Removal of the gaussian stuff from the histogram chapter.

Partial authorship of introduction to the gaussian chapter.

Gaussians.ipynb

@@ -0,0 +1,150 @@
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+ "worksheets": [
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+   "cells": [
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "#Gaussian Probabilities\n",
+      "\n",
+      "#### Introduction\n",
+      "\n",
+      "The histogram filter uses a vector or multidimensional array of probabilities to represent our belief of the system state. This is very powerful, but it has limitations. Let us list a few. First, it is multimodal, and we may not want a multimodal result. In other words, we would not want our GPS telling us that we are 50% likely to be here, 40% likely to be there, and so on. There are plenty of techniques to go from a multimodal histogram to a unimodal result, but they all involve significant processing time. This becomes worse as we extend the problem into multiple dimensions. In general for $d$ dimensions the computational run time will be $O(n^d)$. So these filters become intractable in higher dimensions. \n",
+      "\n",
+      "Second, the results are hard to analyze. Recall the GPS in your car. It probably has a display that tells you how much error is in the filter - it tells you are at a specific location on the earth, with for example 9 meters of error. How would you derive that information from a histogram filter? There is no clear cut way. Most, if not all of the bins in the histogram will be non-zero, and there will usually be several clusters of higher probabilities. Being multimodal, there is no clear way to derive a single error estimate. Certainly one could come up with a heuristic, but often we want mathematically precise answers, not heuristics.\n",
+      "\n",
+      "Third, the histogram is discrete, but we live in a continuous world. The histogram requires that you model the output of your filter as a set of discrete points. In our dog in the hallway example, we used 10 positions, which is obviously far too few positions for anything but a toy problem. For example, for a 100 meter hallway you would need 10,000 positions to model the hallway to 1cm accuracy. So each sense and update operation would entail performing calculations for 10,000 different probabilities. It gets exponentially worse as we add dimensions. If our dog was roaming in a $100x100 m^2$ courtyard, we would need 100,000,000 bins ($10,000^2$) to get 1cm accuracy.\n",
+      "\n",
+      "Finally, the histogram does not represent what happens in the pysical world very well. For example, in the last chapter we had this as a probability distribution: [ 0.2245871   0.06288015  0.06109133  0.0581008   0.09334062  0.2245871\n",
+      "  0.06288015  0.06109133  0.0581008   0.09334062]. The largest probabilities are in position 0 and position 5. This does not fit our physical intuition at all. A dog cannot be in two places at once.\n",
+      "\n",
+      "Consider using a laser rangefinder. It works by shooting a laser beam to a target, which bounces the laser beam back to a sensor in the rangefinder. The device times the round trip time of the beam, and from that calculates the distance. However, if the beam is not in a vacuum the light beam will be affected by the atmosphere. Recall how images waver over hot tarmac. They waver because the light is being bent by heat currents in the air. The same thing happens to a laser beam, though obviously in most cases the effect is much smaller. So suppose that the rangefinder is exactly 100m from an object. The beam might travel 100.01m due to atmospherics. Less often, it might travel 100.1m. It would rarely travel 101m, and probably never travel 200m. Without doing the math we can see that the laser will almost aways have some error, but most of the time the error will be very close to the actual value. Another way to say the same thing is that larger errors happen less frequently than smaller errors. This is not happenstance, but a consequence of how laser rangefinders work physically.  \n",
+      "\n",
+      "So we desire a unimodal, continuous way to represent probabilities that models how the real world works, and that is very computationally efficient to calculate. As you might guess from the chapter name, gaussian distributions provide all of these features.\n",
+      "\n"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "So let us explore how gaussians work. A gaussian is a probability distribution that is completely described with two parameters, the mean and the variance. It is defined as:\n",
+      "$$ \n",
+      "f(x, \\mu, \\sigma) = \\frac{1}{\\sigma\\sqrt{2\\pi}} e^{-0.5*{(x-\\mu)^2}/\\sigma^2 }\n",
+      "$$\n",
+      "\n",
+      "where $\\mu$ is the mean and $\\sigma^2$ is the variance (we will define these in a moment).\n",
+      "Let us plot that with Python. First, we will define a function that computes the gaussian for any x. "
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "import math\n",
+      "\n",
+      "def gaussian (x, mu, sigma):\n",
+      "    ''' compute the gaussian with the specified mean and sigma'''\n",
+      "    return math.exp (-0.5 * (x-mu)**2 / sigma) / math.sqrt(2.*math.pi*sigma)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 1
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Now we will plot a gaussian centered around 5, with a variance of 1."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "import matplotlib.pyplot as plt\n",
