Briefly covered batch processing and RTS.

Added sections on batch processing and using RTS smoothing.

10_Unscented_Kalman_Filters/Unscented_Kalman_Filter.ipynb

@@ -1,7 +1,7 @@
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-  "signature": "sha256:69a64babdd622c8d8baa2d202fe7e15207c4b74ceb33e933c23742815922808f"
+  "signature": "sha256:2a8444d5bb1c578519436371d5b6499027efd9beb16ee407d09eeb71499ea867"
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  "nbformat": 3,
  "nbformat_minor": 0,
@@ -259,13 +259,13 @@
        ],
        "metadata": {},
        "output_type": "pyout",
-       "prompt_number": 1,
+       "prompt_number": 2,
        "text": [
-        "<IPython.core.display.HTML at 0x7fd8c0d15128>"
+        "<IPython.core.display.HTML at 0x7f65c5720e10>"
        ]
       }
      ],
-     "prompt_number": 1
+     "prompt_number": 2
     },
     {
      "cell_type": "markdown",
@@ -309,13 +309,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd8a48af278>"
+        "<matplotlib.figure.Figure at 0x7f65a88b5080>"
        ]
       }
      ],
-     "prompt_number": 2
+     "prompt_number": 3
     },
     {
      "cell_type": "markdown",
@@ -347,13 +347,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd8a488d8d0>"
+        "<matplotlib.figure.Figure at 0x7f65a88c02b0>"
        ]
       }
      ],
-     "prompt_number": 3
+     "prompt_number": 4
     },
     {
      "cell_type": "markdown",
@@ -396,11 +396,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd8a487ab00>"
+        "<matplotlib.figure.Figure at 0x7f65a88aafd0>"
        ]
       }
      ],
-     "prompt_number": 4
+     "prompt_number": 5
     },
     {
      "cell_type": "markdown",
@@ -431,7 +431,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 5
+     "prompt_number": 6
     },
     {
      "cell_type": "markdown",
@@ -494,20 +494,20 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAs4AAAF2CAYAAAB+nR6pAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd89ff99160>"
+        "<matplotlib.figure.Figure at 0x7f65a86a3438>"
        ]
       },
       {
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "Difference in mean x=0.192, y=43.977\n"
+        "Difference in mean x=0.076, y=42.562\n"
        ]
       }
      ],
-     "prompt_number": 6
+     "prompt_number": 7
     },
     {
      "cell_type": "markdown",
@@ -542,11 +542,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd8a4554128>"
+        "<matplotlib.figure.Figure at 0x7f65a8840b38>"
        ]
       }
      ],
-     "prompt_number": 7
+     "prompt_number": 8
     },
     {
      "cell_type": "markdown",
@@ -582,11 +582,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd89ffb8400>"
+        "<matplotlib.figure.Figure at 0x7f65a0386eb8>"
        ]
       }
      ],
-     "prompt_number": 8
+     "prompt_number": 9
     },
     {
      "cell_type": "markdown",
@@ -637,11 +637,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd89fd809e8>"
+        "<matplotlib.figure.Figure at 0x7f65a02f5710>"
        ]
       }
      ],
-     "prompt_number": 9
+     "prompt_number": 10
     },
     {
      "cell_type": "markdown",
@@ -846,7 +846,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 10
+     "prompt_number": 11
     },
     {
      "cell_type": "markdown",
@@ -885,7 +885,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 11
+     "prompt_number": 12
     },
     {
      "cell_type": "markdown",
@@ -978,13 +978,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7fd8a007ee48>"
+        "<matplotlib.figure.Figure at 0x7f65a861d9e8>"
        ]
       }
      ],
-     "prompt_number": 12
+     "prompt_number": 13
     },
     {
      "cell_type": "markdown",
@@ -1010,13 +1010,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd89ffd65f8>"
+        "<matplotlib.figure.Figure at 0x7f65a02eff60>"
        ]
       }
      ],
-     "prompt_number": 13
+     "prompt_number": 14
