如果你也在 怎样代写多元统计分析Multivariate Statistical Analysis这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。多元统计分析Multivariate Statistical Analysis在此重定向。在数学上的用法，见多变量微积分。多变量统计是统计学的一个分支，包括同时观察和分析一个以上的结果变量。多变量统计涉及到理解每一种不同形式的多变量分析的不同目的和背景，以及它们之间的关系。多变量统计在特定问题上的实际应用可能涉及几种类型的单变量和多变量分析，以了解变量之间的关系以及它们与所研究问题的相关性。
多元统计分析Multivariate Statistical Analysis通常情况下，希望使用多变量分析的研究会因为问题的维度而停滞。这些问题通常通过使用代理模型来缓解，代理模型是基于物理学的代码的高度精确的近似。由于代用模型采取方程的形式，它们可以被快速评估。这成为大规模MVA研究的一个有利因素：在基于物理学的代码中，整个设计空间的蒙特卡洛模拟是困难的，而在评估代用模型时，它变得微不足道，代用模型通常采取响应面方程的形式。
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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:
非线性方法 nonlinear method functional analysis
变分法 Calculus of Variations
统计代写|多元统计分析作业代写Multivariate Statistical Analysis代考|The Orthogonal Factor Model
The aim of factor analysis is to explain the outcome of $p$ variables in the data matrix $\mathcal{X}$ using fewer variables, the so-called factors. Ideally all the information in $\mathcal{X}$ can be reproduced by a smaller number of factors. These factors are interpreted as latent (unobserved) common characteristics of the observed $x \in \mathbb{R}^{p}$. The case just described occurs when every observed $x=\left(x_{1}, \ldots, x_{p}\right)^{\top}$ can be written as
$$
x_{j}=\sum_{\ell=1}^{k} q_{j \ell} f_{\ell}+\mu_{j}, j=1, \ldots, p
$$
Here $f_{\ell}$, for $\ell=1, \ldots, k$ denotes the factors. The number of factors, $k$, should always be much smaller than $p$. For instance, in psychology $x$ may represent $p$ results of a test measuring intelligence scores. One common latent factor explaining $x \in \mathbb{R}^{p}$ could be the overall level of “intelligence”. In marketing studies, $x$ may consist of $p$ answers to a survey on the levels of satisfaction of the customers. These $p$ measures could be explained by common latent factors like the attraction level of the product or the image of the brand, and so on. Indeed
252
10 Factor Analysis
it is possible to create a representation of the observations that is similar to the one in (10.1) by means of principal components, but only if the last $p-k$ eigenvalues corresponding to the covariance matrix are equal to zero. Consider a $p$-dimensional random vector $\mathrm{X}$ with mean $\mu$ and covariance matrix $\operatorname{Var}(X)=\Sigma$. A model similar to (10.1) can be written for $X$ in matrix notation, namely
$$
X=\mathcal{Q} F+\mu
$$
where $F$ is the $k$-dimensional vector of the $k$ factors. When using the factor model (10.2) it is often assumed that the factors $F$ are centered, uncorrelated and standardized: $E(F)=0$ and $\operatorname{Var}(F)=\mathcal{I}_{k}$. We will now show that if the last $p-k$ eigenvalues of $\Sigma$ are equal to zero, we can easily express $X$ by the factor model (10.2).
统计代写|多元统计分析作业代写Multivariate Statistical Analysis代考|Estimation of the Factor Model
In practice, we have to find estimates $\widehat{\mathcal{Q}}$ of the loadings $\mathcal{Q}$ and estimates $\widehat{\Psi}$ of the specific variances $\Psi$ such that analogously to (10.7)
$$
\mathcal{S}=\widehat{\mathcal{Q}} \widehat{\mathcal{Q}}^{\top}+\widehat{\Psi}
$$
where $\mathcal{S}$ denotes the empirical covariance of $\mathcal{X}$. Given an estimate $\widehat{\mathcal{Q}}$ of $\mathcal{Q}$, it is natural to set
$$
\widehat{\psi}{j j}=s{X_{j} X_{j}}-\sum_{\ell=1}^{k} \widehat{q}{j \ell}^{2} . $$ We have that $\widehat{h}{j}^{2}=\sum_{\ell=1}^{k} \widehat{q}{j \ell}^{2}$ is an estimate for the communality $h{j}^{2}$.
