daviddoji/Kalman-and-Bayesian-Filters-in-Python
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Bug in dog animation cell.

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We were adding the movement error in the wrong place, causing the
filter output to never vary dispite changes in sensor noise.

Also some PEP-8 cleanup.
This commit is contained in:
Roger Labbe committed 2014-11-28 10:33:10 -08:00
1 parent 22375983c2
commit f9af363792
1 file changed
+231 -209
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05_Kalman_Filters/Kalman_Filters.ipynb
+231 -209
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@@ -1,7 +1,7 @@
{
"metadata": {
"name": "",
"signature": "sha256:c5e57a44310de3ce74648ad1f5a4bf5751547875bc0c61e507e3c7704eebb9d0"
"signature": "sha256:a7ff3c98bbd62de00e4ac578a1ebc6606df5c76dd2d2e757a71b513d92a423c5"
},
"nbformat": 3,
"nbformat_minor": 0,
@@ -257,13 +257,13 @@
],
"metadata": {},
"output_type": "pyout",
"prompt_number": 2,
"prompt_number": 1,
"text": [
"<IPython.core.display.HTML at 0x7f00445d3898>"
"<IPython.core.display.HTML at 0x7f167808a978>"
]
}
],
"prompt_number": 2
"prompt_number": 1
},
{
"cell_type": "heading",
@@ -310,14 +310,14 @@
"\n",
"class DogSensor(object):\n",
" \n",
" def __init__(self, x0=0, velocity=1, noise=0.0):\n",
" def __init__(self, x0=0, velocity=1, variance=0.0):\n",
" \"\"\" x0 - initial position\n",
" velocity - (+=right, -=left)\n",
" noise - scaling factor for noise, 0== no noise\n",
" variance - variance in measurement\n",
" \"\"\"\n",
" self.x = x0\n",
" self.velocity = velocity\n",
" self.noise = math.sqrt(noise)\n",
" self.noise = math.sqrt(variance)\n",
"\n",
" def sense(self):\n",
" self.x = self.x + self.velocity\n",
@@ -326,13 +326,15 @@
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 3
"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."
"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. and a standard deviation of 1.0. *Variance* is defined as the standard deviation squared, so in `__init()__` we take the square root of the variance to get the standard deviation. Therefore the expression `self.x + random.randn() * self.noise` computes a number with the variance that we desire. \n",
"\n",
"Let's look at some example output for that."
]
},
{
@@ -351,14 +353,14 @@
"output_type": "stream",
"stream": "stdout",
"text": [
"-0.4981\t-0.0555\t-1.0808\t 0.3023\t 2.7174\t\n",
" 0.7017\t 0.1831\t 0.3057\t-0.5140\t 0.0210\t\n",
" 0.0418\t-0.1166\t-0.5818\t 1.1765\t 0.9700\t\n",
" 1.1401\t-1.0132\t-0.4607\t 1.7878\t 1.6513\t\n"
"-0.2367\t 0.2690\t 0.2938\t 1.7591\t-0.3278\t\n",
"-0.3011\t-0.2756\t-1.0273\t-0.4946\t 1.4776\t\n",
" 1.7101\t-0.7903\t-1.3902\t 0.1243\t-1.4785\t\n",
" 0.3633\t-2.3127\t-0.5851\t-1.6827\t-1.1897\t\n"
]
}
],
"prompt_number": 4
"prompt_number": 3
},
{
"cell_type": "markdown",
@@ -366,7 +368,7 @@
"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"
"Okay, so lets look at the output of the `DogSensor` class. We will start by setting the variance to 0 to check that the class does what we think it does. Zero variance means there is no noise in the signal, so the results should be a straight line."
