Read Online Now

Online PDF Version

Introductory textbook for Kalman filters and Bayesian filters. All code is written in Python, and the book itself is written in IPython Notebook so that you can run and modify the code in the book in place, seeing the results inside the book. What better way to learn?

Reading Online

You may access this book via nbviewer at any time by using this address: Read Online Now

The quickest way to get starting with reading the book is to read it online using the link above. The book is written as a collection of IPython Notebooks, an interactive, browser based system that allows you to combine text, Python, and math into your brower. The website http://nbviewer.org provides an IPython Notebook server that renders notebooks stored at github (or elsewhere). The rendering is done in real time when you load the book. If you read my book today, and then I make a change tomorrow, when you go back tomorrow you will see that change. Perhaps more importantly, the book uses animations to demonstrate how the algorithms perform over time. The PDF version of the book, discussed in the next paragraph, cannot show the animations.

The preface available from the link above has all the information in this README and more, so feel free to follow the link now.

I periodically generate a PDF of the book from the Notebooks. I do not do this for every check in, so the PDF will usually lag the content in github and on nbviewer.org. However, I do generate it whenever I make a substantial change. Of course, you will not be able to run and modify the code in the notebooks, nor will you be able to see the animations.

PDF Version of the book

Companion Software

All of the filters used in this book as well as others not in this book are implemented in my Python library FilterPy, available here. You do not need to download or install this to read the book, but you will likely want to use this library to write your own filters. It includes Kalman filters, Fading Memory filters, H infinity filters, Extended and Unscented filters, least square filters, and many more. It also includes helper routines that simplify the designing the matrices used by some of the filters, and other code such as Kalman based smoothers.

Downloading the book

However, this book is intended to be interactive and I recommend using it in that form. If you install IPython on your computer and then clone this book you will be able to run all of the code in the book yourself. You can perform experiments, see how filters react to different data, see how different filters react to the same data, and so on. I find this sort of immediate feedback both vital and invigorating. You do not have to wonder "what happens if". Try it and see!

The github pages for this project are at https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python You can clone it to your hard drive with the command

git clone https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python.git

This will create a directory named Kalman-and-Bayesian-Filters-in-Python. Navigate to the directory, and run IPython notebook with the command

ipython notebook

This will open a browswer window showing the contents of the base directory. The book is organized into chapters. To read Chapter 2, click on the link for chapter 2. This will cause the browwer to open that subdirectory. In each subdirectory there will be one or more IPython Notebooks (all notebooks have a .ipynb file extension). The chapter contents are in the notebook with the same name as the chapter name. There are sometimes supporting notebooks for doing things like generating animations that are displayed in the chapter. These are not intended to be read by the end user, but of course if you are curious as to how an animation is made go ahead and take a look.

This is admittedly a somewhat cumbersome interface to a book; I am following in the footsteps of several other projects that are somewhat repurposing IPython Notebook to generate entire books. I feel the slight annoyances have a huge payoff - instead of having to download a separate code base and run it in an IDE while you try to read a book, all of the code and text is in one place. If you want to alter the code, you may do so and immediately see the effects of your change. If you find a bug, you can make a fix, and push it back to my repository so that everyone in the world benefits. And, of course, you will never encounter a problem I face all the time with traditional books - the book and the code are out of sync with each other, and you are left scratching your head as to which source to trust.

Version 0.0 - not ready for public consumption. In development.

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 sememster of math into 4 pages' approash 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 editting 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.

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.

Finally, nothing has been spell checked or proof read yet. I wish IPython Notebook had spell check, but it doesn't seem to.

Motivation

This is a book for programmers that have a need or interest in Kalman filtering. The motivation for this book came out of my desire for a gentle introduction to Kalman filtering. I'm a software engineer that spent almost two decades in the avionics field, and so I have always been 'bumping elbows' with the Kalman filter, but never implemented one myself. As I moved into solving tracking problems with computer vision the need became urgent. There are classic textbooks in the field, such as Grewal and Andrew's excellent Kalman Filtering. But sitting down and trying to read many of these books is a dismal and trying experience if you do not have the background. Typcially the first few chapters fly through several years of undergraduate math, blithely referring you to textbooks on, for example, Itō calculus, and presenting an entire semester's worth of statistics in a few brief paragraphs. These books are good textbooks for an upper undergraduate course, and an invaluable reference to researchers and professionals, but the going is truly difficult for the more casual reader. Symbology is introduced without explanation, different texts use different words and variables names for the same concept, and the books are almost devoid of examples or worked problems. I often found myself able to parse the words and comprehend the mathematics of a defition, but had no idea as to what real world phenomena these words and math were attempting to describe. "But what does that mean?" was my repeated thought.

