Added animation for voltage.

Implemented an animation showing how the filter responds to
residuals as its confidence (P) gets better.

Chapter05_Kalman_Filters/Kalman_Filters.ipynb

@@ -1,7 +1,7 @@
 {
  "metadata": {
   "name": "",
-  "signature": "sha256:64e8545f18bda8e1f5d616e6ae32bb4de6b9d2facb49965254214130a9a105c2"
+  "signature": "sha256:eb5b8609726afd3ea3868dd795d6475e74e63d38a908e6972ad35ee3c6f0b2f1"
  },
  "nbformat": 3,
  "nbformat_minor": 0,
@@ -257,7 +257,7 @@
        "output_type": "pyout",
        "prompt_number": 1,
        "text": [
-        "<IPython.core.display.HTML at 0x7f9a41e44828>"
+        "<IPython.core.display.HTML at 0x7f209db34f98>"
        ]
       }
      ],
@@ -349,10 +349,10 @@
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "-1.8386\t 0.7546\t-0.5046\t-2.9576\t-0.2675\t\n",
-        " 1.4816\t-0.9532\t-0.8862\t-0.3574\t-0.4382\t\n",
-        " 0.1608\t 1.1827\t 0.4886\t 2.0902\t-0.2451\t\n",
-        "-0.4182\t 0.5224\t 0.1251\t 2.1121\t-0.6231\t\n"
+        " 0.0276\t 1.7666\t-0.6031\t 0.0895\t-0.9401\t\n",
+        " 0.2449\t-0.4231\t 0.2825\t 1.7339\t-2.7824\t\n",
+        " 1.1785\t-1.0472\t-0.3156\t 1.3697\t 1.1010\t\n",
+        " 0.8988\t-0.6833\t-0.1195\t-2.1915\t-2.8026\t\n"
        ]
       }
      ],
@@ -397,9 +397,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f9a28e15f98>"
+        "<matplotlib.figure.Figure at 0x7f2083ecbc18>"
        ]
       }
      ],
@@ -442,9 +442,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f9a28db9400>"
+        "<matplotlib.figure.Figure at 0x7f2083b76710>"
        ]
       }
      ],
@@ -476,9 +476,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f9a28bfb7f0>"
+        "<matplotlib.figure.Figure at 0x7f2083ecb470>"
        ]
       }
      ],
@@ -496,9 +496,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f9a28da1e48>"
+        "<matplotlib.figure.Figure at 0x7f2083ab2208>"
        ]
       }
      ],
@@ -539,9 +539,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f9a28d9ee10>"
+        "<matplotlib.figure.Figure at 0x7f2083a4d5c0>"
        ]
       }
      ],
@@ -578,9 +578,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f9a21695d30>"
+        "<matplotlib.figure.Figure at 0x7f2083ecbcf8>"
        ]
       }
      ],
@@ -675,9 +675,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7f9a215e8c50>"
+        "<matplotlib.figure.Figure at 0x7f207c43cf28>"
        ]
       }
      ],
@@ -723,9 +723,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f9a21634f28>"
+        "<matplotlib.figure.Figure at 0x7f207ab86a20>"
        ]
       }
      ],
@@ -773,9 +773,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f9a28dd1908>"
+        "<matplotlib.figure.Figure at 0x7f207ab044e0>"
        ]
       }
      ],
@@ -898,26 +898,26 @@
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "time: 0 \tposition = 0.827 \tvariance = 2.500\n",
-        "time: 1 \tposition = 0.427 \tvariance = 1.667\n",
-        "time: 2 \tposition = 0.192 \tvariance = 1.250\n",
-        "time: 3 \tposition = 0.548 \tvariance = 1.000\n",
-        "time: 4 \tposition = 0.416 \tvariance = 0.833\n",
-        "time: 5 \tposition = 0.409 \tvariance = 0.714\n",
-        "time: 6 \tposition = 0.297 \tvariance = 0.625\n",
-        "time: 7 \tposition = 0.490 \tvariance = 0.556\n",
-        "time: 8 \tposition = 0.407 \tvariance = 0.500\n",
