Added labels to the charts.

08_Designing_Kalman_Filters/Designing_Kalman_Filters.ipynb

@@ -1,7 +1,7 @@
 {
  "metadata": {
   "name": "",
-  "signature": "sha256:7218b6288b0001ce5f7fcd6a56f34c39179f46962e827a90476ae20391d95362"
+  "signature": "sha256:ef59d8c8332c0a051e5c4f822df55c70336f15032f80a2ebc3511a42b20b9ddc"
  },
  "nbformat": 3,
  "nbformat_minor": 0,
@@ -259,13 +259,13 @@
        ],
        "metadata": {},
        "output_type": "pyout",
-       "prompt_number": 2,
+       "prompt_number": 1,
        "text": [
-        "<IPython.core.display.HTML at 0x7f979bfc7a58>"
+        "<IPython.core.display.HTML at 0x7f89ce782128>"
        ]
       }
      ],
-     "prompt_number": 2
+     "prompt_number": 1
     },
     {
      "cell_type": "heading",
@@ -354,9 +354,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd561729780>"
+        "<matplotlib.figure.Figure at 0x7f89ce772dd8>"
        ]
       }
      ],
@@ -707,9 +707,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd5617297f0>"
+        "<matplotlib.figure.Figure at 0x7f89ce7820b8>"
        ]
       }
      ],
@@ -788,9 +788,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd56137d358>"
+        "<matplotlib.figure.Figure at 0x7f89b1816eb8>"
        ]
       },
       {
@@ -875,7 +875,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd55d1ea320>"
+        "<matplotlib.figure.Figure at 0x7f89a9ec8cc0>"
        ]
       }
      ],
@@ -960,17 +960,17 @@
       "        self.x = x0\n",
       "        self.vel = vel\n",
       "        self.noise_scale = noise_scale\n",
-      "    \n",
+      "\n",
+      "\n",
       "    def update(self):\n",
       "        self.vel += randn()*self.noise_scale\n",
       "        self.x += self.vel\n",
       "        return (self.x, self.vel)\n",
-      "    \n",
+      "\n",
       "\n",
       "def sense(x, noise_scale=1.):\n",
       "    return x[0] + randn()*noise_scale\n",
-      "    \n",
-      "    \n",
+      "\n",
       "\n",
       "obj = ConstantVelocityObject()\n",
       "\n",
@@ -981,8 +981,8 @@
       "    z = sense(x)\n",
       "    xs.append(x)\n",
       "    zs.append(z)\n",
-      "    \n",
-      "    \n",
+      "\n",
+      "\n",
       "xs = np.asarray(xs)\n",
       "plt.plot(xs[:,0], label='track')\n",
       "plt.plot(zs, label='measurements')\n",
@@ -997,7 +997,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd5611d8240>"
+        "<matplotlib.figure.Figure at 0x7f89b1821f28>"
        ]
       }
      ],
@@ -1186,14 +1186,14 @@
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "def SecondOrderKF(R, Q, dt):\n",
+      "def SecondOrderKF(R_std, Q, dt):\n",
       "    \"\"\" Create zero order Kalman filter. Specify R and Q as floats.\"\"\"\n",
       "    kf = KalmanFilter(dim_x=3, dim_z=1)\n",
       "    kf.x = np.zeros(3)\n",
       "    kf.P[0,0] = 100\n",
       "    kf.P[1,1] = 1\n",
       "    kf.P[2,2] = 1\n",
-      "    kf.R *=R\n",
+      "    kf.R *= R_std**2\n",
       "    kf.Q = Q_discrete_white_noise(3, dt, Q)\n",
       "    kf.F = np.array([[1., dt, .5*dt*dt],\n",
       "                     [0., 1.,       dt],\n",
@@ -1204,7 +1204,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 17
+     "prompt_number": 75
     },
     {
      "cell_type": "heading",
@@ -1247,7 +1247,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 18
+     "prompt_number": 76
     },
     {
      "cell_type": "markdown",
@@ -1280,7 +1280,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 19
+     "prompt_number": 77
     },
     {
