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

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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|Methods for Estimation

In this section, we describe several methods to estimate the parameters in a FA model. In FA model (3.2), the loading matrix $\Lambda$ and the factors $f$ and error $\eta$ are all unobservable. Since $\boldsymbol{f}$ and $\eta$ are assumed to be independent, based on FA model (3.2) and its assumptions, we can obtain the following factor analysis (FA) equation
$$\Sigma=\Lambda \Lambda^{\mathrm{T}}+\Psi .$$
The above FA equation leads to the following partition of variance for variable $x_{j}$ :
$$\sigma_{j j}=\sum_{k=1}^{m} \lambda_{j k}^{2}+\psi_{j}, \quad j=1, \cdots, p$$

i.e., the total variance (variation) of the original variable $x_{j}$ can be partitioned into the contributions from the common factors and the random variation. The communality of variable $x_{j}$, defined as
$$c_{j}=\sum_{k=1}^{m} \lambda_{j k}^{2} / \sigma_{j j}, \quad j=1, \cdots, p$$
is the proportion of the variance of $x_{j}$ that is explained by the common factors $\left(f_{1}, \cdots, f_{m}\right)$. Therefore, the communality $c_{j}$ indicates the importance of the common factors to variable $x_{j}, j=1,2, \cdots, p$. In other words, the communality $c_{j}$ indicates how much variation in the original variable $x_{j}$ can be explained by the common factors $\left(f_{1}, \cdots, f_{m}\right)$. The variance of random error $\psi_{i}$ is called uniqueness or specificity.

The FA equations can be solved using different methods, such as the the maximum likelihood method and the principal factor method. For the maximum likelihood method, we assume that the error $\eta$ follows the multivariate normal distribution $N(0, \Phi)$, and then we maximize the likelihood to obtain the maximum likelihood estimates of $\Gamma$ and $f$. The principal factor method uses a different approach and does not require the normality assumption for $\eta$. We omit the technical details here. Interested readers can find the technical details in Johnson and Wichern (2007). Although different methods are available for solving the FA equations, each method has its own advantages and limitations. Thus, in data analysis, a good strategy is to use different methods to analyze the same dataset and then compare the results. If all the results are similar, the conclusions may be reliable. If the results based on different methods lead to different conclusions, we should do a further investigation and try to gain some insights as to why the results differ.

Note that the factor loading matrix $\Lambda$ is not unique, i.e., different loading matrices may satisfy the same FA equations. In fact, for any orthogonal matrix $Q$, the new matrix $\Lambda^{}$ obtained by the following orthogonal transformation $$A^{}=A Q$$
is also a loading matrix, since $\Lambda^{} \Lambda^{ \mathrm{~T}}=\Lambda Q Q^{\mathrm{T}} \Lambda=\Lambda \Lambda^{\mathrm{T}}$. The matrix $\Lambda^{}$ is called a rotation of the loading matrix $\Lambda$, since $\Lambda^{}$ is an orthogonal transformation of $\Lambda$.

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

In practice, we often wish to separate individuals into different groups based on observed multivariate data. For example, a bank may wish to separate good customers from bad customers based on their credit history, education, and income, or a teacher may wish to separate good students from bad students based on their grades, motivation, attitude, and other performances. In other words, given observed multivariate data, we wish to decide which group (or population) an individual in the sample belongs to. This type of analysis is called discriminant analysis or cluster analysis. Such analysis would be easy if the separation is only based on one variable, such as students’ grades or customers’ income. However, when we have multivariate data, the analysis may not be easy. For example, a student may have an average grade but excellent attitude, or a customer may have high income but poor credit history and low education, then it is not clear which group the student or customer belong to.

Discriminant analysis and cluster analysis are important in many areas. The difference between discriminant analysis and cluster analysis depends on whether the groups (or populations) are known in advance or not. If the groups are known in advance, e.g., we already know that there are good and bad students in a class or there are good customers and bad customers in a bank and we just want to separate them, then the analysis is called discriminant analysis. If the groups are not known in advance, e.g., we don’t know whether there are good students or bad students in a class and we wish to determine if all students in a class can be separated into two groups (sometimes maybe all students in a class are good students), then the analysis is called cluster analysis.

In the following section, we first introduce the basic idea of discriminant analysis using simple examples, and then we describe cluster analysis and discuss the differences and similarities between these two analyses. We first consider continuous data, and then we discuss how to treat discrete or categorical data.

