PEP-8 compliance, better variable names.

05_Kalman_Filters/Kalman_Filters.ipynb

@@ -1,7 +1,7 @@
 {
  "metadata": {
   "name": "",
-  "signature": "sha256:a7ff3c98bbd62de00e4ac578a1ebc6606df5c76dd2d2e757a71b513d92a423c5"
+  "signature": "sha256:83a0cc97c2553a428390210b3c7ca528383ef83c162d206cb853248bfe8bc04e"
  },
  "nbformat": 3,
  "nbformat_minor": 0,
@@ -259,7 +259,7 @@
        "output_type": "pyout",
        "prompt_number": 1,
        "text": [
-        "<IPython.core.display.HTML at 0x7f167808a978>"
+        "<IPython.core.display.HTML at 0x7fce9a598978>"
        ]
       }
      ],
@@ -319,7 +319,7 @@
       "        self.velocity = velocity\n",
       "        self.noise = math.sqrt(variance)\n",
       "\n",
-      "    def sense(self):\n",
+      "    def sense_position(self):\n",
       "        self.x = self.x + self.velocity\n",
       "        return self.x + random.randn() * self.noise"
      ],
@@ -332,7 +332,7 @@
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "The constructor `_init()__` initializes the DogSensor class with an initial position `x0`, velocity `vel`, and an noise scaling factor. The `sense()` function has the dog move by the set velocity and returns its new position, with noise added. If you look at the code for `sense()` you will see a call to `numpy.random.randn()`. This returns a number sampled from a normal distribution with a mean of 0.0. and a standard deviation of 1.0. *Variance* is defined as the standard deviation squared, so in `__init()__` we take the square root of the variance to get the standard deviation. Therefore the expression `self.x + random.randn() * self.noise` computes a number with the variance that we desire. \n",
+      "The constructor `_init()__` initializes the DogSensor class with an initial position `x0`, velocity `vel`, and an noise scaling factor. The `sense_position()` function has the dog move by the set velocity and returns its new position, with noise added. If you look at the code for `sense_position()` you will see a call to `numpy.random.randn()`. This returns a number sampled from a normal distribution with a mean of 0.0. and a standard deviation of 1.0. *Variance* is defined as the standard deviation squared, so in `__init()__` we take the square root of the variance to get the standard deviation. Therefore the expression `self.x + random.randn() * self.noise` computes a number with the variance that we desire. \n",
       "\n",
       "Let's look at some example output for that."
      ]
@@ -342,7 +342,7 @@
      "collapsed": false,
      "input": [
       "for i in range(20):\n",
-      "    print('{: 5.4f}'.format(random.randn()),end='\\t')\n",
+      "    print('{: 5.4f}'.format(random.randn()), end='\\t')\n",
       "    if (i+1) % 5 == 0:\n",
       "        print ('')"
      ],
@@ -353,10 +353,10 @@
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "-0.2367\t 0.2690\t 0.2938\t 1.7591\t-0.3278\t\n",
-        "-0.3011\t-0.2756\t-1.0273\t-0.4946\t 1.4776\t\n",
-        " 1.7101\t-0.7903\t-1.3902\t 0.1243\t-1.4785\t\n",
-        " 0.3633\t-2.3127\t-0.5851\t-1.6827\t-1.1897\t\n"
+        " 1.2921\t 0.1444\t 0.7473\t 0.3516\t 0.4305\t\n",
+        " 1.8350\t-0.1108\t-0.1800\t 0.3818\t-0.2313\t\n",
+        "-0.2557\t 1.3211\t 0.1475\t 0.9038\t-0.1875\t\n",
+        "-0.3380\t-1.1683\t 0.6501\t-1.3499\t-0.9251\t\n"
        ]
       }
      ],
@@ -378,10 +378,10 @@
       "import matplotlib.pyplot as plt\n",
       "import matplotlib.pylab as pylab\n",
       "\n",
-      "dog = DogSensor (variance=0.0)\n",
+      "dog = DogSensor(variance=0.0)\n",
       "xs = []\n",
       "for i in range(10):\n",
-      "    x = dog.sense()\n",
+      "    x = dog.sense_position()\n",
       "    xs.append(x)\n",
       "    print(\"%.4f\" % x, end=' '),\n",
       "plt.plot(xs, label='dog position')\n",
@@ -401,9 +401,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f1669269240>"
+        "<matplotlib.figure.Figure at 0x7fce7e0e9978>"
        ]
       }
      ],
@@ -427,10 +427,11 @@
       "\n",
       "    xs = []\n",
       "    for i in range(100):\n",
-      "        x = dog.sense()\n",
