Copy edit of chapter.
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Roger Labbe committed
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@@ -120,8 +120,9 @@
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" }\n",
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" div.text_cell_render{\n",
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" font-family: 'Fira sans', verdana,arial,sans-serif;\n",
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" line-height: 125%;\n",
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" }\n",
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@@ -267,6 +268,8 @@
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"%autoreload 2 \n",
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"from __future__ import division, print_function\n",
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"import sys\n",
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"import warnings\n",
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"warnings.filterwarnings('ignore')\n",
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"sys.path.insert(0,'./code')\n",
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"from book_format import load_style\n",
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"load_style()"
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@@ -326,23 +329,23 @@
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"source": [
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" In other words, covariance is a measure for how much variables change together. A couple of examples. Generally speaking as the square footage of a house increases the price increases. These variables are *correlated*, which is also sometimes called *positively correlated*. As the temperature of an automobile's engine increases its life expectancy lowers. These are *inversely correlated* because as one variable gets larger the other variable lowers. This is also called *negatively correlated*. The price of tea and the number of tail wags my dog makes have no relation to each other, and we say they are not correlated - each can change independent of the other.\n",
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"\n",
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"Correlation implies *prediction*. If our houses are in the same neighborhood, and yours has twice the square footage of my house I can predict that your house has a higher market price than mine. This is not guaranteed as there are other factors such as proximity to garbage dumps which also affect the price. If my car engine significantly overheats I start planning on replacing it soon. If my dog wags his tail more I don't conclude that tea prices will be increasing.\n",
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"Correlation implies *prediction*. If our houses are in the same neighborhood, and yours has twice the square footage of my house I can predict that your house has a higher market price than mine. This is not guaranteed as there are other factors which also affect the price. If my car engine significantly overheats I start planning on replacing it soon. If my dog wags his tail more I don't conclude that tea prices will be increasing.\n",
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"\n",
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"For example, here is a plot of of floor plan size vs price for a neighborhood. If a new house comes on the market with a floor plan size of 225 $m^2$ we can estimate its price to be somewhere around $425,000. The correspondence between floor plan area and price is not perfect, but clearly there is enough correlation between them for us to make a reasonable prediction."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 45,
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"execution_count": 2,
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"metadata": {
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"collapsed": false
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},
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"outputs": [
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{
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"data": {
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truncated
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"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8e0b7d62b0>"
|
||||
"<matplotlib.figure.Figure at 0x71fb908>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -375,15 +378,15 @@
|
||||
"\n",
|
||||
"If you haven't seen this before it is probably a bit confusing. Instead of starting with the mathematical definition I will build your intuition with thought experiments. At this point, note that the diagonal contains the variance for each variable, and that all off-diagonal elements (covariances) represent how much the $i$th (horizontal row) and $j$th (vertical column) variables are linearly correlated to each other. So $\\sigma_3^2$ is the variance of the third variable, and $\\sigma_{13}$ is the covariance between the first and third variables.\n",
|
||||
"\n",
|
||||
"A covariance of 0 indicates no correlation. So, for example, if the variance for x is 10, the variance for y is 4, and there is no linear correlation between x and y, then we would write\n",
|
||||
"A covariance of 0 indicates no correlation. If the variance for $x$ is 10, the variance for $y$ is 4, and there is no linear correlation between $x$ and $y$, then we would write\n",
|
||||
"\n",
|
||||
"$$\\Sigma = \\begin{bmatrix}10&0\\\\0&4\\end{bmatrix}$$\n",
|
||||
"\n",
|
||||
"If there was a small amount of correlation between x and y we might have\n",
|
||||
"If there was a small amount of correlation between $x$ and $y$ we might have\n",
|
||||
"\n",
|
||||
"$$\\Sigma = \\begin{bmatrix}10&1.2\\\\1.2&4\\end{bmatrix}$$\n",
|
||||
"\n",
|
||||
"where 1.2 is the covariance between x and y. Note that this is always symmetric - the covariance between x and y is always equal to the covariance between y and x. That is, $\\sigma_{xy}=\\sigma_{yx}$ for any x and y."
|
||||
"where 1.2 is the covariance between $x$ and $y$. Note that this is always symmetric. Logically the covariance between $x$ and $y$ is always equal to the covariance between $y$ and $x$. That is, $\\sigma_{xy}=\\sigma_{yx}$ for any $x$ and $y$."
