### 统计代写|回归分析作业代写Regression Analysis代考|Principal components analysis: A reconnaissance

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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|Principal components analysis: A reconnaissance

Eigendecomposition (i.e., rewriting a matrix in terms of its eigenvalues and eigenvectors) of a correlation matrix produces a set of eigenvalues that can be arranged in descending order, along with their corresponding eigenvectors. These eigenvectors are referred to as principals components (PCs) and have the intriguing property that the amount of variance they capture follows the same order as their magnitudes. In other words, the first eigenvector sum$\mathrm{~ m a n i z e s ~ t h e ~ m u s l ~ v a n i a n e , ~ t h e ~ s e c o n d ~ c i g c u v e l t e n ~ s u m u a r e s s ~ t h e ~ s e c u a}$ largest amount of varianec, and so forth. Because the first few cigenvectors often are sufficient for representing most of the information contained in their correlation matrix, principal components analysis (PCA) commonly is used for variable selection. However, here we use PCA to illustrate linkages between eigenvectors and SA.

Griffith (1984) began to explore the application of PCA to the binary SWM C, initially making little headway because the analysis was of $n$ indicator variable correlation (i.e., phi) coefficients. Later, a simple modification of matrix $\mathbf{C}$ illuminated a breakthrough: adding an identity matrix to $\mathbf{C}$ [i.e., $(\mathbf{C}+\mathbf{I})$ ] renders a SWM mimicking a correlation matrix, allowing the application of conventional PCA to it in a way that yields interpretable results (see Appendix 2.A). Furthermore, based on the mathematical properties of eigenvalues and eigenvectors, matrices $(\mathbf{C}+\mathbf{I})$ and $\mathbf{C}$ have the same eigenvectors and the same eigenvalues if one is added to the eigenvalues of matrix $\mathrm{C}$ (i.e., $\lambda_{j}+1$ ). The resulting PCA gives an important insight into the asymptotic relationship between the $\mathrm{SWM} \mathrm{C}$ eigenvectors and $\mathrm{SA}$ : the second $\mathrm{PC}$ corresponds to the maximum positive SA (PSA), and the last $\mathrm{PC}$ to the maximum negative SA (NSA). More specifically, PCA made possible the discovery of, and now makes obvious, the fundamental theorem of MESF, which pertains to matrix $\left(\mathrm{I}-11^{\mathrm{T}} / \mathrm{n}\right) \mathbf{C}\left(\mathbf{I}-11^{\mathrm{T}} / \mathrm{n}\right)$ .

## 统计代写|回归分析作业代写Regression Analysis代考|The spectral decomposition of a modified SWM

The MC can be written in matrix form as
The modified matrix $\left(\mathbf{I}-11^{\mathrm{T}} / \mathrm{n}\right) \mathrm{C}\left(\mathrm{I}-11^{\mathrm{T}} / \mathrm{n}\right)$ in its numerator is the matrix that relates to SA. Accordingly, the pair of eigenfunction problems becomes
$$\operatorname{Det}\left[\left(\mathbf{I}-11^{\mathrm{T}} / \mathrm{n}\right) \mathbf{C}\left(\mathbf{I}-11^{\mathrm{T}} / \mathrm{n}\right)-\lambda \mathbf{I}\right]=0$$
and
$$\left[\left(\mathbf{I}-11^{\mathrm{T}} / \mathrm{n}\right) \mathbf{C}\left(\mathbf{I}-11^{\mathrm{T}} / \mathrm{n}\right)-\lambda_{\mathrm{j}} \mathbf{I}\right] \mathbf{E}_{\mathrm{j}}=0, \mathrm{j}=1,2, \ldots, \mathrm{n}$$

subject to $\mathbf{E}{j}^{\mathrm{T}} \mathbf{E}{j}=1$ and $\mathbf{E}{j}^{\mathrm{T}} \mathbf{E}{\mathrm{k}}=0, j \neq k$. Because matrix $\left(\mathbf{I}-11^{\mathrm{T}} / \mathrm{n}\right) \times$ $\mathbf{C}\left(\mathbf{I}-11^{\mathrm{T}} / \mathrm{n}\right)$ is square and symmetric, it can be decomposed into a series of rank 1 matrices consisting of the products of the eigenvalues and eigenvectors; that is,
$$\left(\mathbf{I}-\frac{11^{\mathrm{T}}}{n}\right) \mathbf{C}\left(\mathbf{I}-\frac{11^{\mathrm{T}}}{n}\right)=\sum_{j=1}^{n} \lambda_{j} \mathbf{E}{j} \mathbf{E}{j}^{\mathrm{T}}$$
This right-hand summation expression is a matrix expansion of the standard eigendecomposition $\left(\mathbf{I}-11^{\mathrm{T}} / \mathrm{n}\right) \mathbf{C}\left(\mathbf{I}-11^{\mathrm{T}} / \mathrm{n}\right)=\mathbf{E} \Lambda \mathbf{E}^{\mathrm{T}}$, where $\Lambda$ is the diagonal matrix containing the n eigenvalues $\lambda_{\mathrm{j}}$