     },
     {
      "cell_type": "markdown",
@@ -1059,11 +1059,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd89ff61320>"
+        "<matplotlib.figure.Figure at 0x7f65a8875e10>"
        ]
       }
      ],
-     "prompt_number": 14
+     "prompt_number": 15
     },
     {
      "cell_type": "markdown",
@@ -1182,11 +1182,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd8a431a128>"
+        "<matplotlib.figure.Figure at 0x7f65a032d630>"
        ]
       }
      ],
-     "prompt_number": 15
+     "prompt_number": 16
     },
     {
      "cell_type": "markdown",
@@ -1214,11 +1214,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd89ff68048>"
+        "<matplotlib.figure.Figure at 0x7f65a02ef4e0>"
        ]
       }
      ],
-     "prompt_number": 16
+     "prompt_number": 17
     },
     {
      "cell_type": "markdown",
@@ -1245,7 +1245,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 17
+     "prompt_number": 18
     },
     {
      "cell_type": "markdown",
@@ -1302,11 +1302,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd8a45edbe0>"
+        "<matplotlib.figure.Figure at 0x7f65a0291c88>"
        ]
       }
      ],
-     "prompt_number": 18
+     "prompt_number": 19
     },
     {
      "cell_type": "markdown",
@@ -1347,11 +1347,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd8a436e5f8>"
+        "<matplotlib.figure.Figure at 0x7f65a88a2208>"
        ]
       }
      ],
-     "prompt_number": 19
+     "prompt_number": 20
     },
     {
      "cell_type": "markdown",
@@ -1373,7 +1373,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 20
+     "prompt_number": 21
     },
     {
      "cell_type": "markdown",
@@ -1405,11 +1405,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd8a42db9e8>"
+        "<matplotlib.figure.Figure at 0x7f65a88c90f0>"
        ]
       }
      ],
-     "prompt_number": 21
+     "prompt_number": 22
     },
     {
      "cell_type": "markdown",
@@ -1432,7 +1432,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 22
+     "prompt_number": 23
     },
     {
      "cell_type": "markdown",
@@ -1468,7 +1468,7 @@
        ]
       }
      ],
-     "prompt_number": 23
+     "prompt_number": 24
     },
     {
      "cell_type": "markdown",
@@ -1504,11 +1504,11 @@
        "output_type": "display_data",
        "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7fd8a006c6d8>"
+        "<matplotlib.figure.Figure at 0x7f65a02c2160>"
        ]
       }
      ],
-     "prompt_number": 24
+     "prompt_number": 25
     },
     {
      "cell_type": "markdown",
@@ -1563,11 +1563,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd8a4316e10>"
+        "<matplotlib.figure.Figure at 0x7f65a0255c88>"
        ]
       }
      ],
-     "prompt_number": 25
+     "prompt_number": 26
     },
     {
      "cell_type": "markdown",
@@ -1663,11 +1663,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd8a43a8ac8>"
+        "<matplotlib.figure.Figure at 0x7f65a8849f60>"
        ]
       }
      ],
-     "prompt_number": 26
+     "prompt_number": 27
     },
     {
      "cell_type": "markdown",
@@ -1698,11 +1698,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd8a005d400>"
+        "<matplotlib.figure.Figure at 0x7f65a8570438>"
        ]
       }
      ],
-     "prompt_number": 27
+     "prompt_number": 28
     },
     {
      "cell_type": "markdown",
@@ -1779,7 +1779,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 28
+     "prompt_number": 29
     },
     {
      "cell_type": "markdown",
@@ -1866,13 +1866,13 @@
       {
        "metadata": {},
        "output_type": "pyout",
-       "prompt_number": 29,
+       "prompt_number": 30,
        "text": [
         "array([410, 520, 630])"
        ]
       }
      ],
-     "prompt_number": 29
+     "prompt_number": 30
     },
     {
      "cell_type": "markdown",
@@ -1957,7 +1957,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 30
+     "prompt_number": 31