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -396,13 +399,14 @@
|
||||
"f(\\mathbf{x},\\, \\mu,\\,\\Sigma) = \\frac{1}{\\sqrt{(2\\pi)^n|\\Sigma|}}\\, \\exp \\Big [{ -\\frac{1}{2}(\\mathbf{x}-\\mu)^\\mathsf{T}\\Sigma^{-1}(\\mathbf{x}-\\mu) \\Big ]}\n",
|
||||
"$$\n",
|
||||
"\n",
|
||||
"I urge you to not try to remember this equation. We will program it in a Python function and then call it if we need to compute a specific value. Plus, the Kalman filter equations compute this for us automatically; we never have to explicitly compute it. However, note that it has the same form as the univariate normal distribution. It uses matrices instead of scalar values, and the root of $\\pi$ is scaled by $n$. If you set n=1 then it turns into the univarate equation. Here is the univariate equation for reference:\n",
|
||||
"I urge you to not try to remember this equation. We will program it in a Python function and then call it if we need to compute a specific value. Plus, the Kalman filter equations compute this for us automatically; we never have to explicitly compute it. However, note that it has the same form as the univariate normal distribution. It uses matrices instead of scalar values, and the root of $\\pi$ is scaled by $n$. If you set \n",
|
||||
"$n=1$ then it turns into the univarate equation. Here is the univariate equation for reference:\n",
|
||||
"\n",
|
||||
"$$ \n",
|
||||
"f(x, \\mu, \\sigma) = \\frac{1}{\\sigma\\sqrt{2\\pi}} \\exp \\Big [{-\\frac{1}{2}}{(x-\\mu)^2}/\\sigma^2 \\Big ]\n",
|
||||
"$$\n",
|
||||
"\n",
|
||||
"The multivariate version merely replaces the scalars of the univariate equations with matrices. If you are reasonably well-versed in linear algebra this equation should look quite manageable; if not, don't worry! Let's plot it and see what it looks like."
|
||||
"The multivariate version merely replaces the scalars of the univariate equations with matrices. If you are reasonably well-versed in linear algebra this equation should look quite manageable; if not, don't worry, we have code to compute it for us! Let's plot it and see what it looks like."
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -414,9 +418,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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||||
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truncated
|
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"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8e09f110b8>"
|
||||
"<matplotlib.figure.Figure at 0x8539128>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -432,7 +436,7 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"This is a plot of two dimensional multivariate Gaussian with a mean of $\\mu=[\\begin{smallmatrix}2\\\\17\\end{smallmatrix}]$ and a covariance of $\\Sigma=[\\begin{smallmatrix}10&0\\\\0&4\\end{smallmatrix}]$. The three dimensional shape shows the probability density of for any value of (x,y) in the z-axis. I have projected the variance for x and y onto the walls of the chart - you can see that they take on the normal Gaussian bell curve shape. The curve for x is wider than the curve for y, which is explained by $\\sigma_x^2=10$ and $\\sigma_y^2=4$. The highest point of the curve is centered over (2, 17), the means for x and y. \n",
|
||||
"This is a plot of multivariate Gaussian with a mean of $\\mu=[\\begin{smallmatrix}2\\\\17\\end{smallmatrix}]$ and a covariance of $\\Sigma=[\\begin{smallmatrix}10&0\\\\0&4\\end{smallmatrix}]$. The three dimensional shape shows the probability density of for any value of (x,y) in the z-axis. I have projected the variance for x and y onto the walls of the chart - you can see that they take on the normal Gaussian bell curve shape. The curve for $x$ is wider than the curve for $y$, which is explained by $\\sigma_x^2=10$ and $\\sigma_y^2=4$. The highest point of the curve is at (2, 17), the means for $x$ and $y$. \n",
|
||||
"\n",
|
||||
"All multivariate Gaussians have this shape. If we think of this as a the Gaussian for the position of a dog, the z-value at each point of (x, y) is the probability density of it being at that position. Strictly speaking this is the *joint probability density function*, which I will define soon. So, he has the highest probability of being near (2, 17), a modest probability of being near (5, 14), and a very low probability of being near (10, 10). As with the univariate case this is a *probability density*, not a *probability*. Continuous distributions have an infinite range, and so the probability of being exactly at (2, 17), or any point, is 0%. We can compute the probability of being within a given range by computing the area under the curve. \n",
