Wholesale changes to connect chapters together.

I made a lot of changes so that each chapter makes clear that
they are all implementing the same basic bayesian algorithm.

This required a lot of editting, and it doesn't make sense to
try to do that atomically, hence this huge check in.

I made a lot of edits, and haven't copy editted anything. i'm
sure I introduced a lot of problems and discontinuities.

00_Preface.ipynb

@@ -570,7 +570,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.4.3"
+   "version": "3.4.1"
   }
  },
  "nbformat": 4,

01_g-h_filter.ipynb

@@ -30,7 +30,7 @@
        "@import url('http://fonts.googleapis.com/css?family=Arimo');\n",
        "\n",
        "    div.cell{\n",
-       "        width: 850px;\n",
+       "        width: 900px;\n",
        "        margin-left: 0% !important;\n",
        "        margin-right: auto;\n",
        "    }\n",
@@ -237,7 +237,10 @@
        "                },\n",
        "                displayAlign: 'center', // Change this to 'center' to center equations.\n",
        "                \"HTML-CSS\": {\n",
-       "                    scale:85,\n",
+       "                    scale:100,\n",
+       "                    availableFonts: [\"Neo-Euler\"],\n",
+       "                    preferredFont: \"Neo-Euler\",\n",
+       "                    webFont: \"Neo-Euler\",\n",
        "                    styles: {'.MathJax_Display': {\"margin\": 4}}\n",
        "                }\n",
        "        });\n",
@@ -321,7 +324,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7489208>"
+       "<matplotlib.figure.Figure at 0x6e67400>"
       ]
      },
      "metadata": {},
@@ -356,7 +359,7 @@
      "data": {
       "image/png": "iVBORw0KGgoAAAANSUhEUgAAAxsAAAB6CAYAAADXnk6rAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x75479e8>"
+       "<matplotlib.figure.Figure at 0x6e67dd8>"
       ]
      },
      "metadata": {},
@@ -388,7 +391,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7c57d30>"
+       "<matplotlib.figure.Figure at 0x74b3f60>"
       ]
      },
      "metadata": {},
@@ -424,7 +427,7 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Average of measurements is 165.0539\n"
+      "Average of measurements is 164.9881\n"
      ]
     }
    ],
@@ -455,7 +458,7 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Average of measurements is 165.0597\n"
+      "Average of measurements is 165.0003\n"
      ]
     }
    ],
@@ -488,7 +491,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x560b240>"
+       "<matplotlib.figure.Figure at 0x74d9240>"
       ]
      },
      "metadata": {},
@@ -526,7 +529,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x560b400>"
+       "<matplotlib.figure.Figure at 0x74d9128>"
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      },
      "metadata": {},
@@ -555,7 +558,7 @@
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x560b438>"
+       "<matplotlib.figure.Figure at 0x762ec50>"
       ]
      },
      "metadata": {},
@@ -586,7 +589,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x55fcc88>"
+       "<matplotlib.figure.Figure at 0x76a04a8>"
       ]
      },
      "metadata": {},
@@ -616,9 +619,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA48AAAEtCAYAAACoIlXDAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7e1e320>"
+       "<matplotlib.figure.Figure at 0x7716400>"
       ]
      },
      "metadata": {},
@@ -653,7 +656,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7e1bfd0>"
+       "<matplotlib.figure.Figure at 0x76a00f0>"
       ]
      },
      "metadata": {},
@@ -682,7 +685,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7e1bcf8>"
+       "<matplotlib.figure.Figure at 0x7515c50>"
       ]
      },
      "metadata": {},
@@ -724,7 +727,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7e1b780>"
+       "<matplotlib.figure.Figure at 0x74d7e48>"
       ]
      },
      "metadata": {},
@@ -782,7 +785,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x751def0>"
+       "<matplotlib.figure.Figure at 0x6ee7908>"
       ]
      },
      "metadata": {},
@@ -859,7 +862,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8152ba8>"
+       "<matplotlib.figure.Figure at 0x7607668>"
       ]
      },
      "metadata": {},
@@ -901,7 +904,7 @@
      "data": {
       "image/png": 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       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7cbac88>"
+       "<matplotlib.figure.Figure at 0x88669e8>"
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      "metadata": {},
@@ -980,7 +983,7 @@
    "source": [
     "This filter is the basis for a huge number of filters, including the Kalman filter. In other words, the Kalman filter is a form of the g-h filter, which I will prove later in the book. So is the Least Squares filter, which you may have heard of, and so is the Benedict-Bordner filter, which you probably have not. Each filter has a different way of assigning values to *g* and *h*, but otherwise the algorithms are identical. For example, the $\\alpha$-$\\beta$ filter just assigns a constant to *g* and *h*, constrained to a certain range of values. Other filters such as the Kalman will vary *g* and *h* dynamically at each time step.\n",
     "\n",
-    "**Let me repeat the key points as they are so important**. If you do not understand these you will not understand the rest of the book. If you do understand them, then the rest of the book will unfold naturally for you as mathematically elaborations to various 'what if' questions we will ask about *g* and *h*.\n",
+    "**Let me repeat the key points as they are so important**. If you do not understand these you will not understand the rest of the book. If you do understand them, then the rest of the book will unfold naturally for you as mathematically elaborations to various 'what if' questions we will ask about *g* and *h*. The math may look profoundly different, but the algorithm will be exactly the same.\n",
     "\n",
     "* Multiple data points are more accurate than one data point, so throw nothing away no matter how inaccurate it is.\n",
     "* Always choose a number part way between two data points to create a more accurate estimate.\n",
@@ -1002,7 +1005,7 @@
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       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7d7bcf8>"
+       "<matplotlib.figure.Figure at 0x75000f0>"
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      "metadata": {},
@@ -1390,7 +1393,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7dd5630>"
+       "<matplotlib.figure.Figure at 0x885b860>"
       ]
      },
      "metadata": {},