      "cell_type": "markdown",
@@ -1298,12 +1298,14 @@
       "    plt.plot(filter_xs[:,0], label='filter')\n",
       "    plt.plot(zs, label='measurements', c='k', alpha=0.5)\n",
       "    plt.legend(loc='best')\n",
+      "    plt.ylabel('meters')\n",
+      "    plt.xlabel('time (sec)')\n",
       "    plt.show()"
      ],
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 20
+     "prompt_number": 78
     },
     {
      "cell_type": "markdown",
@@ -1332,13 +1334,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAzEAAAGkCAYAAAABoqtZAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd5610c8b70>"
+        "<matplotlib.figure.Figure at 0x7f89a8593898>"
        ]
       }
      ],
-     "prompt_number": 21
+     "prompt_number": 79
     },
     {
      "cell_type": "markdown",
@@ -1351,27 +1353,30 @@
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "def plot_residuals(xs, filter_xs, Ps, title):\n",
+      "def plot_residuals(xs, filter_xs, Ps, title, y_label):\n",
       "    res = xs - filter_xs\n",
       "    plt.plot(res)\n",
       "\n",
-      "    third_std = 3*Ps\n",
-      "    plt.plot(-third_std, c='k', ls=':')\n",
-      "    plt.plot(third_std, c='k', ls=':')\n",
+      "    std = np.sqrt(Ps)\n",
+      "    plt.plot(-std, c='k', ls=':')\n",
+      "    plt.plot(std, c='k', ls=':')\n",
       "    plt.title(title)\n",
+      "    plt.ylabel(y_label)\n",
+      "    plt.xlabel('time (sec)')\n",
       "    plt.show()"
      ],
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 22
+     "prompt_number": 80
     },
     {
      "cell_type": "code",
      "collapsed": false,
      "input": [
       "plot_residuals(xs[:,0], fxs1[:,0], ps1[:,0], \n",
-      "               'First Order Position Residuals')   "
+      "               'First Order Position Residuals',\n",
+      "               'meters')   "
      ],
      "language": "python",
      "metadata": {},
@@ -1379,19 +1384,19 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": "iVBORw0KGgoAAAANSUhEUgAAA0AAAAGxCAYAAABLMtbXAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd55d157ba8>"
+        "<matplotlib.figure.Figure at 0x7f89b1848630>"
        ]
       }
      ],
-     "prompt_number": 23
+     "prompt_number": 81
     },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "How do we interpret this plot? The residual is drawn as the jagged line - the difference between the measurement and the actual position. If there was no measurement noise and the Kalman filter prediction was always perfect the residual would always be zero. So the ideal output would be a horizontal line at 0. We can see that the residual is centered around 0, so this gives us confidence that the noise is Gaussian (because the errors fall equally above and below 0). The dotted lines show the error that is allowed for 3 standard deviations. The residual falls within this range, so we see that the filter is performing well, and that it is not diverging.\n",
+      "How do we interpret this plot? The residual is drawn as the jagged line - the difference between the measurement and the actual position. If there was no measurement noise and the Kalman filter prediction was always perfect the residual would always be zero. So the ideal output would be a horizontal line at 0. We can see that the residual is centered around 0, so this gives us confidence that the noise is Gaussian (because the errors fall equally above and below 0). The dotted lines show the theoretical performance of the filter for 1 standard deviations. In other words, approximately 68% of the errors should fall within the dotted lines. The residual falls within this range, so we see that the filter is performing well, and that it is not diverging.\n",