|
||||
"\n",
|
||||
@@ -461,7 +465,7 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"SciPy's `stats` module implements the multivariate normal equation with `multivariate_normal()`. It implements a 'frozen' form where you set the mean and covariance once, and then calculate the probability for any number of values for x over any arbitrary number of calls. This is much more efficient then recomputing everything in each call. So, if you have version 0.14 or later you may want to substitute my function for the built in version. Use `scipy.version.version` to get the version number. I named my function `multivariate_gaussian()` to ensure it is never confused with the SciPy version. I will say that for a single call, where the frozen variables do not matter, mine consistently runs faster as measured by the `timeit` function.\n",
|
||||
"SciPy's `stats` module implements the multivariate normal equation with `multivariate_normal()`. It implements a 'frozen' form where you set the mean and covariance once, and then calculate the probability for any number of values for x over any arbitrary number of calls. This is much more efficient then recomputing everything in each call. I named my function `multivariate_gaussian()` to ensure it is never confused with the SciPy version. I will say that for a single call, where the frozen variables do not matter, mine consistently runs faster as measured by the `timeit` function.\n",
|
||||
"\n",
|
||||
"> The <a href=\"http://docs.scipy.org/doc/scipy/reference/tutorial/stats.html\">tutorial</a>[1] for the `scipy.stats` module explains 'freezing' distributions and other very useful features."
|
||||
]
|
||||
@@ -579,9 +583,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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|
||||
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|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8e09323f98>"
|
||||
"<matplotlib.figure.Figure at 0x727db00>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -599,9 +603,9 @@
|
||||
"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 density quickly drops away in all directions. This plot shows what is known as the *joint probability*. On the sides of the plot I have drawn the *marginal probability* for $x$ and $y$.\n",
|
||||
"The result is clearly a 3D bell shaped curve. We can see that the Gaussian is centered around (2,7), and that the probability density quickly drops away in all directions. This plot shows what is known as the *joint probability*. On the sides of the plot I have drawn the *marginal probability* for $x$ and $y$ (defined below).\n",
|
||||
"\n",
|
||||
"It's time to define some terms. The *joint probability*, denoted $p(x,y)$, is the probability of both $x$ and $y$ happening simultaneously. For example, if you have two die $p(2,5)$ is the probability of the first die rolling a 2 and the second die rolling a 5. Assuming the die are six sided and fair, the probability $p(2,5) = \\frac{1}{6}\\times \\frac{1}{6}=\\frac{1}{36}$. The 3D chart above shows the *joint probability density function*. This means that the area under the curve at any location is proportional to probability of the dog being in that $(x, y)$ position.\n",
|
||||
"It's time to define some terms. The *joint probability*, denoted $P(x,y)$, is the probability of both $x$ and $y$ happening. For example, if you have two die $P(2,5)$ is the probability of the first die rolling a 2 and the second die rolling a 5. Assuming the die are six sided and fair, the probability $P(2,5) = \\frac{1}{6}\\times \\frac{1}{6}=\\frac{1}{36}$. The 3D chart above shows the *joint probability density function*. This means that the area under the curve at any for any range of the random variables is proportional to probability of those random variables taking on those values. For example, if this is the plot for our dog's position, the area under the curve for $x$ in the range 4 to 5 and for $y$ in the range 10 to 12 is equal to the probability of the dog being somewhere within that that location. \n",
|
||||
"\n",
|
||||
"The *marginal probability* is the probability of an event happening without regard of any other event. In the chart above the Gaussian curve drawn to the left is the marginal for $y$. This is the probability for the dog being at any position in $y$ disregarding the value for $x$. In back I've plotted the marginal for $x$, which is the probability of the dog being at any position in $x$ disregarding $y$."
