrYqCr2qExXZN/3REe4va5omOTk5/PLLL2RlZTFlyhQSEhLIzMwkOzubevXqkZ6ezrPP\nPkuvXr0q3OfKlSt55plnmD9/PlBGKC2CCXDXXXdxySWXcOONNwLQqlUrlixZQnZ29km3BZgxYwZf\nfvklU6ZMqaKZOPv4bamh/+AoKSlB13VcLhc2m42ioqLfxB+3RZ7g+AilNdazPU4rklEZRFo7ORwO\nYmJizmiM4XNwpohM9cfExIRIsKWlDU/1Wytu63eWLMQal7VPwzCiRvFO5bxLS+HzzxXefNNGZqao\nSO/XT6dWLZFObdrU5C9/MWjaVNgVrVgh8+67QrvasaNBt24GTqdITy9eLDxTDxyQ2LdPRORUtazw\nKCXF4Oef5RBRlSQYOFBj0SKlXGTONIWVUlGRdIykiEioBVUVdk9XXqlx8cUGPXuKqK8kCZlBy5Yu\n1q/34XCUOVJYhXSqqrJqlZ3Ro09MAleulBk06MQM2TTFfPz738FyC4ry3xHXdcEC0aVJUQwCAS10\nzSMjseefLzN3rk7Pnm6GD6+YTEdixQqhs0xMFPfQo4+KIqV331W5886K97N2rbAX697dIDk5SLdu\norVseroekpjYbArp6Q5atVK49loT0wxwySUORo3SGDxY56mnbOzeLfHkk0EWLZJZuFBh82axsAkG\nOY6oWlHUrVtl4uNNtmyR+fVX8fv8fEE2fT74+Wfxe02D8eP9vP66jdWrFX74QaFTJ+EGYe3z7rvt\ntGgh5AHnn2/w+ut+rr1WtKHdu1c6JgmQyciQmDfPxrZtUsiqy2aDHj0M/v3vAO3aCTJbu7YRusYH\nDkhs2QIzZ6q8955KXp5wdigulvj8c5m1a1XS0gSRbdHC5PLLNYYODbJggcLbb8ei6wZDhwZp317I\nCbZuVZgzRxDZ3FyZ5GSdlBQhMSnTx5o0aCCjqtHvqXACG97iOPyeqil97J893mXNsyRJ1K9fn0su\nuYSLL76YadOmMXPmTCRJQtd19uzZQ0ZGBsnJySfc3759+2jatGno58TERFavXn3S7+zbt4/9+/ef\ndFuADz74gKFDh57O6dYYzpHVakRMTExIBgBVS5DOFMFgkEAgEIpQVod+MxyVmYtIEmhFQqvioXym\n1yJaqj+8YCtaVX842YlM9Xs8nlCqPzJlqOt6SFJimuZJ09EgOhlNnKgyZYpKt246L7wQIDXVZM8e\n4cu5e7fMihUKu3eLyOq+fVKoGCc52aR/f53ERBObzYpcGTRoILowffWVwrRpCg0amFx5pU6PHgZ+\nP9x2mx2fT6T1g0FBYL76quyRo6ri/wWDIhoHQrsaGytS1888E+Cuu3QuvdTBk08GufTS43PP33+v\n0L27jml6KSoSzQ/cbndocRAIiCKhHj2ip+LFtYGVKxVefPHEhDY7W3RXatas4vvEWjwsX27jkkvM\ncoUXkbICawESE2OiKC5mzpTp3Flj8GCtwutoYfZshauvLiPXNhtMnBjg0kudXHyxaBMbDVOmKNx0\nk4YkCQ3pI48EGD1aZerUfOx2W+i+K39O8MQTQR57zE7HjgaTJqmsWuWjSRMRLbQaL4wcaadpU5Mx\nY4JkZkps2iTz3XcyS5YIj11Ng0OHpOP2LcsmvXoZ5OVJbNkiCOmwYU7S00VL3Y0bZTZulLjjjiDv\nvWcLLXS2b5fp318nJ0di1CgHDz8sPGNbthQkUsgDxPVdv17lsstEp646dcTfw/ffK0ycKJOZKeF2\nC4/e5GSTwkJYs0YmPh6efz7IX/+qH9P0intAEGGFjRsVpk2T2bJFpaREFPCdd16Qa6/1kZRkUL++\nwYUXmtSpo5STFbzzjo3Jk+3HdMTw4Yd2MjPlYx62xrFCMaGvTUkxaNlSIjY2+rM4MgIb/mwAokZj\nzxaR/S0EXqob1mKhIlhzoigKycnJJyWq4dtU5ting7Fjx2K32xk2bNhpbV9TOEdWqxGyLJcjq5E/\nVzcsghQIBJAkCbfbfcYRytNFRWTxZCTwbB//ZDiRLVbki8Q6TnjBlBXJ1nU9FIWNPDdJklAUJeri\nIZzEWhoqy2EAJJYvdzJxopslS4TZfXKyuN/mzVNITBSazfR0k759NXJzhU3Vpk0qAwboXHutTvPm\nJjk5gmQcPixI7Nq1Mr/+KiJpXq+InMmy8ATdsEFBls1ykVOrij7yVjdNSEgQ3Zq2b5cYNkxj0yZh\nQ/Xddz6Sk4UBf1aWTJ8+RsS2Yu7mz5fo2VNEoaPN3c8/y6SmHl8MYxhCnpCTI7Fhg9hGWERVjOXL\nZS680Dhhmt3C0qUKEyf6y/2/8OsYrs3+5huFXr0MnnnGxzXXxFCrVgn9+vlC19EiGWXkVWbOHCdf\nfFFe35qebvLPfwYZMcLOd9/5iVTU+HwwfbrKihW+kJvEsGEaU6bUY/78OgwZUvGz6NJLDWrXNrnt\nNjv33RekSZPj5+pf/wpy4YVORo4M0q6dSbt2OsOG6UCQkhJo29bFSy/5yc+XWLlSZtUqmd27ZUpL\nJZYtE/e25Yag64JQulxisVNQIDFhgu24uV+4UGH+fB+dOgmrsW3bxL6nT1fYvFk0aJBlSEsTuuu8\nPIn69U0GDtRJS9NwOMoi5m+9pTJ9ukLDhkJScuSIxN132/nvf8ukBKJISyM93Y/fb2PhQg9duhiM\nHKnRpInQx2ZkqCxYIDFunJAVxMQYNGumoesSmZkqSUkG//xnKYMH67hcZfrYnByxfWamzObNMq+8\n4iAzUzSyiIszj0Vzwz1kTVJSJByO458N0YqErOh+RYVepysrsGQKf0ZUdO4+nw+n03la+0xISGDP\nnj2hn/fs2UNiYuIJv7N3714SExMJBoMn3Hby5MnMnTuXRYsWndbYahLnyGoNoiYiq5EWWZbtlKVL\nrSlEzoVFAsPboFZFx67KHv9kiCw6O1mqP5KkWqQSwG6343a7T+vcoqWji4th4kSFKVNUFAVuv93H\n+PFFlJYa7NkjsXevzIEDKvv3C93e5s029uyRQyb4DRsK4/5PP1WpW1eY6MfHm3i9onOT1bZzyBCN\n884zUFWhP/X5YOFCkRaOPlZBDDp0MJgwIUDbtuVT+Z99ppKUZLJokS9U0DVzpsJf/qJjcTtr7ixn\njaVLPdxzj4nLdfyjbO9eiVdeUdF1k7/9zU5OjsTRoxJHjohCo1q1hH2TqJA3SUtz0bixSceOBuef\nLz4dOxohortihcKFF558cSkq+CXat6/c/TRvnsKVVxq0by/zxRcBrrsuhilTbFx0UVmqN1wTu2GD\niSwbJCYWUFRUPpo+fLjG11+7ef55laefLi8HmD1bpn17jTp1xL3gcDioU8fFG29oDB1q54orfFE7\nQIGIgA4apPHkk3ZmzfJH/U7TpiY336zx/PM23nyzfJTa44H77w8ye7bKRx8FGDlSRGN1XeiArcYH\nGRlSKMqu6+JetiDLkJhosm+fRGysSX6+kJ5cdpkTRRHXE8R92KOHwauvBhg8WMfvh4wMma1bJbZt\nk5k6VWb7dhs7d4r2sLoudLi9e+t88IGf3r3LrrnXy7ECK5lt20xmzpRYu9ZJTo4bu11YqDVtKhph\nKIpJ584GgwcboYXCoUPw8ss2pkyx07y5weWXaxQUSLz8sou//12mSRO9nO1W/foG69erTJtm59JL\ndT74oJQ2bUz27YOMjDJ97PLlKllZCnv2yLhcJh07arRsaRxzLAjXx0YnspHSgvBMzakWep0jq8ef\ne05ODvXq1TutfV5wwQVkZGSwc+dOmjRpwtSpU/n000/LfWfgwIG89dZbDBkyhFWrVlG7dm0aNmxI\nfHx8hdvOnz+fl156iaVLl542ka5JnCuwqkZEtly1ooXRul9UNUzTDEUAIy2yLOIaYzGEGkB4kVQ4\nka7qjl0VobS0FEmSovrlWQiP8hqGcVzRWbRUf3VW9e/YITFhgsr//qeGXvJW9DQhQXwSE00SEgxU\nVeebbxSmT7eTlqZxyy2lXHqpl9JSmcJClYIClfx8hQMHZH74wcaKFcJDtXlzUfVcq1aZr+fOnYIM\n7N9//Hk4HDB4sHhBZ2VJHDoks2WLl/Auw08/bePNN1XGjAnyyCMa4dz7ssscPPRQkP799eP0qLt2\n2ejf30lmpi80lq1bJWbPVpg1SxjBy7JwHujd26BePZN69QTxjo8nRICvv97O9dfr/PWvOtu2Saxb\nJ7N+vcy6daK5QaNGJuefb7B4scKUKT569z7xI/OzzxRmzlT49NPACb8HgpA1b+5i2TIfSUliv0uW\nyAwf7mDGDD+dOh1PjseOtVFcDM8/H4ha4HXggMlll9Vj0qQCunUzQpG766+P4brrvPztbxynQx81\nyo7DYfLKK9GlEKYJV1zhoKQE+vc3+Oc/o38vNxfOP9/FggW+46QIJSVw3nku5szx0bZt2e+ysiR6\n9XLyl7/oLF8uNLVJSWao+1ng5NOIJMF11+nHMgciVb9li3CnSEkxadPGoE0b4eOraWKOZ8xQSU83\nuOACgzp1TDIzRVvb7dtlYmNNWrQwjxFAjdJSjaVL7WzcaOOWWzTuuENHVc1juli5nEZ23z6Jhg3F\n+R06JDqijRih0aePXs6j2JIVCKcBmDPHRmamHCpYS0vTSEnRQvrY9HSTFi2gbl2JvXsVXn/dxhdf\n2OjfP0i/fkEOHw4v9FLIz5dIThbb1qtn0K6dmIO0NIMGDSr2j43mVHCiRghWDUF1vMd+a6jIBWHd\nunVMnTqVt99++7T2O2/evJD91IgRI3j88ceZMGECAHfeeScAo0aNYv78+Xg8HiZNmkSnTp0q3BYg\nPT2dQCAQ6tTYo0cPxo0bd1rjqwmcI6vViEiyWh0kMVznqaoqTqfzOJ2nRQJiwxlENcI0RX/lYDBY\nrmq+Om1QvF4vpmlGtReJTPWHd+mqjIF/ZKrfbrdXmR7YNGH6dJnJk1U2bFAYPlxj5EiNpCRh/bRv\nnzDnt/79+WeZn36SOXJEVPhbhvFxcWXdkmrVMjEMEV3dvVshPt6gcWNhmB4ISAQCMvn5Ejk5EoWF\nZf3eLaiqiM76fBJLlngZNcqBx2NSp44gIU8+GQyN/a23VJ580sYDD2g880x5AnTgAFxwgYvNm3OR\nZaFHtdvtocXLu++Kdq533qkxa5bC7NmiIvzqq3UGDtRp3Vp0vcrO9lJRIMHng+RkF1u2eInWbVfX\nRdODhQtlnn7ajssFQ4dqjB6thchlJO6+206HDgZ33XXygqlVq2RGj7azZk35lP6cOQqjR9v5+msf\nrVuXP0737k5efTVQYZTXNE1mzRK+tt99l4fTGeTAAZl+/eqzdu0hPB7pOJ1zfr5C165upk/3c/75\nx5/X4sUyDz4ooqoXXeSMSkYtvPaayurVMp99djzLfO01lXXrZD76SPzu0CH4+9/tLF6s0LatwVtv\niTa7kgRPPWVj+3aJTz4JsHKlxG232dm79/+zd9ZhVtXr2/+s2DFFd8MwwNBIqJSAAiJKSYgCKraC\nLRwVxDgqgoGSAgIiLVjEoHSXSOfQXRIzzMyOFe8fz6w9PagH8fzOy3Nd+5qB2bH23ivu7/PcoWbZ\n39JX/vxQu7ZFrVrSFa9c2eLCBeFKr12rcfCgTAEURWgTtWtbISBbtaos5mwbTpyADRtsZszQWLvW\nTTCYpuJ3OLGVK1sh14EKFYTPvX69OA2sXq3RrJlJTIzFmTNqKj1AgLjT/axUSTxlV64UvvjDDxv0\n7WtQvLjwZg8eVFLBsxKiB8THq5im8H9jYgxatQpQu7ZJpUqSIhcZmcaPTUqCX35RGTXKzbZtGo0a\nBbl8Wc3Cj3VoBTExFuXLW+TPn/15Nyfbrf9fgxCc61Z2LgiLFi0bA85HAAAgAElEQVRi+/btvP32\n2//Mxv0P1k2wegPLsYNyKhgMkpKSQh5nfnWdX8cBSJkjPDOXYRgkJSXdcIPg9M4DzrblyZPnHzm5\nZdflzjzqd4A+ZDxxpx8F5Tbqv540hqQkmD5dZ/RonUuXZIyZkpJm4VSypNhJlSwp3cRdu1QWLNAI\nBODRRw0eecSgcGF5rkBABE1nzij88IPOnDkaly4pNGxocvvtZsik/soVm2XLVDZtEkGJY1/knEEU\nBYoVs2jRwmT3bo0uXQy++cbFHXeYvPCCQcOGXrZtS6FgQenAPf20m0OHxP9z3z4fTgPdAfijR0v2\n+7hxKVkWL8eOKdxzj4fLl6FQIWjXTgDqLbdYoc7slCkaCxZoTJuWc2vul19UhgxxsXhx9qNtp778\nUmfzZpV33gkwapSLSZN0Wrc2eemlYIYuIUC1al5mz/ZnAZnZ1aBB0t7NDNQBpk/XGDTIxaJF/pCo\n68gRhTvu8HLoUEq2FkiQtrjq21c6PqNHBxk2zMPhwyrDh/uzAA2nKzt9upfJkyOIi7uM262lAxsq\nLVuG8dRTBt26mYwapfPTTxpxcdnbZPl8UKuWl0mTAlmEbU53de5cH3v2qLz2mpuePQ2eey5IkyZe\nJkwI0KSJPMbvh0aNvLz2WpBu3STN7M47PbRvb5KQAJ984soWuFaoYKGqQvlISHDieaFGDRnD33mn\nuCgcPKiye7dQW3bvlltyMhQtahEM2pw9q3HbbQZPP23Stq3sVxcvirBr3z6hFOzfL/SC48fFD1hV\nxWmhQweDmjUFkKY/rf7+u0wh4uKks3v8uFjBJSYqFClih3xjY2LSkrhKlRKLrk8+cbFqlUaPHkEa\nNTI5fToNxB44IPzfwoUlzStvXotDh3ROnNB4+GE/L70UoGDB7PmxBw6obNyosXWrht+vUKCAnQpi\nnY6uw4+VSUnmCgaDoSTDa5no/y8FIThgNbtm04wZM/D7/fTp0+cf2LL/zboJVm9gZQar1xskWpaF\nzyfCDE3TMnQAcyvTNElMTCRfToS161jpQVx6b1TLsv4W4P5Hy+lyR0REhID+Xx31OxZAmqaFOoHX\n66R89KjCu++6+PlnjUaNTJ59VkbciiKg88wZ6aKePKmwa5fC4sUaO3eqRERI59SyJAUpIUE6oFFR\nYuKekiI8TsdP1TQVkpLkQu90YH0+4R663SJmSV9RUaLqfvnlFDweiylTvFy4oPLMM8k8/bSPQYOi\n0DQZXW/apPPoo17atTO5elWhbFmL/v2NLHzULl0K8OyzJu3apQGenTsVPvvMxcKF4rsZF+fj1lvt\nbEFT9+5u2rY16dEjZ0uqV15xUayYzWuv5d4F7dDBw8MPG3TsKM91+TKMH68zapSLunVNXnnF4Lbb\nLI4cUWjeXMDkH/nKGzTw8sUXOQcHjBkjYQ2//OKjeHHpRO/erTJqVFYA7iz+DMPA5XIRCHho1Cic\nDz4I8vbbLkaNyvl15PE2rVt76Nw5wCOP+EL7+aJFOu++G8Xy5ZdwuVRsW6VVqzw89VSQnj2tbPft\nKVM0JkzQWbIkK6D98EOdCRN08uSBL78MUK+ebNP8+Rqvv+5i/XofzoBjyxaFjh29rFuXQvHiQnVp\n3tzL3Lk+ihSxufVWL6oqQrn0Aj5dl/22Xj2LW281KVxYbNt27VLZtk3EgVWqWNSqJeDQMGy2bIEl\nS1yULm0SE2MTGSlew3v2SCeyalWL2FiL2Fib2FgZrf/wg4sJEzRiYmzuvVfS2w4cECArYQQqkZHS\njY2JkaSyDRs0rl6Fl1826NnTIDxcjrNjx5zwAnnsvn0Cpi9eFBeKKlUsWrSQLrAj9ipYMO09Gwb8\n8IPGsGE6hw6pVKli4nLZHDmicuGCSpky6fmx0k29dEllwoQwDh5UefllEdwJICeDd6zDj82f36Jo\nUYvbbkujJpQp46dUKYvw8KxINrOJfmYwmx60/qdCrxtdzjUrO/rDyJEjqVChAl27dv0Htux/s26C\n1RtcTqcNZGe/cuVKKGrzr1RmY3xHMPVnxszXYzv+yGuk90ZNb5AP/1x316n0YQ2Zgf5/w6h/7VqV\nkSN1Vq7U0HW5MF29Kp3UEiXSOqklSsj4b/lyjR07VNq2Nend26BaNYuoKELdOMuChQtVRo92sWWL\nSsuWBnfcYVG0qIxBfT5Ytkxj3jzhvjlOAhcuSCfU8dEsU8bCNOX/vv46wC23mNSpE4aqwogRftq1\nC3L2rEX9+lEsX36Z775zM2pUGB9/nECjRgb16hVk3borFC4sljsOzeLSJZ1atcI5dCiFsDB5/598\nItv6zDNBCha0mT9fZ86c7DuiPp9wQXfsSCEnnYNtQ/XqXmbM8FOjRs6nwatXITo6jPj4FDKvpVJS\nYOpUnWHDdEqUsKlVy+L8eYVJk65NtDx2TKFJk9y7pABDhuh8+63Ozz/76N7dw4svBmnTxvEFzbjv\nOR18Z3G1bp1Kly4e8ua12bnTd00AvXOnQtu2XjZuTKFoUfmMmjb18OKLAdq3D4Y6sVu2qDz4YB6W\nLTtPwYIZwyw0TcO2VRo1CmfAgCDt2qUtFq5ehfvu87B7t8q4cf4MCxGARx5xU7y4zYcfpnWa33vP\nxfbtCrNmBVAU6TgPHepi1Sof+/crtG/v5emngwwZ4iIiQoRXTuXJY1OypHRpT51SKF9eAFbBggLi\ndu6UfVc8ehVKl7aoW1css2rVsqhZ06JQITh3DnbvFg7z4sUqv/2mcfmydBpjY4X3WqWKWLpVqWKF\nuKm2LeP8MWPkOwSbIkVksXPpkkJ0tJ0KZAU0V6pkER1ts3q1xscf61y4oPDss0Hq1rU4elRstvbv\nV0OiL10XWkFkpLxOSorCY48F6dPHyCCWS04WoO+kea1aJX6xSUngctlUqiTuCA6Qlc6uTb580ln/\n5ReVjz7ycO6cQpcugRDH98ABhYMHNS5eVClbVgBshQoWhQvb1Ksnz1Ws2B/jx/5fC0JwjrvsqGPv\nvvsubdq0oUWLFv/Alv1v1k2weoPL6RqBHKyXLl0if/78f/rgy84Y3+Px/KWD+D/ZjmtVZm/UnKJa\nTdMkISHhbwXMOW2fz+cL2XdFRUX914z6/X4YNUq8Uf1+6NPHoEcPI6SU9/tFeX7qlKRM/fKLxvLl\nKklJCiVK2ERFiQ+qE6fpRJY6xv22TUjAEhYGkZESqen3S6cqLEwAqstls2OHht8vY86ICHnefv2C\nzJypc/aswuLFPmrWtGnSxMPevSrz5vm59VYBIu++KybyFy86cZ9+Spe2GDlSZd06ldGjL2X4TFVV\nZerUcFaudNO1a5AvvvBy/rzKCy8E6dHDxOuFzp093H+/Qffu2XdNFy5U+ewzFz//nPN4f+9ehXbt\nPOzblzuImzdPY/Ronfnzc34uw4DvvxeeadGiNj/84KdcudxPrWPH6mzcqDJ+fO7A1rbhjTdcrFwp\nXS7h4Ka5eti2neu+d/vtXhITYds2X66g2Kk33nBx/rzCuHEB5s/XePddF+vW+TII3wBefdVFUhKM\nGOEjO1rBkiVu3n47D6tWXcbtVvH5VLp1iyQ62qZLF5MnnxSubvpD/vx5aNAgjFmz/NSvL/tPIABN\nmnh54YVgqhUWPP64G68XRowIMGeOxoABLr7+2k/Hjh58PgknSJ+CpqSmneXJI9xSw4BLl6BwYYtG\njQwaN4a6daVDL76uAky3bxe1fenS4hqwf79KxYpiVdWpk8nVqxJ2sGeP0AGc321b6AjBoDgFVK9u\n0bevQfv2ZuhzTEwkBD4dasHGjSJU1DTh1DZsKJ65MTFCCShd2g59h6YJ06ZpfPaZiytXxGVD0+T1\njhwRkVfFinZojF+xosXJkwoTJwqN57XXgnTubKbyY51OMKFY2gMHNNxuC8MQisOddwZo1y6Q6kdr\noarBkItMIKATH6/w7bc6s2a5UVURch46JNSC6GgzNZo2zbGgQgWTAgVy5sf+WaGXA2JvFJA1DINg\nMJitKPf555/n5ZdfpkaNGjdkW/5/qJtg9QZXesNmkFz4P+MbmplHmRP4+7P1Z7cjt8rMmfV6vdeM\nCnUAc4HsVC7XuZztSx8lq2kaPp+PPHny/OOj/nPnxO907FgXUVE2Pp90YUB8SZ1OasmSNvny2Wzb\nJnZR5cpZPPywQdu2JvnyCTB1NuncOfjySxfjx+vUqGHRubNB3brCw7MsEWKNHSvRq2FhAmZr1LAI\nBGR0qqrQsqXBkiU6efJI52vAABfJyQobN6ZgmgoPP+xm61aV1at9VK8up5WEBLGlCguzefBBk7fe\nCqCqJj6fn8aN8/PJJ8k0b55mvyXfjUWjRuFcuSLWUn36JNGmTTK6LhenhASNW27Jz65diaEc98yf\nfd++LipWtHnhhZzH+zIuVfjii9zDAJ57zk1srEWfPrlTBXw+iUB95pkg48e7eOutIL17GzkC4Y4d\nPfToYXD//dfOlrVtaNXKQ3y8yvbtl1GUQGiRmtu+l5gon3+VKhbNm+es4k9fV69KJOvYsQH693cz\nYECQe+/Nuo0JCVCvnvBMGzfOSi+wLJu2bT106BCga1cfDz0USeHCJsOGXUbTFN58Mw9Xr6qMGZOS\nAXTMmaMzeLCLNWt8IY7k1q3SQV23TsIIEhOhYUMv774bpGNHk/fec7FihcqYMQHuuceD1ytKeyd5\nCmShVbKkALpLlxSaNg1SowZ4vQp79qhs2KBy8aKo92NiZFF06JDKsmUSNVuihI3bbXP4sMq5c0Ij\nqFHDpnp1K3TLmxeWLlUZPlxnwwZJfStZUmJm9+5VSUqCmBjpwjoCrVKlLFas0Bg1SqdSJZsXXwwS\nE2OHRFkOnSA+Xni45crZhIXZHD2qEhFh8/jjwkHPTAk4dkwJPXbxYpV167SQs0J0dJp3bFon1aJw\nYdnXfvxR46OPXASD0LFjkCJF7NRurvgenzihUbSo0AoqVDBJSVFZu9ZNeDi8/rqPdu0sdF34sb//\nbofAsPPzt990TpxQKVAgs3+snRpRKwvo7I+F7LuxNzoIIRgMhq5vmatHjx6MHTuWokWLXtfX/P+5\nboLVG1yZwerly5eJiorKdWScHbi6njnxf3Q7rlV/lTMLf293N/32pfduTb99wWCQpKSk0Egn8yrd\nGbc6KV/Xe9QPws/797/FJqpjR5Pnnsso3klIIMRH3bJFZe5cGfUXLmxTpIiNZUkXNSFB4coVeUxk\npFy4kpKks6SqMrrWNPmb1yt81YQESaq6806D4sVhxw6V+fM1VBXuusvgxAmxc7rzThOPR/LVIyNh\n9WofkyfrfP65JAtNnOindWvZv1NS4P77PaxfrzJtmp8WLXyhycLGjeH06xfBr7+mdTVNE779VuO9\n91ycOKEwfbqfNm2s1HFqWqdl8mSdn3/WmDAhIXSRSn+BApVq1fKycGEKMTE5J8K0aeOhb1+De+7J\nGSxaFsTEePnlFz/R0bmfKuPiVIYNk27unj0KTz7pJl8+ybF3YlGdSkoSYLtvX0qO3qYZt8OiTRsP\nSUkiuPn6az9u97VtzyZM0Fi0SGPYsACNG3sZPjzA3Xdf2yv2xx81XnvNReHCNqtXZy+kcu7ndF4z\nhxCAhDJ07uymWjWbAgVsJkwIoKryXSYmWjRqFME77yRx991pwi9FUendOx9Vq5q8+aY/RDH48EM3\nmzdrzJkj27N5s0qnTh5Wr/ZRsqR46ebLB/36BWnTxkPduibz5unougDnQCCNvuJyCYXGtuH8eYXI\nSHGwSE4WuoDXKxOIpCQoUcLmtttk1F+vnlhA+f1ptICdOxU2bZKOqmlKNPCtt5p06GBSp44AUgfP\nXL4M+/ZJF3bTJgHCx47Jh1umjNAPnPStKlUEUDrUk5QU4Ul//rlO3rxQp46JYYglXHy8isdDBgAa\nHW1z8KDCN9/o5Mtn079/kLvvtkhJEcqAM8Z3aAX796sEArLPh4fL4rRNG+Hali0bwOXyp/PjdnH4\nsMrkyRrTp+uYpgjTLl1SQpQAhx/ruA1UqGCzfr2boUM9REZC//4BatQwOXAgjR/r2G45YrE8eWxG\njbpKrVryGWRO/stcOXVjcwpC+E+EXs65LDt/8rZt27JkyZIM4R836z+rm2D1BpdhSE64UwkJCYSF\nhWW7U6dXy+u6jsfj+cPg78/WlStXQvGlf6ayCxn4q96o17O7m77Sd6Mzb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LpUC42nnQte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\nzMzcvBcqQA0ND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- "text": [ - "" - ] - } - ], - "prompt_number": 7 - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The result is clearly a 3D bell shaped curve. We can see that the gaussian is centered around (2,7), and that the probability quickly drops away in all directions.\n", - "\n", - "As beautiful as this is, it is perhaps a bit hard to get useful information. For example, it is not easy to tell if x and y both have the same variance or not. So for most of the rest of this book we will display multidimensional gaussian using contour plots. I will use some helper functions in gaussian.py to plot them. If you are interested in linear algebra go ahead and look at the code used to produce these contours, otherwise feel free to ignore it." - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "import gaussian as g\n", - "pylab.rcParams['figure.figsize'] = 12,4\n", - "\n", - "cov = array([[2,0],[0,2]])\n", - "e = g.sigma_ellipse (cov, 2, 7)\n", - "subplot(131)\n", - "g.plot_sigma_ellipse(e, '|2 0|\\n|0 2|')\n", - "\n", - "\n", - "cov = array([[2,0],[0,9]])\n", - "e = g.sigma_ellipse (cov, 2, 7)\n", - "subplot(132)\n", - "g.plot_sigma_ellipse(e, '|2 0|\\n|0 9|')\n", - "\n", - "subplot(133)\n", - "cov = array([[2,3],[1,2]])\n", - "e = g.sigma_ellipse (cov, 2, 7)\n", - "g.plot_sigma_ellipse(e,'|2 3|\\n|1 2|')\n", - "show()\n", - "pylab.rcParams['figure.figsize'] = 6,4" - ], - "language": "python", - "metadata": {}, - "outputs": [ - { - "metadata": {}, - "output_type": "display_data", - "png": 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w4osvYu/evQCAd999FyVLlsR7772HMWPGID4+PssFbmyvjrVhA/DRR8DZs0BICNCjB1Cm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\nE08ATz8tjwYNgMKF1WTTNFnAsG2bPNaula2cgoJ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- "text": [ - "" - ] - } - ], - "prompt_number": 8 - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "From a mathematical perspective these display the values that the multivariate gaussian takes for a specific sigma (in this case $\\sigma^2=1$. Think of it as taking a horizontal slice through the 3D surface plot we did above. However, thinking about the physical interpretation of these plots clarifies their meaning.\n", - "\n", - "The first plot uses mean and the covariance matrices $\n", - "\\mu =\\begin{bmatrix}2\\\\7\\end{bmatrix}, cov = \\begin{bmatrix}2&0\\\\0&2\\end{bmatrix}$. Let this be our current belief about the position of our dog in a field. In other words, we believe that he is positioned at (2,7) with a variance of $\\sigma^2=2$ for both x and y. The contour plot shows where we believe the dog is located with the '+' in the center of the ellipse. The ellipse shows the boundary for the $1\\sigma^2$ probability - points where the dog is quite likely to be based on our current knowledge. Of course, the dog might be very far from this point, as Gaussians allow the mean to be any value. For example, the dog could be at (3234.76,189989.62), but that has vanishing low probability of being true. Generally speaking displaying the $1\\sigma^2$ to $2\\sigma^2$ contour captures the most likely values for the distribution. An equivelent way of thinking about this is the circle/ellipse shows us the amount of error in our belief. A tiny circle would indicate that we have a very small error, and a very large circle indicates a lot of error in our belief. We will use this throughout the rest of the book to display and evaluate the accuracy of our filters at any point in time. \n", - "\n", - "The second plot uses mean and the covariance matrices $\n", - "\\mu =\\begin{bmatrix}2\\\\7\\end{bmatrix}, cov = \\begin{bmatrix}2&0\\\\0&9\\end{bmatrix}$. This time we use a different variance for x (2) vs y (9). The result is an ellipse. When we look at it we can immediately tell that we have a lot more uncertainty in the y value vs the x value. Our belief that the value is (2,7) is the same in both cases, but errors are different. This sort of thing happens naturally as we track objects in the world - one sensor has a better view of the object, or is closer, than another sensor, and so we end up with different error rates in the different axis.\n", - "\n", - "\n", - "The third plot uses mean and the covariance matrices $\n", - "\\mu =\\begin{bmatrix}2\\\\7\\end{bmatrix}, cov = \\begin{bmatrix}2&3\\\\1&2\\end{bmatrix}$. This is the first contour that has values in the off-diagonal elements of $cov$, and this is the first contour plot with a slanted ellipse. This is not a coincidence. The two facts are telling use the same thing. A slanted ellipse tells us that the x and y values are somehow **correlated**. We denote that in the covariance matrix with values off the diagonal. What does this mean in physical terms? Think of trying to park your car in a parking spot. You can not pull up beside the spot and then move sideways into the space because most cars cannot go purely sideways. $x$ and $y$ are not independent. This is a consequence of the steering system in a car. When your tires are turned the car rotates around its rear axle while moving forward. Or think of a horse attached to a pivoting exercise bar in a corral. The horse can only walk in circles, he cannot vary $x$ and $y$ independently, which means he cannot walk straight forward to to the side. If $x$ changes, $y$ must also change in a defined way. \n", - "\n", - "So when we see this ellipse we know that $x$ and $y$ are correlated, and that the correlation is \"strong\". I will not prove it here, but a 45 $^{\\circ}$ angle denotes complete correlation between $x$ and $y$, whereas $0$ and $90$ denote no correlation at all. Those who are familiar with this math will be objecting quite strongly, as this is actually quite sloppy language that does not adress all of the mathematical issues. They are right, but for now this is a good first approximation to understanding these ellipses from a physical interpretation point of view. The size of the ellipse shows how much error we have in each axis, and the slant shows how strongly correlated the values are.