+      "        x = dog.sense_position()\n",
       "        xs.append(x)\n",
+      "\n",
       "    plt.plot(xs, label='sensor')\n",
-      "    plt.plot([0,99],[1,100], 'r--', label='actual')\n",
+      "    plt.plot([0, 99], [1, 100], 'r--', label='actual')\n",
       "    plt.xlabel('time')\n",
       "    plt.ylabel('pos')\n",
       "    plt.ylim([0,100])\n",
@@ -446,9 +447,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f1668dae2e8>"
+        "<matplotlib.figure.Figure at 0x7fce7dc2d3c8>"
        ]
       }
      ],
@@ -480,9 +481,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f1668f914a8>"
+        "<matplotlib.figure.Figure at 0x7fce7e0e9278>"
        ]
       }
      ],
@@ -500,9 +501,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
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+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f1668efb4e0>"
+        "<matplotlib.figure.Figure at 0x7fce7dd79470>"
        ]
       }
      ],
@@ -536,9 +537,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f1668e3a898>"
+        "<matplotlib.figure.Figure at 0x7fce7dae7518>"
        ]
       }
      ],
@@ -563,7 +564,7 @@
       "xs = range(100)\n",
       "ys = []\n",
       "for i in xs:\n",
-      "    ys.append(dog.sense())\n",
+      "    ys.append(dog.sense_position())\n",
       "               \n",
       "plt.plot(xs,ys, label='dog position')\n",
       "plt.legend(loc='best')\n",
@@ -575,9 +576,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f16692693c8>"
+        "<matplotlib.figure.Figure at 0x7fce79cf12b0>"
        ]
       }
      ],
@@ -678,9 +679,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f1668150978>"
+        "<matplotlib.figure.Figure at 0x7fce79d0dc18>"
        ]
       }
      ],
@@ -726,9 +727,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f166802a438>"
+        "<matplotlib.figure.Figure at 0x7fce79c932e8>"
        ]
       }
      ],
@@ -776,9 +777,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f1668019e10>"
+        "<matplotlib.figure.Figure at 0x7fce74a743c8>"
        ]
       }
      ],
@@ -882,7 +883,7 @@
       "\n",
       "pos,s = 2, 5\n",
       "for i in range(20):\n",
-      "    pos,s = update(pos, s, dog.sense(), 5)\n",
+      "    pos,s = update(pos, s, dog.sense_position(), 5)\n",
       "    print('time:', i, \n",
       "          '\\tposition =', \"%.3f\" % pos, \n",
       "          '\\tvariance =', \"%.3f\" % s)"
@@ -894,26 +895,26 @@
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "time: 0 \tposition = 1.529 \tvariance = 2.500\n",
-        "time: 1 \tposition = 1.416 \tvariance = 1.667\n",
-        "time: 2 \tposition = 1.260 \tvariance = 1.250\n",
-        "time: 3 \tposition = 0.756 \tvariance = 1.000\n",
-        "time: 4 \tposition = 0.854 \tvariance = 0.833\n",
-        "time: 5 \tposition = 0.728 \tvariance = 0.714\n",
-        "time: 6 \tposition = 0.531 \tvariance = 0.625\n",
-        "time: 7 \tposition = 0.376 \tvariance = 0.556\n",
-        "time: 8 \tposition = 0.306 \tvariance = 0.500\n",
-        "time: 9 \tposition = 0.353 \tvariance = 0.455\n",
-        "time: 10 \tposition = 0.357 \tvariance = 0.417\n",
-        "time: 11 \tposition = 0.268 \tvariance = 0.385\n",
-        "time: 12 \tposition = 0.423 \tvariance = 0.357\n",
-        "time: 13 \tposition = 0.455 \tvariance = 0.333\n",
-        "time: 14 \tposition = 0.481 \tvariance = 0.312\n",
-        "time: 15 \tposition = 0.399 \tvariance = 0.294\n",
-        "time: 16 \tposition = 0.387 \tvariance = 0.278\n",
-        "time: 17 \tposition = 0.432 \tvariance = 0.263\n",
-        "time: 18 \tposition = 0.463 \tvariance = 0.250\n",
-        "time: 19 \tposition = 0.489 \tvariance = 0.238\n"
+        "time: 0 \tposition = 1.848 \tvariance = 2.500\n",
+        "time: 1 \tposition = 1.440 \tvariance = 1.667\n",
+        "time: 2 \tposition = 1.032 \tvariance = 1.250\n",
+        "time: 3 \tposition = 0.551 \tvariance = 1.000\n",
+        "time: 4 \tposition = 0.504 \tvariance = 0.833\n",
+        "time: 5 \tposition = 0.379 \tvariance = 0.714\n",
+        "time: 6 \tposition = 0.244 \tvariance = 0.625\n",