@@ -1482,9 +1485,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA5EAAAEPCAYAAADbIUClAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA5EAAAEUCAYAAACyMLL9AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\nAAALEgAACxIB0t1+/AAAIABJREFUeJzs3XlcV3W+x/HX78cuICAKiOyyoyIIqbmvLU63zTana9ak\nd2aclqmZyqZxaTPLabrTVDM102bpzayptDK1XFI0UcENEFAWcUFUQFH237l//JJCVhVQ8f18PHgE\n53y/3/M5HQXenu/5HpNhGAYiIiIiIiIirWC+2AWIiIiIiIjI5UMhUkRERERERFpNIVJERERERERa\nTSFSREREREREWk0hUkRERERERFpNIVJERERERERaTSFSREREREREWq3ZEPnaa68RGxuLm5sbbm5u\nXH311Xz11Vf12syePZtevXrRpUsXRo0aRVpaWrsWLCIiIiIiIhdPsyHS39+fF198kZSUFLZu3cro\n0aO56aab2L59OwDz5s3j5Zdf5u9//zvJycl4eXkxbtw4ysrKOqR4ERERERER6VgmwzCMc+ng6enJ\nCy+8wP3334+vry8PPvggM2bMAKCiogIvLy/mz5/PtGnT2qVgERERERERuXhsW9uwtraWjz/+mIqK\nCoYPH05OTg6FhYWMHz++ro2joyPDhw8nKSmpQYgsLS1tu6pFRERERDqYm5vbxS5B5JLQYojcuXMn\ngwcPprKyEicnJxYvXkxERARJSUkAeHt712vv5eXFwYMH26daERERERERuahaDJGRkZHs2LGD0tJS\nPv74Y+68805Wr17dbB+TydRmBYqIiIiIiMilo8UQaWdnR0hICABxcXEkJyfz2muvMXPmTAAKCwvx\n8/Ora19YWIiPj0+zY2ZlZV1IzSIiIiJyhbE7fBi/v/+d8t69OXzvvR123LCwsA47lsjlotXPRJ5R\nW1uLxWIhODgYHx8fVqxYwYABAwDrwjrr169n/vz5zY6RkJBwftXKJWHLli2ArmNnoGvZOeg6dh66\nlp2HrmUbOn0aXnoJ5s2D8nLw9MRv/nxwcuqQw2tdD5GGmg2RTzzxBL/4xS/w8/Pj5MmTLFy4kLVr\n17J8+XIAHn74YZ5//nkiIyMJCwvj2WefxdXVlUmTJnVI8SIiIiLSSRkG/N//weOPw/791m233QYv\nvthhAVJEGtdsiCwsLOTuu+/m8OHDuLm5ERsby/Llyxk3bhwAjz32GOXl5UyfPp3i4mIGDRrEihUr\ncHZ27pDiRURERKSTMpng/fetATIuDl55BYYPv9hViQgthMh33nmnxQFmzZrFrFmz2qwgEREREREA\nXn4ZJk6EKVPAxuZiVyMiPzrnZyJFRERERNqUYVjvPJ4tKsr6ISKXFPPFLkBERERErlCGAZ98AjEx\nkJt7XkOUlB2j+GRR29YlIs1SiBQRERGRjrd9O4webZ2ump4Or756Tt3zDmfy3vKXmf3ONL7e9H/t\nVKSINEbTWUVERESk4xw9Cn/6E/zrX2CxgKcnPP00TJvWYtdaSy3bszeyJnUpuYf21G3fsmcdNwyZ\njGsXt/asXER+pBApIiIiIh2npATefRfMZnjgAZg1Czw8mu1yquIkG3et5PvtX1FcdrTB/praapJ2\nfcM1V93eTkWLyM8pRIqIiIhIxwkNhbfegquugsjIZpsWHi9gbeoyNqevpqqmssl2Xu6+dOvq1daV\nikgTFCJFREREpGNNntzkLsMwyMhPZU3KUtLztjU7TGRAf0bG3UBkYBxmk5b6EOkoCpEiIiIictFV\nVVeSnLGGNalLKTxe0GQ7O1t7roocxfD+v6Cnp38HVigiZyhEioiIiEj72LwZMjKavfNYfLKI77d/\nTdKuFZyuLGuynZuLJ8P7Xc/Vfcbh7NS1PaoVkVZSiBQRERGRtnf0qPX1Hfv3g6Mj3F5/0ZucQ3tY\nm7qU1KwkLIalyWECfcIZ2f8G+ocOxsZGv7qKXAr0N1FERERE2lZtLUyaZA2QgwfDTTf9uLmG1B9f\n0ZF3OLPJ7maTmf5hQxjR/xcE94zoqKpFpJUUIkVERESkbc2eDStXQo8esHgxJ6pPs2n7F3y/42tK\ny4412a2LgwtX972GYf2uxcO1R8fVKyLnRCFSRERERNrOsmXw7LMYZjN5rzzNt6kfsDMnGYultsku\n3t38GNn/BhIjR2Jv59CBxYrI+VCIFBEREZE2U9rDDZuAnmxM9GNp0XIoarptdGA8I+JuIDKgPyaT\nqeOKFJELohApIiIiIhektraG3blb2bhrJWl527CbHk+1nU2jbe1tHbgqejQjYifg3c2vgysVkbag\nECkiIiIi5+Vo6WE27lrJD+nfceJUcd32KoeGv2J6e/gxuM84BkWPoYujS0eWKSJtTCFSRERERFqt\nuqaanft+IGnXCjL372i2rZ2NPXHhQxgcM44Q3yhNWRXpJBQiRURERKRFh4/vZ+OulWzOWMOp8hN1\n253LKjnlUn8xnF7dgxjcZzwJkcPp4qC7jiKdjUKkiIiIiDSqqrqS1OwkknatYN/B9Ab7ncsq+eNf\n1rInogdf3HkVsX1GcXWf8fh79dZdR5FOTCFSREREROopKNpH0q6VbM1YS3nV6UbbmCwGkxdspVtx\nOX1OORF/31s4uLp3cKUicjEoRIqIiIhc4SqrKzh4NI/8wiyS09eQfyS72fZODs7ck1RK1J4i6N4d\n1y9XggKkyBVDIVJERETkclJbCzaNvz6jJYZhUFJ2jANFORw4msuBozkcLMqlqOQQBkaL/Xv3iuHq\nPuPon1mC3YKbwWyGRYvA3/+86hGRy5NCpIiIiMilzGKBHTtg1Srrx+bNcOAAODk1262mtprC4wUU\n/BgYDxblUHA0l9MVJ8/p8C5OblwVNYrBfcbh7dELDAPuH2bd+cwzMHbs+Z6ZiFymFCJFRERELlW/\n/jV88gkcPVp/+7ZtMGRI3Zdl5Sfq7i4ePJpLQVEOhccLqLXUnNdhTZiICIhlcJ/x9A1JxNbG7mc7\nTfD11/Dmm/D735/X+CJyeVOIFBEREblUFRVZA6SfH4wbB2PHUpzYjxyjhINJH9TdZSwtO4bJYmCY\nsIa8c2TCRHf3nvTqHoS/V2/iw4fi6ebddAdXV3j00fM/LxG5rClEioiIiHS08nLYsME6PfXbb2H6\ndJgypWG7OXNg7lxO+fuQkpVEcsYacr5a3OiQ41ZlEp51lM/+K4YC/6YXubG3dcC3exC9ugfRq0cw\nvXoE4esZiIN989NjRUTOUIgUERER6SirVsELL8D69VBZ+dP2lSsbhMia2mrSnMpITl/DrhVbqK1t\nemqqudbC4I15eBaX89hf1pI8wI9lE6IwAvzp1T24Liz26h5EdzcfzObzW5hHRAQUIkVEREQ6TkWF\n9c4jQFycdVGasWNh6FDAunpq7uFMkjPWkJK5nlOtXATHZGfPghcmM35lJpGff0/i1gISdhVhevBB\nuOcJsL2AX/mOH4fnn4enn4YuXc5/HBHpNJr9jjJ37lw+/fRTMjMzcXBwYNCgQcydO5eYmJi6NlOm\nTOH999+v12/QoEEkJSW1T8UiIiIil7LKSpy3b+dUbGzDfSNHwkcfwahR0KNH3eZjpYUk/7CU5Iy1\nFJUcbPEQQT4RBPeMsN5h7B6Md7de1sVvpgG5ufCnP2FauBB27rywAGmxwN13WxfSOXYM3nnn/McS\nkU6j2e8qa9eu5Xe/+x2JiYlYLBZmzpzJ2LFjSUtLw8PDAwCTycS4ceNYsGBBXT97e/v2rVpERETk\nUmSxwD33ELFkCTlPPw0JCfX3u7jA7bcDcLqyjNSsJJLT17D3YFqLQ/dw60lC1EgSI0fQ3c2n6YZB\nQfDhh9aVU11dL+BksL7C4+uvwdPT+nymiAgthMjly5fX+3rBggW4ubmRlJTEhAkTAOu0C3t7e7y8\nvNqvShEREZFLnWHAI4/ARx9hODtTERjYoEltbQ1pedtIzljDrn3J1NRWNztkFwcX4sOHkhg1kiCf\nCEznsvLq2QH25/bvB3//5vsvX24NjiYTLFwIAQGtP7aIdGrnNL/hxIkTWCyWuruQYL0TuX79ery9\nvXF3d2fEiBE899xz9PjZFA0RERGRTu/FF+F//xfs7cl+6SXKIyIA6z+45xdmk5yxhq2Z33Oq/ESz\nw9iYbYkJTiAxciTRQQOws7Vrtv05y8uDiAi47jrrIj8/1tmgzS9/aQ3GzzwD48e3bQ0iclkzGYZh\ntLbx7bffzt69e9myZUvdv4R99NFHODs7ExwcTE5ODk899RS1tbVs3bq13rTW0tLSus+zsrLa8BRE\nRERELi7PZcsInjMHw2Ri33PPUTxuHGUVJeQU7WJv0U5OlB9rcYwern6E9OhLYPcoHO3abwEb99Wr\nCZ45E5uKCgwbG4puvpmDU6dS061bXRtzeTmBc+dic/Ik2X/5C5jN7VbPpS4sLKzuczc3t4tYicil\no9Uh8pFHHmHx4sWsX7+eoKCgJtsdOnSIwMBAPvroI26++ea67QqRIiIi0lm5bNtG6KOPsn/a/Wwe\n05fswu0UnshruZ+DOyFefQnp0ZeuTt1abN9W7I4exfef/6T7F19gsliodXYmZ/ZsSkaO/KmRYWCq\nqsJwcOiwui5FCpEiDbVqOuvvf/97Fi9ezOrVq5sNkAA9e/bEz8+P7OzsJtskNDdHXy55W7ZsAXQd\nOwNdy85B17Hz0LW8PBmGQW4vV5Z2e5Kkoh1UZuU0297JwZm4sCEkRo4kxDfq3J5zbEvXXgu7d8Pj\nj2OzYgWht9wCISEXp5ZL2M9vhIiIVYsh8qGHHuLjjz9m9erVhIeHtzhgUVERBw4coGfPnm1SoIiI\niFzB/vUvWL0a5s695BZ2OXGqhOSMNWxKW0Xh8YJm25rNNkQHDSAxciR9ghOws71EVLine truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7d65e10>"
+       "<matplotlib.figure.Figure at 0x746ae80>"
       ]
      },
      "metadata": {},
@@ -1542,9 +1545,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA5kAAAEWCAYAAADsL1MCAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA5kAAAEUCAYAAACh5/IJAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x752cb00>"
+       "<matplotlib.figure.Figure at 0x6ec1b00>"
       ]
      },
      "metadata": {},
@@ -1605,9 +1608,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA6UAAAEPCAYAAAC6OWLnAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7519c18>"
+       "<matplotlib.figure.Figure at 0x6ebb9e8>"
       ]
      },
      "metadata": {},
@@ -1672,7 +1675,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9316390>"
+       "<matplotlib.figure.Figure at 0x76d5be0>"
       ]
      },
      "metadata": {},
@@ -1750,7 +1753,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9383128>"
+       "<matplotlib.figure.Figure at 0x89c33c8>"
       ]
      },
      "metadata": {},
@@ -1792,7 +1795,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9383d30>"
+       "<matplotlib.figure.Figure at 0x8c31eb8>"
       ]
      },
      "metadata": {},
@@ -1837,7 +1840,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x99b77f0>"
+       "<matplotlib.figure.Figure at 0x8c4d550>"
       ]
      },
      "metadata": {},
@@ -1884,7 +1887,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7cb1da0>"
+       "<matplotlib.figure.Figure at 0x766eef0>"
       ]
      },
      "metadata": {},
@@ -1943,7 +1946,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x751cc88>"
+       "<matplotlib.figure.Figure at 0x77211d0>"
       ]
      },
      "metadata": {},
@@ -2010,7 +2013,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7d7ab00>"
+       "<matplotlib.figure.Figure at 0x6eaa748>"
       ]
      },
      "metadata": {},
@@ -2063,7 +2066,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9994828>"
+       "<matplotlib.figure.Figure at 0x8c461d0>"
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      },
      "metadata": {},
@@ -2094,7 +2097,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9994ac8>"
+       "<matplotlib.figure.Figure at 0x7680198>"
       ]
      },
      "metadata": {},
@@ -2127,7 +2130,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9994c88>"
+       "<matplotlib.figure.Figure at 0x6eab160>"
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      },
      "metadata": {},
@@ -2177,7 +2180,7 @@
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9994c18>"
+       "<matplotlib.figure.Figure at 0x74eb2b0>"
       ]
      },
      "metadata": {},
@@ -2208,7 +2211,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7d69dd8>"
+       "<matplotlib.figure.Figure at 0x8be1940>"
       ]
      },
      "metadata": {},

