Added example of KF without using FilterPy.

06-Multivariate-Kalman-Filters.ipynb

@@ -369,9 +369,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAtYAAADlCAYAAAB3YbDsAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x4617b70>"
+       "<matplotlib.figure.Figure at 0x1cef5568128>"
       ]
      },
      "metadata": {},
@@ -572,7 +572,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 25,
+   "execution_count": 7,
    "metadata": {
     "collapsed": false
    },
@@ -584,7 +584,7 @@
        "       [   0.,   49.]])"
       ]
      },
-     "execution_count": 25,
+     "execution_count": 7,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -603,7 +603,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 26,
+   "execution_count": 8,
    "metadata": {
     "collapsed": false
    },
@@ -615,7 +615,7 @@
        "       [   0.,   49.]])"
       ]
      },
-     "execution_count": 26,
+     "execution_count": 8,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -722,7 +722,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 27,
+   "execution_count": 9,
    "metadata": {
     "collapsed": false
    },
@@ -752,7 +752,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 28,
+   "execution_count": 10,
    "metadata": {
     "collapsed": false
    },
@@ -785,7 +785,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 29,
+   "execution_count": 11,
    "metadata": {
     "collapsed": false
    },
@@ -822,7 +822,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 30,
+   "execution_count": 12,
    "metadata": {
     "collapsed": false
    },
@@ -834,7 +834,7 @@
        "       [  14.7 ,   49.  ]])"
       ]
      },
-     "execution_count": 30,
+     "execution_count": 12,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -854,7 +854,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 34,
+   "execution_count": 13,
    "metadata": {
     "collapsed": false,
     "scrolled": true
@@ -862,9 +862,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7de6a58>"
+       "<matplotlib.figure.Figure at 0x1cef83d8860>"
       ]
      },
      "metadata": {},
@@ -926,7 +926,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 35,
+   "execution_count": 14,
    "metadata": {
     "collapsed": false,
     "scrolled": true
@@ -939,7 +939,7 @@
        "       [ 0.001,  0.024]])"
       ]
      },
-     "execution_count": 35,
+     "execution_count": 14,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1027,9 +1027,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7e47940>"
+       "<matplotlib.figure.Figure at 0x1cef8364da0>"
       ]
      },
      "metadata": {},
@@ -1097,7 +1097,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 36,
+   "execution_count": 17,
    "metadata": {
     "collapsed": true
    },
@@ -1138,7 +1138,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 37,
+   "execution_count": 18,
    "metadata": {
     "collapsed": true
    },
@@ -1178,7 +1178,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 38,
+   "execution_count": 19,
    "metadata": {
     "collapsed": false
    },
@@ -1197,7 +1197,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 43,
+   "execution_count": 20,
    "metadata": {
     "collapsed": false
    },
@@ -1239,7 +1239,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 44,
+   "execution_count": 21,
    "metadata": {
     "collapsed": true
    },
@@ -1276,7 +1276,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 45,
+   "execution_count": 22,
    "metadata": {
     "collapsed": false
    },
@@ -1360,7 +1360,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 46,
+   "execution_count": 23,
    "metadata": {
     "collapsed": false,
     "scrolled": false
@@ -1368,9 +1368,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x99ec400>"
+       "<matplotlib.figure.Figure at 0x1cef9419da0>"
       ]
      },
      "metadata": {},
@@ -1378,9 +1378,9 @@
     },
     {
      "data": {
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truncated
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9a9d278>"
+       "<matplotlib.figure.Figure at 0x1cef95211d0>"
       ]
      },
      "metadata": {},
@@ -1421,7 +1421,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 47,
+   "execution_count": 24,
    "metadata": {
     "collapsed": false,
     "scrolled": false
@@ -1429,9 +1429,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x88d3630>"
+       "<matplotlib.figure.Figure at 0x1cef9627e48>"
       ]
      },
      "metadata": {},
@@ -1520,16 +1520,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 48,
+   "execution_count": 25,
    "metadata": {
     "collapsed": false
    },
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9a042e8>"
+       "<matplotlib.figure.Figure at 0x1cef83f5fd0>"
       ]
      },
      "metadata": {},
@@ -1568,7 +1568,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 49,
+   "execution_count": 26,
    "metadata": {
     "collapsed": false,
     "scrolled": true
@@ -1576,9 +1576,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7dc8390>"
+       "<matplotlib.figure.Figure at 0x1cef7aa5d30>"
       ]
      },
      "metadata": {},
@@ -1632,7 +1632,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 50,
+   "execution_count": 27,
    "metadata": {
     "collapsed": false,
     "scrolled": true
@@ -1640,9 +1640,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x88f37f0>"
+       "<matplotlib.figure.Figure at 0x1cef9577278>"
       ]
      },
      "metadata": {},
@@ -1752,6 +1752,107 @@
     "This equation can be numerically unstable and I don't use it in FilterPy. Later I'll share more complicated but numerically stable forms of this equation."