       "\n",
       "But that is just for position. Let's look at the residuals for velocity. "
      ]
@@ -1401,7 +1406,8 @@
      "collapsed": false,
      "input": [
       "plot_residuals(xs[:,1], fxs1[:,1], ps1[:,1], \n",
-      "               'First Order Velocity Residuals')   "
+      "               'First Order Velocity Residuals',\n",
+      "               'meters/sec')   "
      ],
      "language": "python",
      "metadata": {},
@@ -1409,13 +1415,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7fd56145dcf8>"
+        "<matplotlib.figure.Figure at 0x7f89a9ed5128>"
        ]
       }
      ],
-     "prompt_number": 24
+     "prompt_number": 82
     },
     {
      "cell_type": "markdown",
@@ -1442,13 +1448,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd55d14d470>"
+        "<matplotlib.figure.Figure at 0x7f89a84faef0>"
        ]
       }
      ],
-     "prompt_number": 25
+     "prompt_number": 83
     },
     {
      "cell_type": "markdown",
@@ -1464,7 +1470,8 @@
      "collapsed": false,
      "input": [
       "plot_residuals(xs[:,0], fxs0[:,0], ps0[:,0], \n",
-      "               'Zero Order Position Residuals')   "
+      "               'Zero Order Position Residuals',\n",
+      "               'meters')"
      ],
      "language": "python",
      "metadata": {},
@@ -1472,13 +1479,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd55d0b5f60>"
+        "<matplotlib.figure.Figure at 0x7f89a8504b70>"
        ]
       }
      ],
-     "prompt_number": 26
+     "prompt_number": 84
     },
     {
      "cell_type": "markdown",
@@ -1511,13 +1518,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7fd5586f7d30>"
+        "<matplotlib.figure.Figure at 0x7f89a84e3240>"
        ]
       }
      ],
-     "prompt_number": 27
+     "prompt_number": 85
     },
     {
      "cell_type": "markdown",
@@ -1542,6 +1549,8 @@
       "plt.plot(ps2[:,0]*3, c='k', ls=':')\n",
       "plt.title('Second Order Position Residuals')\n",
       "plt.legend()\n",
+      "plt.ylabel('meters')\n",
+      "plt.xlabel('time (sec)')\n",
       "plt.show()"
      ],
      "language": "python",
@@ -1550,13 +1559,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd55d0af0f0>"
+        "<matplotlib.figure.Figure at 0x7f89a8358550>"
        ]
       }
      ],
-     "prompt_number": 28
+     "prompt_number": 86
     },
     {
      "cell_type": "markdown",
@@ -1580,6 +1589,8 @@
       "plt.plot(ps2[:,1]*3, c='k', ls=':')\n",
       "plt.title('Second Order Velocity Residuals')\n",
       "plt.legend()\n",
+      "plt.ylabel('meters/sec')\n",
+      "plt.xlabel('time (sec)')\n",
       "plt.show()"
      ],
      "language": "python",
@@ -1588,13 +1599,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd55d1457f0>"
+        "<matplotlib.figure.Figure at 0x7f89b1518dd8>"
        ]
       }
      ],
-     "prompt_number": 29
+     "prompt_number": 87
     },
     {
      "cell_type": "markdown",
@@ -1625,13 +1636,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd558738f98>"
+        "<matplotlib.figure.Figure at 0x7f89a8358240>"
        ]
       }
      ],
-     "prompt_number": 30
+     "prompt_number": 88
     },
     {
      "cell_type": "markdown",
@@ -1654,7 +1665,8 @@
       "\n",
       "plot_track(xs500, fxs2, zs500)\n",
       "plot_residuals(xs500[:,0], fxs2[:,0], ps2[:,0], \n",
