## 统计代写|主成分分析代写Principal Component Analysis代考|STAT6020

statistics-lab™ 为您的留学生涯保驾护航 在代写主成分分析Principal Component 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。我们的专家在代写主成分分析Principal Component Analysis代写方面经验极为丰富，各种代写主成分分析Principal Component Analysis相关的作业也就用不着说。

• 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|主成分分析代写Principal Component Analysis代考|Lymphoma data sets

The lymphoma data set comes from a study of gene expression of three prevalent lymphoid malignancies: B-cell chronic lymphocytic leukemia (B-CLL), follicular lymphoma (FL) and diffuse large B-cell lymphoma (DLCL). Among 96 samples we took 62 samples 4026 genes in three classes: 11 cases of B-CLL, 9 cases of FL and 42 cases of DLCL. Gene expression levels were measured using 2-channel cDNA microarrays.

After preprocessing, all gene expression profiles were base 10 log-transformed and, in order to prevent single arrays from dominating the analysis, standardized to zero mean and unit variance. Finally, we complete the preprocessing of the gene expression data with gene centring.

In this example we perform the KPCA, as detailed in the previous section, we compute the kernel matrix with using the radial basis kernel with $c=0.01$, this value is set heuristically. The resulting plot is given in Figure 6. It shows the projection onto the two leading kernel principal components of microarrays. In this figure we can see that KPCA detect the group structure in reduced dimension. DLCL, FL and B-CLL are fully separated by KPCA.

To validate our procedure we select a list of genes differentially expressed proposed by (Reverter et al. (2010)) and a list of genes that are not differentially expressed. In particular, in Figures 7, 8,9 and 10 we show the results in the case of genes: $139009,1319066,1352822$ and 1338456 , respectively. The three first genes belong to the list of genes differentially expressed and the last gene is not differentially expressed.

Figure 7 (top) shows the tangent vectors associated with 139009 gene attached at each sample point. This vector field reveals upper expression towards DLCL cluster as is expected from references above mentioned. This gene is mainly represented by the first principal component. The length of the arrows indicate the influence strength of the gene on the sample position despite the dimension reduction. Figure 7 (bottom) shows the expression profile of 139009 gene. We can observe that 139009 gene is up regulated in DLCL cluster. This profile is agree with our procedure because the direction in which the expression of the 139009 gene increases points to the DLCL cluster.

## 统计代写|主成分分析代写Principal Component Analysis代考|Definitions of major «omics» in molecular biology and their goals

The nomicsm era, also called classically the post-genomic era, is described as the period of time which extends the first publication of the human genome sequence draft in 2001 (International Human Genome Sequencing Consortium, 2001; Venter et al., 2001). Ten years after that milestone, extensive use of high-throughput analytical technologies, high performance computing power and large advances in bioinformatics have been applied to solve fundamental molecular biology questions as well as to find clues concerning human diseases (cancers) and aging. Principal nomicsw, such as Gen-omics, Transcript-omics, Proteomics and Metabol-omics, are biology disciplines whose main and extremely ambitious objective is to describe as extensively as possible the complete class-specific molecular components of the cell. In the a omics sciences, the catalog of major cell molecular components, respectively, genes, messenger RNAs and small interfering and regulatory RNAs, proteins, and metabolites of living organisms, is recorded qualitatively as well as quantitatively in response to environmental changes or pathological situations. Various research communities, organized in institutions both at the academic and private levels and working in the nomicsm fields, have spent large amounts of effort and money to reach. standardization in the different experimental and data processing steps. Some of these “omics” specific steps basically include the following: the optimal experimental workflow design, the technology-dependent data acquisition and storage, the pre-processing methods and the post-processing strategies in order to extract some level of relevant biological knowledge from usually large data sets. Just like Perl (Practical Extraction and Report Language) has been recognized to have saved the Human Genome project initiative (Stein, 1996), by using accurate rules to parse genomic sequence data, other web-driven. programming languages and file formats such as XML have also facilitated nomics” data dissemination among scientists and helped rationalize and integrate molecular biology data.
Data resulting from different womicsw have several characteristics in common, which are summarized in Figure 1: (a) the number of measured variables $\mathrm{n}$ ( $\mathrm{SNP}$, gene expression, proteins, peptides, metabolites) is quite large in size (from 100 to 10000), (b) the number of samples or experiments $\mathrm{p}$ where these variables are measured associated with factors such as the pathological status, environmental conditions, drug exposure or kinetic points (temporal experiments) is rather large $(10$ to 1000$)$ and (c) the measured variables are organized in a matrix of $\mathrm{n} \times \mathrm{p}$ dimensions. The cell contents of such a matrix usually record a metric (or numerical code) related to the abundance of the measured variables. The observed data are acquired keeping the lowest amount of possible technical and analytical variability. Exploring these womicsw data requires fast computers and state-of-the-art data visualization and statistical multivariate tools to extract relevant knowledge, and among these tools PCA is a tool of choice in order to perform initial exploratory data analysis (EDA).

## 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。

## 统计代写|主成分分析代写Principal Component Analysis代考|ENVX2001

statistics-lab™ 为您的留学生涯保驾护航 在代写主成分分析Principal Component 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。我们的专家在代写主成分分析Principal Component Analysis代写方面经验极为丰富，各种代写主成分分析Principal Component Analysis相关的作业也就用不着说。

• 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|主成分分析代写Principal Component Analysis代考|Validation

In this section we illustrate our procedure with data from the leukemia data set of Golub et al. (1999) and the lymphoma data set Alizadeh et al. (2000).

In these examples our aim is to validate our procedure for adding input variables information into KPCA representation. We follow the following steps. First, in each data set, we build a list of genes that are differentially expressed. This selection is based in accordance with previous studies such as (Golub et al. (1999), Pittelkow \& Wilson (2003), Reverter et al. (2010)). In addition we compute the expression profile of each gene selected, this profile confirm the evidence of differential expression.

Second, we compute the curves through each sample point associated with each gene in the list. These curves are given by the $\phi$-image of points of the form:
$$\mathbf{y}(s)=\mathbf{x}{i}+s \mathbf{e}{k}$$
where $x_{i}$ is the $1 \times n$ expression vector of the $i$-th sample, $i=1, \ldots, m, k$ denotes the index in the expression matrix of the gene selected to be represented, $\mathbf{e}{k}=(0, \ldots, 1, \ldots, 0)$ is a $1 \times n$ vector with zeros except in the $k$-th. These curves describe locally the change of the sample $x{i}$ induced by the change of the gene expression.

