### 数学代写|matlab代写|Square-Root Methods and All That

matlab是一个编程和数值计算平台，被数百万工程师和科学家用来分析数据、开发算法和创建模型。

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|matlab代写|Numerical Stability Problems

Numerical Stability Problems. The great success of Kalman filtering was not without its problems, not the least of which was marginal stability of the numerical solution of the associated Riccati equation. In some applications, small roundoff errors tended to accumulate and eventually degrade the performance of the filter. In the decades immediately following the introduction of the Kalman filter, there appeared several better numerical implementations of the original formulas. Many of these were adaptations of methods previously derived for the least squares problem.
Early ad hoc Fixes. It was discovered early on ${ }^{13}$ that forcing symmetry on the solution of the matrix Riccati equation improved its apparent numerical stability-a phenomenon that was later given a more theoretical basis by Verhaegen and Van Dooren [232]. It was also found that the influence of roundoff errors could be ameliorated by artificially increasing the covariance of process noise in the Riccati equation. A symmetrized form of the discrete-time Riccati equation was developed by Joseph [15] and used by R. C. K. Lee at Honeywell in 1964. This “structural” reformulation of the Kalman filter equations improved robustness against roundoff errors in some applications, although later methods have performed better on some problems [125].

Square-Root Filtering. These methods can also be considered as “structural” reformulations of the Riccati equation, and they predate the Bucy-Joseph form. The first of these was the “square-root” implementation by Potter and Stern [208], first published in 1963 and successfully implemented for space navigation on the Apollo manned lunar exploration program. Potter and Stern introduced the idea of factoring the covariance matrix into Cholesky factors, ${ }^{14}$ in the format
$$P=C C^{\mathrm{T}}$$
and expressing the observational update equations in terms of the Cholesky factor $C$, rather than $P$. The result was better numerical stability of the filter implementation at the expense of added computational complexity. A generalization of the Potter and Stern method to handle vector-valued measurements was published by one of the authors [130] in 1968 , but a more efficient implementation-in terms of triangular Cholesky factors – was published by Bennet in 1967 [138].

## 数学代写|matlab代写|Beyond Kalman Filtering

Extended Kalman Filtering and the Kalman-Schmidt Filter. Although it was originally derived for a linear problem, the Kalman filter is habitually applied with impunity-and considerable success – to many nonlinear problems. These extensions generally use partial derivatives as linear approximations of nonlinear relations. Schmidt [118] introduced the idea of evaluating these partial derivatives at the estimated value of the state variables. This approach is generally called the extended Kalman filter, but it was called the Kalman-Schmidt filter in some early publications. This and other methods for approximate linear solutions to nonlinear problems are discussed in Chapter 5 , where it is noted that these will not be adequate for all nonlinear problems. Mentioned here are some investigations that have addressed estimation problems from a more general perspective, although they are not covered in the rest of the book.

Nonlinear Filtering Using Higher Order Approximations. Approaches using higher order expansions of the filter equations (i.e., beyond the linear terms) have been derived by Stratonovich [78], Kushner [191], Bucy [147], Bass et al. [134], and others for quadratic nonlinearities and by Wiberg and Campbell [235] for terms through third order.

Nonlinear Stochastic Differential Equations. Problems involving nonlinear and random dynamic systems have been studied for some time in statistical mechanics. The propagation over time of the probability distribution of the state of a nonlinear dynamic system is described by a nonlinear partial differential equation called the Fokker-Planck equation. It has been studied by Einstein [157], Fokker [160], Planck [207], Kolmogorov [187], Stratonovich [78], Baras and Mirelli [52], and others. Stratonovich modeled the effect on the probability distribution of information obtained through noisy measurements of the dynamic system, an effect called conditioning. The partial differential equation that includes these effects is called the conditioned Fokker-Planck equation. It has also been studied by Kushner [191], Bucy [147], and others using the stochastic calculus of Kiyosi Itô-also called the “Itô calculus.” It is a non-Riemannian calculus developed specifically for stochastic differential systems with noise of infinite bandwidth. This general approach results in a stochastic partial differential equation describing the evolution over time of the probability distribution over a “state space” of the dynamic system under study. The resulting model does not enjoy the finite representational characteristics of the Kalman filter, however. The computational complexity of obtaining a solution far exceeds the already considerable burden of the conventional Kalman filter. These methods are of significant interest and utility but are beyond the scope of this book.

## 数学代写|matlab代写|Point Processes and the Detection Problem

Point Processes and the Detection Problem. A point process is a type of random process for modeling events or objects that are distributed over time or space, such as the arrivals of messages at a communications switching center or the locations of stars in the sky. It is also a model for the initial states of systems in many estimation problems, such as the locations of aircraft or spacecraft under surveillance by a radar installation or the locations of submarines in the ocean. The dection problem for these surveillance applications must usually be solved before the estimation problem (i.e., tracking of the objects with a Kalman filter) can begin. The Kalman filter requires an initial state for each object, and that initial state estimate must be obtained by detecting it. Those initial states are distributed according to some point process, but there are no technically mature methods (comparable to the Kalman filter) for estimating the state of a point process. A unified approach combining detection and tracking into one optimal estimation method was derived by Richardson [214] and specialized to several applications. The detection and tracking problem for a single object is represented by the conditioned Fokker-Planck equation. Richardson derived from this one-object model an infinite hierarchy of partial differential equations representing object densities and truncated this hierarchy with a simple closure assumption about the relationships between orders of densities. The result is a single partial differential equation approximating the evolution of the density of objects. It can be solved numerically. It provides a solution to the difficult problem of detecting dynamic objects whose initial states are represented by a point process.

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。