Added explanation on why covariance ellipse shape changes with R.

Kalman_Filters.ipynb

@@ -1,7 +1,7 @@
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@@ -299,7 +299,7 @@
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "The constructor (__init__()) initializes the DogSensor class with an initial position (x0), velocity (vel), and an noise scaling factor. The *sense()* function has the dog move by the set velocity and returns its new position, with noise added. If you look at the code for *sense()* you will see a call to *numpy.random.randn()*. This returns a number sampled from a normal distribution with a mean of 0.0. Let's look at some example output for that.\n",
+      "The constructor *\\_\\_init()\\_\\_* initializes the DogSensor class with an initial position (x0), velocity (vel), and an noise scaling factor. The *sense()* function has the dog move by the set velocity and returns its new position, with noise added. If you look at the code for *sense()* you will see a call to *numpy.random.randn()*. This returns a number sampled from a normal distribution with a mean of 0.0. Let's look at some example output for that.\n",
       "\n"
      ]
     },
@@ -471,7 +471,7 @@
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "You may not have a full understanding of the exact *meaning* of a noise value of 100.0, but as it turns out if you multiply *randn()* with a number $n$, the result is just a normal distribution with $\\sigma = \\sqrt{n}$. So the example with noise = 100 is using the normal distribution  $\\mathcal{N}(0,100)$. Recall the notation for a normal distribution is $\\mathcal{N}(\\mu,\\sigma^2)$. If the square root is confusing, recall that normal distributions use $\\sigma^2$ for the variance, and $\\sigma$ is the standard deviation, which we do not use in this book. *dog_sensor.__init__()* takes the square root of the noise setting so that the *noise * randn()* call properly computes the normal distribution. "
+      "You may not have a full understanding of the exact *meaning* of a noise value of 100.0, but as it turns out if you multiply *randn()* with a number $n$, the result is just a normal distribution with $\\sigma = \\sqrt{n}$. So the example with noise = 100 is using the normal distribution  $\\mathcal{N}(0,100)$. Recall the notation for a normal distribution is $\\mathcal{N}(\\mu,\\sigma^2)$. If the square root is confusing, recall that normal distributions use $\\sigma^2$ for the variance, and $\\sigma$ is the standard deviation, which we do not use in this book. DogSensor.<code><code>\\_\\_init__()</code> takes the square root of the noise setting so that the *noise &ast; randn()* call properly computes the normal distribution."
      ]
     },
     {

Multidimensional_Kalman_Filters.ipynb

@@ -1,7 +1,7 @@
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   "name": "",
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@@ -237,13 +237,13 @@
        ],
        "metadata": {},
        "output_type": "pyout",
-       "prompt_number": 1,
+       "prompt_number": 2,
        "text": [
-        "<IPython.core.display.HTML at 0x1a45850>"
+        "<IPython.core.display.HTML at 0xfebb90>"
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@@ -336,7 +336,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 2
+     "prompt_number": 3
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     {
      "cell_type": "markdown",
@@ -359,7 +359,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
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     {
      "cell_type": "markdown",
@@ -377,7 +377,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
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+     "prompt_number": 5
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     {
      "cell_type": "markdown",
@@ -395,7 +395,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
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     },
     {
      "cell_type": "markdown",
@@ -421,7 +421,7 @@
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@@ -450,7 +450,7 @@
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     {
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@@ -501,7 +501,7 @@
       {
        "metadata": {},
        "output_type": "pyout",
-       "prompt_number": 8,
+       "prompt_number": 9,
        "text": [
         "(-10, 20)"
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@@ -511,11 +511,11 @@
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@@ -555,11 +555,11 @@
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        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x32d8510>"
+        "<matplotlib.figure.Figure at 0x2252790>"
        ]
       }
      ],
-     "prompt_number": 9
+     "prompt_number": 10
     },
     {
      "cell_type": "markdown",
@@ -639,11 +639,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2ffacd0>"