Third, we project the tangent vector of each curve at $s=0$, that is, at the sample points $\mathbf{x}_{i}$, $i=1, \ldots, m$, onto the KPCA subspace spanned by the eigenvectors (9). This representation capture the direction of maximum variation induced in the samples when the expression of gene increases.

By simultaneously displaying both the samples and the gene information on the same plot it is possible both to visually detect genes which have similar profiles and to interpret this pattern by reference to the sample groups.

## 统计代写|主成分分析代写Principal Component Analysis代考|Leukemia data sets

The leukemia data set is composed of 3051 gene expressions in three classes of leukemia: 19 cases of B-cell acute lymphoblastic leukemia (ALL), 8 cases of T-cell ALL and 11 cases of acute myeloid leukemia (AML). Gene expression levels were measured using Affymetrix high-density oligonucleotide arrays.

The data were preprocessed according to the protocol described in Dudoit et al. (2002). In addition, we complete the preprocessing of the gene expression data with a microarray standardization and gene centring.

In this example we perform the KPCA, as detailed in the previous section, we compute the kernel matrix with using the radial basis kernel with $c=0.01$, this value is set heuristically. The resulting plot is given in Figure 1. It shows the projection onto the two leading kernel principal components of microarrays. In this figure we can see that KPCA detect the group structure in reduced dimension. AML, T-cell ALL and B-cell ALL are fully separated by KPCA.

To validate our procedure we select a list of genes differentially expressed proposed by (Golub et al. (1999), Pittelkow \& Wilson (2003), Reverter et al. (2010)) and a list of genes that are not differentially expressed. In particular, in Figures 2, 3,4 and 5 we show the results in the case of genes: X76223_s_at, X82240_rna1_at, Y00787_s_at and D50857_at, respectively. The three first genes belong to the list of genes differentially expressed and the last gene is not differentially expressed.

Figure 2 (top) shows the tangent vectors associated with $\mathrm{X} 76223_{\text {_s_at gene, attached at }}$ each sample point. This vector field reveals upper expression towards T-cell cluster as is expected from references above mentioned. This gene is well represented by the second principal component. The length of the arrows indicate the strength of the gene on the sample position despite the dimension reduction. Figure 2 (bottom) shows the expression profile of X76223_s_at gene. We can observe that X76223_s_at gene is up regulated in T-cell class. This profile is agree with our procedure because the direction in which the expression of the $\mathrm{x} 76223$ _s_at gene increases points to the T-cell cluster.

## 统计代写|主成分分析代写Principal Component Analysis代考|Validation

$$\mathbf{y}(s)=\mathbf{x} i+s \mathbf{e} k$$

## 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。

## 统计代写|主成分分析代写Principal Component Analysis代考|STAT3888

statistics-lab™ 为您的留学生涯保驾护航 在代写主成分分析Principal Component 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。我们的专家在代写主成分分析Principal Component Analysis代写方面经验极为丰富，各种代写主成分分析Principal Component Analysis相关的作业也就用不着说。

• 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|主成分分析代写Principal Component Analysis代考|Kernel PCA methodology

KPCA is a nonlinear equivalent of classical PCA that uses methods inspired by statistical learning theory. We describe shortly the KPCA method from Scholkopf et al. (1998).

Given a set of observations $\mathbf{x}{i} \in \mathbb{R}^{n}, i=1, \ldots, m$. Let us consider a dot product space $F$ related to the input space by a map $\phi: \mathbb{R}^{n} \rightarrow F$ which is possibly nonlinear. The feature space $F$ could have an arbitrarily large, and possibly infinite, dimension. Hereafter upper case characters are used for elements of $F$, while lower case characters denote elements of $\mathbb{R}^{n}$. We assume that we are dealing with centered data $\sum{i=1}^{m} \phi\left(\mathbf{x}{i}\right)=0$. In $F$ the covariance matrix takes the form $$\mathrm{C}=\frac{1}{m} \sum{j=1}^{m} \phi\left(\mathbf{x}{j}\right) \phi\left(\mathbf{x}{j}\right)^{\top} .$$
We have to find eigenvalues $\lambda \geq 0$ and nonzero eigenvectors $V \in F \backslash{0}$ satisfying
$$\mathbf{C V}=\lambda \mathrm{V} \text {. }$$

As is well known all solutions $\mathbf{V}$ with $\lambda \neq 0$ lie in the span of $\left{\phi\left(\mathbf{x}{i}\right)\right}{i=1}^{m}$. This has two consequences: first we may instead consider the set of equations
$$\left\langle\phi\left(\mathbf{x}{k}\right), \mathbf{C V}\right\rangle=\lambda\left\langle\phi\left(\mathbf{x}{k}\right), \mathbf{V}\right\rangle,$$
for all $k=1, \ldots, m$, and second there exist coefficients $\alpha_{i}, i=1, \ldots, m$ such that
$$\mathbf{V}=\sum_{i=1}^{m} \alpha_{i} \phi\left(\mathbf{x}{i}\right)$$ Combining (1) and (2) we get the dual representation of the eigenvalue problem $$\frac{1}{m} \sum{i=1}^{m} \alpha_{i}\left\langle\phi\left(\mathbf{x}{k}\right), \sum{j=1}^{m} \phi\left(\mathbf{x}{j}\right)\left\langle\phi\left(\mathbf{x}{j}\right), \phi\left(\mathbf{x}{i}\right)\right\rangle\right\rangle=\lambda \sum{i=1}^{m} \alpha_{i}\left\langle\phi\left(\mathbf{x}{k}\right), \phi\left(\mathbf{x}{i}\right)\right\rangle,$$
for all $k=1, \ldots m$. Defining a $m \times m$ matrix $K$ by $K_{i j}:=\left\langle\phi\left(\mathbf{x}{i}\right), \phi\left(\mathbf{x}{j}\right)\right\rangle$, this reads
$$K^{2} \alpha=m \lambda K \alpha,$$

## 统计代写|主成分分析代写Principal Component Analysis代考|Adding input variable information into Kernel PCA

In order to get interpretability we add supplementary information into KPCA representation. We have developed a procedure to project any given input variable onto the subspace spanned by the eigenvectors (9).

