Added variance to voltage animation.

Voltage animation now displays the Gaussion as it innovates.

Chapter05_Kalman_Filters/Kalman_Filters.ipynb

@@ -1,7 +1,7 @@
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@@ -255,13 +255,13 @@
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@@ -349,14 +349,14 @@
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-        " 0.0276\t 1.7666\t-0.6031\t 0.0895\t-0.9401\t\n",
-        " 0.2449\t-0.4231\t 0.2825\t 1.7339\t-2.7824\t\n",
-        " 1.1785\t-1.0472\t-0.3156\t 1.3697\t 1.1010\t\n",
-        " 0.8988\t-0.6833\t-0.1195\t-2.1915\t-2.8026\t\n"
+        " 0.3063\t 0.0614\t 0.7215\t 0.8443\t-0.6958\t\n",
+        " 0.3727\t 1.0889\t 0.2490\t 0.1126\t-1.0971\t\n",
+        " 2.5001\t 1.0345\t 1.0200\t 0.2969\t-1.5525\t\n",
+        " 0.9133\t-1.1189\t 0.9056\t-0.8069\t-0.5581\t\n"
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truncated
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@@ -442,13 +442,13 @@
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+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f2083b76710>"
+        "<matplotlib.figure.Figure at 0x7feec3fbfb38>"
        ]
       }
      ],
-     "prompt_number": 5
+     "prompt_number": 6
     },
     {
      "cell_type": "markdown",
@@ -476,13 +476,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f2083ecb470>"
+        "<matplotlib.figure.Figure at 0x7feec428f4a8>"
        ]
       }
      ],
-     "prompt_number": 6
+     "prompt_number": 7
     },
     {
      "cell_type": "code",
@@ -496,13 +496,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f2083ab2208>"
+        "<matplotlib.figure.Figure at 0x7feec3f060b8>"
        ]
       }
      ],
-     "prompt_number": 7
+     "prompt_number": 8
     },
     {
      "cell_type": "markdown",
@@ -541,11 +541,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f2083a4d5c0>"
+        "<matplotlib.figure.Figure at 0x7feec3eb3c50>"
        ]
       }
      ],
-     "prompt_number": 8
+     "prompt_number": 9
     },
     {
      "cell_type": "markdown",
@@ -578,13 +578,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f2083ecbcf8>"
+        "<matplotlib.figure.Figure at 0x7feec4299940>"
        ]
       }
      ],
-     "prompt_number": 9
+     "prompt_number": 10
     },
     {
      "cell_type": "markdown",
@@ -677,11 +677,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f207c43cf28>"
+        "<matplotlib.figure.Figure at 0x7feec0193ba8>"
        ]
       }
      ],
-     "prompt_number": 10
+     "prompt_number": 11
     },
     {
      "cell_type": "markdown",
@@ -725,11 +725,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f207ab86a20>"
+        "<matplotlib.figure.Figure at 0x7feec00e27f0>"
        ]
       }
      ],
-     "prompt_number": 11
+     "prompt_number": 12
     },
     {
      "cell_type": "markdown",
@@ -775,11 +775,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f207ab044e0>"
+        "<matplotlib.figure.Figure at 0x7feec00479e8>"
        ]
       }
      ],
-     "prompt_number": 12
+     "prompt_number": 13
     },
     {
      "cell_type": "markdown",
@@ -850,7 +850,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 13
+     "prompt_number": 14
     },
     {
      "cell_type": "markdown",
@@ -869,7 +869,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 14
+     "prompt_number": 15
     },
     {
      "cell_type": "markdown",
@@ -898,30 +898,30 @@
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "time: 0 \tposition = 1.519 \tvariance = 2.500\n",
-        "time: 1 \tposition = 0.769 \tvariance = 1.667\n",
-        "time: 2 \tposition = 1.024 \tvariance = 1.250\n",
-        "time: 3 \tposition = 0.897 \tvariance = 1.000\n",
-        "time: 4 \tposition = 0.705 \tvariance = 0.833\n",
-        "time: 5 \tposition = 0.665 \tvariance = 0.714\n",
-        "time: 6 \tposition = 0.479 \tvariance = 0.625\n",
-        "time: 7 \tposition = 0.486 \tvariance = 0.556\n",
-        "time: 8 \tposition = 0.528 \tvariance = 0.500\n",
