Added interaction, better explanation

Added interactive plot of 2D gaussian, and explained the
position-velociy Gaussian better.

05-Multivariate-Gaussians.ipynb

@@ -28,6 +28,7 @@
        "@import url('http://fonts.googleapis.com/css?family=Source+Code+Pro');\n",
        "@import url('http://fonts.googleapis.com/css?family=Vollkorn');\n",
        "@import url('http://fonts.googleapis.com/css?family=Arimo');\n",
+       "@import url('http://fonts.googleapis.com/css?family=Fira_sans');\n",
        "\n",
        "    div.cell{\n",
        "        width: 900px;\n",
@@ -118,9 +119,9 @@
        "        white-space: nowrap;\n",
        "    }\n",
        "    div.text_cell_render{\n",
-       "        font-family: 'Arimo',verdana,arial,sans-serif;\n",
+       "        font-family: 'Fira sans', verdana,arial,sans-serif;\n",
        "        line-height: 125%;\n",
-       "        font-size: 120%;\n",
+       "        font-size: 115%;\n",
        "        text-align:justify;\n",
        "        text-justify:inter-word;\n",
        "    }\n",
@@ -338,7 +339,7 @@
     "\n",
     "A couple of examples. Generally speaking as the square footage of a house increases the price increases. These variables are correlated. As the temperature of an engine increases its life expectancy lowers. These are **inversely 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",
     "\n",
-    "Correlation implies *prediction*. If our houses are in the same neighborhood, and you have twice the square footage I can predict that the price is likely to be higher. 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 grocery I don't conclude that tea prices will be increasing.\n",
+    "Correlation implies *prediction*. If our houses are in the same neighborhood, and you have twice the square footage I can predict that the price is likely to be higher. 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 I don't conclude that tea prices will be increasing.\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 say\n",
     "\n",
@@ -381,7 +382,7 @@
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x439c2e8>"
+       "<matplotlib.figure.Figure at 0x436c8d0>"
       ]
      },
      "metadata": {},
@@ -545,7 +546,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x439c198>"
+       "<matplotlib.figure.Figure at 0x436c7f0>"
       ]
      },
      "metadata": {},
@@ -563,7 +564,7 @@
    "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 $x$ 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. On the sides of the plot I have drawn the Gaussians for several different variances for $x$ in greens and for $y$ in orange."
    ]
   },
   {
@@ -582,9 +583,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x853f1d0>"
+       "<matplotlib.figure.Figure at 0x798eac8>"
       ]
      },
      "metadata": {},
@@ -621,7 +622,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8980128>"
+       "<matplotlib.figure.Figure at 0x7c12748>"
       ]
      },
      "metadata": {},
@@ -663,7 +664,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x91b9710>"
+       "<matplotlib.figure.Figure at 0x833b898>"
       ]
      },
      "metadata": {},
@@ -696,7 +697,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8980320>"
+       "<matplotlib.figure.Figure at 0x831deb8>"
       ]
      },
      "metadata": {},
@@ -712,7 +713,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Thinking about the physical interpretation of these plots clarifies their meaning. The mean and covariance of the fist plot is\n",
+    "Thinking about the physical interpretation of these plots clarifies their meaning. The mean and covariance of the first plot is\n",
     "\n",
     "$$\n",
     "\\mathbf{\\mu} =\\begin{bmatrix}2\\\\7\\end{bmatrix},\\, \\,\n",
@@ -776,7 +777,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7cceb38>"
+       "<matplotlib.figure.Figure at 0x7b39438>"
       ]
      },
      "metadata": {},
@@ -811,7 +812,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x89edc88>"
+       "<matplotlib.figure.Figure at 0x7b24cc0>"
       ]
      },
      "metadata": {},
@@ -854,7 +855,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x91c7e80>"
+       "<matplotlib.figure.Figure at 0x8213f98>"
       ]
      },
      "metadata": {},
@@ -901,14 +902,15 @@
    "cell_type": "code",
    "execution_count": 16,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "scrolled": true
    },
    "outputs": [
     {
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8107f28>"
+       "<matplotlib.figure.Figure at 0x822ee48>"
       ]
      },
      "metadata": {},
@@ -939,7 +941,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x89b6fd0>"
+       "<matplotlib.figure.Figure at 0x87f5ba8>"
       ]
      },
      "metadata": {},
@@ -972,7 +974,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x941f1d0>"
+       "<matplotlib.figure.Figure at 0x9008748>"
       ]
      },
      "metadata": {},
@@ -1010,7 +1012,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x89b26d8>"
+       "<matplotlib.figure.Figure at 0x92ab588>"
       ]
      },
      "metadata": {},
@@ -1041,7 +1043,7 @@
      "data": {
       "image/png": "iVBORw0KGgoAAAANSUhEUgAAAuAAAAEWCAYAAAAn/SKQAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa6eebe0>"
+       "<matplotlib.figure.Figure at 0x9319630>"
       ]
      },
      "metadata": {},
@@ -1079,9 +1081,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa716be0>"
+       "<matplotlib.figure.Figure at 0x9006080>"
       ]
      },
      "metadata": {},
@@ -1089,10 +1091,58 @@
     }
    ],
    "source": [
-    "P1 = [[2, 1.9], [1.9, 8]]\n",
+    "P1 = [[2, 1.9], [1.9, 8.4]]\n",
