Explained biased vs unbiased estimators.
I had glossed over this difference, and so it would be confusing to understand why my equations for VAR and COV are incorrect for samples. I distinguished between the two and gave the correct computations for each.
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Roger Labbe committed
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@@ -346,9 +342,11 @@
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"\n",
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"$$\\sigma^2 = \\begin{bmatrix}10\\\\4\\end{bmatrix}$$ \n",
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"This is incomplete because it does not consider the more general case. In the **Gaussians** chapter we computed the variance in the heights of students. We can do the same for their weights. But there is also a relationship between height and weight. In general, a taller person weighs more than a shorter person. Height and weight are *correlated*. We want a way to express not only what we think the variance is in the height and the weight, but also the degree to which they are correlated. \n",
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"This is incomplete because it does not consider the more general case. In the **Gaussians** chapter we computed the variance in the heights of students. That is a measure of how the weights vary relative to each other. If all students are the same height the variance is 0, and if their heights are wildly different the variance will be large. \n",
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"\n",
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"Before we can understand multivariate normal distributions we need to understand the mathematics behind correlations."
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"There is also a relationship between height and weight. In general, a taller person weighs more than a shorter person. Height and weight are *correlated*. We want a way to express not only what we think the variance is in the height and the weight, but also the degree to which they are correlated. In other words, we want to know how weight varies compared to the heights. We call that the *covariance*. \n",
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"\n",
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"Before we can understand multivariate normal distributions we need to understand the mathematics behind correlations and covariances."
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"iVBORw0KGgoAAAANSUhEUgAAAv0AAADxCAYAAABYmwhGAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8e3ba58>"
|
||||
"<matplotlib.figure.Figure at 0x1a19a636f28>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -397,7 +395,6 @@
|
||||
"source": [
|
||||
"In this book we only consider *linear correlation*. We assume that the relationship between variables is linear. That is, a straight line is a good fit for the data. I've fit a straight line through the data in the above chart. The concept of *nonlinear correlation* exists, but we will not be using it.\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"The equation for the covariance between $X$ and $Y$ is\n",
|
||||
"\n",
|
||||
"$$ COV(X, Y) = \\sigma_{xy} = \\mathbb E\\big[(X-\\mu_x)(Y-\\mu_y)\\big]$$\n",
|
||||
@@ -406,6 +403,11 @@
|
||||
"\n",
|
||||
"$$\\mathbb E[X] = \\begin{cases} \\sum_{i=1}^n p_ix_i & \\mbox{discrete}\\\\ \\int_{-\\infty}^\\infty x\\, f(x) & \\mbox{continuous}\\end{cases}$$\n",
|
||||
"\n",
|
||||
"We assume each data point is equally likely, so the probability of each is $\\frac{1}{N}$, yielding:\n",
|
||||
"\n",
|
||||
"$$\\mathbb E[X] = \\frac{1}{N}\\sum_{i=1}^n x_i$$\n",
|
||||
"\n",
|
||||
"for the discrete case we will be considering.\n",
|
||||
"\n",
|
||||
"Compare the covariance equation to the equation for the variance. As you can see they are very similar:\n",
|
||||
"\n",
|
||||
@@ -441,7 +443,6 @@
|
||||
"If there was a large amount of negative correlation between between $x$ and $y$ we might have\n",
|
||||
"$$\\Sigma = \\begin{bmatrix}10&-9.7\\\\-9.7&4\\end{bmatrix}$$\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"The covariance matrix is symmetric. After all, 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 $y$."
|
||||
]
|
||||
},
|
||||
@@ -449,12 +450,121 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Let's work through a few examples. `numpy.cov` computes the covariance matrix given a list of values for two random variables. Lets create data with perfectly correlated data. By this I mean that the data perfectly fits on a line - there is no variance from the line."