]
},
{
@@ -376,7 +378,7 @@
"import matplotlib.pyplot as plt\n",
"import matplotlib.pylab as pylab\n",
"\n",
"dog = DogSensor (noise=0.0)\n",
"dog = DogSensor (variance=0.0)\n",
"xs = []\n",
"for i in range(10):\n",
" x = dog.sense()\n",
@@ -401,11 +403,11 @@
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f002d8fe208>"
"<matplotlib.figure.Figure at 0x7f1669269240>"
]
}
],
"prompt_number": 5
"prompt_number": 4
},
{
"cell_type": "markdown",
@@ -420,8 +422,8 @@
"cell_type": "code",
"collapsed": false,
"input": [
"def test_sensor(noise_scale):\n",
" dog = DogSensor(noise=noise_scale)\n",
"def test_sensor(variance):\n",
" dog = DogSensor(variance=variance)\n",
"\n",
" xs = []\n",
" for i in range(100):\n",
@@ -432,11 +434,11 @@
" plt.xlabel('time')\n",
" plt.ylabel('pos')\n",
" plt.ylim([0,100])\n",
" plt.title('noise = ' + str(noise_scale))\n",
" plt.title('variance = ' + str(variance))\n",
" plt.legend(loc='best')\n",
" plt.show()\n",
" \n",
"test_sensor(4.0)"
"test_sensor(variance=4.0)"
],
"language": "python",
"metadata": {},
@@ -444,13 +446,13 @@
{
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f002d3dca90>"
"<matplotlib.figure.Figure at 0x7f1668dae2e8>"
]
}
],
"prompt_number": 6
"prompt_number": 5
},
{
"cell_type": "markdown",
@@ -470,7 +472,7 @@
"cell_type": "code",
"collapsed": false,
"input": [
"test_sensor(100.0)"
"test_sensor(variance=100.0)"
],
"language": "python",
"metadata": {},
@@ -478,19 +480,19 @@
{
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f002d3e49b0>"
"<matplotlib.figure.Figure at 0x7f1668f914a8>"
]
}
],
"prompt_number": 7
"prompt_number": 6
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"test_sensor(0.5)"
"test_sensor(variance=0.5)"
],
"language": "python",
"metadata": {},
@@ -498,20 +500,13 @@
{
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f002d4c66d8>"
"<matplotlib.figure.Figure at 0x7f1668efb4e0>"
]
}
],
"prompt_number": 8
},
{
"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 $\\mathcal{N}(0,100)$. Recall the notation for a normal distribution is $\\mathcal{N}(\\mu,\\sigma^2)$. 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. `DogSensor.__init__()` takes the square root of the noise setting so that the `noise * randn()` call properly computes the normal distribution."
]
"prompt_number": 7
},
{
"cell_type": "heading",
@@ -533,7 +528,7 @@
"collapsed": false,
"input": [
"import stats\n",
"stats.plot_gaussian(23, 5)"
"stats.plot_gaussian(mean=23, variance=5)"
],
"language": "python",
"metadata": {},
@@ -543,11 +538,11 @@
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f002d4bae48>"
"<matplotlib.figure.Figure at 0x7f1668e3a898>"
]
}
],
"prompt_number": 9
"prompt_number": 8
},
{
"cell_type": "markdown",
@@ -564,7 +559,7 @@
"cell_type": "code",
"collapsed": false,
"input": [
"dog = DogSensor(23, 0, 5)\n",
"dog = DogSensor(x0=23, velocity=0, variance=5)\n",
"xs = range(100)\n",
"ys = []\n",
"for i in xs:\n",
@@ -580,13 +575,13 @@
{
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"png": "iVBORw0KGgoAAAANSUhEUgAAAsYAAAF2CAYAAABtSl5dAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
"text": [
"<matplotlib.figure.Figure at 0x7f002d8fe080>"
"<matplotlib.figure.Figure at 0x7f16692693c8>"
]
}
],
"prompt_number": 10
"prompt_number": 9
},
{
"cell_type": "markdown",
@@ -653,19 +648,25 @@
"import numpy as np\n",
"\n",
"def multiply(mu1, var1, mu2, var2):\n",
" if var1 == 0.0:\n",
" var1=1.e-80\n",
" \n",
" if var2 == 0:\n",
" var2 = 1e-80\n",
" \n",
" mean = (var1*mu2 + var2*mu1) / (var1+var2)\n",
