### 统计代写|回归分析作业代写Regression Analysis代考|Summary

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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|回归分析作业代写Regression Analysis代考|Summary

This chapter presents findings supporting the motivating contention that recognizing and efficiently and effectively accounting for and handling SA matters in georeferenced data analyses. Its discussion begins with a treatment of various subtly different definitional perspectives about $\mathrm{SA}$, emphasizing this concept in terms of map pattern. A mathematical formularization of Tobler’s First Law of Geography follows defining SA. A critical element of quantifying SA is specification of a SWM, whose two most popular forms are denoted by matrix $\mathbf{C}$, defined in terms of $\mathrm{n}$ binary zero-one indicators based upon the topological structure of a set of areal units, and matrix $\mathbf{W}$, the row-standardized version of matrix C. A SWM enables the positing of different indices to quantify SA that are specific to different data measurement scale types. Of these indices, the most popular is the $\mathrm{MC}$, which was developed for interval/ratio data measurement scales but has been extended to nominal and ordinal data measurement scales. Accordingly, this chapter presents the basics of the $\mathrm{MC}$ sampling distribution theory, which is essential for statistical inference purposes. The next section (Section 1.2) addresses impacts of SA on statistical analyses, stressing various ways it affects variance. Then discussion turns to SA contextualized as a violation of the IID assumption of classical statistics, and how it can be visualized with maps and with a Moran scatterplot, as well as with other graphical tools. Although the SA literature focuses on its impact upon variance, the last substantive/pedagogic background section furnishes illustrations showing how SA more generally can modify the shape of histograms.

## 统计代写|回归分析作业代写Regression Analysis代考|The mean and variance of the MC for linear

Following Clift and Ord (1973), Upton and Fingleton (1985, p. 338) show that the mean and variance of the $\mathrm{MC}$ for linear regression residuals, for
Spatial autocorrelation 23
which the $S W M C$ is symmetric and the number of covariates is $\mathrm{P}$, are as follows, where $\mathbf{M}=\mathbf{I}-\mathbf{X}\left(\mathbf{X}^{\mathrm{T}} \mathbf{X}\right)^{-1} \mathbf{X}^{\mathrm{T}}$ is the standard projection matrix from linear regression theory:
$$\begin{gathered} E(M C)=\frac{n}{\mathbf{1}^{T} \mathbf{C 1}} \frac{\operatorname{TR}\left{\left[\mathbf{I}-\mathbf{X}\left(\mathbf{X}^{\mathrm{T}} \mathbf{X}\right)^{-1} \mathbf{X}^{\mathrm{T}}\right] \mathbf{C}\right}}{\mathrm{n}-\mathrm{k}}, \text { and } \ \operatorname{VAR}(\mathrm{MC})=2\left(\frac{\mathrm{n}}{\mathbf{1}^{\mathrm{T}} \mathbf{C} 1}\right)^{2}\left[\frac{\operatorname{TR}\left{\left[\mathbf{I}-\mathbf{X}\left(\mathbf{X}^{\mathrm{T}} \mathbf{X}\right)^{-1} \mathbf{X}^{\mathrm{T}}\right] \mathbf{C}\left[\mathbf{I}-\mathbf{X}\left(\mathbf{X}^{\mathrm{T}} \mathbf{X}\right)^{-1} \mathbf{X}^{\mathrm{T}}\right] \mathbf{C}\right}}{(\mathrm{n}-\mathrm{k})(\mathrm{n}-\mathrm{k}+2)}\right. \ \left.-\frac{\left.\left(\operatorname{TR}\left{\left[\mathbf{I}-\mathbf{X}\left(\mathbf{X}^{\mathrm{T}} \mathbf{X}\right)^{-1} \mathbf{X}^{\mathrm{T}}\right] \mathbf{C}\right}\right)^{2}\right]}{(\mathrm{n}-\mathrm{k})^{2}}\right] \end{gathered}$$
where $\mathrm{k}=\mathrm{p}+1$.
For the following cases, suppose $\mathbf{X}{j}$ is proportional to the principal eigenvector of matrix $\left(\mathbf{I}-11^{\mathrm{T}} / \mathrm{n}\right) \mathrm{C}(\mathbf{I}-\mathbf{1 1} / \mathrm{T})$, whose eigenvalue is $\lambda{1}$, which results in the maximum degree of PSA (a worse-case scenario). The maximum principal eigenvalue for a planar surface partitioning is such that $\lambda_{1} \leq \mathrm{a}+\sqrt{2 \mathrm{n}-\mathrm{b}}$, for appropriately calculated real numbers a and $\mathrm{b}$ (Griffith \& Sone, 1995 , p. 170). However, most empirical surface partitionings relate more to a mixture of a regular square and a regular hexagonal tessellation. In addition, most empirical surfaces do not have any areal units whose number of neighbors is a function of n; rather, areal units with rook adjacency-defined neighbors of 30 or more are rare. Accordingly, their principal eigenvalues are much closer to 6 and not a function of $\mathrm{n}$.