-        "time: 9 \tposition = 0.353 \tvariance = 0.455\n",
-        "time: 10 \tposition = 0.218 \tvariance = 0.417\n",
-        "time: 11 \tposition = 0.280 \tvariance = 0.385\n",
-        "time: 12 \tposition = 0.344 \tvariance = 0.357\n",
-        "time: 13 \tposition = 0.506 \tvariance = 0.333\n",
-        "time: 14 \tposition = 0.394 \tvariance = 0.312\n",
-        "time: 15 \tposition = 0.358 \tvariance = 0.294\n",
-        "time: 16 \tposition = 0.338 \tvariance = 0.278\n",
-        "time: 17 \tposition = 0.334 \tvariance = 0.263\n",
-        "time: 18 \tposition = 0.346 \tvariance = 0.250\n",
-        "time: 19 \tposition = 0.306 \tvariance = 0.238\n"
+        "time: 0 \tposition = 1.650 \tvariance = 2.500\n",
+        "time: 1 \tposition = 1.081 \tvariance = 1.667\n",
+        "time: 2 \tposition = 0.837 \tvariance = 1.250\n",
+        "time: 3 \tposition = 0.305 \tvariance = 1.000\n",
+        "time: 4 \tposition = 0.540 \tvariance = 0.833\n",
+        "time: 5 \tposition = 0.393 \tvariance = 0.714\n",
+        "time: 6 \tposition = 0.301 \tvariance = 0.625\n",
+        "time: 7 \tposition = 0.262 \tvariance = 0.556\n",
+        "time: 8 \tposition = 0.257 \tvariance = 0.500\n",
+        "time: 9 \tposition = 0.240 \tvariance = 0.455\n",
+        "time: 10 \tposition = 0.204 \tvariance = 0.417\n",
+        "time: 11 \tposition = 0.147 \tvariance = 0.385\n",
+        "time: 12 \tposition = 0.261 \tvariance = 0.357\n",
+        "time: 13 \tposition = 0.330 \tvariance = 0.333\n",
+        "time: 14 \tposition = 0.297 \tvariance = 0.312\n",
+        "time: 15 \tposition = 0.287 \tvariance = 0.294\n",
+        "time: 16 \tposition = 0.269 \tvariance = 0.278\n",
+        "time: 17 \tposition = 0.168 \tvariance = 0.263\n",
+        "time: 18 \tposition = 0.173 \tvariance = 0.250\n",
+        "time: 19 \tposition = 0.116 \tvariance = 0.238\n"
        ]
       }
      ],
-     "prompt_number": 15
+     "prompt_number": 16
     },
     {
      "cell_type": "markdown",
@@ -988,13 +988,13 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 16
+     "prompt_number": 17
     },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "What is left? Just calling these functions. The histogram did nothing more than loop over the `sense()` and `update()` functions, so let's do the same. "
+      "What is left? Just calling these functions. The histogram did nothing more than loop over the `update()` and `predict()` functions, so let's do the same. "
      ]
     },
     {
@@ -1037,28 +1037,28 @@
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "PREDICT:     1.0000   502.0000\tUPDATE:    -0.2825     9.8047\n",
-        "PREDICT:     0.7175    11.8047\tUPDATE:     1.3165     5.4138\n",
-        "PREDICT:     2.3165     7.4138\tUPDATE:     1.6450     4.2574\n",
-        "PREDICT:     2.6450     6.2574\tUPDATE:     3.7906     3.8490\n",
-        "PREDICT:     4.7906     5.8490\tUPDATE:     3.6658     3.6904\n",
-        "PREDICT:     4.6658     5.6904\tUPDATE:     5.7559     3.6267\n",
-        "PREDICT:     6.7559     5.6267\tUPDATE:     7.8662     3.6007\n",
-        "PREDICT:     8.8662     5.6007\tUPDATE:     8.4671     3.5900\n",
-        "PREDICT:     9.4671     5.5900\tUPDATE:    12.2074     3.5856\n",
-        "PREDICT:    13.2074     5.5856\tUPDATE:    11.9462     3.5838\n"
+        "PREDICT:     1.0000   502.0000\tUPDATE:     5.9025     9.8047\n",
+        "PREDICT:     6.9025    11.8047\tUPDATE:     5.2123     5.4138\n",
+        "PREDICT:     6.2123     7.4138\tUPDATE:     4.9745     4.2574\n",
+        "PREDICT:     5.9745     6.2574\tUPDATE:     5.2439     3.8490\n",
+        "PREDICT:     6.2439     5.8490\tUPDATE:     5.8732     3.6904\n",
+        "PREDICT:     6.8732     5.6904\tUPDATE:     5.6334     3.6267\n",
+        "PREDICT:     6.6334     5.6267\tUPDATE:     7.1423     3.6007\n",
+        "PREDICT:     8.1423     5.6007\tUPDATE:     9.0740     3.5900\n",