+        "time: 7 \tposition = 0.321 \tvariance = 0.556\n",
+        "time: 8 \tposition = 0.381 \tvariance = 0.500\n",
+        "time: 9 \tposition = 0.305 \tvariance = 0.455\n",
+        "time: 10 \tposition = 0.435 \tvariance = 0.417\n",
+        "time: 11 \tposition = 0.423 \tvariance = 0.385\n",
+        "time: 12 \tposition = 0.445 \tvariance = 0.357\n",
+        "time: 13 \tposition = 0.404 \tvariance = 0.333\n",
+        "time: 14 \tposition = 0.397 \tvariance = 0.312\n",
+        "time: 15 \tposition = 0.282 \tvariance = 0.294\n",
+        "time: 16 \tposition = 0.239 \tvariance = 0.278\n",
+        "time: 17 \tposition = 0.161 \tvariance = 0.263\n",
+        "time: 18 \tposition = 0.204 \tvariance = 0.250\n",
+        "time: 19 \tposition = 0.160 \tvariance = 0.238\n"
        ]
       }
      ],
@@ -999,23 +1000,23 @@
      "input": [
       "# assume dog is always moving 1m to the right\n",
       "movement = 1\n",
-      "movement_error = 2\n",
-      "sensor_error = 10\n",
+      "movement_variance = 2\n",
+      "sensor_variance = 10\n",
       "pos = (0, 500)   # gaussian N(0,50)\n",
       "\n",
-      "dog = DogSensor(pos[0], velocity=movement, variance=sensor_error)\n",
+      "dog = DogSensor(pos[0], velocity=movement, variance=sensor_variance)\n",
       "\n",
       "zs = []\n",
       "ps = []\n",
       "\n",
       "for i in range(10):\n",
-      "    pos = predict(pos[0], pos[1], movement, movement_error)\n",
+      "    pos = predict(pos[0], pos[1], movement, movement_variance)\n",
       "    print('PREDICT: {: 10.4f} {: 10.4f}'.format(pos[0], pos[1]),end='\\t')\n",
       "    \n",
-      "    Z = dog.sense()\n",
+      "    Z = dog.sense_position()\n",
       "    zs.append(Z)\n",
       "    \n",
-      "    pos = update(pos[0], pos[1], Z, sensor_error)\n",
+      "    pos = update(pos[0], pos[1], Z, sensor_variance)\n",
       "    ps.append(pos[0])\n",
       "    \n",
       "    print('UPDATE: {: 10.4f} {: 10.4f}'.format(pos[0], pos[1]))\n",
@@ -1033,24 +1034,24 @@
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "PREDICT:     1.0000   502.0000\tUPDATE:    -1.2045     9.8047\n",
-        "PREDICT:    -0.2045    11.8047\tUPDATE:    -1.4522     5.4138\n",
-        "PREDICT:    -0.4522     7.4138\tUPDATE:     3.2771     4.2574\n",
-        "PREDICT:     4.2771     6.2574\tUPDATE:     6.7788     3.8490\n",
-        "PREDICT:     7.7788     5.8490\tUPDATE:     7.2769     3.6904\n",
-        "PREDICT:     8.2769     5.6904\tUPDATE:     7.7228     3.6267\n",
-        "PREDICT:     8.7228     5.6267\tUPDATE:     7.7554     3.6007\n",
-        "PREDICT:     8.7554     5.6007\tUPDATE:     8.6251     3.5900\n",
-        "PREDICT:     9.6251     5.5900\tUPDATE:     9.8390     3.5856\n",
-        "PREDICT:    10.8390     5.5856\tUPDATE:     9.2933     3.5838\n"
+        "PREDICT:     1.0000   502.0000\tUPDATE:    -2.4457     9.8047\n",
+        "PREDICT:    -1.4457    11.8047\tUPDATE:     0.7765     5.4138\n",
+        "PREDICT:     1.7765     7.4138\tUPDATE:     1.4225     4.2574\n",
+        "PREDICT:     2.4225     6.2574\tUPDATE:     2.3347     3.8490\n",
+        "PREDICT:     3.3347     5.8490\tUPDATE:     4.8032     3.6904\n",
+        "PREDICT:     5.8032     5.6904\tUPDATE:     5.4486     3.6267\n",
+        "PREDICT:     6.4486     5.6267\tUPDATE:     6.5119     3.6007\n",
+        "PREDICT:     7.5119     5.6007\tUPDATE:     8.9309     3.5900\n",
+        "PREDICT:     9.9309     5.5900\tUPDATE:     9.5451     3.5856\n",
+        "PREDICT:    10.5451     5.5856\tUPDATE:     9.8194     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 0x7f16680dd160>"
+        "<matplotlib.figure.Figure at 0x7fce79c8bfd0>"
        ]
       }
      ],
@@ -1063,15 +1064,15 @@
       "There is a fair bit of arbitrary constants code above, but don't worry about it. What does require explanation are the first few lines:\n",
       "\n",
       "    movement = 1 \n",
-      "    movement_error = 2\n",
+      "    movement_variance = 2\n",
       "    \n",
