### 统计代写|回归分析作业代写Regression Analysis代考| Interpreting an ESF and its parameter estimates

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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|Comparisons between ESF and SAR model specification

The simplest version of MESF accounts for $\mathrm{SA}$ by including a nonconstant mean in a regression model. The spatial SAR specification does this as well by including the term $\left[(1-\rho) \beta_{0} 1+\rho \mathbf{W Y}\right]$, where $\beta_{0}$ denotes the intercept term. The pure SA SAR model is specified as
$$\mathbf{Y}=(1-\rho) \beta_{0} 1+\rho \mathbf{W}+\boldsymbol{\varepsilon},$$
employing the row-standardized version of matrix $\mathbf{C}$, namely, matrix $\mathbf{W}$. For the Box-Cox transformed PD studied in this chapter, the maximum likelihood estimate of the SA parameter is $\hat{\rho}=0.70120$. This $\mathrm{SA}$ term

accounts for about $48.1 \%$ of the variance in the Box-Cox transformed PD across Texas. This percentage is less than the $62 \%$ for the ESF specification, in part because the SAR specification includes all, not only the relevant subset of, eigenvectors, introducing some noise into its estimation. Meanwhile, the SAR residual Shapiro-Wilk statistic, $0.96204$, is statistically significant $(p<0.0001)$. Both Getis and Griffith $(2002)$ and Thayn and Simanis (2013) present comparisons of spatial autoregressive and ESF analyses. An ESF specification frequently outperforms a spatial autoregressive specification.
Perhaps one of the greatest advantages MESF has vis-à-vis spatial autoregression is its ability to visualize the SA latent in a georeferenced attribute variable. It also has implementation advantages for generalized linear models (GLMs; see Chapter 5 ).

## 统计代写|回归分析作业代写Regression Analysis代考|Simulation experiments based upon ESFs

Griffith (2017) argues that MESF is superior to spatial autoregression for spatial statistical simulation experiments because it preserves an underlying map pattern and is characterized by constant variance; in other words, it supports conditional geospatial simulations. A spatial analyst can undertake a simulation experiment employing MESF in one of the following three ways: (1) draw a random error term from a normal distribution with mean zero and variance equal to the linear regression mean squared error; (2) randomly permute the n residuals calculated with linear regression estimation; and, (3) randomly sample, with replacement, the n residuals from the linear regression estimation (similar to bootstrapping). Each of these three strategies was used to perform a sensitivity analysis simulation for the ESF constructed in Section 3.2.2. Each simulation experiment involved 10,000 replications (to profit from the Law of Large Numbers).

The first simulation experiment added random noise $\varepsilon_{i} \sim \mathrm{N}\left(0,1.24350^{2}\right)$, $\mathrm{i}=1,2, \ldots, 254$, to the ESF + intercept tern (i.e., $4.40986$ ). The simulation mean of the map averages (based upon sets of $254 \varepsilon_{i}$ ) is $-0.00045$; the simulation mean of the map variances is $1.15826$. Fig. $3.5 \mathrm{~A}$ portrays the simulated mean map pattern for the simulated log-transformed PD values; it essentially is identical to the map pattern in Fig. 3.1B. The variances for the individual county simulations span the range from $1.13704^{2}$ to $1.18137^{2}$; the F-ratio for these two extreme variances is $1.08$, which is not statistically significant, yielding a single variance class (Fig. 3.5B). One important advantage of MESF vis-à-vis spatial autoregression-based simulation experiments is that the variance is constant across a geographic landscape, which is not the case

for spatial autoregression (see Griffith, 2017). The simulation mean $\mathrm{R}^{2}$ value is $0.6699$, which is somewhat greater than the actual $\mathrm{R}^{2}$ value. Meanwhile, the simulation mean Shapiro-Wilk probability is $0.50136 .$

Table $3.1$ tabulates the eigenvector selection significance level probabilities, Psig, as well as the eigenvector selection simulation probabilities, psimUsing a $10 \%$ level of significance selection criterion renders roughly a $10 \%$ chance that some of the 52 eigenvectors not selected in the original analysis are selected in a simulation analysis. The relationship between these two selection probabilities may be described as follows:
$$0.24\left(\mathrm{c}^{-3.35 p_{u}^{2.2}}-\mathrm{e}^{-3.35}\right), \mathrm{pscudo}^{-\mathrm{R}^{2}} \approx 1.0000$$

