Minor edits.

changed variable name, cleaned up a few expressions.
2015-05-24 17:00:26 -07:00
1 parent a30013c00d
commit 163052bcb1
@@ -16,7 +16,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 1,
+   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
@@ -237,7 +237,10 @@
       "                },\n",
       "                displayAlign: 'center', // Change this to 'center' to center equations.\n",
       "                \"HTML-CSS\": {\n",
-       "                    scale:85,\n",
+       "                    scale:100,\n",
+       "                    availableFonts: [\"Neo-Euler\"],\n",
+       "                    preferredFont: \"Neo-Euler\",\n",
+       "                    webFont: \"Neo-Euler\",\n",
       "                    styles: {'.MathJax_Display': {\"margin\": 4}}\n",
       "                }\n",
       "        });\n",
@@ -247,7 +250,7 @@
       "<IPython.core.display.HTML object>"
      ]
     },
-     "execution_count": 1,
+     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
@@ -375,7 +378,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 2,
+   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
@@ -384,7 +387,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x5de9240>"
+       "<matplotlib.figure.Figure at 0x627f4e0>"
      ]
     },
     "metadata": {},
@@ -435,7 +438,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 3,
+   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
@@ -457,7 +460,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 4,
+   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
@@ -475,7 +478,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 5,
+   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
@@ -493,7 +496,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 6,
+   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
@@ -512,7 +515,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 7,
+   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
@@ -545,7 +548,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 8,
+   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
@@ -554,7 +557,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x5de91d0>"
+       "<matplotlib.figure.Figure at 0x87a34e0>"
      ]
     },
     "metadata": {},
@@ -582,16 +585,16 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 9,
+   "execution_count": 10,
   "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 0x5de90f0>"
+       "<matplotlib.figure.Figure at 0x627f3c8>"
      ]
     },
     "metadata": {},
@@ -618,7 +621,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 10,
+   "execution_count": 11,
   "metadata": {
    "collapsed": false,
    "scrolled": true
@@ -628,7 +631,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0xa3ea4a8>"
+       "<matplotlib.figure.Figure at 0xa3da128>"
      ]
     },
     "metadata": {},
@@ -657,7 +660,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 11,
+   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
@@ -666,7 +669,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x8f37080>"
+       "<matplotlib.figure.Figure at 0xb4afe48>"
      ]
     },
     "metadata": {},
@@ -742,7 +745,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 12,
+   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
@@ -751,7 +754,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x9d0b3c8>"
+       "<matplotlib.figure.Figure at 0x8f063c8>"
      ]
     },
     "metadata": {},
@@ -778,7 +781,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 13,
+   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
@@ -787,7 +790,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x8e7ff28>"
+       "<matplotlib.figure.Figure at 0x9d57d68>"
      ]
     },
     "metadata": {},
@@ -968,7 +971,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 14,
+   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
@@ -999,7 +1002,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 15,
+   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
@@ -1016,7 +1019,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0xa3423c8>"
+       "<matplotlib.figure.Figure at 0x8ec2278>"
      ]
     },
     "metadata": {},
@@ -1041,7 +1044,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 16,
+   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
@@ -1050,7 +1053,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x9cc3ba8>"
+       "<matplotlib.figure.Figure at 0x9c8bb70>"
      ]
     },
     "metadata": {},
@@ -1071,7 +1074,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 17,
+   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