However, as I began to finally understand the Kalman filter I realized the underlying concepts are quite straightforward. A few simple probability rules, some intuition about how we integrate disparate knowledge to explain events in our everyday life and the core concepts of the Kalman filter are accessible. Kalman filters have a reputation for difficulty, but shorn of much of the formal terminology the beauty of the subject and of their math became clear to me, and I fell in love with the topic.

As I began to understand the math and theory more difficulties itself. A book or paper's author makes some statement of fact and presents a graph as proof. Unfortunately, why the statement is true is not clear to me, nor is the method by which you might make that plot obvious. Or maybe I wonder "is this true if R=0?" Or the author provides pseudocode - at such a high level that the implementation is not obvious. Some books offer Matlab code, but I do not have a license to that expensive package. Finally, many books end each chapter with many useful exercises. Exercises which you need to understand if you want to implement Kalman filters for yourself, but excercises with no answers. If you are using the book in a classroom, perhaps this is okay, but it is terrible for the independent reader. I loathe that an author witholds information from me, presumably to avoid 'cheating' by the student in the classroom.

None of this necessary, from my point of view. Certainly if you are designing a Kalman filter for a aircraft or missile you must thoroughly master of all of the mathematics and topics in a typical Kalman filter textbook. I just want to track an image on a screen, or write some code for my Arduino project. I want to know how the plots in the book are made, and chose different parameters than the author chose. I want to run simulations. I want to inject more noise in the signal and see how a filter performs. There are thousands of opportunities for using Kalman filters in everyday code, and yet this fairly straightforward topic is the provence of rocket scientists and academics.

I wrote this book to address all of those needs. This is not the book for you if you program avionics for Boeing or design radars for Ratheon. Go get a degree at Georgia Tech, UW, or the like, because you'll need it. This book is for the hobbiest, the curious, and the working engineer that needs to filter or smooth data.

This book is interactive. While you can read it online as static content, I urge you to use it as intended. It is written using IPython Notebook, which allows me to combine text, Python, and Python output in one place. Every plot, every piece of data in this book is generated from Python that is available to you right inside the notebook. Want to double the value of a parameter? Click on the Python cell, change the parameter's value, and click 'Run'. A new plot or printed output will appear in the book.

This book has exercises, but it also has the answers. I trust you. If you just need an answer, go ahead and read the answer. If you want to internalize this knowledge, try to implement the exercise before you read the answer.

This book has supporting libraries for computing statistics, plotting various things related to filters, and for the various filters that we cover. This does require a strong caveat; most of the code is written for didactic purposes. It is rare that I chose the most efficient solution (which often obscures the intent of the code), and in the first parts of the book I did not concern myself with numerical stability. This is important to understand - Kalman filters in aircraft are carefully designed and implemented to be numerically stable; the naive implemention is not stable in many cases. If you are serious about Kalman filters this book will not be the last book you need. My intention is to introduce you to the concepts and mathematics, and to get you to the point where the textbooks are approachable.

Finally, this book is free. The cost for the books required to learn Kalman filtering is somewhat prohibitive even for a Silicon Valley engineer like myself; I cannot believe the are within the reach of someone in a depressed economy, or a financially struggling student. I have gained so much from free software like Python, and free books like those from Allen B. Downey here. It's time to repay that. So, the book is free, it is hosted on free servers, and it uses only free and open software such as IPython and mathjax to create the book.

Contents

Preface

Motivation for the book. Where to download, how to use.

• Chapter 1: The g-h Filter

Intuitive introduction to the g-h filter, which is a family of filters that includes the Kalman filter. Not filler - once you understand this chapter you will understand the concepts behind the Kalman filter.

• Chapter 2: The Discrete Bayes Filter

Introduces the Discrete Bayes Filter. From this you will learn the probabilistic reasoning that underpins the Kalman filter in an easy to digest form.

• Chapter 3: Least Squares Filter

Introduces the least squares filter in batch and recursive forms.

• Chapter 4: Gaussian Probabilities

Introduces using Gaussians to represent beliefs. Gaussians allow us to implement the algorithms used in the Discrete Bayes Filter to work in continuous domains.