|
||||
"I fear I might be losing you, so let's work an example. In the **Gaussians** chapter we had a class of students with heights H=[1.8, 2.0, 1.7, 1.9, 1.6] meters. We computed:\n",
|
||||
"\n",
|
||||
"$$\\begin{aligned}\n",
|
||||
"\\mathit{VAR}(H) &= E[(H - \\mu_H)^2] \\\\\n",
|
||||
"&= \\frac{1}{N}\\sum_{i=1}^n (H_i - \\mu_H)^2 \\\\\n",
|
||||
"&= \\frac{1}{5}\\left[(1.8-1.8)^2 + (2-1.8)^2 + (1.7-1.8)^2 + (1.9-1.8)^2 + (1.6-1.8)^2\\right] \\\\\n",
|
||||
"&= 0.02\n",
|
||||
"\\end{aligned}$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Easy, right? If we weigh the students we might find their weights to be W = [70.1, 91.2, 59.5, 93.2, 53.5]. Can we use the covariance equation to create the covariance matrix? Sure. It will look like:\n",
|
||||
"\n",
|
||||
"$$\\Sigma = \\begin{bmatrix}\\sigma_H^2 & \\sigma_{H,W} \\\\\n",
|
||||
"\\sigma_{W,H} & \\sigma_{H}^2\\end{bmatrix}$$\n",
|
||||
"\n",
|
||||
"We just computed the variance of the height, and it will go in the upper left hand corner of the matrix. The lower right corner contains the variance in heights. Using the same equation we get:\n",
|
||||
"\n",
|
||||
"$$\\begin{aligned}\n",
|
||||
"\\mu_W &= \\frac{1}{5}(70.1 + 91.2 + 59.5 + 93.2 + 53.5) = 73.5 \\\\\n",
|
||||
"\\sigma_W^2 &= \\frac{1}{5}\\left[(70.1-73.5) + (91.2-73.5) + (59.5-73.5) + (93.2-73.5) + (53.5-73.5)\\right] \\\\\n",
|
||||
"&= 261.8\n",
|
||||
"\\end{aligned}$$\n",
|
||||
"\n",
|
||||
"Now the covariances. Using the formula above, we compute:\n",
|
||||
"\n",
|
||||
"$$\\begin{aligned}\n",
|
||||
"\\sigma_{H,W} &= \\mathbb E\\big[(H-\\mu_H)(W-\\mu_W)\\big] \\\\\n",
|
||||
"&= \\frac{1}{N}\\sum_{i=1}^n (H_i-\\mu_H)(W_i-\\mu_W) \\\\\n",
|
||||
"&= (1.8-1.8)(70.1-73.5) + (2-1.8)(91.2-73.5) + (1.7-1.8)(59.5-73.5)\\, +\\\\\n",
|
||||
"&\\, \\, \\, \\, \\, (1.9-1.8)(93.2-73.5) + (1.6-1.8)(53.5-73.5) \\\\\n",
|
||||
"&= 2.18\n",
|
||||
"\\end{aligned}$$\n",
|
||||
"\n",
|
||||
"That was tedious, but easy enough. We will never do that again because, of course, NumPy will compute it for you. "
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 33,
|
||||
"execution_count": 3,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"array([[ 0.025 , 2.7275],\n",
|
||||
" [ 2.7275, 327.235 ]])"
|
||||
]
|
||||
},
|
||||
"execution_count": 3,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"import numpy as np\n",
|
||||
"%precision 4\n",
|
||||
"W = [70.1, 91.2, 59.5, 93.2, 53.5]\n",
|
||||
"H = [1.8, 2.0, 1.7, 1.9, 1.6]\n",
|
||||
"np.cov(H, W)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"That doesn't agree with our calculation! What went wrong? Nothing. NumPy applies a correction for small sample sizes; it uses $\\frac{1}{N-1}$ as the normalization term instead of $\\frac{1}{N}$.\n",
|
||||
"\n",
|
||||
"This is a bit beyond the scope of this book. Briefly, suppose the actual class size is 200 students, and we took a sample of 5 students to perform this computation because we couldn't afford to measure and weigh all 200 students. It is nearly certain that there will be some error in our estimator because the sample is unlikely to perfectly represent the class. As our sample size approaches 200 the error will approach 0. We say there is no *bias* in the latter, and that we have an *unbiased estimator*. In contrast, when we take a small sample there is bias (error is nonzero), and we have a *biased estimator*.\n",
|
||||
"\n",
|
||||
"If the error is zero it makes sense to divide by $N$. I will not prove why, but for biased estimators we use $\\frac{1}{N-1}$ to correct for the small sample size. NumPy does this by default because in practice we are almost always working from data samples. If you want the unbiased estimator, which we computed above, use `bias=1` in the call to `np.cov'."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 4,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"array([[ 0.02 , 2.182],\n",
|
||||
" [ 2.182, 261.788]])"
|
||||
]
|
||||
},
|
||||
"execution_count": 4,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"np.cov(H, W, bias=1)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"This agrees with our computation. We will not use `bias=1` again in this book since we are using *random variables* which are sampling from the infinite set of positions of the objects we are tracking. Here we are computing the variance and covariance for the entire population, so `bias=1` is correct.\n",
|
||||
"\n",
|
||||
"What does this matrix tell us? It tells us the variance in heights is 0.02 $m^2$ and the variance in weights is 261.788 $kg^2$. Furthermore, it tells us the weights and heights are positively correlated - as heights increase so do the weights.\n",
|
||||
"\n",
|
||||
"Let's create perfectly correlated data. By this I mean that the data perfectly fits on a line - there is no variance from the line."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 5,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -463,13 +573,12 @@
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"[[ 6.96 6.96]\n",
|
||||
" [ 6.96 6.96]]\n"
|
||||
"[[ 6.9559 6.9559]\n",
|
||||
" [ 6.9559 6.9559]]\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"import numpy as np\n",
|
||||
"X = np.linspace(1, 10, 100)\n",
|
||||
"Y = np.linspace(1, 10, 100)\n",
|
||||
"print(np.cov(X, Y))"
|
||||
@@ -486,16 +595,16 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 34,
|
||||