+        "PREDICT:    10.0740     5.5900\tUPDATE:     9.8442     3.5856\n",
+        "PREDICT:    10.8442     5.5856\tUPDATE:    13.7547     3.5838\n"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f207c48ec18>"
+        "<matplotlib.figure.Figure at 0x7feec01adac8>"
        ]
       }
      ],
-     "prompt_number": 17
+     "prompt_number": 18
     },
     {
      "cell_type": "markdown",
@@ -1190,9 +1190,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f207ab1d828>"
+        "<matplotlib.figure.Figure at 0x7feeb80a87b8>"
        ]
       },
       {
@@ -1200,7 +1200,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f207c444860>"
+        "<matplotlib.figure.Figure at 0x7feeb80817b8>"
        ]
       },
       {
@@ -1211,7 +1211,7 @@
        ]
       }
      ],
-     "prompt_number": 18
+     "prompt_number": 19
     },
     {
      "cell_type": "markdown",
@@ -1347,7 +1347,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 19
+     "prompt_number": 20
     },
     {
      "cell_type": "markdown",
@@ -1366,9 +1366,11 @@
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "Now we need to write the Kalman filter processing loop. As with our previous problem, we need to perform a cycle of sensing and updating. The sensing step probably seems clear - call `volt()` to get the measurement, pass the result into `update()` function, but what about the update step? We do not have a sensor to detect 'movement' in the voltage, and for any small duration we expect the voltage to remain constant. How shall we handle this?\n",
+      "Now we need to write the Kalman filter processing loop. As with our previous problem, we need to perform a cycle of predicting and updating. The sensing step probably seems clear - call `volt()` to get the measurement, pass the result into `update()` function, but what about the predict step? We do not have a sensor to detect 'movement' in the voltage, and for any small duration we expect the voltage to remain constant. How shall we handle this?\n",
       "\n",
-      "As always, we will trust in the math. We have no movement, and no error associated with them, so we will just set both to zero. Let's see what happens. "
+      "As always, we will trust in the math. We have no known movement, so we will set that to zero. However, that means that we are predicting that the temperature will never change over time. If that is true, then over time we should become extremely confident in our results. Once the filter has enough measurements it will become very confident that it can predict the subsequent temperatures, and this will lead it to ignoring measurements that result due to an actual temperature change. This is called a *smug* filter, and is something you want to avoid. So we will add a bit of error to our prediction step to tell the filter not to discount changes in voltage over time. In the code below I set `movement_error = .2`. This is just the expected variance in the change of voltage over each time step. I chose this value merely to be able to show how the variance changes through the update and predict steps. For an real sensor you would set this value for the actual amount of change you expect. For example, this would be an extremely small number if it is a thermometer for ambient air temperature in a house, and a high number if this is a thermocouple in a chemical reaction chamber. We will say more about selecting the actual value in the next chapter. \n",
+      "\n",
+      "Let's see what happens. "
      ]
     },
     {
@@ -1376,7 +1378,7 @@
      "collapsed": false,
      "input": [
       "sensor_error = temp_variance\n",
-      "movement_error = 0\n",
+      "movement_error = .2\n",
       "movement = 0\n",
       "voltage = (25,1000) #who knows what the first value is?\n",
       "\n",
@@ -1412,29 +1414,29 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f207c4272b0>"
+        "<matplotlib.figure.Figure at 0x7feec00ae048>"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f207c410898>"
