diff --git a/00_Preface.ipynb b/00_Preface.ipynb index 8be43e0..0acaa74 100644 --- a/00_Preface.ipynb +++ b/00_Preface.ipynb @@ -349,13 +349,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Not ready for public consumption. In development.\n", - "\n", - "> author's note: The chapter on g-h filters is fairly complete as far as planned content goes. The content for the discrete Bayesian chapter, chapter 2, is also fairly complete. After that I have questions in my mind as to the best way to present the statistics needed to understand the filters. I try to avoid the 'dump a semester of math into 4 pages' approach of most textbooks, but then again perhaps I put things off a bit too long. In any case, the subsequent chapters are due a strong editing cycle where I decide how to best develop these concepts. Otherwise I am pretty happy with the content for the one dimensional and multidimensional Kalman filter chapters. I know the code works, I am using it in real world projects at work, but there are areas where the content about the covariance matrices is pretty bad. The implementation is fine, the description is poor. Sorry. It will be corrected. \n", - "\n", - "> Beyond that the chapters are much more in a state of flux. Reader beware. My writing methodology is to just vomit out whatever is in my head, just to get material, and then go back and think through presentation, test code, refine, and so on. Whatever is checked in in these later chapters may be wrong and not ready for your use. \n", - "\n", - "> Finally, nothing has been spell checked or proof read yet. I with IPython Notebook had spell check, but it doesn't seem to." + "Not ready for public consumption. In development." ] }, { @@ -401,9 +395,11 @@ "source": [ "** author's note**. *The book is still being written, and so I am not focusing on issues like supporting multiple versions of Python. I am staying more or less on the bleeding edge of Python 3 for the time being. If you follow my suggestion of installing Anaconda all of the versioning problems will be taken care of for you, and you will not alter or affect any existing installation of Python on your machine. I am aware that telling somebody to install a specific packaging system is not a long term solution, but I can either focus on endless regression testing for every minor code change, or work on delivering the book, and then doing one sweep through it to maximize compatibility. I opt for the latter. In the meantime I welcome bug reports if the book does not work on your platform.*\n", "\n", - "If you want to run the notebook on your computer, which is what I recommend, then you will have to have IPython 2.4 or later 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", + "If you want to run the notebook on your computer, which is what I recommend, then you will have to have IPython 2.4 or later installed. IPython is an interactive architecture that provides IPython Notebook, the tool used to write this book. Note that the IPython version has nothing to do with the Python version. IPython 2.4 can run Python 3.4, IPython 3.0 can run Python 2.7, and so on. \n", "\n", - "The notebook format was changed as of IPython 3.0. If you are running 2.4 you will still be able to open and run the notebooks, but they will be downconverted for you. If you make changes DO NOT push 2.4 version notebooks to me! I strongly recommend updating to 3.0 as soon as possible, as this format change will just become more frustrating with time.\n", + "I do not cover how to install IPython 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", + "The IPython Notebook format was changed as of IPython 3.0. If you are running 2.4 you will still be able to open and run the notebooks, but they will be downconverted for you. If you make changes DO NOT push 2.4 version notebooks to me! I strongly recommend updating to 3.0 as soon as possible, as this format change will just become more frustrating with time.\n", "\n", "You will need Python 2.7 or later installed. Almost all of my work is done in Python 3.4, but I periodically test on 2.7. I do not promise any specific check in will work in 2.7 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", @@ -427,7 +423,6 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "\n", "I am writing an open source Bayesian filtering Python library called **FilterPy**. It is available on github at (https://github.com/rlabbe/filterpy). To ensure that you have the latest release you will want to grab a copy from github, and follow your Python installation's instructions for adding it to the Python search path.\n", "\n", "I have also made the project available on PyPi, the Python Package Index. I will be honest, I am not updating this as fast as I am changing the code in the library. That will change as the library and this book mature. To install from PyPi, at the command line issue the command\n", @@ -445,7 +440,7 @@ "\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. I try to avoid this because then I always face the issue of code in the directory becoming out of sync with the code in the book. However, IPython Notebook does not give us a way to refer to code cells in other notebooks, so this is the only mechanism I know of to share functionality across notebooks.\n", "\n", - "There is an undocumented directory called **exp**. This is where I write and test code prior to putting it in the book. There is some interesting stuff in there, and feel free to look at it. As the book evolves I plan to create examples and projects, and a lot of this material will end up there. Small experiments will eventually just be deleted. If you are just interested in reading the book you can safely ignore this directory. \n", + "There is an undocumented directory called **experiments**. This is where I write and test code prior to putting it in the book. There is some interesting stuff in there, and feel free to look at it. As the book evolves I plan to create examples and projects, and a lot of this material will end up there. Small experiments will eventually just be deleted. If you are just interested in reading the book you can safely ignore this directory. \n", "\n", "\n", "The directory **styles** contains a css file containing the style guide for the book. The default look and feel of IPython Notebook is rather plain. Work is being done on this. I have followed the examples set by books such as [Probabilistic Programming and Bayesian Methods for Hackers](http://nbviewer.ipython.org/github/CamDavidsonPilon/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers/blob/master/Chapter1_Introduction/Chapter1_Introduction.ipynb). I have also been very influenced by Professor Lorena Barba's fantastic work, [available here](https://github.com/barbagroup/CFDPython). I owe all of my look and feel to the work of these projects. " @@ -574,7 +569,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.4.2" + "version": "3.4.3" } }, "nbformat": 4, diff --git a/01_g-h_filter.ipynb b/01_g-h_filter.ipynb index 09fe9b6..d5e7800 100644 --- a/01_g-h_filter.ipynb +++ b/01_g-h_filter.ipynb @@ -424,7 +424,7 @@ "GgAAkohrAABIIq4BACCJuAYAgCTiGgAAkohrAABI8v8BSJ+bTP4sBwsAAAAASUVORK5CYII=\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -571,7 +571,7 @@ "ACT5/7HUkn4JFkA2AAAAAElFTkSuQmCC\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -709,7 +709,7 @@ "QBJxDQAASf4fTXV4ruVZZG0AAAAASUVORK5CYII=\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -749,7 +749,7 @@ "name": "stdout", "output_type": "stream", "text": [ - "Average of measurements is 165.0365\n" + "Average of measurements is 164.9830\n" ] } ], @@ -787,7 +787,7 @@ "name": "stdout", "output_type": "stream", "text": [ - "Average of measurements is 164.9493\n" + "Average of measurements is 164.9658\n" ] } ], @@ 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write code that uses `gen_data` and `g_h_filter` to filter 100 data points that starts at 5, has a derivative of 2, a noise scaling factor of 10, and uses g=0.2 and h=0.02. Set you initial guess for x to be 100." + "Now write code that uses `gen_data` and `g_h_filter` to filter 100 data points that starts at 5, has a derivative of 2, a noise scaling factor of 1, and uses g=0.2 and h=0.05. Set your initial guess for x to be 100." ] }, { @@ -9059,551 +9144,410 @@ { "data": { "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAugAAAFpCAYAAADOXt1wAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", - "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3Xd4FNXixvHvbnonCaRA6IQOkV6kN6VIURTBAtarV7li\n", - "49quXdSfXi/2dkUE5CIo2Oi9hRaQTugdEkIICQmpu/v7Y2hL2ibZFOD9PE+e2cycOXMG54lvTs6c\n", - "Y7LZbDZERERERKRCMJd3A0RERERE5DIFdBERERGRCkQBXURERESkAlFAFxERERGpQBTQRUREREQq\n", - "ENfybsCVkpOTy7sJIiIiIiLFFhAQUOI61IMuIiIiIlKBKKCLiIiIiFQgFWqIy5Wc8ecBuT7FxMQA\n", - "0Lp163JuiVRUekbEEXpOxBF6Tm4wOTlQuzYcOwZbtkDz5g6d5uxh2upBFxEREZEb2759cOQIuLrC\n", - "4MHGvs8/L7fmKKCLiIiIyI3tX/+CWrVg0iR44glj35QpkJRULs1RQBcRERGRG1dSEsyaZXzu1g0a\n", - "NoReveD8efj++3JpkgK6iIiIiNy4/vc/yMw0QnmNGsa+J580tp9/DlZrmTepwr4kKiIiIiJS6iZM\n", - "MLYPPHB534AB8PDDcPfdYDKVeZMU0EVERETkxrRtG2zcCAEBl18OBXBxgW+/LbdmKaCLiIiIyI0p\n", - "IACeeQa8vIyvCkIBXURERERuTDVqwL//Xd6tyEUviYqIiIiIVCAFBvR3332XNm3aEBAQQEhICAMH\n", - 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"plot_g_h_results(measurements=zs, filtered_data=data)" ] }, @@ -9666,965 +9610,1192 @@ "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAvMAAAFpCAYAAAAREKdvAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", - "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3XdYFOf2B/DvLr0rSrMBKooVsfeuicYaY+xeUzTR9Gba\n", - "NfGmJz9vYoxJboomGnuMJrETFQUUO4oKFlRQRFARKdJ39/fHYdid7Qtb4Xyex2dhdnb2ZV3YM++c\n", - "9xyJQqFQgDHGGGOMMeZwpLYeAGOMMcYYY6xmOJhnjDHGGGPMQXEwzxhjjDHGmIPiYJ4xxhhjjDEH\n", - "xcE8Y4wxxhhjDsrZ1gOoqfz8fFsPgTHGGGOMsRrz8/Or9TF4Zp4xxhhjjDEHxcE8Y4wxxhhjDsph\n", - "02xUmeMSBaubTpw4AQDo3r27jUfC7Bm/T5gx+H3CDOH3iJkcOQL06QP06AEcO2br0ZiduVPFeWae\n", - "McYYY4zZj+xsug0Ksu04HAQH84wxxhhjzH4IwXxwsG3H4SA4mGeMMcYYY/bDywvo3BmIiLD1SBwC\n", - "B/OMMcYYY8x+zJoF/PADcOECsHy5rUdj9ziYZ4wxxhhj9iUzE/jlF2DfPluPxO5xMM8YY4wxxuyL\n", - "kC+fk2PbcTgADuYZY4wxxph9ESrZCIthmU4czDPGGGOMMfvCM/NG42CeMcYYY4zZh7IyIDYWyMoC\n", - "PDyA4mKgqMjWo7JrdaIDLGOMMcYYqwOuXweGDgXCw4Eff6Qylc4crurDrw5jjDHGGLMPQlpNcDAw\n", - 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We not only filter out noise, but legitimate changes in the signal as well. \n", "\n", - "Maybe we need a 'Goldilocks' filter, where is not too large, not too small, but just right? Well, not exactly. As alluded to earlier, different filters choose g and h in different ways depending on the mathematical properties of the problem. For example, the Benedict-Bordner filter was invented to minimize the transient error in this example, where $\\dot{x}$ makes a step jump. We will not discuss this filter in this book, but here are two plots chosen with different allowable pairs of g and h. This filter design minimizes transient errors for step jumps in $\\dot{x}$ at the cost of not being optimal for other types of changes in $\\dot{x}$." + "Maybe we need a 'Goldilocks' filter, where is not too large, not too small, but just right? Well, not exactly. As alluded to earlier, different filters choose g and h in different ways depending on the mathematical properties of the problem. For example, the Benedict-Bordner filter was invented to minimize the transient error in this example, where $\\dot{x}$ makes a step jump. We will not discuss this filter in this book, but here are two plots chosen with different allowable pairs of g and h for that filter design. This filter design minimizes transient errors for step jumps in $\\dot{x}$ at the cost of not being optimal for other types of changes in $\\dot{x}$." ] }, { @@ -14380,415 +14798,398 @@ "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAtIAAAF9CAYAAAApyI8MAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", - "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3XlYVGX/BvD7DPu+CsoOCiIIKuKWabiDGKZlhZVgvWFZ\n", - "vVrZruHrW9qq2c9KrTS0enNNcd9wyxVUzF3cUEFRWWXYZ87vj1FynAEGmGFguD/XxQU9yzlfYILb\n", - "w3OeI4iiKIKIiIiIiOpFou8CiIiIiIhaIgZpIiIiIqIGYJAmIiIiImoABmkiIiIiogZgkCYiIiIi\n", - "agAGaSIiIiKiBmCQJiJqBaZPnw6JRIKkpCR9l0JEZDAYpInI4MhkMsydOxehoaGwtLSEk5MToqOj\n", - "ceDAgXodZ8+ePXjhhRfQuXNnODk5wdzcHL6+voiOjsb69etrnFdaWorp06ejY8eOsLCwgKurK555\n", - "5hmcPXtWZWxeXh5+/vlnPPnkkwgICICVlRXs7OzQu3dvfPvtt6iqqqr3518bQRC0ejxdyc/Px5tv\n", - "vgkfHx+Ym5vD3d0dL730ErKysup9rPp8PwDAx8cHEolE7Vu7du00Oud///vf6jlbtmypd81E1DIY\n", - "67sAIiJtEkURsbGxWLlyJQIDA/HGG28gNzcXy5YtQ//+/bFq1SrExMRodKydO3di586d6N27NwYP\n", - 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We know that h affects how much of we favor the measurement of $\\dot{x}$ vs our prediction. But what does this *mean*? If our signal is changing a lot (quickly relative to the time step of our filter), then a large $h$ will cause us to react to those transient changes rapidly. A smaller $h$ will cause us to react more slowly.