In the ideal case $d=0$, there is an exact solution. However, $d$ is usually greater than zero, therefore we have to find $\widehat{\mathcal{Q}}$ and $\widehat{\Psi}$ such that $S$ is approximated by $\widehat{\mathcal{Q}} \widehat{\mathcal{Q}}^{\top}+\widehat{\Psi}$. As mentioned above, it is often easier to compute the loadings and the specific variances of the standardized model.
Define $\mathcal{Y}=\mathcal{H} \mathcal{X} \mathcal{D}^{-1 / 2}$, the standardization of the data matrix $\mathcal{X}$, where, as usual, $\mathcal{D}=$ $\operatorname{diag}\left(s_{X_{1} X_{1}}, \ldots, s_{X_{p} X_{p}}\right.$ ) and the centering matrix $\mathcal{H}=\mathcal{I}-n^{-1} 1_{n} 1_{n}^{\top}$ (recall from Chapter 2 that $\left.\mathcal{S}=\frac{1}{n} \mathcal{X}^{\top} \mathcal{H} \mathcal{X}\right)$. The estimated factor loading matrix $\widehat{\mathcal{Q}}{Y}$ and the estimated specific variance $\widehat{\Psi}{Y}$ of $\mathcal{Y}$ are
$$
\widehat{\mathcal{Q}}{Y}=\mathcal{D}^{-1 / 2} \widehat{\mathcal{Q}}{X} \quad \text { and } \quad \widehat{\Psi}{Y}=\mathcal{D}^{-1} \widehat{\Psi}{X} \text {. }
$$
For the correlation matrix $\mathcal{R}$ of $\mathcal{X}$, we have that
$$
\mathcal{R}=\widehat{\mathcal{Q}}{Y} \widehat{\mathcal{Q}}{Y}^{\top}+\widehat{\Psi}{Y} $$ The interpretations of the factors are formulated from the analysis of the loadings $\widehat{\mathcal{Q}}{Y}$.
统计代写|多元统计分析作业代写MULTIVARIATE STATISTICAL ANALYSIS代考|Factor Scores and Strategies
Up to now strategies have been presented for factor analysis that have concentrated on the estimation of loadings and communalities and on their interpretations. This was a logical step since the factors $F$ were considered to be normalized random sources of information and were explicitely addressed as nonspecific (common factors). The estimated values of the factors, called the factor scores, may also be useful in the interpretation as well as in the diagnostic analysis. To be more precise, the factor scores are estimates of the unobserved random vectors $F_{l}, l=1, \ldots, k$, for each individual $x_{i}, i=1, \ldots, n$. Johnson and Wichern (1998) describe three methods which in practice yield very similar results. Here, we present the regression method which has the advantage of being the simplest technique and is easy to implement.
The idea is to consider the joint distribution of $(X-\mu)$ and $F$, and then to proceed with the regression analysis presented in Chapter 5 . Under the factor model (10.4), the joint covariance matrix of $(X-\mu)$ and $F$ is:
$$
\operatorname{Var}\left(\begin{array}{c}
X-\mu \
F
\end{array}\right)=\left(\begin{array}{cc}
\mathcal{Q} \mathcal{Q}^{\top}+\Psi & \mathcal{Q} \
\mathcal{Q}^{\top} & \mathcal{I}_{k}
\end{array}\right)
$$
多元统计分析代写
统计代写|多元统计分析作业代写MULTIVARIATE STATISTICAL ANALYSIS代考|THE ORTHOGONAL FACTOR MODEL
因子分析的目的是解释p数据矩阵中的变量X使用较少的变量，即所谓的因子。理想情况下，所有信息都在X可以通过较少数量的因素再现。这些因素被解释为潜在的在n这bs和r在和d观察到的共同特征X∈Rp. 刚刚描述的情况发生在每个观察到的X=(X1,…,Xp)⊤可以写成
Xj=∑ℓ=1ķqjℓFℓ+μj,j=1,…,p
这里Fℓ， 为了ℓ=1,…,ķ表示因子。因素的数量，ķ, 应该总是比p. 例如，在心理学X可能代表p测量智力分数的测试结果。一种常见的潜在因素解释X∈Rp可能是“智能”的整体水平。在营销研究中，X可能包括p对客户满意度调查的回答。这些p措施可以用常见的潜在因素来解释，如产品的吸引力程度或品牌的形象等。确实
252
10 因子分析
可以创建类似于10.1通过主成分，但前提是最后一个p−ķ对应于协方差矩阵的特征值等于零。考虑一个p维随机向量X平均μ和协方差矩阵曾是(X)=Σ. 一个类似的模型10.1可以写成X在矩阵表示法中，即
X=问F+μ
在哪里F是个ķ的维向量ķ因素。使用因子模型时10.2通常认为这些因素F是居中的、不相关的和标准化的：和(F)=0和曾是(F)=一世ķ. 我们现在将证明，如果最后一个p−ķ的特征值Σ等于零，我们可以很容易地表示X因子模型10.2.