\n", - "\n", - "\n", - "\n", - "\n", - "\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "###Kalman Filter Basics\n", - "\n", - "Let's say we are tracking an aircraft and we get the following data for the $x$ coordinate at time $t$=1,2, and 3 seconds. What does your intuition tell you the value of $x$ will be at time $t$=4 seconds?\n" - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "scatter ([1,2,3],[1,2,3]);xlim([0,4]);ylim([0,4]);show()" - ], - "language": "python", - "metadata": {}, - "outputs": [ - { - "metadata": {}, - "output_type": "display_data", - "png": 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- "text": [ - "" - ] - } - ], - "prompt_number": 9 - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "It appears that the aircraft is flying in a straight line because we can draw a line between the three points, and we know that aircraft cannot turn on a dime. The most reasonable guess is that $x$=4 at $t$=4. I will depict that below with a green square to depict the predictions." - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "scatter ([1,2,3],[1,2,3]);xlim([0,5]);ylim([0,5])\n", - "plot([0,5],[0,5],'r')\n", - "scatter ([4], [4], c='g', marker='s',s=200)\n", - "show()" - ], - "language": "python", - "metadata": {}, - "outputs": [ - { - "metadata": {}, - "output_type": "display_data", - "png": 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- "text": [ - "" - ] - } - ], - "prompt_number": 10 - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If this is data from a Kalman filter, then each point has both a mean and variance. Let's try to show that by showing the approximate error for each point. Don't worry about why I am using a covariance matrix to depict the variance at this point, it will become clear in a few paragraphs. The intent at this point is to show that while we have$x$=1,2,3 that there is a lot of error associated with each measurement." - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "cov = array([[0.003,0], [0,12]])\n", - "sigma = sigma=[0.5,1.,1.5,2]\n", - "e1 = g.sigma_ellipses(cov, x=1, y=1, sigma=sigma)\n", - "e2 = g.sigma_ellipses(cov, x=2, y=2, sigma=sigma)\n", - "e3 = g.sigma_ellipses(cov, x=3, y=3, sigma=sigma)\n", - "g.plot_sigma_ellipses([e1, e2, e3], axis_equal=True,x_lim=[0,4],y_lim=[0,15])\n", - "plt.ylim([0,11])\n", - "show()" - ], - "language": "python", - "metadata": {}, - "outputs": [ - { - "metadata": {}, - "output_type": "display_data", - "png": 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RnYphfc1taEAcTqY0iz2n2i5O5k8icqZZMmMHups2bcrrfQYk4c1qlvzRuV1t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\nMY8GinkCcfUJ2bgxmF5hZJ6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- "text": [ - "" - ] - } - ], - "prompt_number": 11 - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can see that there is a lot of error associated with each value of $x$. We could write a 1D Kalman filter as we did in the last chapter, but suppose this is the output of that filter, and not just raw sensor measurements. Are we out of luck?\n", - "\n", - "Let us think about how we predicted that $x$=4 at $t$=4. In one sense we just drew a straight line between the points and saw where it lay at $t$=4. My constant refrain: what is the physical interpretation of that? What is the difference in $x$ over time? What is $\\frac{\\partial x}{\\partial t}$? The derivative, or difference in distance over time is *velocity*. \n", - "\n", - "This is the **key point** in Kalman filters, so read carefully! Our sensor is only detecting the position of the aircraft (how doesn't matter). It does not have any kind of sensor that provides velocity to us. But based on the position estimates we can compute velocity. In Kalman filters we would call the velocity an **unobserved variable**. Unobserved means what it sounds like - there is no sensor that is measuring velocity directly. Since the velocity is based on the position, and the position has error, the velocity will have error as well. What happens if we draw the velocity errors over the positions errors?" - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "from matplotlib.patches import Ellipse\n", - "\n", - "cov = array([[1,1],[1,1.1]])\n", - "ev = g.sigma_ellipses(cov, x=2, y=2, sigma=sigma)\n", - "\n", - "isct = Ellipse(xy=(2,2), width=.2, height=1.2, edgecolor='r', fc='None', lw=4)\n", - "plt.figure().gca().add_artist(isct)\n", - "g.plot_sigma_ellipses([e1, e2, e3, ev], axis_equal=True,x_lim=[0,4],y_lim=[0,15])\n", - "plt.ylim([0,11])\n", - "plt.show()" - ], - "language": "python", - "metadata": {}, - "outputs": [ - { - "metadata": {}, - "output_type": "display_data", - "png": 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+RjCSlwccPUqX5uHDvH3sGI+TJynOnTvTqOrQAbjoImDmTLYcbN/e++eqIujh\ntBWhQmK+YcMGLF26FMuXL0dBQQGysrIwe/ZsfPDBB27nzZ079/fbsbGxiI2NrfhIayglJaZ/URfz\nlBSWbFWFjgBm2m3dytt6ZxyJZjFRlqNecKu0vpV2TSnEMidq41O5SgBTpK1zGhbG9H7APVlIt+4D\nhctFS/rXX81DiXdaGsW6e3cegwcDV10FdOlC8VafoaokLi4OcXFxv3e9qigVEvMFCxZgwYIFAIDV\nq1fjhRde8BBywF3MBXuUVe5wuH9BkpPdrXKAv6t/uHTGsUdlKnqra25FGhB7pzQ3i3VO9XZxav6U\nRe4vN0txMQV6/35GgunCHREB9OrFo29fJuJ1707rWv1tgSI2NhY5ObG45Rbgiy+AK6+cV6HXqbDP\nXEeiWSqO7tvVm1ScOmXfs/LkSd6uX58f3pIS6Yyjo2cq+iLm+qJ4wQWmWImY27tZ1GKpKk0q9KqT\numWurPvKtMxLSija+/aZx/79dJG0bw/07g306QNcdhnw8MOMCKtMd0hlkp4OPP448P33wJIlwOjR\nFX+t8xbzMWPGYMyYMef7MrUWbw2I7XpW6g2IHQ5TiKQzjok3y9xb30q9FrwKSRQxJ7qY1/ktiFkt\nlnrtG8A918G68elwVKytoWHw8753L489e/jz4EF+F/r2pWhPmwb89a8U7fPxOfuToiLg7beBp54C\nrr6aCYG+NKUpjUqxzIWKYxUdZY2npACRke7nKqFXoq8KbImbxUSF0amNUMC9XnxREfDNN8AVV1Bs\nvDUgFjE3xVy5WwBzfq2bdXpElQpJ1BeCssS8sJAW9u7d5rFnD5/brx+Piy8G7rmHAu4tOinYKS4G\n/vMfttrr2RNYuRLo379yXlvEPMCU1rOyVy/3cx0Os29l586mEImbxUSPomjY0AwDU3P8/vvAnXcC\nH38MXHut9wbEEs3iLubKMlcWt149EXDfhFfzp7tZdJ/5mTO0RHftMoX76FGG8w0YwGPKFIpcZCQ/\n99WdnBx+9l58kfkiH38MjBxZue8hYh5gdDHXLci0NM8GxAALGqkdeLHMPVFuFuX71ptUAOwmP2YM\nf157rekeMAwRcyvKRaKLuV6jRbkHAXdXn575mZYGxMUxJnvqVNYbys2lYA8cCIwbR792796ebf1q\nAikpwGuv0aUyZgywaBHDG6sCEfMAozLqAE8xVxXpdJo3N+N5VYch8Zmb6G4Wa5MKgNbgM88wQQOg\n4Ki4dF1KHAKEAAAgAElEQVTMgzUu2p+UZpnroZ8A5zA7m2KVmgrccgv92yEhFG7DAG66CXj1VcZs\nV5a1XVzMqpjZ2WaWpcPB8TRpwn0nf4QX6hgGG1wvXGgaDRs30iKvSkTMA4xumevC461vpV1H+bAw\nscwVSsx1y1xZkIWFDPm89FKKjWHwi6/8vaGhnE+xzImaH30DVC1ySUmcy4ceArZvZwZjURHw5Zec\nx9mzGXfeoAEwfTrn+4orfH/vwkIz01JlXSYnm6nzKSn8HuTnm1mWDRrwu2QY/P9nZtIl2b07cM01\nwKOPVq2wZ2UBixcDb7zB8d99N/DWW94rdlY2IuYBRs9U1MU8Pd2+b2VEhHtnnLw83icbdkSFxNk1\nIE5IYKx+eDivak6fpk9WuVqkM447atPy6FEK1YMPUlSHDePctmlDl8Hf/845nTmT4XVDhgDDh/Nz\nqmr0222AZmYyxPDQIdYhUpmXx4/zf9O2LRN3VLZlv37AxIn8n0VG0g3ZuHHpVn5JCa/Ghg4FPvuM\nVwuVPUcbNwLvvgt89RVwySXAP/9J95G/ff0i5gFGv1y19q20644jbc5KR8VE21VPTEqiKABm38rI\nSHPzrrZ3kzcMCum2bTyWLKE1/PHH3MBr04afye+/pwXarBmtXYBWuDIoVEiiimZJTKQr5IUX3BN5\n8vPpeujRgz8vvpiF5bp0oZCHVII6hYSYIb7333/+r6dISeG8/Pvf/BtvvRV49lnPCDR/ImIeYOw2\nQEtKeNsu0aFZM7MTurQ5c8cwzDA6vWelEvNTp8wvth6zb838rC1ultRUYMsWlohQPxs0oGU9ZAg3\n7Jo0AW6+GbjhBuDPf6YLITzcdGMpHA6K85dfMrHt8ceBX37hFWarVrTsk5NpIV9/PcPyWreueut1\n2TLgtttoLd922/m9VkEBsHQp8N//Ahs20H301lvsAhYMETci5gFG3wBVwpOZST+b8lPqhIcz/hYQ\ny9yKir5wOExR1kPolFsF4M/UVN62inlNdLPk5NC3vWWLeWRl0WUydCj9u0OHumcdP/MMI0+U7xyg\nFXr6NN0iR47QhbF3Ly3twkLGUBsG3S9RUfxs3ngjMGsW8NJL/vt7Dx4E/u//eBXwySdcmCqCy0Xh\n/ugj4PPP2flrzhy6bLxlFQcKEfMAYxXzBg14SeprmzOxzE30utt6VIves1KFe6qelYC9m6U6i7nT\nyQScTZuAzZsp3MeOMapk2DBalAsW0LVRmkVZVMQF7+uvaW2PHUvr+rLLOGfduvG17r+fi+PAgbRc\nR42ib3vDBs6xP63Ww4eBp5+mK+iRR+gqqkjI4y+/0I2yeDFF+7rrGFbZoUPlj7myEDEPMGqjDjAt\n85MnSxdzvZu8sswlmsW+IJS+J5Gezk00wD3EU1nm1bX+9smTFG0l3tu30+c8YgQ3Iu++mwk4ulvE\nSkYGreydO83j4EG69YYM4Rw+9hhw4ABf/+mnuTjcfLP5fDVnasNTF/GKpPOXh40b6ZNfswa47z5e\nNZQ3iiQ+Hvj0U4p4RgavJr75hnMXDG6UshAxDzB62rkS89J6VnoTc7HM3cXcrgGxvqkcEcEoDcAs\nsKUnuwTrBmhBAcV00ybzyM+naA8fzholQ4eWXucjNZWhhNu3U7R37OBVS//+dCPExjLk8OuvKcLT\np9PfPHEi58aaSAS4l7lVjwOeol7Zc/HZZ4znTkvjmD/4oHzuj6Qkuk8+/ZQLwBVXMMln9Gh7N2cw\nI2IeYOws89LE3NqzMjfXTHqp7Vgt85AQzqVaLM+dM0VO9awEzCsbvaYI4ClY/sYwGCWyaRMtz40b\n6T6JiqJPevp0RlB07epdME+domirWPDt27m5PmgQj6uuMl0u1lKw331nfq7U6+uFs/T31MMP1f1V\nZZkfOsQokvff59/wxBNM//e1lG1KCt0vn37K+Zw2DZg7l+GEpV29BDsi5gFGWeYulyns2dm+