Wrote all but the summary of the Gaussian chapter.
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Roger Labbe committed
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@@ -23,24 +23,16 @@
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"\n",
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"Finally, the histogram does not represent what happens in the pysical world very well. For example, in the last chapter we had this as a probability distribution: [ 0.2245871 0.06288015 0.06109133 0.0581008 0.09334062 0.2245871\n",
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" 0.06288015 0.06109133 0.0581008 0.09334062]. The largest probabilities are in position 0 and position 5. This does not fit our physical intuition at all. A dog cannot be in two places at once.\n",
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"\n",
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"Consider using a laser rangefinder. It works by shooting a laser beam to a target, which bounces the laser beam back to a sensor in the rangefinder. The device times the round trip time of the beam, and from that calculates the distance. However, if the beam is not in a vacuum the light beam will be affected by the atmosphere. Recall how images waver over hot tarmac. They waver because the light is being bent by heat currents in the air. The same thing happens to a laser beam, though obviously in most cases the effect is much smaller. So suppose that the rangefinder is exactly 100m from an object. The beam might travel 100.01m due to atmospherics. Less often, it might travel 100.1m. It would rarely travel 101m, and probably never travel 200m. Without doing the math we can see that the laser will almost aways have some error, but most of the time the error will be very close to the actual value. Another way to say the same thing is that larger errors happen less frequently than smaller errors. This is not happenstance, but a consequence of how laser rangefinders work physically. \n",
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" \n",
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" \n",
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"Consider how a bimetallic thermometer works. These thermometers use a strip of bimetallic material bent into a loose coil shape, with a pointer attached to the spring. The two metals expand at different rates, so the strip will either coil tighter or uncoil as the temperature changes, and the pointer indicates the current temperature. It is in some ways a crude system, yet it works well. What kinds of errors might we expect from this system? It will not respond to rapid temperature changes, so it will always lag the current temperature a small amount. There may be frictions in the system causing the pointer to not register the correct value. And so on. Finally, when you read the dial you will rarely be exactly perpindicular to the face of the dial. Viewing the pointer against the dial from an angle will result in a small reading error - 34 might look like 35 from the left of the dial, for example. However, in total all of these effects will generally be low, and we would expect the total errors to always be small. We would probably agree that an error of 2 degrees is reasonable, but an error of 100 degrees would occur extremely rarely, if ever. Furthermore, we would expect the errors to cluster around the correct value. So in most systems we would expect the errors to equally likely to be larger or smaller than the correct value. If the correct temperature was 35 degrees, and we were told the thermometer was accurate within 2 degrees, we wouldn't be suprised to read either 33 degrees or 37 degrees. \n",
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"\n",
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"So we desire a unimodal, continuous way to represent probabilities that models how the real world works, and that is very computationally efficient to calculate. As you might guess from the chapter name, gaussian distributions provide all of these features.\n",
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"\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"So let us explore how gaussians work. A gaussian is a probability distribution that is completely described with two parameters, the mean and the variance. It is defined as:\n",
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"$$ \n",
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"f(x, \\mu, \\sigma) = \\frac{1}{\\sigma\\sqrt{2\\pi}} e^{-0.5*{(x-\\mu)^2}/\\sigma^2 }\n",
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"$$\n",
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"\n",
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"where $\\mu$ is the mean and $\\sigma^2$ is the variance (we will define these in a moment).\n",
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"Let us plot that with Python. First, we will define a function that computes the gaussian for any x. "
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"\n",
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"#### Probability Distributions\n",
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"\n",
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"Before we go into the math, lets look at a graph of the gaussian distribution. Don't bother reading the code yet; it is not important at this stage."
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]
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},
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{
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@@ -50,30 +42,75 @@
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"import math\n",
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"\n",
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"def gaussian (x, mu, sigma):\n",
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" ''' compute the gaussian with the specified mean and sigma'''\n",
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" return math.exp (-0.5 * (x-mu)**2 / sigma) / math.sqrt(2.*math.pi*sigma)"
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" ''' compute the gaussian with the specified mean(mu) and sigma'''\n",
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" return math.exp (-0.5 * (x-mu)**2 / sigma) / math.sqrt(2.*math.pi*sigma)\n"
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],
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"language": "python",
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"metadata": {},
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"outputs": [],
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"prompt_number": 1
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"prompt_number": 7
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"xs = arange(0,10,0.1)\n",
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"ys = [gaussian (x, 5, 3) for x in xs]\n",
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"plot (xs, ys)\n",
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"show()"
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],
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"language": "python",
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"metadata": {},
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"outputs": [
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{
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"metadata": {},
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"output_type": "display_data",
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"png": 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truncated
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"text": [
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"<matplotlib.figure.Figure at 0x3d52450>"
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]
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}
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],
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"prompt_number": 16
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Now we will plot a gaussian centered around 5, with a variance of 1."