+        "<matplotlib.figure.Figure at 0x2249390>"
        ]
       }
      ],
-     "prompt_number": 10
+     "prompt_number": 11
     },
     {
      "cell_type": "markdown",
@@ -666,11 +666,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x1a456d0>"
+        "<matplotlib.figure.Figure at 0x22c2810>"
        ]
       }
      ],
-     "prompt_number": 11
+     "prompt_number": 12
     },
     {
      "cell_type": "markdown",
@@ -693,11 +693,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x32c8ad0>"
+        "<matplotlib.figure.Figure at 0x24f0fd0>"
        ]
       }
      ],
-     "prompt_number": 12
+     "prompt_number": 13
     },
     {
      "cell_type": "markdown",
@@ -724,11 +724,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2fdde90>"
+        "<matplotlib.figure.Figure at 0x22d3a90>"
        ]
       }
      ],
-     "prompt_number": 13
+     "prompt_number": 14
     },
     {
      "cell_type": "markdown",
@@ -770,11 +770,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x3320790>"
+        "<matplotlib.figure.Figure at 0x26b4f90>"
        ]
       }
      ],
-     "prompt_number": 14
+     "prompt_number": 15
     },
     {
      "cell_type": "markdown",
@@ -916,7 +916,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 15
+     "prompt_number": 16
     },
     {
      "cell_type": "markdown",
@@ -940,7 +940,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 16
+     "prompt_number": 17
     },
     {
      "cell_type": "markdown",
@@ -1035,9 +1035,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2fbcb10>"
+        "<matplotlib.figure.Figure at 0x2249110>"
        ]
       },
       {
@@ -1045,11 +1045,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x334cfd0>"
+        "<matplotlib.figure.Figure at 0x2520250>"
        ]
       }
      ],
-     "prompt_number": 17
+     "prompt_number": 18
     },
     {
      "cell_type": "markdown",
@@ -1094,11 +1094,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x3497f90>"
+        "<matplotlib.figure.Figure at 0x233df50>"
        ]
       }
      ],
-     "prompt_number": 18
+     "prompt_number": 19
     },
     {
      "cell_type": "markdown",
@@ -1178,21 +1178,21 @@
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x32c8ad0>"
+        "<matplotlib.figure.Figure at 0x23c4d90>"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x3328910>"
+        "<matplotlib.figure.Figure at 0x2518d50>"
        ]
       }
      ],
-     "prompt_number": 19
+     "prompt_number": 20
     },
     {
      "cell_type": "markdown",
@@ -1223,13 +1223,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x300d190>"
+        "<matplotlib.figure.Figure at 0x2cd9150>"
        ]
       }
      ],
-     "prompt_number": 20
+     "prompt_number": 21
     },
     {
      "cell_type": "markdown",
@@ -1296,11 +1296,11 @@
        "output_type": "display_data",
        "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x2fccd50>"
+        "<matplotlib.figure.Figure at 0x2cc98d0>"
        ]
       }
      ],
-     "prompt_number": 21
+     "prompt_number": 22
     },
     {
      "cell_type": "markdown",
@@ -1325,11 +1325,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2fdc550>"
+        "<matplotlib.figure.Figure at 0x26b7e10>"
        ]
       }
      ],
-     "prompt_number": 22
+     "prompt_number": 23
     },
     {
      "cell_type": "markdown",
@@ -1414,21 +1414,21 @@
       {
        "metadata": {},
        "output_type": "display_data",
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        "text": [
-        "<matplotlib.figure.Figure at 0x2fba950>"
+        "<matplotlib.figure.Figure at 0x282e5d0>"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x300dd90>"
+        "<matplotlib.figure.Figure at 0x281cc10>"
        ]
       }
      ],
-     "prompt_number": 23
+     "prompt_number": 29
     },
     {
      "cell_type": "markdown",
@@ -1440,15 +1440,69 @@
       "\n",
       "The first plot, with $R=5$, finishes up with an ellipse that is wider than it is tall. If that is not clear I have printed out the variances for the last ellipse in the lower right hand corner. The variance for position is 3.85, and the variance for velocity is 3.0. \n",
       "\n",
-      "In contrast, the second plot, with $R=0.5$, has a final ellipse that is taller than wide. The ellipses in the second plot are all much smaller than the ellipses in the first plot. This stands to reason because a small $R$ implies a small amount of noise in our measurements. Small noise means accurate predictions, and thus a strong belief in our position. \n",
-      "\n",
-      "** EXPLAIN WHY SECOND PLOT ELLIPSE IS TALLER THAN WIDE!!!**\n",
-      "\n",
