### 统计代写|贝叶斯统计代写Bayesian statistics代考|CAR models

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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

## 统计代写|贝叶斯统计代写beyesian statistics代考|CAR models

The keyword CAR in CAR models stands for Conditional AutoRegression. This concept is often used in the context of modeling areal data which can be either discrete counts or continuous measurements. However, the CAR models are best described using the assumption of the normal distribution although CAR models for discrete data are also available. In our Bayesian modeling for areal data CAR models are used as prior distributions for spatial effects defined on the areal units. This justifies our treatment of CAR models using the normal distribution assumption.

Assume that we have areal data $Y_{i}$ for the $n$ areal units. The conditional in CAR stands for conditioning based on all the others. For example, we like to think of $Y_{1}$ given $Y_{2}, \ldots, Y_{n}$. The AutoRegression terms stand for regression on itself (auto). Putting these concepts together the CAR models are based on regression of each $Y_{i}$ conditional on the others $Y_{j}$ for $j=1, \ldots n$ but with

$j \neq i$. The constraint $j \neq i$ makes sure that we do not use $Y_{i}$ to define the distribution of $Y_{i}$. Thus, a typical CAR model will be written as
$$Y_{i} \mid y_{j}, j \neq i \sim N\left(\sum_{j} b_{i j} y_{j}, \sigma_{i}^{2}\right)$$
where the $b_{i j}$ ‘s are presumed to be the regression coefficients for predicting $Y_{i}$ based on all the other $Y_{j}$ ‘s. The full distributional specification for $\mathbf{Y}=\left(Y_{1}, \ldots, Y_{n}\right)$ comes from the independent product specification of the distributions $(2.6)$ for each $i=1, \ldots, n$. There are several key points and concepts that we now discuss to understand and we present those below as a bulleted list.

• The models (2.6) can be equivalently rewritten as
$$\mathbf{Y}=B \mathbf{Y}+\epsilon$$
where $\epsilon=\left(\epsilon_{1}, \ldots, \epsilon_{n}\right)$ is a multivariate normal error distribution with zero means. The appearance of $\mathbf{Y}$ on the right hand side of the above emphasizes the keywords AutoRegression in CAR.
• The CAR specification defines a valid multivariate normal probability distribution for $\mathbf{Y}$ under the additional conditions
$$\frac{b_{i j}}{\sigma_{i}^{2}}=\frac{b_{j i}}{\sigma_{j}^{2}}, i, j=1, \ldots, n$$
which are required to ensure that the inverse covariance matrix $\Sigma^{-1}$ in (A.24), is symmetric.

## 统计代写|贝叶斯统计代写beyesian statistics代考|Point processes

Spatial point pattern data arise when an event of interest, e.g. outbreak of a disease, e.g. Covid-19, occurs at random locations inside a study region of interest, $\mathbb{D}$. Often the main interest in such a case lies in discovering any explainable or non-random pattern in a scatter plot of the data locations. Absence of any regular pattern in the data locations is said to correspond to the model of complete spatial randomness, CSR, which is also called a Poisson process. Under CSR, the number of points in any given sub-region will follow the Poisson distribution with a parameter value proportional to the area of the sub-region. Often, researchers are interested in rejecting the model of CSR in favor of their own theories of the evolution or clustering of the points. In this context the researchers have to decide what all type of clustering may possibly explain the clustering pattern of points and which one of those provides the “best” fit to the observed data. There are other obvious investigations to make, for example, are there any suitable covariates which may explain the pattern? To illustrate, a lack of trees in many areas in a city may be explained by a layer of built environment.

Spatio-temporal point process data are naturally found in a number of disciplines, including (human or veterinary) epidemiology where extensive datasets are also becoming more common. One important distinction in practice is between processes defined as a discrete-time sequence of spatial point processes, or as a spatially and temporally continuous point process. See the books by Diggle (2014) and Møller and Waagepetersen (2003) for many examples and theoretical developments.

## 统计代写|贝叶斯统计代写beyesian statistics代考|Conclusion

The main purpose of this chapter has been to introduce the key concepts we need to pursue spatio-temporal modeling in the later chapters. Spatiotemporal modeling, as any other substantial scientific area of research, has its own unique set of keywords and concept dictionary. Not knowing some of these is a barrier to fully understanding, or more appropriately appreciating, what is going on under the hood of modeling equations. Thus, this chapter plugs the knowledge gap a reader may have regarding the typical terminology used while modeling.

It has not been possible to keep the chapter completely notation free. Notations have been introduced to keep the rigor in presentation and also as early and unique reference points for many key concepts assumed in the later chapters. For example, the concepts of Gaussian Process (GP), Kriging, internal and external standardization are defined without the data application overload. Of course, it is possible to skip reading of this chapter until a time when the reader is confronted with an un-familiar jargon.

## 统计代写|贝叶斯统计代写beyesian statistics代考|CAR models

CAR 模型中的关键字 CAR 代表 Conditional AutoRegression。这个概念通常用于建模区域数据的上下文中，这些数据可以是离散计数或连续测量。然而，CAR 模型最好使用正态分布的假设来描述，尽管也可以使用离散数据的 CAR 模型。在我们对面积数据的贝叶斯建模中，CAR 模型被用作在面积单位上定义的空间效应的先验分布。这证明了我们使用正态分布假设处理 CAR 模型的合理性。

j≠一世. 约束j≠一世确保我们不使用是一世定义分布是一世. 因此，典型的 CAR 模型将被写为

• 模型（2.6）可以等效地重写为
是=乙是+ε
在哪里ε=(ε1,…,εn)是具有零均值的多元正态误差分布。的出现是上图右侧强调了 CAR 中的关键字 AutoRegression。
• CAR 规范定义了一个有效的多元正态概率分布是在附加条件下
b一世jσ一世2=bj一世σj2,一世,j=1,…,n
这需要确保逆协方差矩阵Σ−1在 (A.24) 中，是对称的。

## 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。