NSCu\nV4/JRaGhZpOL6nA5WFXYWeb6Yqk2lgEulrqYW9P49fA6f5GfT7HdsMEUcLWZeNFFjMgYPNh7Bci0\nNDOsUB35+XzO4MGMSHn55fPPwLSzzHVrHHAX78r4TOblsdb3u+9yU3POHGD9eiYE+UJyMiNtlixh\nLZjLL2d0zsSJNaeMgIh5gFFio3y7Dkf5xLywkM/Rw/FqK7qYq9u6mOvzqvf+VGLeoIHpKrDW8a4K\nTp6kcKtj715WBBwxghbziy8CHTvai6E1OmXrVrqRBg+mm+W66xg9UhHhLi7m2FR8+Fdf0Zp94gnO\n4ZNP8v1yczneFSvMeHPAveSt9Wd5MAxg7VqGAn75Jefl/vtZ48WXTM6EBC4AS5bQ1z91KuvATJzo\nXlempiBiHmCUZa77dktrQKzqsRiGe6U6vbhUbUUv1arEXN+TsIq5tQFx48bulnllN1TYvZvWthLv\nnBwgJobH//t/3Gi0s7qLiyn0unAfO0Yf97BhdBM88wxdJb5cSRgGwwsPHzb7Xh4/ztc8cYIWfmQk\nz2vUiHOoKk/WqcPwxdBQVj+MieFr/ulPZgncuDjW6F61ii6Ryy6jNe0rBw+yxsuHH3KBnTOH7pA2\nbcp+7qFDFP6vvuLfNW0aF6Fx4/xfo8XfiJgHmKIiirIeolhaz8q6dfmlys93r5YoNbjtC0LpC5y+\nSOqNnO3iy89XzDMz6SpZv55Fl7ZupZUdE0PLcO5cugisVrNh0CJWkSmbNzPKpGNHbnAOG8aa3v36\nlS1OTicFet8+Wqa//sqfBw/yb+7enQtAt27A5MmsxNmxI8U6JIQRK0VF9M3ffjvwt79xs/HOO2n1\nJieb72XNVm7YkPsTLVv6JqInTzIefNEivu4113Bzc/Dg0q8sDIObuF99RRHPyOB458/nRm5tMm5E\nzANMSYnpZtHrmtsV2VI0akRhEjF3pywx1+vF6+3i9J6Vyu9bnvBEw6C4KeFev55W4ZAhrFn9yCP0\neduVZ8jNpdCryJTNm/m+w4fTrTBvHt0m3jbE1fsnJ5sdefbuZZz0wYO0sPv04abpmDEU4p497ev+\n+EpJCV/7l1+Y6amSi1SVz9hYlk2IjqY13ayZZ21+gFcHaiNyzx7gj39kNcuxY0vfzCwp4Rx//TVF\nPDSUz/33v7nYVbcolMpCxDzAeOtZWVpygmpCXK+eGW0gYm5fd1v9dLnMqyDAvV2ctWclUHp4otNJ\nwVy71hTvkhIK98iRzHgcONDTIjUMhr/p0SmHDtFdMnw445pfftm7nxzg37Bvn3tjhz17+Df268fX\nuuQS+pajoiq3I09hIeds9GjefuABtj0LCXH3iesb8dafaWkU4E8/5QbtlCl00UyaVLofOzeX5WS/\n+YZRNh060AL/9lu2j6vNG/8KEfMAYxcPrTd2tkOVvtXjy0XM3cVc3VabwqoUrvrSq4qThuGeLKSL\nuRKo/Hz6qteto4Bv3EiLc9QoitEzz9iHB+blUbA2bKDgb9rE9x0xgpb6DTfQevUmYrm5FOsdO3io\nRJ7OnblYDBjAcr4DBnj2i61M0tIY4ti5M/doXniBm57t2vHxuXO5/2CNblGcOUP3x/jxvAqZNIlX\nCFOmeI/MAbjp+u23FPA1a7jgTZtG908wZ2IGChHzAKPERveZWxsQ5+Twd/VlETG3x1p3W7lZVANi\nXTRDQniu7itXrpWzZ3n/3LmMGNm9m9bf6NEUoQ8/tO8ClZTkHp2ybx+t5ZgYbuL961+mAFrJy6O1\nrUIKt2/npmTv3vQbDxsG3HEHX680AawsnE5ujm7bxoYSF1wA/Pwz/f0XXQT88IN3a1xtsK5bx0Xo\nrbf42D33UMi9jd8weMXz7bc8Dh7k+dddx9oodm4qwUTEPMBYO+MAZqszgBZPs2bAX/7CTR3AdBHo\nYq5iqmszejSLuq3CC60NiAGzto3Kqly3jtZvp07m+fPn0yK0XikpV8v69aavvKDAjE558UWKsF23\neD06ZetWHkeOULiHDmUp2Iceoq/b3xEYhw4xi/LDD/nZHDCAi9CDD5ruDMPwdEOp2wsWUMBvv50b\nq0OH0q8+ezb92lYKC4HVq00BdzgYQvj00/Tx1/QIlMpExDzA6A2I9W7ySgR++olW4OLFppg3aEDr\nPSLCXcyDNQXdX6jsRHVbuVnqhZSg8NNlqJd5EYw2g3AssznW1huPotznMaBbIVKyG6FzZwcmTqSQ\n79lDC/pPfzLdFzk53JxUwr15M10tI0cCEyZwo9KucJVhMKJEFbzasoWWfufOZsXCu+6ixR2o5JXT\np+nDXrSIVwPXXceCWd9+S6OhSRP37E4l5iUldDl98w0jSXJz6Y7p2BF4/XX+zenpZv6E/n7ff8/X\nX7mSi9jUqSxX26eP+L8rioh5gLFzs+iW+datrIvx8su8/I+IMC1KFZ8OiJgDXnzmxxKQ9sNP+G9G\nAtIwFu1yt8CAA2PyVqM+cvHOmSuxquU1aDR6EiZd3wFr15obpsuWAfv3U7z376eVOmoUCzktWmTf\nozUri4KtR6dccAH95MOG0XIdPNh7Upi/yMgwNyI3b6aYPvkk/doqUWrZMtN9oov5jz/Smv7wQ/qu\np00DXnmFrqTXXuOVSWioKfouFz+vTz3FzcuDB/k+l1/O+uitWgVuHmoSIuYBxi7tXPfv/vorEzB6\n9eLtmBh3MVdx0dW9bGtloARnzx6K1f1XJmPNLy3QAH/ASKxHU2QiDmPRBcfgANAFR9EFxxF3JgVn\n3vkKy86Ow4kTfdGzJzf9Fi2i1f3Pf9KCtm5Uulx0S+jp98ePc1NzxAg2MH7zTe9+cn+TkUGL+/PP\nuZE7bhyrHn75pf2GuxLxjAxa2H/8I0Mg33yTC9nll5uNsZOTzUXAMHh1qUIk33uPcffnznExGz1a\n3CdVgYh5gFEbb9bqicrNcuwYIyW6dOHtmBj3BsTKT15bxdzppNti9WpakkeOADNmAAU5xZj4ywto\nh96Iwq8YjbW4BwvRFcfgggP7EYVcNMJ9eA1rMRqG4UDMkg2oH9EVn33WAFOmMI1ctZkDzJhwtcG5\ncSNdEKp2yu2303ovLVHF5eJG6eHDZt/LhAQmzZw+TdFTLrcWLcxGJBXl9GkK+BdfcMzjxjEEctEi\n77HraiMyLo5ivHAhhf2KK+gPX7wYePtt9xKzyt334ou8ipk+nclHnTvTPfj443xMqDpEzAOMtZs8\n4N7qLDHRbGqblMT7rA2IgZrZHccOJd4//0yxWbeOojFmDLMYExOBQ8uPoFOPUPwRX+MEOgEwsBf9\ncCasM6YPOou1e5uiWcMiFJ12YkTxJvTAQTREHq7G55h97mMMaB6OunXbISWFLggVS75/P33bI0fS\n6n77be8p5obB/9eePRTGfft4HDxIEVR9L7t0oYXbpg0TfJo25d8wbVrFmw/Hx5sJNbt3MyJEdcfx\n5t7Jy2P6/Xff8QgJ4byOHs065tddx1DKefO4IIWG8jnLltH/vXQpF46DB7nH8+9/M5ywqIif3dqa\nyONPRMwDjAqhU24TwBTzwkL6YFu04Jf9+HE+bm1zBtRcy1yJd1wcj7VrORexsYyQePddsxXc3r3A\nf/9roPjuB1BgvIu3cDu+wTScRiQiW7pQFNII19wFLBwNtG1bH4MGAVOmj8D/5m5AlhGGeHREmisc\nc0YdxalT7TBxIn3kI0fSqhwyxD46xemkiOnx4Lt305XQvz8XgEsuAe69l4k83izirVtZ0vb77xnN\n8uc/+zZHhsH3/eYbHqdO0Qf+yCP0TXuLY4+PN8V73Tr68i+7jMk5vXqxXkx6Oj9vLhffx+mkUH/+\nORedmBjGi7/+OiOu3nqLbqbGjc0NaX1jWqg6RMwDjAqd0yseqr6Vp0/TyqlTh1EVmzfzcW9iXhM2\nQF0ud/Fes4Z/+9ixbAT8zjueHdCLixkP/cknwImjTjTftxgFqI9UtEYvHMCdf0zFkLuH49lnWfMD\n4HwXFAAfnr0MP7YajuOpDfERbkAuwjDyxMf4sfkIrNtSD126uL+XYTBVf/NmMyZ81y6OafBgCtlf\n/sKkHl86taen023x3nv0Td99NzcRy4qpzs9npNO339I6btSI1vwbb9Bfb5cOX1jIxfD774Hly7mh\nfumlvMpYvNizM4/Dwef8/DMXiK5dTRfRRRfxiuLll3nurl2mH1wZKGoDVP0UqhYR8wCjfOZ6/0rV\ntzI+3kxOadWK4g6YVrtujVfXOHPDoPvi5595mb96Na9Exo6lb/ettzyzG5V4K8HfuJEui969geZ1\nM7ADPTEAuzEPT+LZC19Dk3HDkJ/Pq5xnnuECsWkTXychAegfG4ExP32Be9Lm4ip8jtvxDhYUPQuH\nox7OnnVvybZ1K1Pkhw5ldMq8eWyQUJ6EluxsivAnn3Asl10GPP88/dmliV5SkmlJx8Vx4Zg6lfPW\ns6f9cxISTPGOi+McTZnCSJRBgzzfzzC4mP7vf1w4ExL4t4aEcMwzZwJ//zsXkoQE83l6ez699V4g\n6sLXVkTMA4w1UxEwvxjp6Wb4m7VnZUaGp8+8OljmyrJdtYpHXBxdF5dcYrbsatvW/TklJXQjqPM3\nbKCVGBtLS/aTT5hYtWuHC3s/OY0IZMAFBzZjOLY2HotVrzqQmMj5Sk/ncxYtoiX7pz8BGzbUwZmi\naBhfAYW4AO/iFqTn1MP48QbOnHFgyBBau3fdxe7zFUmdP3eOIvzFFxTCUaMojB9/7N2PXVLCRWT5\ncj43OZn+72uuYZcdu2Js+flcIFas4JGWxr2EmTPpHrELp0xLY7jh//7HIyyMz4mN5f/mwQeZ+dmn\nj3kFWK+eWX4Z4GdWuXPU1abTyc+xuFn8g4h5gFFirtwsTic//CEhvAxWbc6aNaOAA+49K/VaIsHa\nhDgpiQKmBNzlonhPnEgfcefO7ue7XNw4XLWKFvvatdwEHjuW6fSLFnlW/SsoALZ9cQKpzuaIxc84\nhTZ4ts4TcNZrjAkXUzw/+sgzomLXLl4N/Lq/C97BeuShIVZjDC4wCvDKIxm49Lb2Prcjs/u7ly7l\nsWEDN2mvvJKi6q1HZ0oKRXj5cibUdOxIV4g394lh0F+/YgWFeN06ungmT/ZufRcVcTw//MDj8GEK\n98SJLHPbtSvPe/llXh3qny11NaiuDvX5V2KurjKt7hahaqmQmCcmJmL27Nk4ffo0HA4Hbr/9dtx/\n//2VPbZagS7melq/w+HZs1IXc7s2Z8FimaenU4SVgKenU7wvuYQhatY63obBGHrdWm/ZkuI9ezb9\nydZaKEVFdHv8/DOPrVuBLo0awgXgMTyLX9ETyy5diCcvfBv9+9PCdLkoXmvWmAWzzpyhb7t33zp4\npMXTuGPbrfgQs9EFRxGVGIe6dW/2+e92uej++e47uiROnKBL49ZbuWFoZ4EXFXEcypKOj6e7ZcoU\niqldtMzZs5zb//3PrJEyeTLfZ/FiT5ePYVCw1flr1tAtM3EiF7eLLrIPp1Qirqfu62Kel2eem59v\nbg4r16HubhHLvOqpkJiHhobipZdewsCBA5GTk4PBgwdjwoQJiIqKquzx1Xj0DVAVb673rFSRD40b\n88ujLl2DScxzciiOq1ZRZI4coSU8bhyLQ/Xv72mZJSSYYv/TTxSTcePoann1VU9Xi9PJ+inKWt+w\ngYI0diyjNkaPBo6O+xtuPH03JuEH1IGB3JgJSFpLC/nYMbp3srNpIT/+OOOfn3yS1vnJk0DLvv1h\nbKPqOGD8tuNcupinp9NFsXw5xbJFC1OIY2I8284pN9MPP/D81au5uE2ezIiQ4cM9n1NczKGo5xw4\nwL930iTg//6P82AVy/R0zuuPP/IoKeH5119PF42du8WKEnH9s6U+c3oJYcBTzFUWrbLMRcyrngqJ\neevWrdH6N8dhWFgYoqKicPLkSRHzCqAq/dllgmZlmWJepw4jFrKzPbvJG0b5mimcL8XFtIRXruSx\nYwcjOcaNo8976FDPDD9lra9cSZE5d46W+rhxrE7YpYuntX7gAM/96SeKXtu2fI7uJ9cpTM9BNsLw\nBJ5GGlpg4NNXoEk4Qwqvu45W89KldCnMn89NR/39HL17w4A5CONshsffXlLCv135l/fvZ2GsKVOY\nrt6pk+d8ZWRwEfrxRwpyYSGt4muuYWil9apDXan8+CPna/Vquj4mTWLGZUyMZx2XggLGwq9cyecd\nPkzBnzCBYY5RUeUXVPWZ0t15KutYVe5U5OWZJShUzoT6bOu9WYWq47x95vHx8di5cyeGDx9eGeOp\ndejV/ex6Vur+ZNW3Uu+Mo3dKryrLXImLEorVqzmu8eOBv/6VomFNB8/PN8Vl5UqmvY8eTfFWhaWs\n1npysmlNrlxJwRo/nuUM3nzTc+PR6WRMt3qPjcffBWDAgAONkYUt3+XiucUdMWwYLdGdO/m81av5\nc+tWbv6dOEHxyxvSGIfAsJBj6IrcXLOhhBrTzz/Tjz1xIiNjRo70FNbCQrpO1HwdOGAW5Lr3Xvti\nUqdPm+evXMnHJ0wArr2WPnar4LtcjKtXlveGDaxqOGECXScjRpx/yrxys+gJacrg0Ds1Ae5dnKw+\nc72apVB1nJeY5+TkYMaMGXjllVcQZtPSZO7cub/fjo2NRWxs7Pm8XY1EF2KrmFt7gaoOQ9aQRN1H\nWVmcOuUurCEhpri8+65ncSSXi+4KJS6bN9O9Mn48u8QPH+4pLjk59I+r56Sm0vIeP97eWgcovGrj\nbtUqCvz48ex6M3f3dbg37Uk8gyfwPm5CgzB3BVm71rPzzWOP0S+9cyeQcQUHeBIX8rHke/FLR87r\n+PF0Ab3xhueiomLjf/qJc7V+PZNuJkzwbklnZ9N3rZ6TkMArhgkTGKfeo4fn3378uHml8tNPjAuf\nMIGbwp9+Wvn1vnUx1y3z4mL3HqqA+2dVt8yVqMsGqHfi4uIQFxd33q9TYTEvLi7GlVdeieuvvx7T\np0+3PUcXc8EeXcyVMOtNKvQ1Un2BrCGJ1i9cRcjLo7goV0ByMv3REyaYEQ5WcUlMNIV45UpavxMm\nUFjHjPHMdHS56PdWYrx9O2O1J0xgDe3oaE8LLj+flrTaIDx7luerhBXdt74rDECa5Y/SGD2aKe6q\ngNS4cRTz9espjO981gQXoBD9sBcNkYOhTQ7ixR8u9vBJqw1FtWH7888MExw3jvVZ7KJtlN9buZl2\n7qQ7avx4lgUYMsTTV37mjLtrKjeX76EWiY4dy/y3nhdKzHU3i4qkCgvjgqTIyTE3eJWYWzdCBXus\nhu68efMq9DoVEnPDMHDLLbegd+/eePDBByv0xgLRfebW5CHdDwmYfkpdzO02qXxBt6R/+IFlW6Oj\n6T54912Ki/ULmJtLS1qJ8ZkzFKOJE5n6rRelUpw6xXNXrKAotWxJ3+9f/kJfs121vvh4M7Z67VoW\nr5o8mTHZ0dGlWHmNLC1sUlIA9Pj9V1VVccQI+tx//RW45Rb+Ha1bA1cPSMeFyMZRdEMPHMLsjqvR\ntddtAHhFoCJnVq3i64wbx0Xln//0/NudTs6vEvz167nROX48F8dRozw77qiNZGV5HzvGORo3jj09\n/V3rW/9sqStBVa65SRN3Mc/ONq/W1Ca+MlDEzeIfKiTm69evx0cffYT+/fsjOjoaAPDss89i8uTJ\nlTq42oBuVder57uYW6NY9J6V3khONoX4p59oPU6YwKSQ2FjP0DnDYLy32uzbsoUbnRMnMmbbTliL\ni+m//f57CnhCAsVo0iTvgu9y8bVVQ4TUVMZWz5nD9/EWk+1Bt+7APu331athGBcjNZV/x44dFNSz\nZynmDRowEiYri7o/J3ERvsYs1IULxQjBV3WvxP6bKeB5eZyjsWOBJ57wbEShMlmVeK9ezUJVl1xC\na/3jj+1j4zdtMp+zaxcX0UsuYaXCoUNLr8BY1eidmtTnTUVSNWnCaCsVqZKZyasba9SLuFn8R4XE\nfNSoUXAFa4ZKNUO5WfSNJiXmeiIGwNsFBbzEtYq52gjVKSigpafEODmZluGkSYzksGuKq7IBV6yg\n6IeF8fwHHqCQ2cVKJyebKeOrVpmhdm+8QTeK1X0AUBBWraLbY+lSuin+8Ae6HIYNq6AlNzIGrm8c\n2IxhyEEj3PrRGMT9tkj260exWbSIFm6dOnQF9ezJKJesTANLf6iPFLRGVxxBMtrhp4KGuHww8PDD\nTIO3ivfBg6a1HhfHxXbcOOCqq/i3X3ih+/BKSuhaUiGZmzfzdVVET0yMf/p7+ooSYjsxr1ePc6ga\nqZw7R5+9Hlorbhb/IhmgAUYXc93dAnj2rVSlb8PD3cVcf40DB0zxXreOIjZpEutsDB3q+aVyOilm\nypI+cIAW6OTJjMG2FppSz9m82awTkpDA95g+nVEn3jrHFBVxofjsM1rgUVFseLB2LS3dinDypNnV\n58cVl2AfHLgdb8OJurg053OETxuM0ZMaoUMHWrv9+pmx3qdOcVHbsgXIzzUwunAQ6qMA32AaLscy\nLPygNbr05vuo5+jiXbcuF7hLL+XrWMMS9YqPKpO1Y0da3g88QBeKtbhVMKGMC3XV53K5951VWckN\nG/KnVcx1N4tY5lWPiHmAsRNz3TLXoyBUtURrudulS+mT/eor05K+5RZaoXYuijNnKNzff0/r+8IL\nKd4LFtCXaxfSlpnJBULVr27ThgWiFi60T3RRGAbH9uGHrEvSqxfrhCxY4JkYVBbZ2bRst27lYrJ5\nM90fI0bwuPOeELz60DHszh+IdkjEdHyNfUlz4HINQUICffHXXsuNXpeLwvOHP3BRqbtmNW7c/idc\nhc/RF/uA+vVxIrU+Vm2gyyQujuI0diyPp57yjLZxudjMQQn+mjVc2MaOZS3wf/+7erVI06/6lGtP\nfQYBs15Q27ZmHSG94JZY5v5FxDzA6DWfrQkWhYWeYl5QYHanGTOG/t8PPqBoz5rFinbWTTKXi/5i\ntal48CCtQ2VR2vmxAYqfqi2yZQujQS6/nAk3di4anTNnmIb/7rsUgtmzKcS+RmBkZ9OHvGMHrxy2\nbxm5OUgAABsgSURBVOcm5IABvMK44grWddF913v2AK83aYyi/FAUIRRv4g78b3sLfLTfhSbhNA3H\njaMQd+3KBeCqq4CfF6cgY/tWHEcnnEM4rsNHSCpshWuv5fljxnBevfnJlXivXk3rdOxYxsa/8Yb3\n5hXVAbvMT13MW7SgWw7gz+bN3QtuqagWla0sVC0yxQFGb3rrTcwzMhgJsm4d44mbNOEX5NFHKcwf\nf0yBiow0xSYzk1b38uW0pCMimKX47LPerW/DoIB+/TWPU6co3vfey41Su8gTK4mJTKb59FO6XT78\nkJa7tygMl4sL0969dEmoIzmZLpHoaLp9Hn6Yvm67DcHMTLpZvvwSOJbdAhHIQBFCkYgO6IFfcUv4\ndxj55mw89VJj3HKL+b7Z2cCXH+Xi+3eycAx34z+Yg0JcgLHNdiOu/iysW++etGUY7BYUF0fhXr2a\nLoaxY1mB8aWXvC+M1RGrmBcXu1dLVGWZnU4aFS1acMFVBogS87w8EXN/IFMcYOws8zp1zKbEM2fS\n+hs9muFzf/gDe1zecQfFOSTEXBBSUykoy5bRFTFqFF0hf/+7Z2VChdNJN8gXX1DAQ0Ppxy6tyYE3\nFi1iCN1tt3GR0V0K2dm8ojhyhNmgBw/SP79/P63ZPn1odU+fzhrhPXrYC4Bh8Iph/XpGzaxfT1/2\nkCF8TvPmDuz5xzcYcM9IPIW/40X8CY1PHYXjofuREbIQL73U8PdCW9lZLhxM2IhuhZkYjR9wG97B\nbHyIW98Zjqf/VAeGQbeJ3igjLIyLy2WXMTrHLn2/pmBN4y8uNvvPAvw8njrFq7BmzXiO7hrU481F\nzKsemeIAo0K78vNp9a1cSV/w9On8Mjz4IK2+Bg0Ymx0eblpMJSW00OfNA5Ys4YIwcyY318aN825J\nl5TQLbBkCQW8TRu6Lb777vximV9/nVcN8fGsn3LuHBeY5GR+yTt3pquiRw8uNLffTn91aaGHBQV0\ntWzcaDZRBpgeHxMD3HgjS75ecAHnb906IPyuWcDDp7GpYDi2YTB+wAQkHWsPJxw4+PxXuLpHCl5r\n9x0m7v5/eKb4ASzBDDhRFy7UQWHjFngtuR/OnKE7p1kzive0aUyTL8u9VJOwxpcrMVfVO9u1YxGz\npCTeBuzdLPqmvlB1iJgHkCNH6CKYPZs+6c6dKRyDBtHiDA+nX1tFtISE0ML96ScKZmQkE03Cwhj7\nPXIkLWM7SkpoXX72GTdKu3Shha8aPVQG333HcWdl8UscHk7rvG1b+lPLWiQMgy4XFZ2ycSPdL1FR\nLNN65ZXACy/QGra+1rlzFPLUVGDUaAcSC1vh2YbPIDTvHEZgE27AvXgE/8CbqVcAqXxOIV7GAfTC\nOoxEAjrglboPo7CwIfbsdaB+fW4SDx1aOXNTHfFmmSs3S6dONDzi4829EL16ojWtX6haZIr9SHEx\nxW7ZMh6ZmbSsZ8+mYA0YQMs1Pp5ipSya5GQ26v3kE/qkhw6lr3bLFlq4999Pq9EqcIZBQVy0iPW0\n27dnpb6tW6vGPdCsGX3svpKdzc1NFZmyaRPHfNFFdPE8/zzdJ3ZXGAkJnMt163gcO2bGgs+dC9xx\nhwNLvmyDd67eh+aHfkUrnEYB6mEdRmINLsZqjMFxdMHfMB+NkIOBHc/h5tc6469zHXjnHS5M5Y22\nqWnorQhVSKLuZunene6yQ4d4G+Bjdpa5iHnVI1NcxaSncwNy2TJuSHbtSsFTqemdOjGUcNcud9/5\ngQO8ZB07lj7hyy6jiE+bBtx8M6Mw2rVz7wKjMkAPH+bG40cf0X957bUUPPWFCwRFRbSyt241Qwvj\n47mADR9O99DLL9ONYV2UnE76rnXxLizkQjZqFDNFo6M5T9OmMTEqJATIdjZEQvQ0bKs3El8evQG/\n5kXhQbyMi7EGd+MN7EA0Vg16BK/2eA2N+3VGRGvzvaWhgnumsbLM9TrmUVFcRDdsMBtl65a5ijkX\nMfcPMsVVwMGDTIpZupSRGZdcQgF/6SXPrEDlM1dRHXFx3JybOJH3zZ/P54eGsrjSuXPuqftKcIqL\nadl+9hnF/NpraY0PGuR/USopYd0TFRO+dSvFuHNnZncOHQrccw+jVeyiU7Kz+beoTc7Nm+nXj4lh\nVM28eZ5hggDnJiuLfT0TEij0LVoAnTq1wF0LWyD5sRJse/U4kB0ONJ2NG29pgforl8H5rGcNEWl1\nZm+Zq8Q1gBb4kCG8ivnXv3ifbpmrTFERc/8gU1wJlJTQnaFisnNyGHXy2GO0rPWUfB2nk/7Hp55i\nKF/DhswKHDiQgl6nDkVdiZaywvXU/cJCFm768kta3vPnM8rFXzU9Skp4FbFjB8V7+3YuYG3a8Is+\neDBjrgcNcq8AqTAMWncbN/JYv557CYMGmXsAF13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- "text": [ - "" - ] - } - ], - "prompt_number": 12 - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Think about what this plot means. We have a lot of error in our position estimates. We therefore have a lot of error in our velocity estimates. But look at the intersections between the velocity and the positions. Take the intersection at $t$=2. The intersection between the velocity and the position is where our aircraft is most likely to be, which I have roughly depicted with a red ellipse ('roughly' because I set the size via eyeball, not via math). The size of the error is much smaller than the error of the positions, despite the fact that velocity was derived from position. \n", - "\n", - "What makes this possible? Imagine for a moment that we superimposed the velocity from a *different* airplane over the position graph. Cleary the two are not related, and there is no way that combining the two could possibly yield any additional information. In contrast, the velocity of the this airplane tells us something very important - the direction and speed of travel. So long as the aircraft does not alter its velocity the velocity allows us to predict where the next position is. After a relatively small amount of error in velocity the probability that it is a good match with the position is very small. Think about it - if you suddenly change direction your position is also going to change a lot. If the position measurement is not in the direction of the assumed velocity change it is very unlikely to be true. The two are correlated, so if the velocity changes so must the position, and in a predictable way. " - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Kalman Filter Algorithm\n", - "So in general terms we can show how a multidimensional Kalman filter works. In the example above, we compute velocity from the previous position measurements using something called the **measurement function**. Then we predict the next position by using the current estimate and something called the **state transition function**. In our example above, *new_position = old_position + velocity*time*. Next, we take the measurement from the sensor, and compare it to the prediction we just made. In a world with perfect sensors and perfect airplanes the prediction will always match the measured value. In the real world they will always be at least slightly different. We call the difference between the two the **residual**. Finally, we use something called the **Kalman gain** to update our estimate to be somewhere between the measured position and the predicted position. I will not describe how the gain is set, but suppose we had perfect confidence in our measurement - no error is possible. Then, clearly, we would set the gain so that 100% of the position came from the measurement, and 0% from the prediction. At the other extreme, if he have no confidence at all in the sensor (maybe it reported a hardware fault), we would set the gain so that 100% of the position came from the prediction, and 0% from the measurement. In normal cases, we will take a ratio of the two: maybe 53% of the measurement, and 47% of the prediction. The gain is updated on every cycle based on the variance of the variables (in a way yet to be explained). It should be clear that if the variance of the measurement is low, and the variance of the prediction is high we will favor the measurement, and vice versa. \n", - "\n", - "The chart shows a prior estimate of $x=1$ and $\\dot{x}=1$ ($\\dot{x}$ is the shorthand for the derivative of x, which is velocity). Therefore we predict $\\hat{x}=2$. However, the new measurement $x^{'}=1.3$, giving a residual $r=0.7$. Finally, Kalman filter gain $k$ gives us a new estimate of $\\hat{x^{'}}=1.8$.\n", - "\n", - "** CHECK SYMBOLOGY!!!!**" - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "from mkf_internal import *\n", - "show_residual_chart()" - ], - "language": "python", - "metadata": {}, - "outputs": [ - { - "metadata": {}, - "output_type": "display_data", - "png": 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The gaussians represent a mean and variance that are scalars - real numbers. They provide no way to represent multidimensional data, such as the position of a dog in a field. You may retort that you could use two Kalman filters for that case, one tracks the x coordinate and the other tracks the y coordinate. That does work in some cases, but put that thought aside, because soon you will see some enormous benefits to implementing the multidimensional case.\n", + "\n", + "\n", + "## Multivariate Normal Distributions\n", + "\n", + "What might a multivariate (meaning multidimensional) normal distribution look like? Our goal is to be able to represent a normal distribution across multiple dimensions. Consider the 2 dimensional case. Let's say we believe that x = 2 and y = 7. Therefore we can see that for N dimensions, we need N means, like so:\n", + "$$ \\mu = \\begin{bmatrix}{\\mu}_1\\\\{\\mu}_2\\\\ \\vdots \\\\{\\mu}_n\\end{bmatrix} \n", + "$$\n", + "\n", + "Therefore for this example we would have\n", + "$$\n", + "\\mu = \\begin{bmatrix}2\\\\7\\end{bmatrix} \n", + "$$\n", + "\n", + "The next step is representing our variances. At first blush we might think we would also need N variances for N dimensions. We might want to say the variance for x is 10 and the variance for y is 8. While this is possible, it does not consider the more general case. For example, suppose we were tracking house prices vs total $m^2$ of the floor plan. These numbers are *correlated*. It is not an exact correlation, but in general houses in the same neighborhood are more expensive if they have a larger floor plan. We want a way to express not only what we think the variance is in the price and the $m^2$, but also the degree to which they are correlated. It turns out that we use a matrix to denote this:\n", + "\n", + "$$\n", + "\\Sigma = \\begin{pmatrix}\n", + " {\\sigma}_{1,1} & {\\sigma}_{1,2} & \\cdots & {\\sigma}_{1,n} \\\\\n", + " {\\sigma}_{2,1} &{\\sigma}_{2,2} & \\cdots & {\\sigma}_{2,n} \\\\\n", + " \\vdots & \\vdots & \\ddots & \\vdots \\\\\n", + " {\\sigma}_{n,1} & {\\sigma}_{n,2} & \\cdots & {\\sigma}_{n,n}\n", + " \\end{pmatrix}\n", + "$$\n", + "\n", + "This is called the covariance matrix, and is probably a bit confusing at the moment. Rather than explain the math in detail at the moment, we will take our usual tactic of building our intuition first with various physical models. \n", + "\n", + "So here is the full equation for the multivarate normal distribution.