\n", + "Now let's leave g unchanged and investigate the effect of modifying h. We know that h affects how much we favor the measurement of $\\dot{x}$ vs our prediction. But what does this *mean*? If our signal is changing a lot (quickly relative to the time step of our filter), then a large $h$ will cause us to react to those transient changes rapidly. A smaller $h$ will cause us to react more slowly.\n", "\n", "We will look at three examples. We have a noiseless measurement that slowly goes from 0 to 1 in 50 steps. Our first filter uses a nearly correct initial value for $\\dot{x}$ and a small $h$. You can see from the output that the filter output is very close to the signal. The second filter uses the very incorrect guess of $\\dot{x}=2$. Here we see the filter 'ringing' until it settles down and finds the signal. The third filter uses the same conditions but it now sets $h=0.5$. If you look at the amplitude of the ringing you can see that it is much smaller than in the second chart, but the frequency is greater. It also settles down a bit quicker than the second filter, though not by much." ] @@ -15262,430 +15643,413 @@ "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAt8AAAF9CAYAAADcNgS8AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", - "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3Xl8VPW9//H3TPaEEBJCWAOEfd+ysEMgEBZFUTa1Vwva\n", - "WqnVqm3dbn/VWmuvbbXeuqD3tooblcguW0jYSQLZ2HcIO0kgLEkIWWfO7w8uUZoZCJCcbK/n4+Hj\n", - "oed8zsx74hHeHs58j8UwDEMAAAAAqp21pgMAAAAADQXlGwAAADAJ5RsAAAAwCeUbAAAAMAnlGwAA\n", - "ADAJ5RsAAAAwCeUbQL3Svn17hYSE1HQM01it1gb1eQGgrqN8A6h3LBZLTUco98UXX2jgwIHy9fVV\n", - "kyZNNGrUKK1YsaJK36M2fd5bWbFihUaNGiU/Pz/5+vpq0KBB+uKLL+7otZKSknTPPfcoICBA3t7e\n", - "6tu3r/77v/9bdru9wuzcuXNltVqd/vXJJ5/c7UcDgEpxrekAAFBf/eY3v9E777yj4OBgPfnkkyou\n", - "LtY333yjSZMm6f3339fTTz9d0xFN9eGHH+qZZ55RYGCgHnvsMbm7u+vbb7/VzJkztXv3bv3lL3+p\n", - "9GstXbpUU6ZMkbe3t2bMmKGAgAAtW7ZMzz//vBISEhQTE+PwuMjISEVGRlbYHh4efqcfCwBui4Un\n", - "XAKoT9q3by+r1aqMjIwazZGYmKhhw4apU6dOSklJkZ+fnyTpxIkTCg0NVUFBgQ4cOKB27drd1ftY\n", - 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We will eventually learn that the Kalman filter is optimal from a least squared fit perspective under certain conditions. However, `polyfit()` fits a polynomial to the data, not an arbitrary curve, by minimizing the value of this formula:\n", "$$E = \\sum_{j=0}^k |p(x_j) - y_j|^2$$\n", "\n", - "I assumed that my weight gain was constant at 1 lb/day, and so when I tried to fit a polynomial of $n=1$, which is a line, the result very closely matched the actual weight gain. But, of course, no one consistently only gains or loses weight. We fluctuate. Using 'polyfit()' for a longer series of data would yield poor results. In contrast, the g-h filter reacts to changes in the rate - the $h$ term controls how quickly the filter reacts to these changes. If we gain weight, hold steady for awhile, then lose weight, the filter will track that change automatically. 'polyfit()' would not be able to do that unless the gain and loss could be well represented by a polynomial.\n", + "I assumed that my weight gain was constant at 1 lb/day, and so when I tried to fit a polynomial of $n=1$, which is a line, the result very closely matched the actual weight gain. But, of course, no one consistently only gains or loses weight. Our weights fluctuate. Using 'polyfit()' for a longer series of data, such as a year's worth of weighings, would yield poor results. In contrast, the g-h filter reacts to changes in the rate - the $h$ term controls how quickly the filter reacts to these changes. If we gain weight, hold steady for awhile, then lose weight, the filter will track that change automatically. 'polyfit()' would not be able to do that unless the gain and loss could be well represented by a polynomial.\n", "\n", "Another advantage of this form of filter, even if the data fits a *n*-degree polynomial, is that it is *recursive*. That is, we can compute the estimate for this time period knowing nothing more than the estimate and rate from the last time period. In contrast, if you dig into the implementation for `polyfit()` you will see that it needs all of the data before it can produce an answer. Therefore algorithms like `polyfit()` are not well suited for real-time data filtering. In the 60's when the Kalman filter was developed computers were very slow and had extremely limited memory. They were utterly unable to store, for example, thousands of readings from an aircraft's inertial navigation system, nor could they process all of that data in the short period of time needed to provide accurate and up-to-date navigation information. \n", "\n", diff --git a/02_Discrete_Bayes.ipynb b/02_Discrete_Bayes.ipynb index d03dfb1..130d997 100644 --- a/02_Discrete_Bayes.ipynb +++ b/02_Discrete_Bayes.ipynb @@ -290,7 +290,7 @@ "\n", "When I begin listening to the sensor I have no reason to believe that Simon is at any particular position in the hallway. He is equally likely to be in any position. The probability that he is in each position is therefore 1/10. \n", "\n", - "Let us represent our belief of his position at any time in a numpy array." + "Let us represent our belief of his position at any time in a NumPy array." ] }, { @@ -484,7 +484,7 @@ "WLWwiMcZ3gAAAABJRU5ErkJggg==\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -505,7 +505,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "We call this a multimodal distribution because we have multiple beliefs about the position of our dog. Of course we are not saying that we think he is simultaneously in three different locations, merely that so far we have narrowed down our knowledge in his position to these locations. \n", + "We call this a multimodal distribution because we have multiple beliefs about the position of our dog. Of course we are not saying that we think he is simultaneously in three different locations, merely that so far we have narrowed down our knowledge in his position to be one of these three locations. \n", "\n", "I hand coded the `pos_belief` array in the code above. How would we implement this in code? Well, hallway represents each door as a 1, and wall as 0, so we will multiply the hallway variable by the percentage, like so;" ] @@ -549,7 +549,7 @@ " * door\n", " \n", "\n", - "Can we deduce where Simon is at the end of that sequence? Of course! Given the hallway's layout there is only one place where you can be in front of a door, move once to the right, and be in front of another door, and that is at the left end. Therefore we can confidently state that Simon is in front of the second doorway. If this is not clear, suppose Simon had started at the second or third door. After moving to the right, his sensor would have returned 'wall'. Therefore the only possibility is that he is now in front of the second door. We denote this in Python with:" + "Can we deduce where Simon is at the end of that sequence? Of course! Given the hallway's layout there is only one place where you can be in front of a door, move once to the right, and be in front of another door, and that is at the left end. Therefore we can confidently state that Simon is in front of the second doorway. If this is not clear, suppose Simon had started at the second or third door. After moving to the right, his sensor would have returned 'wall'. That doesn't match the sensor readings, so we know he didn't start there. We can continue with that logic for all the remaining starting positions. Therefore the only possibility is that he is now in front of the second door. We denote this in Python with:" ] }, { @@ -576,7 +576,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Obviously I carefully constructed the hallway layout and sensor readings to give us an exact answer quickly. Real problems will not be so clear cut. But this should trigger your intuition - the first sensor reading only gave us very low probabilities (0.333) for Simon's location, but after a position update and another sensor reading we knew much more about where he is. You might suspect, correctly, that if you had a very long hallway with a large number of doors that after several sensor readings and positions updates we would either be able to know where Simon was, or have the possibilities narrowed down to a small number of possibilities. For example, suppose we had a long sequence of \"door, right, door, right, wall, right, wall, right, door, right, door, right, wall, right, wall, right, wall, right, wall, right, door\". Simon could only be located where we had a sequence of [1,1,0,0,1,1,0,0,0,0,1] in the hallway. There might be only one match for that, or at most a few. Either way we will be far more certain about his position then when we started.\n", + "Obviously I carefully constructed the hallway layout and sensor readings to give us an exact answer quickly. Real problems will not be so clear cut. But this should trigger your intuition - the first sensor reading only gave us very low probabilities (0.333) for Simon's location, but after a position update and another sensor reading we knew much more about where he is. You might suspect, correctly, that if you had a very long hallway with a large number of doors that after several sensor readings and positions updates we would either be able to know where Simon was, or have the possibilities narrowed down to a small number of possibilities. For example, suppose we had a long sequence of \"door, right, door, right, wall, right, wall, right, door, right, door, right, wall, right, wall, right, wall, right, wall, right, door\". Simon could only have started in a location where his movements had a door sequence of [1,1,0,0,1,1,0,0,0,0,1] in the hallway. There might be only one match for that, or at most a few. Either way we will be far more certain about his position then when we started.\n", "\n", "We could work through the code to implement this solution, but instead let us consider a real world complication to the problem." ] @@ -596,7 +596,7 @@ "\n", "At first this may seem like an insurmountable problem. If the sensor is noisy it casts doubt on every piece of data. How can we conclude anything if we are always unsure?\n", "\n", - "The key, as with the problem above, is probabilities. We are already comfortable with assigning a probabilistic belief about the location of the dog; now we just have to incorporate the additional uncertainty caused by the sensor noise. Say we think there is a 50% chance that our dog is in front of a specific door and we get a reading of 'door'. Well, we think that is only likely to be true 0.6 of the time, so we multiply: $0.5 * 0.6= 0.3$. Likewise, if we think the chances that our dog is in front of a wall is 0.1, and the reading is 'door', we would multiply the probability by the chances of a miss: $0.1 * 0.2 = 0.02$.\n", + "The key, as with the problem above, is probabilities. We are already comfortable with assigning a probabilistic belief about the location of the dog; now we just have to incorporate the additional uncertainty caused by the sensor noise. Say we think there is a 50% chance that our dog is in front of a specific door and then we get a reading of 'door'. Well, we think that is only likely to be true 0.6 of the time, so we multiply: $0.5 * 0.6= 0.3$. Likewise, if we think the chances that our dog is in front of a wall is 0.1, and the reading is 'door', we would multiply the probability by the chances of a miss: $0.1 * 0.2 = 0.02$.\n", "\n", "However, we more or less chose 0.6 and 0.2 at random; if we multiply the `pos_belief` array by these values the end result will no longer represent a true probability distribution. " ] @@ -764,7 +764,7 @@ "FtXJCJBlAAAAAElFTkSuQmCC\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -959,7 +959,7 @@ "FtPch8/CAAAAAElFTkSuQmCC\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -1201,7 +1201,7 @@ "ACQirgEAIBFxDQAAiYhrAABIRFwDAEAi4hoAABL5fwpmJuNqHGHfAAAAAElFTkSuQmCC\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -1247,7 +1247,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "We want to solve real world problems, and we have already stated that all sensors have noise. Therefore the code above must be wrong. What if the sensor reported that our dog moved one space, but he actually moved two spaces, or zero? Once again this may initially sound like an insurmountable problem, but let's just model it in math. Since this is just an example, we will create a pretty simple noise model for the sensor - later in the book we will handle far more sophisticated errors.\n", + "We want to solve real world problems, and we have already stated that all sensors have noise. Therefore the code above must be wrong since it assumes perfect measurements. What if the sensor reported that our dog moved one space, but he actually moved two spaces, or zero? Once again this may initially sound like an insurmountable problem, but let's just model it in math. Since this is just an example, we will create a pretty simple noise model for the sensor - later in the book we will handle far more sophisticated errors.\n", "\n", "We will say that when the sensor sends a movement update, it is 80% likely to be right, and it is 10% likely to overshoot one position to the right, and 10% likely to undershoot to the left. That is, if we say the movement was 4 (meaning 4 spaces to the right), the dog is 80% likely to have moved 4 spaces to the right, 10% to have moved 3 spaces, and 10% to have moved 5 spaces.\n", "\n", @@ -1405,7 +1405,7 @@ "DQAAiYhrAABIRFwDAEAi4hoAABIR1wAAkIi4BgCARP4/nnsht+19KnYAAAAASUVORK5CYII=\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -1440,7 +1440,8 @@ "cell_type": "code", "execution_count": 11, "metadata": { - "collapsed": false + "collapsed": false, + "scrolled": true }, "outputs": [ { @@ -1588,7 +1589,7 @@ "iGsAAEhEXAMAQCLiGgAAEhHXAACQiLgGAIBExDUAACTy/wCvjTrhgem4kgAAAABJRU5ErkJggg==\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -1756,7 +1757,7 @@ "iGsAAEhEXAMAQCLiGgAAEhHXAACQiLgGAIBExDUAACTy/wCvjTrhgem4kgAAAABJRU5ErkJggg==\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -1771,7 +1772,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "This is not a coincidence, or the result of a carefully chosen example - it is always true of the update step. This is inevitable; if our sensor is noisy we will lose a bit of information on every update. Suppose we were to perform the update an infinite number of times - what would the result be? If we lose information on every step, we must eventually end up with no information at all, and our probabilities will be equally distributed across the `pos_belief` array. Let's try this with say 500 iterations.\n" + "This is not a coincidence, or the result of a carefully chosen example - it is always true of the predict step. This is inevitable; if our sensor is noisy we will lose a bit of information on every prediction. Suppose we were to perform the prediction an infinite number of times - what would the result be? If we lose information on every step, we must eventually end up with no information at all, and our probabilities will be equally distributed across the `pos_belief` array. Let's try this with 500 iterations.\n" ] }, { @@ -1935,7 +1936,7 @@ "/wDmJxpYHvwEtQAAAABJRU5ErkJggg==\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -1977,13 +1978,13 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "The problem of losing information during an update may make it seem as if our system would quickly devolve into no knowledge. However, recall that our process is not an endless series of updates, but of *measure->update->measure->update->measure->update...* The output of the measure step is fed into the update. The update step, with a degraded certainty, is then fed into the measure step. \n", + "The problem of losing information during a prediction may make it seem as if our system would quickly devolve into no knowledge. However, recall that our process is not an endless series of predictions, but of *update->predict->update->predict->update->predict->update...* The output of the update step, where we measure the current position, is fed into the prediction. The prediction step, with a degraded certainty, is then fed back into the update step where we measure the position again.\n", "\n", - "Let's think about this intuitively. After the first measure->update round we have degraded the knowledge we gained by the measurement by a small amount. But now we take another measurement. When we try to incorporate that new measurement into our belief, do we become more certain, less certain, or equally certain. Consider a simple case - you are sitting in your office. A co-worker asks another co-worker where you are, and they report \"in his office\". You keep sitting there while they ask and answer \"has he moved\"? \"No\" \"Where is he\" \"In his office\". Eventually you get up and move, and lets say the person didn't see you move. At that time the questions will go \"Has he moved\" \"no\" (but you have!) \"Where is he\" \"In the kitchen\". Wow! At that moment the statement that you haven't moved conflicts strongly with the next measurement that you are in the kitchen. If we were modeling these with probabilities the probability that you are in your office would lower, and the probability that you are in the kitchen would go up a little bit. But now imagine the subsequent conversation: \"has he moved\" \"no\" \"where is he\" \"in the kitchen\". Pretty quickly the belief that you are in your office would fade away, and the belief that you are in the kitchen would increase to near certainty. The belief that you are in the office will never go to zero, nor will the belief that you are in the kitchen ever go to 1.0 because of the chances of error, but in practice your co-workers would be correct to be quite confident in their system.\n", + "Let's think about this intuitively. After the first update->predict round we have degraded the knowledge we gained by the measurement by a small amount. But now we take another measurement. When we try to incorporate that new measurement into our belief, do we become more certain, less certain, or equally certain? Consider a simple case - you are sitting in your office. A co-worker asks another co-worker where you are, and they report \"in his office\". You keep sitting there while they ask and answer \"has he moved\"? \"No\" \"Where is he\" \"In his office\". Eventually you get up and move, and lets say the person didn't see you move. At that time the questions will go \"Has he moved\" \"no\" (but you have!) \"Where is he\" \"In the kitchen\". Wow! At that moment the statement that you haven't moved conflicts strongly with the next measurement that you are in the kitchen. If we were modeling these with probabilities the probability that you are in your office would lower, and the probability that you are in the kitchen would go up a little bit. But now imagine the subsequent conversation: \"has he moved\" \"no\" \"where is he\" \"in the kitchen\". Pretty quickly the belief that you are in your office would fade away, and the belief that you are in the kitchen would increase to near certainty. The belief that you are in the office will never go to zero, nor will the belief that you are in the kitchen ever go to 1.0 because of the chances of error, but in practice your co-workers would be correct to be quite confident in their system.\n", "\n", "That is what intuition tells us. What does the math tell us?\n", "\n", - "Well, we have already programmed the measure step, and we have programmed the update step. All we need to do is feed the result of one into the other, and we will have programmed our dog tracker!!! Let's see how it performs. We will input data as if the dog started at position 0 and moved right at each update. However, as in a real world application, we will start with no knowledge and assign equal probability to all positions. " + "Well, we have already programmed the update step, and we have programmed the predict step. All we need to do is feed the result of one into the other, and we will have programmed our dog tracker!!! Let's see how it performs. We will input measurements as if the dog started at position 0 and moved right at each update. However, as in a real world application, we will start with no knowledge and assign equal probability to all positions. " ] }, { @@ -2169,7 +2170,7 @@ "QmCC\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -2186,7 +2187,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "So after the first update we have assigned a high probability to each door position, and a low probability to each wall position. The update step shifted these probabilities to the right, smearing them about a bit. Now let's look at what happens at the next sense." + "So after the first update we have assigned a high probability to each door position, and a low probability to each wall position. The predict step shifted these probabilities to the right, smearing them about a bit. Now let's look at what happens at the next sense." ] }, { @@ -2353,7 +2354,7 @@ "YII=\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -2530,7 +2531,7 @@ "kMj/Azv6KDQ33AZlAAAAAElFTkSuQmCC\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -2708,7 +2709,7 @@ "SOT/ATXCA84OAAAAA0lEQVQfKKaMWAKYAAAAAElFTkSuQmCC\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -2749,7 +2750,7 @@ "source": [ "You may be suspicious of the results above because I always passed correct sensor data into the functions. However, we are claiming that this code implements a *filter* - it should filter out bad sensor measurements. Does it do that?