|
||||
]
|
||||
@@ -622,9 +626,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8e0a11c7f0>"
|
||||
"<matplotlib.figure.Figure at 0x7503080>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -646,7 +650,7 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"As beautiful as this is, it is hard to get useful information. For example, it is not easy to tell if $x$ and $y$ both have the same variance. In most of the book I'll display Gaussians using contour plots. Helper functions in FilterPy plot them for us. If you are interested in linear algebra look at the code used to produce these contours, otherwise feel free to ignore it. "
|
||||
"As beautiful as these plots are, it is hard to get useful information from them. For example, it is not easy to tell if $x$ and $y$ both have the same variance. In most of the book I'll display Gaussians using contour plots. Helper functions in FilterPy plot them for us. If you are interested in linear algebra look at the code used to produce these contours, otherwise feel free to ignore it. "
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -659,9 +663,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8e0b80d1d0>"
|
||||
"<matplotlib.figure.Figure at 0x88758d0>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -701,9 +705,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8e0a246cc0>"
|
||||
"<matplotlib.figure.Figure at 0x8745da0>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -734,9 +738,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8e0b7b76d8>"
|
||||
"<matplotlib.figure.Figure at 0x8745198>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -769,9 +773,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8e0a246dd8>"
|
||||
"<matplotlib.figure.Figure at 0x88ceef0>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -815,9 +819,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8e0bf0f3c8>"
|
||||
"<matplotlib.figure.Figure at 0x8981d30>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -859,9 +863,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8e0a2bc748>"
|
||||
"<matplotlib.figure.Figure at 0x8815ac8>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -894,8 +898,8 @@
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"array([[ 2.23611111e+02, 8.80625000e+05],\n",
|
||||
" [ 8.80625000e+05, 3.68125000e+09]])"
|
||||
"array([[ 5.99150000e+03, 4.42500000e+06],\n",
|
||||
" [ 4.42500000e+06, 3.68125000e+09]])"
|
||||
]
|
||||
},
|
||||
"execution_count": 17,
|
||||
@@ -919,9 +923,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8e0a2ab978>"
|
||||
"<matplotlib.figure.Figure at 0x8f15d30>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -952,9 +956,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8e0a1bb1d0>"
|
||||
"<matplotlib.figure.Figure at 0x87d7da0>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -976,7 +980,7 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
" When we see this ellipse we know that $x$ and $y$ are correlated, and that the correlation is \"strong\". The size of the ellipse shows how much error we have in each axis, and the slant shows how the relative sizes of the variance in $x$ and $y$. For example, a very long and narrow ellipse tilted almost to the horizontal has a strong correlation between $x$ and $y$ (because the ellipse is narrow), and the variance of $x$ is much larger than that of $y$ (because the ellipse is much longer in $x$). Note that in this plot the scales of the x-axis and y-axis are not identical, so the appearence of the plot can be deceiving."
|
||||
" When we see this ellipse we know that $x$ and $y$ are correlated, and that the correlation is \"strong\". The size of the ellipse shows how much error we have in each axis, and the slant shows how the relative sizes of the variance in $x$ and $y$. For example, a very long and narrow ellipse tilted almost to the horizontal has a strong correlation between $x$ and $y$ (because the ellipse is narrow), and the variance of $x$ is much larger than that of $y$ (because the ellipse is much longer in $x$). Note that in this plot the scales of the x-axis and y-axis are not identical, so the appearence of the plot can be deceiving. The next section provides us a numerical measure of correlation which removes the need for subjective evaluation."
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -1004,7 +1008,7 @@
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"0.97061425335184115"
|
||||
"0.94221027747659869"
|
||||
]
|
||||
},
|
||||
"execution_count": 20,
|
||||
@@ -1064,9 +1068,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8e0a26dc50>"
|
||||
"<matplotlib.figure.Figure at 0x8808630>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1097,9 +1101,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8e0a0c60f0>"
|
||||
"<matplotlib.figure.Figure at 0x884b710>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1140,9 +1144,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8e0bee14a8>"
|
||||
"<matplotlib.figure.Figure at 0x880c780>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1180,7 +1184,7 @@
|
||||
"\\Sigma &\\approx \\frac{\\Sigma_1\\Sigma_2}{(\\Sigma_1+\\Sigma_2)}\n",
|
||||
"\\end{aligned}$$\n",
|
||||
"\n",
|
||||
"In this form we can surmise that these equations are the linear algebra form of the univariate equations.\n",
|
||||
"In this form the relationship between the univariate and multivariate equations is clear.\n",
|
||||
"\n",
|
||||
"Now let's explore multivariate Gaussians in terms of a concrete example. Suppose that we are tracking an aircraft with two radar systems. I will ignore altitude so I can use two dimensional plots. Radars give us the range and bearing to a target. We start out being uncertain about the position of the aircraft, so the covariance, which is our uncertainty about the position, might look like this. In the language of Bayesian statistics this is our *prior*. "
|
||||
]
|
||||