We can consider that our observations are realizations of the random vector $X=\left(X_{1}, \ldots, X_{n}\right)$. Then to represent the prominence of the input variable $X_{k}$ in the $\mathrm{KPCA}$. We take a set of points of the form $\mathbf{y}=\mathbf{a}+s \mathbf{e}{k} \in \mathbb{R}^{n}$ where $\mathbf{e}{k}=(0, \ldots, 1, \ldots, 0) \in \mathbb{R}^{n}, s \in \mathbb{R}$, where $k$-th component is equal 1 and otherwise are 0 . Then, we can compute the projections of the image of these points $\phi(\mathbf{y})$ onto the subspace spanned by the eigenvectors (9).

Taking into account equation (11) the induced curve in the eigenspace expressed in matrix form is given by the row vector:
$$\sigma(s){1 \times r}=\left(\mathbf{Z}{s}^{\top}-\frac{1}{m} 1_{m}^{\top} K\right)\left(\mathbf{I}{m}-\frac{1}{m} 1{m} \mathbf{1}{m f}^{\top}\right) \tilde{\mathbf{V}}{r}$$
where $\mathbf{Z}{\mathrm{s}}$ is of the form (10). In addition we can represent directions of maximum variation of $\sigma(\mathrm{s})$ associated with the variable $X{k}$ by projecting the tangent vector at $s=0$. In matrix form, we have
$$\left.\frac{d \sigma}{d s}\right|{s=0}=\left.\frac{d \mathbf{Z}{s}^{\top}}{d s}\right|{s=0}\left(\mathbf{I}{m}-\frac{1}{m} \mathbf{1}{m} \mathbf{1}{m}^{\top}\right) \tilde{\mathbf{V}}$$
with
$$\left.\frac{d \mathbf{Z}{s}^{\top}}{d s}\right|{s=0}=\left(\left.\frac{d \mathbf{Z}{s}^{1}}{d s}\right|{s=0} \ldots,\left.\frac{d \mathbf{Z}{s}^{m}}{d s}\right|{s=0}\right)^{\top}$$ and, with
\begin{aligned} \left.\frac{d \mathbf{Z}{s}^{t}}{d s}\right|{s=0} &=\left.\frac{d K\left(\mathbf{y}{,} \mathbf{x}{i}\right)}{d s}\right|{s=0} \ &=\left.\left(\sum{t=1}^{m} \frac{\partial K\left(\mathbf{y}, \mathbf{x}{i}\right)}{\partial y{t}} \frac{d y_{t}}{d s}\right)\right|{s=0} \ &=\left.\sum{t=1}^{m} \frac{\partial K\left(\mathbf{y}, \mathbf{x}{i}\right)}{\partial y{t}}\right|{\mathbf{y}=\mathbf{a}} \delta{t}^{k}=\left.\frac{\partial K\left(\mathbf{y}{,} \mathbf{x}{i}\right)}{\partial y_{k}}\right|_{\mathbf{y}=\mathbf{a}} \end{aligned}

## 统计代写|主成分分析代写Principal Component Analysis代考|Kernel PCA methodology

KPCA 是经典 PCA 的非线性等价物，它使用受统计学习理论启发的方法。我们简要描述了 Scholkopf 等人的 KPCA 方法。（1998 年）。

$$\mathrm{C}=\frac{1}{m} \sum j=1^{m} \phi(\mathbf{x} j) \phi(\mathbf{x} j)^{\top} .$$

$$\mathbf{C V}=\lambda \mathrm{V} \text {. }$$ 我们可以考虑方程组
$$\langle\phi(\mathbf{x} k), \mathbf{C V}\rangle=\lambda\langle\phi(\mathbf{x} k), \mathbf{V}\rangle,$$

$$\mathbf{V}=\sum_{i=1}^{m} \alpha_{i} \phi(\mathbf{x} i)$$

$$\frac{1}{m} \sum i=1^{m} \alpha_{i}\left\langle\phi(\mathbf{x} k), \sum j=1^{m} \phi(\mathbf{x} j)\langle\phi(\mathbf{x} j), \phi(\mathbf{x} i)\rangle\right\rangle=\lambda \sum i=1^{m} \alpha_{i}\langle\phi(\mathbf{x} k), \phi(\mathbf{x} i)\rangle$$

$$K^{2} \alpha=m \lambda K \alpha,$$

## 统计代写|主成分分析代写Principal Component Analysis代考|Adding input variable information into Kernel PCA

$$\sigma(s) 1 \times r=\left(\mathbf{Z} s^{\top}-\frac{1}{m} 1_{m}^{\top} K\right)\left(\mathbf{I} m-\frac{1}{m} 1 m \mathbf{1} m f^{\top}\right) \tilde{\mathbf{V}} r$$

$$\frac{d \sigma}{d s}\left|s=0=\frac{d \mathbf{Z} s^{\top}}{d s}\right| s=0\left(\mathbf{I} m-\frac{1}{m} \mathbf{1} m \mathbf{1} m^{\top}\right) \tilde{\mathbf{V}}$$

$$\frac{d \mathbf{Z} s^{\top}}{d s} \mid s=0=\left(\frac{d \mathbf{Z} s^{1}}{d s}\left|s=0 \ldots, \frac{d \mathbf{Z} s^{m}}{d s}\right| s=0\right)^{\top}$$

$$\frac{d \mathbf{Z} s^{t}}{d s}\left|s=0=\frac{d K(\mathbf{y}, \mathbf{x} i)}{d s}\right| s=0 \quad=\left(\sum t=1^{m} \frac{\partial K(\mathbf{y}, \mathbf{x} i)}{\partial y t} \frac{d y_{t}}{d s}\right) \mid s=0=\sum t=1^{m} \partial K$$

## 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。

## 统计代写|主成分分析代写Principal Component Analysis代考|ICE 2022

statistics-lab™ 为您的留学生涯保驾护航 在代写主成分分析Principal Component 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。我们的专家在代写主成分分析Principal Component Analysis代写方面经验极为丰富，各种代写主成分分析Principal Component Analysis相关的作业也就用不着说。

• 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|主成分分析代写Principal Component Analysis代考|Examples of Mixed Data Modeling