## 统计代写|回归分析作业代写Regression Analysis代考|Representing the MC with eigenfunctions

Let vector $\mathbf{Y}=\mathbf{E}{j}$. Next, substitute $\mathbf{E}{j}$ into Eq. (2.4):
$$\begin{gathered} {\left[\mathrm{n} /\left(\mathbf{1}^{\mathrm{T}} \mathbf{C} 1\right)\right] \mathbf{E}{\mathrm{j}}^{\mathrm{T}}\left(\mathbf{I}-11^{\mathrm{T}} / \mathrm{n}\right) \mathbf{C}(\mathbf{I}-\mathbf{1 1} / \mathrm{T}) \mathbf{E}{\mathrm{j}} /} \ {\left[\mathbf{E}{\mathrm{j}}^{\mathrm{T}}\left(\mathbf{I}-\mathbf{1 1} \mathbf{T}^{\mathrm{T}} / \mathrm{n}\right) \mathbf{E}{\mathrm{j}}\right]=\left[\mathrm{n} /\left(\mathbf{1}^{\mathrm{T}} \mathbf{C} 1\right)\right] \lambda_{\mathrm{j}} / 1} \end{gathered}$$
which is the same as Eq. (2.3). This outcome highlights why the extreme eigenvalues of matrix $\left(\mathrm{I}-11^{\mathrm{T}} / \mathrm{n}\right) \mathbf{C}\left(\mathbf{I}-11^{\mathrm{T}} / \mathrm{n}\right), \lambda_{1}$ and $\lambda_{\mathrm{m}}$, respectively, define the maximum and minimum values of a $\mathrm{MC}$ – they optimize its Rayleigh quotient. This feature of the MC distinguishes it from a Pearson product moment correlation coefficient: it does not have extreme values of $\pm 1$ but rather has an upper bound that often exceeds 1 by as much as $0.15$, or more, and a lower bound that often is closer to $-0.5$ than to $-1$.

Fig. 2.3A portrays the 2010 census tract surface partitioning $(n=1052)$ for the Dallas-Fort Worth (DFW) metroplex, and Fig. 2.3B is ArcMap SA output. The principal eigenvalue for this surface partitioning is $6.2675604$, based on the rook’s contiguity, with $1^{\mathrm{T}} \mathbf{C} 1=5626$. Therefore for PSA, $\mathrm{MC}=(1052 / 5626)(6.2675604)=1.171965$. This maximum PSA value exceeds 1 by $>0.17$. Meanwhile, for $\mathrm{NSA}, \quad \mathrm{MC}=(1052 / 5626) \times$ $(-3.949034)=-0.738426$. This maximum NSA value is closer to $-0.5$ than to $-1$.

## 统计代写|回归分析作业代写Regression Analysis代考|Principal components analysis: A reconnaissance

Griffith (1984) 开始探索将 PCA 应用于二进制 SWM C，最初进展不大，因为分析是n指标变量相关（即phi）系数。后来对矩阵进行了简单的修改C阐明了一个突破：将单位矩阵添加到C[IE，(C+一世)] 呈现模拟相关矩阵的 SWM，允许以产生可解释结果的方式将常规 PCA 应用于它（参见附录 2.A）。此外，基于特征值和特征向量的数学性质，矩阵(C+一世)和C如果将一个添加到矩阵的特征值中，则具有相同的特征向量和相同的特征值C（IE，λj+1）。由此产生的 PCA 提供了一个重要的洞察力之间的渐近关系小号在米C特征向量和小号一种： 第二磷C对应于最大正 SA (PSA)，最后一个磷C到最大负 SA (NSA)。更具体地说，PCA 使MESF 基本定理的发现成为可能，并且现在使之变得显而易见，该定理与矩阵有关(一世−11吨/n)C(一世−11吨/n) .

## 统计代写|回归分析作业代写Regression Analysis代考|The spectral decomposition of a modified SWM

MC 可以写成矩阵形式为

[(一世−11吨/n)C(一世−11吨/n)−λj一世]和j=0,j=1,2,…,n

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