@@ -1195,9 +1199,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8e0a31c9e8>"
|
||||
"<matplotlib.figure.Figure at 0x88db4a8>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1213,7 +1217,7 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Now suppose that there is a radar to the lower left of the aircraft. Further suppose that the radar is very accurate in the bearing measurement, but not very accurate at the range. That covariance, which is the uncertainty in the reading might look like this (plotted in blue):"
|
||||
"Now suppose that there is a radar to the lower left of the aircraft. Further suppose that the radar bearing measurement is accurate, but the range measurement is inaccurate. The covariance for the error in the measurement might look like this (plotted in blue):"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -1226,9 +1230,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8e0b7d8b38>"
|
||||
"<matplotlib.figure.Figure at 0x89e3b70>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1247,21 +1251,22 @@
|
||||
"source": [
|
||||
"Recall that Bayesian statistics calls this the *evidence*. The ellipse points towards the radar. It is very long because the range measurement is inaccurate, and the aircraft could be within a considerable distance of the measured range. It is very narrow because the bearing estimate is very accurate and thus the aircraft must be very close to the bearing estimate.\n",
|
||||
"\n",
|
||||
"We want to find the *posterior* - the mean and covariance of incorporating the evidence into the prior. As in every chapter so far we multiply them together. I have the equations for this and we could use those, but I will use FilterPy's `multivariate_multiply` method."
|
||||
"We want to find the *posterior* - the mean and covariance that results from incorporating the evidence into the prior. As in every chapter so far we multiply them together. Here we use FilterPy's `multivariate_multiply` function, which implements the equations from the above:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 27,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
"collapsed": false,
|
||||
"scrolled": true
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8e0a0fd710>"
|
||||
"<matplotlib.figure.Figure at 0x86e52e8>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1281,9 +1286,9 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Here I have plotted the original estimate (prior) in a very transparent yellow, the radar reading in blue (evidence), and the finale estimate (posterior) in yellow.\n",
|
||||
"I have plotted the original estimate (prior) in a very transparent yellow, the radar reading in blue (evidence), and the finale estimate (posterior) in yellow.\n",
|
||||
"\n",
|
||||
"The Gaussian retained the same shape and position as the radar measurement, but is smaller. We've seen this with one dimensional Gaussians. Multiplying two Gaussians makes the variance smaller because we are incorporating more information, hence we are less uncertain. But the main point I want to make is that the covariance shape reflects the physical layout of the aircraft and the radar system.\n",
|
||||
"The posterior retained the same shape and position as the radar measurement, but is smaller. We've seen this with one dimensional Gaussians. Multiplying two Gaussians makes the variance smaller because we are incorporating more information, hence we are less uncertain. But the main point I want to make is that the covariance shape reflects the physical layout of the aircraft and the radar system. The importance of this will become clear in the next step.\n",
|
||||
"\n",
|
||||
"Now lets say we get a measurement from a second radar, this one to the lower right, which I will plot in blue against the yellow covariance of our current belief. "
|
||||
]
|
||||
@@ -1297,9 +1302,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8e0b77fbe0>"
|
||||
"<matplotlib.figure.Figure at 0x89d4b00>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1328,9 +1333,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8e0cf40400>"
|
||||
"<matplotlib.figure.Figure at 0x8889b00>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1356,7 +1361,7 @@
|
||||
"\n",
|
||||
"$$\\Sigma = \\begin{bmatrix}2&1.9\\\\1.9&2\\end{bmatrix}$$\n",
|
||||
"\n",
|
||||
"Why did this result in a 45 degree ellipse? Think about that before reading on. It was 45 degrees because the values in the diagonal were identical. So if x=10 then y=10, and so on. We can alter the angle by making the variance for x or y different, like so:"
|
||||
"Why did this result in a 45 degree ellipse? Think about that before reading on. It was 45 degrees because the values in the diagonal were identical. So if x=10 then y=10, and so on. We can alter the angle by making the variance for $x$ or $y$ different, like so:"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -1368,9 +1373,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8e0cf4d470>"
|
||||
"<matplotlib.figure.Figure at 0x87af6a0>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1398,9 +1403,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8e0b75a2e8>"
|
||||
"<matplotlib.figure.Figure at 0x88703c8>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1438,7 +1443,7 @@
|
||||
"\n",
|
||||
"$$\\Sigma = \\begin{bmatrix}2&0\\\\0&0.2\\end{bmatrix}$$\n",
|
||||
"\n",
|
||||
"to model the measurement. Incidentally, I invented those values. We haven't learned how to transform a matrix from one coordinate system to another. \n",
|
||||
"to model the measurement. Incidentally, I invented those values. We haven't learned how to transform a covariance matrix from one coordinate system to another. \n",
|
||||
"\n",
|
||||
"In the next graph I plot the original estimate in a very light yellow, the radar measurement in blue, and the new estimate based on multiplying the two Gaussians together in yellow."
