Description of initial value for P.

Added code and text to explain how to choose an initial value for P.

Multidimensional_Kalman_Filters.ipynb

@@ -1,7 +1,7 @@
 {
  "metadata": {
   "name": "",
-  "signature": "sha256:8308d6b4e959bab21d85e3e32855ed8611242c5db237c72712607673761dd677"
+  "signature": "sha256:b799d26a0c11e8a5c71db9928336e67fc5d95fa40acedbb3f1522362e2d3dec0"
  },
  "nbformat": 3,
  "nbformat_minor": 0,
@@ -250,7 +250,7 @@
        "output_type": "pyout",
        "prompt_number": 1,
        "text": [
-        "<IPython.core.display.HTML at 0x2dddcd0>"
+        "<IPython.core.display.HTML at 0x1630c50>"
        ]
       }
      ],
@@ -528,7 +528,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x4450250>"
+        "<matplotlib.figure.Figure at 0x2c9d250>"
        ]
       }
      ],
@@ -572,7 +572,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x4661350>"
+        "<matplotlib.figure.Figure at 0x2eb4350>"
        ]
       }
      ],
@@ -660,7 +660,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x4450110>"
+        "<matplotlib.figure.Figure at 0x2f27b90>"
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@@ -687,7 +687,7 @@
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        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x46d4b90>"
+        "<matplotlib.figure.Figure at 0x2eb39d0>"
        ]
       }
      ],
@@ -714,7 +714,7 @@
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        "text": [
-        "<matplotlib.figure.Figure at 0x4688310>"
+        "<matplotlib.figure.Figure at 0x2edb050>"
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x4653b50>"
+        "<matplotlib.figure.Figure at 0x2e387d0>"
        ]
       }
      ],
@@ -798,7 +798,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x4682310>"
+        "<matplotlib.figure.Figure at 0x32590d0>"
        ]
       }
      ],
@@ -1140,7 +1140,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 17
+     "prompt_number": 16
     },
     {
      "cell_type": "markdown",
@@ -1168,7 +1168,7 @@
       "\n",
       "$$\\begin{bmatrix} 500&0\\\\0&500\\end{bmatrix}$$\n",
       "\n",
-      "This will become much clearer once we look at the covariance matrix in detail in later sessions. For now recognize that each diagonal element $e_{ii}$ is the variance for the $ith$ state variable. \n",
+      "This will become much clearer once we look at the covariance matrix in detail in later sessions. For now recognize that each diagonal element $e_{ii}$ is the variance for the $ith$ state variable, and that the *500* is just a statement that we are very uncertain about our initial value by some *amount*. \n",