" variance = 1 / (1/var1 + 1/var2)\n",
" return (mean, variance)\n",
"\n",
"xs = np.arange(16, 30, 0.1)\n",
"\n",
"m1,v1 = 23, 5\n",
"m, v = multiply(m1,v1,m1,v1)\n",
"mean1, var1 = 23, 5\n",
"mean, var = multiply(mean1, var1, mean1, var1)\n",
"\n",
"ys = [stats.gaussian(x,m1,v1) for x in xs]\n",
"ys = [stats.gaussian(x, mean1, var1) for x in xs]\n",
"plt.plot (xs, ys, label='original')\n",
"\n",
"ys = [stats.gaussian(x,m,v) for x in xs]\n",
"ys = [stats.gaussian(x, mean, var) for x in xs]\n",
"plt.plot (xs, ys, label='multiply')\n",
"\n",
"plt.legend(loc='best')\n",
@@ -679,11 +680,11 @@
"output_type": "display_data",
"png": "iVBORw0KGgoAAAANSUhEUgAAAtEAAAF2CAYAAACoOMTMAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
"text": [
"<matplotlib.figure.Figure at 0x7f00254d8710>"
"<matplotlib.figure.Figure at 0x7f1668150978>"
]
}
],
"prompt_number": 11
"prompt_number": 10
},
{
"cell_type": "markdown",
@@ -704,18 +705,18 @@
"input": [
"xs = np.arange(16, 30, 0.1)\n",
"\n",
"m1, v1 = 23, 5\n",
"m2, v2 = 25, 5\n",
"m, s = multiply(m1,v1,m2,v2)\n",
"mean1, var1 = 23, 5\n",
"mean2, var2 = 25, 5\n",
"mean, var = multiply(mean1, var1, mean2, var2)\n",
"\n",
"ys = [stats.gaussian(x,m1,v1) for x in xs]\n",
"plt.plot (xs, ys, label='measure 1')\n",
"ys = [stats.gaussian(x, mean1, var1) for x in xs]\n",
"plt.plot(xs, ys, label='measure 1')\n",
"\n",
"ys = [stats.gaussian(x,m2,v2) for x in xs]\n",
"plt.plot (xs, ys, label='measure 2')\n",
"ys = [stats.gaussian(x, mean2, var2) for x in xs]\n",
"plt.plot(xs, ys, label='measure 2')\n",
"\n",
"ys = [stats.gaussian(x,m,v) for x in xs]\n",
"plt.plot(xs, ys,label='multiply')\n",
"ys = [stats.gaussian(x, mean, var) for x in xs]\n",
"plt.plot(xs, ys, label='multiply')\n",
"plt.legend()\n",
"plt.show()"
],
@@ -727,11 +728,11 @@
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f00254313c8>"
"<matplotlib.figure.Figure at 0x7f166802a438>"
]
}
],
"prompt_number": 12
"prompt_number": 11
},
{
"cell_type": "markdown",
@@ -754,17 +755,17 @@
"input": [
"xs = np.arange(0, 60, 0.1)\n",
"\n",
"m1, v1 = 10, 5\n",
"m2, v2 = 50, 5\n",
"m, v = multiply(m1,v1,m2,v2)\n",
"mean1, var1 = 10, 5\n",
"mean2, var2 = 50, 5\n",
"mean, var = multiply(mean1, var1, mean2, var2)\n",
"\n",
"ys = [stats.gaussian(x,m1,v1) for x in xs]\n",
"ys = [stats.gaussian(x, mean1, var1) for x in xs]\n",
"plt.plot (xs, ys, label='measure 1')\n",
"\n",
"ys = [stats.gaussian(x,m2,v2) for x in xs]\n",
"ys = [stats.gaussian(x, mean2, var2) for x in xs]\n",
"plt.plot (xs, ys, label='measure 2')\n",
"\n",
"ys = [stats.gaussian(x,m,v) for x in xs]\n",
"ys = [stats.gaussian(x, mean, var) for x in xs]\n",
"plt.plot(xs, ys, label='multiply')\n",
"plt.legend()\n",
"plt.show()"
@@ -777,11 +778,11 @@
"output_type": "display_data",
"png": "iVBORw0KGgoAAAANSUhEUgAAAtEAAAF2CAYAAACoOMTMAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
"text": [
"<matplotlib.figure.Figure at 0x7f00202cdbe0>"
"<matplotlib.figure.Figure at 0x7f1668019e10>"
]
}
],
"prompt_number": 13
"prompt_number": 12
},
{
"cell_type": "markdown",
@@ -852,27 +853,18 @@
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 14
"prompt_number": 13
},
{
"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:"
"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:\n",
"\n",
" def update_dog(dog_pos, dog_variance, measurement, measurement_variance):\n",
" return multiply(dog_pos, dog_sigma, measurement, measurement_variance)"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"def update_dog(dog_pos, dog_variance, measurement, measurement_variance):\n",