In this section, we present several examples to show how to use $R$ to do factor analysis. Example 1. The weekly rates of the return of the following 5 stocks were determined (Johnson and Wichern, 2007): Applied Chemical $\left(x_{1}\right)$, duPont $\left(x_{2}\right)$, Union Carbide $\left(x_{3}\right)$, Exxon $\left(x_{4}\right)$, Texaco $\left(x_{5}\right)$. Observations in 100 successive weeks were obtained. Based on a PCA, the first two PCs explain about $73 \%$ variation. The first two $\mathrm{PCs}$ are given by

\begin{aligned} &y_{1}=0.46 x_{1}+0.46 x_{2}+0.47 x_{3}+0.42 x_{4}+0.42 x_{5} \ &y_{2}=0.24 x_{1}+0.51 x_{2}+0.26 x_{3}-0.53 x_{4}-0.58 x_{5} \end{aligned}
We see that the first $\mathrm{PC} y_{1}$ is an equally weighted sum of individual stocks, so it may be interpreted as representing market component. The second $\mathrm{PC} y_{2}$ is a contrast between the chemical stocks (the first three stocks $x_{1}, x_{2}, x_{3}$ ), and the oil stocks (the last two stocks $x_{4}, x_{5}$ ), so it may be interpreted as representing the industry component. Thus, most of the variation in these five stock returns is due to market activity and uncorrelated industry activity.

We can also do a factor analysis, which may allows us to obtain better interpretations. Based on the above PCA, we consider 2 factors. The following FA results are obtained: one without rotation and one with rotation of the loadings

We see that the rotated factors $f_{1}^{}$ and $f_{2}^{}$ are easier to interpret: the chemical companies Applied Chemical, du Pont, and Union Carbide contribute most to the first factor $f_{1}^{}$ (the corresponding loadings are high), while the oil companies Exxon and Texaco contribute most to the second factor $f_{2}^{}$ (the corresponding loadings are high). Thus, we can interpret the two factors as follows: factor $f_{1}^{}$ represents chemical stocks and factor $f_{2}^{}$ represents oil stocks. Such a meaningful and practical interpretation is unavailable in PCA, and it is an advantage of factor analysis.

Example 2. We return to the job applicant data described in Chapter 2. Here we do a factor analysis on this dataset for comparison. In PCA, it was shown that the first two PCs can explain most variation in the data. Thus, in the following we consider a factor analysis using two factors $(m=2)$. We first repeat the $\mathrm{PCA}$, and then do a factor analysis.

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

In practice, we often wish to separate individuals into different groups based on observed multivariate data. For example, a bank may wish to separate good customers from bad customers based on their credit history, education, and income, or a teacher may wish to separate good students from bad students based on their grades, motivation, attitude, and other performances. In other words, given observed multivariate data, we wish to decide which group (or population) an individual in the sample belongs to. This type of analysis is called discriminant analysis or cluster analysis. Such analysis would be easy if the separation is only based on one variable, such as students’ grades or customers’ income. However, when we have multivariate data, the analysis may not be easy. For example, a student may have an average grade but excellent attitude, or a customer may have high income but poor credit history and low education, then it is not clear which group the student or customer belong to.

Discriminant analysis and cluster analysis are important in many areas. The difference between discriminant analysis and cluster analysis depends on whether the groups (or populations) are known in advance or not. If the groups are known in advance, e.g., we already know that there are good and bad students in a class or there are good customers and bad customers in a bank and we just want to separate them, then the analysis is called discriminant analysis. If the groups are not known in advance, e.g., we don’t know whether there are good students or bad students in a class and we wish to determine if all students in a class can be separated into two groups (sometimes maybe all students in a class are good students), then the analysis is called cluster analysis.

In the following section, we first introduce the basic idea of discriminant analysis using simple examples, and then we describe cluster analysis and discuss the differences and similarities between these two analyses. We first consider continuous data, and then we discuss how to treat discrete or categorical data.

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Methods for Estimation

Σ=ΛΛ吨+Ψ.

σjj=∑ķ=1米λjķ2+ψj,j=1,⋯,p

Cj=∑ķ=1米λjķ2/σjj,j=1,⋯,p

## 有限元方法代写

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## MATLAB代写

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