       "\n",
       "> Summary: For our dog tracking problem, in the 1-D case $\\mu$ was the position, and $\\sigma^2$ was the variance. In the 2-D case $\\mathbf{x}$ is our position and velocity, and $\\mathbf{P}$ is the *covariance* of the position and velocity. It is the same thing, just in higher dimensions!\n",
       "\n",
@@ -1180,6 +1180,7 @@
      "cell_type": "code",
      "collapsed": false,
      "input": [
+      "import numpy as np\n",
       "from DogSensor import DogSensor\n",
       "import filterpy.kalman as kf\n",
       "\n",
@@ -1200,7 +1201,7 @@
       "    return dog_filter\n",
       "\n",
       "\n",
-      "def filter_dog(noise=0, count=0, R=0, Q=0, data=None):\n",
+      "def filter_dog(noise=0, count=0, R=0, Q=0, P=500., data=None, initial_x=None):\n",
       "    \"\"\" Kalman filter 'count' readings from the DogSensor.\n",
       "    'noise' is the noise scaling factor for the DogSensor.\n",
       "    'data' provides the measurements. If set, noise will\n",
@@ -1214,7 +1215,9 @@
       "    else:\n",
       "        zs = data\n",
       "\n",
-      "    dog_filter = dog_tracking_filter(R=R, Q=Q, cov=500.)\n",
+      "    dog_filter = dog_tracking_filter(R=R, Q=Q, cov=P)\n",
+      "    if initial_x is not None:\n",
+      "        dog_filter.x = initial_x\n",
       "\n",
       "    pos = [None] * count\n",
       "    cov = [None] * count\n",
@@ -1233,7 +1236,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 22
+     "prompt_number": 44
     },
     {
      "cell_type": "markdown",
@@ -1273,9 +1276,11 @@
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "def plot_track(noise=None, count=0, R=0, Q=0, data=None, plot_P=True, title='Kalman Filter'):\n",
+      "def plot_track(noise=None, count=0, R=0, Q=0, P=500., initial_x=None,\n",
+      "               data=None, plot_P=True, title='Kalman Filter'):\n",
       "    \n",
-      "    ps, zs, cov = filter_dog(noise=noise, data=data, count=count, R=R, Q=Q)\n",
+      "    ps, zs, cov = filter_dog(noise=noise, data=data, count=count,\n",
+      "                             R=R, Q=Q, P=P, initial_x=initial_x)\n",
       "    \n",
       "    p0, = plt.plot([0,count],[0,count],'g')\n",
       "    p1, = plt.plot(range(1,count+1),zs,c='r', linestyle='dashed')\n",
@@ -1301,7 +1306,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 23
+     "prompt_number": 47
     },
     {
      "cell_type": "markdown",
@@ -1322,9 +1327,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x4a10610>"