" return multiply(dog_pos, dog_sigma, measurement, measurement_variance)"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 15
},
{
"cell_type": "markdown",
"metadata": {},
@@ -886,7 +878,7 @@
"cell_type": "code",
"collapsed": false,
"input": [
"dog = DogSensor(velocity=0, noise=1)\n",
"dog = DogSensor(velocity=0., variance=1.)\n",
"\n",
"pos,s = 2, 5\n",
"for i in range(20):\n",
@@ -902,30 +894,30 @@
"output_type": "stream",
"stream": "stdout",
"text": [
"time: 0 \tposition = 1.137 \tvariance = 2.500\n",
"time: 1 \tposition = 1.010 \tvariance = 1.667\n",
"time: 2 \tposition = 0.593 \tvariance = 1.250\n",
"time: 3 \tposition = -0.018 \tvariance = 1.000\n",
"time: 4 \tposition = -0.040 \tvariance = 0.833\n",
"time: 5 \tposition = -0.093 \tvariance = 0.714\n",
"time: 6 \tposition = -0.078 \tvariance = 0.625\n",
"time: 7 \tposition = -0.180 \tvariance = 0.556\n",
"time: 8 \tposition = -0.200 \tvariance = 0.500\n",
"time: 9 \tposition = -0.165 \tvariance = 0.455\n",
"time: 10 \tposition = -0.142 \tvariance = 0.417\n",
"time: 11 \tposition = -0.220 \tvariance = 0.385\n",
"time: 12 \tposition = -0.174 \tvariance = 0.357\n",
"time: 13 \tposition = -0.094 \tvariance = 0.333\n",
"time: 14 \tposition = -0.187 \tvariance = 0.312\n",
"time: 15 \tposition = -0.147 \tvariance = 0.294\n",
"time: 16 \tposition = -0.169 \tvariance = 0.278\n",
"time: 17 \tposition = -0.082 \tvariance = 0.263\n",
"time: 18 \tposition = -0.083 \tvariance = 0.250\n",
"time: 19 \tposition = -0.060 \tvariance = 0.238\n"
"time: 0 \tposition = 1.529 \tvariance = 2.500\n",
"time: 1 \tposition = 1.416 \tvariance = 1.667\n",
"time: 2 \tposition = 1.260 \tvariance = 1.250\n",
"time: 3 \tposition = 0.756 \tvariance = 1.000\n",
"time: 4 \tposition = 0.854 \tvariance = 0.833\n",
"time: 5 \tposition = 0.728 \tvariance = 0.714\n",
"time: 6 \tposition = 0.531 \tvariance = 0.625\n",
"time: 7 \tposition = 0.376 \tvariance = 0.556\n",
"time: 8 \tposition = 0.306 \tvariance = 0.500\n",
"time: 9 \tposition = 0.353 \tvariance = 0.455\n",
"time: 10 \tposition = 0.357 \tvariance = 0.417\n",
"time: 11 \tposition = 0.268 \tvariance = 0.385\n",
"time: 12 \tposition = 0.423 \tvariance = 0.357\n",
"time: 13 \tposition = 0.455 \tvariance = 0.333\n",
"time: 14 \tposition = 0.481 \tvariance = 0.312\n",
"time: 15 \tposition = 0.399 \tvariance = 0.294\n",
"time: 16 \tposition = 0.387 \tvariance = 0.278\n",
"time: 17 \tposition = 0.432 \tvariance = 0.263\n",
"time: 18 \tposition = 0.463 \tvariance = 0.250\n",
"time: 19 \tposition = 0.489 \tvariance = 0.238\n"
]
}
],
"prompt_number": 16
"prompt_number": 14
},
{
"cell_type": "markdown",
@@ -992,7 +984,7 @@
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 17
"prompt_number": 15
},
{
"cell_type": "markdown",
@@ -1011,7 +1003,7 @@
"sensor_error = 10\n",
"pos = (0, 500) # gaussian N(0,50)\n",
"\n",
"dog = DogSensor(pos[0], velocity=movement, noise=sensor_error)\n",
"dog = DogSensor(pos[0], velocity=movement, variance=sensor_error)\n",
"\n",
"zs = []\n",
"ps = []\n",
@@ -1041,28 +1033,28 @@
"output_type": "stream",
"stream": "stdout",
"text": [
"PREDICT: 1.0000 502.0000\tUPDATE: 0.8389 9.8047\n",
"PREDICT: 1.8389 11.8047\tUPDATE: 1.2876 5.4138\n",
"PREDICT: 2.2876 7.4138\tUPDATE: 2.3493 4.2574\n",
"PREDICT: 3.3493 6.2574\tUPDATE: 1.9995 3.8490\n",
"PREDICT: 2.9995 5.8490\tUPDATE: 2.7867 3.6904\n",
"PREDICT: 3.7867 5.6904\tUPDATE: 3.2452 3.6267\n",
"PREDICT: 4.2452 5.6267\tUPDATE: 4.6676 3.6007\n",
"PREDICT: 5.6676 5.6007\tUPDATE: 7.8293 3.5900\n",
