### 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Check for Multivariate Normality

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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|多元统计分析代写Multivariate Statistical Analysis代考|Check for Multivariate Normality

When we assume a multivariate normal distribution for data analysis, we should check to see if this assumption is supported by the data. Unlike univariate normal distribution, it is not straightforward to check multivariate normal distributional assumption. In the following, we discuss methods that may be used to check multivariate normality.
First, as a simple and naive approach, we may consider methods for checking univariate normality, which may also be useful for checking multivariate normality, Note that, if observations were generated from a multivariate normal distribution, then each univariate distribution will also be normal. In other words, if the univariate data are not normally distributed, then the multivariate data will not be normally distributed either. Boxplots and histograms can be used to check if the univariate data are symmetric or not (note that all normal data are symmetric), but symmetric data are not necessarily normal. If the univariate data are not symmetric, then the normality cannot hold. A more formal method to check univariate normality is normal quantile-quantile (Q-Q) plot. A normal $Q$ – $Q$ plots shows the theoretical quantiles from a normal distribution and the quantiles computed from the data. If a Q-Q plot shows a straight line $(y=x)$, then the univariate data may be considered as normally distributed. A more formal method, called the Shapiro-Wilk test, may also be used to check normality.

A formal method for checking multivariate normality is the Chi-Squared plot. It is a generalization of a Q-Q plot based on the squared Mahalanobis distance
$$d_{j}^{2}=\left(\mathbf{x}{j}-\overline{\mathbf{x}}\right)^{\mathrm{T}} \boldsymbol{S}^{-1}\left(\mathbf{x}{j}-\overline{\mathbf{x}}\right), \quad j=1,2, \cdots, n$$
where $\mathbf{x}{1}, \mathbf{x}{2}, \cdots, \mathbf{x}{n}$ are the sample observations, $\overline{\mathbf{x}}$ is the sample mean vector, and $S$ is the sample covariance matrix. If the population is multivariate normal and $n-p$ is large, each of the squared distances $d{1}^{2}, d_{2}^{2}, \cdots, d_{n}^{2}$ should behave as a chi-square random variable. We can order the squared distances as $d_{(1)}^{2} \leqslant d_{(2)}^{2} \leqslant \cdots \leqslant d_{(n)}^{2}$, and then graph the pairs $\left(q((j-1 / 2) / n), d_{(j)}^{2}\right)$, where $q((j-1 / 2) / n)$ is the $(j-1 / 2) / n$ quantile of the chi-square distribution with $p$ degrees of freedom. Under multivariate normality, the plot should resemble a straight line through the origin with slope 1 . A systematic curved pattern indicates that normality may not hold. A few points far above the line suggests outliers. We give an example below.

To illustrate the idea of the Chi-Squared plot, we simulate a sample of size 100 from the bivariate normal distribution $\left(x_{1}, x_{2}\right) \sim N_{2}(\boldsymbol{\mu}, \Sigma)$, with the mean vector and the covariance matrix given by

$$\mu=\left(\begin{array}{c} -1 \ 1 \end{array}\right), \quad \Sigma=\left(\begin{array}{cc} 10 & -3 \ -3 & 2 \end{array}\right)$$
Then we draw Q-Q plots to verify univariate normalities of the component random variables $x_{1}$ and $x_{2}$ respectively, and draw a Chi-Squared plot to verify the bivariate normality of the random vector $\left(x_{1}, x_{2}\right)$. See Figure $1.7$. The $\mathrm{R}$ code is given below.

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Unsupervised Learning and Supervised Learning

Statistics is about learning from data. In practice, data are often collected on more than one variables, so many statistical methods may be viewed as multivariate analysis in a broad sense. In general, we may classify these methods into two general approaches:

• Unsupervised learning: we treat all variables symmetrically or equally, with the goal of understanding the underlying association structures between these variables. Examples of unsupervised learning include principal components analysis, factor analysis, and cluster analysis.
• Supervised learning: we treat one or more variables as responses and other variables as predictors which are used to partially explain the variations in the responses. Regression models are examples of supervised learning.