\n", "\n", - "To make this easy to program and visualize I will change the layout of the hallway to mostly alternating doors and hallways:" + "To make this easy to program and visualize I will change the layout of the hallway to mostly alternating doors and hallways, and run the algorithm on 5 correct measurements:" ] }, { @@ -2915,7 +2916,7 @@ "rkJggg==\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -3096,7 +3097,7 @@ "gETENQAAJPL/AKVaFt8zsqsBAAAAAElFTkSuQmCC\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -3583,7 +3584,7 @@ "RArh4pqIiIiISCFcXBMRERERKYSLayIiIiIihfx/3pDuDjpTJXEAAAAASUVORK5CYII=\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -3622,9 +3623,9 @@ "\n", "With that said, while this filter is used in industry, it is not used often because it has several limitations. Getting around those limitations is the motivation behind the chapters in the rest of this book.\n", "\n", - "The first problem is scaling. Our dog tracking problem used only one variable, $pos$, to denote the dog's position. Most interesting problems will want to track several things in a large space. Realistically, at a minimum we would want to track our dogs $(x,y)$ coordinate, and probably his velocity $(\\dot{x},\\dot{y})$ as well. We have not covered the multidimensional case, but instead of a histogram we use a multidimensional grid to store the probabilities at each discrete location. Each *sense()* and *update()* step requires updating all values in the grid, so a simple four variable problem would require $O(n^4)$ running time *per time step*. Realistic filters have 10 or more variables to track, leading to exorbitant computation requirements.\n", + "The first problem is scaling. Our dog tracking problem used only one variable, $pos$, to denote the dog's position. Most interesting problems will want to track several things in a large space. Realistically, at a minimum we would want to track our dogs $(x,y)$ coordinate, and probably his velocity $(\\dot{x},\\dot{y})$ as well. We have not covered the multidimensional case, but instead of a histogram we use a multidimensional grid to store the probabilities at each discrete location. Each `update()` and `predict()` step requires updating all values in the grid, so a simple four variable problem would require $O(n^4)$ running time *per time step*. Realistic filters can have 10 or more variables to track, leading to exorbitant computation requirements.\n", "\n", - "The second problem is that 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", + "The second problem is that 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 update and predict 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", "A third problem is that the histogram is multimodal. This is not always a problem - an entire class of filters, the particle filters, are multimodal and are often used because of this property. But imagine if the GPS in your car reported to you that it is 40% sure that you are on D street, but 30% sure you are on Willow Avenue. I doubt that you would find that useful. Also, GPSs report their error - they might report that you are at $(109.878W, 38.326N)$ with an error of $9m$. There is no clear mathematical way to extract error information from a histogram. Heuristics suggest themselves to be sure, but there is no exact determination. You may or may not care about that while driving, but you surely do care if you are trying to send a rocket to Mars or track and hit an oncoming missile.\n", "\n", @@ -3787,7 +3788,7 @@ "rkJggg==\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -3806,7 +3807,7 @@ "source": [ " 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 (my dog Simon certainly tries - his food bowl and my lap often have equal allure to him). We would have to use heuristics to decide how to interpret this distribution, and there is usually no satisfactory answer. This is not always a weakness - a considerable amount of literature has been written on *Multi-Hypothesis Tracking (MHT)*. We cannot always distill our knowledge to one conclusion, and MHT uses various techniques to maintain multiple story lines at once, using backtracking schemes to go *back in time* to correct hypothesis once more information is known. This will be the subject of later chapters. In other cases we truly have a multimodal situation - we may be optically tracking pedestrians on the street and need to represent all of their positions. \n", " \n", - "In practice it is the exponential increase in computation time that leads to the discrete Bayes filter being the least frequently used of all filters in this book. Many problems are best formulated as discrete or multimodal, but we have other filter choices with better performance. With that said, if I had a small problem that this technique could handle I would choose to use it; it is trivial to implement, debug, and understand, all virtues in my book." + "In practice it is the exponential increase in computation time that leads to the discrete Bayes filter being the least frequently used of all filters in this book. Many problems are best formulated as discrete or multimodal, but we have other filter choices with better performance. With that said, if I had a small problem that this technique could handle I would choose to use it; it is trivial to implement, debug, and understand, all virtues." ] }, { @@ -3832,9 +3833,9 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "The code is very small, but the result is huge! We will go into the math more later, but we have implemented a form of a Bayesian filter. It is commonly called a Histogram filter. The Kalman filter is also a Bayesian filter, and uses this same logic to produce its results. The math is a bit more complicated, but not by much. For now, we will just explain that Bayesian statistics compute the likelihood of the present based on the past. If we know there are two doors in a row, and the sensor reported two doors in a row, it is likely that we are positioned near those doors. Bayesian statistics just formalize that example, and Bayesian filters formalize filtering data based on that math by implementing the sense->update->sense->update process. \n", + "The code is very small, but the result is huge! We will go into the math more later, but we have implemented a form of a Bayesian filter. It is commonly called a Histogram filter. The Kalman filter is also a Bayesian filter, and uses this same logic to produce its results. The math is a bit more complicated, but not by much. For now, we will just explain that Bayesian statistics compute the likelihood of some estimate about the present based on imperfect knowledge of the past. If we know there are two doors in a row, and the sensor reported two doors in a row, it is likely that we are positioned near those doors. Bayesian statistics just formalize that example, and Bayesian filters formalize filtering data based on that math by implementing the predict->update->predict->update process. \n", "\n", - "We have learned how to start with no information and derive information from noisy sensors. Even though our sensors are very noisy (most sensors are more than 80% accurate, for example) we quickly converge on the most likely position for our dog. We have learned how the update step always degrades our knowledge, but the addition of another measurement, even when it might have noise in it, improves our knowledge, allowing us to converge on the most likely result.\n", + "We have learned how to start with no information and derive information from noisy sensors. Even though the sensors in this chapter are very noisy (most sensors are more than 80% accurate, for example) we quickly converge on the most likely position for our dog. We have learned how the predict step always degrades our knowledge, but the addition of another measurement, even when it might have noise in it, improves our knowledge, allowing us to converge on the most likely result.\n", "\n", "If you followed the math carefully you will realize that all of this math is exact. The bar charts that we are displaying are not an *estimate* or *guess* - they are mathematically exact results that exactly represent our knowledge. The knowledge is probabilistic, to be sure, but it is exact, and correct.\n", "\n", @@ -3891,7 +3892,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.4.2" + "version": "3.4.3" } }, "nbformat": 4, diff --git a/04_Gaussians.ipynb b/04_Gaussians.ipynb index 1ed8ba9..d226259 100644 --- a/04_Gaussians.ipynb +++ b/04_Gaussians.ipynb @@ -269,7 +269,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "The last chapter ended by discussing some of the drawbacks of the Discrete Bayesian filter. For many tracking and filtering problems our desire is to have a filter that is *unimodal* and *continuous*. That is, we want to model our system using floating point math (continuous) and to have only one belief represented (unimodal). For example, we want to say an aircraft is at (12.34, -95.54, 2389.5) where that is latitude, longitude, and altitude. We do not want our filter to tell us \"it *might be* at (1.65, -78.01, 2100.45) or it *might be* at (34.36, -98.23, 2543.79).\" That doesn't match our physical intuition of how the world works, and as we discussed, it is prohibitively expensive to compute. \n", + "The last chapter ended by discussing some of the drawbacks of the Discrete Bayesian filter. For many tracking and filtering problems our desire is to have a filter that is *unimodal* and *continuous*. That is, we want to model our system using floating point math (continuous) and to have only one belief represented (unimodal). For example, we want to say an aircraft is at (12.34, -95.54, 2389.5) where that is latitude, longitude, and altitude. We do not want our filter to tell us \"it *might be* at (1.65, -78.01, 2100.45) or it *might be* at (34.36, -98.23, 2543.79).\" That doesn't match our physical intuition of how the world works, and as we discussed, it can be prohibitively expensive to compute the multimodal case. \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." ] @@ -282,15 +282,15 @@ "\n", "### Random Variables\n", "\n", - "To understand Gaussians we first need to understand a few simple mathematical definitions and computations\n", + "To understand Gaussians we first need to understand a few simple mathematical definitions and computations.\n", "\n", "To understand Gaussians we first need to understand a few simple mathematical computations. We start with a *random variable* x. Maybe x is the height of the students in a class in meters. We call $x$ *random* because the height of one student does not depend on the height of another student - each is independent. For a specific classrom perhaps the heights are\n", "\n", "$$x= [1.8, 2.0, 1.7, 1.9, 1.6]$$\n", "\n", - "Another example of a random variable would be the result of a coin toss. Each coin toss is independent from all other tosses. Rolling a die would be another example. \n", + "Another example of a random variable would be the result of a coin toss. Each coin toss is independent from all other tosses. Rolling a die would be another example of a random variable.