02_Discrete_Bayes.ipynb

@@ -16,7 +16,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 2,
+   "execution_count": 1,
    "metadata": {
     "collapsed": false
    },
@@ -30,7 +30,7 @@
        "@import url('http://fonts.googleapis.com/css?family=Arimo');\n",
        "\n",
        "    div.cell{\n",
-       "        width: 850px;\n",
+       "        width: 900px;\n",
        "        margin-left: 0% !important;\n",
        "        margin-right: auto;\n",
        "    }\n",
@@ -117,9 +117,6 @@
        "        font-family: 'Arimo',verdana,arial,sans-serif;\n",
        "        line-height: 125%;\n",
        "        font-size: 120%;\n",
-       "        width:740px;\n",
-       "        margin-left:auto;\n",
-       "        margin-right:auto;\n",
        "        text-align:justify;\n",
        "        text-justify:inter-word;\n",
        "    }\n",
@@ -240,7 +237,10 @@
        "                },\n",
        "                displayAlign: 'center', // Change this to 'center' to center equations.\n",
        "                \"HTML-CSS\": {\n",
-       "                    scale:85,\n",
+       "                    scale:100,\n",
+       "                    availableFonts: [\"Neo-Euler\"],\n",
+       "                    preferredFont: \"Neo-Euler\",\n",
+       "                    webFont: \"Neo-Euler\",\n",
        "                    styles: {'.MathJax_Display': {\"margin\": 4}}\n",
        "                }\n",
        "        });\n",
@@ -250,7 +250,7 @@
        "<IPython.core.display.HTML object>"
       ]
      },
-     "execution_count": 2,
+     "execution_count": 1,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -302,7 +302,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 3,
+   "execution_count": 2,
    "metadata": {
     "collapsed": false
    },
@@ -324,7 +324,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 4,
+   "execution_count": 3,
    "metadata": {
     "collapsed": false
    },
@@ -342,7 +342,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 5,
+   "execution_count": 4,
    "metadata": {
     "collapsed": false
    },
@@ -351,7 +351,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7dbf3c8>"
+       "<matplotlib.figure.Figure at 0x7217898>"
       ]
      },
      "metadata": {},
@@ -378,7 +378,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 6,
+   "execution_count": 5,
    "metadata": {
     "collapsed": false
    },
@@ -420,7 +420,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 7,
+   "execution_count": 6,
    "metadata": {
     "collapsed": false
    },
@@ -471,7 +471,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 8,
+   "execution_count": 7,
    "metadata": {
     "collapsed": false,
     "scrolled": true
@@ -489,7 +489,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7e6bb00>"
+       "<matplotlib.figure.Figure at 0x72abd68>"
       ]
      },
      "metadata": {},
@@ -522,7 +522,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 9,
+   "execution_count": 8,
    "metadata": {
     "collapsed": false,
     "scrolled": true
@@ -541,7 +541,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x85f25c0>"
+       "<matplotlib.figure.Figure at 0x79d3748>"
       ]
      },
      "metadata": {},
@@ -600,7 +600,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 10,
+   "execution_count": 9,
    "metadata": {
     "collapsed": false
    },
@@ -617,7 +617,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x85ff978>"
+       "<matplotlib.figure.Figure at 0x72d75c0>"
       ]
      },
      "metadata": {},
@@ -671,7 +671,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 12,
+   "execution_count": 10,
    "metadata": {
     "collapsed": false
    },
@@ -687,7 +687,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x85fb2b0>"
+       "<matplotlib.figure.Figure at 0x7aa2438>"
       ]
      },
      "metadata": {},
@@ -720,7 +720,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 13,
+   "execution_count": 11,
    "metadata": {
     "collapsed": false,
     "scrolled": true
@@ -737,7 +737,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8648fd0>"
+       "<matplotlib.figure.Figure at 0x53cd048>"
       ]
      },
      "metadata": {},
@@ -762,7 +762,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 14,
+   "execution_count": 12,
    "metadata": {
     "collapsed": false
    },
@@ -771,7 +771,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8685898>"
+       "<matplotlib.figure.Figure at 0x7a73390>"
       ]
      },
      "metadata": {},
@@ -807,7 +807,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x79580f0>"
+       "<matplotlib.figure.Figure at 0x72ba978>"
       ]
      },
      "metadata": {},
@@ -981,7 +981,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 18,
+   "execution_count": 17,
    "metadata": {
     "collapsed": false,
     "scrolled": true
@@ -1004,7 +1004,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 19,
+   "execution_count": 18,
    "metadata": {
     "collapsed": false
    },
@@ -1020,7 +1020,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x822b080>"
+       "<matplotlib.figure.Figure at 0x7294e10>"
       ]
      },
      "metadata": {},
@@ -1043,7 +1043,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 20,
+   "execution_count": 19,
    "metadata": {
     "collapsed": false
    },
@@ -1059,7 +1059,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x824c080>"
+       "<matplotlib.figure.Figure at 0x72941d0>"
       ]
      },
      "metadata": {},
@@ -1081,7 +1081,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 21,
+   "execution_count": 20,
    "metadata": {
     "collapsed": false
    },
@@ -1090,7 +1090,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x833f0b8>"
+       "<matplotlib.figure.Figure at 0x7c7e278>"
       ]
      },
      "metadata": {},
@@ -1112,7 +1112,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 22,
+   "execution_count": 21,
    "metadata": {
     "collapsed": false
    },
@@ -1121,7 +1121,7 @@
      "data": {
       "image/png": "iVBORw0KGgoAAAANSUhEUgAAAtcAAACeCAYAAAD5RibQAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8278fd0>"
+       "<matplotlib.figure.Figure at 0x7256358>"
       ]
      },
      "metadata": {},
@@ -1138,7 +1138,39 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Here things have degraded a bit due to the long string of wall positions in the map. We cannot be as sure where we are when there is an undifferentiated line of wall positions, so naturally our probabilities spread out a bit."
+    "Here things have degraded a bit due to the long string of wall positions in the map. We cannot be as sure where we are when there is an undifferentiated line of wall positions, so naturally our probabilities spread out a bit.\n",
+    "\n",
+    "I spread the computation across several cells, but I am just iteratively calling `predict()` and `update()`. This chart from the **g-h Filter** chapter illustrates the algorithm."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 22,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAgAAAAFoCAYAAAAhN0NQAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\nAAALEgAACxIB0t1+/AAAIABJREFUeJzs3XlcE3f+P/DX5CJcCeESEORSQQW0KOKtWNTWE126tdtu\nq631aO2xtrXXtlp7bOv23G7V1bZadf21q1VEt62KAmI9EBUVFVEQBZT7Jgkhyfz+8MusMQEBCZPj\n/Xw8eDyYTyYz7wkfMu/5zOfzGYZlWRaEEEIIsSsCvgMghBBCSM+jBIAQQgixQ5QAEEIIIXbIrAnA\nvHnzIBB0fBcrV66EQCDAjRs3urS/tLQ0CAQC/PDDD116PyGEEGIv2j07t55QP/vssy5tnGEYMAxj\nUJaUlIT33nuvw+t3db+EEEIIaVuHLs+7ekLdsGEDVCqVQVl7CcBf//pXqFQq9OnTp0v7I4QQQkjH\niMy6cZHpzbeVUAiFQgiFQnOGRAghhBB0sg9AYWEhBAIB3nvvPezduxcxMTFwdHSEn58fli9fDp1O\nZ7D+3X0AJkyYgM2bN4NlWQgEAu5n8+bNAEz3Abh58yZeeeUVDBkyBO7u7nB0dMSgQYOwevVq6PX6\n+zl2QgghxG51qQXgl19+wZo1a7BkyRIsWLAASUlJ+PTTT6FQKPDmm28arHvn1f5f//pXvP/++8jI\nyMDWrVu58lGjRrW5r3PnzmHXrl2YM2cOQkND0dLSgl9//RVvvPEGCgoKsG7duq4cAiGEEGLXupQA\nXLx4ERcuXODu1S9atAiRkZH4+uuvjRKAOycajI+Px9atW5GRkYE//elPHdrXhAkTUFBQYFD24osv\n4sknn8S3336LlStXwsfHpyuHQQghhNitLg0DTEhIMOqoN2HCBJSWlkKpVHZLYK2kUin3u0ajQXV1\nNSorKzF58mTo9XqcOnWqW/dHCCGE2IMutQCEhIQYlXl4eAAAqqqq4OTkdH9R3UGr1eLjjz/G5s2b\nkZ+fj7sfXVBTU9Nt+yKEEELsRZcSgPZ66nf3s4WWLVuGf/7zn5g7dy7eeecdeHt7QywW49SpU3j9\n9depIyAhhBDSBWYdBmhKZyf72bJlC8aPH49t27YZlOfl5XV3aIQQQojdMPuzAO4+2bu4uIBl2Q43\n3YtEIqOr/KamJnzxxRfdFiMhhBBib8yeANx9S2DkyJEAgOeeew5btmzBjz/+iMLCwjbfn5iYiIyM\nDMydOxcbNmzABx98gKioKMjlcnOGTQghhNi0brsFYKpp31TZY489hjNnzuDHH3/E9u3bwbIsNm7c\niKCgIJPrf/7553B1dcV//vMf7N69G3369MGiRYswbNgwxMfHm4yDEEIIIe1j2O7utUcIIYQQi2f2\nWwCEEEIIsTyUABBCCCF2iBIAQgghxA5RAkAIIYTYIUoACCGEEDtECQAhhBBihygBIIQQQuwQJQCE\nEEKIHbKZBCAoKAhxcXEGZWlpaRAIBPjhhx86tI3Ors+3nJwciEQiHDx40Czb3717NxwcHHD16lWz\nbJ8QQgh/OpUAKJVKfPnllxg7diw8PDwgkUjg4+ODadOm4YcffoBOpzNXnPfU1lMG7y7Pzs7GypUr\ncf369U5txxItW7YMY8eOxYMPPmiW7c+aNQuRkZF4/fXXzbJ9Qggh/OnwswCuXr2KadOm4cqVK5g0\naRLeeusteHp6ory8HAcOHMD8+fNx8eJFfPLJJ+aMt02mZjQeP348VCoVRKL/HWZ2djZWrVqFiRMn\nIjAw8J7rW6pjx44hJSUFu3fvNut+XnrpJTz11FO4ePEiBg4caNZ9EUII6TkdOtOpVCpMnz4dhYWF\n2LlzJxISEgxef+2115CVlYWsrCyzBNlVDMNAIpGYfM1UwtDe+pZmzZo18PLywtSpU826nzlz5mDJ\nkiVYt24d/vGPf5h1X4QQQnpOh24BfPvtt8jLy8Mrr7xidPJvNWzYMCxevJhbrqysxPPPP4+AgAA4\nODigT58+WLp0Kaqrqw3et2nTJggEAqSmpuLTTz9FaGgopFIpwsLCsHnzZqP9FBUV4Y9//CPkcjnk\ncjlmzpyJ/Px8kzG13tNv3c7KlSvx9NNPAwDi4uIgEAggEAgwf/58g/Xv7gNgjmNRq9VYuXIlwsLC\n4OzsDIVCgaioKCxfvtzksdxJq9UiKSkJ8fHxEAqF91z/fjg7O2Ps2LHYsWOHWfdDCCGkZ3WoBWDH\njh1gGAYLFy7s0Ebr6uowatQo5Ofn45lnnkF0dDROnz6NtWvX4tChQ8jMzISLi4vBe9566y2o1Wos\nWbIEEokEa9euxbx589C3b1+MGjUKAFBbW4tx48ahuLgYS5YswcCBA5GWloaJEydCpVLdM64//OEP\nKC0txfr16/H2229jwIABAIDQ0FCD9e7sA2CuY3n++eexceNGPPXUUxg1ahS0Wi3y8vKQmpp6z+M4\ndeoUmpqaMHz48Huu2x1GjBiBffv24fLlywgLC+uRfRJCCDEztgPc3d1ZNze3jqzKsizLvvXWWyzD\nMOzatWsNyr/55huWYRj2nXfe4co2btzIMgzDRkdHsy0tLVx5SUkJ6+DgwD722GNc2ZtvvskyDMNu\n2rTJYLsvv/wyyzAMGxcXZ1CemprKMgzD/vDDD0b7S09PN4rb1PrmOhaFQsFOmzbNKIaO+P7771mG\nYdg9e/YYvfbhhx+ykZGRLMMwrEwmY7ds2cK+8MILrEAgYPv27cs++eSTnd7fli1bWIZh2J07d3Yp\nXkIIIZanQ7cA6uvr4erq2uGkYteuXfD29jZqMVi0aBG8vLywa9cuo/c899xzBp3v/Pz80L9/f4Mh\naElJSfDx8cGTTz5p8F5z9lI317G4ubkhJycHFy5c6HRMFRUVAAB3d3ej19566y2cOHECISEhUKlU\nGDBgAB588EEMGTIEly5d6tIQRw8PDwBAeXl5p99LCCHEMnUoAZDJZGhoaOjwRq9du4awsDAIBIab\nFwqF6NevH65du2b0npCQEKMyd3d3VFVVccsFBQXo16+f0TA9Hx8fyOXyDsfXGeY6li+//BI1NTWI\njIxE37598eyzzyI5Odlk58S7tR5/W+s6Ojpi/fr10Gq1mDdvHl566SVs2bKly6MbWvdjLcMjCSGE\n3FuHEoCIiAjU1dWZPNl1l7Y6s3XkhGhpOnIsM2fORGFhIbZs2YKJEyfi4MGDSEhIwIQJE9DS0tLu\n9r28vADAqBPinSZOnIinn34aFy5cwIQJE4yG8KWkpOCdd97B+vXr8de//rXd/bXup3W/hBBCrF+H\nEoDExEQAt0cDdERISAhyc3ONJgZq7ehm6gq5o9vNy8uDXq83KL916xbq6uo6tI3OXsWa61gAQKFQ\n4PHHH8f69etRUFCA5cuXIyMj455j+yMjIwEAV65caXe9kSNHAgD+85//GNx+UCqVeOmll/Dee+9h\n4cKFqK+vR1JSUpvbaX1vREREh46LEEKI5etQArBgwQKEhYXh008/RXJyssl1Tp06hbVr1wIAZs+e\njYqKCqOEYcOGDaisrMTs2bO7FGxCQgLKysqMhtR1ZvKh1h77dzbHt8ccx6LX61FbW2tUPmTIEABA\nTU1Nu+8fMmQIZDIZjh071uY6xcXFeO+997B69Wqo1WqDPgzp6ekICQnhbmuMGjWq3aTj+PHj8PHx\nQb9+/dqNixBCiPXo0E1hR0dH7N27F9OmTUNCQgImT56M+Ph4eHh4oKKiAqmpqdi/fz83hn358uXY\nvn07nn/+eZw+fRpDhgzBmTNn8P333yM8PLxDY91b3dlsvnz5cmzbtg3PPvssTp06xQ0DPH78ODw9\nPTt0u2D48OEQCAT48MMPUV1dDWdnZ4SEhLQ5pM4cx1JfXw9fX1/MmjULQ4YMgbe3N65du4a1a9fC\n3d0dM2bMaHc7QqEQc+bMQVJSEjQajdHkRSzLYv78+XjnnXfw7LPP4siRI0hOTsa3336LBQsW4MaN\nG3Bzc+PWl8vluHTpksl9NTY2IiMjAwsWLOjwcRJCCLECnRkyoFQq2S+++IIdM2YMq1AoWLFYzHp7\ne7MPPfQQu3nzZlan03HrVlRUsM899xzr7+/PisViNiAggF26dClbVVVlsM2NGzeyAoHA5LC8CRMm\nsMHBwQZlN27cYBMTE1mZTMbKZDJ25syZbH5+PhsUFGRyGKBAIDAY1seyLPvDDz+wAwcOZCUSCcsw\nDDt//vx21+/uY9FoNOybb77JDh8+nPXw8GAdHBzY4OBg9plnnmGvXr1q8rO/W2ZmJsswDPvzzz8b\nlC9btowdOHAgKxAI2EceeYRlWZZ94IEHWIFAwMpkMnbChAnsqlWr2AULFnDvSUlJYQcOHGhyP5s2\nbWIZhmEvXLjQobgIIYRYB4ZlrbCXHQEAPPzww2hqasLhw4c79b5vvvkGZ86c4W5r7N27F6tXrza5\nnejoaISEhNBMgIQQYmNs5nHA9uizzz7jHgrUGf369UNlZSW3XFVVBX9/f6P1kpKSeH3AEyGEEPOh\nFgA7pFarERERgQsXLsDBwQELFizAQw89xI32IIQQYvsoAbBTO3fuxMmTJxEYGIjCwkJ8/PHHfIdE\nCCGkB1ECQAghhNgh6gNACCGE2CFKAAghhBA7RAkAIYQQYocoASCEEELsECUAPezuBxkRQgghfKAE\noIdotVr8/vvv+PXXX63yEceEEEJsS4ceBkTuT3l5OVJTU7lHFl+4cIEerUsIIYRX1ALQAzQaDXfy\nB4ATJ06YfBwwIYQQ0lMoAegB/v7+GDRoELes0+mQmppK/QEIIYTwhhKAHhIbGwu5XM4tV1RUIDs7\nm8eICCGE2DNKAHqISCRCXFwcGIbhyk6dOmXwVD5CCCGkp1AC0IO8vb0xZMgQbpllWLine truncated
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x7c81898>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import gh_internal\n",
+    "gh_internal.create_predict_update_chart()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "This filter is just a form of the g-h filter. Here we are using the percentages for the errors to implicitly compute the *g* and *h* parameters. We could express the discrete Bayes algorithm as a g-h filter, but that would obscure the logic of this filter."
    ]
   },
   {