@@ -1080,7 +1083,7 @@
     "data": {
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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x9ce6278>"
+       "<matplotlib.figure.Figure at 0x9ccc9b0>"
      ]
     },
     "metadata": {},
@@ -1115,7 +1118,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 18,
+   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
@@ -1124,7 +1127,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x9d2eda0>"
+       "<matplotlib.figure.Figure at 0x9cb89b0>"
      ]
     },
     "metadata": {},
@@ -1189,7 +1192,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 19,
+   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
@@ -1198,7 +1201,7 @@
     "data": {
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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x9cc3be0>"
+       "<matplotlib.figure.Figure at 0x9d332e8>"
      ]
     },
     "metadata": {},
@@ -1219,7 +1222,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 20,
+   "execution_count": 21,
   "metadata": {
    "collapsed": false,
    "scrolled": true
@@ -1229,7 +1232,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x9c468d0>"
+       "<matplotlib.figure.Figure at 0x9d13eb8>"
      ]
     },
     "metadata": {},
@@ -1251,7 +1254,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 21,
+   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
@@ -1260,7 +1263,7 @@
     "data": {
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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0xb48f6a0>"
+       "<matplotlib.figure.Figure at 0xb4f1278>"
      ]
     },
     "metadata": {},
@@ -1290,7 +1293,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 22,
+   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
@@ -1299,7 +1302,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0xb493cc0>"
+       "<matplotlib.figure.Figure at 0xb4f8278>"
      ]
     },
     "metadata": {},
@@ -1321,7 +1324,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 23,
+   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
@@ -1330,7 +1333,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0xb4f1eb8>"
+       "<matplotlib.figure.Figure at 0xb56d1d0>"
      ]
     },
     "metadata": {},
@@ -1362,7 +1365,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 24,
+   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
@@ -1371,7 +1374,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0xb4fcef0>"
+       "<matplotlib.figure.Figure at 0x9cdf748>"
      ]
     },
     "metadata": {},
@@ -1396,7 +1399,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 25,
+   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
@@ -1405,7 +1408,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x86c6fd0>"
+       "<matplotlib.figure.Figure at 0x87af9e8>"
      ]
     },
     "metadata": {},
@@ -1430,7 +1433,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 26,
+   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
@@ -1439,7 +1442,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x8f0ee48>"
+       "<matplotlib.figure.Figure at 0x8fb3e80>"
      ]
     },
     "metadata": {},
@@ -1478,7 +1481,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 27,
+   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
@@ -1487,7 +1490,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0xb4f4780>"
+       "<matplotlib.figure.Figure at 0xa49cda0>"
      ]
     },
     "metadata": {},
@@ -1508,7 +1511,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 28,
+   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
@@ -1517,7 +1520,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x9c5fef0>"
+       "<matplotlib.figure.Figure at 0xa49c908>"
      ]
     },
     "metadata": {},
@@ -1543,7 +1546,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 29,
+   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
@@ -1552,7 +1555,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x8e4a5f8>"
+       "<matplotlib.figure.Figure at 0xb4d2da0>"
      ]
     },
     "metadata": {},
@@ -1572,7 +1575,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 30,
+   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
@@ -1581,7 +1584,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x9c14400>"
+       "<matplotlib.figure.Figure at 0xa3dbb70>"
      ]
     },
     "metadata": {},
@@ -1601,7 +1604,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 31,
+   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
@@ -1610,7 +1613,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x9be27b8>"