\n", "\n", - "The coin toss and die roll are examples of *discrete random variables*. That is, the outcome of any given event comes from a discrete set of values. The die roll can never produce a value of 7 or 3.24, for example. In contrast, the student heights are continuous; they can take on any value (within biological limits, naturally). For example, heights of 1.7, 1.71, 1.711, 1.7111, 1.71111,.... are all possible. Do not confuse the *measurement* of the random variable with the actual value. If we can only measure the height of a person to 0.1 meters we would only record values from 0.1, 0.2, 0.3...2.7, or 27 discrete choices. Nonetheless the heights can vary between any arbitrary real value between those ranges, and so it is continuous. \n", + "The coin toss and die roll are examples of *discrete random variables*. That is, the outcome of any given event comes from a discrete set of values. The roll of a six sided die can never produce a value of 7 or 3.24, for example. In contrast, the student heights are continuous; they can take on any value (within biological limits, naturally). For example, heights of 1.7, 1.71, 1.711, 1.7111, 1.71111,.... are all possible. Do not confuse the *measurement* of the random variable with the actual value. If we can only measure the height of a person to 0.1 meters we would only record values from 0.1, 0.2, 0.3...2.7, or 27 discrete choices. Nonetheless the heights can vary between any arbitrary real value between those ranges, and so it is continuous. \n", "\n", "### The Mean of a Random Variable\n", "\n", @@ -302,7 +302,7 @@ "\n", "$$\\mu_x = 1.8$$\n", "\n", - "We can formalize this computation with the equation\n", + "where the subscript *x* denotes that this is the mean for the vector $\\mathbf{x}$. We can formalize this computation with the equation\n", "\n", "$$ \\mu_x = \\frac{1}{n}\\sum^n_{i=1} x_i$$\n", "\n", @@ -393,7 +393,7 @@ "where $\\sigma$ is the notation for the standard deviation.\n", "\n", "\n", - "Let's just work through that with the data from three classes. We just subtract the mean of x from each value of x, square it, take the average of those, and then take the square root of the result. The mean of $[1.8, 2.0, 1.7, 1.9, 1.6]$ is 1.8, so we compute the standard deviation as\n", + "If this is the first time you have seen this it may not have a lot of meaning for you. But let's just work through that with the data from three classes to be sure we understand the formula. We just subtract the mean of x from each value of x, square it, take the average of those, and then take the square root of the result. The mean of $[1.8, 2.0, 1.7, 1.9, 1.6]$ is 1.8, so we compute the standard deviation as\n", "\n", "$$ \n", "\\begin{aligned}\n", @@ -432,7 +432,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "What does this number *mean* (no pun intended)? *How much* is not a mathematical term. We will be able to define it much more precisely once we introduce the concept of a Gaussian in the next section. For now I'll say that for many things in nature, including the height of people, 68% of all values lie within one standard deviation of the mean. From this we can conclude that for a random class 68% of the students will have heights between 1.66 meters and 1.94 meters. \n", + "What does this number *mean* (no pun intended)? *How much* in \"how much do the heights vary\" is not a mathematical term. We will be able to define it much more precisely once we introduce the concept of a Gaussian in the next section. For now I'll say that for many things in nature, including the height of people, 68% of all values lie within one standard deviation of the mean. In other words we can conclude that for a random class 68% of the students will have heights between 1.66 (1.8-0.1414) meters and 1.94 (1.8+0.1414) meters. \n", "\n", "We can look at this in a plot:" ] @@ -640,7 +640,7 @@ "AxHkAQAAAAMR5AEAAAADEeQBAAAAAxHkAQAAAAP9P3lqIgAYgrVHAAAAAElFTkSuQmCC\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -671,7 +671,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "For only 5 students we obviously will not get exactly 68% within one standard deviation (normally denoted as $1\\sigma$), but we do see that 3 out of 5 students are within $1\\sigma$, or 60%, which is as close as you can get to 68% with only 5 samples. I haven't yet introduced enough math or Python for you to fully understand the next bit of code, but let's look at the results if have a class with 100 students." + "For only 5 students we obviously will not get exactly 68% within one standard deviation (normally denoted as $1\\sigma$), but we do see that 3 out of 5 students are within $1\\sigma$, or 60%, which is as close as you can get to 68% with only 5 samples. I haven't yet introduced enough math or Python for you to fully understand the next bit of code, but let's look at the results for a class with 100 students." ] }, { @@ -685,225 +685,226 @@ "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAvQAAAGHCAYAAADfgsbkAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", - "AAALEgAACxIB0t1+/AAAIABJREFUeJzt3X9cVPWC//H3DPgDTck0kEQFexjX36lghZpayY21tUfZ\n", - "D8XN8GbeWjTF7dZqllqWm27sqollN5Fc29RHrt7auqlpmle7FzKqxdS6Jtq1wRUFF79pVzjfP7pO\n", - "jjDDDDI/PjOv5+PB48Gc8zkz7xk+HN4czhxslmVZAgAAAGAke7ADAAAAAGg8Cj0AAABgMAo9AAAA\n", - "YDAKPQAAAGAwCj0AAABgMAo9AAAAYDAKPQAAAGCwoBb6BQsWKC0tTbGxsYqLi9Po0aNVWlrqcZvD\n", - "hw/LbrfX+di8eXOAUgMAAAChI6iFfseOHZoyZYr27Nmjbdu2KTo6WrfddptOnTrV4LYffPCBHA6H\n", - "82PEiBEBSAwAAACEFlso/afYM2fOKDY2Vps2bdKoUaPqHXP48GF169ZNRUVFGjhwYIATAgAAAKEl\n", - "pM6hP336tGpra9WuXbsGx959992Kj4/XkCFD9PbbbwcgHQAAABB6QuoI/X333ac///nPKi4uls1m\n", - "q3dMRUWF3njjDQ0ePFjR0dHatGmTnn/+eRUWFmr8+PEBTgwAAAAEV8gU+hkzZmjdunXatWuXkpKS\n", - 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"Many texts alternatively use *VAR(x)* to denote the variance of x, and I opted to use that here just to expose you to that convention." ] }, { @@ -1134,7 +1135,7 @@ "AQCkEBkAQAqRAQCk+C9H4O5405jIdgAAAABJRU5ErkJggg==\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -1161,9 +1162,9 @@ "\n", "$$\\frac{(3-0) + (-3-0) + (3-0) + (-3-0)}{4} = 0$$\n", "\n", - "This is clearly incorrect. \n", + "This is clearly incorrect, as there is more than 0 variance in the data. \n", "\n", - "Maybe we can use the absolute value? We can see by inspection that the result is $12/4=3$ which is certainly correct. But what if we change $x=[6, -2, -3, 1]$? In this case we get $12/4=3$. X is clearly more spread out, but we get the same variance, so this cannot be correct. If we use the correct formula we get a variance of 3.5, which correctly reflects the larger variatin in $x$.\n", + "Maybe we can use the absolute value? We can see by inspection that the result is $12/4=3$ which is certainly correct - each value varies by 3 from the mean. But what if we change $x=[6, -2, -3, 1]$? In this case we get $12/4=3$. $x$ is clearly more spread out than in the last example, but we get the same variance, so this cannot be correct. If we use the correct formula we get a variance of 3.5, which correctly reflects the larger variatin in $x$.\n", "\n", "This is clearly not a proof of correctness; indeed, Gauss recognized that is is somewhat arbitrary. If the variable is distributed normally we can prove that the variance is computed by this function. If there are outliers, meaning values that are not normally distributed, than squaring the difference gives disproportinate weight to that term. For example, let's see what happens if we have $x = [1,-1,1,-2,3,2,100]$. I'll use NumPy's `var()` method to compute the variance for me." ] @@ -1192,7 +1193,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Is this *correct*? You tell me. Without the outlier of 100 we get $\\sigma^2=2.89$, which accurately reflects how $x$ absent the outlier is varying. The one outlier swamps the computation. I will not continue down this path; if you are interested you might want to look at the work that James Berger. For Kalman filters, which assume Gaussian distributions, we can prove that squaring the terms produces optimal results. More about that soon." + "Is this *correct*? You tell me. Without the outlier of 100 we get $\\sigma^2=2.89$, which accurately reflects how $x$ absent the outlier is varying. The one outlier swamps the computation. I will not continue down this path; if you are interested you might want to look at the work that James Berger has done on this problem. For Kalman filters, which assume Gaussian distributions, we can prove that squaring the terms produces optimal results. More about that soon." ] }, { @@ -1208,14 +1209,14 @@ "source": [ "We are now ready to learn about Gaussians. Let's remind ourselves of the motivation for this chapter.\n", "\n", - ">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." + ">We desire a unimodal, continuous way to represent probabilities that models how the real world works, and that is very computationally efficient to calculate." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "Before we go into the math, lets just look at a graph of the Gaussian distribution to get a sense of what we are talking about." + "Before we go into the math, lets just look at a graph of a Gaussian distribution to get a sense of what we are talking about." ] }, { @@ -1228,472 +1229,462 @@ { "data": { "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAukAAAGHCAYAAAAN0sy8AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", - "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3XdUVHf+PvBnGBiqoChNUOy9AhZQxALYNWpsGI3GbKIx\n", - "iWVN/Jp1s5uvbrL+3JiYRI27MZbEghuxEwUrFlBRIVijoojCoCKCdJiZ3x/5ZvQTmijwmRme1zme\n", - "s/PMvXeeybk7vBluUeh0Oh2IiIiIiMhgmMkuQEREREREIg7pREREREQGhkM6EREREZGB4ZBORERE\n", - 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variance=15**2, xlabel='IQ')" + "plot_gaussian(mean=1.8, variance=0.1414**2, xlabel='Student Height')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "Probably this is immediately recognizable to you as a 'bell curve'. This curve is ubiquitous because under real world conditions most observations are distributed in such a manner. In fact, this is the bell curve for IQ (Intelligence Quotient). You've probably seen this before, and understand it. It tells us that the average IQ is 100, and that the number of people that have IQs higher or lower than that drops off as they get further away from 100. It's hard to see the exact number, but we can see that very few people have an IQ over 150 or under 50, but a lot have an IQ of 90 or 110. \n", + "Probably this is immediately recognizable to you as a 'bell curve'. This curve is ubiquitous because under real world conditions most observations are distributed in such a manner. In fact, this is the bell curve the student heights given earlier. \n", "\n", - "This curve is not unique to IQ distributions - a vast amount of natural phenomena exhibits this sort of distribution, including the sensors that we use in filtering problems. As we will see, it also has all the attributes that we are looking for - it represents a unimodal belief or value as a probability, it is continuous, and it is computationally efficient. We will soon discover that it also other desirable qualities that we do not yet recognize we need." + "This curve is not unique to heights - a vast amount of natural phenomena exhibits this sort of distribution, including the sensors that we use in filtering problems. As we will see, it also has all the attributes that we are looking for - it represents a unimodal belief or value as a probability, it is continuous, and it is computationally efficient. We will soon discover that it also other desirable qualities that we do not yet recognize we need." ] }, { @@ -2222,7 +2213,7 @@ "90ccRAAAAABJRU5ErkJggg==\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -2239,9 +2230,7 @@ "source": [ "The y-axis depicts the *probability density* - the relative amount of cars that are going the speed at the corresponding x-axis.