Added 2D Gaussian position example.

Position vs velocity confuses people, so I added a 2 position
reading example.

Fixed the Newtonian equation for jerk.

Changed indefinite integral in Newtonian equations to definite.

06_Multivariate_Kalman_Filters.ipynb

@@ -1,7 +1,7 @@
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@@ -257,13 +257,13 @@
        ],
        "metadata": {},
        "output_type": "pyout",
-       "prompt_number": 66,
+       "prompt_number": 2,
        "text": [
-        "<IPython.core.display.HTML at 0x7fd239ba6f60>"
+        "<IPython.core.display.HTML at 0x7f3f104d35c0>"
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@@ -293,16 +293,19 @@
       "\n",
       "If we instead assume constant acceleration we get\n",
       "\n",
-      "$$ x = \\frac{a}{2}t^2 + v_0 t + x_0$$\n",
+      "$$ x = \\frac{1}{2}at^2 + v_0 t + x_0$$\n",
       "\n",
       "And if we assume constant jerk we get\n",
       "\n",
-      "$$ x = \\frac{j}{3}t^3 +  \\frac{a_0}{2}t^2 + v_0 t + x_0$$\n",
+      "$$ x = \\frac{1}{6}jt^3 +  \\frac{1}{2}a_0 t^2 + v_0 t + x_0$$\n",
       "\n",
       "As a reminder, we can generate these equations using basic calculus. Given a constant velocity v we can compute the distance travelled over time with the equation\n",
       "\n",
-      "$$ v = \\frac{dx}{dt}\\\\\n",
-      "x = \\int v \\, \\mathrm{d}t\\\\ x = vt + x_0$$"
+      "$$\\begin{aligned} v &= \\frac{dx}{dt}\\\\\n",
+      "dx &= v\\, dt \\\\\n",
+      "\\int_{x_0}^x\\, dx &= \\int_0^t v\\, dt\\\\\n",
+      "x - x_0 &= vt - 0\\\\\n",
+      "x &= vt + x_0\\end{aligned}$$"
      ]
     },
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truncated
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@@ -445,7 +448,7 @@
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@@ -467,7 +470,7 @@
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@@ -485,7 +488,7 @@
      "language": "python",
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@@ -504,7 +507,7 @@
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      "metadata": {},
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@@ -530,7 +533,7 @@
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@@ -561,11 +564,11 @@
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd22da64550>"
+        "<matplotlib.figure.Figure at 0x7f3f104d3748>"
        ]
       }
      ],
-     "prompt_number": 73
+     "prompt_number": 9
     },
     {
      "cell_type": "markdown",
@@ -593,13 +596,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd22dbaf320>"
+        "<matplotlib.figure.Figure at 0x7f3f104d3be0>"
        ]
       }
      ],
-     "prompt_number": 74
+     "prompt_number": 10
     },
     {
      "cell_type": "markdown",
@@ -654,11 +657,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd22d6a5898>"
+        "<matplotlib.figure.Figure at 0x7f3eedbcdf28>"
        ]
       }
      ],
-     "prompt_number": 75
+     "prompt_number": 11
     },
     {
      "cell_type": "markdown",
@@ -693,11 +696,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd22dbab5f8>"
+        "<matplotlib.figure.Figure at 0x7f3eeda739b0>"
        ]
       }
      ],
-     "prompt_number": 76
+     "prompt_number": 12
     },
     {
      "cell_type": "markdown",
@@ -775,11 +778,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd22da405f8>"
+        "<matplotlib.figure.Figure at 0x7f3eed7edb00>"
        ]
       }
      ],
-     "prompt_number": 77
+     "prompt_number": 13
     },
     {
      "cell_type": "markdown",
@@ -812,11 +815,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd22d8662e8>"
+        "<matplotlib.figure.Figure at 0x7f3eed940208>"
        ]
       }
      ],
-     "prompt_number": 78
+     "prompt_number": 14
     },
     {
      "cell_type": "markdown",
@@ -855,11 +858,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd22d9e0390>"
+        "<matplotlib.figure.Figure at 0x7f3eedb56470>"
        ]
       }
      ],
-     "prompt_number": 79
+     "prompt_number": 15
     },
     {
      "cell_type": "markdown",
@@ -882,11 +885,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd22dc07400>"
+        "<matplotlib.figure.Figure at 0x7f3eedafc828>"
        ]
       }
      ],
-     "prompt_number": 80
+     "prompt_number": 16
     },
     {
      "cell_type": "markdown",
@@ -915,11 +918,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd22db79d30>"
+        "<matplotlib.figure.Figure at 0x7f3eed887f28>"
        ]
       }
      ],
-     "prompt_number": 81
+     "prompt_number": 17
     },
     {
      "cell_type": "markdown",
@@ -942,11 +945,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd22dbc0a20>"
+        "<matplotlib.figure.Figure at 0x7f3eeda73be0>"
        ]
       }
      ],
-     "prompt_number": 82
+     "prompt_number": 18
     },
     {
      "cell_type": "markdown",
@@ -969,11 +972,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fd22d8113c8>"
+        "<matplotlib.figure.Figure at 0x7f3eedba9d30>"
        ]
       }
      ],
-     "prompt_number": 83
+     "prompt_number": 19
     },
     {
      "cell_type": "markdown",
@@ -1014,11 +1017,11 @@
        "output_type": "display_data",
        "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7fd22db794e0>"
+        "<matplotlib.figure.Figure at 0x7f3eedbdd128>"
        ]
       }
      ],
-     "prompt_number": 84
+     "prompt_number": 20
     },
     {
      "cell_type": "markdown",
@@ -1055,11 +1058,11 @@
        "output_type": "display_data",
        "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7fd22e3c3940>"
+        "<matplotlib.figure.Figure at 0x7f3eedaafb70>"
        ]
       }
      ],
-     "prompt_number": 85
+     "prompt_number": 21
     },
     {
      "cell_type": "markdown",
@@ -1081,7 +1084,155 @@
      "source": [