-      "               'Zero Order Position Residuals') "
+      "               'Zero Order Position Residuals',\n",
+      "               'meters') "
      ],
      "language": "python",
      "metadata": {},
@@ -1662,21 +1674,21 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7fd55d097358>"
+        "<matplotlib.figure.Figure at 0x7f89b15a4c18>"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd5614562b0>"
+        "<matplotlib.figure.Figure at 0x7f89b15a4400>"
        ]
       }
      ],
-     "prompt_number": 31
+     "prompt_number": 89
     },
     {
      "cell_type": "markdown",
@@ -1693,15 +1705,16 @@
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "np.random.seed(25944)\n",
-      "xs500, zs500 = simulate_system(Q=0.0001, count=500)\n",
+      "np.random.seed(32594)\n",
+      "xs2000, zs2000 = simulate_system(Q=0.0001, count=2000)\n",
       "\n",
       "kf2 = SecondOrderKF(R, 0, dt=1)\n",
-      "fxs2, ps2 = filter_data(kf2, zs500)\n",
+      "fxs2, ps2 = filter_data(kf2, zs2000)\n",
       "\n",
-      "plot_track(xs500, fxs2, zs500)\n",
-      "plot_residuals(xs500[:,0], fxs2[:,0], ps2[:,0], \n",
-      "               'Zero Order Position Residuals') "
+      "plot_track(xs2000, fxs2, zs2000)\n",
+      "plot_residuals(xs2000[:,0], fxs2[:,0], ps2[:,0], \n",
+      "               'Seceond Order Position Residuals',\n",
+      "               'meters') "
      ],
      "language": "python",
      "metadata": {},
@@ -1709,27 +1722,27 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd561456470>"
+        "<matplotlib.figure.Figure at 0x7f89a8358f60>"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7fd5586dde80>"
+        "<matplotlib.figure.Figure at 0x7f89a844a940>"
        ]
       }
      ],
-     "prompt_number": 32
+     "prompt_number": 96
     },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "Again, the residual plot tells the story. The track looks very good, but the residual plot shows that the filter is diverging. \n",
+      "Again, the residual plot tells the story. The track looks very good, but the residual plot shows that the filter is diverging for significant periods of time.\n",
       "\n",
       "How should you think about all of this? You might argue that the last plot is 'good enough' for your application, and perhaps it is. I warn you however that a diverging filter doesn't always converge. With a different data set, or a physical system that performs differently you can end up with a filter that never converges. \n",
       "\n",
@@ -1752,11 +1765,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd55d125ef0>"
+        "<matplotlib.figure.Figure at 0x7f89a8585128>"
        ]
       }
      ],
-     "prompt_number": 33
+     "prompt_number": 71
     },
     {
      "cell_type": "markdown",
@@ -1771,9 +1784,186 @@
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "So this is the story of Godilocks. You don't want a filter too big, or too small, but one that is just right. In many cases that will be easy to do - if you are designing a Kalman filter to read the thermometer of a freezer it seems clear that a zero order filter is the right choice. But what order should we use if we are tracking a car? Order one will work well while the car is moving in a straight line at a constant speed, but cars turn, speed up, and slow down, in which case a second order filter will perform better. That is the problem addressed in the Adaptive Filters chapter. There we will learn how to design a filter that *adapts* to changing order in the tracked object's behavior."
