### 统计代写|回归分析作业代写Regression Analysis代考|Visualizing map patterns with eigenvectors

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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|Visualizing map patterns with eigenvectors

By convention, a SWM’s rows and columns are ordered with the same sequence of areal units. The eigenvectors of a SWM are n-by-1 vectors, with each of their rows linking to the corresponding areal unit identifier (ID) for the same row in their parent SWM. This one-to-one correspondence

enables mapping the eigenvectors of a SWM. The individual eigenvector elements simply have to be joined to their corresponding polygons in a shapefile file for visualization by ArcMap, for example.

Fig. $2.4$ portrays the extreme eigenvectors for matrix $\left(I-11^{\mathrm{T}} / \mathrm{n}\right) \times$ $\mathbf{C}(\mathbf{I}-\mathbf{1 1} / \mathrm{n})$ representing the 2010 DFW metroplex census tracts. Fig. 2.4A portrays the principal eigenvector map pattern, the maximum PSA case, which depicts a hill/valley (this would change from a hill to a valley, or vice versa, by multiplying the eigenvector by $-1$ ) roughly in the center of the region, surrounded by a trough/hill, with intermediate plateaus in the western and southeastern parts of the region. This hill map pattern is one of the common global geographic trends portrayed by eigenvectors. Another typical map pattern is an east-west gradient trend, adapting to the somewhat rectangular shape of the DFW metroplex. Fig. $2.3 \mathrm{~A}$ reveals that the hill/valley subregion in Fig. 2.4A coincides with a concentration of numerous smaller census tracts located mostly in Dallas and Tarrant Counties. In contrast, Fig. 2.4B portrays a map pattern with considerable neighboring contrasts (e.g., dark red in juxtaposition with dark green), an alternating trend that exemplifies NSA. This map pattern has its intermediate values in its periphery, encircling the stark contrasts created by the alternating trend. A comparison of Fig. 2.4A and B reveals that PSA exhibits more smoothness in its map pattern, and NSA exhibits more fragmentation in its map pattern. These tendencies are less conspicuous in visualizations of the less extreme MC values.

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

A simple one-dimensional geographic landscape (e.g., Fug. $2.5 \mathrm{~A}$ ) illustrates the connection between the spatial frequency and spatial spectral domains, furnishing the basis of the spectral density function equations presented in this section. SA indicates how rapidly a map pattern changes across a geographic landscape, using randomness as its yardstick. SA contains information about the expected frequency content mentioned earlier in this chapter. Meanwhile, the spectral density function furnishes one

mathematical link between $\mathrm{SA}$ and eigenfunctions in the spectral domain (Section 2.1; Bartlett, 1975). For a one-dimensional geographic landscape (e.g., Fig. $2.5 \mathrm{~A}$ ), given a stationary process $\left{\mathrm{X}{v}, \mathrm{t}=1,2, \ldots, \mathrm{T}\right}$ for a unidirectional dependency structure (this situation parallels a time series data structure) of the same form as the simultaneous autoregressive (SAR) model specification, $$\mathrm{X}{\mathrm{t}}=\rho \mathrm{X}{\mathrm{t}-1}+\xi{\mathrm{t}},$$
where $\rho$ is the autocorrelation parameter, $|\rho|<1$, t indexes location in the linear landscape, and $\xi_{t}$ is an independent and identically distributed (IID) random error term. Autocovariance functions, $\gamma(\tau), \tau=0,1, \ldots$, where $\tau$ denotes the number of lags, are established by repeatedly substituting the right-hand expression of an autoregressive equation like the preceding one into it and then applying the calculus of expectations to resulting infinite series forms of covariations. These infinite series are functions of linear combinations of $\xi_{t}^{2}$ together with powers of the SA parameter $\rho$, and hence yield $\sigma^{2}$ as well as a denominator term containing $\rho$. The corresponding atocovariance (i.e., covariation in an autocorrelated mathematical space) function is defined as follows:
$$\gamma_{\mathrm{x}}(\tau)=\frac{\rho^{\tau} \sigma^{2}}{1-\rho^{2}}$$
with $\tau$ being the number of lags (Griffith, 1988 , p- 111).
This preceding autocovariance function can be represented in the fretrency domain with the Fontrier transform
$$f(\theta)=\frac{1}{2 \pi} \sum_{\tau=-\infty}^{\infty} e^{-i \pi \theta_{1}} \gamma_{X}(\tau)$$

## 统计代写|回归分析作业代写Regression Analysis代考|This representation of the autocovariance

This representation of the autocovariance function in the frequency domain is referred to as the spectral density function or the spectrum of the random variable (RV) $\left{X_{t}\right}$. The sum ${ }^{6}$ of the spectrum over all frequencies gives the variance of $\left{X_{t}\right}$,
$$\operatorname{VAR}\left(X_{t}\right)=\int_{0}^{\pi} \frac{\sigma^{2}}{2 \pi\left[1+\rho^{2}-2 \rho \operatorname{COS}(\theta)\right]} d \theta=\frac{\sigma^{2}}{2\left(1-\rho^{2}\right)},|\rho|<1,$$
whereas the covariance at $\operatorname{lng} \tau$ is given by
$$\operatorname{COV}\left(X_{t} X_{t-1}\right)=\int_{0}^{\pi} \frac{\sigma^{2} \operatorname{COS}(\tau \theta)}{2 \pi\left[1+\rho^{2}-2 \rho \operatorname{COS}(\theta)\right]} d \theta=\frac{\rho^{\tau} \sigma^{2}}{2\left(1-\rho^{2}\right)},|\rho|<1 .$$
$\tau=0$ yields Eq. (2.6). In other words, representing the autocovariance function in the frequency domain allows a decomposition of the variance into frequency components that reveals their relative importance. The common effect of variance inflation routinely mentioned as a primary result of SA is apparent here: its arithmetic source is the term $-\rho^{2}$ appearing in the denominator of this expression. When no $S A$ is present, $\rho=0$ and $f(\theta)=\sigma^{2} / 2 \pi$, which is the spectral density for white noise. Accordingly, the variance in this independent observations case reduces to:

$$\operatorname{VAR}\left(X_{\tau}\right)=\int_{0}^{\pi} \frac{\sigma^{2}}{2 \pi} \mathrm{d} \theta=\frac{\sigma^{2}}{2},$$
which is consistent with results from the classical central limit theorem of statistics.

Furnishing asymptotic detaik for Eqs. (2.5) and (2.6), as shown by Berman and Plemmons (1994), the eigenvalues of a binary SWM for linear lattices can be defined as
$$\lambda_{\mathrm{i}}=2 \operatorname{COS}\left(\frac{i \pi}{n+1}\right) .$$
For the spectral density function defined by Eq. (2.5),
$$\begin{gathered} \lim {n \rightarrow \infty} \frac{i}{n+1}=\theta, \ \lambda{\mathrm{i}}=2 \operatorname{COS}(\theta), \text { and } \ f(\theta)=\frac{\sigma^{2}}{2 \pi\left(1+\rho^{2}-\rho \lambda\right)} . \end{gathered}$$
In other words, the spectral density function contains the eigenvalues of its corresponding SWM, which reveals that the autocovariance function for the RV $\left{X_{t}\right}$ is a function of the eigenvalues of the SWM.

To illustrate the relationship between the lagged spatial correlations with the second-order spatial covariance structure matrix $(\mathbf{I}-\rho \mathbf{C})^{-2}$ the SAR model specification-consider the bidirectional and one-dimensional geographic landscape with a symmetric dependency structure; that is,
$$\mathrm{X}{\mathrm{i}}=\rho\left(\mathrm{X}{\mathrm{i}-1}+\mathrm{X}{\mathrm{i}+1}\right)+\xi{\mathrm{i}},|\rho|<\frac{1}{2}$$
Here the autovariances at lags 0 and $\tau$ are
$$\gamma(0)=\mathrm{E}\left(\mathrm{X}{\mathrm{i}} \mathrm{X}{\mathrm{i}}\right)=\frac{\sigma^{2}}{\left(1-4 \rho^{2}\right)^{3 / 2}}, \text { and }$$
$\gamma(\tau)=\mathrm{E}\left(\mathrm{X}{\mathrm{i}} \mathrm{X}{\mathrm{i}-\tau}\right)$
$=\sigma^{2} \frac{\operatorname{SIN}[\tau \pi] \text { Hypergeometric PFQ }\left[\left{\frac{1}{2}, 1,2\right},{1-\tau, 1+\tau} 1-\tau, \frac{4 p}{1+2 p}\right]}{\pi \tau(1+2 p)^{2}}$ $0<\rho<1 / 2$

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

CX(τ)=ρτσ21−ρ2

F(θ)=12圆周率∑τ=−∞∞和−一世圆周率θ1CX(τ)

## 统计代写|回归分析作业代写Regression Analysis代考|This representation of the autocovariance

τ=0产生方程。(2.6)。换句话说，在频域中表示自协方差函数允许将方差分解为频率分量，从而揭示它们的相对重要性。作为 SA 的主要结果，经常提到的方差膨胀的共同影响在这里很明显：它的算术来源是术语−ρ2出现在这个表达式的分母中。没有时小号一种存在，ρ=0和F(θ)=σ2/2圆周率，这是白噪声的频谱密度。因此，这种独立观察情况下的方差减少到：曾是⁡(Xτ)=∫0圆周率σ22圆周率dθ=σ22,

λ一世=2COS⁡(一世圆周率n+1).

$$\begin{gathered} \lim {n \rightarrow \infty} \frac{i}{n+1}=\theta, \ \lambda {\mathrm{i}}=2 \operatorname{COS }(\theta), \text { 和 } \ f(\theta)=\frac{\sigma^{2}}{2 \pi\left(1+\rho^{2}-\rho \lambda\right )} 。 \end{gathered}$$

X一世=ρ(X一世−1+X一世+1)+X一世,|ρ|<12

C(0)=和(X一世X一世)=σ2(1−4ρ2)3/2, 和
C(τ)=和(X一世X一世−τ)
=\sigma^{2} \frac{\operatorname{SIN}[\tau \pi] \text { 超几何 PFQ }\left[\left{\frac{1}{2}, 1,2\right},{ 1-\tau, 1+\tau} 1-\tau, \frac{4 p}{1+2 p}\right]}{\pi \tau(1+2 p)^{2}}=\sigma^{2} \frac{\operatorname{SIN}[\tau \pi] \text { 超几何 PFQ }\left[\left{\frac{1}{2}, 1,2\right},{ 1-\tau, 1+\tau} 1-\tau, \frac{4 p}{1+2 p}\right]}{\pi \tau(1+2 p)^{2}} 0<ρ<1/2

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## MATLAB代写

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