## 统计代写|回归分析作业代写Regression Analysis代考|ESF prediction with linear regression

Prediction is a valuable use of linear regression and is alluded to by the PRESS statistic. Redundant attribute information (i.e., multicollinearity) with the covariates supports the prediction of the response variable; each of these predictions is a conditional mean (i.e., a regression fitted value) based upon the given covariates used to compute it. An extension of this prediction capability is to observations not included in the original sample; a set of estimated regression coefficients enables the calculation of a prediction with covariates measured for out-of-sample observations. These supplemental observations have an additional source of variation affiliated with them, namely, their own stochastic noise, which is not addressed during estimation of the already-calculated regression coefficients.

Cross-validation offers an application of ESF prediction with linear regression. This prediction may be executed with the following modified pure SA linear regression specification when a single attribute variable value, $y_{\mathrm{m}}$, is miscing
$$\left(\begin{array}{c} \mathbf{Y}{\mathrm{o}} \ 0 \end{array}\right)=\beta{0} \mathbf{1}-\mathrm{y}{\mathrm{m}}\left(\begin{array}{c} \mathbf{0}{\mathrm{o}} \ 1 \end{array}\right)+\sum_{\mathrm{k}=1}^{\mathrm{K}}\left(\begin{array}{c} \mathbf{E}{\mathrm{o}, \mathrm{k}} \ \mathbf{E}{\mathrm{m}, \mathrm{k}} \end{array}\right) \boldsymbol{\beta}{\mathrm{E}{\mathrm{K}}}+\left(\begin{array}{c} \boldsymbol{\varepsilon}_{\mathrm{o}} \ 0 \end{array}\right),$$
where the subscript o denotes observed data, the subscript $m$ denotes missing data, and 0 is a vector of zeros. This specification subtracts the unknown data

values, $y_{\mathrm{m}}$, from both sides of the equation and then allows these values to be estimated as regression parameters (i.e., conditional means). In doing so, these conditional means are equivalent to their fitted values and hence have residuals of zero.

Fig. $3.7$ portrays the scatterplot of the log-transformed 2010 Texas PD (vertical axis) versus the corresponding 254 imputed values calculated with Eq. (3.8) but with no covariates (i.e., a pure SA specification); this exercise is similar to kriging. The linear regression equation describing this correspondence may be written as follows:
$$\hat{\mathrm{Y}}=0.98849+0.77990 \mathrm{Y}_{\text {predicted }}, \mathrm{R}^{2}=0.4078$$

## 统计代写|回归分析作业代写Regression Analysis代考|Comparisons between ESF and SAR model specification

MESF 相对于空间自回归的最大优势之一可能是它能够可视化地理参考属性变量中潜在的 SA。它还具有广义线性模型（GLM；见第 5 章）的实施优势。

## 统计代写|回归分析作业代写Regression Analysis代考|Simulation experiments based upon ESFs

Griffith (2017) 认为，MESF 在空间统计模拟实验中优于空间自回归，因为它保留了基础地图模式并且具有恒定方差的特点；换句话说，它支持有条件的地理空间模拟。空间分析师可以通过以下三种方式之一使用 MESF 进行模拟实验： (1) 从均值为零且方差等于线性回归均方误差的正态分布中绘制随机误差项；(2) 随机排列用线性回归估计计算的n个残差；(3) 随机抽取线性回归估计的 n 个残差进行替换（类似于自举）。这三种策略中的每一种都用于对第 3.2.2 节中构建的 ESF 进行敏感性分析模拟。

0.24(C−3.35p在2.2−和−3.35),psC在d这−R2≈1.0000

## 统计代写|回归分析作业代写Regression Analysis代考|ESF prediction with linear regression

(是这 0)=b01−是米(0这 1)+∑ķ=1ķ(和这,ķ 和米,ķ)b和ķ+(e这 0),

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