"execution_count": 6,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x892f3c8>"
|
||||
"<matplotlib.figure.Figure at 0x1a19b70a128>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -505,8 +614,8 @@
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"[[ 6.96 -3.08]\n",
|
||||
" [ -3.08 1.39]]\n"
|
||||
"[[ 6.9559 -3.0843]\n",
|
||||
" [-3.0843 1.3868]]\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
@@ -528,16 +637,16 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 35,
|
||||
"execution_count": 7,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x80ce278>"
|
||||
"<matplotlib.figure.Figure at 0x1a19b76a550>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -547,8 +656,8 @@
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"[[ 10.7 2.95]\n",
|
||||
" [ 2.95 2.24]]\n"
|
||||
"[[ 11.009 3.229 ]\n",
|
||||
" [ 3.229 2.4018]]\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
@@ -571,16 +680,16 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 36,
|
||||
"execution_count": 8,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x9b1e358>"
|
||||
"<matplotlib.figure.Figure at 0x1a19b76ae80>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -590,8 +699,8 @@
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"[[ 1.01 -0.00177]\n",
|
||||
" [ -0.00177 0.998]]\n"
|
||||
"[[ 1.0047 0.0055]\n",
|
||||
" [ 0.0055 1.0021]]\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
@@ -632,16 +741,16 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 37,
|
||||
"execution_count": 9,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x9b1e6a0>"
|
||||
"<matplotlib.figure.Figure at 0x1a19b821048>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -677,7 +786,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 38,
|
||||
"execution_count": 10,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -699,7 +808,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 39,
|
||||
"execution_count": 11,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -717,7 +826,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 40,
|
||||
"execution_count": 12,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -735,7 +844,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 41,
|
||||
"execution_count": 13,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -754,7 +863,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 42,
|
||||
"execution_count": 14,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -765,13 +874,12 @@
|
||||
"0.0315"
|
||||
]
|
||||
},
|
||||
"execution_count": 42,
|
||||
"execution_count": 14,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"%precision 4\n",
|
||||
"multivariate_gaussian(x, mu, P)"
|
||||
]
|
||||
},
|
||||
@@ -793,16 +901,16 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 43,
|
||||
"execution_count": 15,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8928908>"
|
||||
"<matplotlib.figure.Figure at 0x1a19b7da630>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -831,7 +939,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 29,
|
||||
"execution_count": 16,
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"scrolled": true
|
||||
@@ -841,7 +949,7 @@
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0xb3b1518>"
|
||||
"<matplotlib.figure.Figure at 0x1a19b77c710>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -875,7 +983,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 46,
|
||||
"execution_count": 17,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -884,7 +992,7 @@
|
||||
"data": {
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8e01d30>"
|
||||
"<matplotlib.figure.Figure at 0x1a19cc641d0>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -911,7 +1019,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 47,
|
||||
"execution_count": 18,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -920,7 +1028,7 @@
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8e48d68>"
|
||||
"<matplotlib.figure.Figure at 0x1a19b1e3e80>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -946,7 +1054,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 49,
|
||||
"execution_count": 19,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -955,7 +1063,7 @@
|
||||
"data": {
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0xb3dd048>"
|
||||
"<matplotlib.figure.Figure at 0x1a19b7df710>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -991,7 +1099,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 50,
|
||||
"execution_count": 20,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -1000,7 +1108,7 @@
|
||||
"data": {
|
||||
"image/png": 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||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x9b2d5f8>"
|
||||
"<matplotlib.figure.Figure at 0x1a19cc6c400>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1036,7 +1144,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 121,
|
||||
"execution_count": 21,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -1045,7 +1153,7 @@
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x9d95048>"
|
||||
"<matplotlib.figure.Figure at 0x1a19cb8da58>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1066,12 +1174,12 @@
|
||||
"source": [
|
||||
"This is the first contour that has values in the off-diagonal elements of the covariance, and this is the first contour plot with a slanted ellipse. This is not a coincidence. The two facts are telling us the same thing. A slanted ellipse tells us that the $x$ and $y$ values are somehow correlated. The off-diagonal elements in the covariance matrix are non-zero, indicating that a correlation exists.\n",
