Python实现卡尔曼与贝叶斯滤波详解

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"《Python中的卡尔曼滤波与贝叶斯滤波》是一本由Roger R Labbe Jr于2018年5月8日编写的教程，旨在介绍如何在Python环境下应用这两种重要的信号处理和估计技术。该书主要涵盖了卡尔曼滤波（Kalman Filter）和贝叶斯滤波（Bayesian Filtering），并结合实践案例，让读者通过代码实现来理解这两个概念。书中首先从预览部分开始，概述了选择此书的原因，鼓励读者在线阅读或下载PDF版本，同时提到了所需的软件环境，如Jupyter Notebook、SciPy、NumPy和Matplotlib。这些库是进行数值计算和数据可视化的基础工具。章节1介绍了g-h滤波器，这是一种简化版的卡尔曼滤波，作者通过思维实验帮助读者建立直觉，并逐步展示了g-h滤波器的构建过程，包括符号定义和通用算法编写。书中提供了一系列练习，如编写通用算法、选择合适的g和h值、创建测量函数等，每项练习都配以解决方案和讨论，以加深理解。在后续章节中，作者探讨了初始条件对滤波性能的影响，以及极端噪声和加速度变化如何影响滤波结果。此外，还指导读者如何调整滤波器参数g和h以适应不同的应用场景。书中强调了Python编程语言和数学表达式的结合在实际工程中的重要性。通过这本书，读者不仅能够掌握卡尔曼和贝叶斯滤波的基本原理，还能通过实际编程操作掌握如何在Python中实现这些算法，这对于从事信号处理、控制系统设计或机器学习的工程师来说，是一本非常实用的参考资料。"

CONTENTS

You can create arrays of 3 or more dimensions, but we have no need for that here, and so I will not

elaborate.

By default the arrays use the data type of the values in the list; if there are multiple types then it will

choose the type that most accurately represents all the values. So, for example, if your list contains a mix

of int and float the data type of the array would be of type float. You can override this with the dtype

parameter.

In [5]: x = np.array([1, 2, 3], dtype=float)

print(x)

[1. 2. 3.]

You can access the array elements using subscript location:

In [6]: x = np.array([[1, 2, 3],

[4, 5, 6]])

print(x[1,2])

You can access a column or row by using slices. A colon (:) used as a subscript is shorthand for all data

in that row or column. So

x[:,0]

returns an array of all data in the ﬁrst column (the 0 speciﬁes the ﬁrst

column):

In [7]: x[:, 0]

Out[7]: array([1, 4])

We can get the second row with:

In [8]: x[1, :]

Out[8]: array([4, 5, 6])

Get the last two elements of the second row with:

In [9]: x[1, 1:]

Out[9]: array([5, 6])

As with Python lists, you can use negative indexes to refer to the end of the array. -1 refers to the last

index. So another way to get the last two elements of the second (last) row would be:

In [10]: x[-1, -2:]

Out[10]: array([5, 6])

You can perform matrix addition with the + operator, but matrix multiplication requires the dot method

or function. The * operator performs element-wise multiplication, which is not what you want for linear

algebra.

In [11]: x = np.array([[1., 2.],

[3., 4.]])

print('abddition:\n', x + x)

print('\nelement-wise multiplication\n', x * x)

print('\nmultiplication\n', np.dot(x, x))

print('\ndot is also a member of np.array\n', x.dot(x))

CONTENTS

[0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1]

Here is one I haven’t covered yet. The function numpy.asarray() will convert its argument to an ndar-

ray if it isn’t already one. If it is, the data is unchanged. This is a handy way to write a function that can

accept either Python lists or ndarrays, and it is very efﬁcient if the type is already ndarray as nothing new

is created.

In [20]: def one_tenth(x):

x = np.asarray(x)

return x / 10.

print(one_tenth([1, 2, 3])) # I work!

print(one_tenth(np.array([4, 5, 6]))) # so do I!

[0.1 0.2 0.3]

[0.4 0.5 0.6]

0.8 Companion Software

I am writing an open source Bayesian ﬁltering Python library called FilterPy. Installation instructions are

given above.

FilterPy is hosted GitHub at (https://github.com/rlabbe/ﬁlterpy) but the pip installed version should

serve your needs.

Code that is speciﬁc to the book is stored with the book in the subdirectory /kf_book. It contains code

for formatting the book. It also contains python ﬁles with names like xxx_internal.py. I use these to store

functions that are useful for a speciﬁc chapter. This allows me to hide 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 in /kf_book 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, Jupyter 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 /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.

The directory /kf_book contains a css ﬁle containing the style guide for the book. The default look and

feel of Jupyter Notebook is rather plain. I have followed the examples set by books such as Probabilistic

Programming and Bayesian Methods for Hackers [2]. I have also been very inﬂuenced by Professor Lorena

Barba’s fantastic work, available here [3]. I owe all of my look and feel to the work of these projects.

0.9 Thoughts on Python and Coding Math

Most Kalman ﬁltering and other engineering texts are written by mathematicians or academics. When there

is software (rarely), it is not production quality. Take Paul Zarchan’s book Fundamentals of Kalman Filtering

as an example. This is a fantastic book which belongs in your library, and is one of the few books that

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