+       "<matplotlib.figure.Figure at 0xb563d68>"
      ]
     },
     "metadata": {},
@@ -1630,7 +1633,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 32,
+   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
@@ -1639,7 +1642,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0xa3e3898>"
+       "<matplotlib.figure.Figure at 0xb4c1208>"
      ]
     },
     "metadata": {},
@@ -1715,7 +1718,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 33,
+   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
@@ -1724,7 +1727,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x9cc5240>"
+       "<matplotlib.figure.Figure at 0xb4c1320>"
      ]
     },
     "metadata": {},
@@ -1978,7 +1981,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 34,
+   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
@@ -2196,7 +2199,8 @@
    "\n",
    "The Kalman filter generalizes this problem by having you supply a *measurement function*. It is somewhat counterintuitive at first. As I already stated the Kalman filter performs its calculations in *measurement space.*  It needs to do that because it only really makes sense to talk about the residual of the measurement in terms of the measurement. So it does something like this:\n",
    "\n",
-    "    residual = measurement - convert_to_measurement(predicted state)\n",
+    "    CELSIUS_TO_VOLTS = 0.21475 # constant from manufacturer's spec\n",
+    "    residual = measurement - CELSIUS_TO_VOLTS * predicted_state\n",
    "\n",
    "In other words, for the thermometer tracking problem, it would take the filter's current prediction of temperature, convert that to whatever voltage would represent that temperature, and then subtract it from the current thermometer voltage output. This gives it a residual in the correct units (volts).\n",
    "\n",
@@ -2206,7 +2210,12 @@
    "\n",
    "$$\\textbf{y} = \\mathbf{z} - \\mathbf{H x^-} \\;\\;\\;(3)$$\n",
    "\n",
-    "where $\\textbf{y}$ is the residual, $\\mathbf{x^-}$ is the predicted value for $\\mathbf{x}$, $\\textbf{z}$ is the measurement, and $\\textbf{H}$ is the measurement function. It is just a matrix that we multiply the state into to convert it into a measurement.\n",
+    "where $\\textbf{y}$ is the residual, $\\mathbf{x^-}$ is the predicted value for $\\mathbf{x}$, $\\textbf{z}$ is the measurement, and $\\textbf{H}$ is the measurement function. $\\textbf{H}$ is a matrix that we multiply the state into to convert it into a measurement. It is analogous to the constant `TEMPERATURE_TO_VOLTS` in the one dimensional example directly above. Maybe it will help to see the two equations together.\n",
+    "\n",
+    "$$\\begin{aligned}residual &= measurement - CELSIUS\\_TO\\_VOLTS \\times predicted\\_state \\\\\n",
+    "\\textbf{y} &= \\mathbf{z} - \\mathbf{H x^-}\\end{aligned}$$\n",
+    "\n",
+    "\n",
    "\n",
    "For our dog tracking problem we have a sensor that measures position, but no sensor that measures velocity. So for a given state $\\mathbf{x}=\\begin{bmatrix}x & \\dot{x}\\end{bmatrix}^\\mathsf{T}$ we will want to multiply the position $x$ by 1 to get the corresponding measurement of the position, and multiply the velocity $\\dot{x}$ by 0 to get the corresponding measurement of velocity (of which there is none).\n",
    "\n",
@@ -2289,14 +2298,14 @@
    "\n",
    "$$\\mathbf{P} = \\mathbf{FP{F}}^\\mathsf{T} + \\mathbf{Q}$$\n",
    "\n",
-    "In this equation $\\mathbf{FP{F}}^\\mathsf{T}$ is just some linear algebra 'magic' that projects $\\mathbf{P}$ forward to the next time step according to our motion model $\\mathbf{F}$, and $\\mathbf{Q}$ is the *process noise matrix*. We are just adding matrices, so hopefully it is clear that each element in $\\mathbf{Q}$  specifies how much uncertainty is added to the system due to the process noise. \n",
+    "In this equation $\\mathbf{FP{F}}^\\mathsf{T}$ projects $\\mathbf{P}$ forward to the next time step according to our motion model $\\mathbf{F}$, and $\\mathbf{Q}$ is the *process noise matrix*. We are just adding matrices, so hopefully it is clear that each element in $\\mathbf{Q}$  specifies how much uncertainty is added to the system due to the process noise. \n",
    "\n",
    "Let's take a brief moment to look at the behavior of these equation in more detail. I will project $\\mathbf{x}$ forward according to the motion model $\\mathbf{F}$ with the equation $\\mathbf{x} = \\mathbf{Fx}$. Then I will project P to the next timestep with the equation $\\mathbf{P} = \\mathbf{FP{F}}^\\mathsf{T}$. I will draw the original covariance in red and the new covariance in black."