|
||||
"\n",
|
||||
"Recall the plot for height versus weight. It formed a slanted grouping of points. We can use NumPy's `cov()` function to compute the covariance of two or more variables by placing them into a 2D array. Let's do that, then plot the $2\\sigma$ covariance ellipse on top of the data."
|
||||
"Recall the plot for height versus weight. It formed a slanted grouping of points. We can use NumPy's `cov()` function to compute the covariance of two or more variables by placing them into a 2D array. Let's do that, then plot the $2\\sigma$ covariance ellipse on top of the data. We will need to use `bias=1` because the data represents the entire population; it is not a sample."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 122,
|
||||
"execution_count": 22,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -1079,24 +1187,24 @@
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"array([[ 20.6, 1.51e+02],\n",
|
||||
" [ 1.51e+02, 1.21e+03]])"
|
||||
"array([[ 18.5249, 135.701 ],\n",
|
||||
" [ 135.701 , 1092.29 ]])"
|
||||
]
|
||||
},
|
||||
"execution_count": 122,
|
||||
"execution_count": 22,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"import numpy as np\n",
|
||||
"cov_hw = np.cov(np.vstack((height, weight)))\n",
|
||||
"cov_hw = np.cov(np.vstack((height, weight)), bias=1)\n",
|
||||
"cov_hw"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 123,
|
||||
"execution_count": 23,
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"scrolled": true
|
||||
@@ -1104,9 +1212,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0xa3b6b38>"
|
||||
"<matplotlib.figure.Figure at 0x1a19cc2bc88>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1130,16 +1238,16 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 124,
|
||||
"execution_count": 24,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8961320>"
|
||||
"<matplotlib.figure.Figure at 0x1a19b8569b0>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1166,7 +1274,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 125,
|
||||
"execution_count": 25,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -1175,7 +1283,7 @@
|
||||
"data": {
|
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"image/png": 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truncated
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||||
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||||
]
|
||||
},
|
||||
"metadata": {},
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||||
@@ -1224,7 +1332,7 @@
|
||||
},
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||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 126,
|
||||
"execution_count": 26,
|
||||
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||||
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||||
@@ -1232,10 +1340,10 @@
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"execution_count": 126,
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||||
"execution_count": 26,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
@@ -1254,7 +1362,7 @@
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{
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||||
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||||
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@@ -1262,10 +1370,10 @@
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"execution_count": 127,
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"execution_count": 27,
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||||
"metadata": {},
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||||
"output_type": "execute_result"
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||||
}
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@@ -1285,7 +1393,7 @@
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||||
},
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||||
{
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"execution_count": 28,
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@@ -1294,7 +1402,7 @@
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||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0xa2bea58>"
|
||||
"<matplotlib.figure.Figure at 0x1a19b8a5470>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1317,16 +1425,16 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 129,
|
||||
"execution_count": 29,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0xa3b7a20>"
|
||||
"<matplotlib.figure.Figure at 0x1a19cc2bf60>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1362,7 +1470,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 130,
|
||||
"execution_count": 30,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -1371,7 +1479,7 @@
|
||||
"data": {
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0xa392550>"
|
||||
"<matplotlib.figure.Figure at 0x1a19b9a1978>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1421,7 +1529,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 131,
|
||||
"execution_count": 31,
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"scrolled": true
|
||||
@@ -1431,7 +1539,7 @@
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x94f37f0>"
|
||||
"<matplotlib.figure.Figure at 0x1a19b8ae080>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1452,7 +1560,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 132,