06_Kalman_Filter_Math.ipynb

@@ -16,7 +16,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 5,
+   "execution_count": 7,
    "metadata": {
     "collapsed": false
    },
@@ -30,7 +30,7 @@
        "@import url('http://fonts.googleapis.com/css?family=Arimo');\n",
        "\n",
        "    div.cell{\n",
-       "        width: 850px;\n",
+       "        width: 900px;\n",
        "        margin-left: 0% !important;\n",
        "        margin-right: auto;\n",
        "    }\n",
@@ -250,7 +250,7 @@
        "<IPython.core.display.HTML object>"
       ]
      },
-     "execution_count": 5,
+     "execution_count": 7,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -378,6 +378,290 @@
     "In other words the first half of the Discrete Bayes chapter developed Bayes' equation from a thought experiment. I could have just presented Bayes' equation and then given you the Python routine above to implement it, but chances are you would not have understood *why* Bayes' equation works. Presenting the equation first is the normal approach of Kalman filtering texts, and I always found it extremely nonintuitive. "
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Computing Means and Covariances\n",
+    "\n",
+    "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."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Computing the Mean\n",
+    "\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 talk 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",
+    "$$E[x] = (1)(0.8) + (3)(0.15) + (5)(0.05) = 1.5$$\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 the $\\mathbf{x}$ from the beginning of this section.\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",
+    "$$\n",
+    "\\begin{aligned}\n",
+    "VAR(a) &= \\frac{1}{2}[(1-2)^2 + (3-2)^2] \\\\\n",
+    "&= \\frac{1}{2}[1+1] \\\\\n",
+    "&= 1\\end{aligned}$$\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": 8,
+   "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": [
+    "import numpy as np\n",
+    "\n",
+    "x = [[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. If we start with the value $x_0 = (1, 2)$ we can create subsequent values that vary in the same direction with $x_1 = (2, 4)$, $x_2=(3,6)$, and $x_3=(4,8)$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "[[ 1.25  2.5 ]\n",
+      " [ 2.5   5.  ]]\n"
+     ]
+    },
+    {
+     "data": {
+      "image/png": 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truncated
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x813fe10>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "from filterpy.common import plot_covariance_ellipse\n",
+    "\n",
+    "x = [[1, 2, 3, 4], [2, 4, 6, 8]]\n",
+    "cov = np.cov(x, bias=1)\n",
+    "plot_covariance_ellipse([0, 0], cov)\n",
+    "print(cov)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "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": 10,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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truncated
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x8466cc0>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "x = [[1, 2, 3, 4], [2, 4, 6, 8.4]]\n",
+    "plot_covariance_ellipse([0, 0], np.cov(x, bias=1))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Now lets make the second value go away from the first."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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truncated
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x84ebe48>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "x = [[1, 2, 3, 4], [-2, -4, -6, -8.4]]\n",
+    "plot_covariance_ellipse([0, 0], np.cov(x, bias=1))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "As we would hope, the covariance matrix is tilted in the opposite direction, indicating that the variables are inversely correlated. That is, as one gets larger, the other becomes smaller."
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},
@@ -726,6 +1010,74 @@
     "So we have proven that the multivariate equations are equivalent to the univariate equations when we only have one state variable. I'll close this section by recognizing one quibble - I hand waved my assertion that $H=1$ and $F=1$. In general we know this is not true. For example, a digital thermometer may provide measurement in volts, and we need to convert that to temperature, and we use $H$ to do that conversion. I left that issue out to keep the explanation as simple and streamlined as possible. It is very straightforward to add that generalization to the equations above, redo the algebra, and still have the same results."
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Converting Kalman Filter to a g-h Filter\n",
+    "\n",
+    "I've stated that the Kalman filter is a form of the g-h filter. It just takes some algebra to prove this. It's more straightforward to do with the one dimensional case, so I will do that. Recall \n",
+    "\n",
+    "$$\n",
+    "\\mu_{x}=\\frac{\\sigma_1^2 \\mu_2 + \\sigma_2^2 \\mu_1} {\\sigma_1^2 + \\sigma_2^2}\n",
+    "$$\n",
+    "\n",
+    "which I will make more friendly for our eyes as:\n",
+    "\n",
+    "$$\n",
+    "\\mu_{x}=\\frac{ya + xb} {a+b}\n",
+    "$$\n",
+    "\n",
+    "We can easily put this into the g-h form with the following algebra\n",
+    "\n",
+    "$$\n",
+    "\\begin{aligned}\n",
+    "\\mu_{x}&=(x-x) + \\frac{ya + xb} {a+b} \\\\\n",
+    "\\mu_{x}&=x-\\frac{a+b}{a+b}x  + \\frac{ya + xb} {a+b} \\\\ \n",
+    "\\mu_{x}&=x +\\frac{-x(a+b) + xb+ya}{a+b} \\\\\n",
+    "\\mu_{x}&=x+ \\frac{-xa+ya}{a+b}  \\\\\n",
+    "\\mu_{x}&=x+ \\frac{a}{a+b}(y-x)\\\\\n",
+    "\\end{aligned}\n",
+    "$$\n",
+    "\n",
+    "We are almost done, but recall that the variance of estimate is given by \n",
+    "\n",
+    "$${\\sigma_{x}^2} = \\frac{1}{ \\frac{1}{\\sigma_1^2} +  \\frac{1}{\\sigma_2^2}}\\\\\n",
+    "= \\frac{1}{ \\frac{1}{a} +  \\frac{1}{b}}\n",
+    "$$\n",
+    "\n",
+    "We can incorporate that term into our equation above by observing that\n",
+    "\n",
+    "$$ \n",
+    "\\begin{aligned}\n",
+    "\\frac{a}{a+b} &= \\frac{a/a}{(a+b)/a} = \\frac{1}{(a+b)/a}  \\\\\n",
+    " &= \\frac{1}{1 + \\frac{b}{a}} = \\frac{1}{\\frac{b}{b} + \\frac{b}{a}}  \\\\\n",
+    " &= \\frac{1}{b}\\frac{1}{\\frac{1}{b} + \\frac{1}{a}} \\\\\n",
+    " &= \\frac{\\sigma^2_{x'}}{b}\n",
+    " \\end{aligned}\n",
+    "$$\n",
+    "\n",
+    "We can tie all of this together with\n",
+    "\n",
+    "$$\n",
+    "\\begin{aligned}\n",
+    "\\mu_{x}&=x+ \\frac{a}{a+b}(y-x) \\\\\n",
+    "&= x + \\frac{\\sigma^2_{x'}}{b}(y-x) \\\\\n",
+    "&= x + g_n(y-x)\n",
+    "\\end{aligned}\n",
+    "$$\n",
+    "\n",
+    "where\n",
+    "\n",
+    "$$g_n = \\frac{\\sigma^2_{x}}{\\sigma^2_{y}}$$\n",
+    "\n",
+    "The end result is multiplying the residual of the two measurements by a constant and adding to our previous value, which is the *g* equation for the g-h filter. *g* is the variance of the new estimate divided by the variance of the measurement. Of course in this case g is not a constant as it varies with each time step as the variance changes. We can also derive the formula for *h* in the same way but I don't find this a particularly interesting derivation. The end result is\n",
+    "\n",
+    "$$h_n = \\frac{COV (x,\\dot{x})}{\\sigma^2_{y}}$$\n",
+    "\n",
+    "The takeaway point is that *g* and *h* are specified fully by the variance and covariances of the measurement and predictions at time *n*. In other words, we are just picking a point between the measurement and prediction by a scale factor determined by the quality of each of those two inputs. That is all the Kalman filter is. "
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},
@@ -1241,6 +1593,181 @@
     "If you practice this a bit you will become adept at it. Just isolate the highest term, define a new variable and its derivatives, and then substitute."
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "** text ripped from MKF chapter. Needs to be incorporated here somewhere below.**\n",
+    "\n",
+    "Recall what happened to our variance when we performed the predict step in the *Discrete Bayes* chapter and the *Kalman Filter* chapter. Because we are predicting forward in time we lose some certainty about the state. In the *Discrete Bayes* chapter we used this code to implement that:\n",
+    "\n",
+    "    sigma = sigma + movement_sigma\n",
+    "    \n",
+    "That equation did not take the loss of knowledge due to the process noise into account. We can model the process noise as additional uncertainty, so we could write\n",
+    "\n",
+    "In pseudocode we might express this equation as:\n",
+    "\n",
+    "    sigma = sigma + movement_sigma + process_noise\n",
+    "\n",
+    "\n",
+    "In this chapter we do not have multiple state variables, so we have to use linear algebra to perform this computation. \n",
+    "  \n",
+    "In pseudocode we might express this equation as:\n",
+    "    \n",
+    "    P = project_forward(P) + process_noise\n",
+    "\n",
+    "where $P$ is the covariance matrix. The Kalman filter equation corresponding to this pseudocode is\n",
+    "\n",
+    "$$\\mathbf{P} = \\mathbf{FP{F}}^\\mathsf{T} + \\mathbf{Q}$$\n",
+    "\n",
+    "In this equation $\\mathbf{FP{F}}^\\mathsf{T}$ projects $\\mathbf{P}$ forward to the next time step according to our motion model $\\mathbf{F}$, and $\\mathbf{Q}$ is the *process noise matrix*. We are just adding matrices, so hopefully it is clear that each element in $\\mathbf{Q}$  specifies how much uncertainty is added to the system due to the process noise. \n",