+        "<matplotlib.figure.Figure at 0x324abd0>"
        ]
       },
       {
@@ -1332,11 +1337,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x4534b50>"
+        "<matplotlib.figure.Figure at 0x3392850>"
        ]
       }
      ],
-     "prompt_number": 24
+     "prompt_number": 48
     },
     {
      "cell_type": "markdown",
@@ -1446,7 +1451,8 @@
      ],
      "language": "python",
      "metadata": {},
-     "outputs": []
+     "outputs": [],
+     "prompt_number": 20
     },
     {
      "cell_type": "heading",
@@ -1479,7 +1485,25 @@
      ],
      "language": "python",
      "metadata": {},
-     "outputs": []
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x3384590>"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x3384810>"
+       ]
+      }
+     ],
+     "prompt_number": 21
     },
     {
      "cell_type": "markdown",
@@ -1584,7 +1608,17 @@
      ],
      "language": "python",
      "metadata": {},
-     "outputs": []
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAsYAAAF9CAYAAAAHja6eAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\nAAALEgAACxIB0t1+/AAAIABJREFUeJzs3Xd4U2X7wPFvRtPdAmWP2rKhLbZMQURBoAICAopseFEB\nRQUVZSn4Q9yiLyooQ0DWy5IhKkvAgaxSECgFfBURsNCW1d2mSc7vj9q+QFfaJk1Oc3+uiwuTnPOc\nO7mN3n16n+fRnD17VkEIIYQQQggXp3V0AEIIIYQQQjgDKYyFEEIIIYRACmMhhBBCCCEAKYyFEEII\nIYQApDAWQgghhBACkMJYCCGEEEIIQApjIYQQQgghACmMhRBOomnTpnl/2rZty6BBg9i1a5dDYlmw\nYAGDBw8mPDycpk2bFnjMhg0b6Nq1Ky1atOCxxx7jxIkTt72emprK1KlTad26Na1bt2batGlkZGTc\ndszx48cZMGAALVq0oFu3bmzatMku7+fHH3+kT58+hIWF0aVLF5YsWWKX6xTHms+kOL///jsTJkzg\nwQcfpGnTpnz66ad2ilYI4YqkMBZCOI3p06fzyy+/sHr1aho3bsyECRPyFZzlISMjg8jISAYPHoxG\no8n3+v79+3nttdcYMWIEmzZton79+jz11FMkJSXlHfP6669z+PBhFi5cyIIFCzhw4ACzZs3Ke/36\n9es8+eSTNG7cmE2bNjF06FCmTZvGoUOHbPpe/vvf/zJ+/HjatGnDli1bGDduHHPmzOGrr76y6XWs\nUdxnYo309HRq167Niy++SNWqVe0UqRDCVUlhLIRwGr6+vgQEBNCwYUNeeeUVLBYLR48eLfc4Jk6c\nyKhRo2jUqBGKkn9z0FWrVtGhQwdGjBhBgwYNmDVrFkajka1btwJw8+ZNtm3bxosvvkjLli1p1aoV\nEyZMYOvWraSkpACwdetWLBYLs2bNokGDBowaNYq2bduyatUqm76XtWvXUqlSJV577TXq16/PwIED\n6dq1q82vUxxrPhNrtGjRgsmTJ9OrVy8MBoMdIxZCuCK9owMQQohcuUVodnY2X331FVqtlvDwcKvO\njYuLo1evXoW+vnjxYlq1amWTOGNiYhg0aFDeY3d3d1q0aJE3ux0bG4vZbKZ169Z5x7Rt2xaTycSp\nU6e45557OHnyJC1atMDNzS3vmDZt2rBhwwabxHhrrKGhobc9Fxoayvfff092dvZt1y9Mr169iIuL\nK/C1p59+mjFjxhQ7hjWfiRBCOJoUxkIIpzFz5kxmzZpFVlYWwcHBLF++3OrCuEaNGnz99deFvl69\nenVbhcm1a9eoXLky33//PdOmTWPJkiVUrlyZa9eu5b0OUKlSJZ5//nmuXLnCihUrgJwWity/K1eu\nzJUrV3jkkUd47rnnqFKlSt65tnL9+nXuuuuu256rVKkSZrOZpKQkq9oRFi9ejMlkKvA1f39/q+Kw\n5jMRQghHk8JYCOE0JkyYQLdu3Thz5gxTp04lNjb2thnGouh0OurVq2fnCG/n7e1N7dq18fT0BCiw\nH7l69epotYV3rbm5uVGnTh0qVap