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

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我们提供的多元统计分析Multivariate Statistical Analysis及其相关学科的代写,服务范围广, 其中包括但不限于:

  • 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|Factor Analysis

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|Factor Analysis

多元统计分析代考

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

在本节中,我们描述了几种估计 FA 模型中参数的方法。在FA模型(3.2)中,加载矩阵Λ和因素F和错误这都是不可观察的。自从F和这假设是独立的,基于FA模型(3.2)及其假设,我们可以得到以下因子分析(FA)方程

Σ=ΛΛ吨+Ψ.
上面的 FA 方程导致以下变量的方差划分Xj :

σjj=∑ķ=1米λjķ2+ψj,j=1,⋯,p

即原始变量的总方差(variation)Xj可以分为共同因素的贡献和随机变化的贡献。变量的公共性Xj, 定义为

Cj=∑ķ=1米λjķ2/σjj,j=1,⋯,p
是方差的比例Xj这是由共同因素解释的(F1,⋯,F米). 因此,社区Cj表示公因子对变量的重要性Xj,j=1,2,⋯,p. 换句话说,社区Cj表示原始变量有多少变化Xj可以用公因数来解释(F1,⋯,F米). 随机误差的方差ψ一世称为唯一性或特异性。

可以使用不同的方法来求解 FA 方程,例如最大似然法和主因子法。对于最大似然法,我们假设误差这遵循多元正态分布ñ(0,披),然后我们最大化似然来获得最大似然估计Γ和F. 主因子法使用不同的方法,不需要正态性假设这. 我们在这里省略了技术细节。感兴趣的读者可以在 Johnson and Wichern (2007) 中找到技术细节。尽管有不同的方法可用于求解 FA 方程,但每种方法都有其自身的优点和局限性。因此,在数据分析中,一个好的策略是使用不同的方法来分析相同的数据集,然后比较结果。如果所有结果都相似,则结论可能是可靠的。如果基于不同方法的结果导致不同的结论,我们应该做进一步的调查,并尝试获得一些关于为什么结果不同的见解。

注意因子加载矩阵Λ不是唯一的,即不同的加载矩阵可能满足相同的 FA 方程。事实上,对于任何正交矩阵问, 新矩阵Λ通过以下正交变换获得

一个=一个问
也是一个加载矩阵,因为ΛΛ 吨=Λ问问吨Λ=ΛΛ吨. 矩阵Λ称为加载矩阵的旋转Λ, 自从Λ是一个正交变换Λ.

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

在实践中,我们经常希望根据观察到的多元数据将个体分成不同的组。例如,银行可能希望根据他们的信用记录、教育和收入来区分好客户和坏客户,或者老师可能希望根据他们的成绩、动机、态度和其他表现来区分好学生和坏学生。换句话说,给定观察到的多变量数据,我们希望确定样本中的个体属于哪个组(或总体)。这种类型的分析称为判别分析或聚类分析。如果分离仅基于一个变量,例如学生的成绩或客户的收入,那么这种分析将很容易。但是,当我们拥有多变量数据时,分析可能并不容易。例如,一个学生可能成绩一般,但态度很好,

判别分析和聚类分析在许多领域都很重要。判别分析和聚类分析之间的区别取决于是否事先知道组(或总体)。如果这些组是预先知道的,例如,我们已经知道一个班有好学生和坏学生,或者银行有好客户和坏客户,我们只是想将它们分开,那么这种分析称为判别分析。如果事先不知道这些组,例如,我们不知道一个班级中是否有好学生或坏学生,我们希望确定一个班级中的所有学生是否可以分成两组(有时可能是班级中的所有学生)一个班级都是好学生),那么这种分析就叫做聚类分析。

在下一节中,我们首先通过简单的例子介绍判别分析的基本思想,然后描述聚类分析并讨论这两种分析之间的异同。我们首先考虑连续数据,然后讨论如何处理离散或分类数据。

在本节中,我们将提供几个示例来展示如何使用R做因子分析。示例 1. 确定了以下 5 只股票的每周收益率(Johnson 和 Wichern,2007): 应用化学(X1), 杜邦(X2), 联合碳化物(X3), 埃克森(X4), 德士古(X5). 获得连续 100 周的观察结果。基于 PCA,前两台 PC 解释了73%变化。前两个磷Cs由

是1=0.46X1+0.46X2+0.47X3+0.42X4+0.42X5 是2=0.24X1+0.51X2+0.26X3−0.53X4−0.58X5
我们看到第一个磷C是1是个股的等权重总和,因此可以解释为代表市场成分。第二磷C是2是化工股之间的对比(前三只股票X1,X2,X3),以及石油股(最后两只股票X4,X5),因此可以解释为代表行业成分。因此,这五种股票收益的大部分变化是由于市场活动和不相关的行业活动造成的。