-        "time: 9 \tposition = 0.291 \tvariance = 0.455\n",
-        "time: 10 \tposition = 0.120 \tvariance = 0.417\n",
-        "time: 11 \tposition = -0.043 \tvariance = 0.385\n",
-        "time: 12 \tposition = -0.082 \tvariance = 0.357\n",
-        "time: 13 \tposition = -0.130 \tvariance = 0.333\n",
-        "time: 14 \tposition = -0.208 \tvariance = 0.312\n",
-        "time: 15 \tposition = -0.144 \tvariance = 0.294\n",
-        "time: 16 \tposition = -0.075 \tvariance = 0.278\n",
-        "time: 17 \tposition = -0.073 \tvariance = 0.263\n",
-        "time: 18 \tposition = -0.030 \tvariance = 0.250\n",
-        "time: 19 \tposition = -0.045 \tvariance = 0.238\n"
+        "time: 0 \tposition = 1.519 \tvariance = 2.500\n",
+        "time: 1 \tposition = 0.769 \tvariance = 1.667\n",
+        "time: 2 \tposition = 1.024 \tvariance = 1.250\n",
+        "time: 3 \tposition = 0.897 \tvariance = 1.000\n",
+        "time: 4 \tposition = 0.705 \tvariance = 0.833\n",
+        "time: 5 \tposition = 0.665 \tvariance = 0.714\n",
+        "time: 6 \tposition = 0.479 \tvariance = 0.625\n",
+        "time: 7 \tposition = 0.486 \tvariance = 0.556\n",
+        "time: 8 \tposition = 0.528 \tvariance = 0.500\n",
+        "time: 9 \tposition = 0.353 \tvariance = 0.455\n",
+        "time: 10 \tposition = 0.218 \tvariance = 0.417\n",
+        "time: 11 \tposition = 0.280 \tvariance = 0.385\n",
+        "time: 12 \tposition = 0.344 \tvariance = 0.357\n",
+        "time: 13 \tposition = 0.506 \tvariance = 0.333\n",
+        "time: 14 \tposition = 0.394 \tvariance = 0.312\n",
+        "time: 15 \tposition = 0.358 \tvariance = 0.294\n",
+        "time: 16 \tposition = 0.338 \tvariance = 0.278\n",
+        "time: 17 \tposition = 0.334 \tvariance = 0.263\n",
+        "time: 18 \tposition = 0.346 \tvariance = 0.250\n",
+        "time: 19 \tposition = 0.306 \tvariance = 0.238\n"
        ]
       }
      ],
@@ -1037,24 +1037,24 @@
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "PREDICT:     1.0000   502.0000\tUPDATE:    -1.9271     9.8047\n",
-        "PREDICT:    -0.9271    11.8047\tUPDATE:     1.4652     5.4138\n",
-        "PREDICT:     2.4652     7.4138\tUPDATE:     0.8840     4.2574\n",
-        "PREDICT:     1.8840     6.2574\tUPDATE:     2.2629     3.8490\n",
-        "PREDICT:     3.2629     5.8490\tUPDATE:     5.3172     3.6904\n",
-        "PREDICT:     6.3172     5.6904\tUPDATE:     4.8154     3.6267\n",
-        "PREDICT:     5.8154     5.6267\tUPDATE:     6.7095     3.6007\n",
-        "PREDICT:     7.7095     5.6007\tUPDATE:     7.0910     3.5900\n",
-        "PREDICT:     8.0910     5.5900\tUPDATE:     9.0888     3.5856\n",
-        "PREDICT:    10.0888     5.5856\tUPDATE:    11.0984     3.5838\n"
+        "PREDICT:     1.0000   502.0000\tUPDATE:    -0.2825     9.8047\n",
+        "PREDICT:     0.7175    11.8047\tUPDATE:     1.3165     5.4138\n",
+        "PREDICT:     2.3165     7.4138\tUPDATE:     1.6450     4.2574\n",
+        "PREDICT:     2.6450     6.2574\tUPDATE:     3.7906     3.8490\n",
+        "PREDICT:     4.7906     5.8490\tUPDATE:     3.6658     3.6904\n",
+        "PREDICT:     4.6658     5.6904\tUPDATE:     5.7559     3.6267\n",
+        "PREDICT:     6.7559     5.6267\tUPDATE:     7.8662     3.6007\n",
+        "PREDICT:     8.8662     5.6007\tUPDATE:     8.4671     3.5900\n",