## 统计代写|回归分析作业代写Regression Analysis代考|Some multicollinearity among the covariates

Convergence on the asymptotic variance presented here tends to be slower as the number of covariates increases. Nevertheless, it is a good

approximation for cases where $n-k$ is of a practical magnitude (e.g., at least 100$)$.

Table 1.A.1 summarizes results for a numerical example in which the number of areal units and the geographic configuration vary. One modification indicated by Case II is that multiple covariates that are correlated are replaced by their principal components, in a dimension-reducing way, which decreases the value of $k$ here in the denominator (an argument could be made not to decrease $k$ ). Furthermore, the constrained maximum hexagonal tessellation has areal units labeled 1 and $\mathrm{n}$ that wrap around the outside of a complete rectangular region, but constrained so that their maximum numbers of neighbors are 30 . Results in this table corroborate the contention that the asymptotic standard error of the $\mathrm{MC}$ for regression residuals is a very good approximation as long as the principal eigenvalue of matrix $C$ is not a function of $n$, no areal units have numbers of neighbors that are a function of $n$, and the number of covariates is not a function of $n$. Griffith (2010) demonstrates this same result for the case of no covariates. One important index here is $n-k$, which needs to be relatively large for the asymptotic standard error to be a good approximation. The smallest value in Table 1.A.1 for this expression is 97 , for which the approximation is quite good. Results for $\mathrm{n}<50$ tend to be poor in the linear regression context, whereas results for $\mathrm{n}<25$ tend to be poor in the univariate context (i.e., no covariates).

The expected value of the $\mathrm{MC}$ is easier to calculate than its standard error; this quantity goes to zero as $\mathrm{n}$ goes to infinity.

## 统计代写|回归分析作业代写Regression Analysis代考|The mean and variance of the MC for linear

\begin{聚集} E(M C)=\frac{n}{\mathbf{1}^{T} \mathbf{C 1}} \frac{\operatorname{TR}\left{\left[\mathbf{I }-\mathbf{X}\left(\mathbf{X}^{\mathrm{T}} \mathbf{X}\right)^{-1} \mathbf{X}^{\mathrm{T}}\右] \mathbf{C}\right}}{\mathrm{n}-\mathrm{k}}, \text { 和 } \ \operatorname{VAR}(\mathrm{MC})=2\left(\frac {\mathrm{n}}{\mathbf{1}^{\mathbf{T}} \mathbf{C} 1}\right)^{2}\left[\frac{\operatorname{TR}\left{\左[\mathbf{I}-\mathbf{X}\left(\mathbf{X}^{\mathrm{T}} \mathbf{X}\right)^{-1} \mathbf{X}^{\ mathrm{T}}\right] \mathbf{C}\left[\mathbf{I}-\mathbf{X}\left(\mathbf{X}^{\mathrm{T}} \mathbf{X}\right )^{-1} \mathbf{X}^{\mathrm{T}}\right] \mathbf{C}\right}}{(\mathrm{n}-\mathrm{k})(\mathrm{n }-\mathrm{k}+2)}\对。\ \left.-\frac{\left.\begin{gathered} E(M C)=\frac{n}{\mathbf{1}^{T} \mathbf{C 1}} \frac{\operatorname{TR}\left{\left[\mathbf{I}-\mathbf{X}\left(\mathbf{X}^{\mathrm{T}} \mathbf{X}\right)^{-1} \mathbf{X}^{\mathrm{T}}\right] \mathbf{C}\right}}{\mathrm{n}-\mathrm{k}}, \text { and } \ \operatorname{VAR}(\mathrm{MC})=2\left(\frac{\mathrm{n}}{\mathbf{1}^{\mathrm{T}} \mathbf{C} 1}\right)^{2}\left[\frac{\operatorname{TR}\left{\left[\mathbf{I}-\mathbf{X}\left(\mathbf{X}^{\mathrm{T}} \mathbf{X}\right)^{-1} \mathbf{X}^{\mathrm{T}}\right] \mathbf{C}\left[\mathbf{I}-\mathbf{X}\left(\mathbf{X}^{\mathrm{T}} \mathbf{X}\right)^{-1} \mathbf{X}^{\mathrm{T}}\right] \mathbf{C}\right}}{(\mathrm{n}-\mathrm{k})(\mathrm{n}-\mathrm{k}+2)}\right. \ \left.-\frac{\left.\left(\operatorname{TR}\left{\left[\mathbf{I}-\mathbf{X}\left(\mathbf{X}^{\mathrm{T}} \mathbf{X}\right)^{-1} \mathbf{X}^{\mathrm{T}}\right] \mathbf{C}\right}\right)^{2}\right]}{(\mathrm{n}-\mathrm{k})^{2}}\right] \end{gathered}

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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。