diff --git a/06_Multivariate_Kalman_Filters.ipynb b/06_Multivariate_Kalman_Filters.ipynb index aff80ff..f611e40 100644 --- a/06_Multivariate_Kalman_Filters.ipynb +++ b/06_Multivariate_Kalman_Filters.ipynb @@ -2587,7 +2587,7 @@ "olGD4ZeIiIiIRo3/AmG9eXrpxkR3AAAAAElFTkSuQmCC\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -5089,7 +5089,7 @@ "UCgUCoVCeWig4pdCoVAoFAqF8tBAxS+FQqFQKBQK5aHh/wFu3DUzZpLwOQAAAABJRU5ErkJggg==\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -5510,901 +5510,900 @@ "frWWmhaXJ2P9uBOlwTMChmHI4pciHv8A1jZ/jPaYdGZj3MW4bdIyFNiHLPk9gQH8eNtfsDsmsA0A\n", "rpvyOVw76XwwYBKKAS2oERV6WoxzlbrCSvzznCvw4PYnI20tAx1Yd+hN3Dr9Utm2Zc48nFM1ERs7\n", "Dkfa3mk/hAVl41DlKoQVUZN6MFYQ6+XWoMbX2IxWY0IOid8MkI0BLy5rB4NBVRXHpBH7Xq9XthRu\n", - "FQHCMAwcA4fkmR4CiGR68FbOwKDXm/AYonvH/2fvvOPkKuv9/z5t+mzvJWU3nZCERAihSFGQoALS\n", - "hYuAoMJVr+BPL6LiFUSvgGADBEUBL14QQeWqdDRAECGU9J5Nstnep7dTfn/MzuzM7rTdzLawn1fm\n", - 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -6963,7 +6962,7 @@ "RER06P8HjaCfvOLiLbcAAAAASUVORK5CYII=\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -7386,7 +7385,7 @@ "rkJggg==\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -7988,7 +7987,7 @@ "7Hs8ht76sgAAAABJRU5ErkJggg==\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -8612,7 +8611,7 @@ "kL/1NHYURTAj6QAAMCPpAAAwI+kAADAj6QAAMCPpAAAwE0HqO/33yhpfAAAAAElFTkSuQmCC\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -8640,6 +8639,913 @@ "A word about **correlation** and **independence**. If variables are **independent** they can vary separately. If you walk in an open field, you can move in the $x$ direction (east-west), the $y$ direction(north-south), or any combination thereof. Independent variables are always also **uncorrelated**. Except in special cases, the reverse does not hold true. Variables can be uncorrelated, but dependent. For example, consider the pair$(x,y)$ where $y=x^2$. Correlation is a linear measurement, so $x$ and $y$ are uncorrelated. However, they are obviously dependent on each other. " ] }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Computing Means and Covariances" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "You will not need to do this by hand for the rest of the book, but it is important to understand the computations behind these values. It only takes a few minutes to learn.\n", + "\n", + "### Computing the Mean\n", + "So let's start with 3 variables: $a, b, c$. Let's say we have two values for each: $a = [1, 3]$, $b = [8, 7]$, and $c = [3,7]$. Another, equivalent way of writing this is that you will sometimes see is\n", + "\n", + "$$\\mathbf{x} = \\begin{bmatrix}a\\\\b\\\\c\\end{bmatrix} = \\begin{bmatrix}1 & 3\\\\8&7\\\\3&7\\end{bmatrix}$$\n", + "\n", + "You will not see that often in the Kalman filter literature, so I will adopt the alternative notation of using a subscript to indicate the $i^{th}$ value for $\\mathbf{x}$\n", + "\n", + "$$\\mathbf{x}_0 = \\begin{bmatrix}1 \\\\8\\\\3\\end{bmatrix},\\ \\ \\mathbf{x}_1 = \\begin{bmatrix}3\\\\7\\\\7\\end{bmatrix}$$\n", + "\n", + "We compute the mean just as for the univariate case, which we presented in the *Gaussians* chapter. Sum the values and divide by the number of values. Formally\n", + "\n", + "$$ \\mu_x = \\frac{1}{n}\\sum^n_{i=1} x_i$$\n", + "\n", + "Therefore