+      "import numpy as np\n",
+      "xs = np.arange(0,10,0.1)\n",
+      "plt.plot (xs,[gaussian(x, 5, 1) for x in xs])\n",
+      "plt.axvline (5) \n",
+      "plt.show()"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x3aaa810>"
+       ]
+      }
+     ],
+     "prompt_number": 2
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "As we expected the curve is centered around 5. We can see why this is by looking at the equation for the gaussian. When x=5, x-\\mu is 0, and thus e^0 is 1. any other value for x will result in a smaller value for exp function.\n",
+      "\n",
+      "The width of the curve is defined by the variance. If the variance is large than the curve will be wide, and if the variance is small the curve will be narrow.\n",
+      "\n",
+      "Also, since this is a probability distribution it is required that the area under the curve always equals one. This should be intuitively clear - the area under the curve represents all possible occurances, which must sum to one.\n",
+      "\n",
+      "This leads to an important insight. If the variance is small the curve will be narrow. To keep the area == 1, the curve must also be tall. On the other hand if the variance is large the curve will be wide, and thus it will also have to be short to make the area == 1.\n",
+      "\n",
+      "Let us look at that:"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "plt.plot (xs,[gaussian(x, 5, .2) for x in xs],'b')\n",
+      "plt.plot (xs,[gaussian(x, 5, 1) for x in xs],'g')\n",
+      "plt.plot (xs,[gaussian(x, 5, 5) for x in xs],'r')\n",
+      "plt.show()"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAXUAAAEACAYAAABMEua6AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x3ad3710>"
+       ]
+      }
+     ],
+     "prompt_number": 5
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "So what is this telling us? The blue gaussian is very narrow. It is saying that we believe x=5, and that we are very sure about that. In contrast, the red gaussian also believes that x=5, but we are much less sure about that. Our believe that x=5 is lower, and our belief about the likely possible values for x is spread out - we think it is quite likely that x=2 or x=8, for example. The blue gaussian has almost completely eliminated 2 or 8 as possible value - their probably is almost 0.0.\n"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [],
+     "language": "python",
+     "metadata": {},
+     "outputs": []
+    }
+   ],
+   "metadata": {}
+  }
+ ]
+}

histogram_filter.ipynb

@@ -42,7 +42,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 3
+     "prompt_number": 4
     },
     {
      "cell_type": "markdown",
@@ -60,7 +60,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 4
+     "prompt_number": 5
     },
     {
      "cell_type": "markdown",
@@ -92,11 +92,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x4037f90>"
+        "<matplotlib.figure.Figure at 0x3f12b90>"
        ]
       }
      ],
-     "prompt_number": 61
+     "prompt_number": 6
     },
     {
      "cell_type": "markdown",
@@ -125,7 +125,7 @@
        ]
       }
      ],
-     "prompt_number": 6
+     "prompt_number": 7
     },
     {
      "cell_type": "markdown",
@@ -160,7 +160,7 @@
        ]
       }
      ],
-     "prompt_number": 7
+     "prompt_number": 8
     },
     {
      "cell_type": "markdown",
@@ -217,7 +217,7 @@
        ]
       }
      ],
-     "prompt_number": 76
+     "prompt_number": 9
     },
     {
      "cell_type": "markdown",
@@ -267,7 +267,7 @@
        ]
       }
      ],
-     "prompt_number": 78
+     "prompt_number": 10
     },
     {
      "cell_type": "markdown",
@@ -323,7 +323,7 @@
        ]
       }
      ],
-     "prompt_number": 35
+     "prompt_number": 11
     },
     {
      "cell_type": "markdown",
@@ -369,7 +369,7 @@
        ]
       }
      ],
-     "prompt_number": 53
+     "prompt_number": 12
     },
     {
      "cell_type": "markdown",
@@ -397,7 +397,7 @@
        ]
       }
      ],
-     "prompt_number": 57
+     "prompt_number": 13
     },
     {
      "cell_type": "markdown",
@@ -422,11 +422,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x4580050>"
+        "<matplotlib.figure.Figure at 0x3f6f850>"
        ]
       }
      ],
-     "prompt_number": 65
+     "prompt_number": 14
     },
     {
      "cell_type": "markdown",
@@ -460,11 +460,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x405cd10>"
+        "<matplotlib.figure.Figure at 0x3f12590>"
        ]
       }
      ],
-     "prompt_number": 74
+     "prompt_number": 15
     },
     {
      "cell_type": "markdown",
@@ -499,11 +499,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x405d390>"
+        "<matplotlib.figure.Figure at 0x3f80dd0>"
        ]
       }
      ],
-     "prompt_number": 72
+     "prompt_number": 16
     },
     {
      "cell_type": "markdown",
@@ -551,11 +551,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x45fe8d0>"
+        "<matplotlib.figure.Figure at 0x3f12c90>"
        ]
       }
      ],
-     "prompt_number": 85
+     "prompt_number": 17
     },
     {
      "cell_type": "markdown",
@@ -588,11 +588,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x3f82810>"
+        "<matplotlib.figure.Figure at 0x3fbf950>"
        ]
       }
      ],
-     "prompt_number": 86
+     "prompt_number": 18
     },
     {
      "cell_type": "markdown",
@@ -615,13 +615,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x4057490>"
+        "<matplotlib.figure.Figure at 0x3efe410>"