     },
     {
      "cell_type": "markdown",
@@ -1997,7 +1997,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 31
+     "prompt_number": 32
     },
     {
      "cell_type": "markdown",
@@ -2066,7 +2066,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 32
+     "prompt_number": 33
     },
     {
      "cell_type": "markdown",
@@ -2375,7 +2375,176 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 33
+     "prompt_number": 34
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "## Batch Processing\n",
+      "\n",
+      "The Kalman filter is designed as a recursive algorithm - as new measurements come in we immediately create a new estimate. But it is very common to have a set of data that have been already collected which we want to filter. Kalman filters can always be run in a *batch* mode, where all of the measurements are filtered at once. We have implemented this in `KalmanFilter.batch_filter()`. Internally, all the function does is loop over the measurements and collect the resulting state and covariance estimates in arrays. It may seem a bit trivial, but you will almost always want to do this if you can for several reasons. First, it will execute a bit quicker than if you implement the loop yourself. Second, the logic in your code will be a bit cleaner, and you will have a reduced chance of bugs. Third, and most importantly, it you batch process your data you can then use an extremely powerful techique to generate far smoother results than you have seen so far. We will present that technique in the next section; here I will show you how to use `KalmanFilter.batch_filter()`.\n",
+      "\n",
+      "All you have to do is collect your measurements into an array or list. Maybe it is in a CSV file, for example.\n",
+      "\n",
+      "    zs = read_altitude_from_csv()\n",
+      "\n",
+      "Or maybe you will generate it using a generator:\n",
+      "\n",
+      "    zs = [some_func(i) for i in range(1000)]\n",
+      "    \n",
+      "Just do whatever it takes to get your measurements in an array.\n",
+      "\n",
+      "Now we will just call the `batch_filter()` method.\n",
+      "\n",
+      "    Xs, Ps, Xs_pred, Ps_pred = kfilter.batch_filter(zs)\n",
+      "    \n",
+      "The funtion takes the list/array of measurements, filters it, and returns a list of state estimates (Xs), covariance matrices (Ps), and the predictions for the same (Xs_pred, Ps_pred).\n",
+      "\n",
+      "Here is a complete example drawing from the code above"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "import numpy.random as random\n",
+      "\n",
+      "class RadarSim(object):\n",
+      "    def __init__(self, dt):\n",
+      "        self.x = 0\n",
+      "        self.dt = dt\n",
+      "\n",
+      "    def get_range(self):\n",
+      "        vel = 100  + 5*randn()\n",
+      "        alt = 1000 + 10*randn()\n",
+      "        self.x += vel*self.dt\n",
+      "\n",
+      "        v = self.x * 0.05*randn()\n",
+      "        rng = (self.x**2 + alt**2)**.5 + v\n",
+      "        return rng\n",
+      "    \n",
+      "def fx(x, dt):\n",
+      "    A = np.eye(3) + dt * np.array ([[0, 1, 0],\n",
+      "                                    [0, 0, 0],\n",
+      "                                    [0, 0, 0]])\n",
+      "    return A.dot(x)\n",
+      "\n",
+      "def hx(x):\n",
+      "    return np.sqrt (x[0]**2 + x[2]**2)\n",
+      "\n",
+      "dt = 0.05\n",
+      "\n",
+      "kf = UKF(3, 1, dt, fx=fx, hx=hx, kappa=0.)\n",
+      "\n",
+      "kf.Q *= 0.01\n",
+      "kf.R = 10\n",
+      "kf.x = np.array([0., 90., 1100.])\n",
+      "kf.P *= 100.\n",
+      "radar = RadarSim(dt)\n",
+      "\n",
+      "t = np.arange(0,20+dt, dt)\n",
+      "n = len(t)\n",
+      "\n",
+      "random.seed(200)\n",
+      "rs = [radar.get_range() for i in range(n)]\n",
+      "xs, covs = kf.batch_filter(rs)\n",
+      "\n",
