### 统计代写|回归分析作业代写Regression Analysis代考|The spectral analysis of three-dimensional data

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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|The spectral analysis of three-dimensional data

In many cases, the spectral analysis of three-dimensional georeferenced data involves a sequence of maps, one for each point in a specified time series, rather than supplementing planar surfaces with elevation. Griffith and Heurclink (2012) extend the preeeding spectral analysis conceptualizations to this situation. Now the spectral density-based space-time $(\tau, \eta, \nu)-$ lag correlation function becomes, for a regular square tessellation and the rook adjacency definition, and uniformly spaced points in time,
$\frac{\int_{0}^{\pi} \int_{0}^{\pi} \int_{0}^{\pi} \frac{\operatorname{Cos}(\tau \theta) \operatorname{Cos}(\eta \varphi) \operatorname{Cos}(v t)}{\left[1-\operatorname{COS}(t)\left{\rho_{s}[\operatorname{Cos}(\theta)+\operatorname{Cos}(\varphi)]+\rho_{T}\right}\right]^{k}} d \theta d \varphi d t}{\int_{0}^{\pi} \int_{0}^{\pi} \int_{0}^{\pi} \frac{1}{\left[1-\operatorname{COS}(t)\left{\rho_{s}[\operatorname{CoS}(\theta)+\operatorname{CoS}(\varphi)]+\rho_{T}\right}\right]^{k}} \mathrm{~d} \theta \mathrm{d} \varphi \mathrm{dt}}, \boldsymbol{\kappa}=1,2$,
where $t$ denotes the twe argument, $\rho_{\mathrm{s}}$ denotes the SA parameter, and $\rho_{\mathrm{r}}$ denotes the temporal autocorrelation parameter. This specification represents a contemporaneous space-time process, which is additive, whose matrix representation is given by

$$\mathbf{C}=\mathbf{I}{\mathrm{T}} \otimes \mathbf{I}{\mathrm{s}}-\rho_{\mathrm{s}} \mathbf{C}{\mathrm{T}} \otimes \mathbf{C}{\mathrm{s}}-\rho_{\mathrm{T}} \mathbf{C}{\mathrm{T}} \otimes \mathbf{I}{\mathrm{s}},$$
where $\otimes$ denotes the Kronecker product mathematical matrix operation, $\mathbf{C}{\mathrm{s}}$ denotes the SWM, $\mathbf{C}{\mathrm{T}}$ denotes the time-series connectivity matrix, $\mathbf{I}{\mathrm{T}}$ denotes the $T$-by-T identify matrix, $\mathbf{I}{s}$ denotes the $\mathrm{n}-\mathrm{by}-\mathrm{n}$ identity matrix, and $1-\operatorname{COS}(\mathrm{t})\left{\rho_{\mathrm{s}}[\operatorname{COS}(\mathrm{u})+\operatorname{COS}(\mathrm{v})]+\rho_{\mathrm{T}}\right}$ are the limiting eigenvalues of the space time connectivity matrix $C$.

An alternative specification is multiplicative and hence describes a space-time lagged process; its matrix representation is given by
$$\mathbf{C}=\mathbf{I}{\mathrm{T}} \otimes \mathbf{I}{\mathrm{s}}-\rho_{\mathrm{s}} \mathbf{I}{\mathrm{T}} \otimes \mathbf{C}{\mathrm{s}}-\rho_{\mathrm{T}} \mathbf{C}{\mathrm{T}} \otimes \mathbf{I}{\mathrm{s}} \text {, }$$
and its spectral density-based $(\tau, \eta, \nu)$-lag correlations are given by
For a regular square lattice forming a complete $P-b y-Q$ rectangular region,
$$\mathbf{C}{\mathrm{s}}=\mathbf{C}{\mathrm{P}} \otimes \mathbf{I}{\mathrm{Q}}+\mathbf{C}{\mathrm{Q}} \otimes \mathbf{I}{\mathrm{r}},$$ where $C{p}$ and $C_{Q}$, respectively, are $S W M s$ for a $P$ length and a $Q$ length linear landscape, and $\mathbf{I}{\mathrm{P}}$ and $\mathbf{I}{\mathrm{Q}}$, respectively, are $\mathrm{P}-\mathrm{by}-\mathrm{P}$ and $\mathrm{Q}-\mathrm{by}-\mathrm{Q}$ identity matrices.

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

This chapter reviews articulations among SWMs, eigenfunctions, and spectral functions, all three of which relate to $\mathrm{SA}$. In doing so, it also links them to geostatistics. The eigenvalues of a SWM index the nature and degree of SA in the eigenvectors of a modified SWM and also appear in the complex fraction spectral density functions used to calculate lagged spatial correlations. The cells of standardized inverse spatial covariance structures, illustrated here with the popular first- and second-order ones, contain spectral density function results. These notions interlace with concepts for PCA. Although this chapter focuses on the $\mathrm{MC}$ index of $\mathrm{SA}$, similar results may be established for both the Geary ratio (GR) and the join count statistics that are applicable to nominal measurement scale data. The linear geographic landscape furnishes many relatively simple illustrations of the connections of interest here. The two-dimensional geographic landscape furnishes more relevant, albeit more complicated, contexts and highlights map pattern visualizations, one of the most important topics of this chapter.