+        "PREDICT:     9.4671     5.5900\tUPDATE:    12.2074     3.5856\n",
+        "PREDICT:    13.2074     5.5856\tUPDATE:    11.9462     3.5838\n"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f9a2169e208>"
+        "<matplotlib.figure.Figure at 0x7f207c48ec18>"
        ]
       }
      ],
@@ -1176,7 +1176,7 @@
       "#plt.subplot(121)    \n",
       "p1, = plt.plot(zs,c='r', linestyle='dashed')\n",
       "p2, = plt.plot(ps,)\n",
-      "plt.legend([p1,p2], ['measurement', 'filter'], 2)\n",
+      "plt.legend([p1,p2], ['measurement', 'filter'], loc=2)\n",
       "plt.show()\n",
       "\n",
       "plt.plot(vs)\n",
@@ -1190,17 +1190,17 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f9a28d8c780>"
+        "<matplotlib.figure.Figure at 0x7f207ab1d828>"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f9a2161a278>"
+        "<matplotlib.figure.Figure at 0x7f207c444860>"
        ]
       },
       {
@@ -1412,17 +1412,17 @@
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f9a28dbdeb8>"
+        "<matplotlib.figure.Figure at 0x7f207c4272b0>"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
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        "text": [
-        "<matplotlib.figure.Figure at 0x7f9a2169e048>"
+        "<matplotlib.figure.Figure at 0x7f207c410898>"
        ]
       },
       {
@@ -1430,7 +1430,7 @@
        "stream": "stdout",
        "text": [
         "Variance converges to 0.09072976736236911\n",
-        "Last voltage is 16.160958715064073\n"
+        "Last voltage is 16.200756458088527\n"
        ]
       }
      ],
@@ -1444,7 +1444,33 @@
       "\n",
       "The second plot shows how the variance converges over time. Compare this plot to the variance plot for the dog sensor. While this does converge to a very small value, it is much slower than the dog problem. The next section **Explaining the Results - Multi-Sensor Fusion** explains why this happens.\n",
       "\n",
-      "##### Exercise(optional):\n",
+      "For those reading this in IPython Notebook, here is an animation showing the filter working. The animation draws a green line for the predicted next voltage, then a red '+' for the actual measurement, draws a light red line to show the residual, and then draws a blue line to the filter's output. You can see that when the filter starts the corrections made are quite large, but after only a few updates the filter only adjusts its output by a small amount even when the measurement is far from it. \n",
+      "\n",
+      "The plot above this text explains why this happens. When the filter is initialized we are not sure of the voltage, so the variance is high. So when we get a measurement the filter is willing to make a large adjustment to its output. However, as it keeps reading new values it quickly becomes much more confident in its output, and thus it will adjust its output only by a small amount, assuming that the large variance in the measurement is due to noise, not the filter's bad prediction.\n",
+      "\n",
+      "<img src='volt_animate.gif'>"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Think of this animation in terms of the g-h filter. At each step the g-h filter makes a prediction, takes a measurement, computes the residual (the difference between the prediction and the measurement, and then selects a point on the residual line based on the scaling factor *g*. The Kalman filter is doing exactly the same thing, except that the scaling factor *g* varies with time. As the filter becomes more confident in its state the scaling factor favors the filter's prediction over the measurement. \n",
