### 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Examples in R

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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|Examples in R

In this section, we consider several real data examples for PCA and illustrate the use of software R for PCA.

Example 1. In this study (Johnson and Wichern, 2007), 48 individuals who had applied for a job with a large firm were interviewed and rated on 15 criteria. Individuals were rated on the form of their letter of application (FL), their appearance (APP), academic ability (AA), likability (LA), self-confidence (SC), lucidity (LC), honesty(HON), salesmanship (SMS), experience (EXP), drive (DRV), ambition (AMB), ability to grasp concepts (GSP), potential (POT), keenness to join (KJ), and their suitability (SUIT). Each criterion was evaluated on a scale ranging from 0 to 10 , with 0 being a very low and very unsatisfactory rating, and 10 being a very high rating. In this example, there are 15 variables, so the dimension of the data is 15 . So it is difficult to graphically display the data, and to check any outliers and multivariate normality of the data. We consider a PCA to reduce the dimension.

We see that all 15 variables are correlated and some are highly correlated (e.g., $\mathrm{POT}$ and GSP have a correlation of $0.9, \mathrm{LC}$ and $\mathrm{GSP}$ also have a correlation of $0.9$, etc). Since all variables here have comparable scales, we may perform PCA on the sample covariance matrix $S=\hat{\Sigma}$.

In the above results, the “Cumulative Proportion” row gives the cumulative proportions of variation explained by the first $\mathrm{PC}$, the first two $\mathrm{PCs}$, the first three $\mathrm{PCs}$, etc. We see that the first $2 \mathrm{PCs}$ explain $69 \%$ variations (the first two eigenvalues of $\hat{\Sigma}$ are $66.54$ and $18.18$ respectively): the first $\mathrm{PC}$ explains $54 \%$ variation and the second $\mathrm{PC}$ explains $15 \%$ variation. The first $3 \mathrm{PC}$ ‘s explain about $78 \%$ of total variations. Thus, the 15 -dimensional original data may be reduced to 2 or 3 dimensions!

The “new data” (i.e., the PC scores) can be obtained from the original data. To show graphical displays of the new data, we consider PC scores from the first two PCs and then we graphically display the PC scores to check normality of the new data and any possible outliers.

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|The Basic Idea

In principal components analysis (PCA), we try to reduce the dimension of multivariate data to simplify multivariate analysis. A PCA gives us some idea about the minimal number of variables which contain most information in the original set of variables. A disadvantage of PCA is that the principal components may not always have practical interpretations. That is, sometimes the principal components do not have meaningful interpretations in practice, which is a disadvantage in real data analysis. In this section, we try to determine the minimal set of variables which also have meaningful interpretations in practice. Such analysis is called factor analysis, and the factors usually have practical interpretations.

Factor analysis (FA) tries to describe the variance-covariance relationship among variables in terms of a smaller set of unobservable and uncorrelated new random variables called factors. These factors cannot be directly observed, but they have practical meanings and can be used for data analysis. The essential idea of factor analysis is to group the original variables so that all variables in the same group are highly correlated. Each group then represents a factor (a new variable) that explains the variation and correlation in the original variables in the group. The original set of variables may then be replaced by these factors in data analysis. Thus, factor analysis is closely related to PCA. A main difference is that the factors have practical meanings while the principal components may not have practical meanings.

For example, suppose that a multivariate dataset contains exam scores on mathematics, physics, computer science, English, Chinese, French, income, education, and professionals. We wish to perform a multivariate analysis on this dataset. We see that the dimension of the original data is 9 , and we hope to reduce the dimension in data analysis. Note that exam scores on mathematics $\left(x_{1}\right)$, physics $\left(x_{2}\right)$, and computer science $\left(x_{3}\right)$ may be represented by an unobservable factor called intelligence $\left(f_{1}\right)$. Exam scores on English $\left(x_{4}\right)$, Chinese $\left(x_{5}\right)$, and French $\left(x_{6}\right)$ may be represented by an unobservable factor called verbal ability $\left(f_{2}\right)$. Data on income $\left(x_{7}\right)$, education $\left(x_{8}\right)$, and professionals $\left(x_{9}\right)$ may be represented by an unobservable factor called social status $\left(f_{3}\right)$. Thus, we have grouped the original 9 variables that are highly correlated, and obtain three factors $\left(f_{1}, f_{2}, f_{3}\right)$ : intelligence, verbal ability, and social status. These factors are unobservable and uncorrelated, and they have practical meanings. Therefore, the original 9 variables may be represented by three factors, which is a big reduction of data dimension. Although these three factors may not completely represent the original 9 variables, the three factors should contain most information in the original variables or explain most variability in the original variables. Compared to $\mathrm{PCA}$, an attractive feature of factor analysis is that the factors have practical interpretation, since intelligence, verbal ability, and social status are meaningful variables. This is the idea behind factor analysis.