-      "Keep looking at these plots until you grasp how the covariance matrix $P$ has a real, physical interpretation. When you start dealing with a, say, $9\\times 9$ matrix it may seem overwhelming - there are 81 numbers to interpret. Just break it down - the diagonal contains the variance for each state variable, and all off diagonal elements are the product of two variances and a scaling factor $p$. You will not be able to plot a $9\\times 9$ matrix on the screen because it would require living in 10-D space, so you have to develop your intution and understanding in this simple, 2-D case. \n",
-      "\n",
-      "> **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.\n",
-      "\n"
+      "In contrast, the second plot, with $R=0.5$, has a final ellipse that is taller than wide. The ellipses in the second plot are all much smaller than the ellipses in the first plot. This stands to reason because a small $R$ implies a small amount of noise in our measurements. Small noise means accurate predictions, and thus a strong belief in our position. "
      ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "##### Question: Explain Ellipse Differences\n",
+      "\n",
+      "Why are the ellipses for $R=5$ shorter, and more tilted than the ellipses for $R=0.5$. Hint: think about this in the context of what these ellipses mean physically, not in terms of the math. If you aren't sure about the answer,change R to truly large and small numbers such as 100 and 0.1, observe the changes, and think about what this means. \n",
+      "\n",
+      "##### Solution\n",
+      "\n",
+      "The $x$ axis is for position, and $y$ is velocity. An ellipse that is vertical, or nearly so, says there is no correlation between position and velocity, and an ellipse that is diagnal says that there is a lot of correlation. Phrased that way, it sounds unlikely - either they are correlated or not. But this is a measure of the *output of the filter*, not a description of the actual, physical world. When $R$ is very large we are telling the filter that there is a lot of noise in the measurements. In that case the Kalman gain $K$ is set to favor the prediction over the measurement, and the prediction comes from the velocity state variable. So, there is a large correlation between $x$ and $\\dot{x}$. Conversely, if $R$ is small, we are telling the filter that the measurement is very trustworthy, and $K$ is set to favor the measurement over the prediction. Why would the filter want to use the prediction if the measurement is nearly perfect? If the filter is not using much from the prediction there will be very little correlation reported. \n",
+      "\n",
+      "**This is a critical point to understand!**. The Kalman filter is just a mathematical model for a real world system. A report of little correlation *does not mean* there is no correlation in the physical system, just that there was no correlation in the mathematical model. It's just a report of how much measurement vs prediction was incorporated into the model.  \n",
+      "\n",
+      "Let's bring that point home with a truly large measurement error. We will set $R=500$. Think about what the plot will look like before scrolling down. To emphasize the issue, I will set the amount of noise injected into the measurements to 0, so the measurement will exactly equal the actual position. "
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "plot_track (noise=0, R=500, Q=5, count=7, title='R = 500')"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x22c7fd0>"
+       ]
+      }
+     ],
+     "prompt_number": 36
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "I hope the result was what you were expecting. The ellipse quickly became very wide and not very tall. It did this because the Kalman filter mostly used the prediction vs the measurement to produce the filtered result. We can also see how the filter output is slow to acquire the track. The Kalman filter assumes that the measurements are extremely noisy, and so it is very slow to update its estimate for $\\dot{x}$. "
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Keep looking at these plots until you grasp how to interpret the covariance matrix $P$. When you start dealing with a, say, $9\\times 9$ matrix it may seem overwhelming - there are 81 numbers to interpret. Just break it down - the diagonal contains the variance for each state variable, and all off diagonal elements are the product of two variances and a scaling factor $p$. You will not be able to plot a $9\\times 9$ matrix on the screen because it would require living in 10-D space, so you have to develop your intution and understanding in this simple, 2-D case. \n",
+      "\n",
+      "> **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": "code",
+     "collapsed": false,
+     "input": [],
+     "language": "python",
+     "metadata": {},
+     "outputs": []
     }
    ],
    "metadata": {}

Untitled0.ipynb

@@ -1,7 +1,7 @@
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@@ -711,111 +711,15 @@
