## 统计代写|回归分析作业代写Regression Analysis代考|AH7722

2022年10月17日

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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
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## 统计代写|回归分析作业代写Regression Analysis代考|Conditional Distributions of the Bivariate Normal Distribution

The term “regression” appeared in the late 19 th century in Sir Francis Galton’s writings. Using statistical analysis of data on $Y=$ son’s adult height and $X=$ father’s adult height, Galton noticed that, among fathers of above-average height, their sons tended to be shorter than their fathers, but still taller than the general average male. Conversely, among fathers of below-average height, their sons tended to be taller than their fathers, but still shorter than the general average male. Galton coined the phrase “regression to the mean” to describe this phenomenon, because the sons’ heights tended away from their fathers’ heights, toward the overall mean height.

Galton’s observations can be illustrated in terms of the bivariate normal distribution, a joint distribution of $(X, Y)$ pairs that (i) requires that both $X$ and $Y$ have marginal normal distributions, and (ii) allows that $X$ and $Y$ are correlated. Thus, the bivariate normal distribution has five parameters, namely $\mu_x, \sigma_x, \mu_y, \sigma_y$, and $\rho$, where $\rho$ is the correlation parameter.
The joint probability density function of a bivariate normal $(X, Y)$ pair gives the relative likelihood of the various $(x, y)$ combinations, and its mathematical form is
$$p(x, y)=\frac{1}{2 \pi \sigma_x \sigma_y \sqrt{1-\rho^2}} \exp \left{-\frac{1}{2\left(1-\rho^2\right)}\left(z_x^2+z_y^2-2 \rho z_x z_y\right)\right}$$
where $z_x=\left(x-\mu_x\right) / \sigma_x$, and $z_y=\left(y-\mu_y\right) / \sigma_y$.

To illustrate the meaning of this distribution using some realistic numbers, suppose fathers and sons’ heights both have the same mean, $\mu_x=\mu_y=175 \mathrm{~cm}$, as well as the same standard deviation $\sigma_x=\sigma_y=8 \mathrm{~cm}$. The correlation parameter $\rho$ governs similarity between fathers’ and sons’ heights, with $\rho=1$ meaning sons have exactly the same height as their fathers, and $\rho=0$ meaning sons’ heights are completely unrelated to fathers’ heights. Clearly, there is some relationship, but not a perfect one, so $0<\rho<1$ in this case. Figure A.1 displays the bivariate normal density where $\rho=0.4$, indicating a moderately weak relationship between $X$ and $Y$.

## 统计代写|回归分析作业代写Regression Analysis代考|Estimating Regression Model Parameters

So, LOESS is good but not necessarily best. Sometimes linear models are good even though they are almost always wrong. How can you know which estimates to use?

Besides simulation, another guiding principle we will use throughout this book is likelihood. Methods based on likelihood are usually excellent. While not infallible, they can be considered as a “gold standard” of statistical methods: If your data come from particular models $p(y \mid x)$, and if you analyze your data using maximum likelihood that assumes those same particular models, then your analysis will be nearly ideal.

Least squares estimation, the most common method for analyzing regression data, is itself motivated by likelihood, since the least squares estimates are in fact the maximum likelihood estimates that you get when you assume $p(y \mid x)$ is a normal distribution. This fact can be viewed as a lucky coincidence: If the normal distribution did not have a squared term in its exponent, then you would not use least squares. Instead, you would use least absolute deviations, or some other method, as the default for regression analysis.

In addition, likelihood-based methods are an essential first step toward Bayesian methods, which are rapidly becoming an essential statistical tool for all scientists. Finally, standard methods for regression with non-normal distributions, such as logistic regression and Poisson regression use likelihood-based analyses by default, so you need to understand likelihood in order to read the computer output.

We begin this chapter by reviewing likelihood-based methods, with special attention to their use in regression.

# 回归分析代写

## 统计代写|回归分析作业代写回归分析代考|二元正态分布的条件分布

“回归”一词出现在19世纪晚期弗朗西斯·高尔顿爵士的著作中。通过对$Y=$儿子成年身高和$X=$父亲成年身高的数据进行统计分析，高尔顿注意到，在身高高于平均水平的父亲中，他们的儿子往往比父亲矮，但仍然比男性的平均身高高。相反，在身高低于平均水平的父亲中，他们的儿子往往比父亲高，但仍然比男性平均水平矮。高尔顿创造了“回归均值”一词来描述这一现象，因为儿子的身高倾向于远离父亲的身高，向整体平均身高靠拢

Galton的观察可以用二元正态分布来说明，$(X, Y)$对的联合分布(i)要求$X$和$Y$都具有边际正态分布，(ii)允许$X$和$Y$是相关的。因此，二元正态分布有五个参数，分别是$\mu_x, \sigma_x, \mu_y, \sigma_y$和$\rho$，其中$\rho$是相关参数。二元正态$(X, Y)$对的联合概率密度函数给出了各种$(x, y)$组合的相对似然，其数学形式为
$$p(x, y)=\frac{1}{2 \pi \sigma_x \sigma_y \sqrt{1-\rho^2}} \exp \left{-\frac{1}{2\left(1-\rho^2\right)}\left(z_x^2+z_y^2-2 \rho z_x z_y\right)\right}$$

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。