+        "<matplotlib.figure.Figure at 0x7feec01d3550>"
        ]
       },
       {
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "Variance converges to 0.09072976736236911\n",
-        "Last voltage is 16.200756458088527\n"
+        "Variance converges to 0.8577995630075655\n",
+        "Last voltage is 17.438686650938127\n"
        ]
       }
      ],
-     "prompt_number": 20
+     "prompt_number": 22
     },
     {
      "cell_type": "markdown",
@@ -1442,12 +1444,30 @@
      "source": [
       "The first plot shows the individual sensor measurements marked with '+'s vs the filter output. Despite a lot of noise in the sensor we quickly discover the approximate voltage of the sensor. In the run I just completed at the time of authorship, the last voltage output from the filter is $16.213$, which is quite close to the $16.4$ used by the `volt()` function. On other runs I have gotten up to around $16.9$ as an output and also as low as 15.5 or so.\n",
       "\n",
-      "The second plot shows how the variance converges over time. Compare this plot to the variance plot for the dog sensor. While this does converge to a very small value, it is much slower than the dog problem. The next section **Explaining the Results - Multi-Sensor Fusion** explains why this happens.\n",
-      "\n",
-      "For those reading this in IPython Notebook, here is an animation showing the filter working. The animation draws a green line for the predicted next voltage, then a red '+' for the actual measurement, draws a light red line to show the residual, and then draws a blue line to the filter's output. You can see that when the filter starts the corrections made are quite large, but after only a few updates the filter only adjusts its output by a small amount even when the measurement is far from it. \n",
-      "\n",
-      "The plot above this text explains why this happens. When the filter is initialized we are not sure of the voltage, so the variance is high. So when we get a measurement the filter is willing to make a large adjustment to its output. However, as it keeps reading new values it quickly becomes much more confident in its output, and thus it will adjust its output only by a small amount, assuming that the large variance in the measurement is due to noise, not the filter's bad prediction.\n",
+      "The second plot shows how the variance converges over time. Compare this plot to the variance plot for the dog sensor. While this does converge to a very small value, it is much slower than the dog problem. The section **Explaining the Results - Multi-Sensor Fusion** explains why this happens."
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Animation"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "For those reading this in IPython Notebook, here is an animation showing the filter working. The top plot in the animation draws a green line for the predicted next voltage, then a red '+' for the actual measurement, draws a light red line to show the residual, and then draws a blue line to the filter's output. You can see that when the filter starts the corrections made are quite large, but after only a few updates the filter only adjusts its output by a small amount even when the measurement is far from it. \n",
       "\n",
+      "The lower plot shows the Gaussian belief as the filter innovates. When the filter starts the Gaussian curve is centered over 25, our initial guess for the voltage, and is very wide and short due to our inital uncertainty. But as the filter innovates, the Gaussian quickly moves to about 16.0 and becomes taller, reflecting the growing confidence that the filter has in it's estimate for the voltage. You will also note that the Gaussian's height bounces up and down a little bit. If you watch closely you will see that the Gaussian becomes a bit shorter and more spread out during the prediction step, and becomes taller and narrower as the filter incorporates another measurement (the innovation step)."