\n", "\n", - "> *Random variable* will be precisely defined later. For now just think of it as a variable that can 'freely' and 'randomly' vary. A dog's position in a hallway, air temperature, and a drone's height above the ground are all random variables. The position of the North Pole is not, nor is a sin wave (a sin wave is anything but 'free').\n", - "\n", - "You may object that human IQs or automobile speeds cannot be less than zero, let alone $-\\infty$. This is true, but this is a common limitation of mathematical modeling. \"The map is not the territory\" is a common expression, and it is true for Bayesian filtering and statistics. The Gaussian distribution above very closely models the distribution of IQ test results, but being a model it is necessarily imperfect. The difference between model and reality will come up again and again in these filters. \n", + "You may object that human heights or automobile speeds cannot be less than zero, let alone $-\\infty$. This is true, but this is a common limitation of mathematical modeling. \"The map is not the territory\" is a common expression, and it is true for Bayesian filtering and statistics. The Gaussian distribution above very closely models the distribution of the measured automobile speeds, but being a model it is necessarily imperfect. The difference between model and reality will come up again and again in these filters. \n", "\n", "You will see these distributions called *Gaussian distributions*, *normal distributions*, and *bell curves*. Bell curve is ambiguous because there are other distributions which also look bell shaped but are not Gaussian distributions, so we will not use it further in this book. But *Gaussian* and *normal* both mean the same thing, and are used interchangeably. I will use both throughout this book as different sources will use either term, and so I want you to be used to seeing both. Finally, as in this paragraph, it is typical to shorten the name and just talk about a *Gaussian* or *normal* - these are both typical shortcut names for the *Gaussian distribution*. " ] @@ -2277,8 +2266,8 @@ " \"\"\"returns normal distribution for x given a \n", " gaussian with the specified mean and variance. \n", " \"\"\"\n", - " return math.exp((-0.5*(x-mean)**2)/var) / \\\n", - " math.sqrt(_two_pi*var)" + " return np.exp((-0.5*(x-mean)**2)/var) / \\\n", + " np.sqrt(_two_pi*var)" ] }, { @@ -2726,7 +2715,7 @@ "RK5CYII=\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -3435,7 +3424,7 @@ "4AAAAABJRU5ErkJggg==\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -3460,9 +3449,7 @@ "source": [ "So what is this telling us? The blue gaussian is very narrow. It is saying that we believe $x=23$, and that we are very sure about that. In contrast, the red gaussian also believes that $x=23$, but we are much less sure about that. Our believe that $x=23$ is lower, and so our belief about the likely possible values for $x$ is spread out - we think it is quite likely that $x=20$ or $x=26$, for example. The blue gaussian has almost completely eliminated $22$ or $24$ as possible value - their probability is almost $0\\%$, whereas the red curve considers them nearly as likely as $23$.\n", "\n", - "If we think back to the thermometer, we can consider these three curves as representing the readings from three different thermometers. The blue curve represents a very accurate thermometer, and the red one represents a fairly inaccurate one. Green of course represents one in between the two others. Note the very powerful property the Gaussian distribution affords us - we can entirely represent both the reading and the error of a thermometer with only two numbers - the mean and the variance.\n", - "\n", - "The standard notation for a normal distribution for a random variable $X$ is just $X \\sim\\ \\mathcal{N}(\\mu,\\sigma^2)$ where $\\mu$ is the mean and $\\sigma^2$ is the variance. It may seem odd to use $\\sigma$ squared - why not just $\\sigma$? We will not go into great detail about the math at this point, but in statistics $\\sigma$ is the *standard deviation* of a normal distribution. *Variance* is defined as the square of the standard deviation, hence $\\sigma^2$.\n", + "If we think back to the thermometer, we can consider these three curves as representing the readings from three different thermometers. The blue curve represents a very accurate thermometer, and the black one represents a fairly inaccurate one. Green of course represents one in between the two others. Note the very powerful property the Gaussian distribution affords us - we can entirely represent both the reading and the error of a thermometer with only two numbers - the mean and the variance.\n", "\n", "It is worth spending a few words on standard deviation now. The standard deviation is a measure of how much variation from the mean exists. For Gaussian distributions, 68% of all the data falls within one standard deviation($1\\sigma$) of the mean, 95% falls within two standard deviations ($2\\sigma$), and 99.7% within three ($3\\sigma$). This is often called the 68-95-99.7 rule. So if you were told that the average test score in a class was 71 with a standard deviation of 9.4, you could conclude that 95% of the students received a score between 52.2 and 89.8 if the distribution is normal (that is calculated with $71 \\pm (2 * 9.4)$). " ] @@ -3839,7 +3826,7 @@ "fyIiIiKiMsDgT0RERERUBhj8iYiIiIjKAIM/EREREVEZ+B/H1tUZSdQuawAAAABJRU5ErkJggg==\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -4224,7 +4211,7 @@ "iIiIyOdYaBARERERkc+x0CAiIiIiIp9joUFERERERD73Px0L4HlryPN+AAAAAElFTkSuQmCC\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -4272,10 +4259,10 @@ "f(x, \\mu, \\sigma) = c_1 * e^{-c_2{(x-\\mu)^2} }\n", "$$\n", "\n", - "Let's just simplify it further by assuming all the constants are one (not true for Gaussians, but the general shape will be the same).\n", + "Let's just simplify it further by assuming the constants are one(not true for Gaussians, but the general shape will be the same) and the mean is zero.\n", "\n", "$$ \n", - "f(x, \\mu, \\sigma) = e^{-x^2}\n", + "f(x)=e^{-x^2}\n", "$$\n", "\n", "Raising a value to a negative power is the same as the reciprocal of the value raised to a positive power. I.e.\n", @@ -4664,7 +4651,7 @@ "iBSPjQsRERERESne/wN8nGNQWDoh4wAAAABJRU5ErkJggg==\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -5083,7 +5070,7 @@ "M4YFCBERERERzRgWIERERERENGP+H+6pG+1HBqllAAAAAElFTkSuQmCC\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -5100,7 +5087,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "We know that $1/e^x = e^{-x}, so we can equivalently calculate" + "We know that $1/e^x = e^{-x}$, so we can equivalently calculate" ] }, { @@ -5495,7 +5482,7 @@ "iIiIiGbN/wNdf7Zt9OdzAwAAAABJRU5ErkJggg==\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -5973,7 +5960,7 @@ "kUo0rwAAAABJRU5ErkJggg==\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -6571,7 +6558,7 @@ "CD61GQAAAABJRU5ErkJggg==\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -6595,11 +6582,11 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Recall how our discrete Bayesian filter worked. We had a vector implemented as a numpy array representing our belief at a certain moment in time. When we performed another measurement using the `update()` function we had to multiply probabilities together, and when we performed the motion step using the `predict()` function we had to shift and add probabilities. I've promised you that the Kalman filter uses essentially the same process, and that it uses Gaussians instead of histograms, so you might reasonable expect that we will be multiplying, adding, and shifting Gaussians in the Kalman filter.\n", + "Recall how our discrete Bayesian filter worked. We had a vector implemented as a NumPy array representing our belief at a certain moment in time. When we performed another measurement using the `update()` function we had to multiply probabilities together, and when we performed the motion step using the `predict()` function we had to shift and add probabilities. I've promised you that the Kalman filter uses essentially the same process, and that it uses Gaussians instead of histograms, so you might reasonable expect that we will be multiplying, adding, and shifting Gaussians in the Kalman filter.\n", "\n", "A typical textbook would directly launch into a multi-page proof of the behavior of Gaussians under these operations, but I don't see the value in that right now. I think the math will be much more intuitive and clear if we just start developing a Kalman filter using Gaussians. I will provide the equations for multiplying and shifting Gaussians at the appropriate time. You will then be able to develop a physical intuition for what these operations do, rather than be forced to digest a lot of fairly abstract math.\n", "\n", - "The key point, which I will only assert for now, is that all the operations are very simple, and that they preserve the properties of the Gaussian. This is somewhat remarkable, in that the Gaussian is a nonlinear function, and typically if you multiply a nonlinear equation with itself you end up with a different equation. For example, the shape of `sin(x)sin(x)` is very different from `sin(x)`. But the result of multiplying two Gaussians is yet another Gaussian. This is a fundamental property, and a key reason why Kalman filters are possible." + "The key point, which I will only assert for now, is that all the operations are very simple, and that they preserve the properties of the Gaussian. This is somewhat remarkable, in that the Gaussian is a nonlinear function, and typically if you multiply a nonlinear equation with itself you end up with a different equation. For example, the shape of `sin(x)sin(x)` is very different from `sin(x)`. But the result of multiplying two Gaussians is yet another Gaussian. This is a fundamental property, and a key reason why Kalman filters are computationally feasible." ] }, { @@ -6613,7 +6600,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "In this chapter I have used by custom written code for computing Gaussians, plotting, and so on. I chose to do that to give you a chance to look at the code and see how these functions are implemented. However, Python comes with \"batteries included\" as the saying goes, and it comes with a wide range of statistics functions in the module `scipy.stats`. I find the performance of some of the functions rather slow (the scipy.stats documentation contains a warning to this effect), but this is offset by the fact that this is standard code available to everyone, and it is well tested. So let's walk through how to use scipy.stats to compute various things.\n", + "In this chapter I have used custom code for computing Gaussians, plotting, and so on. I chose to do that to give you a chance to look at the code and see how these functions are implemented. However, Python comes with \"batteries included\" as the saying goes, and it comes with a wide range of statistics functions in the module `scipy.stats`. I find the performance of some of the functions rather slow (the scipy.stats documentation contains a warning to this effect), but this is offset by the fact that this is standard code available to everyone, and it is well tested. So let's walk through how to use scipy.stats to compute various things.