\n", + "\n", + "$$\\mathcal{N}(\\mu,\\,\\Sigma) = (2\\pi)^{-\\frac{n}{2}}|\\Sigma|^{-\\frac{1}{2}}\\, e^{ -\\frac{1}{2}(\\mathbf{x}-\\mu)'\\Sigma^{-1}(\\mathbf{x}-\\mu) }$$\n", + "\n", + "I urge you to not try to remember this function. We will program it once in a function and then call it when we need to compute a specific value. However, if you look at it briefly you will note that it looks quite similar to the univarate normal distribution except it uses matrices instead of scalar values. If you are reasonably well-versed in linear algebra this equation should look quite managable; if not, don't worry, the python is coming up next!\n" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import numpy as np\n", + "import math\n", + "def multivariate_gaussian(x, mu, cov):\n", + " n = len(x)\n", + " det = np.sqrt(np.prod(np.diag(cov)))\n", + " frac = (2*math.pi)**(-n/2.) * (1./det)\n", + " fprime = (x - mu)**2\n", + " return frac * np.exp(-0.5*np.dot(fprime, 1./np.diag(cov)))\n" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Let's use it to compute a few values just to make sure we know how to call and use the function, and then move on to more interesting things.\n", + "\n", + "First, let's find the probability for our dog being at (2.5, 7.3) if we believe he is at (2,7) with a variance of 8 for x and a variance of 10 for y. This function requires us to pass everything in as numpy arrays (we will soon provide a more robust version that works with numpy matrices, numpy arrays, and/or scalars in any combinations. That code contains a lot of boilerplate which obscures the algorithm).\n", + "\n", + "So we set x to (2.5,7.3)" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "x = np.array([2.5, 7.3])" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Next, we set the mean of our belief:" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "mu = np.array([2,7])" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Finally, we have to define our covariance matrix. In the problem statement we did not mention any correlation between x and y, and we will assume there is none. This makes sense; a dog can choose to independently wander in either the x direction or y direction without affecting the other. If there is no correlation between the values you just fill in the diagonal of the covariance matrix with the variances:" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "cov = np.array([[8.,0],[0,10.]])" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Now just call the function" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "print multivariate_gaussian(x,mu,cov)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Let's check the probability for the dog being at exactly (2,7)" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "x = np.array([2,7])\n", + "print multivariate_gaussian(x,mu,cov)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "These numbers are not easy to interpret. Let's plot this in 3D, with the z coordinate being the probability." + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "%matplotlib inline\n", + "pylab.rcParams['figure.figsize'] = 12,6\n", + "\n", + "from mpl_toolkits.mplot3d import Axes3D\n", + "\n", + "xs, ys = arange(-8, 13, .75), arange(-8, 20, .75)\n", + "xv, yv = meshgrid (xs, ys)\n", + "\n", + "zs = np.array([multivariate_gaussian(np.array([x,y]),mu,cov) \n", + " for x,y in zip(np.ravel(xv), np.ravel(yv))])\n", + "zv = zs.reshape(xv.shape)\n", + "\n", + "ax = plt.figure().add_subplot(111, projection='3d')\n", + "ax.plot_wireframe(xv, yv, zv)\n", + "show()\n", + "pylab.rcParams['figure.figsize'] = 6,4" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The result is clearly a 3D bell shaped curve. We can see that the gaussian is centered around (2,7), and that the probability quickly drops away in all directions.\n", + "\n", + "As beautiful as this is, it is perhaps a bit hard to get useful information. For example, it is not easy to tell if x and y both have the same variance or not. So for most of the rest of this book we will display multidimensional gaussian using contour plots. I will use some helper functions in gaussian.py to plot them. If you are interested in linear algebra go ahead and look at the code used to produce these contours, otherwise feel free to ignore it." + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import gaussian as g\n", + "pylab.rcParams['figure.figsize'] = 12,4\n", + "\n", + "cov = array([[2,0],[0,2]])\n", + "e = g.sigma_ellipse (cov, 2, 7)\n", + "subplot(131)\n", + "g.plot_sigma_ellipse(e, '|2 0|\\n|0 2|')\n", + "\n", + "\n", + "cov = array([[2,0],[0,9]])\n", + "e = g.sigma_ellipse (cov, 2, 7)\n", + "subplot(132)\n", + "g.plot_sigma_ellipse(e, '|2 0|\\n|0 9|')\n", + "\n", + "subplot(133)\n", + "cov = array([[2,3],[1,2]])\n", + "e = g.sigma_ellipse (cov, 2, 7)\n", + "g.plot_sigma_ellipse(e,'|2 3|\\n|1 2|')\n", + "show()\n", + "pylab.rcParams['figure.figsize'] = 6,4" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "From a mathematical perspective these display the values that the multivariate gaussian takes for a specific sigma (in this case $\\sigma^2=1$. Think of it as taking a horizontal slice through the 3D surface plot we did above. However, thinking about the physical interpretation of these plots clarifies their meaning.\n", + "\n", + "The first plot uses mean and the covariance matrices $\n", + "\\mu =\\begin{bmatrix}2\\\\7\\end{bmatrix}, cov = \\begin{bmatrix}2&0\\\\0&2\\end{bmatrix}$. Let this be our current belief about the position of our dog in a field. In other words, we believe that he is positioned at (2,7) with a variance of $\\sigma^2=2$ for both x and y. The contour plot shows where we believe the dog is located with the '+' in the center of the ellipse. The ellipse shows the boundary for the $1\\sigma^2$ probability - points where the dog is quite likely to be based on our current knowledge. Of course, the dog might be very far from this point, as Gaussians allow the mean to be any value. For example, the dog could be at (3234.76,189989.62), but that has vanishing low probability of being true. Generally speaking displaying the $1\\sigma^2$ to $2\\sigma^2$ contour captures the most likely values for the distribution. An equivelent way of thinking about this is the circle/ellipse shows us the amount of error in our belief. A tiny circle would indicate that we have a very small error, and a very large circle indicates a lot of error in our belief. We will use this throughout the rest of the book to display and evaluate the accuracy of our filters at any point in time. \n", + "\n", + "The second plot uses mean and the covariance matrices $\n", + "\\mu =\\begin{bmatrix}2\\\\7\\end{bmatrix}, cov = \\begin{bmatrix}2&0\\\\0&9\\end{bmatrix}$. This time we use a different variance for x (2) vs y (9). The result is an ellipse. When we look at it we can immediately tell that we have a lot more uncertainty in the y value vs the x value. Our belief that the value is (2,7) is the same in both cases, but errors are different. This sort of thing happens naturally as we track objects in the world - one sensor has a better view of the object, or is closer, than another sensor, and so we end up with different error rates in the different axis.\n", + "\n", + "\n", + "The third plot uses mean and the covariance matrices $\n", + "\\mu =\\begin{bmatrix}2\\\\7\\end{bmatrix}, cov = \\begin{bmatrix}2&3\\\\1&2\\end{bmatrix}$. This is the first contour that has values in the off-diagonal elements of $cov$, and this is the first contour plot with a slanted ellipse. This is not a coincidence. The two facts are telling use the same thing. A slanted ellipse tells us that the x and y values are somehow **correlated**. We denote that in the covariance matrix with values off the diagonal. What does this mean in physical terms? Think of trying to park your car in a parking spot. You can not pull up beside the spot and then move sideways into the space because most cars cannot go purely sideways. $x$ and $y$ are not independent. This is a consequence of the steering system in a car. When your tires are turned the car rotates around its rear axle while moving forward. Or think of a horse attached to a pivoting exercise bar in a corral. The horse can only walk in circles, he cannot vary $x$ and $y$ independently, which means he cannot walk straight forward to to the side. If $x$ changes, $y$ must also change in a defined way. \n", + "\n", + "So when we see this ellipse we know that $x$ and $y$ are correlated, and that the correlation is \"strong\". I will not prove it here, but a 45 $^{\\circ}$ angle denotes complete correlation between $x$ and $y$, whereas $0$ and $90$ denote no correlation at all. Those who are familiar with this math will be objecting quite strongly, as this is actually quite sloppy language that does not adress all of the mathematical issues. They are right, but for now this is a good first approximation to understanding these ellipses from a physical interpretation point of view. The size of the ellipse shows how much error we have in each axis, and the slant shows how strongly correlated the values are.