+      "So this is the story of Godilocks. You don't want a filter too big, or too small, but one that is just right. In many cases that will be easy to do - if you are designing a Kalman filter to read the thermometer of a freezer it seems clear that a zero order filter is the right choice. But what order should we use if we are tracking a car? Order one will work well while the car is moving in a straight line at a constant speed, but cars turn, speed up, and slow down, in which case a second order filter will perform better. That is the problem addressed in the Adaptive Filters chapter. There we will learn how to design a filter that *adapts* to changing order in the tracked object's behavior.\n",
+      "\n",
+      "With that said, a lower order filter can track a higher order process so long as you add enough process noise. The results will not be optimal, but they can still be very good, and I always reach for this tool first before trying an adaptive filter. Let's look at an example with acceleration. First, the simulation."
      ]
     },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "class ConstantAccelerationObject(object):\n",
+      "    def __init__(self, x0=0, vel=1., acc=0.1, acc_noise_scale=.1):\n",
+      "        self.x = x0\n",
+      "        self.vel = vel\n",
+      "        self.acc = acc\n",
+      "        self.acc_noise_scale = acc_noise_scale\n",
+      "    \n",
+      "    def update(self):\n",
+      "        self.acc += randn()*self.acc_noise_scale       \n",
+      "        self.vel += self.acc\n",
+      "        self.x += self.vel\n",
+      "        return (self.x, self.vel, self.acc)\n",
+      "\n",
+      "   \n",
+      "R = 6.\n",
+      "Q = 0.02\n",
+      "def simulate_acc_system(R, Q, count):\n",
+      "    obj = ConstantAccelerationObject(acc_noise_scale=Q)\n",
+      "    zs = []\n",
+      "    xs = []\n",
+      "    for i in range(count):\n",
+      "        x = obj.update()\n",
+      "        z = sense(x, R)\n",
+      "        xs.append(x)\n",
+      "        zs.append(z)\n",
+      "    return np.asarray(xs), zs\n",
+      "\n",
+      "np.random.seed(124)\n",
+      "xs,zs = simulate_acc_system(R=R, Q=Q, count=80)\n",
+      "plt.plot(xs[:,0])\n",
+      "plt.show()"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7f89a8623908>"
+       ]
+      }
+     ],
+     "prompt_number": 97
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Now we will filter the data using a second order filter. "
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "kf2 = SecondOrderKF(R, Q, dt=1)\n",
+      "fxs2, ps2 = filter_data(kf2, zs)\n",
+      "\n",
+      "plot_track(xs, fxs2, zs)\n",
+      "plot_residuals(xs[:,0], fxs2[:,0], ps2[:,0], \n",
+      "               'Second Order Position Residuals',\n",
+      "               'meters') "
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAzsAAAGkCAYAAAAWgDuQAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\nAAALEgAACxIB0t1+/AAAIABJREFUeJzs3Xl8jOf+//HXTLbJZkhkI0RogtCqhtYalYQqautiqZ5S\nRVuUaqvVOpbWoZweVafo6aIcaj+q1dqVkKa21Fp77SFpRIRENpn790d/5iu11BIm4v18PObxMPdc\n9z2f++rQec913ddtMgzDQEREREREpIQxO7oAERERERGR20FhR0RERERESiSFHRERERERKZEUdkRE\nREREpERS2BERERERkRJJYUdEREREREokhR0RERERESmRil3YGT16NGazmX79+hXaPnz4cMqXL4+H\nhwdNmzZl165dhV7Pzc2lX79++Pn54eXlRdu2bUlKSrqTpYuIiIiISDFSrMLO+vXr+fzzz3nggQcw\nmUz27WPGjGHcuHF88sknbNq0CX9/f5o1a0ZmZqa9zYABA1iwYAGzZ89m3bp1nD17ltatW2Oz2Rxx\nKiIiIiIi4mDFJuxkZGTQtWtXvvrqK8qUKWPfbhgG48ePZ/DgwbRv354aNWowbdo0zp07x8yZM+37\nTpkyhQ8//JCYmBhq167N9OnT2b