### 统计代写|回归分析作业代写Regression Analysis代考|Different measurements for different data types

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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|Quantifying SA

Similar to correlation coefficients in classical statistics, a SA index may be specific to an attribute’s measurement scale (i.e., nominal, ordinal, interval, and ratio). Similar to the Pearson product moment correlation coefficient, $r$, the Moran coefficient $(\mathrm{MC})$, the most widely employed $S \mathrm{~A}$ index, can be used with all measurement scales (Griftith, 2010). The $\mathrm{MC}$, originally designed for interval/ratio data, may be defined as follows:
$\sum_{j=1}^{n} \sum_{j=1}^{n} c_{i j}\left(y_{i}-\bar{y}\right)\left(y_{j}-\bar{y}\right)$
$\frac{\sum_{i=1}^{n} \sum_{j=1}^{n} c_{i j}}{\sum_{j=1}^{n}\left(y_{i}-\bar{y}\right)^{2} / n}=\frac{n}{\sum_{i=1}^{n} \sum_{j=1}^{n} c_{i j}} \frac{\sum_{i=1}^{n}\left(y_{i}-\bar{y}\right)\left[\sum_{j=1}^{n} c_{i j}\left(y_{j}-\bar{y}\right)\right]}{(n-1) s^{2}}$

where $c_{\mathrm{ij}}$ is an entry in the SWM $\mathrm{C}, \bar{\gamma}$ is the arithmetic mean and $s^{2}$ is the sample variance of response variable $Y$, and $z_{i}$ is the $z$-score for attribute value $y_{i}$ – The numerator of the $M C$ contains the pairs of values $z_{i}$ and $\sum_{j=1}^{n} c_{i j} z_{j}$, whose graphic portrayal is the Moran scatterplot (see Section 1.2.2). Like r, the $\mathrm{MC}$ is a covariation-based index. Unlike $\mathrm{r}-$ whose extremes are $-1$ (a perfect indirect relationship) and one (a perfect direct relationship), and for which zero denotes no correlation-the MC’s extreme values essentially are a function of the smallest and second largest eigenvalues (a topic treated in ensuing sections) of the employed SWM, and for which $-1 /(n-1)$ denotes no SA for a single RV. Frequently, the smallest possible $\mathrm{MC}$ is closer to $-0.5$, whereas the largest possible $\mathrm{MC}$ is closer to 1.15. For example, the extreme MCs for the 254-county Texas $\mathrm{~ ร ु W M ~ 1 m a n 1 ~ ” I n ~ a ~ a m и}$ and based upon a paired comparisons perspective are $-0.43024$ and $0.89546$ (see de Jong, Sprenger, \& van Veen, 1984).

The Geary ratio (GR) is a second popular $\mathrm{SA}$ index formulated for interval/ratio data. Rather than being based upon cross-products (i.e., covariation), its basis is paired comparisons, or squared differences between those pairs of attribute values whose corresponding row and column entries in the SWM are a positive value. The GR may be defined as follows:
$$\frac{\sum_{i=1}^{n} \sum_{j=1}^{n} r_{i i}\left(y_{i}-y_{j}\right)^{2} / \sum_{i=1}^{n} \sum_{j=1}^{n} r_{i i}}{2 \sum_{i=1}^{n}\left(y_{i}-\bar{y}\right)^{2} /(n-1)}=\frac{n-1}{\sum_{i=1}^{n} \sum_{j=1}^{n} c_{i j}} \frac{\sum_{i=1}^{n}\left(\sum_{j=1}^{n} r_{i i}\right)\left(y_{i}-\bar{y}\right)^{2}}{\sum_{i=1}^{n}\left(y_{i}-\bar{y}\right)^{2}}$$

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

Distributional theory refers to the sampling distribution of a statistic, which a researcher needs for hypothesis-testing purposes. Here the common null hypothesis is zero SA. Cliff and Ord (1981) present this distributional theory for both randomly sampling an attribute from a normal RV (i.e., the normality assumption) and randomization of a given set of attribute values. They extend this former case to linear regression error terms estimated with ordinary least squares (OLS).

For the case of a single RV, the expected value of the sampling distribution of the MC for either the random sampling or randomization inferential basis is $-1 /(n-1)$, signifying zero $\mathrm{SA}$. The asymptotic standard error for this case is $\sqrt{\sum_{i=1}^{n} \sum_{j=1}^{n} c_{y}}$; this asymptotic result is extremely good by $\mathrm{n}>25$ when no covariates are included in an analysis. As $n$ goes to infinity, the sampling distribution of $\mathrm{MC}$ converges on a normal distribution. Accordingly, the test statistic is given by
$$z=\sqrt{\sum_{i=1}^{\infty} \sum_{j=1}^{n} c_{i j}} \frac{M C+{ }_{11=1}^{1}}{\sqrt{2}} .$$
Although the most natural alternate hypothesis is that SA is not equal to zero (i.e., a two-tailed test), because almost all geographic phenomena exhibit PSA, spatial researchers almost always could argue for an alternate hypothesis of PSA.
For linear regression residuals, the variance approximation remains useful when the number of covariates is not a function of $n$ and the spatial structure is far from a geographic maximum connectivity case (these two characteristics generally hold in practice; see Appendix 1.A). Similarly, although the expected value is a function of the covariates included in a linear regression equation specification, it tends to converge to zero from a negative value as $n$ increases, given that the number of covariates remains the same.