   ]
  },
  {
   "cell_type": "code",
-   "execution_count": 35,
+   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
@@ -2305,7 +2314,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0xb43df28>"
+       "<matplotlib.figure.Figure at 0x8755668>"
      ]
     },
     "metadata": {},
@@ -2339,7 +2348,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 36,
+   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
@@ -2348,7 +2357,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0xa411828>"
+       "<matplotlib.figure.Figure at 0x8fb0400>"
      ]
     },
     "metadata": {},
@@ -2376,7 +2385,7 @@
   "source": [
    "You can see that over time our uncertainty in the position grows. However, our uncertainty in the velocity does not. Why is that? Well, our process model states that the velocity is constant, and we do not model the possibility that there is error in the model (so far!). We assume the model is perfect, so the variance of the velocity does not change.\n",
    "\n",
-    "But of course no model is perfect. Except perhaps in a perfect vacuum velocity will never remain constant. If we are tracking a car, for example, even if we set the cruise control the car will slightly change velocity due to the wind, hills, bumps in the road, imperfections in the cruise control, and so on. If the Kalman filter used the equation above it would quickly become over-certain in its computations. Recall that in the unidimensional Kalman filter chapter every predict cycle increased our uncertainty, and every update decreased it. The same needs to happen here.\n",
+    "So I have demonstrated the reason for the $\\mathbf{FP{F}}^\\mathsf{T}$ part of the equation, which assumes a perfect model. But of course no model is perfect. Except perhaps in a perfect vacuum velocity will never remain constant. If we are tracking a car, for example, even if we set the cruise control the car will slightly change velocity due to the wind, hills, bumps in the road, imperfections in the cruise control, and so on. If the Kalman filter used the equation above it would quickly become over-certain in its computations. Recall that in the unidimensional Kalman filter chapter every predict cycle increased our uncertainty, and every update decreased it. The same needs to happen here.\n",
    "\n",
    "This is where $\\mathbf{Q}$ comes in. It is a covariance matrix that describes how much *process noise* is in the system. That is just a fancy way of saying how much uncertainty gets added to the system by things we haven't modeled.\n",
    "\n",
@@ -2385,7 +2394,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 37,
+   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
@@ -2394,7 +2403,7 @@
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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x8dfa320>"
+       "<matplotlib.figure.Figure at 0xa3cd198>"
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     "metadata": {},
@@ -2463,14 +2472,14 @@
   "source": [
    "It general it is not true that acceleration will be constant and uncorrelated, but this is still a useful approximation for moderate time period, and will suffice for this chapter. Fortunately you can get a long way with approximations and simulation. Let's think about what these matrices are implying. We are trying to model the effects of *process noise*, such as the wind buffeting the flight of a thrown ball. Variations in wind will cause changes in acceleration, and so the effect on the acceleration is large. However, the effects on velocity and position are proportionally smaller. In the matrices, the acceleration term is in the lower right, and this is the largest value. **A good rule of thumb is to set $\\sigma$ somewhere from $\\frac{1}{2}\\Delta a$ to $\\Delta a$, where $\\Delta a$ is the maximum amount that the acceleration will change between sample periods**. In practice we pick a number, run simulations on data, and choose a value that works well. \n",
    "\n",
-    "The filtered result will not be optimal, but in my opinion the promise of optimal results from Kalman filters is mostly wishful thinking. Consider, for example, tracking a car. In that problem the process noise would include things like potholes, wind gusts, changing drag due to turning, rolling down windows, and many more factors. We cannot realistically model that analytically, and so in practice we work out a simplified model, compute $\\mathbf{Q}$ based on that simplified model, and then add *a bit more* to  $\\small\\mathbf{Q}$ in hopes of taking the incalculable factors into account. Then we use a lot of simulations and trial runs to see if the filter behaves well; if it doesn't we adjust $\\small\\mathbf{Q}$ until the filter performs well. In this chapter we will focus on forming an intuitive understanding on how adjusting $\\small\\mathbf{Q}$ affects the output of the filter. In the Kalman Filter Math chapter we will discuss the analytic computation of $\\small\\mathbf{Q}$, and also provide code that will compute it automatically for you.\n",
+    "The filtered result will not be optimal, but in my opinion the promise of optimal results from Kalman filters is mostly wishful thinking. Consider, for example, tracking a car. In that problem the process noise would include things like potholes, wind gusts, changing drag due to turning, rolling down windows, and many more factors. We cannot realistically model that analytically, and so in practice we work out a simplified model, compute $\\mathbf{Q}$ based on that simplified model, and then add *a bit more* to  $\\small\\mathbf{Q}$ in hopes of taking the incalculable factors into account. Then we use a lot of simulations and trial runs to see if the filter behaves well; if it doesn't we adjust $\\small\\mathbf{Q}$ until the filter performs well. In this chapter we will focus on forming an intuitive understanding on how adjusting $\\small\\mathbf{Q}$ affects the output of the filter. In the Kalman Filter Math chapter we will discuss the analytic computation of $\\small\\mathbf{Q}$, and also provide code that will compute it for you.\n",
    "\n",
    "For now we will just import the code from the `FilterPy` module, where it is already implemented. I will import it and call help on it so you can see the documentation for it."