The problem of modeling mixed data is quite representative of many data sets that one often encounters in practical applications. To further motivate the importance of modeling mixed data, we give below a few real-world problems that arise in image processing and computer vision. Most of these problems will be revisited later in this book, and more detailed and principled solutions will be given.
Face Clustering under Varying Illumination
The first example arises in the context of image-based face clustering. Given a collection of unlabeled images $\left{I_{j}\right}_{j=1}^{N}$ of several different faces taken under varying illumination, we would like to cluster the images corresponding to the face of the same person. For a Lambertian object, ${ }^{10}$ it has been shown that the set of all images taken under all lighting conditions forms a cone in the image space, which can be well approximated by a low-dimensional subspace called the “illumination subspace” (Belhumeur and Kriegman 1998; Basri and Jacobs 2003).” For example, if $I_{j}$ is the $j$ th image of a face and $d$ is the dimension of the illumination subspace associated with that face, then there exists a mean face $\mu$ and $d$ eigenfaces $\boldsymbol{u}{1}, \boldsymbol{u}{2}, \ldots, \boldsymbol{u}{d}$ such that $I{j} \approx \boldsymbol{\mu}+\boldsymbol{u}{1} y{1 j}+\boldsymbol{u}{2} y{2 j}+\cdots+\boldsymbol{u}{d j} y{d j}$. Now, since the images of different faces will live in different “illumination subspaces,” we can cluster the collection of images by estimating a basis for each one of those subspaces. As we will see later, this is a special case of the subspace clustering problem addressed in Part II of this book. In the example shown in Figure 1.3, we use a subset of the Yale Face Database B consisting of $n=64 \times 3$ frontal views of three faces (subjects 5,8 and 10 ) under 64 varying lighting conditions. For computational efficiency, we first down-sample each image to a size of $30 \times 40$ pixels. We then project the data onto their first three principal components using PCA, as shown in Figure $1.3$ (a). $.^{12}$ By modeling the projected data with a mixture model of linear subspaces in $\mathbb{R}^{3}$, we obtain three affine subspaces of dimension 2, 1, and 1, respectively. Despite the series of down-sampling and projection, the subspaces lead to a perfect clustering of the face images, as shown in Figure 1.3(b).

## 统计代写|主成分分析代写Principal Component Analysis代考|Mathematical Representations of Mixture Models

The examples presented in the previous subsection argue forcefully for the development of modeling and estimation techniques for mixture models. Obviously, whether the model associated with a given data set is mixed depends on the class of primitive models considered. In this book, the primitives are normally chosen to be simple classes of geometric models or probabilistic distributions.

For instance, one may choose the primitive models to be linear subspaces. Then one can use an arrangement of linear subspaces $\left{S_{i}^{}_{i=1}^{n}} \subset \mathbb{R}^{D}\right.$,
$$Z \doteq S_{1} \cup S_{2} \cup \cdots \cup S_{n},$$
also called a piecewise linear model, to approximate many nonlinear manifolds or piecewise smooth topological spaces. This is the standard model considered in geometric approaches to generalized principal component analysis (GPCA), which will be studied in Part II of this book.

The statistical counterpart to the geometric model in (1.7) is to assume instead that the sample points are drawn independently from a mixture of (near singular) Gaussian distributions $\left{p_{\theta_{i}}(\boldsymbol{x}){i=1}^{n}\right.$, where $\boldsymbol{x} \in \mathbb{R}^{D}$ but each distribution has mass concentrated near a subspace. The overall probability density function can be expressed as a sum: $$q{\theta}(x) \doteq \pi_{1} p_{\theta_{1}}(x)+\pi_{2} p_{\theta_{2}}(x)+\cdots+\pi_{n} p_{\theta_{n}}(x),$$
where $\theta=\left(\theta_{1}, \ldots, \theta_{n}, \pi_{1}, \ldots, \pi_{n}\right)$ are the model parameters and $\pi_{i}>0$ are mixing weights with $\pi_{1}+\pi_{2}+\cdots+\pi_{n}=1$. This is the typical model studied in mixtures of probabilistic principal component analysis (PPCA) (Tipping and Bishop $1999 \mathrm{a}$ ), where each component distribution $p_{\theta_{i}}(\boldsymbol{x})$ is a nearly degenerate Gaussian distribution. A classical way of estimating such a mixture model is the expectation maximization (EM) algorithm, where the membership of each sample is represented as a hidden random variable. Appendix B reviews the general EM method, and Chapter 6 shows how to apply it to the case of multiple subspaces.

## 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。

## 统计代写|主成分分析代写Principal Component Analysis代考|ENVX2001

statistics-lab™ 为您的留学生涯保驾护航 在代写主成分分析Principal Component 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。我们的专家在代写主成分分析Principal Component Analysis代写方面经验极为丰富，各种代写主成分分析Principal Component Analysis相关的作业也就用不着说。

• 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|主成分分析代写Principal Component Analysis代考|Statistical Models versus Geometric Models

There are essentially two main categories of models and approaches for modeling a data set. Methods of the first category model the data as random samples from a probability distribution and try to learn this distribution from the data. We call such models statistical models. Models of the second category model the overall geometric shape of the data set with deterministic models such as subspaces, smooth manifolds, or topological spaces. ${ }^{6}$ We call such models geometric models.
Statistical Learning
In the statistical paradigm, one typically assumes that each data point $\boldsymbol{x}_{j}$ in the data set $\mathcal{X}$ is drawn independently from a common probability distribution $p(\boldsymbol{x})$. Such a probability distribution gives a generative description of the samples and can be used to generate new samples or predict the outcome of new observations. Within this context, the task of learning a model from the data becomes one of inferring the most likely probability distribution within a family of distributions of interest (for example, the Gaussian distributions). Normally, the family of distributions is parameterized and denoted by $\mathcal{M} \doteq{p(x \mid \theta): \theta \in \Theta}$, where $p(x \mid \theta)$ is a probability density function parameterized by $\theta \in \Theta$, and $\Theta$ is the space of parameters. Consequently, one popular criterion for choosing a statistical model $p\left(x \mid \theta^{*}\right)$ is the maximum likelihood (ML) estimate given by ${ }^{7}$ $$\theta_{M L}^{} \doteq \underset{\theta \in \Theta}{\arg \max } \prod_{j=1}^{N} p\left(x_{j} \mid \theta\right)$$ If a prior distribution (density) $p(\theta)$ of the parameter $\theta$ is also given, then, following the Bayesian rule, the maximum a posteriori (MAP) estimate is given by $$\theta_{M A P}^{} \doteq \underset{\theta \in \Theta}{\arg \max } \prod_{j=1}^{N} p\left(\boldsymbol{x}_{j} \mid \theta\right) p(\theta)$$