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "#[Kalman Filters and Random Signals in Python](http://github.com/rlabbe/Kalman-Filters-and-Random-Signals-in-Python)\n",
-      "\n",
-      "### Version 0.0 - not ready for public consumption. In development.\n",
-      "\n",
-      "\n",
-      "## Motivation\n",
-      "\n",
-      "This is a book for programmers that have a need or interest in Kalman filtering. The motivation for this book came out of my desire for a gentle introduction to Kalman filtering. I'm a software engineer that spent almost two decades in the avionics field, and so I have always been 'bumping elbows' with the Kalman filter, but never had the need to implement one myself. As I moved into solving tracking problems with computer vision I needed to start implementing them. There are classic textbooks in the field, such as Grewal and Andrew's excellent *Kalman Filtering*. But sitting down and trying to read these books is a dismal and trying experience if you do not have the background. Typcially the first few chapters fly through several years of undergraduate math, blithely referring you to textbooks on, for example, It\u014d calculus, and presenting an entire semester's worth of statistics in a few brief paragraphs. These books are good textbooks for an upper undergraduate course, and an invaluable reference to researchers and professionals, but the going is truly difficult for the more casual reader. Symbology is introduced without explanation, different texts use different words and variables names for the same concept, and the books are almost devoid of examples or worked problems. I often found myself able to parse the words and comprehend the mathematics of a defition, but had no idea as to what real world phenomena these words and math were attempting to describe. \"But what does that *mean?*\" was my repeated thought.\n",
-      "\n",
-      "However, as I began to finally understand the Kalman filter I realized the underlying concepts are quite straightforward. A few simple probability rules, some intuition about how we integrate disparate knowledge to explain events in our everyday life and the core concepts of the Kalman filter are accessible. Kalman filters have a reputation for difficulty, but shorn of much of the formal terminology the beauty of the subject and of their math became clear to me, and I fell in love with the topic. \n",
-      "\n",
-      "As I began to understand the math and theory more difficulties itself. A book or paper's author makes some statement of fact and presents a graph as proof.  Unfortunately, why the statement is true is not clear to me, nor is the method by which you might make that plot obvious. Or maybe I wonder \"is this true if R=0?\"  Or the author provide pseudocode - at such a high level that the implementation is not obvious. Some books offer Matlab code, but I do not have a license to that expensive package. Finally, many books end each chapter with many useful exercises. Exercises which you need to understand, but excercises with no answers. If you are using the book in a classroom, perhaps this is okay, but it is terrible for the independent reader. I loathe that an author witholds information from me, presumably to avoid 'cheating'. \n",
-      "None of this necessary, from my point of view. Certainly if you are designing a Kalman filter for a aircraft or missile you must be a thorough master of all of the mathematics and topics in a typical Kalman filter textbook. I just want to track an image on a screen, or write some code for my Arduino project. I want to know how the plots in the book are made, and chose different parameters than the author chose. I want to run simulations. I want to inject more noise in the signal and see how a filter performs. \n",
-      "\n",
-      "I wrote this book to address all of those needs. This is not the book for you if you program avionics for Boeing or design radars for Ratheon. Go get a degree at Georgia Tech, UW, or the like, because you'll need it for that kind of work. This book is for the hobbiest, the curious, and the working engineer that needs to filter or smooth data. \n",
-      "\n",
-      "This book is interactive. While you can read it online as static content, I urge you to use it as intended. It is written using IPython Notebook, which allows you to combine text, python, and python output in one place. Every plot, every piece of data in this book is generated from Python that is available to you right inside the notebook. Want to double the value of a parameter. Click on the Python cell, change the parameter's value, and click 'Run'. A new plot or printed output will appear in the book. \n",
-      "\n",
-      "This book has exercises, but it also has the answers. I trust you. If you just need an answer, go ahead and read the answer. If you want to internalize this knowledge, try to implement the exercise before you read the answer. \n",