+    "\n",
+    "Let's take a brief moment to look at the behavior of these equation in more detail. I will project $\\mathbf{x}$ forward according to the motion model $\\mathbf{F}$ with the equation $\\mathbf{x} = \\mathbf{Fx}$. Then I will project P to the next timestep with the equation $\\mathbf{P} = \\mathbf{FP{F}}^\\mathsf{T}$. I will draw the original covariance in red and the new covariance in black."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": [
+    "t = 0.2\n",
+    "x = np.array([0., 5.])\n",
+    "F = np.array([[1., t], [0, 1.]])\n",
+    "P = np.array([[1.5, 0.2], [0.2, 3.]])\n",
+    "plot_covariance_ellipse(x, P, edgecolor='r')\n",
+    "\n",
+    "x = np.dot(F, x)\n",
+    "P = np.dot(F, P).dot(F.T)\n",
+    "plt.xlabel('position')\n",
+    "plt.ylabel('velocity')\n",
+    "plot_covariance_ellipse(x, P, edgecolor='k')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Think about what this means. The covariance for $\\mathbf{x}$ started at position 0, velocity 5. In 1/5 of a second the position naturally moves to 1, and the velocity is unchanged at 5. The black covariance ellipse is centered at (1, 5), as we would expect. But the ellipse has changed shape. Why? Is it due to the effects of wind, or wheel slippage that I mentioned earlier? No. Think hard about this before you read the answer - this is important, and it will be rewarding to figure it out yourself.\n",
+    "\n",
+    "The covariance matrix is ($\\begin{smallmatrix}1.5& 0.2\\\\ 0.2& 3\\end{smallmatrix}$). This means that the variance for the velocity if $\\sigma^2_{vel}=3.0$. In other words, we think the velocity is 5, but we have some uncertainty about it. We compute the next position as $position_t = position_{t-1} + .2_{sec} * 5$. But we are uncertain about the velocity 5, so we must also be uncertain about the results of this computation! After all, the velocity might actually be 6, or 4.7, or some other value. Our uncertainty about the position must grow from the initial value of $\\sigma^2_{pos}=1.5$. We can see from the graph that it has - the ellipse is longer in the position axis.\n",
+    "\n",
+    "Let's see this over a longer time period. I will run it for 10 iterations.b"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": [
+    "t = 0.2\n",
+    "x = np.array([0., 5.])\n",
+    "F = np.array([[1., t], [0, 1.]])\n",
+    "P = np.array([[1.5, 0.2], [0.2, 3.]])\n",
+    "plot_covariance_ellipse(x, P, edgecolor='r')\n",
+    "\n",
+    "for _ in range(10):\n",
+    "    x = np.dot(F, x)\n",
+    "    P = np.dot(F, P).dot(F.T)\n",
+    "    plt.xlabel('position')\n",
+    "    plt.ylabel('velocity')\n",
+    "    plot_covariance_ellipse(x, P, edgecolor='k')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "You can see that over time our uncertainty in the position grows. However, our uncertainty in the velocity does not. Why is that? Well, our process model states that the velocity is constant, and we do not model the possibility that there is error in the model (so far!). We assume the model is perfect, so the variance of the velocity does not change.\n",
+    "\n",
+    "So I have demonstrated the reason for the $\\mathbf{FP{F}}^\\mathsf{T}$ part of the equation, which assumes a perfect model. But of course no model is perfect. Except perhaps in a perfect vacuum velocity will never remain constant. If we are tracking a car, for example, even if we set the cruise control the car will slightly change velocity due to the wind, hills, bumps in the road, imperfections in the cruise control, and so on. If the Kalman filter used the equation above it would quickly become over-certain in its computations. Recall that in the unidimensional Kalman filter chapter every predict cycle increased our uncertainty, and every update decreased it. The same needs to happen here.\n",
+    "\n",
+    "This is where $\\mathbf{Q}$ comes in. It is a covariance matrix that describes how much *process noise* is in the system. That is just a fancy way of saying how much uncertainty gets added to the system by things we haven't modeled.\n",
+    "\n",
+    "Let's rerun this simulation, but this time by using the full equation $\\mathbf{P} = \\mathbf{FP{F}}^\\mathsf{T} + \\mathbf{Q}$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": [
+    "t = 0.2\n",
+    "x = np.array([0., 5.])\n",
+    "F = np.array([[1., t], [0, 1.]])\n",
+    "P = np.array([[1.5, 0.2], [0.2, 3.]])\n",
+    "Q = np.array([[.0004, .004], [.004, .04]]) * 8.\n",
+    "plot_covariance_ellipse(x, P, edgecolor='r')\n",
+    "\n",
+    "for _ in range(10):\n",
+    "    x = np.dot(F, x)\n",
+    "    P = np.dot(F, P).dot(F.T) + Q\n",
+    "    plt.xlabel('position')\n",
+    "    plt.ylabel('velocity')\n",
+    "    plot_covariance_ellipse(x, P, edgecolor='k')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "**author's note: I think I go WAY TO MUCH into Q here -- get the poor reader to the implementation!**\n",
+    "\n",
+    "I'll address the computation of $\\mathbf{Q}$ in the next paragraph. For now you can see that as time passes our uncertainty in velocity is growing slowly. It is a bit harder to see, but if you compare this graph to the previous one the uncertainty in position also increased.\n",
+    "\n",
+    "We have not given the math for computing the elements of $\\mathbf{Q}$ yet, but if you suspect the math is sometimes difficult you would be correct. One of the problems is that we are usually modeling a *continuous* system - the behavior of the system is changing at every instant, but the Kalman filter is *discrete*. That means that we have to somehow convert the continuous noise of the system into a discrete value, which usually involves calculus. There are other difficulties I will not mention now.\n",
+    "\n",
+    "However, for the class of problems we are solving in this chapter (*discretized continuous-time kinematic filters*) we can directly compute the state equations for moving objects by using Newton's equations.\n",
+    "\n",
+    "For these kinds of problems we can rely on precomputed forms for $\\mathbf{Q}$. We will learn how to derive these matrices in the next chapter. For now I present them without proof. If we assume that for each time period the acceleration due to process noise is constant and uncorrelated, we get the following.\n",
+    "\n",
+    "For constant velocity the form is\n",
+    "\n",
+    "$$\\begin{bmatrix}\n",
+    "\\frac{1}{4}{\\Delta t}^4 & \\frac{1}{2}{\\Delta t}^3 \\\\\n",
+    "\\frac{1}{2}{\\Delta t}^3 & \\Delta t^2\n",
+    "\\end{bmatrix}\\sigma^2\n",
+    "$$\n",
+    "\n",
+    "and for constant acceleration we have\n",
+    "\n",
+    "$$\\begin{bmatrix}\n",
+    "\\frac{1}{4}{\\Delta t}^4 & \\frac{1}{2}{\\Delta t}^3 & \\frac{1}{2}{\\Delta t}^2 \\\\\n",
+    "\\frac{1}{2}{\\Delta t}^3 & {\\Delta t}^2 & \\Delta t \\\\\n",
+    "\\frac{1}{2}{\\Delta t}^2 & \\Delta t & 1\n",
+    "\\end{bmatrix} \\sigma^2\n",
+    "$$\n",
+    "\n",
+    "It general it is not true that acceleration will be constant and uncorrelated, but this is still a useful approximation for moderate time period, and will suffice for this chapter. Fortunately you can get a long way with approximations and simulation. Let's think about what these matrices are implying. We are trying to model the effects of *process noise*, such as the wind buffeting the flight of a thrown ball. Variations in wind will cause changes in acceleration, and so the effect on the acceleration is large. However, the effects on velocity and position are proportionally smaller. In the matrices, the acceleration term is in the lower right, and this is the largest value. **A good rule of thumb is to set $\\sigma$ somewhere from $\\frac{1}{2}\\Delta a$ to $\\Delta a$, where $\\Delta a$ is the maximum amount that the acceleration will change between sample periods**. In practice we pick a number, run simulations on data, and choose a value that works well. \n",
+    "\n",
+    "The filtered result will not be optimal, but in my opinion the promise of optimal results from Kalman filters is mostly wishful thinking. Consider, for example, tracking a car. In that problem the process noise would include things like potholes, wind gusts, changing drag due to turning, rolling down windows, and many more factors. We cannot realistically model that analytically, and so in practice we work out a simplified model, compute $\\mathbf{Q}$ based on that simplified model, and then add *a bit more* to  $\\small\\mathbf{Q}$ in hopes of taking the incalculable factors into account. Then we use a lot of simulations and trial runs to see if the filter behaves well; if it doesn't we adjust $\\small\\mathbf{Q}$ until the filter performs well. In this chapter we will focus on forming an intuitive understanding on how adjusting $\\small\\mathbf{Q}$ affects the output of the filter. In the Kalman Filter Math chapter we will discuss the analytic computation of $\\small\\mathbf{Q}$, and also provide code that will compute it for you.\n",
+    "\n",
+    "For now we will just import the code from the `FilterPy` module, where it is already implemented. I will import it and call help on it so you can see the documentation for it."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": []
+  },
   {
    "cell_type": "markdown",
    "metadata": {},
@@ -2668,7 +3195,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.4.3"
+   "version": "3.4.1"
   }
  },
  "nbformat": 4,