## 统计代写|主成分分析代写Principal Component Analysis代考|Modeling Mixed Data with a Mixture Model

As we alluded to earlier, many data sets $\mathcal{X}$ cannot be modeled well by a single primitive model $M$ in a pre-chosen or preferred model class $\mathcal{M}$. Nevertheless, it is often the case that if we group such a data set $\mathcal{X}$ into multiple disjoint subsets,
$$\mathcal{X}=\mathcal{X}{1} \cup \mathcal{X}{2} \cup \cdots \cup \mathcal{X}{n}, \quad \text { with } \mathcal{X}{l} \cap \mathcal{X}{m}=\emptyset, \text { for } l \neq m$$ then each subset $\mathcal{X}{i}$ can be modeled sufficiently well by a model in the chosen model class:
$$M_{i}^{}=\underset{M \in \mathcal{M}}{\arg \min } \operatorname{Error}\left(\mathcal{X}{i}, M\right), \quad i=1,2, \ldots, n,$$ where $\operatorname{Error}\left(\mathcal{X}{i}, M\right)$ represents some measure of the error incurred by using the model $M$ to fit the data set $\mathcal{X}{i}$. Each model $M{i}^{}$ is called a primitive or a component model. Precisely in this sense, we call the data set $\mathcal{X}$ mixed (with respect to the chosen model class $\mathcal{M}$ ) and call the collection of primitive models $\left{M_{i}^{*}\right}_{i=1}^{n}$ a mixture model for $\mathcal{X}$. For instance, suppose we are given a set of sample points as shown in Figure 1.2. These points obviously cannot be fit well by any single line, plane, or smooth surface in $\mathbb{R}^{3}$; however, once they are grouped into three subsets, each subset can be fit well by a line or a plane. Note that in this example, the topology of the data is “hybrid”: two of the subspaces are of dimension one, and the other is of dimension two.

## 统计代写|主成分分析代写Principal Component Analysis代考|Statistical Models versus Geometric Models

$$\theta_{M A P} \doteq \underset{\theta \in \Theta}{\arg \max } \prod_{j=1}^{N} p\left(\boldsymbol{x}_{j} \mid \theta\right) p(\theta)$$

## 统计代写|主成分分析代写Principal Component Analysis代考|Modeling Mixed Data with a Mixture Model

$$\mathcal{X}=\mathcal{X} 1 \cup \mathcal{X} 2 \cup \cdots \cup \mathcal{X} n, \quad \text { with } \mathcal{X} l \cap \mathcal{X} m=\emptyset, \text { for } l \neq m$$

$$M_{i}=\underset{M \in \mathcal{M}}{\arg \min } \operatorname{Error}\left(\mathcal{X}_{i}, M\right), \quad i=1,2, \ldots, n,$$

Uleft{M_{i}^{*}}right}_{i=1}^{n}}混合模型 $\mathcal{X}$. 例如，假设给定一组样本点，如图 $1.2$ 所示。这些点显然不能被任何单 一的线、平面或光滑表面很好地拟合 $\mathbb{R}^{3}$; 但是，一旦将它们分成三个子集，每个子集就可以很好地被一条线或一 个平面拟合。注意，在这个例子中，数据的拓扑是“混合的”：两个子空间是一维的，另一个是二维的。

## 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。

## 统计代写|主成分分析代写Principal Component Analysis代考|STAT3888

statistics-lab™ 为您的留学生涯保驾护航 在代写主成分分析Principal Component 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。我们的专家在代写主成分分析Principal Component Analysis代写方面经验极为丰富，各种代写主成分分析Principal Component Analysis相关的作业也就用不着说。

• 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|主成分分析代写Principal Component Analysis代考|Modeling Data with a Parametric Model

The primary goal of this book is to study theory and methods for modeling highdimensional data with one or more low-dimensional subspaces or manifolds. To a large extent, the methods presented in this book aim to generalize the classical principal component analysis (PCA) method (Jolliffe 1986, 2002) to address two major challenges presented by current applications.

One challenge is to generalize the classical PCA method to data with significant amounts of missing entries, errors, outliers, or even a certain level of nonlinearity. Since the very beginning of PCA nearly a century ago (Pearson 1901; Hotelling 1933), researchers have been aware of PCA’s vulnerability to missing data and corruption. Strictly speaking, estimating a subspace from incomplete or corrupted data is an inherently difficult problem, which is generally NP-hard. Nevertheless, due to the practical importance of this problem, many extensions to PCA have been proposed throughout the years in different practical domains to handle imperfect data, even though many of these methods have been largely heuristic, greedy, or even ad hoc. Recent advances in high-dimensional statistics and convex optimization have begun to provide provably correct ${ }^{1}$ and efficient methods for finding the optimal subspace from highly incomplete or corrupted data.

In science and engineering, one is frequently called upon to infer (or learn) a quantitative model $M$ for a given set of sample points $\mathcal{X}=\left{x_{1}, x_{2}, \ldots, x_{N}\right} \subset \mathbb{R}^{D}$. For instance, Figure $1.1$ shows a simple example in which one is given a set of four sample points in a two-dimensional plane. Obviously, these points can be fit perfectly by a (one-dimensional) straight line $L$. The line can then be called a “model” for the given points. The reason for inferring such a model is that it serves many useful purposes. On the one hand, the model can reveal information encoded in the data or underlying mechanisms from which the data were generated. In addition, it can simplify the representation of the given data set and help predict future samples. In the case of the four points shown in Figure $1.1$, the line model gives a more compact one-dimensional representation than the original twodimensional plane $P$. It also suggests that any new point (if generated with a similar mechanism as the existing points) will likely fall on the same line.

## 统计代写|主成分分析代写Principal Component Analysis代考|The Choice of a Model Class

A first important consideration to keep in mind is that inferring the “correct” model for a given data set is an elusive, if not impossible, task. The fundamental difficulty is that if we are not specific about what we mean by a “correct” model, there could easily be many different models that fit the given data set “equally well.” For instance, in the example shown in Figure 1.1, any smooth curve that passes through the sample points would seem to be as valid a model as the straight line. Furthermore, if there were noise in the given sample points, then any curve, including the line, passing through the points exactly would unlikely be the “true model.”