   ]
  },
  {
   "cell_type": "code",
-   "execution_count": 38,
+   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
@@ -2512,7 +2521,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 39,
+   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
@@ -2524,7 +2533,7 @@
       "       [ 0.001175  ,  0.0235    ]])"
      ]
     },
-     "execution_count": 39,
+     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
@@ -2550,7 +2559,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 40,
+   "execution_count": 41,
   "metadata": {
    "collapsed": false
   },
@@ -2605,7 +2614,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 41,
+   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
@@ -2644,7 +2653,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 42,
+   "execution_count": 43,
   "metadata": {
    "collapsed": false
   },
@@ -2739,7 +2748,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 43,
+   "execution_count": 44,
   "metadata": {
    "collapsed": false
   },
@@ -2806,16 +2815,16 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 44,
+   "execution_count": 45,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0xa312fd0>"
+       "<matplotlib.figure.Figure at 0x8fa5ba8>"
      ]
     },
     "metadata": {},
@@ -2825,7 +2834,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0xa38c080>"
+       "<matplotlib.figure.Figure at 0xb4aa908>"
      ]
     },
     "metadata": {},
@@ -2858,66 +2867,11 @@
   "cell_type": "markdown",
   "metadata": {},
   "source": [
-    "## Compare to Univariate Kalman Filter"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "The equations in this chapter look very different from the equations in the last chapter, yet I claimed the last chapter implemented a full 1-D (univariate) Kalman filter. \n",
+    "## Exercise: Show Effect of Unobserved Variables\n",
    "\n",
-    "Recall that the univariate equations for the update step are:\n",
+    "In our filter above velocity is the unobserved variable. How would a filter perform if we did not use velocity in the state?\n",
    "\n",
-    "$$\n",
-    "\\begin{aligned}\n",
-    "\\mu &=\\frac{\\sigma_1^2 \\mu_2 + \\sigma_2^2 \\mu_1} {\\sigma_1^2 + \\sigma_2^2}, \\\\\n",
-    "\\sigma^2 &= \\frac{1}{\\frac{1}{\\sigma_1^2} + \\frac{1}{\\sigma_2^2}}\n",
-    "\\end{aligned}\n",
-    "$$\n",
-    "\n",
-    "and that the 1-D equations for the predict step are:\n",
-    "\n",
-    "$$\n",
-    "\\begin{aligned}\n",
-    "\\mu &= \\mu_1+\\mu_2, \\\\ \\sigma^2 &= \\sigma_1^2 + \\sigma_2^2\n",
-    "\\end{aligned}\n",
-    "$$\n",
-    "\n",
-    "Let's implement a simple 1-D kalman filter using the Kalman filter from this chapter, and compare its output to the kalman filter from the previous chapter by plotting it. We will use a simple model of tracking an object that starts at x=0 and moves by 1 at each step. We will assume the arbitrary value 5 for the measurement noise and .02 for the process noise.\n",
-    "\n",
-    "First, let's implement the filter from the last chapter:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 45,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [],
-   "source": [
-    "from __future__ import division\n",
-    "import numpy as np\n",
-    "from numpy.random import randn\n",
-    "from filterpy.kalman import KalmanFilter\n",
-    "\n",
-    "# 1-D Kalman filter equations\n",
-    "def predict(pos, variance, movement, movement_variance):\n",
-    "    return (pos + movement, variance + movement_variance)\n",
-    "\n",
-    "\n",
-    "def update (mu1, var1, mu2, var2):\n",
-    "    mean = (var1*mu2 + var2*mu1) / (var1+var2)\n",
-    "    variance = 1 / (1/var1 + 1/var2)\n",
-    "    return (mean, variance)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Now, let's implement the same thing using the kalman filter. I will implement it as a function that returns a KalmanFilter object so that you can run the analysis code several times with the KalmanFilter initialized to the same starting conditions each time."