|
||||
"execution_count": 32,
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"scrolled": true
|
||||
@@ -1462,7 +1570,7 @@
|
||||
"data": {
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8f7ad68>"
|
||||
"<matplotlib.figure.Figure at 0x1a19b841cc0>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1486,7 +1594,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 133,
|
||||
"execution_count": 33,
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"scrolled": true
|
||||
@@ -1496,7 +1604,7 @@
|
||||
"data": {
|
||||
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|
||||
"text/plain": [
|
||||
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|
||||
"<matplotlib.figure.Figure at 0x1a19b7c85f8>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1526,7 +1634,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 134,
|
||||
"execution_count": 34,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -1535,7 +1643,7 @@
|
||||
"data": {
|
||||
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truncated
|
||||
"text/plain": [
|
||||
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|
||||
"<matplotlib.figure.Figure at 0x1a19cc73278>"
|
||||
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|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1557,7 +1665,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 135,
|
||||
"execution_count": 35,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -1566,7 +1674,7 @@
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x9e1feb8>"
|
||||
"<matplotlib.figure.Figure at 0x1a19ccb71d0>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1591,16 +1699,16 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 136,
|
||||
"execution_count": 36,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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|
||||
"text/plain": [
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||||
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|
||||
"<matplotlib.figure.Figure at 0x1a19b7ba860>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1624,7 +1732,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 137,
|
||||
"execution_count": 37,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -1633,7 +1741,7 @@
|
||||
"data": {
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|
||||
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||||
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|
||||
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|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1657,7 +1765,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 138,
|
||||
"execution_count": 38,
|
||||
"metadata": {
|
||||
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|
||||
},
|
||||
@@ -1666,7 +1774,7 @@
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x7d2ee48>"
|
||||
"<matplotlib.figure.Figure at 0x1a19fff5b00>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1692,7 +1800,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 139,
|
||||
"execution_count": 39,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -1701,7 +1809,7 @@
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x9e0b048>"
|
||||
"<matplotlib.figure.Figure at 0x1a19b862080>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1733,16 +1841,16 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 51,
|
||||
"execution_count": 40,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0xb2e1c88>"
|
||||
"<matplotlib.figure.Figure at 0x1a19b9539e8>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1763,16 +1871,16 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 52,
|
||||
"execution_count": 41,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8936470>"
|
||||
"<matplotlib.figure.Figure at 0x1a19ba6d470>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1799,7 +1907,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 142,
|
||||
"execution_count": 42,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -1808,7 +1916,7 @@
|
||||
"data": {
|
||||
"image/png": 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truncated
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||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x897f5f8>"
|
||||
"<matplotlib.figure.Figure at 0x1a19cbe3240>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1828,7 +1936,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 143,
|
||||
"execution_count": 43,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -1837,7 +1945,7 @@
|
||||
"data": {
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8a3c5f8>"
|
||||
"<matplotlib.figure.Figure at 0x1a19b91efd0>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1857,7 +1965,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 144,
|
||||
"execution_count": 44,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -1866,7 +1974,7 @@
|
||||
"data": {
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8b0a5f8>"
|
||||
"<matplotlib.figure.Figure at 0x1a19b221c18>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1888,7 +1996,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 145,
|
||||
"execution_count": 45,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -1897,7 +2005,7 @@
|
||||
"data": {
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0xa39b240>"
|
||||
"<matplotlib.figure.Figure at 0x1a19ccaccc0>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
|
||||
Reference in new issue
Block a user