       "For the moment we are assuming that we have some other sensor that detects how the dog is moving. For example, there could be an inertial sensor clipped onto the dog's collar, and it reports how far the dog moved each time it is triggered. The details don't matter. The upshot is that we have a sensor, it has noise, and so we represent it with a Gaussian. Later we will learn what to do if we do not have a sensor for the `predict()` step.\n",
       "\n",
       "For now let's walk through the code and output bit by bit.\n",
       "\n",
       "    movement = 1\n",
-      "    movement_error = 2\n",
-      "    sensor_error = 10\n",
+      "    movement_variance = 2\n",
+      "    sensor_variance = 10\n",
       "    pos = (0, 500)   # gaussian N(0,500)\n",
       "    \n",
       "  \n",
@@ -1084,7 +1085,7 @@
      "source": [
       "Next we initialize the RFID simulator with\n",
       "\n",
-      "    dog = DogSensor(pos[0], velocity=movement, noise=sensor_error)\n",
+      "    dog = DogSensor(pos[0], velocity=movement, noise=sensor_variance)\n",
       "\n",
       "It may seem very 'convenient' to set the simulator to the same position as our guess, and it is. Do not fret. In the next example we will see the effect of a wildly inaccurate guess for the dog's initial position.\n",
       "\n",
@@ -1103,7 +1104,7 @@
       "Now we just enter our `update() ... predict()` loop.\n",
       "\n",
       "    for i in range(10):\n",
-      "        pos = predict(pos[0], pos[1], movement, sensor_error)\n",
+      "        pos = predict(pos[0], pos[1], movement, sensor_variance)\n",
       "        print 'PREDICT:', \"%.4f\" %pos[0], \", %.4f\" %pos[1]\n",
       "\n",
       "Wait, why `predict()` before `update()`? It turns out the order does not matter once, but the first call to `,DogSensor.sense()` assumes that the dog has already moved, so we start with the update step. In practice you will order these calls based on the details of your sensor, and you will very typically do the `sense()` first.\n",
@@ -1114,14 +1115,14 @@
       "\n",
       "What is this saying? After the prediction, we believe that we are at 1.0, and the variance is now 502.0. Recall we started at 500.0. The variance got worse, which is always what happens during the prediction step.\n",
       "\n",
-      "    Z = dog.sense()\n",
+      "    Z = dog.sense_position()\n",
       "    zs.append(Z)\n",
       "    \n",
       "Here we sense the dog's position, and store it in our array so we can plot the results later.\n",
       "\n",
       "Finally we call the update function of our filter, save the result in our *ps* array, and print the updated position belief:\n",
       "\n",
-      "    pos = update(pos[0], pos[1], Z, movement_error)\n",
+      "    pos = update(pos[0], pos[1], Z, movement_variance)\n",
       "    ps.append(pos[0])\n",
       "    print 'UPDATE:', \"%.4f\" %pos[0], \", %.4f\" %pos[1]\n",
       "    \n",
@@ -1150,20 +1151,20 @@
       "%precision 2\n",
       "# assume dog is always moving 1m to the right\n",
       "movement = 1\n",
-      "movement_error = 2\n",
-      "sensor_error = 4.5\n",
+      "movement_variance = 2\n",
+      "sensor_variance = 4.5\n",
       "pos = (0, 100)   # gaussian N(0,50)\n",
       "\n",
-      "dog = DogSensor(pos[0], velocity=movement, variance=sensor_error)\n",
+      "dog = DogSensor(pos[0], velocity=movement, variance=sensor_variance)\n",
       "\n",
       "zs, ps, vs = [], [], []\n",
       "for i in range(50):\n",
-      "    pos = predict(pos[0], pos[1], movement, movement_error)    \n",
-      "    Z = dog.sense()\n",
+      "    pos = predict(pos[0], pos[1], movement, movement_variance)    \n",
+      "    Z = dog.sense_position()\n",
       "    zs.append(Z)\n",
       "    vs.append(pos[1])\n",
       "    \n",
-      "    pos = update(pos[0], pos[1], Z, sensor_error)\n",
+      "    pos = update(pos[0], pos[1], Z, sensor_variance)\n",
       "    ps.append(pos[0])\n",
       "    \n",
       "#plt.subplot(121)    \n",
@@ -1185,17 +1186,17 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f1653bd9a20>"
+        "<matplotlib.figure.Figure at 0x7fce74a4a710>"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f1668c71128>"