\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "###Kalman Filter Basics\n", + "\n", + "Let's say we are tracking an aircraft and we get the following data for the $x$ coordinate at time $t$=1,2, and 3 seconds. What does your intuition tell you the value of $x$ will be at time $t$=4 seconds?\n" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "scatter ([1,2,3],[1,2,3]);xlim([0,4]);ylim([0,4]);show()" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "It appears that the aircraft is flying in a straight line because we can draw a line between the three points, and we know that aircraft cannot turn on a dime. The most reasonable guess is that $x$=4 at $t$=4. I will depict that below with a green square to depict the predictions." + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "scatter ([1,2,3],[1,2,3]);xlim([0,5]);ylim([0,5])\n", + "plot([0,5],[0,5],'r')\n", + "scatter ([4], [4], c='g', marker='s',s=200)\n", + "show()" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "If this is data from a Kalman filter, then each point has both a mean and variance. Let's try to show that by showing the approximate error for each point. Don't worry about why I am using a covariance matrix to depict the variance at this point, it will become clear in a few paragraphs. The intent at this point is to show that while we have$x$=1,2,3 that there is a lot of error associated with each measurement." + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "cov = array([[0.003,0], [0,12]])\n", + "sigma = sigma=[0.5,1.,1.5,2]\n", + "e1 = g.sigma_ellipses(cov, x=1, y=1, sigma=sigma)\n", + "e2 = g.sigma_ellipses(cov, x=2, y=2, sigma=sigma)\n", + "e3 = g.sigma_ellipses(cov, x=3, y=3, sigma=sigma)\n", + "g.plot_sigma_ellipses([e1, e2, e3], axis_equal=True,x_lim=[0,4],y_lim=[0,15])\n", + "plt.ylim([0,11])\n", + "show()" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can see that there is a lot of error associated with each value of $x$. We could write a 1D Kalman filter as we did in the last chapter, but suppose this is the output of that filter, and not just raw sensor measurements. Are we out of luck?\n", + "\n", + "Let us think about how we predicted that $x$=4 at $t$=4. In one sense we just drew a straight line between the points and saw where it lay at $t$=4. My constant refrain: what is the physical interpretation of that? What is the difference in $x$ over time? What is $\\frac{\\partial x}{\\partial t}$? The derivative, or difference in distance over time is *velocity*. \n", + "\n", + "This is the **key point** in Kalman filters, so read carefully! Our sensor is only detecting the position of the aircraft (how doesn't matter). It does not have any kind of sensor that provides velocity to us. But based on the position estimates we can compute velocity. In Kalman filters we would call the velocity an **unobserved variable**. Unobserved means what it sounds like - there is no sensor that is measuring velocity directly. Since the velocity is based on the position, and the position has error, the velocity will have error as well. What happens if we draw the velocity errors over the positions errors?" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "from matplotlib.patches import Ellipse\n", + "\n", + "cov = array([[1,1],[1,1.1]])\n", + "ev = g.sigma_ellipses(cov, x=2, y=2, sigma=sigma)\n", + "\n", + "isct = Ellipse(xy=(2,2), width=.2, height=1.2, edgecolor='r', fc='None', lw=4)\n", + "plt.figure().gca().add_artist(isct)\n", + "g.plot_sigma_ellipses([e1, e2, e3, ev], axis_equal=True,x_lim=[0,4],y_lim=[0,15])\n", + "plt.ylim([0,11])\n", + "plt.show()" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Think about what this plot means. We have a lot of error in our position estimates. We therefore have a lot of error in our velocity estimates. But look at the intersections between the velocity and the positions. Take the intersection at $t$=2. The intersection between the velocity and the position is where our aircraft is most likely to be, which I have roughly depicted with a red ellipse ('roughly' because I set the size via eyeball, not via math). The size of the error is much smaller than the error of the positions, despite the fact that velocity was derived from position. \n", + "\n", + "What makes this possible? Imagine for a moment that we superimposed the velocity from a *different* airplane over the position graph. Cleary the two are not related, and there is no way that combining the two could possibly yield any additional information. In contrast, the velocity of the this airplane tells us something very important - the direction and speed of travel. So long as the aircraft does not alter its velocity the velocity allows us to predict where the next position is. After a relatively small amount of error in velocity the probability that it is a good match with the position is very small. Think about it - if you suddenly change direction your position is also going to change a lot. If the position measurement is not in the direction of the assumed velocity change it is very unlikely to be true. The two are correlated, so if the velocity changes so must the position, and in a predictable way. " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Kalman Filter Algorithm\n", + "So in general terms we can show how a multidimensional Kalman filter works. In the example above, we compute velocity from the previous position measurements using something called the **measurement function**. Then we predict the next position by using the current estimate and something called the **state transition function**. In our example above, *new_position = old_position + velocity*time*. Next, we take the measurement from the sensor, and compare it to the prediction we just made. In a world with perfect sensors and perfect airplanes the prediction will always match the measured value. In the real world they will always be at least slightly different. We call the difference between the two the **residual**. Finally, we use something called the **Kalman gain** to update our estimate to be somewhere between the measured position and the predicted position. I will not describe how the gain is set, but suppose we had perfect confidence in our measurement - no error is possible. Then, clearly, we would set the gain so that 100% of the position came from the measurement, and 0% from the prediction. At the other extreme, if he have no confidence at all in the sensor (maybe it reported a hardware fault), we would set the gain so that 100% of the position came from the prediction, and 0% from the measurement. In normal cases, we will take a ratio of the two: maybe 53% of the measurement, and 47% of the prediction. The gain is updated on every cycle based on the variance of the variables (in a way yet to be explained). It should be clear that if the variance of the measurement is low, and the variance of the prediction is high we will favor the measurement, and vice versa. \n", + "\n", + "The chart shows a prior estimate of $x=1$ and $\\dot{x}=1$ ($\\dot{x}$ is the shorthand for the derivative of x, which is velocity). Therefore we predict $\\hat{x}=2$. However, the new measurement $x^{'}=1.3$, giving a residual $r=0.7$. Finally, Kalman filter gain $k$ gives us a new estimate of $\\hat{x^{'}}=1.8$.\n", + "\n", + "** CHECK SYMBOLOGY!!!!**" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "from mkf_internal import *\n", + "show_residual_chart()" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "code", + "collapsed": false, + "input": [], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + } + ], + "metadata": {} + } + ] } \ No newline at end of file diff --git a/Untitled0.ipynb b/Untitled0.ipynb index 7c9c161..c1b5900 100644 --- a/Untitled0.ipynb +++ b/Untitled0.ipynb @@ -1,190 +1,174 @@ -{ - "metadata": { - "name": "", - "signature": "sha256:72a139de073c5d503980bcd0a85ebae9c6b611ce8aba196ba5cf862647b7cad0" - }, - "nbformat": 3, - "nbformat_minor": 0, - "worksheets": [ - { - "cells": [ - { - "cell_type": "code", - "collapsed": false, - "input": [ - "def to_array(x):\n", - " try:\n", - " x.shape\n", - " try:\n", - " if type(x) != numpy.ndarray:\n", - " x=asarray(x)[0]\n", - " return x\n", - " except:\n", - " pass\n", - "\n", - " except:\n", - " return array(mat(x)).reshape(1)\n", - "\n", - "def to_cov(x,n):\n", - " try:\n", - " x.shape\n", - " return x\n", - " except:\n", - " return eye(n) * x\n", - " \n", - "def multivariate_gaussian (x, mu, cov):\n", - " \"\"\" This is designed to work the same as scipy.stats.multivariate_normal\n", - " which is available before version 0.14. You may either pass in a \n", - " multivariate set of data:\n", - " multivariate_gaussian (array([1,1]), array([3,4]), eye(2)*1.4)\n", - " or unidimensional data:\n", - " multivariate_gaussian(1, 3, 1.4)\n", - " \n", - " In the multivariate case if cov is a scalar it is interpreted as eye(n)*cov\n", - " \"\"\"\n", - " \n", - " # force all to numpy.array type\n", - " x = to_array(x)\n", - " mu = to_array(mu)\n", - " n = mu.size\n", - " cov = to_cov (cov, n)\n", - "\n", - " det = numpy.sqrt(numpy.prod(numpy.diag(cov)))\n", - " frac = (2*numpy.pi)**(-n/2.0) * (1.0/det)\n", - " fprime = x - mu\n", - " fprime **= 2\n", - " m = frac * numpy.exp(-0.5*numpy.dot(fprime, 1/numpy.diag(cov)))\n", - " return m\n", - "\n" - ], - "language": "python", - "metadata": {}, - "outputs": [] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "print mvg (array([1,1]), array([1,1]), eye(2))\n", - "print mvg (mat([1,1]), mat([1,1]), eye(2))\n", - "print mvg (2,3,1)" - ], - "language": "python", - "metadata": {}, - "outputs": [] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "print numpy.array(3)\n", - "print pylab.rcParams['figure.figsize']" - ], - "language": "python", - "metadata": {}, - "outputs": [] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "class KalmanFilter1D(object):\n", - "\n", - " def __init__ (self, x0, var):\n", - " self.mean = x0\n", - " self.variance = var\n", - "\n", - " def estimate(self, z, z_variance):\n", - " self.mean = \\\n", - " (self.variance*z + z_variance*self.mean) / \\\n", - " self.variance + z_variance\n", - " self.variance = 1. / (1./self.variance+ 1./z_variance)\n", - "\n", - "f = KalmanFilter1D(1,2)\n", - "f.estimate(z = 2,\n", - " z_variance = 3)\n", - "\n", - "print f.mean, f.variance" - ], - "language": "python", - "metadata": {}, - "outputs": [] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "%%file test.py\n", - "print 'hi'" - ], - "language": "python", - "metadata": {}, - "outputs": [] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "%run test.py\n" - ], - "language": "python", - "metadata": {}, - "outputs": [] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "import math\n", - "from IPython.html.widgets import interact, interactive, fixed\n", - "#from IPython.html import widgets\n", - "#from IPython.display import clear_output, display, HTML\n", - "\n", - "def gaussian (x, mu, sigma):\n", - " ''' compute the gaussian with the specified mean(mu) and sigma'''\n", - " return math.exp (-0.5 * (x-mu)**2 / sigma) / math.sqrt(2.*math.pi*sigma)\n", - "\n", - "def plt_g (mu,gamma):\n", - " xs = arange(0,10,0.15)\n", - " ys = [gaussian (x, mu,gamma) for x in xs]\n", - " plot (xs, ys)\n", - " show()\n", - "\n", - " \n", - "interact (plt_g, mu=(0,10), gamma=(0.01,6))" - ], - "language": "python", - "metadata": {}, - "outputs": [ - { - "metadata": {}, - "output_type": "display_data", - "png": 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- "text": [ - "" - ] - }, - { - "metadata": {}, - "output_type": "pyout", - "prompt_number": 2, - "text": [ - "" - ] - } - ], - "prompt_number": 2 - }, - { - "cell_type": "code", - "collapsed": false, - "input": [], - "language": "python", - "metadata": {}, - "outputs": [] - } - ], - "metadata": {} - } - ] +{ + "metadata": { + "name": "", + "signature": "sha256:590f0bf7162a3bfaae7a3f473dc035e859b9b29d9c8a7e0a72698e7589560fdc" + }, + "nbformat": 3, + "nbformat_minor": 0, + "worksheets": [ + { + "cells": [ + { + "cell_type": "code", + "collapsed": false, + "input": [ + "def to_array(x):\n", + " try:\n", + " x.shape\n", + " try:\n", + " if type(x) != numpy.ndarray:\n", + " x=asarray(x)[0]\n", + " return x\n", + " except:\n", + " pass\n", + "\n", + " except:\n", + " return array(mat(x)).reshape(1)\n", + "\n", + "def to_cov(x,n):\n", + " try:\n", + " x.shape\n", + " return x\n", + " except:\n", + " return eye(n) * x\n", + " \n", + "def multivariate_gaussian (x, mu, cov):\n", + " \"\"\" This is designed to work the same as scipy.stats.multivariate_normal\n", + " which is available before version 0.14. You may either pass in a \n", + " multivariate set of data:\n", + " multivariate_gaussian (array([1,1]), array([3,4]), eye(2)*1.4)\n", + " or unidimensional data:\n", + " multivariate_gaussian(1, 3, 1.4)\n", + " \n", + " In the multivariate case if cov is a scalar it is interpreted as eye(n)*cov\n", + " \"\"\"\n", + " \n", + " # force all to numpy.array type\n", + " x = to_array(x)\n", + " mu = to_array(mu)\n", + " n = mu.size\n", + " cov = to_cov (cov, n)\n", + "\n", + " det = numpy.sqrt(numpy.prod(numpy.diag(cov)))\n", + " frac = (2*numpy.pi)**(-n/2.0) * (1.0/det)\n", + " fprime = x - mu\n", + " fprime **= 2\n", + " m = frac * numpy.exp(-0.5*numpy.dot(fprime, 1/numpy.diag(cov)))\n", + " return m\n", + "\n" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "print mvg (array([1,1]), array([1,1]), eye(2))\n", + "print mvg (mat([1,1]), mat([1,1]), eye(2))\n", + "print mvg (2,3,1)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": "" + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "print numpy.array(3)\n", + "print pylab.rcParams['figure.figsize']" + ], + "language": "python", + "metadata": {}, + "outputs": [] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "class KalmanFilter1D(object):\n", + "\n", + " def __init__ (self, x0, var):\n", + " self.mean = x0\n", + " self.variance = var\n", + "\n", + " def estimate(self, z, z_variance):\n", + " self.mean = \\\n", + " (self.variance*z + z_variance*self.mean) / \\\n", + " self.variance + z_variance\n", + " self.variance = 1. / (1./self.variance+ 1./z_variance)\n", + "\n", + "f = KalmanFilter1D(1,2)\n", + "f.estimate(z = 2,\n", + " z_variance = 3)\n", + "\n", + "print f.mean, f.variance" + ], + "language": "python", + "metadata": {}, + "outputs": [] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "%%file test.py\n", + "print 'hi'" + ], + "language": "python", + "metadata": {}, + "outputs": [] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "%run test.py\n" + ], + "language": "python", + "metadata": {}, + "outputs": [] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "from IPython.html.widgets import interact, interactive, fixed\n", + "#from IPython.html import widgets\n", + "#from IPython.display import clear_output, display, HTML\n", + "\n", + "def gaussian (x, mu, sigma):\n", + " ''' compute the gaussian with the specified mean(mu) and sigma'''\n", + " return math.exp (-0.5 * (x-mu)**2 / sigma) / math.sqrt(2.*math.pi*sigma)\n", + "\n", + "def plt_g (mu,gamma):\n", + " xs = arange(0,10,0.15)\n", + " ys = [gaussian (x, mu,gamma) for x in xs]\n", + " plot (xs, ys)\n", + " show()\n", + "\n", + " \n", + "interact (plt_g, mu=(0,10), gamma=(0.01,6))" + ], + "language": "python", + "metadata": {}, + "outputs": [] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [], + "language": "python", + "metadata": {}, + "outputs": [] + } + ], + "metadata": {} + } + ] } \ No newline at end of file diff --git a/histogram_filter.ipynb b/histogram_filter.ipynb index 7c32d01..36c0bf4 100644 --- a/histogram_filter.ipynb +++ b/histogram_filter.ipynb @@ -1,7 +1,7 @@ { "metadata": { "name": "", - "signature": "sha256:8d7cf224140d3802499c78af56da276f1e1e580fa08934a23e11b1ea2301209a" + "signature": "sha256:1a6b28b567f1c5631b9b36052e0f65cf80787c42b18cc448bbc07857f94a982a" }, "nbformat": 3, "nbformat_minor": 0, @@ -17,22 +17,6 @@ "The Kalman filter belongs to a family of filters called *bayesian filters*. Without going into" ] }, - { - "cell_type": "code", - "collapsed": false, - "input": [], - "language": "python", - "metadata": {}, - "outputs": [] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [], - "language": "python", - "metadata": {}, - "outputs": [] - }, { "cell_type": "markdown", "metadata": {}, @@ -59,7 +43,7 @@ "language": "python", "metadata": {}, "outputs": [], - "prompt_number": 1 + "prompt_number": "" }, { "cell_type": "markdown", @@ -77,7 +61,7 @@ "language": "python", "metadata": {}, "outputs": [], - "prompt_number": 2 + "prompt_number": "" }, { "cell_type": "markdown", @@ -105,17 +89,8 @@ ], "language": "python", "metadata": {}, - "outputs": [ - { - "metadata": {}, - "output_type": "display_data", - "png": 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