09_Unscented_Kalman_Filter.ipynb

@@ -289,9 +289,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x541ce80>"
+       "<matplotlib.figure.Figure at 0x6f07cf8>"
       ]
      },
      "metadata": {},
@@ -302,7 +302,7 @@
     "from numpy.random import normal\n",
     "import numpy as np\n",
     "import matplotlib.pyplot as plt\n",
-    "from nonlinear_plots import plot_transfer_func\n",
+    "from nonlinear_plots import plot_nonlinear_func\n",
     "\n",
     "gaussian=(0., 1.)\n",
     "data = normal(loc=gaussian[0], scale=gaussian[1], size=500000)\n",
@@ -311,7 +311,7 @@
     "    return (np.cos(4*(x/2 + 0.7))) - 1.3*x\n",
     "\n",
     "with book_format.figsize(y=5):\n",
-    "    plot_transfer_func(data, g, gaussian=gaussian)"
+    "    plot_nonlinear_func(data, g, gaussian=gaussian)"
    ]
   },
   {
@@ -351,7 +351,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x5427358>"
+       "<matplotlib.figure.Figure at 0x440d278>"
       ]
      },
      "metadata": {},
@@ -397,14 +397,14 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Difference in mean x=0.042, y=44.008\n"
+      "Difference in mean x=0.057, y=43.363\n"
      ]
     },
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x541cf98>"
+       "<matplotlib.figure.Figure at 0x84d2588>"
       ]
      },
      "metadata": {},
@@ -489,7 +489,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x541ccf8>"
+       "<matplotlib.figure.Figure at 0x8499780>"
       ]
      },
      "metadata": {},
@@ -527,7 +527,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x541cf60>"
+       "<matplotlib.figure.Figure at 0x84dfa90>"
       ]
      },
      "metadata": {},
@@ -572,7 +572,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x54272b0>"
+       "<matplotlib.figure.Figure at 0x85a73c8>"
       ]
      },
      "metadata": {},
@@ -613,7 +613,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8841e80>"
+       "<matplotlib.figure.Figure at 0x7e67c50>"
       ]
      },
      "metadata": {},
@@ -726,7 +726,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x54270f0>"
+       "<matplotlib.figure.Figure at 0x85f3b00>"
       ]
      },
      "metadata": {},
@@ -765,14 +765,14 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Difference in mean x=-0.019, y=0.088\n"
+      "Difference in mean x=-0.019, y=-0.004\n"
      ]
     },
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x83bf048>"
+       "<matplotlib.figure.Figure at 0x8a796d8>"
       ]
      },
      "metadata": {},
@@ -991,7 +991,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa404748>"
+       "<matplotlib.figure.Figure at 0x8a793c8>"
       ]
      },
      "metadata": {},
@@ -1103,7 +1103,7 @@
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa2e9eb8>"
+       "<matplotlib.figure.Figure at 0x8b9e588>"
       ]
      },
      "metadata": {},
@@ -1188,7 +1188,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa40e438>"
+       "<matplotlib.figure.Figure at 0x8a65ba8>"
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      "metadata": {},
@@ -1366,7 +1366,7 @@
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8ec2400>"
+       "<matplotlib.figure.Figure at 0x8bbbe80>"
       ]
      },
      "metadata": {},
@@ -1376,7 +1376,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa476e10>"
+       "<matplotlib.figure.Figure at 0x8563e48>"
       ]
      },
      "metadata": {},
@@ -1386,7 +1386,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7e2dba8>"
+       "<matplotlib.figure.Figure at 0x85b0e80>"
       ]
      },
      "metadata": {},
@@ -1492,7 +1492,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x83bf6d8>"
+       "<matplotlib.figure.Figure at 0x83e60b8>"
       ]
      },
      "metadata": {},
@@ -1565,7 +1565,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x76a0898>"
+       "<matplotlib.figure.Figure at 0x8b6e9e8>"
       ]
      },
      "metadata": {},
@@ -1664,7 +1664,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa6ec080>"
+       "<matplotlib.figure.Figure at 0x84aec18>"
       ]
      },
      "metadata": {},
@@ -1724,7 +1724,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa3730b8>"
+       "<matplotlib.figure.Figure at 0x85dc208>"
       ]
      },
      "metadata": {},
@@ -1820,7 +1820,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa37c940>"
+       "<matplotlib.figure.Figure at 0x86114e0>"
       ]
      },
      "metadata": {},
@@ -1872,7 +1872,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa321908>"
+       "<matplotlib.figure.Figure at 0xa421208>"
       ]
      },
      "metadata": {},
@@ -1974,7 +1974,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa6c2080>"
+       "<matplotlib.figure.Figure at 0x7eb54e0>"
       ]
      },
      "metadata": {},
@@ -2086,7 +2086,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa6c2d30>"
+       "<matplotlib.figure.Figure at 0xa466f98>"
       ]
      },
      "metadata": {},
@@ -2127,7 +2127,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa6c2908>"
+       "<matplotlib.figure.Figure at 0x7e71f28>"
       ]
      },
      "metadata": {},
@@ -2201,7 +2201,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x826b0b8>"
+       "<matplotlib.figure.Figure at 0x84fb048>"
       ]
      },
      "metadata": {},
@@ -2320,7 +2320,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa37cac8>"
+       "<matplotlib.figure.Figure at 0x8445710>"
       ]
      },
      "metadata": {},
@@ -2383,7 +2383,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa399ac8>"
+       "<matplotlib.figure.Figure at 0x7e71fd0>"
       ]
      },
      "metadata": {},
@@ -2439,7 +2439,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa4f69b0>"
+       "<matplotlib.figure.Figure at 0x8537780>"
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@@ -2535,7 +2535,7 @@
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7e33b70>"
+       "<matplotlib.figure.Figure at 0x7e8c6a0>"
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      },
      "metadata": {},
@@ -2886,7 +2886,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa2e9320>"
+       "<matplotlib.figure.Figure at 0x8485a90>"
       ]
      },
      "metadata": {},
@@ -2896,7 +2896,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8ec4400>"
+       "<matplotlib.figure.Figure at 0x8ae12b0>"
       ]
      },
      "metadata": {},
@@ -2906,7 +2906,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa347f60>"
+       "<matplotlib.figure.Figure at 0xa3e7a58>"
       ]
      },
      "metadata": {},
@@ -2976,7 +2976,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa7700b8>"
+       "<matplotlib.figure.Figure at 0x8bb8cc0>"
       ]
      },
      "metadata": {},
@@ -3024,7 +3024,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa770320>"
+       "<matplotlib.figure.Figure at 0xa4b3c18>"
       ]
      },
      "metadata": {},
@@ -3034,7 +3034,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa6a9278>"
+       "<matplotlib.figure.Figure at 0x8504358>"
       ]
      },
      "metadata": {},
@@ -3044,7 +3044,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x826b668>"
+       "<matplotlib.figure.Figure at 0x841d240>"
       ]
      },
      "metadata": {},
@@ -3343,7 +3343,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa2e4d30>"
+       "<matplotlib.figure.Figure at 0x84ed208>"
       ]
      },
      "metadata": {},
@@ -3740,9 +3740,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa3c9f98>"
+       "<matplotlib.figure.Figure at 0x84c0ef0>"
       ]
      },
      "metadata": {},
@@ -3825,9 +3825,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa3c9fd0>"
+       "<matplotlib.figure.Figure at 0x84c07b8>"
       ]
      },
      "metadata": {},
@@ -3864,9 +3864,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x83b35f8>"
+       "<matplotlib.figure.Figure at 0xa3d0240>"
       ]
      },
      "metadata": {},