\n", "\n", "The scipy.stats module contains a number of objects which you can use to compute attributes of various probability distributions. The full documentation for this module is here: http://http://docs.scipy.org/doc/scipy/reference/stats.html. However, we will focus on the norm variable, which implements the normal distribution. Let's look at some code that uses `scipy.stats.norm` to compute a Gaussian, and compare its value to the value returned by the `gaussian()` function." ] @@ -6691,9 +6678,9 @@ "name": "stdout", "output_type": "stream", "text": [ - "[ 3.77999795 -0.18956982 2.69579025 2.56248639 4.57891564 3.00244391\n", - " 3.0723681 7.75724739 -0.25014797 0.85577169 3.94158436 5.14455528\n", - " 6.62881127 1.93136557 1.61332113]\n" + "[ 1.27260045 4.2238164 -2.42780752 1.88734677 8.10098857 6.17644384\n", + " 2.94186917 -1.4252601 2.73244031 2.65367357 1.35429458 1.26983037\n", + " 4.86749902 -0.23136419 2.56313679]\n" ] } ], @@ -6778,7 +6765,7 @@ "source": [ "Earlier I spoke very briefly about the *central limit theorem*, which states that under certain conditions the arithmetic sum of **any** independent random variables will be normally distributed, regardless of how the random variables are distributed. This is extremely important for (at least) two reasons. First, nature is full of distributions which are not normal, but when we apply the central limit theorem over large populations we end up with normal distributions. Second, Gaussians are mathematically *tractable*. We will see this more as we develop the Kalman filter theory, but there are very nice closed form solutions for operations on Gaussians that allow us to use them analytically.\n", "\n", - "However, a key part of the proof is \"under certain conditions\". These conditions often do not hold for the physical world. The resulting distributions are called *fat tailed*. Let's consider a trivial example. We think of things like test scores as being normally distributed. If you have ever had a professor 'grade on a curve' you have been subject to this assumption. But of course test scores cannot follow a normal distribution. This is because the distribution assigns a nonzero probability distribution for *any* value, no matter how far from the mean. So, for example, say your mean is 90 and the standard deviation is 13. The normal distribution assumes that there is a large chance of somebody getting a 90, and a small chance of somebody getting a 40. However, it also assumes that there is a tiny chance of somebody getting a -10, or of 150. It assumes that there is a infinitesimal chance of getting a score of -1e300, or 4e50. In other words, the *tails* of the distribution are infinite. \n", + "However, a key part of the proof is \"under certain conditions\". These conditions often do not hold for the physical world. The resulting distributions are called *fat tailed*. Let's consider a trivial example. We think of things like test scores as being normally distributed. If you have ever had a professor 'grade on a curve' you have been subject to this assumption. But of course test scores cannot follow a normal distribution. This is because the distribution assigns a nonzero probability distribution for *any* value, no matter how far from the mean. So, for example, say your mean is 90 and the standard deviation is 13. The normal distribution assumes that there is a large chance of somebody getting a 90, and a small chance of somebody getting a 40. However, it also implies that there is a tiny chance of somebody getting a -10, or of 150. It assumes that there is a infinitesimal chance of getting a score of -1e300, or 4e50. In other words, the *tails* of the distribution are infinite. \n", "\n", "But for a test we know this is not true. You cannot get less than 0, or more than 100. Let's plot this using a normal distribution." ] @@ -7155,7 +7142,7 @@ "IiIiIiKfxMkKERERERH5JE5WiIiIiIjIJ/0XkMJRB/+WilkAAAAASUVORK5CYII=\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -7175,7 +7162,7 @@ "source": [ "So the problem with this chart is that the area under the drawn curve cannot equal 1, so it is not a probability distribution. What actually happens is that a few more students get scores nearer the upper end of the range (for example), and that tail becomes 'fat'. Also, the test is probably not able to perfectly distinguish incredibly minute differences in skill in the students, so the distribution to the left of the mean is also probably a bit bunched up in places. The resulting distribution is called a *fat tail distribution*.\n", "\n", - "We are not going to be applying Kalman filters to grades of student papers, but we are using physical sensors to measure the world. The errors in sensor's measurements are rarely truly Gaussian. And, of course, the signal itself will have time dependent correlations. For example, consider a sensor mounted on a helicopter. The rotating blades cause vibrations in the air frame, which will cause small deviations in the flight path. This deviation will be related to the frequency of the vibration, hence the values at t and t+1 are not *independent*. The vibration will also effect sensors such as the INS. No physical object can react instantly to changes, so the INS will somewhat lag the vibrations in a perhaps quite complex way, and so the noise introduced by the vibration will be anything but independent. \n", + "We are not going to be applying Kalman filters to grades of student papers, but we are using physical sensors to measure the world. The errors in sensor's measurements are rarely truly Gaussian. And, of course, the signal itself will have time dependent correlations. For example, consider a sensor mounted on a helicopter. The rotating blades cause vibrations in the air frame, which will cause small deviations in the flight path. This deviation will be related to the frequency of the vibration, hence the values at t and t+1 are not *independent*. The vibration will effect sensors such as the INS. No physical object can react instantly to changes, so the INS will somewhat lag the vibrations in a perhaps quite complex way, and so the noise introduced by the vibration will be anything but independent. \n", "\n", "It is far too early to be talking about the difficulties that this presents to the Kalman filter designer. It is worth keeping in the back of your mind the fact that the Kalman filter math is based on a somewhat idealized model of the world. For now I will present a bit of code that I will be using later in the book to form fat tail distributions to simulate various processes and sensors. 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"nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.4.2" + "version": "3.4.3" } }, "nbformat": 4, diff --git a/05_Kalman_Filters.ipynb b/05_Kalman_Filters.ipynb index 5dc0029..a69d4cb 100644 --- a/05_Kalman_Filters.ipynb +++ b/05_Kalman_Filters.ipynb @@ -282,7 +282,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "As in the histogram chapter we will be tracking a dog in a long hallway at work. However, in our latest hackathon someone created an RFID tracker that provides a reasonable accurate position for our dog. Suppose the hallway is 100m long. The sensor returns the distance of the dog from the left end of the hallway. So, 23.4 would mean the dog is 23.4 meters from the left end of the hallway.\n", + "As in the histogram chapter we will be tracking a dog in a long hallway at work. However, in our latest hackathon someone created an RFID tracker that provides a reasonably accurate position for our dog. Suppose the hallway is 100m long. The sensor returns the distance of the dog from the left end of the hallway. So, 23.4 would mean the dog is 23.4 meters from the left end of the hallway.\n", "\n", "Naturally, the sensor is not perfect. A reading of 23.4 could correspond to a real position of 23.7, or 23.0. However, it is very unlikely to correspond to a real position of say 47.6. Testing during the hackathon confirmed this result - the sensor is reasonably accurate, and while it had errors, the errors are small. Furthermore, the errors seemed to be evenly distributed on both sides of the measurement; a true position of 23m would be equally likely to be measured as 22.9 as 23.1.\n", "\n", @@ -346,10 +346,10 @@ "name": "stdout", "output_type": "stream", "text": [ - " 0.4293\t-2.4606\t 0.5406\t-1.0854\t 0.8824\t\n", - " 0.0225\t-1.0454\t 0.7260\t-0.2236\t-0.4063\t\n", - " 0.1890\t 1.3218\t-0.2226\t 1.7665\t-1.4184\t\n", - "-0.5187\t 0.1318\t-0.9789\t 0.0092\t 0.0671\t\n" + "-0.4386\t-0.1743\t-0.7493\t-2.0268\t-0.0036\t\n", + "-0.0845\t 0.6183\t 0.6811\t 2.4236\t-0.2784\t\n", + "-1.5221\t-0.7595\t 0.5254\t 2.8830\t-1.2800\t\n", + " 0.4620\t-0.2230\t-1.0736\t-0.6136\t 0.9257\t\n" ] } ], @@ -744,7 +744,7 @@ "WkRERETEA1SsRUREREQ84H8Buu08rSD7Bp4AAAAASUVORK5CYII=\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -787,592 +787,617 @@ "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAvIAAAGUCAYAAABX3saWAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", - 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The numbers I see as I write this are unlikely to be the ones that you see. If you run the cell above multiple times, you should get a slightly different result each time. I could use `numpy.random.seed(some_value)` to force the results to be the same each time. This would simplify my explanations in some cases, but would ruin the interactive nature of this chapter. To get a real feel for how normal distributions and Kalman filters work you will probably want to run cells several times, observing what changes, and what stays roughly the same.\n", + "> **Note**: NumPy uses a random number generator to generate the normal distribution samples. The numbers I see as I write this are unlikely to be the ones that you see. If you run the cell above multiple times, you should get a slightly different result each time. I could use `numpy.random.seed(some_value)` to force the results to be the same each time. This would simplify my explanations in some cases, but would ruin the interactive nature of this chapter. To get a real feel for how normal distributions and Kalman filters work you will probably want to run cells several times, observing what changes, and what stays roughly the same.\n", "\n", - "So the output of the sensor should be a wavering blue line drawn over a dotted red line. The dotted red line shows the actual position of the dog, and the blue line is the noise signal produced by the simulated RFID sensor. Please note that the red dotted line was manually plotted - we do not yet have a filter that recovers that information! \n", + "So the output of the sensor should be a wavering dotted red line drawn over a straight green line. The green line shows the actual position of the dog, and the dotted red line is the noisy signal produced by the simulated RFID sensor. Please note that the red dotted line was manually plotted - we do not yet have a filter that recovers that information! \n", "\n", - "If you are running this in an interactive IPython Notebook, I strongly urge you to run the script several times in a row. You can do this by putting the cursor in the cell containing the Python code and pressing CTRL+Enter. Each time it runs you should see a different jagged blue line wavering over the top of the dotted red line.\n", + "If you are running this in an interactive IPython Notebook, I strongly urge you to run the script several times in a row. You can do this by putting the cursor in the cell containing the Python code and pressing CTRL+Enter. Each time it runs you should see a different sensor output.