we can write:\n", + "\n", + "$$\\begin{aligned}\n", + "\\mu_a &= \\frac{1+3}{2} = 2 \\\\\n", + "\\mu_b &= \\frac{8+7}{2} = 7.5 \\\\\n", + "\\mu_c &= \\frac{3+7}{2} = 5 \n", + "\\end{aligned}$$\n", + "\n", + "Which I can express in our shorthand matrix notation as\n", + "\n", + "$$\\mu_x = \\begin{bmatrix}2\\\\7.5\\\\5\\end{bmatrix}$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Expected Value\n", + "\n", + "I could just give you the formula for the covariance but it will make more sense if you see how it is derived. To do that we first need to talked about *expected value* of a random variable. The expected value is just the value we expect, on average, for the variable. \n", + "\n", + "The *expected value* of a random variable is just the average value it would have if we took an infinite number of samples of it and then averaged those samples together. Let's say we have $x=[1,3,5]$ and each value is equally probable. What would we *expect* $x$ to have, on average?\n", + "\n", + "It would just be the average of 1, 3, and 5, of course, which is 3. That should make sense; we would expect equal numbers of 1, 3, and 5 to occur, so $(1+3+5)/3=3$ is clearly the average of that infinite series of samples.\n", + "\n", + "Now suppose that each value has a different probability of happening. Say 1 has an 80% chance of occurring, 3 has an 15% chance, and 5 has only a 5% chance. In this case we compute the expected value by multiplying each value of $x$ by the percent chance of it occurring, and summing the result. So for this case we could compute\n", + "\n", + "$$E[x] = 1\\times 0.8 + 3\\times 0.15 + 5 \\times 0.05 = 1.5$$\n", + "\n", + "Here I have introduced the standard notation $E[x]$ for the expected value of $x$.\n", + "\n", + "We can formalize this by letting $x_i$ be the $i^{th}$ value of $x$, and $p_i$ be the probability of its occurrence. This gives us\n", + "\n", + "$$E[X] = \\sum_{i=1}^n p_ix_i$$\n", + "\n", + "The value 1.5 for $x$ makes intuitive sense because x is far more like to be 1 than 3 or 5, and 3 is more likely than 5 as well.\n", + "\n", + "It is probably worth me pointing out now that if $x$ is continuous we substitute the sum for an integral, like so\n", + "\n", + "$$E[X] = \\int_{-\\infty}^\\infty xf(x)$$\n", + "\n", + "where $f(x)$ is the probability distribution function of $x$. We won't be using this equation yet, but we will be using it in the next chapter." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Computing the Covariance\n", + "\n", + "\n", + "Now we are prepared to compute the covariance matrix. The *covariance* measures how much two random variables move in the same direction, and is defined as\n", + "\n", + "$$ COV(x,y) = \\frac{1}{N}\\sum_{i=1}^N (x_i - \\mu_x)(y_i - \\mu_y)$$\n", + "\n", + "If we compare this to the formula for the variance of a single variable we can see where this definition came from.