+      "\n",
+      "> If this is not clear, I urge you to go back and review the g-h chapter. This is the crux of the algorithms in this book. "
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "##### Exercise(optional):"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
       "Write a function that runs the Kalman filter many times and record what value the voltage converges to each time. Plot this as a histogram. After 10,000 runs do the results look normally distributed? Does this match your intuition of what should happen?\n",
       "\n",
       "> use plt.hist(data,bins=100) to plot the histogram. "
@@ -1493,9 +1519,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f9a2163a518>"
+        "<matplotlib.figure.Figure at 0x7f207ab078d0>"
        ]
       }
      ],
@@ -1562,17 +1588,17 @@
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f9a28d8c198>"
+        "<matplotlib.figure.Figure at 0x7f207ab57e80>"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f9a28bd72e8>"
+        "<matplotlib.figure.Figure at 0x7f207c3d1eb8>"
        ]
       }
      ],
@@ -1648,9 +1674,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f9a20599a90>"
+        "<matplotlib.figure.Figure at 0x7f207c47c4e0>"
        ]
       }
      ],
@@ -1712,9 +1738,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7f9a216a1e10>"
+        "<matplotlib.figure.Figure at 0x7f207ab21b70>"
        ]
       }
      ],
@@ -1775,9 +1801,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f9a216e42e8>"
+        "<matplotlib.figure.Figure at 0x7f207ab21cc0>"
        ]
       }
      ],
@@ -1827,9 +1853,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f9a21680588>"
+        "<matplotlib.figure.Figure at 0x7f207a953588>"
        ]
       }
      ],
@@ -1936,9 +1962,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f9a2169e630>"
+        "<matplotlib.figure.Figure at 0x7f207a8827b8>"
        ]
       },
       {
@@ -2027,9 +2053,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f9a41e44780>"
+        "<matplotlib.figure.Figure at 0x7f207a935f60>"
        ]
       }
      ],
@@ -2112,9 +2138,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f9a205b3a90>"
+        "<matplotlib.figure.Figure at 0x7f207aa3ada0>"
        ]
       }
      ],
@@ -2127,7 +2153,7 @@
       "###### Discussion\n",
       "This is terrible! The output is not at all like a sin wave, except in the grossest way. With linear systems we could add extreme amounts of noise to our signal and still extract a very accurate result, but here even modest noise creates a very bad result.\n",
       "\n",
-      "Very shortly after practitioners began implementing Kalman filters they recognized the poor performance of them for nonlinear systems and began devising ways of dealing with it. Much of this book is devoted to this problem and its various solutions."
+      "Very shortly after practitioners began implementing Kalman filters they recognized the poor performance of them for nonlinear systems and began devising ways of dealing with it. Much of the remainder of this book is devoted to this problem and its various solutions."
