Roger R Labbe Jr

Kalman and Bayesian Filters in Python

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Contents

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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?

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

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 browser. 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 cannot show the animations.

Language

English

ISBN

Unknown

Preface

Reading Online

PDF Version

Downloading the book

Version 0.0

Motivation

Installation and Software Requirements

Provided Libraries

Thoughts on Python and Coding Math

License

Contact

Resources

The g-h Filter

Building Intuition via Thought Experiments

The g-h Filter

Notation and Nomenclature

Exercise: Write Generic Algorithm

Solution and Discussion

Choice of g and h

Exercise: create measurement function

Solution

Exercise: Bad Initial Conditions

Solution and Discussion

Exercise: Extreme Noise

Solution and Discussion

Exercise: The Effect of Acceleration

Solution and Discussion

Exercise: Varying g

Solution and Discussion

Varying h

Tracking a Train

g-h Filters with FilterPy

Final Thoughts

Summary

References

Discrete Bayes Filter

Tracking a Dog

Extracting Information from Multiple Sensor Readings

Noisy Sensors

Incorporating Movement Data

Adding Noise to the Prediction

Integrating Measurements and Movement Updates

The Effect of Bad Sensor Data

Drawbacks and Limitations

Generalizing to Multiple Dimensions

Summary

References

Least Squares Filters

Introduction

Gaussian Probabilities

Introduction

Nomenclature

Gaussian Distributions

The Variance

Interactive Gaussians

Computational Properties of the Gaussian

Computing Probabilities with scipy.stats

Fat Tails

Summary and Key Points

References

One Dimensional Kalman Filters

Tracking A Dog

Math with Gaussians

Implementing the Update Step

Implementing Predictions

Animating the Tracking

Implementation in a Class

Relationship to the g-h Filter

Exercise: Modify Variance Values

Introduction to Designing a Filter

Animation

Exercise(optional):

Solution

Discussion

Explaining the Results - Multi-Sensor Fusion

More examples

Example: Extreme Amounts of Noise

Example: Bad Initial Estimate

Example: Large Noise and Bad Initial Estimate

Exercise: Interactive Plots

Solution

Exercise - Nonlinear Systems

Solution

Discussion

Exercise - Noisy Nonlinear Systems

Solution

Discussion

Summary

Multivariate Kalman Filters

Introduction

Multivariate Normal Distributions

Multiplying Multidimensional Gaussians

Unobserved Variables

Kalman Filter Algorithm

The Equations

Kalman Equations Expressed as an Algorithm

Implementation in Python

Tracking a Dog

Step 1: Choose the State Variables and Set Initial Conditions

Step 2: Design the State Transition Function

Step 3: Design the Motion Function

Step 4: Design the Measurement Function

Step 5: Design the Measurement Noise Matrix

Step 6: Design the Process Noise Matrix

Implementing the Kalman Filter

Compare to Univariate Kalman Filter

Discussion

Converting the Multivariate Equations to the Univariate Case

Exercise: Compare to a Filter That Incorporates Velocity

Solution

Discussion

Adjusting the Filter

A Detailed Examination of the Covariance Matrix

Question: Explain Ellipse Differences

Solution

Batch Processing

Smoothing the Results

Walking Through the KalmanFilter Code (Optional)

References

Kalman Filter Math

Bayesian Probability

Bayes' theorem

Modeling a Dynamic System that Has Noise

Modeling Dynamic Systems

Why This is Hard

Finding the Fundamental Matrix for Time Invariant Systems

Forming First Order Equations from Higher Order Equations

Walking Through the Kalman Filter Equations

Design of the Process Noise Matrix

Continuous White Noise Model

Piecewise White Noise Model

Using FilterPy to Compute Q

Simplification of Q

Numeric Integration of Differential Equations

Runge Kutta Methods

Iterative Least Squares for Sensor Fusion

Derivation of ILS Equations (Optional)

Implementing Iterative Least Squares

References

Designing Kalman Filters

Introduction

Tracking a Robot

Step 1: Choose the State Variables

Step 2: Design State Transition Function

Step 3: Design the Motion Function

Step 4: Design the Measurement Function

Step 5: Design the Measurement Noise Matrix

Step 6: Design the Process Noise Matrix

Step 7: Design Initial Conditions

Implement the Filter Code

The Effect of Order

Zero Order Kalman Filter

First Order Kalman Filter

Second Order Kalman Filter

Evaluating the Performance

Sensor Fusion

Exercise: Can you Kalman Filter GPS outputs?

Excercise: Prove that the PS improves the filter

Exercise: Different Data Rates

Tracking a Ball

Step 1: Choose the State Variables

Step 2: Design State Transition Function

Interlude: Test State Transition

Step 3: Design the Motion Function

Step 4: Design the Measurement Function

Step 5: Design the Measurement Noise Matrix

Step 6: Design the Process Noise Matrix

Step 7: Design the Initial Conditions

Tracking a Ball in Air

Implementing Air Drag

Tracking Noisy Data

Nonlinear Filtering

Introduction

The Problem with Nonlinearity

An Intuitive Look at the Problem

The Effect of Nonlinear Functions on Gaussians

A 2D Example

The Algorithms

Summary

Unscented Kalman Filters

Choosing Sigma Points

Handling the Nonlinearities

Sigma Point Algorithm

The Unscented Transform

The Unscented Filter

Predict Step

Update Step

Using the UKF

Tracking a Flying Airplane

First Attempt

Tracking Manuevering Aircraft

Sensor Fusion

Multiple Position Sensors

Exercise: Track a target moving in a circle

Solution

Discussion

Exercise: Sensor Position Effects

Solution

Exercise: Compute Position Errors

Solution

Discussion

Exercise: Explain Filter Performance

Solution

Exercise: Visualize Sigma Points

Solution

Discussion

Implementation of the UKF

Weights

Sigma Points

Predict Step

Update Step

Full Source from FilterPy

Batch Processing

Smoothing the Results

Choosing Values for kappa

The Skewed Unscented Transform

The Scaled Unscented Transform

Nonlinear State Variables

References

The Extended Kalman Filter

The Problem with Nonlinearity

The Effect of Nonlinear Transfer Functions on Gaussians

The Extended Kalman Filter

Example: Tracking a Flying Airplane

Design the State Variables

Design the System Model

Design the Measurement Model

Using SymPy to compute Jacobians

Designing Q

Example: A falling Ball

Designing Nonlinear Kalman Filters

Introduction

Kalman Filter with Air Drag

Realistic 2D Position Sensors

Linearizing the Kalman Filter

References

Smoothing

Introduction

Types of Smoothers

Fixed Point Smoothing

Fixed Lag Smoothing

Fixed Interval Smoothing

References

Adaptive Filtering

Introduction

Maneuvering Targets

Detecting a Maneuver

Adjustable Process Noise

Continuous Adjustment

Continuous Adjustment - Standard Deviation Version

Fading Memory Filter

Noise Level Switching

Variable State Dimension

Multiple Model Estimation

References

H filter

Ensemble Kalman Filters

The Algorithm

Initialize Step

Predict Step

Update Step

Implementation and Example

Outstanding Questions

References

Installation, Python, Numpy, and filterpy

Installing the SciPy Stack

Manual Install of the SciPy stack

Installing/downloading and running the book

Using IPython Notebook

SymPy

Symbology

State

State at step n

Prediction

measurement

control transition Matrix

Nomenclature

Equations

Brookner

Gelb

Brown

Zarchan

Wikipedia

Labbe

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