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"Probably this is immediately recognizable to you as a 'bell curve'. This curve is ubiquitious because under real world conditions most observations are distributed in such a manner. We will not prove the math here, but the **central limit theorem** proves that under certain conditions the arithmetic mean of independent observations will be distributed in this manner, even if the observations themselves do not have this distribution. In nonmathematical terms, this means that if you take a bunch of measurements from a sensor and use them in a filter, they are very likely to create this distribution.\n",
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"\n",
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"Before we go further, a moment for terminology. This is variously called a normal distribution, a Gaussian distribution, or a bell curve. However, other distributions also have a bell shaped curve, so that name is somewhat ambiguous, and we will not use it again."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"#### Gaussian Distributions\n",
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"\n",
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"So let us explore how gaussians work. A gaussian is a continuous probability distribution that is completely described with two parameters, the mean ($\\mu$) and the variance ($\\sigma^2$). It is defined as:\n",
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"$$ \n",
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"f(x, \\mu, \\sigma) = \\frac{1}{\\sigma\\sqrt{2\\pi}} e^{-0.5*{(x-\\mu)^2}/\\sigma^2 }\n",
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"$$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"You will not need to understand how this equation comes about, or remember it for this book, but it is useful to look at it to see how it works. Specifically, notice the term for $e$. When $x==\\mu$, the term reduces to $e^0=1$. Any other value of x will result in a smaller value for the exponent term due to the negative sign, so the curve will always be highest at $x==\\mu$.\n",
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"\n",
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"Now we will plot a gaussian centered around 23 ($\\mu=23$), with a variance of 1 ($\\sigma^2=1$)."
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]
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"import matplotlib.pyplot as plt\n",
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"import numpy as np\n",
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"xs = np.arange(0,10,0.1)\n",
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"plt.plot (xs,[gaussian(x, 5, 1) for x in xs])\n",
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"plt.axvline (5) \n",
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"xs = arange(16,30,0.1)\n",
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"ys = [gaussian (x,23,1) for x in xs]\n",
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"plt.plot (xs,ys, 'r')\n",
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"plt.axvline(23); plt.axvline(24) \n",
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"plt.show()"
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],
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"language": "python",
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@@ -82,23 +119,23 @@
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{
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"metadata": {},
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"output_type": "display_data",
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"png": 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||||
"text": [
|
||||
"<matplotlib.figure.Figure at 0x3aaa810>"
|
||||
"<matplotlib.figure.Figure at 0x41ff2d0>"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 2
|
||||
"prompt_number": 34
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"As we expected the curve is centered around 5. We can see why this is by looking at the equation for the gaussian. When x=5, x-\\mu is 0, and thus e^0 is 1. any other value for x will result in a smaller value for exp function.\n",
|
||||
"As we expected the curve is centered around 23. The width of the curve is defined by the variance. If the variance is large than the curve will be wide, and if the variance is small the curve will be narrow.\n",
|
||||
"\n",
|
||||
"The width of the curve is defined by the variance. If the variance is large than the curve will be wide, and if the variance is small the curve will be narrow.\n",
|
||||
"So what does this curve *mean*? Assume for a moment that we have a themometer that reads 23$\\,^{\\circ}$. We have asserted (without proof) that the distribution will be *normal*. If that is true, then this chart can be interpreted as a continuous curve depicting our belief that the temperature is any given temperature. In this curve, we assign a probability of the temperature being exactly 23$\\,^{\\circ}$ is 40%. Looking to the right, we assign the probability that the temperature is 24$\\,^{\\circ}$ is about 25%. We find 20$\\,^{\\circ}$ and 26$\\,^{\\circ}$ quite unlikely, and temperatures beyond that range extremely rare. \n",
|
||||
"\n",
|
||||