      ]
     },
     {

Chapter05_Kalman_Filters/Kalman_Filters_Animations.ipynb

@@ -0,0 +1,363 @@
+{
+ "metadata": {
+  "name": "",
+  "signature": "sha256:6ae833f8c8c8b1ebf95d6a8830c42247f5580e91d57a284055fb49d7d1488489"
+ },
+ "nbformat": 3,
+ "nbformat_minor": 0,
+ "worksheets": [
+  {
+   "cells": [
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "from __future__ import division, print_function\n",
+      "import matplotlib.pyplot as plt\n",
+      "%matplotlib inline\n",
+      "import sys\n",
+      "sys.path.insert(0,'../code') # allow us to format the book\n",
+      "import book_format\n",
+      "book_format.load_style()"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "html": [
+        "<style>\n",
+        "@import url('http://fonts.googleapis.com/css?family=Source+Code+Pro');\n",
+        "\n",
+        "    div.cell{\n",
+        "        width: 850px;\n",
+        "        margin-left: 0% !important;\n",
+        "        margin-right: auto;\n",
+        "    }\n",
+        "    div.text_cell code {\n",
+        "        background: transparent;\n",
+        "        color: #000000;\n",
+        "        font-weight: 600;\n",
+        "        font-size: 11pt;\n",
+        "        font-style: bold;\n",
+        "        font-family:  'Source Code Pro', Consolas, monocco, monospace;\n",
+        "   }\n",
+        "    h1 {\n",
+        "        font-family: 'Open sans',verdana,arial,sans-serif;\n",
+        "\t}\n",
+        "\t\n",
+        "    div.input_area {\n",
+        "        background: #F6F6F9;\n",
+        "        border: 1px solid #586e75;\n",
+        "    }\n",
+        "\n",
+        "    .text_cell_render h1 {\n",
+        "        font-weight: 200;\n",
+        "        font-size: 30pt;\n",
+        "        line-height: 100%;\n",
+        "        color:#c76c0c;\n",
+        "        margin-bottom: 0.5em;\n",
+        "        margin-top: 1em;\n",
+        "        display: block;\n",
+        "        white-space: wrap;\n",
+        "    } \n",
+        "    h2 {\n",
+        "        font-family: 'Open sans',verdana,arial,sans-serif;\n",
+        "    }\n",
+        "    .text_cell_render h2 {\n",
+        "        font-weight: 200;\n",
+        "        font-size: 20pt;\n",
+        "        font-style: italic;\n",
+        "        line-height: 100%;\n",
+        "        color:#c76c0c;\n",
+        "        margin-bottom: 0.5em;\n",
+        "        margin-top: 1.5em;\n",
+        "        display: block;\n",
+        "        white-space: nowrap;\n",
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+        "        line-height: 100%;\n",
+        "        color:#d77c0c;\n",
+        "        margin-bottom: 0.5em;\n",
+        "        margin-top: 2em;\n",
+        "        display: block;\n",
+        "        white-space: nowrap;\n",
+        "    }\n",
+        "    h4 {\n",
+        "        font-family: 'Open sans',verdana,arial,sans-serif;\n",
+        "    }\n",
+        "    .text_cell_render h4 {\n",
+        "        font-weight: 300;\n",
+        "        font-size: 16pt;\n",
+        "        color:#d77c0c;\n",
+        "        margin-bottom: 0.5em;\n",
+        "        margin-top: 0.5em;\n",
+        "        display: block;\n",
+        "        white-space: nowrap;\n",
+        "    }\n",
+        "    h5 {\n",
+        "        font-family: 'Open sans',verdana,arial,sans-serif;\n",
+        "    }\n",
+        "    .text_cell_render h5 {\n",
+        "        font-weight: 300;\n",
+        "        font-style: normal;\n",
+        "        color: #1d3b84;\n",
+        "        font-size: 16pt;\n",
+        "        margin-bottom: 0em;\n",
+        "        margin-top: 1.5em;\n",
+        "        display: block;\n",
+        "        white-space: nowrap;\n",
+        "    }\n",
+        "    div.text_cell_render{\n",
+        "        font-family: 'Open sans',verdana,arial,sans-serif;\n",
+        "        line-height: 135%;\n",