-      "\n",
-      "This book has supporting libraries for computing statistics, plotting various things related to filters, and for the various filters that we cover. This does require a strong caveat; most code is written for didactic purposes. It is rare that I chose the most efficient solution (which often obscures the intent of the code), and I mostly did not concern myself with numerical stability. This is important to understand - Kalman filters in aircraft are carefully designed and implemented to be numerically stable; the naive implemention is not stable in many cases. If you are serious about Kalman filters this book will not be the last book you need. My intention is to introduce you to the concepts and mathematics, and to get you to the point where the textbooks are approachable.\n",
-      "\n",
-      "Finally, this book is free. The cost for the books required to learn Kalman filtering is somewhat prohibitive even for a Silicon Valley engineer like myself; I cannot believe the are within the reach of someone in a depressed economy, or a financially struggling student. I have gained so much from free software like Python, and free books like those from Allen B. Downey [here](http://www.greenteapress.com/). It's time to repay that. So, the book is free, it is hosted on free servers, and it uses free software for all of the code. \n",
-      "\n",
-      "\n",
-      "\n",
-      "\n",
-      "Contents\n",
-      "-----\n",
-      "\n",
-      "* [**Introduction**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-Filters-and-Random-Signals-in-Python/master/Introduction.ipynb)\n",
-      " \n",
-      "    Introduction to the Kalman filter. Explanation of the idea behind this book.\n",
-      "\n",
-      "\n",
-      "* [**Chapter 1: The g-h Filter**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-Filters-and-Random-Signals-in-Python/master/g-h_filter.ipynb)\n",
-      "\n",
-      "    Intuitive introduction to the g-h filter, which is a family of filters that includes the Kalman filter. Not filler - once you understand this chapter you will understand the concepts behind the Kalman filter. \n",
-      "\n",
-      "\n",
-      "* [**Chapter 2: The Discrete Bayes Filter**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-Filters-and-Random-Signals-in-Python/master/histogram_filter.ipynb)\n",
-      "    Introduces the Discrete Bayes Filter. From this you will learn the probabilistic reasoning that underpins the Kalman filter in an easy to digest form.\n",
-      "\n",
-      "* [**Chapter 3: Gaussian Probabilities**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-Filters-and-Random-Signals-in-Python/master/Gaussians.ipynb)\n",
-      "    Introduces using Gaussians to represent beliefs. Gaussians allow us to implement the algorithms used in the Discrete Bayes Filter to work in continuous domains.\n",
-      "\n",
-      "* [**Chapter 4: One Dimensional Kalman Filters**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-Filters-and-Random-Signals-in-Python/master/Kalman_Filters.ipynb)\n",
-      "    Implements a Kalman filter by modifying the Discrete Bayesian Filter to use Gaussians. This is a full featured Kalman filter, albeit only useful for 1D problems. \n",
-      "\n",
-      "* [**Chapter 5: Multidimensional Kalman Filter**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-Filters-and-Random-Signals-in-Python/master/Multidimensional_Kalman_Filters.ipynb)\n",
-      "    We extend the Kalman filter developed in the previous chapter to the full, generalized filter. \n",
-      "\n",
-      "\n",
-      "Reading the book\n",
-      "-----\n",
-      "There are multiple ways to read this book. However, it is intended to be interactive and I recommend using it in that form. If you install IPython on your computer and then clone this book you will be able to run all of the code in the book yourself. You can perform experiments, see how filters react to different data, see how different filters react to the same data, and so on. I find this sort of immediate feedback both vital and invigorating. You do not have to wonder \"what happens if\". Try it and see!\n",
-      "\n",
-      "If you do not want to do that you can read this book online. the website [nbviewer]http://nbviewer.org provides an IPython Notebook server that renders a notebook stored at github (or elsewhere). The rendering is done in real time when you load the book. If you read my book today, and then I make a change tomorrow, when you go back tomorrow you will see that change. \n",
-      "\n",
-      "You may access this book via nbviewer at any by using this address:\n",
-      "http://nbviewer.ipython.org/github/rlabbe/Kalman-Filters-and-Random-Signals-in-Python/blob/master/Introduction.ipynb\n",
-      "\n",