59OytXrnTUKYmIiIiIiAMVm7DTq1cvnn76aZo0aYJhGPbthw4d\nIiUlhebNm9u3WSwWoqKiSEhIACAxMZH8/PxCbYKDg6levbq9jYiIiIiI3FucHV0AwOeff87Bgwft\nIzWXTmFLTk4GICAgoNA+/v7+nDhxwt7GyckJX1/fQm0CAgJISUkptC0jI6PI6xcRERERkdvParXe\nUHuHh529e/fy7rvvEh8fj5OTE/DH1LVLR3eu5tJQJCIiIiIicimHT2P7+eefOXXqFDVq1MDFxQUX\nFxfWrl3LpEmTcHV1pWzZsgCXjdCkpKQQGBgIQGBgIAUFBaSlpRVqk5ycbG8jIiIiIiL3FoeP7LRv\n356HH37Y/twwDLp37054eDjvvPMOYWFhBAYGsnz5ciIjIwHIyckhPj6eDz/8EIDIyEhcXFxYvnw5\nnTt3BuD48ePs2bOHBg0aXPW9b3QYTG7N5s2bqVOnjqPLuOeo3x1D/e4Y6nfHUL87hvrdMdTvt87I\nOkfa+Pcps2kFuwNrEDJ+Ct4W16u2v5XLUBwedqxW62Whw8PDgzJlyhAREQH8saz0qFGjqFatGmFh\nYYwcORJvb2+6dOliP0aPHj0YNGgQ/v7++Pj4MHDgQGrVqkVsbOwdPycREREREbmcsW8Huf98mzKp\nJzjv5MZvkS2IcL19kcThYedKTCZToetxBg0aRHZ2Nn369CE9PZ169eqxfPlyPD097W3Gjx+Ps7Mz\nHTt2JDs7m9jYWGbMmKHrekREREREHMyw2TC+mYpt5iRcCgrYUzqU79oNZlCbOpjNt+/7erEMO6tX\nr75s27Bhwxg2bNhV93F1dWXChAlMmDDhdpYmIiIiIiI3wDidiu3jv8O2DZiAGWGtWR/9AuObh+B0\nG4MOFNOwIyIiIiIid68ffviBAwcO4JF6Ao+4RXjm5WDy8GZ6jafJrPQIH1axcfZ0GqVKlcLNze22\n1aGwIyIiIiIiRWbv3r1sXL8etm8gbe92AE6VCmSpV12cU1Npa13H/Jk/AdCwYUOaNWt222pR2BER\nERERkSKRn5/P4u+/x1jzPY8amYSG+rPs/tbM9XyI0kYe/Wp44GHLJSsri6ysLEqXLn1b61HYuQab\nzUZeXp6jyygxQkJCyMnJcXQZ95xL+93V1RWz2eG31xIREZES6qeffuLMmiUEnD1F44eqMavt3/nP\nuSDKupj5vFl5qvta7mg9CjtXYbPZyM3NxWKxaEW3ImKx3NkPt/zhYr8bhkFOTg5ubm4KPCIiIlLk\n0tPTWTd9Csb+nTweGsDiZ0bw0Sk/XMwmxjUJuuNBB0DfeK4iLy9PQUdKFJPJhMVi0WiliIiI3BZL\npn9F/vofeaCUO2c6v8HwU36YgH80CuDhIA+H1KSRnWtQ0JGSRp9pERERuR32bvmFPf/9D26GjZot\nO/Cs6REA3n7Yj2Yh3g6rSyM7IiIiIiJy0/Lz8lj894GQmcEjEdXpWfkFbJjo/YAPz1S9vQsQ/BWF\nHRERERERuWnxo4eQfmAPZb29mNRsGJkmVzpVtdL7AR9Hl6awIyIiIiIiN+f0uuWsmzMDgF2PDeCQ\nRxCtQr15s65fsZg+r7Ajd9Tw4cMxm838/vvvji5FRERERG6BkZrM0ncGcMFm40zNaDZUfYyo8p4M\naxCAuRgEHVDYueckJCQwYsQIMjIyHF2KiIiIiNyljPw89g7uzZ7U06RZy7O22es85O/OmKhAXMzF\nI+iAws49R2FHRERERG5V/mcfsHRDIuddPUlo2pfqwX6MbxqExbl4xYviVY3cMYZh/GWb7OzsO1CJ\niIiIiNxNbKsX8dOs6Zy6YGNN1RZUfDiKidHl8HZ1cnRpl1HYuYcMHz6cQYMGARAaGorZbMZsNhMX\nF0elSpV4/PHHWbVqFY888gju7u6MHTsWgO+++44nnniCChUqYLFYqFSpEoMGDSI3N/ey99i3bx+d\nO3fG398fd3d3wsPDee21165Z14kTJ4iIiCA8PJzjx48X/YmLiIiISJEwdm/l9McjiE/LJD6wNh5N\nOzK5eUV83Ivn7TuLZ1VyWzz55JPs37+fWbNmMX78eMqWLQtA9erVMZlMHDhwgKeffppevXrRs2dP\nKlasCMDUqVNxd3enf//+WK1Wfv75Zz766COOHTvGrFmz7Mf/9ddfadiwIc7OzvTq1YvKlStz6NAh\n5s6dy0cffXTFmo4cOUJMTAwWi4V169YREBBw+ztCRERERG6YsWsLtvf6sCLpFDuslUh9sAWLnq1H\nOS8XR5d2VQo7RaD29P239fhbngsrkuPcf//91K5dm1mzZtGuXTt7mIE/prX99ttvfPfdd7Ru3brQ\nfl9//TXu7u725z179iQsLIwhQ4bwz3/+k+DgYAD69OmDzWYjMTGRkJAQe/t//OMfV6znwIEDxMTE\n4Ovry4oVK/D19S2S8xQRERGRonU+MYFtb/dma+oZ1roGsbFiPea82pHKVldHl3ZNCjtiV6FChcuC\nDmAPOjabjXPnzpGfn0/Dhg0xDIMtW7YQHBxMamoqa9eupW/fvoWCztXs2rWLLl26ULFiRZYtW4bV\nai3y8xERERGRm2cYBgcPHuSX7+aza8pECvLz2G8N4afQxnzQ6xkeCfV3dIl/SWGnCBTVyIujVa5c\n+Yrbd+7cyaBBg4iLi7ts0YKLq7odPHgQgJo1a17Xe7Vp0wZ/f39WrlyJl5fXLVQtIiIiIkUpPT2d\nrVu3snXrVs7s340RtxhTQQHnKj7AzqavMrVzI6IrlXZ0mddFYUfsLp2qdlFGRgZNmzbF29ubUaNG\ncd999+Hu7s7x48fp1q0bNpvtpt7r6aef5quvvmLq1Kn07dv3VksXERERkVt08uRJVqxYYf8R20hJ\nwrp+BQ+WcedkzVg+ajCAD6PKEV3J28GVXj+FnXuM6QbvZrt69WrS0tJYsGABjRs3tm9fsWJFoXZV\nqlQBYMeOHdd13NGjR2OxWOjfvz9eXl5069bthuoSERERkaKTlZXF119/TWZmJs7OzlT3cKHWps1U\nqliaRZWaMi7yJYbUD+SxuyjogJaevud4enoCcPr06etq7+T0x3rpl47g2Gw2xo0bV6hd2bJladKk\nCVOnTuXw4cOFXrvaPX0mTpzIc889R8+ePZk3b971noKIiIiIFCHDMPjuu+/IzMwkJCSE15tH0W79\nQkJdYFFoNO9HvsTAuv60D7v7rrHWyM49pm7dugAMHjyYzp074+rqSnR09FXbN2rUCF9fX55//nn6\n9euHs7Mz8+fPJysr67K2//73v2nUqBGRkZH07t2b0NBQjh49ypw5c9i3b98Vjz9lyhQyMzPp2rUr\nnp6etGzZsmhOVERERESuS2JiInv37sVisdDuvmBcPxwEebl8GxrNyNq96P1gWZ6tXsbRZd4Ujezc\nYyIjIxk9ejS7du3ihRde4Nlnn2X37t1Xnd5WpkwZfvjhBypUqMCwYcP44IMPqFWrFv/9738