"PREDICT: 8.8293 5.5900\tUPDATE: 8.5649 3.5856\n",
"PREDICT: 9.5649 5.5856\tUPDATE: 10.8447 3.5838\n"
"PREDICT: 1.0000 502.0000\tUPDATE: -1.2045 9.8047\n",
"PREDICT: -0.2045 11.8047\tUPDATE: -1.4522 5.4138\n",
"PREDICT: -0.4522 7.4138\tUPDATE: 3.2771 4.2574\n",
"PREDICT: 4.2771 6.2574\tUPDATE: 6.7788 3.8490\n",
"PREDICT: 7.7788 5.8490\tUPDATE: 7.2769 3.6904\n",
"PREDICT: 8.2769 5.6904\tUPDATE: 7.7228 3.6267\n",
"PREDICT: 8.7228 5.6267\tUPDATE: 7.7554 3.6007\n",
"PREDICT: 8.7554 5.6007\tUPDATE: 8.6251 3.5900\n",
"PREDICT: 9.6251 5.5900\tUPDATE: 9.8390 3.5856\n",
"PREDICT: 10.8390 5.5856\tUPDATE: 9.2933 3.5838\n"
]
},
{
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f002543f2e8>"
"<matplotlib.figure.Figure at 0x7f16680dd160>"
]
}
],
"prompt_number": 18
"prompt_number": 16
},
{
"cell_type": "markdown",
@@ -1162,12 +1154,9 @@
"sensor_error = 4.5\n",
"pos = (0, 100) # gaussian N(0,50)\n",
"\n",
"dog = DogSensor(pos[0], velocity=movement, noise=sensor_error)\n",
"\n",
"zs = []\n",
"ps = []\n",
"vs = []\n",
"dog = DogSensor(pos[0], velocity=movement, variance=sensor_error)\n",
"\n",
"zs, ps, vs = [], [], []\n",
"for i in range(50):\n",
" pos = predict(pos[0], pos[1], movement, movement_error) \n",
" Z = dog.sense()\n",
@@ -1178,18 +1167,17 @@
" ps.append(pos[0])\n",
" \n",
"#plt.subplot(121) \n",
"p1, = plt.plot(zs,c='r', linestyle='dashed')\n",
"p2, = plt.plot(ps,)\n",
"plt.legend([p1,p2], ['measurement', 'filter'], loc=2)\n",
"plt.plot(zs,c='r', linestyle='dashed', label='measurement')\n",
"plt.plot(ps, label='filter')\n",
"plt.legend(loc=2)\n",
"plt.show()\n",
"\n",
"plt.plot(vs)\n",
"plt.title('Variance')\n",
"plt.show()\n",
"\n",
"\n",
"for i in range(0,len(vs), 5):\n",
" print ('{:.4f} {:.4f} {:.4f} {:.4f} {:.4f}'.format(*[v for v in vs[i:i+5]]))"
" print('{:.4f} {:.4f} {:.4f} {:.4f} {:.4f}'.format(*[v for v in vs[i:i+5]]))"
],
"language": "python",
"metadata": {},
@@ -1197,9 +1185,9 @@
{
"metadata": {},
"output_type": "display_data",
"png": 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"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f0020276d68>"
"<matplotlib.figure.Figure at 0x7f1653bd9a20>"
]
},
{
@@ -1207,7 +1195,7 @@
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f00201e3278>"
"<matplotlib.figure.Figure at 0x7f1668c71128>"
]
},
{
@@ -1227,7 +1215,7 @@
]
}
],
"prompt_number": 26
"prompt_number": 17
},
{
"cell_type": "markdown",
@@ -1273,7 +1261,7 @@
"level": 2,
"metadata": {},
"source": [
"Implementation in a Class (Optional)\n"
"Implementation in a Class"
]
},
{
@@ -1342,7 +1330,7 @@
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 19
"prompt_number": 18
},
{
"cell_type": "heading",
@@ -1356,7 +1344,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"In the first chapter I stated that the Kalman filter is a form of g-h filter. However, we have been reasoning about the probability of Gaussians, and not used any of the reasoning or equations of the first chapter. A trivial amount of algebra will reveal the relationship, so let's do that now. It's not particularly illuminating algebra, so feel free to skip to the bottom to see the final equation that relates *g* to the variances.\n",
"In the first chapter I stated that the Kalman filter is a form of g-h filter. However, we have been reasoning about the probability of Gaussians, and not used any of the reasoning or equations of the first chapter. A trivial amount of algebra will reveal the relationship, so let's do that now. It's not particularly illuminating algebra, so feel free to skip to the bottom to see the final equation that relates *g* and *h* to the variances.\n",