0W4+yrRQUU66i4rpVrVq1bBUOUPxnIoQQjiKFsRDCaQQEBBAY\nGEhgYCBXrlxh7ty59OnTh0qVKhV7bnm2UgQEBHDjxg0GDRrE5s2bgZxZz9wCMiAgAMjpq3311Vfz\n4gOoUqVK3jGJiYkEBATk3Qj36aef2vyGsipVqnDz5s3bnrt58yZardbq2V5btFJY85kIIYSjSWEs\nhHBKgwYN4rPPPmP58uU8//zzxR5fnq0UYWFhHDlyJO9xZmYmJ06cIDIyEoBmzZqh0+mIiori4Ycf\nBuDw4cPo9XpCQkLyxvj3v/+N0WjMu4ns8OHDhIWF2SxOyOkn3rZtG4qi5M0enzx5ktDQUHQ6nVVj\nFNVK4efnZ9UY1nwmuYxGIwkJCfj7++Pr62vV+EIIYQvyuywhhFMyGAwMHDiQVatWWbXWbW4rRWF/\n3N3drb52XFwcp0+fzpvRPHPmDKdPnyY7OxuAIUOGsH//fpYvX87vv//OzJkz8fDwyCv4KleuTM+e\nPfnwww+Jjo7myJEjzJ07l969e+cVer1790av1zNz5kx+//13li1bRlRUFMOGDcsXzyeffELTpk2J\nioqy+j3kevzxx7l58yZvvPEGf/zxB2vXrmX37t0MHTrU6jFq1apV6Odq7ayzNZ9JrmPHjtG1a1e+\n/PLLfONkZ2dz+vRpTp8+jdFoJDExkdOnT3P58mWr348QQhRGCmMhhNMaOnQoaWlprF+/vlyv+/HH\nH9OvXz8+/fRTNBoNjzzyCP379ycxMRGADh06MHv2bJYvX06/fv04d+4cixYtuq1IfP3112nXrh1j\nx45l3LhxtG/fnhkzZuS9XrlyZRYtWsR///tf+vXrx6pVq3j77bdp27ZtvnjS09PRaDR57Qgl0bBh\nQ+bNm0dUVBR9+/Zl5syZjB49mr59+5bikymb4j6TWxXWGx0fH0+/fv3o168f165dY+3atfTr14+P\nP/7YnqELIVyE5uzZs/kX6bRSamoqDz30EKNHj2b06NF89913zJ07F4ApU6bQuXNnmwUqhBCuasCA\nAdSpU8cmxd/UqVM5ePAgS5YsITg42AbRCSFExVGmHuPPP/+c0NBQNBoNRqOROXPmsH79erKyshgx\nYoQUxkIIUUZJSUmcPXuW9957zybjvfXWWyxatAgvLy+bjCeEEBVJqQvjc+fOcf36dUJDQ1EUhRMn\nTtCoUaO8u4tr1qzJmTNnaNq0qc2CFUIIV+Pv709MTIzNxtNoNFatIiGEEK6o1D3GH374Ic8991ze\n46tXr1KtWjXWrFnDtm3bqFatGgkJCTYJUgghhBBCCHsr1Yzxnj17CAoKolatWnlbuObK3SZ1165d\nRS4sL4QQQgghhDMpVWF84sQJdu7cye7du7lx4wZarZYhQ4bk3bENkJiYSLVq1fKde+zYMZsvYC+E\nEEIIIcSdUlJSaN68udXHl6ownjhxIhMnTgRydmry9vZm2LBhPPTQQ1y/fp2srCzi4+ML7C+uWrUq\nLVu2LM1lhRPYunUrvXv3dnQYohQkd+om+VMvyZ26Sf7U7ejRoyU63mY737m5ufHSSy8xePBgAKZN\nm2aroYUTuXDhgqNDEKUkuVM3yZ96Se7UTfLnWspcGD/77LN5/9yzZ0969uxZ1iGFEEIIIYQod7Lz\nnSiRrl27OjoEUUqSO3WT/KmX5E7dJH+uRQpjUSJNmjRxdAiilCR36ib5Uy/JnbpJ/lyLzXqMy0pR\nFBISEjCbzY4ORRQhKSkJf3//Ep+n0+moXr26LOHnQPv27aNjx46ODkOUkuRPvSR36ib5cy1OUxgn\nJCTg6+sr25Q6udq1a5fqvPT0dBISEqhRo4aNIxJCCCGEsA2naaUwm81SFFdgXl5e8tsAB5MZD3WT\n/KmX5E7dJH+uxWkKYyGEEEIIIRxJCmMhXMS+ffscHYIoA8mfeknu1E3y51qkMFaZ++67j3r16hEQ\nEIDFYnF0OEIIIYQQFYYUxnYWEBDA+fPnbTbezz//zP79+202nnAd0ienbpI/9ZLcqZvkz7VIYWxH\niqLc9retxxVCCCGEELYjhbEVPv74Y1q2bEndunVp3bo1mzdvznvtl19+oWvXrgQFBdGuXbu82dzH\nHnuMu+66C4BOnToRGBjI9OnTgZx9129thdi3bx+hoaFWXU+I0pI+OXWT/KmX5E7dJH+uxWnWMXZm\nlSpVYv369TRo0ICdO3cycuRIOnXqREpKCkOGDGHhwoV0796dP/74g2vXrgGwfv16IKeV4ueffyYo\nKKjM16tSpYo93p4QQgghhEAKY6uMGDEi75+7d++Ov78/Z8+e5cCBA3Tp0oXIyEgAGjZsSMOGDe1y\nvd9++4177rmnzGML1yV9cuom+VMvyZ26Sf5ci2oK4yof22a29Prz10t8ztq1a5k3bx5///03FouF\nlJQUsrOziYuLo169ejaJy5rrCSGEEEII+1FNYVyagtYWLl68yMSJE/n6669p06YNAA0aNEBRFOrU\nqcPx48eLPF+j0eR7zt3dHQCTyYTBYCAlJcWq6+Vyc3MDcnYL1GqlTVxYZ9++fTLzoWKSP/WS3Kmb\n5M+1SFVVjLS0NDQaDVWrVsVkMvHJJ5+QlJSERqNhwIAB7Nmzh+3bt2M2mzl37hyHDh267fwaNWoQ\nGxt723NVq1bFz8+P6OhoAL7++utir3er6tWr4+fnxy+//GKndy2EEEII4XqkMC5G06ZNGT9+PF27\ndqV58+akpaXltU8EBgayatUqPvjgA+rXr8/gwYPzbbrx6quv8sorrxASEsLs2bMB0Ol0zJo1izFj\nxtCrVy+qVq2aN7Nc1PVy6XQ63n//fcaNG0dgYCDbtm0rh09CqJ3MeKib5E+9JHfqJvlzLZqzZ8+W\n66K4Fy9epGXLlvmej4uLo3bt2uUZiihnkmMhhBBClKejR4+W6H4wmTEWwkXIWpzqJvlTL8mdukn+\nXIsUxkIIIYQQQiCFsRAuQ/rk1E3yp16SO3WT/LkWKYyFEEIIIYRACmMhXIb0yamb5E+9JHfqJvlz\nLVIYCyGEEEIIgRTGQrgM6ZNTN8mfeknu1E3y51qkMBZCCCGEEAIpjIVwGdInp26SP/WS3Kmb5M+1\nSGEsbCIgIIDz5887OgwhhBBCuDpFwWPWLMjKKvGpUhiLMlMU5ba/hXOSPjl1k/ypl+RO3SR/6qOJ\nj8d9+XIwGEp8rhTGxVi9ejVdunQhJCSE0aNHM3jwYJo1a0ZsbCwWi4V3332X8PBwmjZtypQpUzCZ\nTAD89ddf9O3bl/r163PXXXfxr3/9i+Tk5Lxxd+zYQdu2bQkMDKRNmzbs2bMn77W7776bH3Line truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x2db6b50>"
+       ]
+      }
+     ],
+     "prompt_number": 22
     },
     {
      "cell_type": "markdown",
@@ -1643,7 +1677,17 @@
      ],
      "language": "python",
      "metadata": {},
-     "outputs": []
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x3392450>"
+       ]
+      }
+     ],
+     "prompt_number": 23
     },
     {
      "cell_type": "markdown",
@@ -1661,7 +1705,17 @@
      ],
      "language": "python",
      "metadata": {},
-     "outputs": []
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAr0AAAGNCAYAAAActsnoAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x2da7290>"
+       ]
+      }
+     ],
+     "prompt_number": 24
     },
     {
      "cell_type": "markdown",
@@ -1692,18 +1746,164 @@
       "\n",
       "In general, we can state that $\\small\\mathbf{P}_{i,j}=\\small\\mathbf{P}_{j,i}$.\n",
       "\n",
-      "So for my example I multiplied the diagonals, 2 and 6, to get 12, and then scaled that with the arbitrarily chosen $p=.2$ to get 2.4.\n",
-      "\n",
-      "Let's get back to concrete terms. Lets do another Kalman filter for our dog, and this time plot the covariance ellipses on the same plot as the position."
+      "So for my example I multiplied the diagonals, 2 and 6, to get 12, and then scaled that with the arbitrarily chosen $p=.2$ to get 2.4."
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Let's get back to concrete terms. Let's start by revisting plotting a track. I will hard code the data and noise to avoid being at the mercy of the random number generator, which might generate data that does not illustrate what I want to talk about. I will start by using the same parameters as a chart above: `R=5`, `Q=.02`, and `P=500`."