|
||||
]
|
||||
@@ -1452,9 +1457,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8e0b7a8fd0>"
|
||||
"<matplotlib.figure.Figure at 0x87a2978>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1485,9 +1490,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8e0a0baa58>"
|
||||
"<matplotlib.figure.Figure at 0x876bba8>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1506,7 +1511,7 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Our estimate is not as accurate as the previous example. The two radar stations are no longer orthogonal to each other relative to the aircraft's position so the triangulation is not optimal. Imagine standing on the ground and trying to triangulate on an aircraft in the sky with a transit. If you took a measurement, moved the transit 5 meters and took a second measurement the tiny change in angle between the two measurements would result in a very poor measurement because a very small error in either measurement would give a wildly different result. Think of the measurements as two nearly parallel lines. Changing the angle between them slightly will move the intersection between the two by a large amount. If you were to take the measurements from positions 100 km apart the lines might be nearly perpendicular to each other, in which case a small measurement error would result in a very small shift in the intersection point."
|
||||
"Our estimate is not as accurate as the previous example. The two radar stations are no longer orthogonal to each other relative to the aircraft's position so the triangulation is not optimal. Imagine standing on the ground and trying to triangulate on an aircraft in the sky with a transit. If you took a measurement, moved the transit 5 meters and took a second measurement the tiny change in angle between the two measurements would result in a very poor estimate because a very small error in either measurement would give a wildly different result. Think of the measurements as two nearly parallel lines. Changing the angle between them slightly will move the intersection between the two by a large amount. If you were to take the measurements from positions 100 km apart the lines might be nearly perpendicular to each other, in which case a small measurement error would result in a very small shift in the intersection point."
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -1520,7 +1525,7 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"You can probably already see why a multivariate Kalman filter can perform better than a univariate one. The last section demonstrated how we can use correlations between variables to significantly improve our estimates. We can take this much further. This section contains the key insight to this chapter, so read carefully.\n",
|
||||
"You can already see why a multivariate Kalman filter can perform better than a univariate one. The last section demonstrated how we can use correlations between variables to significantly improve our estimates. We can take this much further. This section contains the key insight to this chapter, so read carefully.\n",
|
||||
"\n",
|
||||
"Let's say we are tracking an aircraft and we get the following data for the $x$ and $y$ coordinates 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?"