"\n",
"The equation for our estimate is:\n",
"\n",
@@ -1450,20 +1438,20 @@
"source": [
"So far we have developed our filter based on the dog sensors introduced in the Discrete Bayesian filter chapter. We are used to this problem by now, and may feel ill-equipped to implement a Kalman filter for a different problem. To be honest, there is still quite a bit of information missing from this presentation. The next chapter will fill in the gaps. Still, lets get a feel for it by designing and implementing a Kalman filter for a thermometer. The sensor for the thermometer outputs a voltage that corresponds to the temperature that is being measured. We have read the manufacturer's specifications for the sensor, and it tells us that the sensor exhibits white noise with a standard deviation of 2.13.\n",
"\n",
"We do not have a real sensor to read, so we will simulate the sensor with the following function. We have hard-coded the voltage to 16.3 - obviously the voltage will differ based on the temperature, but that is not important to our filter design."
"We do not have a real sensor to read, so we will simulate the sensor with the following function. "
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"def volt(temp_variance):\n",
" return random.randn()*temp_variance + 16.3"
"def volt(voltage, temp_variance):\n",
" return random.randn()*temp_variance + voltage"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 20
"prompt_number": 19
},
{
"cell_type": "markdown",
@@ -1493,18 +1481,19 @@
"cell_type": "code",
"collapsed": false,
"input": [
"temp_variance = 2.13**2\n",
"variance = 2.13**2\n",
"movement_error = .2\n",
"actual_voltage = 16.3\n",
"\n",
"N=50\n",
"zs = [volt(temp_variance) for i in range(N)]\n",
"zs = [volt(actual_voltage, variance) for i in range(N)]\n",
"ps = []\n",
"estimates = []\n",
"\n",
"kf = KalmanFilter1D(x0=25, # initial state\n",
" P = 1000, # initial variance \n",
" # large says 'who knows?'\n",
" R=temp_variance, # sensor noise\n",
" R=variance, # sensor noise\n",
" Q=movement_error) # movement noise\n",
"\n",
"for i in range(N):\n",
@@ -1532,9 +1521,9 @@
{
"metadata": {},
"output_type": "display_data",
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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f5ff2721198>"
"<matplotlib.figure.Figure at 0x7f1668055668>"
]
},
{
@@ -1542,7 +1531,7 @@
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f5ffc053cc0>"
"<matplotlib.figure.Figure at 0x7f165c042e80>"
]
},
{
@@ -1553,7 +1542,7 @@
]
}
],
"prompt_number": 21
"prompt_number": 20
},
{
"cell_type": "markdown",
@@ -1622,7 +1611,7 @@
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 22
"prompt_number": 21
},
{
"cell_type": "markdown",
@@ -1635,13 +1624,15 @@
"cell_type": "code",
"collapsed": false,
"input": [
"sensor_error = temp_variance\n",
"variance = 2.13**2\n",
"actual_voltage = 16.3\n",
"\n",
"\n",
"def VKF():\n",
" voltage=(14,1000)\n",
" for i in range(N):\n",
" Z = volt(temp_variance)\n",
" voltage = update(voltage[0], voltage[1], Z, sensor_error)\n",
" Z = volt(actual_voltage, variance)\n",
" voltage = update(voltage[0], voltage[1], Z, variance)\n",
" return voltage[0]\n",
"\n",
"vs = []\n",
@@ -1656,13 +1647,13 @@
{
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"png": "iVBORw0KGgoAAAANSUhEUgAAAsoAAAF2CAYAAAB3dr7TAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
"text": [
"<matplotlib.figure.Figure at 0x7f5ff26b3a20>"