• Chapter 5: One Dimensional Kalman Filters

Implements a Kalman filter by modifying the Discrete Bayesian Filter to use Gaussians. This is a full featured Kalman filter, albeit only useful for 1D problems.

• Chapter 6: Multivariate Kalman Filter

We extend the Kalman filter developed in the previous chapter to the full, generalized filter.

• Chapter 7: Kalman Filter Math

We gotten about as far as we can without forming a strong mathematical foundation. This chapter is optional, especially the first time, but if you intend to write robust, numerically stable filters, or to read the literature, you will need to know this.

• Chapter 8: Designing Kalman Filters

Building on material in Chapter 6, walks you through the design of several Kalman filters. Discusses, but does not solve issues like numerical stability.

• Chapter 9: Extended Kalman Filters

Kalman filter as covered only work for linear problems. Extended Kalman filters (EKF) are the most common approach to linearizing non-linear problems.

• Chapter 10: Unscented Kalman Filters

Unscented Kalman filters (UKF) are a recent development in Kalman filter theory. They allow you to filter nonlinear problems without requiring a closed form solution like the Extended Kalman filter requires.

Chapter 11: Ensemble Kalman Filters

Discusses the ensemble Kalman Filter, which uses a Monte Carlo approach to deal with very large Kalman filter states in nonlinear systems.

• Chapter 12: Designing Nonlinear Kalman Filters

EKF and UKF are linear approximations of nonlinear problems. Unless programmed carefully, they are not numerically stable. We discuss some common approaches to this problem.

• Chapter 13: H-Infinity Filters

H-inifinity filters are a form of filter that is very robust in the presence of non-Gaussian noise. They do not perform as well as Kalman filters, but are less likely to diverge.

• Chapter 14: Smoothing

Kalman filters are recursive, and thus very suitable for real time filtering. However, they work well for post-processing data. We discuss some common approaches.

• [Chapter XX: Numerical Stability](not implemented)

Not written yet.

• [Chapter XX: Particle Filters](not implemented)

Not written yet

• [Chapter XX: Multihypothesis Tracking](not implemented)

Not written yet.

• Appendix: Installation, Python, NumPy, and FilterPy

Brief introduction of Python and how it is used in this book. Description of the companion library FilterPy.

• Appendix: Symbols and Notations

Symbols and notations used in this book. Comparison with notations used in the literature.

Installation and Software Requirements

** author's note**. The book is still being written, and so I am not focusing on issues like supporting multipe 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 off 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.

If you want to run the notebook on your computer, which is what I recommend, then you will have to have IPython installed. I do not cover how to do that in this book; requirements change based on what other Python installations you may have, whether you use a third party package like Anaconda Python, what operating system you are using, and so on.

To use all features you will have to have IPython 2.0 installed, which is released and stable as of April 2014. Most of the book does not require that recent of a version, but I do make use of the interactive plotting widgets introduced in this release. A few cells will not run if you have an older version installed. This is merely a minor annoyance.

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.

You will need a recent version of NumPy, SciPy, and Matplotlib installed. I don't really know what the minimal version might be. I have numpy 1.71, SciPy 0.13.0, and Matplotlib 1.4.0 installed on my machines.

Personally, I use the Anaconda Python distribution in all of my work, available here. I am not selecting them out of favoritism, I am merely documenting my environment. Should you have trouble running any of the code, perhaps knowing this will help you.

Provided Libraries and Modules

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.

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

pip install filterpy

If you do not have pip, you may follow the instructions here: https://pip.pypa.io/en/latest/installing.html.

Code that is specific to the book is stored with the book in the subdirectory ./code. This code is in a state of flux; I do not wish to document it here yet. I do mention in the book when I use code from this directory, so it should not be a mystery.

In the code subdirectory there are Python files with a name like xxx_internal.py. I use these to store functions that are useful for a specific chapter. This allows me to hide away Python code that is not particularly interesting to read - I may be generating a plot or chart, and I want you to focus on the contents of the chart, not the mechanics of how I generate that chart with Python. If you are curious as to the mechanics of that, just go and browse the source.

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.

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.

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. I have also been very influenced by Professor Lorena Barba's fantastic work, available here. I owe all of my look and feel to the work of these projects.

License

Kalman Filters and Random Signals in Python by Roger Labbe is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
Based on a work at https://github.com/rlabbe/Kalman-Filters-and-Random-Signals-in-Python.

Contact

rlabbejr at gmail.com