Added a class to implement the Kalman filter.

Not sure i am crazy about it. But it needs to either be removed,
or added to all the examples in the chapter.

Chapter05_Kalman_Filters/Kalman_Filters.ipynb

@@ -1,7 +1,7 @@
 {
  "metadata": {
   "name": "",
-  "signature": "sha256:6bbf3b69472f466f37cf43314d0391a512ece3f8f9592b3b02ec292d425f2c17"
+  "signature": "sha256:40612f6bc9924dbda2c3d14cf0077f63a2ba9109782c950a0523fc7bd1192850"
  },
  "nbformat": 3,
  "nbformat_minor": 0,
@@ -257,7 +257,7 @@
        "output_type": "pyout",
        "prompt_number": 2,
        "text": [
-        "<IPython.core.display.HTML at 0x7feedcb960f0>"
+        "<IPython.core.display.HTML at 0x7f634c5fa588>"
        ]
       }
      ],
@@ -349,10 +349,10 @@
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        " 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"
+        " 0.7405\t 0.7229\t-0.5326\t 1.2238\t-0.1799\t\n",
+        " 0.0760\t-0.5359\t 1.0663\t 0.5362\t-0.6931\t\n",
+        " 1.9148\t 0.5687\t 1.1368\t-0.7581\t 1.7235\t\n",
+        "-1.1808\t 0.9561\t 0.4169\t 0.1081\t-1.5491\t\n"
        ]
       }
      ],
@@ -399,7 +399,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7feec4299b70>"
+        "<matplotlib.figure.Figure at 0x7f633596c3c8>"
        ]
       }
      ],
@@ -442,9 +442,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7feec3fbfb38>"
+        "<matplotlib.figure.Figure at 0x7f63356150b8>"
        ]
       }
      ],
@@ -476,9 +476,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7feec428f4a8>"
+        "<matplotlib.figure.Figure at 0x7f6335619d68>"
        ]
       }
      ],
@@ -496,9 +496,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
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+       "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7feec3f060b8>"
+        "<matplotlib.figure.Figure at 0x7f63355a0630>"
        ]
       }
      ],
@@ -541,7 +541,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7feec3eb3c50>"
+        "<matplotlib.figure.Figure at 0x7f63354cd8d0>"
        ]
       }
      ],
@@ -578,9 +578,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7feec4299940>"
+        "<matplotlib.figure.Figure at 0x7f633596c550>"
        ]
       }
      ],
@@ -677,7 +677,7 @@
        "output_type": "display_data",
        "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7feec0193ba8>"
+        "<matplotlib.figure.Figure at 0x7f6335563630>"
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@@ -725,7 +725,7 @@
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        "text": [
-        "<matplotlib.figure.Figure at 0x7feec00e27f0>"
+        "<matplotlib.figure.Figure at 0x7f63205d3d30>"
        ]
       }
      ],
@@ -775,7 +775,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7feec00479e8>"
+        "<matplotlib.figure.Figure at 0x7f632057f208>"
        ]
       }
      ],
@@ -898,26 +898,26 @@
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "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"
+        "time: 0 \tposition = 1.616 \tvariance = 2.500\n",
+        "time: 1 \tposition = 0.501 \tvariance = 1.667\n",
+        "time: 2 \tposition = 0.388 \tvariance = 1.250\n",
+        "time: 3 \tposition = 0.105 \tvariance = 1.000\n",
+        "time: 4 \tposition = 0.222 \tvariance = 0.833\n",
+        "time: 5 \tposition = 0.154 \tvariance = 0.714\n",
+        "time: 6 \tposition = 0.194 \tvariance = 0.625\n",
+        "time: 7 \tposition = 0.024 \tvariance = 0.556\n",
+        "time: 8 \tposition = -0.019 \tvariance = 0.500\n",
+        "time: 9 \tposition = -0.079 \tvariance = 0.455\n",
+        "time: 10 \tposition = -0.064 \tvariance = 0.417\n",
+        "time: 11 \tposition = -0.046 \tvariance = 0.385\n",
+        "time: 12 \tposition = -0.108 \tvariance = 0.357\n",
+        "time: 13 \tposition = -0.138 \tvariance = 0.333\n",
+        "time: 14 \tposition = -0.159 \tvariance = 0.312\n",
+        "time: 15 \tposition = -0.151 \tvariance = 0.294\n",
+        "time: 16 \tposition = -0.212 \tvariance = 0.278\n",
+        "time: 17 \tposition = -0.212 \tvariance = 0.263\n",
+        "time: 18 \tposition = -0.266 \tvariance = 0.250\n",
+        "time: 19 \tposition = -0.241 \tvariance = 0.238\n"
        ]
       }
      ],
@@ -1037,24 +1037,24 @@
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "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"
+        "PREDICT:     1.0000   502.0000\tUPDATE:     0.0324     9.8047\n",
+        "PREDICT:     1.0324    11.8047\tUPDATE:     1.4467     5.4138\n",