      ]
     },
     {
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "def plot_track(noise, count, R, Q=0, plot_P=True, title='Kalman Filter'):\n",
+      "# guarantee the noise is the same each time so I can be sure of\n",
+      "# what the graphs look like.\n",
+      "zs = [-6.947, 12.467, 6.899, 2.643, 6.980, 5.820, 5.788, 10.614, 5.210, \n",
+      "      14.338, 11.401, 19.138, 14.169, 19.572, 25.471, 13.099, 27.090,\n",
+      "      12.209, 14.274, 21.302, 14.678, 28.655, 15.914, 28.506, 23.181, \n",
+      "      18.981, 28.197, 39.412, 27.640, 31.465, 34.903, 28.420, 33.889, \n",
+      "      46.123, 31.355, 30.473, 49.861, 41.310, 42.526, 38.183, 41.383, \n",
+      "      41.919, 52.372, 42.048, 48.522, 44.681, 32.989, 37.288, 49.141, \n",
+      "      54.235, 62.974, 61.742, 54.863, 52.831, 61.122, 61.187, 58.441, \n",
+      "      47.769, 56.855, 53.693, 61.534, 70.665, 60.355, 65.095, 63.386]\n",
+      "\n",
+      "plot_track (data=zs, R=5, Q=.02, P=500., count=len(zs), plot_P=False, title='P=500')"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAsYAAAF9CAYAAAAHja6eAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x2dc88d0>"
+       ]
+      }
+     ],
+     "prompt_number": 42
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Looking at the output we see a very large spike in the filter output at the beginning. If you look at the data (dotted red line) you will see a corresponding, smaller spike in the beginning of the data. We set `P=500`, which corresponds to $\\text{P}=[\\begin{smallmatrix}500&0\\\\0&500\\end{smallmatrix}]$. We now have enough information to understand what this means, and how the Kalman filter treats it. The 500 in the upper left hand corner corresponds to $\\sigma^2_x$; therefore we are saying the standard deviation of `x` is $\\sqrt{500}$, or roughly 22.36 meters, assuming our measurements are in meters. If we recall how standard deviations work, roughly 99% of the samples occur withing $3\\sigma$, therefore `P=500` is telling the Kalman filter that the initial estimate could be up to 67 meters off. That is a pretty large error, so when the measurement spikes the Kalman filter distrusts its own estimate and jumps wildly to try to incorporate the measurement. Then, as the filter evolves $\\mathbf{P}$ quickly converges to a more realistic value.\n",
+      "\n",
+      "Now let us see the effect of a smaller initial value for `P`."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "plot_track (data=zs, R=5, Q=.02, P=1., count=len(zs), plot_P=False, title='P=1')"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAsYAAAF9CAYAAAAHja6eAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x324acd0>"
+       ]
+      }
+     ],
+     "prompt_number": 45
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "This plot does not have the spike that the former plot did; the filter starts tracking the measurements and doesn't take any time to 'settle' to the signal.\n",
+      "\n",
+      "Do not conclude from this that the 'magic' is to just use a small $\\text{P}$. Yes, this will avoid having the Kalman filter take time to accurately track the signal, but if we are truly uncertain about the initial measurements this can cause the filter to generate very bad results. If we are tracking a living object we are probably very uncertain about where it is before we start tracking it. On the other hand, if we are filtering the output of a thermometer, we are just as certain about the first measurement as the 1000th. For your Kalman filter to perform well you must set $\\text{P}$ to a value that truly reflects your knowledge about the data. \n",
+      "\n",
+      "\n",
+      "Let's see the result of a bad initial estimate coupled with a very small $\\text{P}$ We will set our initial estimate at 100m (whereas the dog starts at 0m), but set $\\text{P=1}$.\n"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "x = np.array([[100,0]]).T\n",
+      "plot_track(data=zs, R=5, Q=.02, P=1., initial_x=x,count=len(zs),\n",
+      "           plot_P=False, title='P=1')"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAsYAAAF9CAYAAAAHja6eAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x2ea89d0>"
+       ]
+      }
+     ],
+     "prompt_number": 55
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "We can see that the initial estimates are terrible, up to around step 15 to 20. This is because we told the Kalman filter that we strongly believe in our initial estimate of 100m.\n",