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

Consider the geographic landscape in Fig. $2.5 \mathrm{C}$. Its rook adjacency SWM C is as follows:
$$\left[\begin{array}{llll} 0 & 1 & 1 & 0 \ 1 & 0 & 0 & 1 \ 1 & 0 & 0 & 1 \ 0 & 1 & 1 & 0 \end{array}\right]$$
The Perron-Frobenius theorem states that the principal eigenvalue is contained in the interval defined by the largest and smallest row sums; therefore here $\lambda_{1}=2$. For each pair of rows or columns that is identical, an eigenvalue equals zero; therefore because the first and fourth rows/columns are identical, and the second and third rows/columns are identical, two eigenvalues equal zero. Finally the trace of this matrix equals the sum of its four eigenvalues; therefore $2+0+0+\lambda=0$, and hence an eigenvalue equals $-2$.

Eq. (2.1) for this SWM is $\lambda^{2}\left(\lambda^{2}-4\right)=0$. The first $\lambda^{2}$ term is for the two roots of zero, whereas the second term factors into $(\lambda+2)(\lambda-2)$, which is for the two roots $\pm 2$. Ord (1975) also states that the eigenvalues for this particular type of geographic surface partitioning and SWM are given by $\lambda=2\left[\operatorname{COS}\left(\frac{\mathrm{h} \pi}{2+1}\right)+\operatorname{COS}\left(\frac{\mathrm{k} \pi}{2+1}\right)\right], \mathrm{h}=1,2$ and $\mathrm{k}=1,2$. This equation yields $2(0.5+0.5)=2 ; 2(0.5-0.5)=0 ; 2(-0.5+0.5)=0$; and, $2(-0.5-0.5)=-2$.

Griffith (2000, p. 98) proves that the solution to Eq. (2.2) for this particular type of geographic surface partitioning and SWM are the eigenvectors given by
$$\frac{2}{\sqrt{(2+1)(2+1)}}\left[\operatorname{SIN}\left(\frac{h \pi}{2+1}\right) \times \operatorname{SIN}\left(\frac{k \pi}{2+1}\right)\right]$$
This expression produces the 4-by-4 eigenvector matrix
$$\left[\begin{array}{rrrr} 0.5 & 0.5 & 0.5 & 0.5 \ 0.5 & -0.5 & 0.5 & -0.5 \ 0.5 & 0.5 & -0.5 & -0.5 \ 0.5 & -0.5 & -0.5 & 0.5 \end{array}\right]$$

## 统计代写|回归分析作业代写Regression Analysis代考|The spectral analysis of three-dimensional data

\frac{\int_{0}^{\pi} \int_{0}^{\pi} \int_{0}^{\pi} \frac{\operatorname{Cos}(\tau \theta) \operatorname{ Cos}(\eta \varphi) \operatorname{Cos}(v t)}{\left[1-\operatorname{COS}(t)\left{\rho_{s}[\operatorname{Cos}(\theta)+ \operatorname{Cos}(\varphi)]+\rho_{T}\right}\right]^{k}} d \theta d \varphi d t}{\int_{0}^{\pi} \int_{0 }^{\pi} \int_{0}^{\pi} \frac{1}{\left[1-\operatorname{COS}(t)\left{\rho_{s}[\operatorname{CoS}( \theta)+\operatorname{CoS}(\varphi)]+\rho_{T}\right}\right]^{k}} \mathrm{~d} \theta \mathrm{d} \varphi \mathrm{dt }}, \boldsymbol{\kappa}=1,2\frac{\int_{0}^{\pi} \int_{0}^{\pi} \int_{0}^{\pi} \frac{\operatorname{Cos}(\tau \theta) \operatorname{ Cos}(\eta \varphi) \operatorname{Cos}(v t)}{\left[1-\operatorname{COS}(t)\left{\rho_{s}[\operatorname{Cos}(\theta)+ \operatorname{Cos}(\varphi)]+\rho_{T}\right}\right]^{k}} d \theta d \varphi d t}{\int_{0}^{\pi} \int_{0 }^{\pi} \int_{0}^{\pi} \frac{1}{\left[1-\operatorname{COS}(t)\left{\rho_{s}[\operatorname{CoS}( \theta)+\operatorname{CoS}(\varphi)]+\rho_{T}\right}\right]^{k}} \mathrm{~d} \theta \mathrm{d} \varphi \mathrm{dt }}, \boldsymbol{\kappa}=1,2,

C=一世吨⊗一世s−ρs一世吨⊗Cs−ρ吨C吨⊗一世s,

Cs=C磷⊗一世问+C问⊗一世r,在哪里Cp和C问，分别是小号在米s为一个磷长度和一个问长度线性景观，和一世磷和一世问，分别是磷−b是−磷和问−b是−问身份矩阵。

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

[0110 1001 1001 0110]
Perron-Frobenius 定理指出，主特征值包含在由最大和最小行和定义的区间内；因此在这里λ1=2. 对于每对相同的行或列，特征值等于 0；因此，由于第一和第四行/列相同，并且第二和第三行/列相同，因此两个特征值为零。最后这个矩阵的迹等于它的四个特征值之和；所以2+0+0+λ=0，因此特征值等于−2.

Griffith (2000, p. 98) 证明了方程的解。(2.2) 对于这种特殊类型的地理表面划分和 SWM 是由下式给出的特征向量
2(2+1)(2+1)[罪⁡(H圆周率2+1)×罪⁡(ķ圆周率2+1)]

[0.50.50.50.5 0.5−0.50.5−0.5 0.50.5−0.5−0.5 0.5−0.5−0.50.5]

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