+      "plt.figure()\n",
+      "plt.subplot(311)\n",
+      "plt.plot(t, xs[:,0])\n",
+      "plt.xlabel('time(sec)')\n",
+      "plt.ylabel('position')\n",
+      "\n",
+      "plt.subplot(312)\n",
+      "plt.plot(t, xs[:,1])\n",
+      "plt.xlabel('time(sec)')\n",
+      "plt.ylabel('velocity')\n",
+      "\n",
+      "plt.subplot(313)\n",
+      "plt.plot(t, xs[:,2])\n",
+      "plt.xlabel('time(sec)')\n",
+      "plt.ylabel('altitude')\n",
+      "plt.show()"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7f659ae26b70>"
+       ]
+      }
+     ],
+     "prompt_number": 52
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "## Smoothing the Results\n",
+      "\n",
+      "I have an entire chapter on using the Kalman filter to smooth data; I will not repeat the chapter's information here. However, it is so easy to use, and offers such a profoundly improved output that I will tease you will a few examples. The smoothing chapter is not especially difficult; you are sufficiently prepared to read it now.\n",
+      "\n",
+      "Briefly, the recursive form of Kalman filtering that we have been using up to now use information from the past to generate the current estimate. Recall the last section, where we used batch processing to filter a collection of measurements. This implies that we know now only the past, but the future! This is a key insight.\n",
+      "\n",
+      "Let's assume that  we are tracking a car. Suppose we get a noisy measurement that implies that the car is starting to turn to the left, but the state function has predicted that the car is moving straight. The Kalman filter has no choice but to move the state estimate somewhat towards the noisy measurement, as it cannot judge whether this is just a particularly noisy measurement or the true start of a turn. \n",
+      "\n",
+      "However, if we have measurements from the future we can often figure out if a turn was made or not. Suppose the subsequent measurements all continue turning left. We can then be sure that the measurement was not very noisy, but instead a true indication that a turn was initiated. On the other hand, if the subsequent measurements continued on in a straight line we would know that the measurement was noisy and should be mostly ignored.\n",
+      "\n",
+      "We will not develop the math or algorithm here, I will just show you how to call the algorithm in `filterpy`. The algorithm that we have implemented is called an *RTS smoother*, after the three inventors of the algorithm: Rauch, Tung, and Striebel.\n",
+      "\n",
+      "The routine is `UnscentedKalmanFilter.rts_smoother()`. Using it is trivial; we pass in the means and covariances computed from the `batch_filter` step, and receive back the smoothed means, covariances, and Kalman gain."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Ms,_,_ = kf.rts_smoother(xs, covs)\n",
+      "plt.figure()\n",
+      "plt.subplot(311)\n",
+      "plt.plot(t[0:100], xs[0:100,0])\n",
+      "plt.plot(t[0:100], Ms[0:100,0],c='k')\n",
+      "plt.xlabel('time(sec)')\n",
+      "plt.ylabel('position')\n",
+      "\n",
+      "plt.subplot(312)\n",
+      "plt.plot(t[0:100], xs[0:100,1])\n",
+      "plt.plot(t[0:100], Ms[0:100,1],c='k')\n",
+      "plt.xlabel('time(sec)')\n",
+      "plt.ylabel('velocity')\n",
+      "\n",
+      "plt.subplot(313)\n",
+      "plt.plot(t[0:100], xs[0:100,2])\n",
+      "plt.plot(t[0:100], Ms[0:100,2],c='k')\n",
+      "plt.xlabel('time(sec)')\n",
+      "plt.ylabel('altitude')\n",
+      "plt.show()"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7f65a001b7b8>"
+       ]
+      }
+     ],
+     "prompt_number": 55
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "It was difficult to see the improvement in the position if I plotted all the data, so I have only displayed the first 5 seconds of output. "
+     ]
     },
     {
      "cell_type": "markdown",