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|The Factor Analysis Model

The idea of factor analysis can be formally stated as follows. Let $\mathbf{x}=\left(x_{1}, \cdots, x_{p}\right)^{\mathrm{T}} \sim$ $(\mu, \Sigma)$ be the original set of variables with mean vector $\mu=\left(\mu_{1}, \mu_{2}, \cdots, \mu_{p}\right)^{\mathrm{T}}$ and covariance matrix $\Sigma=\left(\sigma_{i j}\right){p \times p}$. Note that $\mathbf{x} \sim(\boldsymbol{\mu}, \Sigma)$ means that the random vector $\mathrm{x}$ has a mean vector $\mu$ and covariance matrix $\Sigma$, without a distributional assumption. The factor analysis (FA) model can be written as $$x{i}=\mu_{i}+\lambda_{i 1} f_{1}+\cdots+\lambda_{i m} f_{m}+\eta_{i}, \quad i=1,2, \cdots, p, \quad m \leqslant p,$$
where the $f_{j}$ ‘s are random variables called factors or common factors, the quantities $\lambda_{i j}$ ‘s are called loadings, $m$ is an positive integer smaller than the original number of variables $p$, and $\eta_{i}$ ‘s are random errors.

In the factor analysis model (3.1), the original set of variables $x_{j}$ ‘s are written as linear combinations of the common factors $f_{j}$ ‘s plus random errors. In other words, the variation in the original set of variables can be partially explained by the variation in the common factors. Typically, the number of factors is less than the number of original variables, i.e., $m<p$. The factors are unobservable. The loading $\lambda_{i j}$ ‘s represent the contribution (importance) of factor $f_{j}$ to variable $x_{i}$. The random errors $\eta_{i}$ ‘s represent variations that cannot be explained by the factors.

Note that the FA model (3.1) is different from a regression model. In a regression model, both the responses and the predictors are observed (or known). In the FA model, however, the common factors $f_{j}$ ‘s are not observed (or unknown). Thus, statistical methods for FA models are different from those for regression models.
To estimate the unknown parameters and factors based on given data, we must make some assumptions for the FA model. The common assumptions for the FA model (3.1) are

• the factors $f_{j}$ ‘s are i.i.d, $\sim(0,1)$, i.e., the factors are independently and identically distributed with mean 0 and variance 1 ;
• the random errors $\eta_{j}$ ‘s $\sim\left(0, \psi_{j}\right)$, and are independent, i.e., the random errors are independent with mean 0 and variances $\psi_{j}$ ‘s;
• the factor $f_{k}$ and the random error $\eta_{j}$ are independent for any $k, j$.
These assumptions are needed for a standard factor analysis.
The FA model and its assumptions can be written in the following compact matrix form:
\begin{aligned} &\mathbf{x}=\boldsymbol{\mu}+\Lambda \mathbf{f}+\eta \ &\mathbf{f} \sim(0, I), \quad \eta \sim(0, \Psi), \quad \mathbf{f} \text { and } \eta \text { are independent, } \end{aligned}
where $\Lambda=\left(\lambda_{i j}\right){p \times m}$ is the loading matrix, $\mathbf{f}=\left(f{1}, \cdots, f_{m}\right)^{\mathrm{T}}, \eta=\left(\eta_{1}, \cdots, \eta_{p}\right)^{\mathrm{T}}$, and $\Psi=\operatorname{diag}\left(\psi_{1}, \cdots, \psi_{p}\right)$ is a diagonal matrix. This matrix form is convenient for presentation and mathematical arguments. Like $\mathrm{PCA}$, the covariance matrix $\Sigma$ of $\mathbf{x}$ plays a key role in factor analysis, since the covariance matrix measures the variation and correlation in the data. The mean vector $\mu$ simply measures the location of the data, so it does not contain any information about the variation and correlation in the data.

In both PCA and FA, the variation in the original set of variables are partially explained by the variation in a smaller set of new and uncorrelated variables (principal components and factors). However, PCA mostly focuses on dimension reduction and the principal components may or may not have meaningful practical interpretation, while in factor analysis the factors typically have meaningful practical interpretation. Often, a PCA can be used to roughly determine the number $m$ of factors needed in factor analysis.

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Examples in R

“新数据”（即PC分数）可以从原始数据中获得。为了显示新数据的图形显示，我们考虑前两台 PC 的 PC 分数，然后我们以图形方式显示 PC 分数以检查新数据的正态性和任何可能的异常值。

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|The Factor Analysis Model

X一世=μ一世+λ一世1F1+⋯+λ一世米F米+这一世,一世=1,2,⋯,p,米⩽p,

• 因素Fj是 iid，∼(0,1)，即因子独立同分布，均值为0，方差为1；
• 随机误差这j的∼(0,ψj), 并且是独立的，即随机误差独立于均值 0 和方差ψj的;
• 因素Fķ和随机误差这j是独立的任何ķ,j.
标准因子分析需要这些假设。
FA 模型及其假设可以写成以下紧凑矩阵形式：
X=μ+ΛF+这 F∼(0,我),这∼(0,Ψ),F 和 这 是独立的，
在哪里Λ=(λ一世j)p×米是加载矩阵，F=(F1,⋯,F米)吨,这=(这1,⋯,这p)吨， 和Ψ=诊断⁡(ψ1,⋯,ψp)是对角矩阵。这种矩阵形式便于表示和数学论证。喜欢磷C一个, 协方差矩阵Σ的X在因子分析中起着关键作用，因为协方差矩阵衡量了数据中的变化和相关性。平均向量μ只是测量数据的位置，因此它不包含任何有关数据变化和相关性的信息。

## 有限元方法代写

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## MATLAB代写

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