我们还可以做一个因子分析,这可以让我们获得更好的解释。基于上述 PCA,我们考虑 2 个因素。得到以下 FA 结果:一个没有旋转,一个有旋转载荷

我们看到旋转的因子F1和F2更容易解释:化学公司 Applied Chemical、du Pont 和 Union Carbide 对第一个因素的贡献最大F1(相应的负载很高),而石油公司埃克森和德士古对第二个因素的贡献最大F2(相应的负载很高)。因此,我们可以将这两个因素解释如下:F1代表化学股票和因子F2代表石油股。这种有意义和实用的解释在 PCA 中是不可用的,它是因子分析的优势。

示例 2. 我们回到第 2 章中描述的求职者数据。这里我们对这个数据集进行因子分析以进行比较。在 PCA 中,表明前两个 PC 可以解释数据中的大多数变化。因此,在下文中,我们考虑使用两个因子进行因子分析(米=2). 我们首先重复磷C一个,然后进行因子分析。

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

在实践中,我们经常希望根据观察到的多元数据将个体分成不同的组。例如,银行可能希望根据他们的信用记录、教育和收入来区分好客户和坏客户,或者老师可能希望根据他们的成绩、动机、态度和其他表现来区分好学生和坏学生。换句话说,给定观察到的多变量数据,我们希望确定样本中的个体属于哪个组(或总体)。这种类型的分析称为判别分析或聚类分析。如果分离仅基于一个变量,例如学生的成绩或客户的收入,那么这种分析将很容易。但是,当我们拥有多变量数据时,分析可能并不容易。例如,一个学生可能成绩一般,但态度很好,

判别分析和聚类分析在许多领域都很重要。判别分析和聚类分析之间的区别取决于是否事先知道组(或总体)。如果这些组是预先知道的,例如,我们已经知道一个班有好学生和坏学生,或者银行有好客户和坏客户,我们只是想将它们分开,那么这种分析称为判别分析。如果事先不知道这些组,例如,我们不知道一个班级中是否有好学生或坏学生,我们希望确定一个班级中的所有学生是否可以分成两组(有时可能是班级中的所有学生)一个班级都是好学生),那么这种分析就叫做聚类分析。

在下一节中,我们首先通过简单的例子介绍判别分析的基本思想,然后描述聚类分析并讨论这两种分析之间的异同。我们首先考虑连续数据,然后讨论如何处理离散或分类数据。

统计代写|多元统计分析代写Multivariate Statistical Analysis代考 请认准statistics-lab™

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

金融工程是使用数学技术来解决金融问题。金融工程使用计算机科学、统计学、经济学和应用数学领域的工具和知识来解决当前的金融问题,以及设计新的和创新的金融产品。

非参数统计代写

非参数统计指的是一种统计方法,其中不假设数据来自于由少数参数决定的规定模型;这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型(GLM)归属统计学领域,是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

术语 广义线性模型(GLM)通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归,以及方差分析和方差分析(仅含固定效应)。

有限元方法代写

有限元方法(FEM)是一种流行的方法,用于数值解决工程和数学建模中出现的微分方程。典型的问题领域包括结构分析、传热、流体流动、质量运输和电磁势等传统领域。

有限元是一种通用的数值方法,用于解决两个或三个空间变量的偏微分方程(即一些边界值问题)。为了解决一个问题,有限元将一个大系统细分为更小、更简单的部分,称为有限元。这是通过在空间维度上的特定空间离散化来实现的,它是通过构建对象的网格来实现的:用于求解的数值域,它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统,以模拟整个问题。然后,有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。

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随机分析代写


随机微积分是数学的一个分支,对随机过程进行操作。它允许为随机过程的积分定义一个关于随机过程的一致的积分理论。这个领域是由日本数学家伊藤清在第二次世界大战期间创建并开始的。

时间序列分析代写

随机过程,是依赖于参数的一组随机变量的全体,参数通常是时间。 随机变量是随机现象的数量表现,其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值(如1秒,5分钟,12小时,7天,1年),因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中,往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录,以得到其自身发展的规律。

回归分析代写

多元回归分析渐进(Multiple Regression Analysis Asymptotics)属于计量经济学领域,主要是一种数学上的统计分析方法,可以分析复杂情况下各影响因素的数学关系,在自然科学、社会和经济学等多个领域内应用广泛。

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。

R语言代写问卷设计与分析代写
PYTHON代写回归分析与线性模型代写
MATLAB代写方差分析与试验设计代写
STATA代写机器学习/统计学习代写
SPSS代写计量经济学代写
EVIEWS代写时间序列分析代写
EXCEL代写深度学习代写
SQL代写各种数据建模与可视化代写

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