The question now is this: in what sense can we say that a model is correct or optimal for a given data set? To make the model inference problem well posed, i.e., to guarantee that there is a unique optimal model for the given data, we need to impose additional assumptions or restrictions on the class of models considered. To this end, we should not be looking for just any model that can describe the data. Instead, we should look for a model $M^{*}$ that is the best among a restricted class of models $\mathcal{M} .^{4}$ In addition, to make the model inference problem computationally tractable, we need to specify how restricted the class of models needs to be. A common strategy, known as the principle of Occam’s razor, ${ }^{5}$ is to try to get away with the simplest possible class of models that is just necessary to describe the data or solve the problem at hand. More precisely, the model class should be rich enough to contain at least one model that can fit the data to a desired accuracy and yet be restricted enough that it is relatively simple to find the best model for the given data.
Thus, in engineering practice, the most popular strategy is to start from the simplest class of models and increase the complexity of the models only when the simpler models become inadequate. For instance, to fit a set of sample points, one may first try the simplest class of models, namely linear models, followed by the class of hybrid (piecewise) linear models (subspaces), and then followed by the class of (piecewise) nonlinear models (submanifolds). One of the goals of this book is to demonstrate that among them, piecewise linear models can already achieve an excellent balance between expressiveness and simplicity for many important practical data sets and problems.

## 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。

## 机器学习代写|主成分分析作业代写PCA代考| Matrix Analysis

statistics-lab™ 为您的留学生涯保驾护航 在代写主成分分析PCA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。我们的专家在代写主成分分析PCA代写方面经验极为丰富，各种代写主成分分析PCA相关的作业也就用不着说。

• 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

## 机器学习代写|主成分分析作业代写PCA代考|Gradient of Real Function with Respect to Real Vector

Define gradient operator $\nabla_{x}$ of an $n \times 1$ vector $x$ as
$$\nabla_{x}=\left[\frac{\partial}{\partial x_{1}}, \quad \frac{\partial}{\partial x_{2}}, \quad \cdots, \quad \frac{\partial}{\partial x_{n}}\right]^{\mathrm{T}}=\frac{\partial}{\partial x},$$
Then the gradient of a real scalar quantity function $f(\boldsymbol{x})$ with respect to $\boldsymbol{x}$ is a $n \times 1$ column vector, which is defined as
$$\nabla_{x} f(\boldsymbol{x})=\left[\frac{\partial f(\boldsymbol{x})}{\partial x_{1}}, \quad \frac{\partial f(\boldsymbol{x})}{\partial x_{2}}, \quad \cdots, \quad \frac{\partial f(\boldsymbol{x})}{\partial x_{n}}\right]^{\mathrm{T}}=\frac{\partial f(\boldsymbol{x})}{\partial \boldsymbol{x}}$$
The negative direction of the gradient direction is called as the gradient flow of variable $\boldsymbol{x}$, written as
$$\dot{x}=-\nabla_{x} f(x)$$
The gradient of $m$-dimensional row vector function $f(\boldsymbol{x})=$ $\left[f_{1}(\boldsymbol{x}), f_{2}(\boldsymbol{x}), \ldots, f_{m}(\boldsymbol{x})\right]$ with respect to the $n \times 1$ real vector $x$ is an $n \times m$ matrix, defined as
$$\frac{\partial f(\boldsymbol{x})}{\partial \boldsymbol{x}}=\left[\begin{array}{lll} \frac{\partial f_{1}(\boldsymbol{x})}{\partial x_{1}} & \frac{\partial y_{2}(\boldsymbol{x})}{\partial x_{1}} & \frac{\partial f_{w}(\boldsymbol{x})}{\partial x_{1}} \ \frac{\partial f_{1}(\boldsymbol{x})}{\partial x_{2}} & \frac{\partial f_{2}(\boldsymbol{x})}{\partial x_{2}} & \frac{\partial f_{w}(\boldsymbol{x})}{\partial x_{2}} \ \frac{\partial f_{1}(\boldsymbol{x})}{\partial x_{n}} & \frac{\partial f_{2}(\boldsymbol{x})}{\partial x_{n}} & \frac{\partial f_{w}(\boldsymbol{x})}{\partial x_{n}} \end{array}\right]=\nabla_{x} \boldsymbol{f}(\boldsymbol{x}) .$$
Some properties of gradient operations can be summarized as follows:
(1) If $f(\boldsymbol{x})=c$ is a constant, then gradient $\frac{\partial c}{\partial \boldsymbol{x}}=\boldsymbol{O}$.
(2) Linear principle: If $f(\boldsymbol{x})$ and $g(\boldsymbol{x})$ are real functions of vector $\boldsymbol{x}$, and $c_{1}$ and $c_{2}$ are real constants, then
$$\frac{\partial\left[c_{1} f(\boldsymbol{x})+c_{2} g(\boldsymbol{x})\right]}{\partial \boldsymbol{x}}=c_{1} \frac{\partial f(\boldsymbol{x})}{\partial \boldsymbol{x}}+c_{2} \frac{\partial g(\boldsymbol{x})}{\partial \boldsymbol{x}}$$
(3) Product principle: If $f(\boldsymbol{x})$ and $g(\boldsymbol{x})$ are real functions of vector $\boldsymbol{x}$, then
$$\frac{\partial f(\boldsymbol{x}) g(\boldsymbol{x})}{\partial \boldsymbol{x}}=g(\boldsymbol{x}) \frac{\partial f(\boldsymbol{x})}{\partial \boldsymbol{x}}+f(\boldsymbol{x}) \frac{\partial g(\boldsymbol{x})}{\partial \boldsymbol{x}}$$
(4) Quotient principle: If $g(x) \neq 0$, then
$$\frac{\partial f(\boldsymbol{x}) / g(\boldsymbol{x})}{\partial \boldsymbol{x}}=\frac{1}{g^{2}(\boldsymbol{x})}\left[g(\boldsymbol{x}) \frac{\partial f(\boldsymbol{x})}{\partial \boldsymbol{x}}-f(\boldsymbol{x}) \frac{\partial g(\boldsymbol{x})}{\partial \boldsymbol{x}}\right] .$$
(5) Chain principle: If $\boldsymbol{y}(\boldsymbol{x})$ is a vector-valued function of $\boldsymbol{x}$, then
$$\frac{\partial f(\boldsymbol{y}(\boldsymbol{x}))}{\partial \boldsymbol{x}}=\frac{\partial \boldsymbol{y}^{T}(\boldsymbol{x})}{\partial \boldsymbol{x}} \frac{\partial f(\boldsymbol{y})}{\partial \boldsymbol{y}}$$
where $\frac{\partial y^{\mathrm{T}}(x)}{\partial x}$ is an $n \times n$ matrix.