+        "<matplotlib.figure.Figure at 0x7fce79d0d320>"
        ]
       },
       {
@@ -1270,17 +1271,17 @@
      "source": [
       "For many purposes the code above suffices. However, if you write enough of these filters the functions will become a bit annoying. For example, having to write\n",
       "\n",
-      "    pos = predict(pos[0], pos[1], movement, movement_error) \n",
+      "    pos = predict(pos[0], pos[1], movement, movement_variance) \n",
       "    \n",
       "is a bit cumbersome and error prone. Let's investigate how we might implement this in a form that makes our lives easier.\n",
       "\n",
       "First, values for the movement error and the measurement errors are typically constant for a given problem, so we only want to specify them once. We can store them in instance variables in the class. Second, it is annoying to have to pass in the state (pos in the code snippet above) and then remember to assign the output of the function back to that state, so the state should also be an instance variable. Our first attempt might look like:\n",
       "\n",
       "    class KalmanFilter1D:\n",
-      "        def __init__(self, initial_state, measurement_error, movement_error):\n",
+      "        def __init__(self, initial_state, measurement_variance, movement_variance):\n",
       "            self.state = initial_state\n",
-      "            self.measurement_error = measurement_error\n",
-      "            self.movement_error = movement error\n",
+      "            self.measurement_variance = measurement_variance\n",
+      "            self.movement_variance = movement_variance\n",
       "\n",
       "That works, but I am going to use different naming. The Kalman filter literature has settled on one letter notations for each of these concepts, and so you might as well start getting exposed to it now. At first it seems impossiblely terse, but as you become familiar with the nomenclature you'll see that the math formulas in the textbooks will have an exact one-to-one correspondance with the code. Unfortunately there is not a lot of meaning behind the names chosen; you will just have to memorize them.\n",
       "\n",
@@ -1413,15 +1414,15 @@
      "metadata": {},
      "source": [
       "#####Exercise:\n",
-      "Modify the values of `movement_error` and `sensor_error` and note the effect on the filter and on the variance. Which has a larger effect on the value that variance converges to. For example, which results in a smaller variance:\n",
+      "Modify the values of `movement_variance` and `sensor_variance` and note the effect on the filter and on the variance. Which has a larger effect on the value that variance converges to. For example, which results in a smaller variance:\n",
       "\n",
-      "    movement_error = 40\n",
-      "    sensor_error = 2\n",
+      "    movement_variance = 40\n",
+      "    sensor_variance = 2\n",
       "    \n",
       "or:\n",
       "\n",
-      "    movement_error = 2\n",
-      "    sensor_error = 40  "
+      "    movement_variance = 2\n",
+      "    sensor_variance = 40"
      ]
     },
     {
@@ -1472,7 +1473,7 @@
      "source": [
       "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 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",
+      "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_variance = .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. "
      ]
@@ -1482,7 +1483,7 @@
      "collapsed": false,
      "input": [
       "variance = 2.13**2\n",
-      "movement_error = .2\n",
+      "movement_variance = .2\n",
       "actual_voltage = 16.3\n",
       "\n",
       "N=50\n",
@@ -1494,7 +1495,7 @@
       "                    P = 1000,         # initial variance \n",
       "                                      # large says 'who knows?'\n",
       "                    R=variance,       # sensor noise\n",
-      "                    Q=movement_error) # movement noise\n",
+      "                    Q=movement_variance) # movement noise\n",
       "\n",
       "for i in range(N):\n",