        ]
       }
      ],
-     "prompt_number": 91
+     "prompt_number": 19
     },
     {
      "cell_type": "markdown",
@@ -644,13 +644,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAXYAAAEACAYAAACnJV25AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x3f875d0>"
+        "<matplotlib.figure.Figure at 0x3fbf390>"
        ]
       }
      ],
-     "prompt_number": 92
+     "prompt_number": 20
     },
     {
      "cell_type": "markdown",
@@ -681,7 +681,8 @@
       "for m in measurements:\n",
       "    pos = sense (pos, m, .6, .2)\n",
       "    pos = update (pos, 1, .8, .1, .1)\n",
-      "bar_plot(pos)"
+      "bar_plot(pos)\n",
+      "print pos"
      ],
      "language": "python",
      "metadata": {},
@@ -691,11 +692,19 @@
        "output_type": "display_data",
        "png": "iVBORw0KGgoAAAANSUhEUgAAAXYAAAEACAYAAACnJV25AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x4580390>"
+        "<matplotlib.figure.Figure at 0x3f6fc90>"
+       ]
+      },
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "[ 0.2245871   0.06288015  0.06109133  0.0581008   0.09334062  0.2245871\n",
+        "  0.06288015  0.06109133  0.0581008   0.09334062]\n"
        ]
       }
      ],
-     "prompt_number": 97
+     "prompt_number": 25
     },
     {
      "cell_type": "markdown",
@@ -720,11 +729,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x44d0f10>"
+        "<matplotlib.figure.Figure at 0x3f5b250>"
        ]
       }
      ],
-     "prompt_number": 98
+     "prompt_number": 22
     },
     {
      "cell_type": "markdown",
@@ -750,13 +759,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x45fe590>"
+        "<matplotlib.figure.Figure at 0x3f5c390>"
        ]
       }
      ],
-     "prompt_number": 100
+     "prompt_number": 23
     },
     {
      "cell_type": "markdown",
@@ -765,6 +774,14 @@
       "As you can see we quickly filtered out the bad sensor reading and converged on the most likely positions for our dog."
      ]
     },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "####Generalizing to Multiple Dimensions\n",
+      "\n"
+     ]
+    },
     {
      "cell_type": "markdown",
      "metadata": {},
@@ -781,147 +798,37 @@
       "\n",
       "Finally, the bar charts may strike you as being a bit less certain than we would want. A 25% certaintly may not give you a lot of confidence in the anwser. Of course, what is important here is the ratio of this probability to the other probabilities in your vector. If the next largest bar is 23% then we are not very knowledgable about our position, whereas if the next largest is 3% we are in fact quite certain. But this is not clear or intuitive. However, there is an extremely important insight that Kalman filters implement that will signficantly improve our accuracy from the same data.\n",
       "\n",
+      "Do not be mislead by the simplicity of the example I chose. This is a robust and complete implementation of a histogram filter, and you may use the code in complicated, real world solutions. If you need a multimodal, discrete filter, this filter works.\n",
+      "\n",
       "**If you can understand this chapter you will be able to understand and implement Kalman filters** I cannot stress this enough. If anything is murky, go back and reread this chapter and play with the code. the rest of this book will build on the algorithms that we use here. If you don't intuitively understand why this histogram filter works, and can at least work through the math, you will have little success with the rest of the material. However, if you grasp the fundamental insight - multiplying probabilities when we measure, and shifting probabilities when we update leads to a converging solution - then you understand everything important you need to grasp the Kalman filter. "
      ]
     },
-    {
-     "cell_type": "heading",
-     "level": 2,
-     "metadata": {},
-     "source": [
-      "Gaussian Probabilities"
-     ]
-    },
-    {
-     "cell_type": "code",
-     "collapsed": false,
-     "input": [],
-     "language": "python",
-     "metadata": {},
-     "outputs": []
-    },
-    {
-     "cell_type": "code",
-     "collapsed": false,
-     "input": [],
-     "language": "python",
-     "metadata": {},
-     "outputs": []
-    },
     {
      "cell_type": "markdown",
      "metadata": {},
-     "source": [
-      "So let us explore how gaussians work. A gaussian is a probability distribution that is completely described with two parameters, the mean and the variance. It is defined as:\n",
-      "$$ \n",
-      "f(x, \\mu, \\sigma) = \\frac{1}{\\sigma\\sqrt{2\\pi}} e^{-0.5*{(x-\\mu)^2}/\\sigma^2 }\n",
-      "$$\n",
-      "\n",
-      "where $\\mu$ is the mean and $\\sigma^2$ is the variance (we will define these in a moment).\n",
-      "Let us plot that with Python. First, we will define a function that computes the gaussian for any x. "
-     ]
+     "source": []
     },
     {
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "import math\n",
-      "\n",
-      "def gaussian (x, mu, sigma):\n",
-      "    ''' compute the gaussian with the specified mean and sigma'''\n",
-      "    return math.exp (-0.5 * (x-mu)**2 / sigma) / math.sqrt(2.*math.pi*sigma)"
-     ],
-     "language": "python",
-     "metadata": {},
-     "outputs": [],
-     "prompt_number": 10
-    },
-    {
-     "cell_type": "markdown",
-     "metadata": {},
-     "source": [
-      "Now we will plot a gaussian centered around 5, with a variance of 1."