"<matplotlib.figure.Figure at 0x7f1653b8c4a8>"
]
}
],
"prompt_number": 23
"prompt_number": 22
},
{
"cell_type": "markdown",
@@ -1697,7 +1688,7 @@
"movement_sensor = 30\n",
"pos = (0,500)\n",
"\n",
"dog = DogSensor(0, velocity=movement, noise=sensor_error)\n",
"dog = DogSensor(0, velocity=movement, variance=sensor_error)\n",
"\n",
"zs = []\n",
"ps = []\n",
@@ -1727,9 +1718,9 @@
{
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f5ff2759c18>"
"<matplotlib.figure.Figure at 0x7f1668f3e160>"
]
},
{
@@ -1737,11 +1728,11 @@
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f5ff25b72b0>"
"<matplotlib.figure.Figure at 0x7f166815b3c8>"
]
}
],
"prompt_number": 24
"prompt_number": 23
},
{
"cell_type": "markdown",
@@ -1787,7 +1778,7 @@
"movement_error = 2\n",
"pos = (0,500)\n",
"\n",
"dog = DogSensor(pos[0], velocity=movement, noise=sensor_error)\n",
"dog = DogSensor(pos[0], velocity=movement, variance=sensor_error)\n",
"\n",
"zs = []\n",
"ps = []\n",
@@ -1813,13 +1804,13 @@
{
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f5ffe138780>"
"<matplotlib.figure.Figure at 0x7f166802d748>"
]
}
],
"prompt_number": 25
"prompt_number": 24
},
{
"cell_type": "markdown",
@@ -1852,7 +1843,7 @@
"movement_error = 2\n",
"pos = (1000,500)\n",
"\n",
"dog = DogSensor(0, velocity=movement, noise=sensor_error)\n",
"dog = DogSensor(0, velocity=movement, variance=sensor_error)\n",
"\n",
"zs = []\n",
"ps = []\n",
@@ -1877,13 +1868,13 @@
{
"metadata": {},
"output_type": "display_data",
"png": 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"text": [
"<matplotlib.figure.Figure at 0x7f5ff60bacc0>"
"<matplotlib.figure.Figure at 0x7f1653a7f978>"
]
}
],
"prompt_number": 26
"prompt_number": 25
},
{
"cell_type": "markdown",
@@ -1916,7 +1907,7 @@
"pos = (1000,500)\n",
"\n",
"\n",
"dog = DogSensor(0, velocity=movement, noise=sensor_error) \n",
"dog = DogSensor(0, velocity=movement, variance=sensor_error) \n",
"zs = []\n",
"ps = []\n",
"\n",
@@ -1940,13 +1931,13 @@
{
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f5ff24ffbe0>"
"<matplotlib.figure.Figure at 0x7f16539c9128>"
]
}
],
"prompt_number": 27
"prompt_number": 26
},
{
"cell_type": "markdown",
@@ -1965,7 +1956,7 @@
"movement_error = 2\n",
"pos = None\n",
"\n",
"dog = DogSensor(0, velocity=movement, noise=sensor_error)\n",
"dog = DogSensor(0, velocity=movement, variance=sensor_error)\n",
"\n",
"zs = []\n",
"ps = []\n",
@@ -1992,13 +1983,13 @@
{
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f5ff2569438>"
"<matplotlib.figure.Figure at 0x7f1653abbd30>"
]
}
],
"prompt_number": 28
"prompt_number": 27
},
{
"cell_type": "markdown",
@@ -2035,8 +2026,8 @@
"def plot_kalman_filter(start_pos, \n",
" sensor_noise, \n",
" movement, \n",
" movement_noise, n\n",
" oise_scale):\n",
" movement_noise,\n",
" noise_scale):\n",
" # your code goes here\n",
" pass\n",
"\n",
@@ -2051,65 +2042,76 @@
"metadata": {},
"outputs": [
{
"ename": "SyntaxError",
"evalue": "invalid syntax (<ipython-input-29-83e872123bef>, line 9)",
"output_type": "pyerr",
"traceback": [
"\u001b[1;36m File \u001b[1;32m\"<ipython-input-29-83e872123bef>\"\u001b[1;36m, line \u001b[1;32m9\u001b[0m\n\u001b[1;33m oise_scale):\u001b[0m\n\u001b[1;37m ^\u001b[0m\n\u001b[1;31mSyntaxError\u001b[0m\u001b[1;31m:\u001b[0m invalid syntax\n"
"metadata": {},
"output_type": "pyout",
"prompt_number": 28,
"text": [
"<function __main__.plot_kalman_filter>"
]
}
],
"prompt_number": 29
"prompt_number": 28
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"######Solution\n",
"One possible solution follows."