va1uz\nZk3Wr19PdHQ0//nPf+jfvz/z5s2jTZs29jYmk6nQe5nNZmbNmkVMTAxPP/00a9asKfJzFhEREZEr\nO3XqFMuWLQOgVVgIpf799z+CTuUYRtbuRdcaPvS638fBVd48k3G1OUYl1MXVw4BrLneck5ODxWK5\nEyWJ3FH6bN8Zmzdvpk6dOo4u456jfncM9btjqN8doyT1e0FBAV988QUnT57kgdAQ2q6cBucy+LZK\nM0bW6kH7sNIMqed/w9d8F7Xr/f5+JRrZERERERG5B/3444+cPHmS0tZStNi5Bs5lsCHoQUbW6sFj\noaV45xHHB51bpbAjIiIiInKPOXToEAkJCZhMJtqbs3Dbv51T7j68G9mXxhW8ea9hIE7muzvogMKO\niIiIiMg9JTs7m2+++QbDMIgqV5bya76hwGTmnbqvUjUkgLFRgbiUgKADCjsiIiIiIvcMwzBYtGgR\nZ8+epXwZKw3WzsdkGHxW/SmMGg/x0aPlcHMqORGhWJzJxIkTqVWrFlarFavVSoMGDVi8eLH99W7d\numE2mws9GjRoUOgYubm59OvXDz8/P7y8vGjbti1JSUl3+lRERERERIqtbdu2sWvXLlycnWl9eAvO\n586wwe9+NjXszL+jy+PuUiziQZEpFmdToUIFxo4dy5YtW0hMTCQ6Opp27dqxbds24I/lips1a0Zy\ncrL9cWkYAhgwYAALFixg9uzZrFu3jrNnz9K6detCN8MUEREREblXnT592v4dOsaUjf+hHaLine truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7f89b1656400>"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAzMAAAGxCAYAAABWTOhMAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7f89a9ef39e8>"
+       ]
+      }
+     ],
+     "prompt_number": 98
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "We can see that the filter is perfoming within the theoretical limits of the filter. \n",
+      "\n",
+      "Now let's use a lower order filter. As already demonstrated the lower order filter will lag the signal because it is not modelling the acceleration. However, we can account for that (to an extent) by increasing the size of the process noise. The filter will treat the acceration as noise in the process model. The result will be suboptimal, but if designed well it will not diverge. Choosing the amount of extra process noise is not an exact science. You will have to experiment with representative data. Here, I've multiplied it by 10, and am getting good results."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "kf2 = FirstOrderKF(R, Q*10, dt=1)\n",
+      "fxs2, ps2 = filter_data(kf2, zs)\n",
+      "\n",
+      "plot_track(xs, fxs2, zs)\n",
+      "plot_residuals(xs[:,0], fxs2[:,0], ps2[:,0], \n",
+      "               'Second Order Position Residuals',\n",
+      "               'meters') "
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7f89b15af6d8>"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7f89b1880160>"
+       ]
+      }
+     ],
+     "prompt_number": 100
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Think about what will happen if you make the process noise many times larger than it needs to be. A large process noise tells the filter to favor the measurements, so we would expect the filter to closely mimic the noise in the measurements. Let's find out."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "kf2 = FirstOrderKF(R, Q*10000, dt=1)\n",
+      "fxs2, ps2 = filter_data(kf2, zs)\n",
+      "\n",
+      "plot_track(xs, fxs2, zs)\n",
+      "plot_residuals(xs[:,0], fxs2[:,0], ps2[:,0], \n",
+      "               'Second Order Position Residuals',\n",
+      "               'meters') "
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7f89a835f438>"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7f89a835f400>"
+       ]
+      }
+     ],
+     "prompt_number": 101
+    },
     {
      "cell_type": "heading",
      "level": 2,
@@ -1893,20 +2083,31 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f9781a1ba58>"
+        "<matplotlib.figure.Figure at 0x7f89b18a5780>"
        ]
       }
      ],
-     "prompt_number": 4
+     "prompt_number": 38
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "We can see the result of the Kalman filter in blue."
