### 统计代写|贝叶斯分析代写Bayesian Analysis代考|Dynamic Linear Model

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|Dynamic Linear Model

The dynamic linear model consists of two equations: the observation equation and the system equation.

The first expresses the observations as a linear function of the states of the system, while the second expresses the current state as a linear function of previous states. Usually, the system equation is a first-order autoregressive series, and various expressions of the dynamic linear model depend on what is assumed about the parameters of the model. First to be considered is the dynamic model in discrete time and the various phases of inference are explained. The various phases are as follows: (1) estimation, (2) filtering, (3) smoothing, and (4) prediction. Filtering is the posterior distribution of the present state at time $t$ induced by the prior distribution of the previous state at time $t-1$ in a sequential manner, and this recursive Bayesian algorithm is called the Kalman Filter. An important generalization of the dynamic linear model is where the system equation is amended to include a control variable, which is quite useful in navigation problems. Briefly, the control problem is as follows: at time $t$, the position of the object (satellite, missile, etc.) is measured as the observation at $t$, then the next position of the object is predicted so that the posterior mean of the next state is equal to the target value (the desired position) at time $t+1$. These ideas are illustrated with various examples beginning with a model where the observation and system vectors are univariate with known coefficients. For this case, the Kalman filter estimates the current state in a recursive manner as described earlier. As usual, $R$ generates the observations and states of the system via the observation and system equations of the dynamic linear model, which is followed by a Bayesian analysis via WinBUGS code that estimates the current state (the Kalman filter) of the system. $\mathrm{R}$ includes a powerful package dlm that will generate the observations and states of the process. One must download the TSA package and dlm package into R! The last section of the chapter deals with the control problem, where the choice of the control variable is described by a six-step algorithm. Finally, the problem of adaptive estimation is explained; this a generalization of the dynamic linear model, where the precision parameters of the observation and system equations are unknown. An example involving bivariate observations and states and unknown $2 \times 2$ precision matrices illustrates the Bayesian approach to the Kalman filter. The bivariate observations and states are generated with the dlm R package. There are 14 exercises and 21 references.

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|The Shift Point Problem

Shift point models are defined in general, followed by an example involving a univariate normal sequence of normal random variables with one shift where the time when the shift occurs is known. For such a case, the posterior distribution of the parameters of the two sequences is derived, assuming noninformative prior distributions for the parameters. There are three parameters: the mean of the first and second phases, plus the common precision parameter of both. Next to be considered is a twophase AR(1) model with one shift but unknown time where the shift occurs. $R$ generates the observations from this model where the time when the shift occurs is known but where the shift point is a random variable. A prior distribution is specified for the distribution of the shift point (where the shift occurs), the common precision of the errors, and the two AR(1) parameters. Of course, the parameters of the model are assumed to be known when generating the observations via $R$, and the posterior analysis executed with WinBUGS. The next series to be considered with one shift is the MA(1) and a similar scenario of generating observations via $R$ and posterior analysis via WinBUGS. A large part of the chapter focuses on examples taken from econometrics. For example, a regression model with AR(1) errors and one gradual shift (modeled with a transition function) is presented and the posterior distribution of the continuous shiftpoint is derived. This is repeated for a linear regression model with MA(1) errors and one unknown shift point, followed by a section on testing hypotheses concerning the moving average parameter of the two phases. An important subject in shift point models is that of threshold autoregressive models, where the model changes parameters depending on when the observation exceeds a given value, the threshold. The chapter ends with 10 problems and 28 references.

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|Residuals and Diagnostic Tests

Residuals of a time series model are defined and their use as diagnostic checks for how well the model fits the data is described. Procedures for calculating residuals and the corresponding standardized residuals are explained followed by many examples that illustrate those procedures. $R$ generates observations from an $\mathrm{AR}(1$ ) model with known parameters, and using these as data, a Bayesian analysis computes the posterior characteristics of the residuals. Given the posterior means of the residuals, the sample mean and standard deviation of the residuals are computed. Based on the mean and standard deviation of the residuals, the posterior analysis is repeated but this time the posterior means of the standardized residuals are computed. A normal $\mathrm{q}-\mathrm{q}$ plot of the standardized residuals is evaluated to see how well the model fits the data. If the model fits the data well, the normal $\mathrm{q}-\mathrm{q}$ plot should appear as linear. The more pronounced the linearity, the more confident is one of the models fit. The above scenario of diagnostic checks is repeated for a linear regression model with AR(1) errors, and a linear regression model with MA(1) errors.

The chapter concludes with comments and conclusions, four problems, and eight references.

It is hoped that this brief summary of the remaining chapters will initiate enough interest in the reader so that they will continue to learn how the Bayesian approach is used to analyze time series.

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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。