+    "Write a Kalman filter that uses the state $\\mathbf{x}=\\begin{bmatrix}x & \\dot{x}\\end{bmatrix}^\\mathsf{T}$ and compare it against the filter in the last exercise which used the state $\\mathbf{x}=\\begin{bmatrix}x\\end{bmatrix}$."
   ]
  },
  {
@@ -2927,122 +2881,6 @@
    "collapsed": false
   },
   "outputs": [],
-   "source": [
-    "from filterpy.kalman import KalmanFilter\n",
-    "\n",
-    "def mkf_filter(R, Q):\n",
-    "    f = KalmanFilter(dim_x=1, dim_z=1, dim_u=1)\n",
-    "    f.P = 500.\n",
-    "    f.H = np.array([[1.]])\n",
-    "    f.F = np.array([[1.]])\n",
-    "    f.B = np.array([[1.]])\n",
-    "    f.Q = Q\n",
-    "    f.R = R\n",
-    "    \n",
-    "    return f"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Finally, let's compare the two. I will plot the data from the 1-D Kalman filter as a blue line, and the output of the filter from this chapter as red dots. I wrote it as a function so you can easily modify the parameters and regenerate the plots."
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 47,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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-       "<matplotlib.figure.Figure at 0xa3af9b0>"
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-     "metadata": {},
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-      "text/plain": [
-       "<matplotlib.figure.Figure at 0x9bbecf8>"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "def plot_kf_compare(x0, p0, R, Q, move):\n",
-    "    # storage for filter output\n",
-    "    x1, x2 = [], []\n",
-    "    p1, p2 = [], []\n",
-    "\n",
-    "    # initialize the filters\n",
-    "    f = mkf_filter(R, Q)\n",
-    "    f.x[0] = 0.\n",
-    "    f.P[0, 0] = p0\n",
-    "    pos = (x0, p0)\n",
-    "    for i in range(50):\n",
-    "        z = i*move + randn()\n",
-    "        pos = update(pos[0], pos[1], z, R)\n",
-    "        f.update(z)\n",
-    "\n",
-    "        x1.append(pos[0])\n",
-    "        x2.append(f.x[0])\n",
-    "\n",
-    "        p1.append(pos[1])\n",
-    "        p2.append(f.P[0, 0])\n",
-    "\n",
-    "        u = move + randn()\n",
-    "        pos = predict(pos[0], pos[1], u, Q)\n",
-    "        f.predict(u=u)\n",
-    "\n",
-    "    plt.scatter(range(len(x2)), x2, c='r')\n",
-    "    plt.title('State')\n",
-    "    plt.plot(x1)\n",
-    "    plt.xlim([0, 50])\n",
-    "\n",
-    "    plt.figure()\n",
-    "    plt.plot(p1)\n",
-    "    plt.scatter(range(len(x2)), p2, c='r')\n",
-    "    plt.title('Variance')\n",
-    "    plt.xlim([0, 50])\n",
-    "    plt.show()\n",
-    "    \n",
-    "plot_kf_compare(x0=0., p0=500., R=5., Q=.2, move=1.)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Discussion\n",
-    "\n",
-    "As you can see, both filters produce the same results. Feel free to vary the initial guess, the measurement noise, and the process noise; so long as you make the same changes to both filters the output should be the same. This is a solid demonstration, albeit not a rigorous proof, that both filters in fact implement the same math for the 1-D case. "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Exercise: Compare to a Filter That Incorporates Velocity\n",
-    "\n",
-    "The last example did not use one of the fundamental insights of this chapter, unobserved variables. In this example velocity would the the unobserved variable. Write a Kalman filter that uses the state $\\mathbf{x}=\\begin{bmatrix}x & \\dot{x}\\end{bmatrix}^\\mathsf{T}$ and compare it against the filter in the last exercise which used the state $\\mathbf{x}=\\begin{bmatrix}x\\end{bmatrix}$."