"One possible solution follows. We have sliders for the start position, the amount of noise in the sensor, the amount we move in each time step, and how much movement error there is. Movement error is perhaps the least clear - it models how much the dog wanders off course at each time step, so we add that into the dog's position at each step. I set the random number generator seed so that each redraw uses the same random numbers, allowing us to compare the graphs as we move the sliders."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"zs = np.zeros(100)\n",
"ps = np.zeros(100)\n",
"\n",
"def plot_kalman_filter(start_pos, \n",
" sensor_noise, \n",
" movement,\n",
" movement_noise,\n",
" noise_scale):\n",
" dog = DogSensor(start_pos, velocity=movement, noise=sensor_noise)\n",
" movement_noise):\n",
" n = 20\n",
" zs = []\n",
" ps = []\n",
" \n",
" dog = DogSensor(start_pos, velocity=movement, variance=sensor_noise)\n",
" random.seed(303)\n",
" pos = (0,100)\n",
" pos = (0., 1000.) # mean and variance\n",
"\n",
" for i in range(100):\n",
" Z = dog.sense() + random.randn()*noise_scale\n",
" zs[i] = Z\n",
" for _ in range(n): \n",
" move_error = random.randn()*movement_noise\n",
" dog.x += move_error\n",
" \n",
" z = dog.sense()\n",
" zs.append(z)\n",
"\n",
" pos = update(pos[0], pos[1], Z, sensor_error)\n",
" ps[i] = pos[0]\n",
" pos = update(pos[0], pos[1], z, sensor_noise)\n",
" ps.append(pos[0])\n",
"\n",
" noise = random.randn()*movement_noise\n",
" pos = predict(pos[0], pos[1], movement + noise, movement_noise)\n",
" pos = predict(pos[0], pos[1], movement, movement_noise)\n",
"\n",
" plt.plot(zs,c='r', linestyle='dashed', label='measurement')\n",
" plt.plot(ps, c='#004080', label='filter')\n",
" plt.plot(ps, c='#004080', alpha=0.7, label='filter')\n",
" plt.legend(loc='best')\n",
" plt.show()\n",
"\n",
"interact(plot_kalman_filter,\n",
" start_pos=(-10,10), \n",
" sensor_noise=widgets.IntSliderWidget(value=5,min=0,max=100), \n",
" sensor_noise=widgets.FloatSliderWidget(value=5,min=0.,max=100), \n",
" movement=widgets.FloatSliderWidget(value=1,min=-2.,max=2.), \n",
" movement_noise=widgets.FloatSliderWidget(value=2,min=0,max=100.),\n",
" noise_scale=widgets.FloatSliderWidget(value=1,min=0,max=20.))"
" movement_noise=widgets.FloatSliderWidget(value=.1,min=0,max=.5))"
],
"language": "python",
"metadata": {},
"outputs": []
"outputs": [
{
"metadata": {},
"output_type": "display_data",
"png": "iVBORw0KGgoAAAANSUhEUgAAAsoAAAF2CAYAAAB3dr7TAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
"text": [
"<matplotlib.figure.Figure at 0x7f1653b22048>"
]
}
],
"prompt_number": 34
},
{
"cell_type": "heading",
@@ -2181,7 +2183,17 @@
],
"language": "python",
"metadata": {},
"outputs": []
"outputs": [
{
"metadata": {},
"output_type": "display_data",
"png": "iVBORw0KGgoAAAANSUhEUgAAAsYAAAF2CAYAAABtSl5dAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
"text": [
"<matplotlib.figure.Figure at 0x7f1653b3b278>"
]
}
],
"prompt_number": 35
},
{
"cell_type": "markdown",
@@ -2255,7 +2267,17 @@
],
"language": "python",
"metadata": {},
"outputs": []
"outputs": [
{
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f165c03de10>"
]
}
],
"prompt_number": 36
},
{
"cell_type": "markdown",