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Excercise: Prove that the GPS improves the filter"
+     ]
     },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "We can see the result of the Kalman filter in blue.\n",
-      "\n",
-      "##### Excercise: Prove that the GPS improves the filter\n",
-      "\n",
       "Devise a way to prove that fusing the GPS and wheel measurements yields a better result than using the wheel alone.\n",
       "\n",
       "##### Solution 1\n",
@@ -1933,7 +2134,7 @@
        ]
       }
      ],
-     "prompt_number": 35
+     "prompt_number": 39
     },
     {
      "cell_type": "markdown",
@@ -2012,11 +2213,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd561396e48>"
+        "<matplotlib.figure.Figure at 0x7f89b1821518>"
        ]
       }
      ],
-     "prompt_number": 36
+     "prompt_number": 40
     },
     {
      "cell_type": "markdown",
@@ -2121,7 +2322,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 37
+     "prompt_number": 41
     },
     {
      "cell_type": "markdown",
@@ -2166,7 +2367,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd561400ef0>"
+        "<matplotlib.figure.Figure at 0x7f89b18db048>"
        ]
       },
       {
@@ -2174,11 +2375,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd561400400>"
+        "<matplotlib.figure.Figure at 0x7f89b18f4080>"
        ]
       }
      ],
-     "prompt_number": 38
+     "prompt_number": 42
     },
     {
      "cell_type": "markdown",
@@ -2315,7 +2516,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 39
+     "prompt_number": 43
     },
     {
      "cell_type": "markdown",
@@ -2359,11 +2560,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd558660198>"
+        "<matplotlib.figure.Figure at 0x7f89a859e390>"
        ]
       }
      ],
-     "prompt_number": 40
+     "prompt_number": 44
     },
     {
      "cell_type": "markdown",
@@ -2504,7 +2705,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 41
+     "prompt_number": 45
     },
     {
      "cell_type": "markdown",
@@ -2555,11 +2756,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd5584c9cf8>"
+        "<matplotlib.figure.Figure at 0x7f89a8358fd0>"
        ]
       }
      ],
-     "prompt_number": 42
+     "prompt_number": 46
     },
     {
      "cell_type": "markdown",
@@ -2698,11 +2899,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd5586a9908>"
+        "<matplotlib.figure.Figure at 0x7f89a84f2be0>"
        ]
       }
      ],
-     "prompt_number": 43
+     "prompt_number": 47
     },
     {
      "cell_type": "markdown",
@@ -2776,7 +2977,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 44
+     "prompt_number": 48
     },
     {
      "cell_type": "markdown",
@@ -2836,11 +3037,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd561197c50>"
+        "<matplotlib.figure.Figure at 0x7f89b1852240>"
        ]
       }
      ],
-     "prompt_number": 45
+     "prompt_number": 49
     },
     {
      "cell_type": "markdown",
@@ -2904,7 +3105,7 @@
        "output_type": "display_data",
        "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7fd5586a9978>"
+        "<matplotlib.figure.Figure at 0x7f89a84f2cc0>"
        ]
       },
       {
@@ -2912,11 +3113,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd561191a20>"
+        "<matplotlib.figure.Figure at 0x7f89b272d940>"
        ]
       }
      ],
-     "prompt_number": 46
+     "prompt_number": 50
     },
     {
      "cell_type": "markdown",
@@ -2942,7 +3143,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd5610cceb8>"
+        "<matplotlib.figure.Figure at 0x7f89a8593780>"
        ]
       },
       {
@@ -2950,11 +3151,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd561396898>"
+        "<matplotlib.figure.Figure at 0x7f89a84ad278>"
        ]
       }
      ],
-     "prompt_number": 47
+     "prompt_number": 51
     },
     {
      "cell_type": "markdown",