       "    kf.predict(movement)\n",
@@ -1521,17 +1522,17 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f1668055668>"
+        "<matplotlib.figure.Figure at 0x7fce7dd40780>"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f165c042e80>"
+        "<matplotlib.figure.Figure at 0x7fce74adaa90>"
        ]
       },
       {
@@ -1647,9 +1648,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f1653b8c4a8>"
+        "<matplotlib.figure.Figure at 0x7fce74aac3c8>"
        ]
       }
      ],
@@ -1684,25 +1685,25 @@
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "sensor_error = 30\n",
+      "sensor_variance = 30\n",
       "movement_sensor = 30\n",
       "pos = (0,500)\n",
       "\n",
-      "dog = DogSensor(0, velocity=movement, variance=sensor_error)\n",
+      "dog = DogSensor(0, velocity=movement, variance=sensor_variance)\n",
       "\n",
       "zs = []\n",
       "ps = []\n",
       "vs = []\n",
       "\n",
       "for i in range(100):\n",
-      "    Z = dog.sense()\n",
+      "    Z = dog.sense_position()\n",
       "    zs.append(Z)\n",
       "    \n",
-      "    pos = update(pos[0], pos[1], Z, sensor_error)\n",
+      "    pos = update(pos[0], pos[1], Z, sensor_variance)\n",
       "    ps.append(pos[0])\n",
       "    vs.append(pos[1])\n",
       "\n",
-      "    pos = predict(pos[0], pos[1], movement+ random.randn(), movement_error)\n",
+      "    pos = predict(pos[0], pos[1], movement+ random.randn(), movement_variance)\n",
       "\n",
       "plt.plot(ps, label='filter')\n",
       "plt.plot(zs, linestyle='dashed', c='r', label='measurement')\n",
@@ -1718,17 +1719,17 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f1668f3e160>"
+        "<matplotlib.figure.Figure at 0x7fce74aa2358>"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f166815b3c8>"
+        "<matplotlib.figure.Figure at 0x7fce79d25080>"
        ]
       }
      ],
@@ -1774,26 +1775,26 @@
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "sensor_error = 30000\n",
-      "movement_error = 2\n",
+      "sensor_variance = 30000\n",
+      "movement_variance = 2\n",
       "pos = (0,500)\n",
       "\n",
-      "dog = DogSensor(pos[0], velocity=movement, variance=sensor_error)\n",
+      "dog = DogSensor(pos[0], velocity=movement, variance=sensor_variance)\n",
       "\n",
       "zs = []\n",
       "ps = []\n",
       "\n",
       "for i in range(1000):\n",
-      "    pos = predict(pos[0], pos[1], movement, movement_error)\n",
+      "    pos = predict(pos[0], pos[1], movement, movement_variance)\n",
       "    \n",
-      "    Z = dog.sense()\n",
+      "    Z = dog.sense_position()\n",
       "    zs.append(Z)\n",
       "    \n",
-      "    pos = update(pos[0], pos[1], Z, sensor_error)\n",
+      "    pos = update(pos[0], pos[1], Z, sensor_variance)\n",
       "    ps.append(pos[0])\n",
       "\n",
       "\n",
-      "p1, = plt.plot(zs,c='r', linestyle='dashed', label='measurement')\n",
+      "p1, = plt.plot(zs, c='r', linestyle='dashed', label='measurement')\n",
       "p2, = plt.plot(ps, c='#004080', label='filter')\n",
       "plt.legend(loc='best')\n",
       "plt.show()"
@@ -1804,9 +1805,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f166802d748>"
+        "<matplotlib.figure.Figure at 0x7fce74a748d0>"
        ]
       }
      ],
@@ -1839,26 +1840,26 @@
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "sensor_error = 30\n",
-      "movement_error = 2\n",
+      "sensor_variance = 30\n",
+      "movement_variance = 2\n",
       "pos = (1000,500)\n",
       "\n",
-      "dog = DogSensor(0, velocity=movement, variance=sensor_error)\n",
+      "dog = DogSensor(0, velocity=movement, variance=sensor_variance)\n",
       "\n",
       "zs = []\n",
       "ps = []\n",
       "\n",
       "for i in range(100):\n",
-      "    pos = predict(pos[0], pos[1], movement, movement_error)\n",
+      "    pos = predict(pos[0], pos[1], movement, movement_variance)\n",
       "    \n",
-      "    Z = dog.sense()\n",
+      "    Z = dog.sense_position()\n",
       "    zs.append(Z)\n",
       "    \n",
-      "    pos = update(pos[0], pos[1], Z, sensor_error)\n",