-      "Finally, you may generate output in a variety of formats. I will not cover how to do that, other than to point you to [IPython nbconvert](http://ipython.org/ipython-doc/rel-1.0.0/interactive/nbconvert.html). You can convert this book into static HTML pages, latex, or PDF. While I don't recommend it particularly, it is useful for those that don't want to program and/or are working offline.\n",
-      "\n",
-      "\n",
-      "Installation and Software Requirements\n",
-      "-----\n",
-      "If you want to run the notebook on your computer, which is what I recommend, then you will have to have IPython installed. I do not cover how to do that in this book; requirements change based on what other python installations you may have, whether you use a third party package like Anaconda Python, what operating system you are using, and so on. \n",
-      "\n",
-      "To use all features you will have to have Ipython 2.0 installed, which is released and stable as of April 2014. Most of the book does not require that, but I do make use of the interactive plotting widgets introduced in this release. A few cells will not run if you have an older version installed.\n",
-      "\n",
-      "You will need Python 2.7 or later installed. Almost all of my work is done in Python 2.7, but I periodically test on 3.3. I do not promise any specific check in will work in 3.X, however. I do use Python's \"from __future__ import ...\" statement to help with compatibility. For example, all prints need to use parenthesis. If you try to add, say, \"print 3.14\" into the book your script will fail; you must write \"print (3.4)\" as in Python 3.X.\n",
-      "\n",
-      "You will need a recent version of NumPy, SciPy, and Matplotlib installed. I don't really know what the minimal might be. \n",
-      "I have numpy 1.71, SciPy 0.13.0, and Matplotlib 1.3.1 installed on my machines.\n",
-      "\n",
-      "Personally, I use the Anaconda Python distribution in all of my work, [available here](https://store.continuum.io/cshop/anaconda/). I am not selecting them out of favoritism, I am merely documenting my environment. Should you have trouble running any of the code, perhaps knowing this will help you.\n",
-      "\n",
-      "Provided Libraries\n",
-      "-----\n",
-      "\n",
-      "I've not structured anything nicely yet. For now just look for any .py files in the base directory. As I pull everything together I will turn this into a python library, and probably create a separate git project just for the python code.\n",
-      "\n",
-      "There are python files with a name like *xxx*_internal.py. I use these to store functions that are useful for the book, but not of general interest. Often the Python is the point and focus of what I am talking about, but sometimes I just want to display a chart. IPython Notebook does not allow you to collapse the python code, and so it sometimes gets in the way. Some IPython books just incorporate .png files for the image, but I want to ensure that everything is open - if you want to look at the code you can. \n",
-      "\n",
-      "Some chapters introduce functions that are useful for the rest of the book. Those functions are initially defined within the Notebook itself, but the code is also stored in a Python file that is imported if needed in later chapters. I do document when I do this where the function is first defined. But this is still a work in progress.\n",
-      "\n",
-      "\n",
-      "License\n",
-      "-----\n",
-      "\n",
-      "<a rel=\"license\" href=\"http://creativecommons.org/licenses/by-nc-sa/4.0/\"><img alt=\"Creative Commons License\" style=\"border-width:0\" src=\"http://i.creativecommons.org/l/by-nc-sa/4.0/88x31.png\" /></a><br /><span xmlns:dct=\"http://purl.org/dc/terms/\" property=\"dct:title\">Kalman Filters and Random Signals in Python</span> by <a xmlns:cc=\"http://creativecommons.org/ns#\" href=\"https://github.com/rlabbe/Kalman-Filters-and-Random-Signals-in-Python\" property=\"cc:attributionName\" rel=\"cc:attributionURL\">Roger Labbe</a> is licensed under a <a rel=\"license\" href=\"http://creativecommons.org/licenses/by-nc-sa/4.0/\">Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License</a>.<br />Based on a work at <a xmlns:dct=\"http://purl.org/dc/terms/\" href=\"https://github.com/rlabbe/Kalman-Filters-and-Random-Signals-in-Python\" rel=\"dct:source\">https://github.com/rlabbe/Kalman-Filters-and-Random-Signals-in-Python</a>.\n",
-      "\n",
-      "\n",
-      "Contact\n",
-      "-----\n"
+      "dogsensor.<code>\\_\\_init__()</code>"
      ]
     },
     {
-     "cell_type": "code",
-     "collapsed": false,
-     "input": [],
-     "language": "python",
+     "cell_type": "markdown",
      "metadata": {},
-     "outputs": []
+     "source": [
+      "dog\\_sensor.<code>\\_\\_init__()</code> takes the square root of the noise setting so that the *noise &ast; randn()* call properly computes the normal distribution."