"Also, since this is a probability distribution it is required that the area under the curve always equals one. This should be intuitively clear - the area under the curve represents all possible occurances, which must sum to one.\n",
|
||||
"Since this is a probability distribution it is required that the area under the curve always equals one. This should be intuitively clear - the area under the curve represents all possible occurances, which must sum to one.\n",
|
||||
"\n",
|
||||
"This leads to an important insight. If the variance is small the curve will be narrow. To keep the area == 1, the curve must also be tall. On the other hand if the variance is large the curve will be wide, and thus it will also have to be short to make the area == 1.\n",
|
||||
"\n",
|
||||
@@ -109,9 +146,9 @@
|
||||
"cell_type": "code",
|
||||
"collapsed": false,
|
||||
"input": [
|
||||
"plt.plot (xs,[gaussian(x, 5, .2) for x in xs],'b')\n",
|
||||
"plt.plot (xs,[gaussian(x, 5, 1) for x in xs],'g')\n",
|
||||
"plt.plot (xs,[gaussian(x, 5, 5) for x in xs],'r')\n",
|
||||
"plt.plot (xs,[gaussian(x, 23, .2) for x in xs],'b')\n",
|
||||
"plt.plot (xs,[gaussian(x, 23, 1) for x in xs],'g')\n",
|
||||
"plt.plot (xs,[gaussian(x, 23, 5) for x in xs],'r')\n",
|
||||
"plt.show()"
|
||||
],
|
||||
"language": "python",
|
||||
@@ -120,28 +157,38 @@
|
||||
{
|
||||
"metadata": {},
|
||||
"output_type": "display_data",
|
||||
"png": 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truncated
|
||||
"png": 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truncated
|
||||
"text": [
|
||||
"<matplotlib.figure.Figure at 0x3ad3710>"
|
||||
"<matplotlib.figure.Figure at 0x4478790>"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 5
|
||||
"prompt_number": 35
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"So what is this telling us? The blue gaussian is very narrow. It is saying that we believe x=5, and that we are very sure about that. In contrast, the red gaussian also believes that x=5, but we are much less sure about that. Our believe that x=5 is lower, and our belief about the likely possible values for x is spread out - we think it is quite likely that x=2 or x=8, for example. The blue gaussian has almost completely eliminated 2 or 8 as possible value - their probably is almost 0.0.\n"
|
||||
"So what is this telling us? The blue gaussian is very narrow. It is saying that we believe x=23, and that we are very sure about that. In contrast, the red gaussian also believes that x=23, but we are much less sure about that. Our believe that x=23 is lower, and so our belief about the likely possible values for x is spread out - we think it is quite likely that x=2 or x=8, for example. The blue gaussian has almost completely eliminated 22 or 24 as possible value - their probably is almost 0.0, whereas the red curve considers them nearly as likely as 23.\n",
|
||||
"\n",
|
||||
"If we think back to the thermometer, we can consider these three curves as representing 3 thermometers. The blue curve represents a very accurate thermometer, and the red one represents a fairly inaccurate one. Green of course represents one in between the two others. Note the very powerful property the Gaussian distribution affords us - we can entirely represent both the reading and the error of a thermometer with only two numbers - the mean and the variance.\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"collapsed": false,
|
||||
"input": [],
|
||||
"language": "python",
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"outputs": []
|
||||
"source": [
|
||||
"#### Computational Properties of the Gaussian\n",
|
||||
"\n",
|
||||
"Recall how our histogram filter worked. We had a vector (Python array) representing our belief at a certain moment in time. When we performed another measurement using the *sense()* function we had to multiply probabilities together, and when we performed the motion step using the *update()* function we had to shift and add probabilities. I've promised you that the Kalman filter uses essentially the same process, and that it uses Gaussians instead of histograms, so you might reasonable expect that we will be multipling, adding, and shifting Gaussians in the Kalman filter.\n",
|
||||
"\n",
|
||||
"A typical math book would directly launch into a multipage proof of the behavior of Gaussians under these operations, but I don't see the value in that unless you plan to do statistics. I think the math will be much more intuitive and clear if we just start developing a Kalman filter using Gaussians, and I will provide the equations for multiplying and shifting Gaussians at the appropriate time. You will then be able to develop a physical intuition for what these operations do, rather than be forced to digest a lot of fairly abstract math.\n",
|
||||
"\n",
|
||||
"The key point, which I will only assert for now, is that all the operations are very simple, and that they preserve the properties of the Gaussian. This is somewhat remarkable, in that the Gaussian is a nonlinear function, and typically if you multiply a nonlinear equation with itself you end up with a different equation. For example, the shape of $sin(x)sin(x)$ is very different from $sin(x)$. But the result of multiplying two Gaussians is yet another Gaussian. This is a fundamental discovery, and the key reason why Kalman filters are possible\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"#### Summary and Key Points"
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {}
|
||||
|
||||
Reference in new issue
Block a user