+      "\n",
+      "Now, let's provide a more reasonable value for P and see the difference."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "x = np.array([[100,0]]).T\n",
+      "plot_track(data=zs, R=5, Q=.02, P=500., initial_x=x,count=len(zs),\n",
+      "           plot_P=False, title='P=500')"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAsYAAAF9CAYAAAAHja6eAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x15a11d0>"
+       ]
+      }
+     ],
+     "prompt_number": 58
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "In this case the Kalman filter is very uncertain about the initial state, so it converges onto the signal much faster. It is producing good output after only 5 to 6 evolutions. With the theory we have developed so far this is about as good as we can do. However, this scenario is a bit artificial; if we do not know where the object is when we start tracking we do not initialize the filter to some arbitrary value, such as 0 or 100. Instead, we would normally take the first measurement, use that to initialize the Kalman filter, and proceed from there. But this is an engineering decision. You really need to understand the domain in which you are working and initialize your filter on the best available information. For example, suppose we were trying to track horses in a horse race. The initial measurements might be very bad, and provide you with a position far from the starting gate. We know that the horse must of started at the starting gate; initializing the filter to the initial measurement would lead to suboptimal results. In this scenario we would want to always initialize the Kalman filter with the starting gate position. \n",
+      "\n",
+      "If we have the luxury of not needing to perform the filtering in real time, as the data comes in, we can take advantage of other techniques. We can 'eyeball' the data and see that  the initial measurements are giving us reasonable values for the dog's position because we can see all of the data at once. A `fixed lag smoother` will look N steps ahead before computing the state, and other filters will do things like first run forwards, than backwards over the data. This will be the subject of later chapters. It is worthwhile to keep in mind that whenever possible we should prefer this sort of batch processing because it takes advantage of all available information. It does incur cost of additional processing time and increased storage due to the requirement to store some or all of the measurements. And, of course, batch processing does not work if we need real time results, such as when using GPS in our car."
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Lets do another Kalman filter for our dog, and this time plot the covariance ellipses on the same plot as the position."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "def plot_track_ellipses(noise, count, R, Q=0, P=20., plot_P=True, title='Kalman Filter'):\n",
       "    dog = DogSensor(velocity=1, noise=noise)\n",
-      "    f = dog_tracking_filter(R=R, Q=Q, cov=20.)\n",
+      "    f = dog_tracking_filter(R=R, Q=Q, cov=P)\n",
       "\n",
       "    ps = []\n",
       "    zs = []\n",
@@ -1737,12 +1937,30 @@
       "    plt.show()\n",
       "\n",
       "\n",
-      "plot_track (noise=5, R=5, Q=5, count=20, title='R = 5')\n",
-      "plot_track (noise=5, R=.5, Q=5, count=20, title='R = 0.5')"
+      "plot_track_ellipses (noise=5, R=5, Q=5, count=20, title='R = 5')\n",
+      "plot_track_ellipses (noise=5, R=.5, Q=5, count=20, title='R = 0.5')"
      ],
      "language": "python",
      "metadata": {},
-     "outputs": []
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x2edb310>"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAsAAAAF9CAYAAAAKk97ZAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x32c1990>"
+       ]
+      }
+     ],
+     "prompt_number": 59
     },
     {
      "cell_type": "markdown",
@@ -1795,11 +2013,21 @@
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "plot_track (noise=0, R=500, Q=5, count=7, title='R = 500')"
+      "plot_track_ellipses (noise=0, R=500, Q=5, count=7, title='R = 500')"
      ],
      "language": "python",
      "metadata": {},
-     "outputs": []
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": "iVBORw0KGgoAAAANSUhEUgAAA3wAAAF9CAYAAABf12quAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x331bcd0>"
+       ]
+      }
+     ],
+     "prompt_number": 60
     },
     {
      "cell_type": "markdown",