       "What makes this possible? Imagine for a moment that we superimposed the velocity from a *different* airplane over the position graph. Clearly the two are not related, and there is no way that combining the two could possibly yield any additional information. In contrast, the velocity of the this airplane tells us something very important - the direction and speed of travel. So long as the aircraft does not alter its velocity the velocity allows us to predict where the next position is. After a relatively small amount of error in velocity the probability that it is a good match with the position is very small. Think about it - if you suddenly change direction your position is also going to change a lot. If the position measurement is not in the direction of the assumed velocity change it is very unlikely to be true. The two are correlated, so if the velocity changes so must the position, and in a predictable way. \n",
       "\n",
-      "We are going to see this again and again in the rest of the book, so don't worry if this feels a bit magical to you. But it is important to understand that we are taking advantage of the fact that velocity and position are correleated coupled with the ability to infer velocity by looking at the distance and time between two measurement, to produce very accurate estimates after only a few observations. Please reread this section if you have any doubts. If you grasp this point the rest is straightforward. If you do not you will quickly find it impossible to reason about what you will learn in the rest of this chapter."
+      "We are going to see this again and again in the rest of the book, so don't worry if this feels a bit magical to you. But it is important to understand that we are taking advantage of the fact that velocity and position are correleated coupled with the ability to infer velocity by looking at the distance and time between two measurement, to produce very accurate estimates after only a few observations. Please reread this section if you have any doubts. If you grasp this point the rest is straightforward. If you do not you will quickly find it impossible to reason about what you will learn in the rest of this chapter.\n",
+      "\n",
+      "A lot of people find the reasoning about the velocity somewhat magical at first, so lets think about a more geometric example. Suppose that we are tracking an aircraft with a couple of radar systems. Radars give us the range and bearing to a target. We start out being uncertain about the position of the aircraft, so the covariance, which is just our uncertainty about the position, might look like this."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "P0 = np.array([[6, 0], [0, 6]])\n",
+      "stats.plot_covariance_ellipse((10, 10), P0, facecolor='y', alpha=0.6)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAxEAAAGNCAYAAABuXPx3AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7f3ef585cc88>"
+       ]
+      }
+     ],
+     "prompt_number": 34
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Now suppose that there is a radar to the lower right of the aircraft. Further suppose that the radar is very accurate in the bearing measurement, but not very accurate at the range. That covariance (uncertainty in the reading) might look like this (plotted in blue):"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "P1 = np.array([[2, 1.9], [1.9, 2]])\n",
+      "stats.plot_covariance_ellipse((10, 10), P0, facecolor='y', alpha=0.6)\n",
+      "stats.plot_covariance_ellipse((10, 10), P1, facecolor='b', alpha=0.6)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAxEAAAGNCAYAAABuXPx3AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7f3eed9f6978>"
+       ]
+      }
+     ],
+     "prompt_number": 41
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "But to combine the information we multiply the two Gaussians together. Let's see the result of that."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "P3 = stats.multivariate_multiply((10,10), P0, (10,10), P1)[1]\n",
+      "stats.plot_covariance_ellipse((10, 10), P0, facecolor='y', alpha=0.6)\n",
+      "stats.plot_covariance_ellipse((10, 10), P3, facecolor='b', alpha=0.6)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7f3eedb66fd0>"
+       ]
+      }
+     ],
+     "prompt_number": 40
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "It retained the same shape and position as the radar measurement, but is smaller. We've seen this with the one dimensional Gaussians - multiplying two Gaussians makes the variance smaller because we are incorporating more information, hence we are less uncertain.  But the main point I want to make is that the covariance shape reflects the physical layout of the aircraft and the radar system.\n",
+      "\n",
+      "Now lets say we get a measurement from a second radar bank, this one to the lower right, which I will plot in blue against the yellow covariance of our current belief. "
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "P4 = np.array([[2, -1.9], [-1.9, 2]])\n",
+      "stats.plot_covariance_ellipse((10, 10), P3, facecolor='y', alpha=0.6)\n",
+      "stats.plot_covariance_ellipse((10, 10), P4, facecolor='b', alpha=0.6)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAx4AAAGNCAYAAACfV6f6AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7f3eedf404a8>"
+       ]
+      }
+     ],
+     "prompt_number": 45
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "This looks a lot like the velocity charts I showed you earlier, and the math is identical. Let's multiply the two covariances together and see the result."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "P5 = stats.multivariate_multiply((10,10), P3, (10,10), P4)[1]\n",
+      "stats.plot_covariance_ellipse((10, 10), P4, facecolor='y', alpha=0.6)\n",
+      "stats.plot_covariance_ellipse((10, 10), P5, facecolor='b', alpha=0.6)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7f3eed650be0>"
+       ]
+      }
+     ],
+     "prompt_number": 46
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "We have taken advantage of the geometry of the system to produce a very accurate position estimate. We did the same thing with the position vs velocity problem. The math does not care whether we are working with two positions, or a position and a correlated velocity. If we can express our uncertainties as a multidimensional velocity we can then multiply our uncertainty in position with our uncertainty in velocity and get a much more accurate result, in exactly the same way we can multipy the uncertainty in two position estimates to get a more accurate result."
      ]
     },
     {