+      "    pos = update(pos[0], pos[1], Z, sensor_variance)\n",
       "    ps.append(pos[0])\n",
       "\n",
       "plt.plot(ps, label='filter')\n",
-      "plt.plot(zs,c='r', linestyle='dashed', label='measurement')\n",
+      "plt.plot(zs, c='r', linestyle='dashed', label='measurement')\n",
       "plt.legend(loc='best')\n",
       "plt.show()"
      ],
@@ -1868,9 +1869,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f1653a7f978>"
+        "<matplotlib.figure.Figure at 0x7fce749750b8>"
        ]
       }
      ],
@@ -1902,25 +1903,25 @@
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "sensor_error = 30000\n",
-      "movement_error = 2\n",
+      "sensor_variance = 30000\n",
+      "movement_variance = 2\n",
       "pos = (1000,500)\n",
       "\n",
       "\n",
-      "dog = DogSensor(0, velocity=movement, variance=sensor_error) \n",
+      "dog = DogSensor(0, velocity=movement, variance=sensor_variance) \n",
       "zs = []\n",
       "ps = []\n",
       "\n",
       "for i in range(1000):\n",
-      "    pos = predict(pos[0], pos[1], movement, movement_error)\n",
+      "    pos = predict(pos[0], pos[1], movement, movement_variance)\n",
       "    \n",
-      "    Z = dog.sense()\n",
+      "    Z = dog.sense_position()\n",
       "    zs.append(Z)\n",
       "    \n",
-      "    pos = update(pos[0], pos[1], Z, sensor_error)\n",
+      "    pos = update(pos[0], pos[1], Z, sensor_variance)\n",
       "    ps.append(pos[0])\n",
       "\n",
-      "plt.plot(zs,c='r', linestyle='dashed', label='measurement')\n",
+      "plt.plot(zs, c='r', linestyle='dashed', label='measurement')\n",
       "plt.plot(ps, c='#004080', label='filter')\n",
       "plt.legend(loc='best')\n",
       "plt.show()"
@@ -1931,9 +1932,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f16539c9128>"
+        "<matplotlib.figure.Figure at 0x7fce74943a90>"
        ]
       }
      ],
@@ -1952,27 +1953,27 @@
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "sensor_error = 30000\n",
-      "movement_error = 2\n",
+      "sensor_variance = 30000\n",
+      "movement_variance = 2\n",
       "pos = None\n",
       "\n",
-      "dog = DogSensor(0, velocity=movement, variance=sensor_error)\n",
+      "dog = DogSensor(0, velocity=movement, variance=sensor_variance)\n",
       "\n",
       "zs = []\n",
       "ps = []\n",
       "\n",
       "for i in range(1000):\n",
-      "    Z = dog.sense()\n",
+      "    Z = dog.sense_position()\n",
       "    zs.append(Z)\n",
       "    if pos == None:\n",
       "        pos = (Z, 500)\n",
       "    \n",
-      "    pos = update(pos[0], pos[1], Z, sensor_error)\n",
+      "    pos = update(pos[0], pos[1], Z, sensor_variance)\n",
       "    ps.append(pos[0])\n",
       "\n",
-      "    pos = predict(pos[0], pos[1], movement, movement_error)\n",
+      "    pos = predict(pos[0], pos[1], movement, movement_variance)\n",
       "\n",
-      "plt.plot(zs,c='r', linestyle='dashed', label='measurement')\n",
+      "plt.plot(zs, c='r', linestyle='dashed', label='measurement')\n",
       "plt.plot(ps, c='#004080', label='filter')\n",
       "plt.legend(loc='best')\n",
       "plt.show()"
@@ -1983,9 +1984,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f1653abbd30>"
+        "<matplotlib.figure.Figure at 0x7fce748d07b8>"
        ]
       }
      ],
@@ -2033,10 +2034,10 @@
       "\n",
       "interact(plot_kalman_filter,\n",
       "         start_pos=(-10,10), \n",
-      "         sensor_noise=widgets.IntSliderWidget(value=5,min=0,max=100), \n",
-      "         movement=widgets.FloatSliderWidget(value=1,min=-2.,max=2.), \n",
-      "         movement_noise=widgets.FloatSliderWidget(value=5,min=0,max=100.),\n",
-      "         noise_scale=widgets.FloatSliderWidget(value=1,min=0,max=2.))"
+      "         sensor_noise=widgets.IntSliderWidget(value=5, min=0, max=100), \n",
+      "         movement=widgets.FloatSliderWidget(value=1, min=-2., max=2.), \n",
+      "         movement_noise=widgets.FloatSliderWidget(value=5, min=0, max=100.),\n",
+      "         noise_scale=widgets.FloatSliderWidget(value=1, min=0, max=2.))"