+     ]
     },
     {
      "cell_type": "markdown",

g-h_filter.ipynb

@@ -1,7 +1,7 @@
 {
  "metadata": {
   "name": "",
-  "signature": "sha256:9d7effb7c567759e4ab0db4f358203d9616dc390d53db949437bedefba93d7f4"
+  "signature": "sha256:6c60ac864bcb53244749d19ec802eea1dda3eff6515e5eb3cd0d3d0672d84403"
  },
  "nbformat": 3,
  "nbformat_minor": 0,
@@ -239,13 +239,13 @@
        ],
        "metadata": {},
        "output_type": "pyout",
-       "prompt_number": 2,
+       "prompt_number": 1,
        "text": [
-        "<IPython.core.display.HTML at 0x1e49c10>"
+        "<IPython.core.display.HTML at 0x29af750>"
        ]
       }
      ],
-     "prompt_number": 2
+     "prompt_number": 1
     },
     {
      "cell_type": "markdown",
@@ -338,11 +338,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2477a90>"
+        "<matplotlib.figure.Figure at 0x29ab4d0>"
        ]
       }
      ],
-     "prompt_number": 3
+     "prompt_number": 2
     },
     {
      "cell_type": "markdown",
@@ -371,11 +371,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2e34ed0>"
+        "<matplotlib.figure.Figure at 0x3aa7dd0>"
        ]
       }
      ],
-     "prompt_number": 4
+     "prompt_number": 3
     },
     {
      "cell_type": "markdown",
@@ -405,11 +405,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2e22110>"
+        "<matplotlib.figure.Figure at 0x3a95f50>"
        ]
       }
      ],
-     "prompt_number": 5
+     "prompt_number": 4
     },
     {
      "cell_type": "markdown",
@@ -439,11 +439,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2e4f910>"
+        "<matplotlib.figure.Figure at 0x3e12990>"
        ]
       }
      ],
-     "prompt_number": 6
+     "prompt_number": 5
     },
     {
      "cell_type": "markdown",
@@ -466,11 +466,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2e3a250>"
+        "<matplotlib.figure.Figure at 0x3aa5c90>"
        ]
       }
      ],
-     "prompt_number": 7
+     "prompt_number": 6
     },
     {
      "cell_type": "markdown",
@@ -502,11 +502,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2e224d0>"
+        "<matplotlib.figure.Figure at 0x3a99c10>"
        ]
       }
      ],
-     "prompt_number": 8
+     "prompt_number": 7
     },
     {
      "cell_type": "markdown",
@@ -553,11 +553,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x3043890>"
+        "<matplotlib.figure.Figure at 0x3aa93d0>"
        ]
       }
      ],
-     "prompt_number": 9
+     "prompt_number": 8
     },
     {
      "cell_type": "markdown",
@@ -608,11 +608,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2e3f050>"
+        "<matplotlib.figure.Figure at 0x3107dd0>"
        ]
       }
      ],
-     "prompt_number": 10
+     "prompt_number": 9
     },
     {
      "cell_type": "markdown",
@@ -667,11 +667,11 @@
        "output_type": "display_data",
        "png": "iVBORw0KGgoAAAANSUhEUgAAAq0AAAH8CAYAAAD7ZpZkAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2e3ab10>"
+        "<matplotlib.figure.Figure at 0x3abc410>"
        ]
       }
      ],
-     "prompt_number": 11
+     "prompt_number": 10
     },
     {
      "cell_type": "markdown",
@@ -714,7 +714,7 @@
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "def g_h_filter (data, x0, dx, g, h, dt=1.):\n",
+      "def g_h_filter (data, x0, dx, g, h, dt=1., pred=None):\n",
       "    \"\"\"\n",
       "    Performs g-h filter on 1 state variable with a fixed g and h.\n",
       "    \n",
@@ -724,6 +724,8 @@
       "    'g' is the g-h's g scale factor\n",
       "    'h' is the g-h's h scale factor\n",
       "    'dt' is the length of the time step \n",
+      "    'pred' is an optional list. If provided, each prediction will\n",
+      "    be stored in it\n",
       "    \"\"\"\n",
       "    \n",
       "    x = x0\n",
@@ -733,6 +735,9 @@
       "        x_est = x + (dx*dt)\n",
       "        dx = dx\n",
       "        \n",
+      "        if pred is not None:\n",
+      "            pred.append(x_est)\n",
+      "        \n",
       "        # update step\n",
       "        residual = z - x_est\n",
       "        dx = dx    + h * (residual) / dt\n",
@@ -763,11 +768,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2e3fed0>"
+        "<matplotlib.figure.Figure at 0x3ab7290>"
        ]
       }
      ],
-     "prompt_number": 12
+     "prompt_number": 11
     },
     {
      "cell_type": "markdown",
@@ -817,7 +822,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 13
+     "prompt_number": 12
     },
     {
      "cell_type": "markdown",
@@ -843,13 +848,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x3063410>"
+        "<matplotlib.figure.Figure at 0x3e12150>"
        ]
       }
      ],
-     "prompt_number": 14
+     "prompt_number": 13
     },
     {
      "cell_type": "markdown",
@@ -869,7 +874,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 15
+     "prompt_number": 14
     },
     {
      "cell_type": "markdown",
@@ -892,13 +897,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2e4f190>"
+        "<matplotlib.figure.Figure at 0x3ac4250>"
        ]
       }
      ],
-     "prompt_number": 16
+     "prompt_number": 15
     },
     {
      "cell_type": "markdown",
@@ -921,7 +926,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 17
+     "prompt_number": 16
     },
     {
      "cell_type": "markdown",
@@ -944,13 +949,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2e22590>"
+        "<matplotlib.figure.Figure at 0x3f8e3d0>"
        ]
       }
      ],
-     "prompt_number": 18
+     "prompt_number": 30
     },
     {
      "cell_type": "markdown",
@@ -958,7 +963,7 @@
      "source": [
       "This doesn't look so wonderful to me. We can see that perhaps the filtered signal varies less than the noisey signal, but it is far from the straight line. If we were to plot just the filtered result no one would guess that the signal with no noise starts at 5 and increments by 2 at each time step. And while in locations the filter does seem to reduce the noise, in other places it seems to overshoot and undershoot.\n",
       "\n",
-      "At this poit we don't know enough to really judge this. We added **a lot** of noise; maybe this is as good as filtering can get. However, the existance of the multitude of chapters beyond this one should suggest that we can do much better than this."