-     ]
-    },
-    {
-     "cell_type": "code",
-     "collapsed": false,
-     "input": [
-      "import matplotlib.pyplot as plt\n",
-      "import numpy as np\n",
-      "xs = np.arange(0,10,0.1)\n",
-      "plt.plot (xs,[gaussian(x, 5, 1) for x in xs])\n",
-      "plt.axvline (5) \n",
-      "plt.show()"
+      "Author notes:\n",
+      "    Do I want to go to the multidimensional case? At least describe it, but why not implement it as well?\n",
+      "    "
      ],
      "language": "python",
      "metadata": {},
      "outputs": [
       {
-       "metadata": {},
-       "output_type": "display_data",
-       "png": 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truncated
-       "text": [
-        "<matplotlib.figure.Figure at 0x3ed6390>"
+       "ename": "SyntaxError",
+       "evalue": "invalid syntax (<ipython-input-24-5eaa20e350b9>, line 1)",
+       "output_type": "pyerr",
+       "traceback": [
+        "\u001b[1;36m  File \u001b[1;32m\"<ipython-input-24-5eaa20e350b9>\"\u001b[1;36m, line \u001b[1;32m1\u001b[0m\n\u001b[1;33m    Author notes:\u001b[0m\n\u001b[1;37m               ^\u001b[0m\n\u001b[1;31mSyntaxError\u001b[0m\u001b[1;31m:\u001b[0m invalid syntax\n"
        ]
       }
      ],
-     "prompt_number": 11
-    },
-    {
-     "cell_type": "markdown",
-     "metadata": {},
-     "source": [
-      "As we expected the curve is centered around 5. We can see why this is by looking at the equation for the gaussian. When x=5, x-\\mu is 0, and thus e^0 is 1. any other value for x will result in a smaller value for exp function.\n",
-      "\n",
-      "The width of the curve is defined by the variance. If the variance is large than the curve will be wide, and if the variance is small the curve will be narrow.\n",
-      "\n",
-      "Also, since this is a probability distribution it is required that the area under the curve always equals one. This should be intuitively clear - the area under the curve represents all possible occurances, which must sum to one.\n",
-      "\n",
-      "This leads to an important insight. If the variance is small the curve will be narrow. To keep the area == 1, the curve must also be tall. On the other hand if the variance is large the curve will be wide, and thus it will also have to be short to make the area == 1.\n",
-      "\n",
-      "Let us look at that:"
-     ]
-    },
-    {
-     "cell_type": "code",
-     "collapsed": false,
-     "input": [
-      "plt.plot (xs,[gaussian(x, 5, .2) for x in xs],'b')\n",
-      "plt.plot (xs,[gaussian(x, 5, 1) for x in xs],'g')\n",
-      "plt.plot (xs,[gaussian(x, 5, 5) for x in xs],'r')\n",
-      "\n",
-      "plt.show()"
-     ],
-     "language": "python",
-     "metadata": {},
-     "outputs": [
-      {
-       "metadata": {},
-       "output_type": "display_data",
-       "png": 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truncated
-       "text": [
-        "<matplotlib.figure.Figure at 0x3ed6a50>"
-       ]
-      }
-     ],
-     "prompt_number": 13
-    },
-    {
-     "cell_type": "markdown",
-     "metadata": {},
-     "source": [
-      "So what is this telling us? The blue gaussian is very narrow. It is saying that we believe x=5, and that we are very sure about that. In contrast, the red gaussian also believes that x=5, but we are much less sure about that. Our believe that x=5 is lower, and our belief about the likely possible values for x is spread out - we think it is quite likely that x=2 or x=8, for example. The blue gaussian has almost completely eliminated 2 or 8 as possible value - their probably is almost 0.0.\n",
-      "\n"
-     ]
-    },
-    {
-     "cell_type": "code",
-     "collapsed": false,
-     "input": [],
-     "language": "python",
-     "metadata": {},
-     "outputs": []
+     "prompt_number": 24
     }
    ],
    "metadata": {}