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### An Example not using FilterPy\n",
+    "\n",
+    "FilterPy hides the details of the implementation from us. Normally you will appreciate this, but let's implement the last filter without FilterPy. To do so we need to define our matrices as variables, and then implement the Kalman filter equations explicitly.\n",
+    "\n",
+    "Here we initialize our matrices:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 36,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [],
+   "source": [
+    "dt = 1.\n",
+    "R_var = 10\n",
+    "Q_var = 0.01\n",
+    "x = np.array([[10.0, 4.5]]).T\n",
+    "P = np.diag([500, 49])\n",
+    "F = np.array([[1, dt],\n",
+    "              [0,  1]])\n",
+    "H = np.array([[1., 0.]])\n",
+    "R = np.array([[R_var]])\n",
+    "Q = Q_discrete_white_noise(dim=2, dt=dt, var=Q_var)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 37,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x1cefb0cc978>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "image/png": 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truncated
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x1cef95a4b00>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "from numpy import dot\n",
+    "from scipy.linalg import inv\n",
+    "\n",
+    "count = 50\n",
+    "data = compute_dog_data(R_var, Q_var, count)\n",
+    "xs, cov = [], []\n",
+    "for z in data[:, 1]:\n",
+    "    # predict\n",
+    "    x = dot(F, x)\n",
+    "    P = dot(F, P).dot(F.T) + Q\n",
+    "    \n",
+    "    #update\n",
+    "    S = dot(H, P).dot(H.T) + R\n",
+    "    K = dot(P, H.T).dot(inv(S))\n",
+    "    y = z - dot(H, x)\n",
+    "    x += dot(K, y)\n",
+    "    P = P - dot(K, H).dot(P)\n",
+    "    \n",
+    "    xs.append(x)\n",
+    "    cov.append(P)\n",
+    "    \n",
+    "xs, cov = np.array(xs), np.array(cov)\n",
+    "\n",
+    "plot_track(xs[:, 0], data[:, 0], data[:, 1], cov, dt=dt)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The results are identical to the FilterPy version. Which you prefer is up to you. If you do not like object oriented programming FilterPy might be off-putting. I prefer not polluting my namespace with variables such as `x`, `P`, and so on; `dog_filter.x` is, to me, more readable.\n",
+    "\n",
+    "More importantly, this example requires you to remember and program the equations for the Kalman filter. Sooner or later you will make a mistake. FilterPy's version ensures that your code will be correct. On the other hand, if you make a mistake in your definitions, such as making $\\mathbf{H}$ a column vector instead of a row vector, FilterPy's error message will be harder to debug than this explicit code. \n",
+    "\n",
+    "FilterPy's KalmanFilter class provides additional functionality such as smoothing, batch processing, faded memory filtering, computation of the maximum likelihood function, and more. You get all of this functionality without having to explicitly program it.\n",
+    "\n",
+    "Finally, there are multiple forms of some of the Kalman filter equations, some which we will present later. It is cumbersome to change the equations using FilterPy, whereas it is trivial if you explicitly code the equations yourself.\n",
+    "\n",
+    "In the rest of the book I will continue using FilterPy - I want you to focus on issues of design, not implementation of linear algebra. If the linear algebra interests you I encourage you to rewrite the examples without FilterPy."
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},