统计代写|多元统计分析作业代写MULTIVARIATE STATISTICAL ANALYSIS代考|ESTIMATION OF THE FACTOR MODEL
在实践中，我们必须找到估计问^载荷问和估计Ψ^具体差异Ψ这样类似于10.7$$
\mathcal{S}=\widehat{\mathcal{Q}} \widehat{\mathcal{Q}}^{\top}+\widehat{\Psi}
$$
where $\mathcal{S}$ denotes the empirical covariance of $\mathcal{X}$. Given an estimate $\widehat{\mathcal{Q}}$ of $\mathcal{Q}$, it is natural to set
$$
\widehat{\psi}{j j}=s{X_{j} X_{j}}-\sum_{\ell=1}^{k} \widehat{q}{j \ell}^{2} . $$ We have that $\widehat{h}{j}^{2}=\sum_{\ell=1}^{k} \widehat{q}{j \ell}^{2}$ is an estimate for the communality $h{j}^{2}$.
In the ideal case $d=0$, there is an exact solution. However, $d$ is usually greater than zero, therefore we have to find $\widehat{\mathcal{Q}}$ and $\widehat{\Psi}$ such that $S$ is approximated by $\widehat{\mathcal{Q}} \widehat{\mathcal{Q}}^{\top}+\widehat{\Psi}$.。如上所述，计算标准化模型的载荷和特定方差通常更容易。
定义$\mathcal{Y}=\mathcal{H} \mathcal{X} \mathcal{D}^{-1 / 2}$, the standardization of the data matrix $\mathcal{X}$, where, as usual, $\mathcal{D}=$ $\operatorname{diag}\left(s_{X_{1} X_{1}}, \ldots, s_{X_{p} X_{p}}\right.$ ) and the centering matrix $\mathcal{H}=\mathcal{I}-n^{-1} 1_{n} 1_{n}^{\top}$ (recall from Chapter 2 that $\left.\mathcal{S}=\frac{1}{n} \mathcal{X}^{\top} \mathcal{H} \mathcal{X}\right)$. The estimated factor loading matrix $\widehat{\mathcal{Q}}{Y}$ and the estimated specific variance $\widehat{\Psi}{Y}$ of $\mathcal{Y}$ are
$$
\widehat{\mathcal{Q}}{Y}=\mathcal{D}^{-1 / 2} \widehat{\mathcal{Q}}{X} \quad \text { and } \quad \widehat{\Psi}{Y}=\mathcal{D}^{-1} \widehat{\Psi}{X} \text {. }
$$
For the correlation matrix $\mathcal{R}$ of $\mathcal{X}$, we have that
$$
\mathcal{R}=\widehat{\mathcal{Q}}{Y} \widehat{\mathcal{Q}}{Y}^{\top}+\widehat{\Psi}{Y} $$ The interpretations of the factors are formulated from the analysis of the loadings $\widehat{\mathcal{Q}}{Y}$.
统计代写|多元统计分析作业代写MULTIVARIATE STATISTICAL ANALYSIS代考|FACTOR SCORES AND STRATEGIES
到目前为止，已经提出了因子分析的策略，这些策略集中在载荷和公共性的估计及其解释上。这是一个合乎逻辑的步骤，因为这些因素F被认为是归一化的随机信息来源，并被明确处理为非特异性C这米米这nF一种C吨这rs. 因子的估计值，称为因子分数，也可用于解释和诊断分析。更准确地说，因子分数是对未观察到的随机向量的估计Fl,l=1,…,ķ, 对于每个人X一世,一世=1,…,n. 约翰逊和威彻恩1998描述在实践中产生非常相似结果的三种方法。在这里，我们介绍回归方法，该方法具有技术最简单且易于实现的优点。
这个想法是考虑联合分布(X−μ)和F，然后继续进行第 5 章中介绍的回归分析。因子模型下10.4，联合协方差矩阵(X−μ)和F是：
曾是(X−μ F)=(问问⊤+Ψ问 问⊤一世ķ)
统计代写|多元统计分析作业代写Multivariate Statistical Analysis代考 请认准UprivateTA™. UprivateTA™为您的留学生涯保驾护航。