10_Extended_Kalman_Filters.ipynb

@@ -312,7 +312,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x6f67e80>"
+       "<matplotlib.figure.Figure at 0x6ed8eb8>"
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      },
      "metadata": {},
@@ -361,7 +361,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x6f67908>"
+       "<matplotlib.figure.Figure at 0x6ed8e48>"
       ]
      },
      "metadata": {},
@@ -436,7 +436,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x451e2b0>"
+       "<matplotlib.figure.Figure at 0x6ed8668>"
       ]
      },
      "metadata": {},
@@ -772,9 +772,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAwYAAAEtCAYAAACyMncIAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x81d6630>"
+       "<matplotlib.figure.Figure at 0x8126f60>"
       ]
      },
      "metadata": {},
@@ -782,9 +782,9 @@
     },
     {
      "data": {
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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8202780>"
+       "<matplotlib.figure.Figure at 0x815ee10>"
       ]
      },
      "metadata": {},
@@ -792,9 +792,9 @@
     },
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAwMAAAEtCAYAAABUG7xMAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x81463c8>"
+       "<matplotlib.figure.Figure at 0x8044400>"
       ]
      },
      "metadata": {},
@@ -1046,7 +1046,7 @@
      "data": {
       "image/png": "iVBORw0KGgoAAAANSUhEUgAAAhQAAAEWCAYAAAA+Wg5zAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x81e0a58>"
+       "<matplotlib.figure.Figure at 0x80c57f0>"
       ]
      },
      "metadata": {},
@@ -1640,9 +1640,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x6ff1240>"
+       "<matplotlib.figure.Figure at 0x9ed1a20>"
       ]
      },
      "metadata": {},
@@ -1652,7 +1652,7 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[ 0.02479667  0.03958041  0.00220384]\n"
+      "[ 0.02404007  0.04230372  0.002213  ]\n"
      ]
     }
    ],
@@ -1674,9 +1674,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa2c0668>"
+       "<matplotlib.figure.Figure at 0x803f160>"
       ]
      },
      "metadata": {},
@@ -1686,7 +1686,7 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[ 0.02026067  0.0203502   0.00153694]\n"
+      "[ 0.01886792  0.02163098  0.00153879]\n"
      ]
     }
    ],
@@ -1719,9 +1719,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x6ff12b0>"
+       "<matplotlib.figure.Figure at 0x803f080>"
       ]
      },
      "metadata": {},
@@ -1731,7 +1731,7 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[ 0.02037772  0.04521487  0.00022534]\n"
+      "[ 0.0198982   0.04518314  0.00022389]\n"
      ]
     }
    ],
@@ -1758,9 +1758,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa2b1278>"
+       "<matplotlib.figure.Figure at 0xa07ff28>"
       ]
      },
      "metadata": {},
@@ -1770,7 +1770,7 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[ 0.2560811   0.81614092  0.00355988]\n"
+      "[ 0.28419616  0.80806564  0.00361808]\n"
      ]
     }
    ],
@@ -1797,9 +1797,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa26b9b0>"
+       "<matplotlib.figure.Figure at 0xa233748>"
       ]
      },
      "metadata": {},
@@ -1809,7 +1809,7 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[ 0.00865132  0.00887946  0.00077272]\n"
+      "[ 0.00893561  0.0085525   0.00077497]\n"
      ]
     }
    ],
@@ -1859,9 +1859,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xa257c50>"
+       "<matplotlib.figure.Figure at 0xa233908>"
       ]
      },
      "metadata": {},
@@ -1871,7 +1871,7 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "actual mean=1.30, std=1.12\n",
+      "actual mean=1.30, std=1.14\n",
       "EKF    mean=1.00, std=0.95\n"
      ]
     }
@@ -1899,9 +1899,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x43aeda0>"
+       "<matplotlib.figure.Figure at 0x74fff60>"
       ]
      },
      "metadata": {},
@@ -1911,7 +1911,7 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "actual mean=1.29, std=1.12\n",
+      "actual mean=1.30, std=1.13\n",
       "UKF    mean=1.30, std=1.08\n"
      ]
     }

12_Smoothing.ipynb

@@ -16,7 +16,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 4,
+   "execution_count": 2,
    "metadata": {
     "collapsed": false
    },
@@ -250,7 +250,7 @@
        "<IPython.core.display.HTML object>"
       ]
      },
-     "execution_count": 4,
+     "execution_count": 2,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -281,7 +281,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 5,
+   "execution_count": 3,
    "metadata": {
     "collapsed": false
    },
@@ -290,7 +290,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7044cc0>"
+       "<matplotlib.figure.Figure at 0x4516320>"
       ]
      },
      "metadata": {},
@@ -326,7 +326,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 6,
+   "execution_count": 4,
    "metadata": {
     "collapsed": false
    },
@@ -335,7 +335,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x704d320>"
+       "<matplotlib.figure.Figure at 0x7002320>"
       ]
      },
      "metadata": {},
@@ -359,7 +359,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 7,
+   "execution_count": 5,
    "metadata": {
     "collapsed": false
    },
@@ -368,7 +368,7 @@
      "data": {
       "image/png": "iVBORw0KGgoAAAANSUhEUgAAAugAAAEWCAYAAAA0KmJkAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7bc4668>"
+       "<matplotlib.figure.Figure at 0x7591b00>"
       ]
      },
      "metadata": {},
@@ -502,7 +502,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 10,
+   "execution_count": 15,
    "metadata": {
     "collapsed": false
    },
@@ -511,7 +511,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7bc4b38>"
+       "<matplotlib.figure.Figure at 0x9734438>"
       ]
      },
      "metadata": {},
@@ -525,7 +525,7 @@
     "from filterpy.kalman import KalmanFilter\n",
     "import book_plots as bp\n",
     "\n",
-    "def plot_rts(noise, Q=0.001):\n",
+    "def plot_rts(noise, Q=0.001, show_velocity=False):\n",
     "    random.seed(123)\n",
     "    fk = KalmanFilter(dim_x=2, dim_z=1)\n",
     "\n",
@@ -547,10 +547,17 @@
     "    M,P,C = fk.rts_smoother(mu, cov)\n",
     "\n",
     "    # plot data\n",
-    "    bp.plot_measurements(zs, lw=1)\n",
-    "    plt.plot(M[:, 0], c='b', label='RTS')\n",
-    "    plt.plot(mu[:, 0], c='g', linestyle='--', label='KF output')\n",
-    "    plt.plot([0, len(zs)], [0, len(zs)], 'k', linewidth=2, label='track') \n",
+    "    if show_velocity:\n",
+    "        index = 1\n",
+    "        print('gu')\n",
+    "    else:\n",
+    "        index = 0\n",
+    "    if not show_velocity:\n",
+    "        bp.plot_measurements(zs, lw=1)\n",
+    "    plt.plot(M[:, index], c='b', label='RTS')\n",
+    "    plt.plot(mu[:, index], c='g', linestyle='--', label='KF output')\n",
+    "    if not show_velocity:\n",
+    "        plt.plot([0, len(zs)], [0, len(zs)], 'k', linewidth=2, label='track') \n",
     "    plt.legend(loc=4)\n",
     "    plt.show()\n",
     "    \n",
@@ -568,16 +575,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 6,
+   "execution_count": 7,
    "metadata": {
     "collapsed": false
    },
    "outputs": [
     {
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7109400>"
+       "<matplotlib.figure.Figure at 0x703a0b8>"
       ]
      },
      "metadata": {},
@@ -597,7 +604,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 11,
+   "execution_count": 8,
    "metadata": {
     "collapsed": false
    },
@@ -606,7 +613,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7bdfac8>"
+       "<matplotlib.figure.Figure at 0x820dcc0>"
       ]
      },
      "metadata": {},
@@ -621,7 +628,47 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "This underscores the fact that these filters are not *smoothing* the data in colloquial sense of the term. The filter is making an optimal estimate based on previous measurements, future measurements, and what you tell it about the behavior of the system and the noise in the system and measurements."
+    "This underscores the fact that these filters are not *smoothing* the data in colloquial sense of the term. The filter is making an optimal estimate based on previous measurements, future measurements, and what you tell it about the behavior of the system and the noise in the system and measurements.\n",
+    "\n",
+    "Let's wrap this up by looking at the velocity estimates of Kalman filter vs the RTS smoother."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 16,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "gu\n"
+     ]
+    },
+    {
+     "data": {
+      "image/png": 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truncated
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x9440588>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "plot_rts(7.,show_velocity=True)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The improvement in the velocity, which is an unobserved variable, is even more dramatic. \n",
+    "\n",
+    "Since we are plotting velocities let's look at what the the 'raw' velocity is, which we can compute by subtracting subsequent measurements. We see below that the noise swamps the signal, causing the computed values to be essentially worthless. I show this to reiterate the importance of using Kalman filters to compute velocities, accelerations, and even higher order values. I use a Kalman filter even when my measurements are so accurate that I am willing to use them unfiltered if I am also interested in the velocities and/or accelerations. Even a very small error in a measurement gets magnified when computing velocities. "
    ]
   },
   {
@@ -789,7 +836,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.4.3"
+   "version": "3.4.1"
   }
  },
  "nbformat": 4,

