Fixed path issues, improved gaussian discussion.

Chapter was not adding ../code to the search path. Multivarite discusion about the ellipses
was a bit unclear about the relationship of the standard deviation.

Chapter06_Multivariate_Kalman_Filter/Multivariate_Kalman_Filters.ipynb

@@ -1,7 +1,7 @@
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@@ -32,7 +32,7 @@
       "from __future__ import division, print_function\n",
       "import matplotlib.pyplot as plt\n",
       "import sys\n",
-      "sys.path.insert(0,'../') # allow us to import book_format\n",
+      "sys.path.insert(0,'../code') # allow us to import book_format\n",
       "import book_format\n",
       "book_format.load_style()"
      ],
@@ -259,7 +259,7 @@
        "output_type": "pyout",
        "prompt_number": 1,
        "text": [
-        "<IPython.core.display.HTML at 0x7f45ed9179e8>"
+        "<IPython.core.display.HTML at 0x7fcc5c06a9e8>"
        ]
       }
      ],
@@ -274,7 +274,16 @@
      ]
     },
     {
-     "cell_type": "markdown",
+     "cell_type": "raw",
+     "metadata": {},
+     "source": [
+      "The techniques in the last chapter are very powerful, but they only work in one dimension. The gaussians represent a mean and variance that are scalars - real numbers. They provide no way to represent multidimensional data, such as the position of a dog in a field. You may retort that you could use two Kalman filters for that case, one tracks the x coordinate and the other tracks the y coordinate. That does work in some cases, but put that thought aside, because soon you will see some enormous benefits to implementing the multidimensional case.\n",
+      "\n",
+      "In this chapter I am purposefully glossing over many aspects of the mathematics behind Kalman filters. If you are familiar with the topic you will read statements that you disagree with because they contain simplifications that do not necessarily hold in more general cases. If you are not familiar with the topic, expect some paragraphs to be somewhat 'magical' - it will not be clear how I derived a certain result. I prefer that you develop an intuition for how these filters work through several worked examples. If I started by presenting a rigorous mathematical formulation you would be left scratching your head about what all these terms mean and how you might apply them to your problem. In later chapters I will provide a more rigorous mathematical foundation, and at that time I will have to either correct approximations that I made in this chapter or provide additional information that I did not cover here. "
+     ]
+    },
+    {
+     "cell_type": "raw",
      "metadata": {},
      "source": [
       "The techniques in the last chapter are very powerful, but they only work in one dimension. The gaussians represent a mean and variance that are scalars - real numbers. They provide no way to represent multidimensional data, such as the position of a dog in a field. You may retort that you could use two Kalman filters for that case, one tracks the x coordinate and the other tracks the y coordinate. That does work in some cases, but put that thought aside, because soon you will see some enormous benefits to implementing the multidimensional case.\n",
@@ -370,7 +379,7 @@
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+        "<matplotlib.figure.Figure at 0x7fcc5c06a9b0>"
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@@ -565,7 +574,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f45ccc52eb8>"
+        "<matplotlib.figure.Figure at 0x7fcc3b36ee10>"
        ]
       }
      ],
@@ -612,7 +621,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f45ccf14898>"
+        "<matplotlib.figure.Figure at 0x7fcc3b60b048>"
        ]
       }
      ],
@@ -622,7 +631,7 @@
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "From a mathematical perspective these display the values that the multivariate gaussian takes for a specific sigma (in this case $\\sigma^2=1$. Think of it as taking a horizontal slice through the 3D surface plot we did above. However, thinking about the physical interpretation of these plots clarifies their meaning.\n",
+      "From a mathematical perspective these display the values that the multivariate gaussian takes for a specific standard deviation (in this case $\\sigma=1$). Think of it as taking a horizontal slice through the 3D surface plot we did above. However, thinking about the physical interpretation of these plots clarifies their meaning.\n",
       "\n",
       "The first plot uses the mean and covariance matrices of\n",
       "$$\n",
@@ -632,7 +641,9 @@
       "\\end{aligned}\n",
       "$$ \n",
       "\n",
-      "Let this be our current belief about the position of our dog in a field. In other words, we believe that he is positioned at (2,7) with a variance of $\\sigma^2=2$ for both x and y. The contour plot shows where we believe the dog is located with the '+' in the center of the ellipse. The ellipse shows the boundary for the $1\\sigma^2$ probability - points where the dog is quite likely to be based on our current knowledge. Of course, the dog might be very far from this point, as Gaussians allow the mean to be any value. For example, the dog could be at (3234.76,189989.62), but that has vanishing low probability of being true. Generally speaking displaying  the $1\\sigma^2$ to $2\\sigma^2$ contour captures the most likely values for the distribution. An equivalent way of thinking about this is the circle/ellipse shows us the amount of error in our belief. A tiny circle would indicate that we have a very small error, and a very large circle indicates a lot of error in our belief. We will use this throughout the rest of the book to display and evaluate the accuracy of our filters at any point in time. "