|
||||
]
|
||||
@@ -1534,9 +1539,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8e0b744320>"
|
||||
"<matplotlib.figure.Figure at 0x8905198>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1564,9 +1569,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8e0a1d7898>"
|
||||
"<matplotlib.figure.Figure at 0x9421908>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1581,9 +1586,10 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"You made this inference because you *inferred* a constant velocity for the airplane. The *reasonable* assumption is that the aircraft is moving one unit each in *x* and *y* per time step.\n",
|
||||
"You made this inference because you *inferred* a constant velocity for the airplane. The reasonable\n",
|
||||
"assumption is that the aircraft is moving one unit each in *x* and *y* per time step.\n",
|
||||
"\n",
|
||||
"Think back to the g-h filter chapter when we were trying to improve the weight predictions of a noisy scale. We incorporated *weight gain* into the equations because it allowed us to make a better prediction of the weight the next day. The g-h filter uses the *g* parameter to scale the amount of significance given to the current weight measurement, and the *h* parameter scaled the amount of significance given to the weight gain.\n",
|
||||
"Think back to the **g-h Filter** chapter when we were trying to improve the weight predictions of a noisy scale. We incorporated *weight gain* into the equations because it allowed us to make a better prediction of the weight the next day. The g-h filter uses the $g$ parameter to scale the amount of significance given to the current weight measurement, and the $h$ parameter scaled the amount of significance given to the weight gain.\n",
|
||||
"\n",
|
||||
"We are going to do the same thing with our Kalman filter. After all, the Kalman filter is a form of a g-h filter. In this case we are tracking an airplane, so instead of weight and weight gain we need to track position and velocity. Weight gain is the *derivative* of weight, and of course velocity is the derivative of position. It's impossible to plot and understand the 4D chart that would be needed to plot *x* and *y* and their respective velocities so let's do it for $x$, knowing that the math generalizes to more dimensions.\n",
|
||||
"\n",
|
||||
@@ -1599,9 +1605,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8e0cf5b3c8>"
|
||||
"<matplotlib.figure.Figure at 0x896b9b0>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1628,9 +1634,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8e0a1fe4a8>"
|
||||
"<matplotlib.figure.Figure at 0x946e6d8>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1657,9 +1663,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8e0be72cf8>"
|
||||
"<matplotlib.figure.Figure at 0x7271b38>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1688,9 +1694,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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|
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"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8e0bedd358>"
|
||||
"<matplotlib.figure.Figure at 0x8870240>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1712,7 +1718,7 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"This is a key point in Kalman filters, so read carefully! Our sensor is only detecting the position of the aircraft (how doesn't matter). This is called an *observed variable*. It does not have a sensor that provides velocity. But based on the position estimates we can compute velocity. In Kalman filters we would call the velocity a *hidden variable*. Hidden means what it sounds like - there is no sensor that is measuring velocity, thus its value is hidden from us. We are able to use the correlation between position and velocity to infer its value very accurately.\n",
|
||||
"This is a key point in Kalman filters, so read carefully! Our sensor is only detecting the position of the aircraft (how doesn't matter). This is called an *observed variable*. It does not have a sensor that provides velocity. But based on the position estimates we can compute velocity. We call the velocity a *hidden variable*. Hidden means what it sounds like - there is no sensor that is measuring velocity, thus its value is hidden from us. We are able to use the correlation between position and velocity to infer its value very accurately.\n",
|
||||
"\n",
|
||||
"To round out the terminology there are also *unobserved variables*. For example, the aircraft's state includes things such as as heading, engine RPM, weight, color, the first name of the pilot, and so on. We cannot sense these directly using the position sensor so they are not *observed*. There is no way to *infer* them from the sensor measurements and correlations (red planes don't go faster than white planes), so they are not *hidden*. Instead, they are *unobservable*. If you include an unobserved variable in your filter state the estimate for that variable will be nonsense."
|
||||
]
|
||||
@@ -1730,7 +1736,9 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"In summary we have taken advantage of the geometry and correlations of the system to produce a very accurate estimate. The math does not care whether we are working with two positions, or a position and a correlated velocity, or if these are spatial dimensions. If floor space is correlated to house price you can write a Kalman filter to track house prices. If age is correlated to disease incidence you can write a Kalman filter to track diseases. If the zombie population is inversely correlated with the number of shotguns then you can write a Kalman filter to track zombies. I showed you this in terms of geometry and talked about *triangulation*. That was just to build your intuition. Get used to thinking of these as Gaussians with correlations. If we can express our uncertainties as a multidimensional Gaussian we can then multiply the prior with the likelihood and get a much more accurate result. "
|
||||
"## Summary\n",
|
||||
"\n",
|
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"We have taken advantage of the geometry and correlations of the system to produce a very accurate estimate. The math does not care whether we are working with two positions, or a position and a correlated velocity, or if these are spatial dimensions. If floor space is correlated to house price you can write a Kalman filter to track house prices. If age is correlated to disease incidence you can write a Kalman filter to track diseases. If the zombie population is inversely correlated with the number of shotguns then you can write a Kalman filter to track zombie populations. I showed you this in terms of geometry and talked about *triangulation*. That was just to build your intuition. You can write a Kalman filter for state variables that have no geometric representation, such as filters for stock prices or milk production of cows (I received an email from someone tracking milk production!) Get used to thinking of these as Gaussians with correlations. If we can express our uncertainties as a multidimensional Gaussian we can then multiply the prior with the likelihood and get a much more accurate result. "
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]
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},
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{
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Reference in new issue
Block a user