## 机器学习代写|主成分分析作业代写PCA代考|Gradient Matrix of Real Function

The gradient of a real function $f(\boldsymbol{A})$ with respect to an $m \times n$ real matrix $\boldsymbol{A}$ is an $m \times n$ matrix, called as gradient matrix, defined as
$$\frac{\partial f(\boldsymbol{A})}{\partial \boldsymbol{A}}=\left[\begin{array}{cccc} \frac{\partial f(\boldsymbol{A})}{\partial A_{11}} & \frac{\partial f(\boldsymbol{A})}{\partial A_{12}} & \cdots & \frac{\partial f(\boldsymbol{A})}{\partial A_{L E}} \ \frac{\partial f(\boldsymbol{A})}{\partial A_{21}} & \frac{\partial f(\boldsymbol{A})}{\partial A_{22}} & \cdots & \frac{\partial f(\boldsymbol{A})}{\partial A_{2 c}} \ \vdots & \vdots & & \vdots \ \frac{\partial f(\boldsymbol{A})}{\partial A_{m 1}} & \frac{\partial f(\boldsymbol{A})}{\partial A_{\mathrm{m} 2}} & \cdots & \frac{\partial f(\boldsymbol{A})}{\partial A_{m n}} \end{array}\right]=\nabla_{\boldsymbol{A}} f(\boldsymbol{A})$$
where $A_{i j}$ is the element of matrix $A$ on its $i$ th row and $j$ th column.
Some properties of the gradient of a real function with respect to a matrix can be summarized as follows:
(1) If $f(\boldsymbol{A})=c$ is a constant, where $\boldsymbol{A}$ is an $m \times n$ matrix, then $\frac{\partial c}{\partial A}=\boldsymbol{O}{m \times n}$. (2) Linear principle: If $f(\boldsymbol{A})$ and $g(\boldsymbol{A})$ are real functions of matrix $\boldsymbol{A}$, and $c{1}$ and $c_{2}$ are real constants, then
$$\frac{\partial\left[c_{1} f(\boldsymbol{A})+c_{2} g(\boldsymbol{A})\right]}{\partial \boldsymbol{A}}=c_{1} \frac{\partial f(\boldsymbol{A})}{\partial \boldsymbol{A}}+c_{2} \frac{\partial g(\boldsymbol{A})}{\partial \boldsymbol{A}} .$$
(3) Product principle: If $f(\boldsymbol{A})$ and $g(\boldsymbol{A})$ are real functions of matrix $\boldsymbol{A}$, then
$$\frac{\partial f(\boldsymbol{A}) g(\boldsymbol{A})}{\partial \boldsymbol{A}}=g(\boldsymbol{A}) \frac{\partial f(\boldsymbol{A})}{\partial \boldsymbol{A}}+f(\boldsymbol{A}) \frac{\partial g(\boldsymbol{A})}{\partial \boldsymbol{A}}$$
(4) Quotient principle: If $g(\boldsymbol{A}) \neq 0$, then
$$\frac{\partial f(\boldsymbol{A}) / g(\boldsymbol{A})}{\partial(\boldsymbol{A})}=\frac{1}{g^{2}(\boldsymbol{A})}\left[g(\boldsymbol{A}) \frac{\partial f(\boldsymbol{A})}{\partial \boldsymbol{A}}-f(\boldsymbol{A}) \frac{\partial g(\boldsymbol{A})}{\partial \boldsymbol{A}}\right]$$
(5) Chain principle: Let $\boldsymbol{A}$ be an $m \times n$ matrix, and $y=f(\boldsymbol{A})$ and $g(y)$ are real functions of matrix $A$ and scalar $y$, respectively. Then
$$\frac{\partial g(f(A))}{\partial A}=\frac{d g(y)}{d y} \frac{\partial f(A)}{\partial A}$$
(6) If $\boldsymbol{A} \in \Re^{m \times n}, \boldsymbol{x} \in \Re^{m \times 1}, \boldsymbol{y} \in \Re^{n \times 1}$, then
$$\frac{\partial x^{\mathrm{T}} A y}{\partial A}=A y^{\mathrm{T}}$$
(7) If $A \in \Re^{n \times n}$ is nonsingular $\boldsymbol{x} \in \Re^{n \times 1}, \boldsymbol{y} \in \mathcal{K}^{n \times 1}$, then
$$\frac{\partial \boldsymbol{x}^{\mathrm{T}} \boldsymbol{A}^{-1} \boldsymbol{y}}{\partial \boldsymbol{A}}=-\boldsymbol{A}^{-\mathrm{T}} \boldsymbol{A} \boldsymbol{y}^{\mathrm{T}} \boldsymbol{A}^{-\mathrm{T}}$$