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
       "<img src='volt_animate.gif'>"
      ]
     },

Chapter05_Kalman_Filters/Kalman_Filters_Animations.ipynb

@@ -1,7 +1,7 @@
 {
  "metadata": {
   "name": "",
-  "signature": "sha256:6ae833f8c8c8b1ebf95d6a8830c42247f5580e91d57a284055fb49d7d1488489"
+  "signature": "sha256:e4bf6588bd5504bdeff5802ced00bcc98d4cad46767b3d4f649b0fe63a2976cf"
  },
  "nbformat": 3,
  "nbformat_minor": 0,
@@ -241,13 +241,13 @@
        ],
        "metadata": {},
        "output_type": "pyout",
-       "prompt_number": 15,
+       "prompt_number": 2,
        "text": [
-        "<IPython.core.display.HTML at 0x7f5980a89978>"
+        "<IPython.core.display.HTML at 0x7fc962c0a470>"
        ]
       }
      ],
-     "prompt_number": 15
+     "prompt_number": 2
     },
     {
      "cell_type": "code",
@@ -274,7 +274,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 16
+     "prompt_number": 3
     },
     {
      "cell_type": "code",
@@ -284,6 +284,7 @@
       "\n",
       "import numpy.random as random\n",
       "import math\n",
+      "import stats\n",
       "\n",
       "def update(mu1, var1, mu2, var2):\n",
       "    mean = (var1*mu2 + var2*mu1) / (var1+var2)\n",
@@ -295,11 +296,11 @@
       "\n",
       "\n",
       "sensor_error = 1.2**2\n",
-      "movement_error = 0\n",
+      "movement_error = .2\n",
       "movement = 0\n",
-      "voltage = (19,20) #who knows what the first value is?\n",
+      "voltage = (25,20) #who knows what the first value is?\n",
       "\n",
-      "volts = [14.44, 18.74, 17.21, 15.98, 16.76, 16.25, 16.01, 16.90, 15.41, 17.31,\n",
+      "volts = [14.44, 18.74, 17.21, 15.98, 16.76, 14.8, 16.01, 16.90, 15.41, 17.31,\n",
       "        16.39, 17.96]\n",
       "ps = [voltage[0]]\n",
       "\n",
@@ -310,30 +311,37 @@
       "    global i, ps, voltage, Z, x, pred\n",
       "    \n",
       "    step = frame % 4\n",
-      "    plt.ylim([12,20])\n",
+      "    plt.subplot(211)\n",
+      "    plt.ylim([14,26])\n",
       "\n",
-      "    if step == 1:\n",
-      "        Z = volts[i]\n",
-      "        i += 1\n",
-      "        \n",
-      "        plt.scatter(i, Z, marker='+', s=64, color='r')\n",
-      "\n",
-      "    elif step == 3:\n",
-      "        voltage = update(voltage[0], voltage[1], Z, sensor_error)\n",
-      "        ps.append(voltage[0])\n",
-      "        plt.plot(ps, c='b')\n",
-      "        \n",
-      "    elif step == 2:\n",
-      "        plt.plot ([i,i], [pred, Z], c='r', alpha=0.3)\n",
-      "\n",
-      "    else:\n",
+      "    if step == 0:\n",
       "        prev = voltage[0]\n",
       "        voltage = predict(voltage[0], voltage[1], movement, movement_error)\n",
       "        pred = voltage[0]\n",
       "        plt.plot ([i, i+1], [prev, pred], c='g') \n",
       "        \n",
-      "N=12       \n",
-      "animate('volt_animate.gif', volt_animate, N*3, 600)"
+      "    elif step == 1:\n",
+      "        Z = volts[i]\n",
+      "        i += 1\n",
+      "        \n",
+      "        plt.scatter(i, Z, marker='+', s=64, color='r')\n",
+      "\n",
+      "    elif step == 2:\n",
+      "        plt.plot ([i,i], [pred, Z], c='r', alpha=0.3)\n",
+      "\n",
+      "    else:\n",
+      "        voltage = update(voltage[0], voltage[1], Z, sensor_error)\n",
+      "        ps.append(voltage[0])\n",
+      "        plt.plot(ps, c='b')\n",
+      "        \n",
+      "    plt.subplot(212)\n",
+      "    plt.cla()\n",
+      "    stats.plot_gaussian(voltage[0], voltage[1], xlim=[0,32])\n",
+      "    plt.ylim([0,1])\n",
+      "    plt.tight_layout()\n",
+      "        \n",
+      "N=12\n",
+      "animate('volt_animate.gif', volt_animate, N*3, 500)"