      ],
      "language": "python",
      "metadata": {},
@@ -2080,7 +2081,7 @@
       "        move_error = random.randn()*movement_noise\n",
       "        dog.x += move_error\n",
       "        \n",
-      "        z = dog.sense()\n",
+      "        z = dog.sense_position()\n",
       "        zs.append(z)\n",
       "\n",
       "        pos = update(pos[0], pos[1], z, sensor_noise)\n",
@@ -2088,16 +2089,16 @@
       "\n",
       "        pos = predict(pos[0], pos[1], movement, movement_noise)\n",
       "\n",
-      "    plt.plot(zs,c='r', linestyle='dashed', label='measurement')\n",
+      "    plt.plot(zs, c='r', linestyle='dashed', label='measurement')\n",
       "    plt.plot(ps, c='#004080', alpha=0.7, label='filter')\n",
       "    plt.legend(loc='best')\n",
       "    plt.show()\n",
       "\n",
       "interact(plot_kalman_filter,\n",
-      "         start_pos=(-10,10), \n",
-      "         sensor_noise=widgets.FloatSliderWidget(value=5,min=0.,max=100), \n",
-      "         movement=widgets.FloatSliderWidget(value=1,min=-2.,max=2.), \n",
-      "         movement_noise=widgets.FloatSliderWidget(value=.1,min=0,max=.5))"
+      "         start_pos=(-10, 10), \n",
+      "         sensor_noise=widgets.FloatSliderWidget(value=5, min=0., max=100), \n",
+      "         movement=widgets.FloatSliderWidget(value=1, min=-2., max=2.), \n",
+      "         movement_noise=widgets.FloatSliderWidget(value=.1, min=0, max=.5))"
      ],
      "language": "python",
      "metadata": {},
@@ -2105,13 +2106,21 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f1653b22048>"
+        "<matplotlib.figure.Figure at 0x7fce79c68c88>"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 29,
+       "text": [
+        "<function __main__.plot_kalman_filter>"
        ]
       }
      ],
-     "prompt_number": 34
+     "prompt_number": 29
     },
     {
      "cell_type": "heading",
@@ -2146,7 +2155,8 @@
      ],
      "language": "python",
      "metadata": {},
-     "outputs": []
+     "outputs": [],
+     "prompt_number": 30
     },
     {
      "cell_type": "markdown",
@@ -2159,24 +2169,22 @@
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "sensor_error = 30\n",
-      "movement_error = 2\n",
+      "sensor_variance = 30\n",
+      "movement_variance = 2\n",
       "pos = (100,500)\n",
       "\n",
-      "zs = []\n",
-      "ps = []\n",
-      "\n",
+      "zs, ps = [], []\n",
       "\n",
       "for i in range(100):\n",
-      "    pos = predict(pos[0], pos[1], movement, movement_error)\n",
+      "    pos = predict(pos[0], pos[1], movement, movement_variance)\n",
       "\n",
       "    Z = math.sin(i/3.)*2\n",
       "    zs.append(Z)\n",
       "    \n",
-      "    pos = update(pos[0], pos[1], Z, sensor_error)\n",
+      "    pos = update(pos[0], pos[1], Z, sensor_variance)\n",
       "    ps.append(pos[0])\n",
       "\n",
-      "plt.plot(zs,c='r', linestyle='dashed', label='input')\n",
+      "plt.plot(zs, c='r', linestyle='dashed', label='input')\n",
       "plt.plot(ps, c='#004080', label='filter')\n",
       "plt.legend(loc='best')\n",
       "plt.show()"
@@ -2187,13 +2195,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f1653b3b278>"
+        "<matplotlib.figure.Figure at 0x7fce79c28ba8>"
        ]
       }
      ],
-     "prompt_number": 35
+     "prompt_number": 31
     },
     {
      "cell_type": "markdown",
@@ -2231,7 +2239,8 @@
      ],
      "language": "python",
      "metadata": {},
-     "outputs": []
+     "outputs": [],
+     "prompt_number": 32
     },
     {
      "cell_type": "markdown",
@@ -2244,23 +2253,22 @@
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "sensor_error = 30\n",
-      "movement_error = 2\n",
+      "sensor_variance = 30\n",
+      "movement_variance = 2\n",
       "pos = (100,500)\n",
       "\n",
-      "zs = []\n",
-      "ps = []\n",
+      "zs, ps = [], []\n",
       "\n",
       "for i in range(100):\n",
-      "    pos = predict(pos[0], pos[1], movement, movement_error)\n",
+      "    pos = predict(pos[0], pos[1], movement, movement_variance)\n",
       "\n",
       "    Z = math.sin(i/3.)*2 + random.randn()*1.2\n",
       "    zs.append(Z)\n",
       "    \n",
-      "    pos = update(pos[0], pos[1], Z, sensor_error)\n",
+      "    pos = update(pos[0], pos[1], Z, sensor_variance)\n",
       "    ps.append(pos[0])\n",
       "\n",
-      "p1, = plt.plot(zs,c='r', linestyle='dashed', label='measurement')\n",
+      "p1, = plt.plot(zs, c='r', linestyle='dashed', label='measurement')\n",
       "p2, = plt.plot(ps, c='#004080', label='filter')\n",
       "plt.legend(loc='best')\n",
       "plt.show()"
@@ -2271,13 +2279,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f165c03de10>"
+        "<matplotlib.figure.Figure at 0x7fce749f6128>"
        ]
       }
      ],
-     "prompt_number": 36
+     "prompt_number": 33
     },
     {
      "cell_type": "markdown",