\n", "\n", "I also urge you to adjust the noise setting to see the result of various values. However, since you may be reading this in a read only notebook, I will show several examples. The first plot shows the noise set to 100.0, and the second shows noise set to 0.5." ] @@ -1427,837 +1452,911 @@ "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAvIAAAGUCAYAAABX3saWAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", - "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3XdcU9f7B/DPTSCEKQKiDJmioAiyHKiIDK0KLhx1o37r\n", - "z7q1ddbWUVerVu34qv3Wulptq1hpqyKIExxFceKeiAgoggoyk/P745pISIAAQUCf9+uVV+Dec889\n", - "SS6XJ+ee+xyOMcZACCGEEEIIqVcEtd0AQgghhBBCSOVRIE8IIYQQQkg9RIE8IYQQQggh9RAF8oQQ\n", - "QgghhNRDFMgTQgghhBBSD1EgTwghhBBCSD1EgTwhdcjChQshEAhw7Nix2m4KIaQe8ff3h0BA/9IJ\n", - "ed/QXz0hdQjHcfIHqb6UlBQsXboUAwcORLNmzSAQCCAQCHDz5s1yt8vLy8PChQvRokUL6OrqonHj\n", - "xhg8eDCuX79e7r7Gjh0LS0tLiMVi2NvbY/r06cjOzq50u7OysjB9+nTY2dlBLBbDysoKY8eOxaNH\n", - "jypdF3k/0HlD86pyHiiLRCLBunXr4ObmBj09PZiamqJXr144depUDbScvE84mhCKkLojMzMTmZmZ\n", - "aNq0KXR1dWu7OfXe3r170b9/fwgEAtjb2yMzMxPPnz/HtWvX0Lx5c5XbFBQUICgoCPHx8fDx8UFA\n", - "QACSk5Oxa9cuiEQiHD58GG3btlXY5u7du/D19UVGRgb69u0LZ2dnnDlzBkeOHEGLFi0QHx8PExMT\n", - 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While we believe that the dog is at 23, note that roughly 21 to 25 are quite likely as well. Let's assume for the moment our dog is standing still, and we query the sensor again. This time it returns 23.2 as the position. Can we use this additional information to improve our estimate of the dog's position?\n", + "Notice how this relates to the bar charts from the Discrete Bayes chapter. In that chapter we drews bars of various heights to depict the probability that the dog was at any given position. In many cases those bars took on a shape very similar to this chart. The differences are that the bars depict the probability of a discrete position, whereas this chart depicts the probability distribution of a continuous range of positions. \n", + "\n", + "This graph corresponds to a fairly inexact belief. While we believe that the dog is at 23, note that roughly speaking positions 21 to 25 are quite likely as well. Let's assume for the moment our dog is standing still, and we query the sensor again. This time it returns 23.2 as the position. Can we use this additional information to improve our estimate of the dog's position?\n", "\n", "Intuition suggests 'yes'. Consider: if we read the sensor 100 times and each time it returned a value between 21 and 25, all centered around 23, we should be very confident that the dog is somewhere very near 23. Of course, a different physical interpretation is possible. Perhaps our dog was randomly wandering back and forth in a way that exactly emulated a normal distribution. But that seems extremely unlikely - I certainly have never seen a dog do that. So the only reasonable assumption is that the dog was mostly standing still at 23.0.\n", "\n", @@ -5389,7 +5501,7 @@ "hG8AAADAIoRvAAAAwCKEbwAAAMAihG8AAADAIoRvAAAAwCL/Azk+Ei0MTpsAAAAAAElFTkSuQmCC\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -5449,7 +5561,7 @@ "$$\n", "N(\\mu_1, \\sigma_1^2)*N(\\mu_2, \\sigma_2^2) = N(\\frac{\\sigma_1^2 \\mu_2 + \\sigma_2^2 \\mu_1}{\\sigma_1^2 + \\sigma_2^2},\\frac{1}{\\frac{1}{\\sigma_1^2} + \\frac{1}{\\sigma_2^2}}) $$ \n", "\n", - "In other words the result is a Gaussian with \n", + "In other words the result of multiplying two Gaussians is a Gaussian with \n", "\n", "$$\\begin{aligned}\n", "\\mu &=\\frac{\\sigma_1^2 \\mu_2 + \\sigma_2^2 \\mu_1} {\\sigma_1^2 + \\sigma_2^2}, \\\\\n", @@ -6083,7 +6195,7 @@ "oAshhBBCCGFC/h8i6kd9xLYtrgAAAABJRU5ErkJggg==\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -6948,7 +7060,7 @@ "kiRJsiEyQZckSZIkSZIkG/L/dpZzTgabjwgAAAAASUVORK5CYII=\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -7545,7 +7657,7 @@ "1OruAAAAAElFTkSuQmCC\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -7624,7 +7736,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Recall the histogram filter uses a numpy array to encode our belief about the position of our dog at any time. That array stored our belief of our dog's position in the hallway using 10 discrete positions. This was very crude, because with a 100m hallway that corresponded to positions 10m apart. It would have been trivial to expand the number of positions to say 1,000, and that is what we would do if using it for a real problem. But the problem remains that the distribution is discrete and multimodal - it can express strong belief that the dog is in two positions at the same time.\n", + "Recall the histogram filter uses a NumPy array to encode our belief about the position of our dog at any time. That array stored our belief of our dog's position in the hallway using 10 discrete positions. This was very crude, because with a 100m hallway that corresponded to positions 10m apart. It would have been trivial to expand the number of positions to say 1,000, and that is what we would do if using it for a real problem. But the problem remains that the distribution is discrete and multimodal - it can express strong belief that the dog is in two positions at the same time.\n", "\n", "Therefore, we will use a single Gaussian to reflect our current belief of the dog's position. In other words, we will use $dog_{pos} = \\mathcal{N}(\\mu,\\sigma^2)$. Gaussians extend to infinity on both sides of the mean, so the single Gaussian will cover the entire hallway. They are unimodal, and seem to reflect the behavior of real-world sensors - most errors are small and clustered around the mean. Here is the entire implementation of the update function for a Kalman filter:" ] @@ -7645,7 +7757,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Kalman filters are supposed to be hard! But this is very short and straightforward. All we are doing is multiplying the Gaussian that reflects our belief of where the dog was with the new measurement. Perhaps this would be clearer if we used more specific names:\n", + "Kalman filters are supposed to be hard! But this is very short and straightforward. All we are doing is multiplying the Gaussian that reflects our belief of where the dog is with the new measurement. Perhaps this would be clearer if we used more specific names:\n", "\n", " def update_dog(dog_pos, dog_variance, measurement, measurement_variance):\n", " return multiply(dog_pos, dog_sigma, measurement, measurement_variance)" @@ -7729,7 +7841,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "That is a beautiful result, but it is not yet a filter. We assumed that the dog was sitting still, an extremely dubious assumption. Certainly it is a useless one - who would need to write a filter to track non-moving objects? The histogram used a loop of sense and update functions, and we must do the same to accommodate movement.\n", + "That is a beautiful result, but it is not yet a filter. We assumed that the dog was sitting still, an extremely dubious assumption. Certainly it is a useless one - who would need to write a filter to track non-moving objects? The histogram used a loop of predict and update functions, and we must do the same to accommodate movement.\n", "\n", "How how do we perform the predict function with Gaussians? Recall the histogram method:\n", "\n", @@ -7752,7 +7864,7 @@ "$$ N({\\mu}_1, {{\\sigma}_1}^2)+N({\\mu}_2, {{\\sigma}_2}^2) = N({\\mu}_1 + {\\mu}_2, {{\\sigma}_1}^2 + {{\\sigma}_2}^2)$$\n", "\n", "All we do is add the means and the variance separately! Does that make sense? Think of the physical representation of this abstract equation.\n", - "${\\mu}_1$ is the old position, and ${\\mu}_2$ is the distance moved. Surely it makes sense that our new position is ${\\mu}_1 + {\\mu}_2$. What about the variance? It is perhaps harder to form an intuition about this. However, recall that with the `update()` function for the histogram filter we always lost information - our confidence after the update was lower than our confidence before the update. Perhaps this makes sense - we don't really know where the dog is moving, so perhaps the confidence should get smaller (variance gets larger). I assure you that the equation for Gaussian addition is correct, and derived by basic algebra. Therefore it is reasonable to expect that if we are using Gaussians to model physical events, the results must correctly describe those events.\n", + "${\\mu}_1$ is the old position, and ${\\mu}_2$ is the distance moved. Surely it makes sense that our new position is ${\\mu}_1 + {\\mu}_2$. What about the variance? It is perhaps harder to form an intuition about this. However, recall that with the `predict()` function for the histogram filter we always lost information - our confidence after the update was lower than our confidence before the update. Perhaps this makes sense - we don't really know where the dog is moving, so perhaps the confidence should get smaller (variance gets larger). I assure you that the equation for Gaussian addition is correct, and derived by basic algebra. Therefore it is reasonable to expect that if we are using Gaussians to model physical events, the results must correctly describe those events.\n", "\n", "I recognize the amount of hand waving in that argument. Now is a good time to either work through the algebra to convince yourself of the mathematical correctness of the algorithm, or to work through some examples and see that it behaves reasonably. This book will do the latter.\n", "\n", @@ -7805,585 +7917,601 @@ "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAtYAAAFpCAYAAAC4ZG/7AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", - "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3Xd4FNX6wPHvbnojlQRSDBKKVDH03nsTqQEEBMTCBbwW\n", - "FPX+EGkXUYSLFStKkSLSO1KkJqH3TggtkArpZef3x8jKkgQSsslkk/fzPHl2OTM78+6wSd6cOec9\n", - "OkVRFIQQQgghhBAFotc6ACGEEEIIIUoCSayFEEIIIYQwA0mshRBCCCGEMANJrIUQQgghhDADSayF\n", - "EEIIIYQwA+uiOlFCQkJRnUoIIYQQQgizc3V1feR26bEWQgghhBDCDCSxFkIIIYQQwgyKbCjIgx7X\n", - "jS4gPDwcgHr16mkcieWQa5ZHJ09CrVoYrKw4sXw5tXv21DoiiyGfsfyR65V/cs3yR65X/sk1y7/8\n", - "DGeWHmshSpsPPwRFIbpXL9L9/LSORgghhCgxJLEWojQJDYWVK8HBgZvDh6ttcXGwfLm2cQkhhBAl\n", - "gCZDQYQQGpk0SX0cO5YMLy/IyoJnnoHbt+H0afW5EEIIIZ6I9FgLUZp88w2MHg3jx6v/trKCrl3V\n", - "54sXaxeXEEIIUQJIYi1EaRIQAF98AR4e/7SFhKiPixeDomgTlxBCCFECSGItRGnXujV4e8P583Do\n", - "kNbRCCGEEBZLxlgLUdpZW0O/fmpP9qJFULeu1hEJIYo5g8FAenq61mEQGBgIQGpqqsaRWA65ZqZs\n", - 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -20358,7 +21802,7 @@ "IvKx/w8OGRGpOUpLbgAAAABJRU5ErkJggg==\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, diff --git a/pdf/readme.txt b/pdf/readme.txt new file mode 100644 index 0000000..2aec683 --- /dev/null +++ b/pdf/readme.txt @@ -0,0 +1,22 @@ +This directory contains code to convert the book into the PDF file. The normal +build process is to cd into this directory, and run buil_book from the command +line. If the build is successful (no errors printed), then run clean_book from +the command line. clean_book is not run automatically because if there is an +error you probably need to look at the intermediate output to debug the issue. + +I build the PDF my merging all of the notebooks into one huge one. I strip out +the initial cells for the book formatting and table of contents, and do a few +other things so it renders well in PDF. + +There is some code to do the same from Windows (.bat files), but they are now +a bit out of date. + +There is also some experimental code to convert to html. + +The files with short in the name combine only a couple of notebooks together. +I use this to test the production without having to wait the relatively long +time required to produce the entire book. Mostly this is for testing the +scripts. + +No one but me should need to run this stuff, but if you fork the project and +want to generate a PDF, this is how you do it.