+        "        font-size: 110%;\n",
+        "        width:750px;\n",
+        "        margin-left:auto;\n",
+        "        margin-right:auto;\n",
+        "        text-align:justify;\n",
+        "        text-justify:inter-word;\n",
+        "    }\n",
+        "    div.output_subarea.output_text.output_pyout {\n",
+        "        overflow-x: auto;\n",
+        "        overflow-y: scroll;\n",
+        "        max-height: 300px;\n",
+        "    }\n",
+        "    div.output_subarea.output_stream.output_stdout.output_text {\n",
+        "        overflow-x: auto;\n",
+        "        overflow-y: scroll;\n",
+        "        max-height: 300px;\n",
+        "    }\n",
+        "    code{\n",
+        "      font-size: 70%;\n",
+        "    }\n",
+        "    .rendered_html code{\n",
+        "    background-color: transparent;\n",
+        "    }\n",
+        "    ul{\n",
+        "        margin: 2em;\n",
+        "    }\n",
+        "    ul li{\n",
+        "        padding-left: 0.5em; \n",
+        "        margin-bottom: 0.5em; \n",
+        "        margin-top: 0.5em; \n",
+        "    }\n",
+        "    ul li li{\n",
+        "        padding-left: 0.2em; \n",
+        "        margin-bottom: 0.2em; \n",
+        "        margin-top: 0.2em; \n",
+        "    }\n",
+        "    ol{\n",
+        "        margin: 2em;\n",
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+        "    ol li{\n",
+        "        padding-left: 0.5em; \n",
+        "        margin-bottom: 0.5em; \n",
+        "        margin-top: 0.5em; \n",
+        "    }\n",
+        "    ul li{\n",
+        "        padding-left: 0.5em; \n",
+        "        margin-bottom: 0.5em; \n",
+        "        margin-top: 0.2em; \n",
+        "    }\n",
+        "    a:link{\n",
+        "       font-weight: bold;\n",
+        "       color:#447adb;\n",
+        "    }\n",
+        "    a:visited{\n",
+        "       font-weight: bold;\n",
+        "       color: #1d3b84;\n",
+        "    }\n",
+        "    a:hover{\n",
+        "       font-weight: bold;\n",
+        "       color: #1d3b84;\n",
+        "    }\n",
+        "    a:focus{\n",
+        "       font-weight: bold;\n",
+        "       color:#447adb;\n",
+        "    }\n",
+        "    a:active{\n",
+        "       font-weight: bold;\n",
+        "       color:#447adb;\n",
+        "    }\n",
+        "    .rendered_html :link {\n",
+        "       text-decoration: underline; \n",
+        "    }\n",
+        "    .rendered_html :hover {\n",
+        "       text-decoration: none; \n",
+        "    }\n",
+        "    .rendered_html :visited {\n",
+        "      text-decoration: none;\n",
+        "    }\n",
+        "    .rendered_html :focus {\n",
+        "      text-decoration: none;\n",
+        "    }\n",
+        "    .rendered_html :active {\n",
+        "      text-decoration: none;\n",
+        "    }\n",
+        "    .warning{\n",
+        "        color: rgb( 240, 20, 20 )\n",
+        "    } \n",
+        "    hr {\n",
+        "      color: #f3f3f3;\n",
+        "      background-color: #f3f3f3;\n",
+        "      height: 1px;\n",
+        "    }\n",
+        "    blockquote{\n",
+        "      display:block;\n",
+        "      background: #fcfcfc;\n",
+        "      border-left: 5px solid #c76c0c;\n",
+        "      font-family: 'Open sans',verdana,arial,sans-serif;\n",
+        "      width:680px;\n",
+        "      padding: 10px 10px 10px 10px;\n",
+        "      text-align:justify;\n",
+        "      text-justify:inter-word;\n",
+        "      }\n",
+        "      blockquote p {\n",
+        "        margin-bottom: 0;\n",
+        "        line-height: 125%;\n",
+        "        font-size: 100%;\n",
+        "      }\n",
+        "</style>\n",
+        "<script>\n",
+        "    MathJax.Hub.Config({\n",
+        "                        TeX: {\n",
+        "                           extensions: [\"AMSmath.js\"]\n",
+        "                           },\n",
