A Kalman Filter Primer
A Kalman Filter Primer
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卡尔曼滤波器入门
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发表时间：
2007
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通讯作者：
Kary L Myers
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作者：
Kary L Myers
长期以来，卡尔曼滤波器是工程师和机器学习研究人员的标准工具，是一种强大的算法，可以将问题转换为按时间索引的随机模型。在线性高斯状态空间模型的上下文中，卡尔曼滤波器允许有效的预测、滤波和参数估计。然而，正如Meinhold和Singpurwalla（1983）所指出的，大多数关于卡尔曼滤波器的出版文献都出现在工程期刊上，使用的符号和语言可以掩盖其与统计问题的相关性。近25年后，这种情况仍然存在，只有少数例外（如Meinhold和Singpurwalla的文章）从贝叶斯的角度介绍了卡尔曼滤波器。这本书建议通过提供卡尔曼滤波器的独立探索来解决这个问题。尤班克推导出卡尔曼递归的数学原理应用于一个简化的状态空间模型，然后扩展到一般的状态空间的情况下，机械。每一章都仔细地建立和完善了之前提出的内容。第一章首先描述了Gramm-Schmidt和Cholesky算法在一般信号加噪声模型中的应用，然后讨论了状态空间模型的特殊情况。第二章推导了状态空间模型中所谓的新息（实际观测值与预测观测值之间的差异）与状态向量之间的协方差关系。作者在第3章中使用了这个关系，得到了一个计算效率很高的乔莱斯基分解，这直接导致了第4章和第5章中的前向和后向卡尔曼滤波器递归。有了这个背景，尤班克然后处理特定的问题和情况，例如如何初始化递归的状态向量（第11章）。6），如何利用卡尔曼递归时，底层的状态空间模型是高斯（章。7），以及如何将结果扩展到状态空间模型的一般公式（第7章）。8）。这本书非常适合那些想要“看看卡尔曼滤波器的引擎盖下”的人，以了解它是如何在数学严格的水平上工作的。虽然没有专门设计为教科书，这本书可以用来指导研究生的专题研讨会。在矩阵代数和介绍研究生水平的统计背景应该是足够的先决条件。Eubank包含了本文中提出的许多算法的伪代码，使其更容易实现这些方法。然而，人们寻找一个不太详细的介绍卡尔曼滤波器，以了解和实现它很快可能会做得更好，最初与指南一样，韦尔奇和主教（2003年）。虽然卡尔曼滤波器入门是相当全面的，一些“化妆品”的问题损害其可访问性。特别是，索引是相当多余的，因此不是很有帮助，和穷人排版（包括奇怪的断字决定，如“C-holesky”和“parameter-s”）分散了从文本。总的来说，虽然，这本书肯定达到尤班克的目标，一个“独立的，”没有装饰，“数学严格推导的所有基本卡尔曼滤波器递归从第一原则。”
Long a standard tool of engineers and machine learning researchers, the Kalman filter (Kalman 1960) is a powerful algorithm for problems that can be cast as stochastic models indexed by time. In the context of linear Gaussian state-space models, the Kalman filter allows for efficient prediction, filtering, and parameter estimation. However, as pointed out by Meinhold and Singpurwalla (1983), most published literature about the Kalman filter appears in engineering journals, using notation and language that can mask its relevance to statistical problems. Nearly 25 years later, this remains the case, with a few exceptions (like the Meinhold and Singpurwalla article) that present the Kalman filter from a Bayesian perspective. This book proposes to remedy this by offering a self-contained exploration of the Kalman filter. Eubank derives the Kalman recursions from mathematical principles applied to a simplified state-space model, then extends the machinery to the general state-space case. Each chapter carefully builds on and refines what was presented before. Chapter 1 starts by describing the use of the Gramm–Schmidt and Cholesky algorithms with general signal-plus-noise models, then treats the special case of state-space models. Chapter 2 derives of the covariance relationship between the so-called innovations (the discrepancies between the actual observations and the predicted observations) and the state vectors in state-space models. The author uses this relationship in Chapter 3 to obtain a computationally efficient version of the Cholesky decomposition, which leads directly to the forward and backward Kalman filter recursions in Chapters 4 and 5. With this background, Eubank then treats particular questions and cases, such as how to initialize the state vector for the recursions (Chap. 6), how to take advantage of the Kalman recursions when the underlying state-space model is Gaussian (Chap. 7), and how to extend the results to a general formulation of the state-space model (Chap. 8). The book is well suited for people who want to “look under the hood” of the Kalman filter to understand how it works at a mathematically rigorous level. Although not specifically designed as a textbook, the book could be used to guide a special topics seminar for graduate students. A background in matrix algebra and introductory graduate-level statistics should be sufficient prerequisites. Eubank includes pseudocode for many of the algorithms presented in the text, making it easier to implement the methods. However, people looking for a less detailed introduction to the Kalman filter to understand and implement it quickly might do better initially with a guide like that of Welch and Bishop (2003). Although A Kalman Filter Primer is quite comprehensive, a few “cosmetic” issues impair its accessibility. In particular, the index is quite spare and thus not very helpful, and poor typesetting (including strange hyphenation decisions like “C-holesky” and “parameter-s”) distracts from the text. Overall, though, the book certainly achieves Eubank’s goal of a “self-contained, ‘no frills,’ mathematically rigorous derivation of all the basic Kalman filter recursions from first principles.”