These two approaches are used to answer different questions, so the choice of methods depends on the study objectives. There is a wide variety of statistical methods available for each approach. If our goal is to understand the relationship among all the variables or if we want to reduce the number of variables, we should consider unsupervised learning methods. If our objective is to predict one or more variables using the other variables or to explain the variations in one or more variables using other variables, we should use regression models.

In unsupervised learning, all variables are treated equally and the goal is to understand the covariance structures in these variables or to reduce the dimension of the data space. Commonly used unsupervised learning methods for multivariate continuous data include principal components analysis (PCA), factor analysis, discriminant analysis, and cluster analysis, For example, in PCA and factor analysis, the original set of variables can be replaced by a smaller set of new variables which may explain most of the variation in the original data. These new variables are usually special linear combinations of the original variables, and they allow us to use graphical tools to display the data and to interpret the data more easily than the original sets of variables. In discriminant analysis and cluster analysis, we classify multivariate data into different clusters based on the “distances” between the observations. In these proce-

dures, distributional assumptions for the data may not be needed. The covariance matrices or correlation matrices play the key role.

Regression models are among the most useful statistical methods. There are many types of regression models. The types of regression models are determined based on the types of the response variables, not the types of predictors. For example, if the response is a continuous variable, we may consider a linear regression model, but if the response is a binary (discrete) variable, we may consider a logistic regression model. The following regression models are commonly used in practice; linear models, nonlinear models, generalized linear models, survival models, and models for longitudinal data or clustered data. Linear regression models are often considered when the response variables are continuous and roughly normally distributed. Analysis of variance (ANOVA) models are special linear models in which all predictors are categorical or discrete. Nonlinear regression models may be used when the response variables are continuous and roughly normally distributed, and there is a good understanding of the mechanisms that generate the data. Generalized linear models (GLMs) are often used when the response variables are binary or count or follow distributions in the exponential family. Survival models are used when the response variables are the times to some events of interest, such as times to death or times to accidents. The foregoing regression models are used for independent data. When the data are correlated or clustered, we should use models for clustered data.

In a regression model, if there are more than one response variables, the model is called a multivariate regression model. For example, if there are two or more responses in an ANOVA model, the model is called a multivariate $A N O V A$ (MANOVA) model.

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Statistical Inference

A main goal of statistical inference is to generalize the results obtained from a sample to the general population. To achieve this, the sample has to be representative of the population, and the data are assumed to follow some parametric distributions such as normal distributions. Such an assumption allows us to do probability calculation required in inference (e.g., p-values in hypothesis testing).

Multivariate continuous data are often assumed to follow multivariate normal distributions. Under this distributional assumption, we can perform usual statistical inference, such as confidence regions and hypothesis testing for the unknown population mean vectors or the covariance matrices. For example, for univariate continuous data, the most well-known test is perhaps the $t$-test for the population mean, while for multivariate continuous data, the most well-known test is perhaps the Hotelling’s $T^{2}$-test for the mean vector. A major consideration in multivariate analysis is to incorporate the correlation between the variables. This allows for more efficient inference

than univariate analysis, which ignores the correlation between variables,
Multivariate discrete data are often assumed to follow multinomial distributions. Under this distributional assumption, statistical inference for the unknown parameters can be done using standard methods such as the maximum likelihood method. The simplest and also the most common multivariate discrete data are often summarized by $2 \times 2$ tables. For example, we may want to compare two methods, with the response being either positive or negative. The results can then be summarized by a $2 \times 2$ table. Many statistical methods are available to analyze such $2 \times 2$ tables. More general multivariate discrete data may be summarized by $k \times m$ contingency tables.

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Check for Multivariate Normality

dj2=(Xj−X¯)吨小号−1(Xj−X¯),j=1,2,⋯,n

μ=(−1 1),Σ=(10−3 −32)

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Unsupervised Learning and Supervised Learning

• 无监督学习：我们对称或平等地对待所有变量，目的是了解这些变量之间的潜在关联结构。无监督学习的例子包括主成分分析、因子分析和聚类分析。
• 监督学习：我们将一个或多个变量视为响应，将其他变量视为预测变量，用于部分解释响应的变化。回归模型是监督学习的例子。

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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。