     "plot_covariance_ellipse((10, 10), P1, facecolor='y', alpha=0.6)"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The relationships can be hard to puzzle out by inspection, so here is an interactive plot. (If you are reading this in a static form instructions to run this online are here: http://git.io/vmv6e)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 22,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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truncated
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x906c358>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "from IPython.html.widgets import interact, interactive, fixed\n",
+    "from IPython.html.widgets import FloatSlider\n",
+    "\n",
+    "def plot_covariance(var_x, var_y, cov_xy):\n",
+    "    P1 = [[var_x, cov_xy], [cov_xy, var_y]]\n",
+    "    try:\n",
+    "        plot_covariance_ellipse((10, 10), P1, axis_equal=False,\n",
+    "                                show_semiaxis=True)\n",
+    "    except:\n",
+    "        plot_covariance_ellipse((10, 10), P1, axis_equal=False)\n",
+    "        \n",
+    "    plt.xlim(4, 16)\n",
+    "    plt.gca().set_aspect('equal')\n",
+    "    plt.ylim(4, 16)\n",
+    "    plt.show()\n",
+    "\n",
+    "interact (plot_covariance,           \n",
+    "          var_x=FloatSlider(value=5., min=0, max=20.), \n",
+    "          var_y=FloatSlider(value=5., min=0., max=20.), \n",
+    "          cov_xy=FloatSlider(value=1.5, min=0.0, max=50, step=.2));"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},
@@ -1108,7 +1158,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 22,
+   "execution_count": 23,
    "metadata": {
     "collapsed": false
    },
@@ -1117,7 +1167,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa731208>"
+       "<matplotlib.figure.Figure at 0x92d0a58>"
       ]
      },
      "metadata": {},
@@ -1141,7 +1191,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 23,
+   "execution_count": 24,
    "metadata": {
     "collapsed": false
    },
@@ -1150,7 +1200,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa699668>"
+       "<matplotlib.figure.Figure at 0x7ab5160>"
       ]
      },
      "metadata": {},
@@ -1190,7 +1240,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 24,
+   "execution_count": 25,
    "metadata": {
     "collapsed": false
    },
@@ -1199,7 +1249,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa7560f0>"
+       "<matplotlib.figure.Figure at 0x933b4e0>"
       ]
      },
      "metadata": {},
@@ -1220,7 +1270,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 25,
+   "execution_count": 26,
    "metadata": {
     "collapsed": false
    },
@@ -1229,7 +1279,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa7562e8>"
+       "<matplotlib.figure.Figure at 0x933b4a8>"
       ]
      },
      "metadata": {},
@@ -1255,7 +1305,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 26,
+   "execution_count": 27,
    "metadata": {
     "collapsed": false
    },
@@ -1264,7 +1314,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x853fcf8>"
+       "<matplotlib.figure.Figure at 0x7a1c320>"
       ]
      },
      "metadata": {},
@@ -1284,7 +1334,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 27,
+   "execution_count": 28,
    "metadata": {
     "collapsed": false
    },
@@ -1293,7 +1343,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x91a6fd0>"
+       "<matplotlib.figure.Figure at 0x7a31e48>"
       ]
      },
      "metadata": {},
@@ -1308,12 +1358,12 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "This implies that our velocity is roughly 5 m/s. But of course position and velocity are correlated. If the velocity is 5 m/s the position would be 5, but if the velocity was 10 m/s the position would be 10. So let's draw a velocity covariance matrix in red."
+    "This implies that our velocity is roughly 5 m/s. But of course position and velocity are correlated. If the velocity is 5 m/s the position would be 5, but if the velocity was 10 m/s the position would be 10. So let's draw a covariance matrix in red showing the relationship between the position and velocity."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 28,
+   "execution_count": 29,
    "metadata": {
     "collapsed": false
    },
@@ -1322,7 +1372,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa679470>"
+       "<matplotlib.figure.Figure at 0x83492e8>"
       ]
      },
      "metadata": {},
@@ -1337,12 +1387,14 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
+    "It won't be clear until the next chapter how I calculate this covariance. Ignore the calculation, and think about what this implies. We have no easy way to say where the object really is because we are so uncertain about the velocity. Hence the ellipse stretches very far in the x-axis. Our uncertainty in velocity of course means it is also very spread in the y-axis. But as I said in the last paragraph, position and velocity is correlated. *If* the velocity is 5 m/s the next position would be 5, and *if* the velocity is 10 the next position would be 10. They are extremely correlated, so the ellipse must be very narrow. \n",
+    "\n",
     "This superposition of the two covariances is where the magic happens. The only reasonable estimate at time t=1 (where position=5) is roughly the *intersection* between the two covariance matrices! More exactly, we can use the math from the last section and *multiply* the two covariances together. From a Bayesian point of view we multiply the prior with the evidence to get the posterior. If we multiply the position covariance with the velocity covariance using the Bayesian equations we get the result shown in the next chart."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 29,
+   "execution_count": 30,
    "metadata": {
     "collapsed": false
    },
@@ -1351,7 +1403,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x808f0b8>"
+       "<matplotlib.figure.Figure at 0x7ab1a58>"
       ]
      },
      "metadata": {},