\n", + "\n", + "$$\\sigma^2 = \\frac{1}{N}\\sum_{i=1}^N(x_i - \\mu)^2$$\n", + "\n", + "And indeed, the covariance of a variable with itself is just the variance of the variable, which we can trivially prove with\n", + "\n", + "$$\\begin{aligned}\n", + "COV(x,x) &= \\frac{1}{N}\\sum_{i=1}^N (x_i - \\mu_x)(x_i - \\mu_x) \\\\\n", + "&= \\frac{1}{N}\\sum_{i=1}^N (x_i - \\mu_x)^2 \\\\\n", + "&= VAR(x)\n", + "\\end{aligned}$$\n", + "\n", + "\n", + "Let's build some familiarity with this by calculating the covariance for $\\mathbf{x}$ above. \n", + "\n", + "$$\\mathbf{x}_0 = \\begin{bmatrix}1 \\\\8\\\\3\\end{bmatrix},\\ \\ \\mathbf{x}_1 = \\begin{bmatrix}3\\\\7\\\\7\\end{bmatrix}$$\n", + "\n", + "We already know the mean is\n", + "\n", + "$$\\mu_x = \\begin{bmatrix}2\\\\7.5\\\\5\\end{bmatrix}$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Recall that we named the three variables as $a, b, c$. Let's start by computing the variance of $a$.\n", + "\n", + "$$\n", + "\\begin{aligned}\n", + "COV(a,b) &= \\frac{1}{2}[(1-2)^2 + (3-2)^2] \\\\\n", + "&= \\frac{1}{2}[(1+1] \\\\\n", + "&= 1\\end{aligned}$$\n", + "\n", + "\n", + "Using similar math we can compute that $VAR(a) = 0.25$ and $VAR(c)=4$. This allows us to fill in the covariance matrix with\n", + "\n", + "$$\\Sigma = \\begin{bmatrix}1 & & \\\\ & 0.25 & \\\\ &&4\\end{bmatrix}$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Now we can compute the covariance of a and b.\n", + "\n", + "$$\n", + "\\begin{aligned}\n", + "COV(a,b) &= \\frac{1}{2}[(1-2)(8-7.5) + (3-2)(7-7.5)] \\\\\n", + "&= \\frac{1}{2}[(-1)(.5)+ (1)(-.5)] \\\\\n", + "&= -0.5\\end{aligned}$$\n", + "\n", + "We can fill in the $a^{th}$ row and $b^{th}$ column in the covariance matrix with this value:\n", + "\n", + "$$\\Sigma = \\begin{bmatrix}1 & & \\\\ -0.5 & 0.25 & \\\\ &&4\\end{bmatrix}$$\n", + "\n", + "We already talked about the symmetry of the covariance matrix; $\\sigma_{ab}=\\sigma_{ba}$. We see now why this is true from the equation for the covariance.\n", + "\n", + "$$\\begin{aligned} COV(x,y) &= \\frac{1}{N}\\sum_{i=1}^N (x_i - \\mu_x)(y_i - \\mu_y) \\\\\n", + "&= \\frac{1}{N}\\sum_{i=1}^N (y_i - \\mu_y)(x_i - \\mu_x \\\\\n", + "&= COV(y,x)\\end{aligned}$$\n", + "\n", + "This gives the covariance matrix\n", + "\n", + "$$\\Sigma = \\begin{bmatrix}1 &-0.5 & \\\\ -0.5 & 0.25 & \\\\ &&4\\end{bmatrix}$$.\n", + "\n", + "The arithmetic is a bit tedious, so let's just use NumPy's `cov()` function to compute the entire covariance matrix. To compute the covariance in accordance with the equations above you will need to set the parameter `bias=1`. The meaning of that parameter is not important to this book. If you are interested, wikipedia has a good article on it here http://en.wikipedia.org/wiki/Bias_of_an_estimator." + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[[ 1. -0.5 2. ]\n", + " [-0.5 0.25 -1. ]\n", + " [ 2. -1. 4. ]]\n" + ] + } + ], + "source": [ + "x = np.array([[1, 3],[8, 7], [3, 7]])\n", + "print(np.cov(x, bias=1))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We said above that \"the *covariance* measures how much two random variables move in the same direction\", so let's see what that means in practice. We will do this with two variables so that we can plot the covariance matrix. If we start with the value $x_0 = (1, 2)$ we can create subsequent sames that vary in the same direction with $x_1 = (2, 4)$, $x_2=(3,6)$, and $x_3=(4,8). I don't feel like computing this by hand, so we will use NumPy to compute the covariance." + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[[ 1.25 2.5 ]\n", + " [ 2.5 5. ]]\n" + ] + }, + { + "data": { + "image/png": [ + "iVBORw0KGgoAAAANSUhEUgAAAtUAAAFwCAYAAAB3ioerAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", + "AAALEgAACxIB0t1+/AAAHBtJREFUeJzt3X1wVeXdLuBfEKGgdJfBQuUYxI7RKgUUAj2Gln4o2KC1\n", + "H1apHT+PlcEPitDXGcdi0RHFV2vPVCtgnY460r4j6AxtFSj2iKADokA9Vq2VU9tqB4MKISSZipbk\n", + "/OFrChJCkrV29l57X9cM48ra2et5nGcwd26ftXZFa2trawAAAN3Wq9ATAACArBOqAQAgIaEaAAAS\n", + "EqoBACAhoRoAABISqgEAICGhGgAAEko1VN99990xevToyOVykcvloqamJpYvX57mEAAAUHQq0vzw\n", + "l9/85jfRt2/fqKqqipaWlrj//vvjtttui+eeey5Gjx6d1jAAAFBUUg3V7Rk0aFDceuutcdlll+Vz\n", + "GAAAKJje+brwnj17YunSpfHuu+/GxIkT8zUMAAAUXOqh+o9//GOccsopsXv37ujXr18sWbIkjj/+\n", + 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+ } + ], + "source": [ + "x = np.array([[1, 2, 3, 4],[2, 4, 6, 8]])\n", + "cov=np.cov(x, bias=1)\n", + "stats.plot_covariance_ellipse([0,0], cov)\n", + "print(cov)" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "collapsed": true + }, + "source": [ + "The values for $x$ are perfectly correlated, and the result is a covariance ellipse with zero width. We can see this by making the values not slightly uncorrelated." + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "image/png": [ + "iVBORw0KGgoAAAANSUhEUgAAAtUAAAFwCAYAAAB3ioerAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", + "AAALEgAACxIB0t1+/AAAIABJREFUeJzt3Xd0VGXi//HPTHpCGEIwAST0gICUQAgkFAuCBhCkFxFx\n", + "LYsFERAXFVf3q4gFuyDK/nZ1BZGmURAQV2oILSC99xo6aaTP/P6YMMqCtJlwZybv1zmenblJ7v3s\n", + "uYfJh4fnPo/JZrPZBAAAAOCGmY0OAAAAAHg6SjUAAADgJEo1AAAA4CRKNQAAAOAkSjUAAADgJEo1\n", + "AAAA4CRKNQAAAOAkl5bq8ePHq3HjxrJYLLJYLEpISNDcuXNdeQkAAADA7ZhcufnLjz/+qICAAEVH\n", + "R8tqterLL7/UO++8ozVr1qhx48auugwAAADgVlxaqi8nPDxcb731lh5//PGSvAwAAABgGN+SOnFR\n", + "UZFmzJih3NxctW3btqQuAwAAABjO5aV606ZNio+PV15enoKCgjR9+nTVrVvX1ZcBAAAA3IbLp38U\n", + "FBTo0KFDSk9P14wZM/TJJ59o0aJFio2NlSSlp6e78nIAAADATWWxWC45VuJzqtu3b68qVaro3//+\n", + "tyRKNQAAADzb5Up1ia9TXVRUJKvVWtKXAQAAAAzj0jnVo0aNUufOnVWlShVlZmbqm2++0ZIlSzR/\n", + 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false + }, + "outputs": [ + { + "data": { + "image/png": [ + "iVBORw0KGgoAAAANSUhEUgAAAtUAAAFwCAYAAAB3ioerAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", + "AAALEgAACxIB0t1+/AAAIABJREFUeJzt3Xd4VFXixvF3Jo0khACBAKGXgNQACS00lSaISm8i4O7K\n", + "WlAE6yKW/SlFEAsqWNa10YuCIigqCIQUCL2DdIQECCEN0mbm98fAKCuGMhPuTPL9PA/PZm4yc173\n", + "PsjL8dxzTDabzSYAAAAAN81sdAAAAADA01GqAQAAACdRqgEAAAAnUaoBAAAAJ1GqAQAAACdRqgEA\n", + "AAAnUaoBAAAAJ7m0VL///vuKiIhQcHCwgoODFR0dreXLl7tyCAAAAMDtmFx5+Ms333wjPz8/hYeH\n", + "y2q16rPPPtOUKVO0ceNGRUREuGoYAAAAwK24tFRfTUhIiCZPnqyHHnqoMIcBAAAADONdWB9ssVi0\n", + "cOFCZWdnq0OHDoU1DAAAAGA4l5fqHTt2qE2bNsrJyZG/v78WLFigevXquXoYAAAAwG24fPlHXl6e\n", + "jh8/rrS0NC1cuFDvvvuuVq9eraioKElSWlqaK4cDAAAAbqng4OA/XSv0NdVdunRRlSpV9Omnn0qi\n", + "VAMAAMCzXa1UF/o+1RaLRVartbCHAQAAAAzj0jXVzz//vHr27KkqVaooIyNDc+bM0Zo1a/T9999f\n", + "9eev1vKvJTExUZIcy0ngGbhvnod75pm4b56He+aZuG+ex9l7dq3VFi4t1cnJyRo6dKiSkpIUHBys\n", + "iIgIff/99+rSpYsrhwEAAADciktL9eV10wAAAEBxUuhrqgEAAICijlINAAAAOIlSDQAAADiJUg0A\n", + 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That is, as one gets larger, the other becomes smaller." + ] + }, { "cell_type": "markdown", "metadata": {}, @@ -33726,7 +34632,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.4.2" + "version": "3.4.3" } }, "nbformat": 4,