+      "Let this be our current belief about the position of our dog in a field. In other words, we believe that he is positioned at (2,7) with a variance of $\\sigma^2=2$ for both x and y. The contour plot shows where we believe the dog is located with the '+' in the center of the ellipse. The ellipse shows the boundary for the $1\\sigma^2$ probability - points where the dog is quite likely to be based on our current knowledge. Of course, the dog might be very far from this point, as Gaussians allow the mean to be any value. For example, the dog could be at (3234.76,189989.62), but that has vanishing low probability of being true. Generally speaking displaying the $1\\sigma$ to $2\\sigma$ contour captures the most likely values for the distribution. Recall from the Gaussians chapter the the 68-95-99.7 rule - 68% of all values will fall within 1 standard deviation ($1\\sigma$), 95% within $2\\sigma$, and 99.7% within $3\\sigma$.\n",
+      "\n",
+      "An equivalent way of thinking about this is the circle/ellipse shows us the amount of error in our belief. A tiny circle would indicate that we have a very small error, and a very large circle indicates a lot of error in our belief. We will use this throughout the rest of the book to display and evaluate the accuracy of our filters at any point in time. "
      ]
     },
     {
@@ -700,7 +711,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f45ccdb60b8>"
+        "<matplotlib.figure.Figure at 0x7fcc3b4cf4e0>"
        ]
       }
      ],
@@ -727,7 +738,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f45ccd825c0>"
+        "<matplotlib.figure.Figure at 0x7fcc3b497dd8>"
        ]
       }
      ],
@@ -754,7 +765,7 @@
        "output_type": "display_data",
        "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7f45ccdb60b8>"
+        "<matplotlib.figure.Figure at 0x7fcc3b62ff60>"
        ]
       }
      ],
@@ -785,7 +796,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f45cce35a90>"
+        "<matplotlib.figure.Figure at 0x7fcc3b4e91d0>"
        ]
       }
      ],
@@ -852,7 +863,7 @@
        "output_type": "display_data",
        "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7f45ccdb67b8>"
+        "<matplotlib.figure.Figure at 0x7fcc3b3647b8>"
        ]
       }
      ],
@@ -1846,9 +1857,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f45cce35278>"
+        "<matplotlib.figure.Figure at 0x7fcc3b38fcf8>"
        ]
       },
       {
@@ -1856,7 +1867,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f45cce37a20>"
+        "<matplotlib.figure.Figure at 0x7fcc5c06a6d8>"
        ]
       }
      ],
@@ -2018,9 +2029,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f45ccf11d30>"
+        "<matplotlib.figure.Figure at 0x7fcc3b634160>"
        ]
       },
       {
@@ -2028,7 +2039,7 @@
        "output_type": "display_data",
        "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7f45cce73208>"
+        "<matplotlib.figure.Figure at 0x7fcc3b3517f0>"
        ]
       }
      ],
@@ -2260,9 +2271,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f45cce29be0>"
+        "<matplotlib.figure.Figure at 0x7fcc3b5c55f8>"
        ]
       }
      ],
@@ -2320,17 +2331,17 @@
       {
        "metadata": {},
        "output_type": "display_data",
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        "text": [
-        "<matplotlib.figure.Figure at 0x7f45ccddde48>"
+        "<matplotlib.figure.Figure at 0x7fcc3b53da20>"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f45cce29a58>"
+        "<matplotlib.figure.Figure at 0x7fcc3b53d978>"
        ]
       }
      ],
@@ -2443,9 +2454,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f45ccf10470>"
+        "<matplotlib.figure.Figure at 0x7fcc3b608438>"
        ]
       }
      ],
@@ -2504,7 +2515,7 @@
      "collapsed": false,
      "input": [
       "P = np.array([[2,0],[0,6]])\n",
-      "stats.plot_covariance_ellipse ((0,0), P,  facecolor='g', alpha=0.2, \n",
+      "stats.plot_covariance_ellipse ((0,0), P,  facecolor='g', alpha=0.2,\n",
       "                               title='|2 0|\\n|0 6|')"
      ],
      "language": "python",
@@ -2515,7 +2526,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f45cce4a780>"
+        "<matplotlib.figure.Figure at 0x7fcc3b4cf0b8>"
        ]
       }
      ],
@@ -2544,7 +2555,7 @@
        "output_type": "display_data",
        "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7f45ed917978>"
+        "<matplotlib.figure.Figure at 0x7fcc3b6345f8>"
        ]
       }
      ],
@@ -2614,7 +2625,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f45ccddadd8>"
+        "<matplotlib.figure.Figure at 0x7fcc3b4f0470>"
        ]
       }
      ],
@@ -2643,7 +2654,7 @@
        "output_type": "display_data",
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        "text": [
-        "<matplotlib.figure.Figure at 0x7f45cce0e780>"
+        "<matplotlib.figure.Figure at 0x7fcc3b4fdf98>"
        ]
       }
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@@ -2677,7 +2688,7 @@
        "output_type": "display_data",
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f45cceebd30>"
+        "<matplotlib.figure.Figure at 0x7fcc3b61c0b8>"
        ]
       }
      ],
@@ -2708,7 +2719,7 @@
        "output_type": "display_data",
        "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7f45ccd82908>"
+        "<matplotlib.figure.Figure at 0x7fcc3b5084a8>"
        ]
       }
      ],
@@ -2778,17 +2789,17 @@
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f45cced7b00>"
+        "<matplotlib.figure.Figure at 0x7fcc3b5506a0>"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f45ccf183c8>"
+        "<matplotlib.figure.Figure at 0x7fcc3b550a20>"
        ]
       }
      ],
@@ -2855,7 +2866,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f45cce4a198>"
+        "<matplotlib.figure.Figure at 0x7fcc3b4e9dd8>"
        ]
       }
      ],