+        "PREDICT:     2.4467     7.4138\tUPDATE:     1.8519     4.2574\n",
+        "PREDICT:     2.8519     6.2574\tUPDATE:     3.6222     3.8490\n",
+        "PREDICT:     4.6222     5.8490\tUPDATE:     3.9355     3.6904\n",
+        "PREDICT:     4.9355     5.6904\tUPDATE:     6.1234     3.6267\n",
+        "PREDICT:     7.1234     5.6267\tUPDATE:     8.3287     3.6007\n",
+        "PREDICT:     9.3287     5.6007\tUPDATE:     8.6857     3.5900\n",
+        "PREDICT:     9.6857     5.5900\tUPDATE:     9.6664     3.5856\n",
+        "PREDICT:    10.6664     5.5856\tUPDATE:     9.7450     3.5838\n"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7feec01adac8>"
+        "<matplotlib.figure.Figure at 0x7f6328076a20>"
        ]
       }
      ],
@@ -1190,9 +1190,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7feeb80a87b8>"
+        "<matplotlib.figure.Figure at 0x7f63205351d0>"
        ]
       },
       {
@@ -1200,7 +1200,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7feeb80817b8>"
+        "<matplotlib.figure.Figure at 0x7f63280d25f8>"
        ]
       },
       {
@@ -1252,6 +1252,82 @@
       "When the track first starts you can see that the measurements varies quite a bit from the initial prediction. At this point the Gaussian probability is small (the curve is low and wide) so the filter does not trust its prediction. As a result, the filter adjusts its estimate a large amount. As the filter innovates you can see that as the Gaussian becomes taller, indicating greater certainty in the estimate, the filter's output becomes very close to a straight line. At `x=15` and greater you can see that there is a large amount of noise in the measurement, but the filter does not react much to it compared to how much it changed for the firs noisy measurement."
      ]
     },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "Implementation in a Class (Optional)\n"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "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",
+      "    \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",
+      "            self.state = initial_state\n",
+      "            self.measurement_error = measurement_error\n",
+      "            self.movement_error = movement error\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",
+      "So, we use `x` for the state (estimated value of the filter) and `P` for the variance of the state. `R` is the measurement error, and `Q` is the movement error. This gives us:\n",
+      "\n",
+      "    class KalmanFilter1D:\n",
+      "        def __init__(self, x0, R, Q):\n",
+      "            self.x = x0\n",
+      "            self.R = R\n",
+      "            self.Q = Q\n",
+      "            \n",
+      "Now we can implement the `update()` and `predict()` function. In the literature the measurement is usually named either `z` or `y`; I find `y` is too easy to confuse with the y axis of a plot, so I like `z`. I like to think I can hear a `z` in *measurement*, which helps me remember what `z` stands for. So for the update method we might write:\n",
+      "\n",
+      "    def update(z):\n",
+      "        self.x = (self.P * z + self.x * self.R) / (self.P + self.R)\n",
+      "        self.P = 1 / (1/self.P + 1/self.R)\n",
+      "\n",
+      "Finally, the movement is usually called `u`, and so we will use that. So for the predict function we might write:\n",
+      "\n",
+      "    def predict(self, u):\n",
+      "        self.x += u\n",
+      "        self.P += self.Q\n",
+      "       \n",
+      "That give us the following code. Production code would require signficant comments. However, in the next chapter we will develop Kalman filter code that works for any dimension, including 1, so this class will never be more than a stepping stone for us, since we can, and will use the class developed in the next chapter in the rest of the book."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "class KalmanFilter1D:\n",
+      "    def __init__(self, x0, P, R, Q):\n",
+      "        self.x = x0\n",
+      "        self.P = P\n",
+      "        self.R = R\n",
+      "        self.Q = Q\n",
+      "        \n",
+      "    def update(self, z):\n",
+      "        self.x = (self.P * z + self.x * self.R) / (self.P + self.R)\n",
+      "        self.P = 1. / (1./self.P + 1./self.R)\n",
+      "\n",
+      "\n",
+      "    def predict(self, u=0.0):\n",
+      "        self.x += u\n",
+      "        self.P += self.Q"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 29
+    },
     {