      ],
      "language": "python",
      "metadata": {},
@@ -341,13 +349,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f597c1a7198>"
+        "<matplotlib.figure.Figure at 0x7fc95db75be0>"
        ]
       }
      ],
-     "prompt_number": 40
+     "prompt_number": 38
     },
     {
      "cell_type": "markdown",
@@ -355,6 +363,176 @@
      "source": [
       "<img src='volt_animate.gif'>"
      ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "class DogSensor(object):\n",
+      "    \n",
+      "    def __init__(self, x0=0, velocity=1, noise=0.0):\n",
+      "        \"\"\" x0 - initial position\n",
+      "            velocity - (+=right, -=left)\n",
+      "            noise - scaling factor for noise, 0== no noise\n",
+      "        \"\"\"\n",
+      "        self.x = x0\n",
+      "        self.velocity = velocity\n",
+      "        self.noise = math.sqrt(noise)\n",
+      "\n",
+      "    def sense(self):\n",
+      "        self.x = self.x + self.velocity\n",
+      "        return self.x + random.randn() * self.noise\n",
+      "    \n",
+      "    \n",
+      "# assume dog is always moving 1m to the right\n",
+      "movement = 1\n",
+      "movement_error = .2\n",
+      "sensor_error = 4.5\n",
+      "pos = (0, 100)   # gaussian N(0,50)\n",
+      "\n",
+      "dog = DogSensor(pos[0], velocity=movement, noise=sensor_error)\n",
+      "\n",
+      "zs = []\n",
+      "ps = []\n",
+      "vs = []\n",
+      "\n",
+      "def dog_animate(frame):\n",
+      "    global pos, zs, ps, vs\n",
+      "    pos = predict(pos[0], pos[1], movement, movement_error)    \n",
+      "    Z = dog.sense()\n",
+      "    zs.append(Z)\n",
+      "    vs.append(pos[1])\n",
+      "    print(pos[1])\n",
+      "    \n",
+      "    pos = update(pos[0], pos[1], Z, sensor_error)\n",
+      "    ps.append(pos[0])\n",
+      "\n",
+      "    plt.subplot(211)\n",
+      "    plt.scatter (frame, Z)\n",
+      "    plt.subplot(212)\n",
+      "    plt.cla()\n",
+      "    stats.plot_gaussian(pos[0], pos[1], xlim=[0,32])\n",
+      "    plt.ylim([0,1])\n",
+      "\n",
+      "\n",
+      "N=12\n",
+      "animate('dog_track.gif', dog_animate, N, 200)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "100.2\n",
+        "4.506590257879656"
+       ]
+      },
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "\n",
+        "2.451646358922152"
+       ]
+      },
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "\n",
+        "1.7870209797122434"
+       ]
+      },
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "\n",
+        "1.4790786661369082"
+       ]
+      },
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "\n",
+        "1.3131905715360732"
+       ]
+      },
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "\n",
+        "1.2165428948514319"
+       ]
+      },
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "\n",
+        "1.1576492519214656"
+       ]
+      },
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "\n",
+        "1.1207749370248357"
+       ]
+      },
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "\n",
+        "1.0972939271041793"
+       ]
+      },
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "\n",
+        "1.0821803421932217"
+       ]
+      },
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "\n",
+        "1.072385204587669"
+       ]
+      },
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "\n"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7fc93cdca0b8>"
+       ]
+      }
+     ],
+     "prompt_number": 49
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "<img src='dog_track.gif'>"
+     ]
     }
    ],
    "metadata": {}