+      "At this point we don't know enough to really judge this. We added **a lot** of noise; maybe this is as good as filtering can get. However, the existance of the multitude of chapters beyond this one should suggest that we can do much better than this."
      ]
     },
     {
@@ -979,7 +984,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 19
+     "prompt_number": 18
     },
     {
      "cell_type": "markdown",
@@ -998,9 +1003,10 @@
       "        zs.append (x0 + dx*i + random.randn()*noise_factor)\n",
       "        dx += accel\n",
       "    return zs\n",
-      "               \n",
-      "zs = gen_data (x0=0, dx=0, count=20, noise_factor=0, accel = 2)\n",
-      "data = g_h_filter (data=zs, x0=0, dx=0, g=0.2, h=0.02)\n",
+      "   \n",
+      "predictions = []\n",
+      "zs = gen_data (x0=10, dx=0, count=20, noise_factor=0, accel = 2)\n",
+      "data = g_h_filter (data=zs, x0=10, dx=0, g=0.2, h=0.02, pred=predictions)\n",
       "plt.xlim([0,20])\n",
       "plot_g_h_results  (measurements=zs, filtered_data=data)"
      ],
@@ -1010,13 +1016,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2e329d0>"
+        "<matplotlib.figure.Figure at 0x3aa9b50>"
        ]
       }
      ],
-     "prompt_number": 20
+     "prompt_number": 29
     },
     {
      "cell_type": "markdown",
@@ -1048,7 +1054,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 21
+     "prompt_number": 20
     },
     {
      "cell_type": "markdown",
@@ -1078,29 +1084,29 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x3063dd0>"
+        "<matplotlib.figure.Figure at 0x3abb290>"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2e1bcd0>"
+        "<matplotlib.figure.Figure at 0x3e39e10>"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x33df410>"
+        "<matplotlib.figure.Figure at 0x4402f10>"
        ]
       }
      ],
-     "prompt_number": 22
+     "prompt_number": 21
     },
     {
      "cell_type": "markdown",
@@ -1135,7 +1141,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x3379c90>"
+        "<matplotlib.figure.Figure at 0x3f8c390>"
        ]
       },
       {
@@ -1143,7 +1149,7 @@
        "output_type": "display_data",
        "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x2e3fcd0>"
+        "<matplotlib.figure.Figure at 0x3acfe10>"
        ]
       },
       {
@@ -1151,11 +1157,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x33d8ad0>"
+        "<matplotlib.figure.Figure at 0x43eb110>"
        ]
       }
      ],
-     "prompt_number": 23
+     "prompt_number": 22
     },
     {
      "cell_type": "markdown",
@@ -1186,7 +1192,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2e25650>"
+        "<matplotlib.figure.Figure at 0x3f8eb90>"
        ]
       },
       {
@@ -1194,11 +1200,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x33e7050>"
+        "<matplotlib.figure.Figure at 0x3ac4050>"
        ]
       }
      ],
-     "prompt_number": 24
+     "prompt_number": 23
     },
     {
      "cell_type": "markdown",
@@ -1236,7 +1242,7 @@
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        "text": [
-        "<matplotlib.figure.Figure at 0x3038b10>"
+        "<matplotlib.figure.Figure at 0x3abc110>"
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x304c790>"
+        "<matplotlib.figure.Figure at 0x3aa0890>"
        ]
       },
       {
@@ -1252,11 +1258,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x33cff50>"
+        "<matplotlib.figure.Figure at 0x43ed110>"
        ]
       }
      ],
-     "prompt_number": 40
+     "prompt_number": 24
     },
     {
      "cell_type": "markdown",