## 统计代写|回归分析作业代写Regression Analysis代考|Impacts of SA on attribute statistical distributions

The principal impact of SA is on the variance of a RV: PSA inflates variance. Table $1.1$ summarizes results from a simulation experiment in which the spatially autocorrelated RVs contain moderate PSA (approximately, MC $=0.7$ and $G R=0.3$ ). The four RVs represent the ones most commonly employed in spatial analyses. The mean does not change, whereas the variance substantially increases (c.g., a variance inflation factor of $1.2^{2}$ to $3.0^{2}$ ) once SA is embedded in a RV. Skewness (i.e., symmetry) tends to be impacted less than variance, with the Poisson RV indicating that existing skewness can be exacerbated by the presence of PSA. Finally, kurtosis (peakedness) tends to be noticeably impacted by the presence of PSA. These summary statistics reveal that the general effect of PSA is to shrink the frequencies of the more central values of a RV and to inflate the frequencies of the values located away from that RV distribution’s center (e.g., arithmetic mean): a more platykurtic distribution with fatter tails.

SA is a two-dimensional concept. As such, visualizing it helps to understand it. The nelevant systenatic urganization of geureferenced atwibute values is a map pattern. A variety of tools exist that highlight map patterns associated with SA. An obvious one is a map. Fig. $1.1$ presents five map patterns depicting different natures and degrees of SA. With regard to the preceding discussion of variance inflation, Fig. 1.1A implies that marked PSA decreases within regions variation as well as increases between regions variation. Fig. 1.1E implies that marked NSA increases within regions variation as well as decreases between regions variation.

Fig. $1.2$ presents an example of the variance inflation introduced into a normal RV by PSA. Fig. 1.2A is the histogram for independent and identically distributed (IID) random observations. Its range is roughly $-3$ to 3 . Fig. 1.2B is the histogram for these same data after embedding PSA in them. Its range is roughly $-5$ to 5 . The highest bar in Fig. $1.2 \mathrm{~A}$ is about $35 \%$, whereas the highest bar in Fig. $1.2 \mathrm{~B}$ is about $23 \%$. The tails in Fig. 1.2B are much heavier than those in Fig. 1.2A. Nevertheless, both frequency distributions center on zero, and both are reasonably symmetric.

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

∑j=1n∑j=1nC一世j(是一世−是¯)(是j−是¯)
∑一世=1n∑j=1nC一世j∑j=1n(是一世−是¯)2/n=n∑一世=1n∑j=1nC一世j∑一世=1n(是一世−是¯)[∑j=1nC一世j(是j−是¯)](n−1)s2

Geary比率（GR）是第二个流行的小号一种为区间/比率数据制定的指数。其基础不是基于叉积（即协变），而是基于成对比较，或在 SWM 中对应的行和列条目为正值的那些属性值对之间的平方差。GR 可以定义如下：
∑一世=1n∑j=1nr一世一世(是一世−是j)2/∑一世=1n∑j=1nr一世一世2∑一世=1n(是一世−是¯)2/(n−1)=n−1∑一世=1n∑j=1nC一世j∑一世=1n(∑j=1nr一世一世)(是一世−是¯)2∑一世=1n(是一世−是¯)2

## 统计代写|回归分析作业代写Regression Analysis代考|Impacts of SA on attribute statistical distributions

SA 的主要影响是对 RV 的方差：PSA 夸大了方差。桌子1.1总结了模拟实验的结果，其中空间自相关的 RV 包含中等 PSA（大约 MC=0.7和GR=0.3）。这四个 RV 代表了空间分析中最常用的那些。均值不变，而方差显着增加（cg，方差膨胀因子为1.22到3.02) 一旦 SA 嵌入到 RV 中。偏度（即对称性）受到的影响往往小于方差，泊松 RV 表明存在 PSA 会加剧现有的偏度。最后，峰度（峰度）往往会受到 PSA 的显着影响。这些汇总统计数据表明，PSA 的一般效果是缩小 RV 更中心值的频率，并夸大远离该 RV 分布中心的值的频率（例如，算术平均值）：更扁平的分布与更肥的尾巴。

SA 是一个二维的概念。因此，将其可视化有助于理解它。geureferenced atwibute 值的相关系统化是一种映射模式。存在多种突出与 SA 相关的地图模式的工具。一个明显的就是地图。如图。1.1呈现了五种地图模式，描绘了 SA 的不同性质和程度。关于前面对方差膨胀的讨论，图 1.1A 表明显着的 PSA 在区域变化内降低，而在区域变化之间增加。图 1.1E 意味着显着的 NSA 在区域变化内增加，在区域变化之间减少。

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## MATLAB代写

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