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 48,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [],
   "source": [
    "# your code here"
   ]
@@ -3058,16 +2896,16 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 49,
+   "execution_count": 51,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
-      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAuYAAAEjCAYAAAB+VMTJAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAuYAAAEjCAYAAAB+VMTJAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x8e5fef0>"
+       "<matplotlib.figure.Figure at 0xa3cf630>"
      ]
     },
     "metadata": {},
@@ -3075,6 +2913,7 @@
    }
   ],
   "source": [
+    "from numpy.random import randn\n",
    "def pos_vel_filter(R, Q):\n",
    "    f = KalmanFilter(dim_x=2, dim_z=1)\n",
    "    f.R = R\n",
@@ -3085,6 +2924,16 @@
    "    f.H = np.array([[1, 0]])  # Measurement function\n",
    "    return f\n",
    "\n",
+    "def mkf_filter(R, Q):\n",
+    "    f = KalmanFilter(dim_x=1, dim_z=1, dim_u=1)\n",
+    "    f.P = 50.\n",
+    "    f.H = np.array([[1.]])\n",
+    "    f.F = np.array([[1.]])\n",
+    "    f.B = np.array([[1.]])\n",
+    "    f.Q = Q\n",
+    "    f.R = R\n",
+    "    \n",
+    "    return f\n",
    "def plot_compare_pos_vel(x0, p0, R, Q, move):\n",
    "    # storage for filter output\n",
    "    x1, x2 = [], []\n",
@@ -3118,7 +2967,7 @@
    "    plt.xlim([0,50])\n",
    "    plt.show()\n",
    "    \n",
-    "plot_compare_pos_vel(x0=0., p0=500., R=5., Q=.2, move=1.)    "
+    "plot_compare_pos_vel(x0=0., p0=50., R=5., Q=.2, move=1.)    "
   ]
  },
  {
@@ -3127,7 +2976,7 @@
   "source": [
    "### Discussion\n",
    "\n",
-    "The output of the filter that incorporates velocity into the state produces much better output than the filter that only tracks position - the output is much closer to a straight line. We've already discussed why unobserved variables increase the precision of the filter, so I will not repeat that explanation here. But the last exercise and this one is intended to trigger a train of thought:\n",
+    "The output of the filter that incorporates velocity into the state produces much better output than the filter that only tracks position - the output is much closer to a straight line. We've already discussed why unobserved variables increase the precision of the filter, so I will not repeat that explanation here. But this exercise is intended to trigger a train of thought:\n",
    "\n",
    "1. The equations in this chapter are mathematically equivalent to the equations in the last chapter when we are only tracking one state variable.\n",
    "    \n",
@@ -3759,6 +3608,49 @@
    "> **sidebar**: when plotting covariance ellipses, make sure to always use *plt.axis('equal')* in your code. If the axis use different scales the ellipses will be drawn distorted. For example, the ellipse may be drawn as being taller than it is wide, but it may actually be wider than tall."
   ]
  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## The Update Equations\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": []
+  },
  {
   "cell_type": "markdown",
   "metadata": {},
@@ -4000,7 +3892,7 @@
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
-   "version": "3.4.1"
+   "version": "3.4.3"
  }
 },
 "nbformat": 4,