code/DogSensor.py

@@ -1,26 +0,0 @@
-# -*- coding: utf-8 -*-
-"""
-Created on Sun May 11 13:21:39 2014
-
-@author: rlabbe
-"""
-
-from __future__ import print_function, division
-
-import numpy.random as random
-import math
-
-class DogSensor(object):
-
-    def __init__(self, x0=0, velocity=1, noise_var=0.0):
-        """ x0 - initial position
-            velocity - (+=right, -=left)
-            noise_var - noise variance, 0== no noise
-        """
-        self.x = x0
-        self.velocity = velocity
-        self.noise = math.sqrt(noise_var)
-
-    def sense(self):
-        self.x = self.x + self.velocity
-        return self.x + random.randn() * self.noise

code/DogSimulation.py

@@ -0,0 +1,44 @@
+# -*- coding: utf-8 -*-
+"""
+Created on Sun May 11 13:21:39 2014
+
+@author: rlabbe
+"""
+
+from __future__ import print_function, division
+
+from numpy.random import randn
+import math
+
+class DogSimulation(object):
+
+    def __init__(self, x0=0, velocity=1,
+                 measurement_variance=0.0, process_variance=0.0):
+        """ x0 - initial position
+            velocity - (+=right, -=left)
+            measurement_variance - variance in measurement m^2
+            process_variance - variance in process (m/s)^2
+        """
+        self.x = x0
+        self.velocity = velocity
+        self.measurement_noise = math.sqrt(measurement_variance)
+        self.process_noise = math.sqrt(process_variance)
+
+
+    def move(self, dt=1.0):
+        '''Compute new position of the dog assuming `dt` seconds have
+        passed since the last update.'''
+        # compute new position based on velocity. Add in some
+        # process noise
+        velocity = self.velocity + randn() * self.process_noise
+        self.x += velocity * dt
+
+    def sense_position(self):
+        # simulate measuring the position with noise
+        measurement = self.x + randn() * self.measurement_noise
+        return measurement
+
+    def move_and_sense(self, dt=1.0):
+        self.move(dt)
+
+        return self.sense_position()

code/mkf_internal.py

@@ -12,30 +12,56 @@ from mpl_toolkits.mplot3d import Axes3D
 from numpy.random import multivariate_normal
 import stats
 
-def show_residual_chart():
-    plt.xlim([0.9,2.5])
-    plt.ylim([1.5,3.5])
 
-    plt.scatter ([1,2,2],[2,3,2.3])
-    plt.scatter ([2],[2.8],marker='o')
-    ax = plt.axes()
-    ax.annotate('', xy=(2,3), xytext=(1,2),
+def show_residual_chart():
+    est_y = ((164.2-158)*.8 + 158)
+
+    ax = plt.axes(xticks=[], yticks=[], frameon=False)
+    ax.annotate('', xy=[1,159], xytext=[0,158],
+                arrowprops=dict(arrowstyle='->',
+                                ec='r', lw=3, shrinkA=6, shrinkB=5))
+
+    ax.annotate('', xy=[1,159], xytext=[1,164.2],
+                arrowprops=dict(arrowstyle='-',
+                                ec='k', lw=1, shrinkA=8, shrinkB=8))
+
+    ax.annotate('', xy=(1., est_y), xytext=(0.9, est_y),
                 arrowprops=dict(arrowstyle='->', ec='#004080',
                                 lw=2,
                                 shrinkA=3, shrinkB=4))
-    ax.annotate('prediction', xy=(2.04,3.), color='#004080')
-    ax.annotate('measurement', xy=(2.05, 2.28))
-    ax.annotate('prior estimate', xy=(1, 1.9))
-    ax.annotate('residual', xy=(2.04,2.6), color='#e24a33')
-    ax.annotate('new estimate', xy=(2,2.8),xytext=(2.1,2.8),
-                arrowprops=dict(arrowstyle='->', ec="k", shrinkA=3, shrinkB=4))
-    ax.annotate('', xy=(2,3), xytext=(2,2.3),
-                arrowprops=dict(arrowstyle="-",
-                                ec="#e24a33",
-                                lw=2,
-                                shrinkA=5, shrinkB=5))
-    plt.title("Kalman Filter Predict and Update")
-    plt.axis('equal')
+
+
+    plt.scatter ([0,1], [158.0,est_y], c='k',s=128)
+    plt.scatter ([1], [164.2], c='b',s=128)
+    plt.scatter ([1], [159], c='r', s=128)
+    plt.text (1.0, 158.8, "prediction ($x_t)$", ha='center',va='top',fontsize=18,color='red')
+    plt.text (1.0, 164.4, "measurement ($z$)",ha='center',va='bottom',fontsize=18,color='blue')
+    plt.text (0, 157.8, "prior estimate ($\hat{x}_{t-1}$)", ha='center', va='top',fontsize=18)
+    plt.text (1.02, est_y-1.5, "residual", ha='left', va='center',fontsize=18)
+    plt.text (0.9, est_y, "new estimate ($\hat{x}_{t}$)", ha='right', va='center',fontsize=18)
+    plt.xlabel('time')
+    ax.yaxis.set_label_position("right")
+    plt.ylabel('state')
+    plt.xlim(-0.5, 1.5)
+    plt.show()
+
+
+def plot_gaussian_multiply():
+    xs = np.arange(-5, 10, 0.1)
+
+    mean1, var1 = 0, 5
+    mean2, var2 = 5, 1
+    mean, var = stats.mul(mean1, var1, mean2, var2)
+
+    ys = [stats.gaussian(x, mean1, var1) for x in xs]
+    plt.plot(xs, ys, label='M1')
+
+    ys = [stats.gaussian(x, mean2, var2) for x in xs]
+    plt.plot(xs, ys, label='M2')
+
+    ys = [stats.gaussian(x, mean, var) for x in xs]
+    plt.plot(xs, ys, label='M1 x M2')
+    plt.legend()
     plt.show()
 
 
@@ -329,6 +355,61 @@ def plot_correlation_covariance():
     plt.show()
 
 
+import book_plots as bp
+def plot_track(ps, zs, cov,
+               plot_P=True, y_lim=None,
+               title='Kalman Filter'):
+
+    count = len(zs)
+    actual = np.linspace(0, count - 1, count)
+    cov = np.asarray(cov)
+    std = np.sqrt(cov[:,0,0])
+    std_top = np.minimum(actual+std, [count + 10])
+    std_btm = np.maximum(actual-std, [-50])
+
+    std_top = actual+std
+    std_btm = actual-std
+
+    bp.plot_track(actual)
+    bp.plot_measurements(range(1, count + 1), zs)
+    bp.plot_filter(range(1, count + 1), ps)
+
+    plt.plot(std_top, linestyle=':', color='k', lw=2)
+    plt.plot(std_btm, linestyle=':', color='k', lw=2)
+    plt.fill_between(range(len(std_top)), std_top, std_btm,
+                     facecolor='yellow', alpha=0.2, interpolate=True)
+    plt.legend(loc=4)
+    if y_lim is not None:
+        plt.ylim(y_lim)
+    else:
+        plt.ylim((-50, count + 10))
+
+    plt.xlim((0,count))
+    plt.title(title)
+    plt.show()
+
+    if plot_P:
+        ax = plt.subplot(121)
+        ax.set_title("$\sigma^2_x$")
+        plot_covariance(cov, (0, 0))
+        ax = plt.subplot(122)
+        ax.set_title("$\sigma^2_y$")
+        plot_covariance(cov, (1, 1))
+        plt.show()
+
+def plot_covariance(P, index=(0, 0)):
+    ps = []
+    for p in P:
+        ps.append(p[index[0], index[1]])
+    plt.plot(ps)
+
+
+def test_fill():
+    plt.fill_between([0,1,2,3,4], [1,1,1,1,1], [4,4,4,4,4])
+    plt.plot([0,6], [6,1])
+   # plt.ylim(2,5)
+    plt.show()
+
 if __name__ == "__main__":
     #show_position_chart()
     #plot_3d_covariance((2,7), np.array([[8.,0],[0,4.]]))
@@ -336,5 +417,6 @@ if __name__ == "__main__":
     #show_residual_chart()
 
     #show_position_chart()
-    show_x_error_chart(4)
+    #show_x_error_chart(4)
+    test_fill()
 

code/nonlinear_plots.py

@@ -22,14 +22,10 @@ def plot_nonlinear_func(data, f, gaussian, num_bins=300):
     #b = f(x) - x*m
 
     # compute new mean and variance based on EKF equations
-
-
-
     ys = f(data)
     x0 = gaussian[0]
     in_std = np.sqrt(gaussian[1])
     y = f(x0)
-    #m = np.mean(ys)
     std = np.std(ys)
 
     in_lims = [x0-in_std*3, x0+in_std*3]
@@ -40,7 +36,6 @@ def plot_nonlinear_func(data, f, gaussian, num_bins=300):
     h = np.histogram(ys, num_bins, density=False)
     plt.subplot(2,2,4)
     plt.plot(h[0], h[1][1:], lw=4, alpha=0.5)
-    print(max(h[0]))
     plt.ylim(out_lims[1], out_lims[0])
     plt.gca().xaxis.set_ticklabels([])
     plt.title('output')
@@ -58,8 +53,6 @@ def plot_nonlinear_func(data, f, gaussian, num_bins=300):
     plt.plot(pdf * max(h[0])/max(pdf), xs, lw=1, color='k')
     print(max(norm.pdf(xs)))'''
 
-
-
     # plot transfer function
     plt.subplot(2,2,3)
     x = np.arange(in_lims[0], in_lims[1], 0.1)
@@ -82,7 +75,6 @@ def plot_nonlinear_func(data, f, gaussian, num_bins=300):
     plt.title('input')
 
     plt.show()
-    print("fuck")
 
 
 import math

experiments/test1d.py

@@ -0,0 +1,25 @@
+# -*- coding: utf-8 -*-
+"""
+Created on Thu Jun 18 09:16:54 2015
+
+@author: rlabbe
+"""
+
+
+from filterpy.kalman import KalmanFilter
+
+
+
+kf = KalmanFilter(1, 1)
+kf.x[0,0] = 1.
+kf.P = np.ones((1,1))
+kf.H = np.array([[2.]])
+kf.F = np.ones((1,1))
+kf.R = 1
+kf.Q = 0
+
+
+
+kf.predict()
+kf.update(2)
+