Interactions.ipynb

@@ -53,7 +53,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 1,
+   "execution_count": 2,
    "metadata": {
     "collapsed": false,
     "scrolled": true
@@ -61,9 +61,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAARAAAAEoCAYAAACQI67VAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7129400>"
+       "<matplotlib.figure.Figure at 0x7407940>"
       ]
      },
      "metadata": {},
@@ -80,7 +80,7 @@
     "import matplotlib.pyplot as plt\n",
     "from matplotlib.patches import Ellipse\n",
     "\n",
-    "def plot_covariance_ellipse(x, P, edgecolor='k', ls='solid'):\n",
+    "def plot_covariance_ellipse(x, P, edgecolor='k'):\n",
     "    U,s,v = linalg.svd(P)\n",
     "    angle = math.atan2(U[1,0],U[0,0])\n",
     "    width  = math.sqrt(s[0]) * 2\n",
@@ -89,7 +89,7 @@
     "    ax = plt.gca()\n",
     "    e = Ellipse(xy=(0, 0), width=width, height=height, angle=angle,\n",
     "                edgecolor=edgecolor, facecolor='none',\n",
-    "                lw=2, ls=ls)\n",
+    "                lw=2, ls='solid')\n",
     "    ax.add_patch(e)\n",
     "    ax.set_aspect('equal')\n",
     "    \n",
@@ -118,18 +118,89 @@
     "         covar=widgets.FloatSliderWidget(value=0, min=0, max=1.));"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Covariance Ellipse\n",
+    "\n",
+    "See the effect of varying the variances and covariance of a covariance matrix of the form\n",
+    "\n",
+    "$$\\begin{bmatrix}\\texttt{var}_x & \\texttt{cov}_xy \\\\ \\texttt{cov}_xy & \\texttt{var}_y\\end{bmatrix}$$"
+   ]
+  },
   {
    "cell_type": "code",
-   "execution_count": 5,
+   "execution_count": 24,
    "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 0x7f660f0>"
+       "<matplotlib.figure.Figure at 0x7754cc0>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "from IPython.html.widgets import interact, interactive, fixed\n",
+    "from IPython.html.widgets import FloatSlider\n",
+    "from math import cos, sin, pi\n",
+    "def plot_covariance_ellipse(P):\n",
+    "    U,s,v = linalg.svd(P)\n",
+    "    angle = math.atan2(U[1,0],U[0,0])\n",
+    "    width  = math.sqrt(s[0]) * 2\n",
+    "    height = math.sqrt(s[1]) * 2\n",
+    "\n",
+    "    ax = plt.gca()\n",
+    "    e = Ellipse(xy=(0, 0), width=width, height=height, angle=angle,\n",
+    "                edgecolor='k', facecolor='none',\n",
+    "                lw=2, ls='solid')\n",
+    "    ax.add_patch(e)\n",
+    "    h, w = height/4, width/4\n",
+    "    plt.plot([0, h*cos(angle+pi/2)], [0, h*sin(angle+pi/2)])\n",
+    "    plt.plot([0, w*cos(angle)],      [0, w*sin(angle)])\n",
+    "\n",
+    "def plot_covariance(var_x, var_y, cov_xy):\n",
+    "    P = [[var_x, cov_xy], [cov_xy, var_y]]\n",
+    "    plot_covariance_ellipse(P)\n",
+    "    plt.xlim(-6, 6)\n",
+    "    plt.gca().set_aspect('equal')\n",
+    "    plt.ylim(-6, 6)\n",
+    "    plt.show()\n",
+    "\n",
+    "interact (plot_covariance,           \n",
+    "          var_x=FloatSlider(value=5., min=0, max=20.), \n",
+    "          var_y=FloatSlider(value=5., min=0., max=20.), \n",
+    "          cov_xy=FloatSlider(value=1.5, min=0.0, max=50, step=.2));\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# g-h Filter\n",
+    "\n",
+    "Experiment with various values for g-h filter parameters."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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truncated
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x6e3cc18>"
       ]
      },
      "metadata": {},