+        "                tex2jax: {\n",
+        "                    inlineMath: [ ['$','$'], [\"\\\\(\",\"\\\\)\"] ],\n",
+        "                    displayMath: [ ['$$','$$'], [\"\\\\[\",\"\\\\]\"] ]\n",
+        "                },\n",
+        "                displayAlign: 'center', // Change this to 'center' to center equations.\n",
+        "                \"HTML-CSS\": {\n",
+        "                    styles: {'.MathJax_Display': {\"margin\": 4}}\n",
+        "                }\n",
+        "        });\n",
+        "</script>\n"
+       ],
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 15,
+       "text": [
+        "<IPython.core.display.HTML at 0x7f5980a89978>"
+       ]
+      }
+     ],
+     "prompt_number": 15
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "from matplotlib import animation\n",
+      "import matplotlib.pyplot as plt\n",
+      "\n",
+      "def animate(filename, func, frames, interval, figsize=(6.5, 6.5)):\n",
+      "    def forward(frame):\n",
+      "        # I don't know if it is a bug or what, but FuncAnimation calls twice\n",
+      "        # with the first frame number. That breaks any animation that uses\n",
+      "        # the frame number in computations\n",
+      "        if forward.first:\n",
+      "            forward.first = False\n",
+      "            return\n",
+      "        func(frame)\n",
+      "    \n",
+      "    fig = plt.figure(figsize=figsize)\n",
+      "    forward.first = True\n",
+      "    anim = animation.FuncAnimation(fig, forward, frames=frames, interval=interval)\n",
+      "    anim.save(filename, writer='imagemagick')"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 16
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "from __future__ import print_function, division\n",
+      "\n",
+      "import numpy.random as random\n",
+      "import math\n",
+      "\n",
+      "def update(mu1, var1, mu2, var2):\n",
+      "    mean = (var1*mu2 + var2*mu1) / (var1+var2)\n",
+      "    variance = 1 / (1/var1 + 1/var2)\n",
+      "    return (mean, variance)\n",
+      "\n",
+      "def predict(pos, variance, movement, movement_variance):\n",
+      "    return (pos + movement, variance + movement_variance)\n",
+      "\n",
+      "\n",
+      "sensor_error = 1.2**2\n",
+      "movement_error = 0\n",
+      "movement = 0\n",
+      "voltage = (19,20) #who knows what the first value is?\n",
+      "\n",
+      "volts = [14.44, 18.74, 17.21, 15.98, 16.76, 16.25, 16.01, 16.90, 15.41, 17.31,\n",
+      "        16.39, 17.96]\n",
+      "ps = [voltage[0]]\n",
+      "\n",
+      "\n",
+      "i = 0\n",
+      "pred = 0\n",
+      "def volt_animate(frame):\n",
+      "    global i, ps, voltage, Z, x, pred\n",
+      "    \n",
+      "    step = frame % 4\n",
+      "    plt.ylim([12,20])\n",
+      "\n",
+      "    if step == 1:\n",
+      "        Z = volts[i]\n",
+      "        i += 1\n",
+      "        \n",
+      "        plt.scatter(i, Z, marker='+', s=64, color='r')\n",
+      "\n",
+      "    elif step == 3:\n",
+      "        voltage = update(voltage[0], voltage[1], Z, sensor_error)\n",
+      "        ps.append(voltage[0])\n",
+      "        plt.plot(ps, c='b')\n",
+      "        \n",
+      "    elif step == 2:\n",
+      "        plt.plot ([i,i], [pred, Z], c='r', alpha=0.3)\n",
+      "\n",
+      "    else:\n",
+      "        prev = voltage[0]\n",
+      "        voltage = predict(voltage[0], voltage[1], movement, movement_error)\n",
+      "        pred = voltage[0]\n",
+      "        plt.plot ([i, i+1], [prev, pred], c='g') \n",
+      "        \n",
+      "N=12       \n",
+      "animate('volt_animate.gif', volt_animate, N*3, 600)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7f597c1a7198>"
+       ]
+      }
+     ],
+     "prompt_number": 40
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "<img src='volt_animate.gif'>"
+     ]
+    }
+   ],
+   "metadata": {}
+  }
+ ]
+}