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

statistics-lab™ 为您的留学生涯保驾护航 在代写多元统计分析Multivariate Statistical Analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写多元统计分析Multivariate Statistical Analysis代写方面经验极为丰富，各种代写多元统计分析Multivariate Statistical Analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。