      "cell_type": "heading",
      "level": 2,
@@ -1365,14 +1441,13 @@
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "temp_variance = 2.13**2\n",
-      "def volt():\n",
+      "def volt(temp_variance):\n",
       "    return random.randn()*temp_variance + 16.3"
      ],
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 20
+     "prompt_number": 32
     },
     {
      "cell_type": "markdown",
@@ -1402,6 +1477,8 @@
      "cell_type": "code",
      "collapsed": false,
      "input": [
+      "'''\n",
+      "temp_variance = 2.13**2\n",
       "sensor_error = temp_variance\n",
       "movement_error = .2\n",
       "movement = 0\n",
@@ -1431,6 +1508,41 @@
       "plt.title('Variance')\n",
       "plt.show()\n",
       "print('Variance converges to',vs[-1])\n",
+      "print('Last voltage is',voltage[0])\n",
+      "\n",
+      "'''\n",
+      "\n",
+      "temp_variance = 2.13**2\n",
+      "movement_error = .2\n",
+      "\n",
+      "N=50\n",
+      "zs = [volt(temp_variance) for i in range(N)]\n",
+      "ps = []\n",
+      "estimates = []\n",
+      "\n",
+      "kf = KalmanFilter1D(x0=25,            # initial state\n",
+      "                    P = 1000,         # initial variance - large says 'who knows?'\n",
+      "                    R=temp_variance,  # sensor noise\n",
+      "                    Q=movement_error) # movement noise\n",
+      "\n",
+      "for i in range(N):\n",
+      "    kf.predict(movement)\n",
+      "    kf.update(zs[i])\n",
+      "\n",
+      "    # save for latter plotting\n",
+      "    estimates.append(kf.x)\n",
+      "    ps.append(kf.P)\n",
+      "\n",
+      "# plot the filter output and the variance\n",
+      "plt.scatter(range(N), zs, marker='+', s=64, color='r', label='measurements')\n",
+      "p1, = plt.plot(estimates, label='filter')\n",
+      "plt.legend(loc='best')\n",
+      "plt.xlim((0,N));plt.ylim((0,30))\n",
+      "plt.show()\n",
+      "plt.plot(ps)\n",
+      "plt.title('Variance')\n",
+      "plt.show()\n",
+      "print('Variance converges to',vs[-1])\n",
       "print('Last voltage is',voltage[0])"
      ],
      "language": "python",
@@ -1439,9 +1551,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7feec00ae048>"
+        "<matplotlib.figure.Figure at 0x7f6320161438>"
        ]
       },
       {
@@ -1449,19 +1561,19 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7feec01d3550>"
+        "<matplotlib.figure.Figure at 0x7f6320126ef0>"
        ]
       },
       {
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "Variance converges to 0.8577995630075655\n",
-        "Last voltage is 17.438686650938127\n"
+        "Variance converges to 16.920051100338714\n",
+        "Last voltage is 25\n"
        ]
       }
      ],
-     "prompt_number": 22
+     "prompt_number": 45
     },
     {
      "cell_type": "markdown",
@@ -1548,7 +1660,7 @@
       "def VKF():\n",
       "    voltage=(14,1000)\n",
       "    for i in range(N):\n",
-      "        Z = volt()\n",
+      "        Z = volt(temp_variance)\n",
       "        voltage = update(voltage[0], voltage[1], Z, sensor_error)\n",
       "    return voltage[0]\n",
       "\n",
@@ -1564,13 +1676,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f207ab078d0>"
+        "<matplotlib.figure.Figure at 0x7f63205bf898>"
        ]
       }
      ],
-     "prompt_number": 22
+     "prompt_number": 42
     },
     {
      "cell_type": "markdown",
@@ -1592,6 +1704,8 @@
      "cell_type": "markdown",
      "metadata": {},
      "source": [
+      "**author's note:** I am not overly keen about this explanation. It is true that multiple sensors improve results, but we get good results merely by having an accurate model of the process. I explain this much better in the next chapter. I'll leave this section here while I mull how best to explain this at this stage of learning. For now don't worry if this section is not entirely convincing; it does need work.\n",
+      "\n",
       "So how does the Kalman filter do so well? I have glossed over one aspect of the filter as it becomes confusing to address too many points at the same time. We will return to the dog tracking problem. We used two sensors to track the dog - the RFID sensor that detects position, and the inertial tracker that tracked movement. However, we have focused all of our attention on the position sensor. Let's change focus and see how the filter performs if the inertial tracker is also noisy. This will provide us with an vital insight into the performance of Kalman filters."
      ]
     },