Added def. of marginal and joint distributions
This commit is contained in:
Roger Labbe committed
1 parent
383a3c18d7
commit
b27bc431e3
1 file changed
+29
-24
@@ -16,7 +16,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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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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@@ -255,7 +255,7 @@
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"<IPython.core.display.HTML object>"
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]
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},
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"execution_count": 1,
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"execution_count": 2,
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"metadata": {},
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"output_type": "execute_result"
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}
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@@ -373,16 +373,16 @@
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"execution_count": 3,
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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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"image/png": 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truncated
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"text/plain": [
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"<matplotlib.figure.Figure at 0x436c8d0>"
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"<matplotlib.figure.Figure at 0x48af668>"
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]
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},
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"metadata": {},
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@@ -400,9 +400,9 @@
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"source": [
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"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",
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"\n",
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"All multivariate Gaussians form 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. 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).\n",
|
||||
"All multivariate Gaussians form 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).\n",
|
||||
"\n",
|
||||
"More details are in the *Kalman Filter Math* chapter. Here we need to understand the following.\n",
|
||||
"For now we need to understand the following:\n",
|
||||
"\n",
|
||||
"1. The diagonal of the matrix contains the variance for each variable. This is because the covariance between x and itself is the variance of x: $\\sigma_{xx} = \\sigma_x^2$.\n",
|
||||
"\n",
|
||||
@@ -434,7 +434,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 3,
|
||||
"execution_count": 4,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -449,14 +449,14 @@
|
||||
"source": [
|
||||
"I'll demonstrate using it, and then move on to more interesting things.\n",
|
||||
"\n",
|
||||
"First, let's find the probability density for our dog being at (2.5, 7.3) if we believe he is at (2, 7) with a variance of 8 for $x$ and a variance of 4 for $y$.\n",
|
||||
"First, let's find the probability density for our dog being at (2.5, 7.3) if we believe he is at (2, 7) with a variance of 8 for $x$ and a variance of 3 for $y$.\n",
|
||||
"\n",
|
||||
"Start by setting $x$ to (2.5, 7.3). You can use a tuple, list, or NumPy array."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 4,
|
||||
"execution_count": 5,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -474,7 +474,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 5,
|
||||
"execution_count": 6,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -492,13 +492,14 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 6,
|
||||
"execution_count": 17,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"P = [[8., 0.], [0., 4.]]"
|
||||
"P = [[8., 0.], \n",
|
||||
" [0., 3.]]"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -510,7 +511,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 7,
|
||||
"execution_count": 18,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -519,7 +520,7 @@
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"0.02739\n"
|
||||
"0.03151\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
@@ -536,7 +537,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 8,
|
||||
"execution_count": 19,
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"scrolled": true
|
||||
@@ -544,9 +545,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x436c7f0>"
|
||||
"<matplotlib.figure.Figure at 0x818fa90>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -564,28 +565,32 @@
|
||||
"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. On the sides of the plot I have drawn the Gaussians for several different variances for $x$ in greens and for $y$ in orange."
|
||||
"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",
|
||||
"\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",
|
||||
"\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 $y$."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Let's look at this in a slightly different way. Instead of plotting a surface showing the probability distribution I will generate 1,000 points with the distribution of $[\\begin{smallmatrix}8&0\\\\0&4\\end{smallmatrix}]$."
|
||||
"Let's look at this in a slightly different way. Instead of plotting a surface showing the probability distribution I will generate 1,000 points with the distribution of $[\\begin{smallmatrix}8&0\\\\0&3\\end{smallmatrix}]$."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 9,
|
||||
"execution_count": 21,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x798eac8>"
|
||||
"<matplotlib.figure.Figure at 0x82d3a20>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -600,7 +605,7 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"We can think of the sampled points as being possible locations for our dog given those particular mean and covariances. The contours on the side show the variance in the points for $x$ and $y$ only. We can see that he is far more likely to be at (2, 7) where there are many points, than at (-5, 5) where there are few."
|
||||
"We can think of the sampled points as being possible locations for our dog given those particular mean and covariances. The contours on the side show the marginal probability for $x$ and $y$. We can see that he is far more likely to be at (2, 7) where there are many points, than at (-5, 5) where there are few."
|
||||
]
|
||||
},
|
||||
{
|
||||
|
||||
Reference in new issue
Block a user