## 机器学习代写|主成分分析作业代写PCA代考|Gradient Matrix of Trace Function

Here, we summarize some properties of gradient matrix of trace functions.
(1)-(3) are gradient matrices of the trace of a single matrix.
(1) If $W$ is an $m \times m$ matrix, then
$$\frac{\partial \operatorname{tr}(\boldsymbol{W})}{\partial \boldsymbol{W}}=\boldsymbol{I}_{m}$$
(2) If an $m \times m$ matrix $\boldsymbol{W}$ is invertible, then
$$\frac{\partial \mathrm{tr}\left(\boldsymbol{W}^{-1}\right)}{\partial \boldsymbol{W}}=-\left(\boldsymbol{W}^{-2}\right)^{\mathrm{T}}$$
(3) For the outer product of two vectors, it holds that
$$\frac{\partial \operatorname{tr}\left(x y^{\mathrm{T}}\right)}{\partial \boldsymbol{x}}=\frac{\partial t r\left(\boldsymbol{y} \boldsymbol{x}^{\mathrm{T}}\right)}{\partial \boldsymbol{x}}=\boldsymbol{y}$$
(4)-(7) are gradient matrices of the trace of the product of two matrices.
(4) If $W \in \Re^{m \times n}, A \in \Re^{n \times m}$, then
$$\frac{\partial \operatorname{tr}(\boldsymbol{W} \boldsymbol{A})}{\partial \boldsymbol{W}}=\frac{\partial \mathrm{tr}(\boldsymbol{A} \boldsymbol{W})}{\partial \boldsymbol{W}}=\boldsymbol{A}^{\mathrm{T}} .$$
(5) If $\boldsymbol{W} \in \Re^{m \times n}, \boldsymbol{A} \in \Re^{m \times n}$, then
$$\frac{\partial \mathrm{tr}\left(\boldsymbol{W}^{\mathrm{T}} \boldsymbol{A}\right)}{\partial \boldsymbol{W}}=\frac{\partial \mathrm{tr}\left(\boldsymbol{A} \boldsymbol{W}^{\mathrm{T}}\right)}{\partial \boldsymbol{W}}=\boldsymbol{A} .$$
(6) If $W \in \Re^{m \times n}$, then
$$\frac{\partial \mathrm{tr}\left(\boldsymbol{W} \boldsymbol{W}^{\mathrm{T}}\right)}{\partial \boldsymbol{W}}=\frac{\partial \mathrm{tr}\left(\boldsymbol{W}^{\mathrm{T}} \boldsymbol{W}\right)}{\partial \boldsymbol{W}}=2 \boldsymbol{W}$$
(7) If $W \in \Re^{m \times n}$, then
$$\frac{\partial \mathrm{tr}\left(\boldsymbol{W}^{2}\right)}{\partial \boldsymbol{W}}=\frac{\partial \mathrm{tr}(\boldsymbol{W} \boldsymbol{W})}{\partial \boldsymbol{W}}=2 \boldsymbol{W}^{\mathrm{T}} .$$
(8) If $\boldsymbol{W}, \boldsymbol{A} \in \Re^{m \times m}$ and $\boldsymbol{W}$ is nonsingular, then
$$\frac{\partial \mathrm{tr}\left(\boldsymbol{A} \boldsymbol{W}^{-1}\right)}{\partial \boldsymbol{W}}=-\left(\boldsymbol{W}^{-1} \boldsymbol{A} \boldsymbol{W}^{-1}\right)^{\mathrm{T}} .$$

## 机器学习代写|主成分分析作业代写PCA代考|Gradient of Real Function with Respect to Real Vector

∇X=[∂∂X1,∂∂X2,⋯,∂∂Xn]吨=∂∂X,

∇XF(X)=[∂F(X)∂X1,∂F(X)∂X2,⋯,∂F(X)∂Xn]吨=∂F(X)∂X

X˙=−∇XF(X)

∂F(X)∂X=[∂F1(X)∂X1∂是2(X)∂X1∂F在(X)∂X1 ∂F1(X)∂X2∂F2(X)∂X2∂F在(X)∂X2 ∂F1(X)∂Xn∂F2(X)∂Xn∂F在(X)∂Xn]=∇XF(X).

(1) 如果F(X)=C是一个常数，然后是梯度∂C∂X=这.
(2) 线性原理：如果F(X)和G(X)是向量的实函数X， 和C1和C2是实常数，那么
∂[C1F(X)+C2G(X)]∂X=C1∂F(X)∂X+C2∂G(X)∂X
（3）产品原理：如果F(X)和G(X)是向量的实函数X， 然后
∂F(X)G(X)∂X=G(X)∂F(X)∂X+F(X)∂G(X)∂X
(4) 商数原则：如果G(X)≠0， 然后
∂F(X)/G(X)∂X=1G2(X)[G(X)∂F(X)∂X−F(X)∂G(X)∂X].
(5)链式原理：如果是(X)是一个向量值函数X， 然后
∂F(是(X))∂X=∂是吨(X)∂X∂F(是)∂是

## 机器学习代写|主成分分析作业代写PCA代考|Gradient Matrix of Real Function

∂F(一种)∂一种=[∂F(一种)∂一种11∂F(一种)∂一种12⋯∂F(一种)∂一种大号和 ∂F(一种)∂一种21∂F(一种)∂一种22⋯∂F(一种)∂一种2C ⋮⋮⋮ ∂F(一种)∂一种米1∂F(一种)∂一种米2⋯∂F(一种)∂一种米n]=∇一种F(一种)

(1) 如果F(一种)=C是一个常数，其中一种是一个米×n矩阵，那么∂C∂一种=这米×n. (2) 线性原理：如果F(一种)和G(一种)是矩阵的实函数一种， 和C1和C2是实常数，那么
∂[C1F(一种)+C2G(一种)]∂一种=C1∂F(一种)∂一种+C2∂G(一种)∂一种.
（3）产品原理：如果F(一种)和G(一种)是矩阵的实函数一种， 然后
∂F(一种)G(一种)∂一种=G(一种)∂F(一种)∂一种+F(一种)∂G(一种)∂一种
(4) 商数原则：如果G(一种)≠0， 然后
∂F(一种)/G(一种)∂(一种)=1G2(一种)[G(一种)∂F(一种)∂一种−F(一种)∂G(一种)∂一种]
(5) 链式原理：让一种豆米×n矩阵，和是=F(一种)和G(是)是矩阵的实函数一种和标量是， 分别。然后
∂G(F(一种))∂一种=dG(是)d是∂F(一种)∂一种
(6) 如果一种∈ℜ米×n,X∈ℜ米×1,是∈ℜn×1， 然后
∂X吨一种是∂一种=一种是吨
(7) 如果一种∈ℜn×n是非奇异的X∈ℜn×1,是∈ķn×1， 然后
∂X吨一种−1是∂一种=−一种−吨一种是吨一种−吨

## 机器学习代写|主成分分析作业代写PCA代考|Gradient Matrix of Trace Function

(1)-(3) 是单个矩阵的迹的梯度矩阵。
(1) 如果在是一个米×米矩阵，那么
∂tr⁡(在)∂在=一世米
(2) 如果一个米×米矩阵在是可逆的，那么
∂吨r(在−1)∂在=−(在−2)吨
(3) 对于两个向量的外积，有
∂tr⁡(X是吨)∂X=∂吨r(是X吨)∂X=是
(4)-(7) 是两个矩阵乘积的迹的梯度矩阵。
(4) 如果在∈ℜ米×n,一种∈ℜn×米， 然后
∂tr⁡(在一种)∂在=∂吨r(一种在)∂在=一种吨.
(5) 如果在∈ℜ米×n,一种∈ℜ米×n， 然后
∂吨r(在吨一种)∂在=∂吨r(一种在吨)∂在=一种.
(6) 如果在∈ℜ米×n， 然后
∂吨r(在在吨)∂在=∂吨r(在吨在)∂在=2在
(7) 如果在∈ℜ米×n， 然后
∂吨r(在2)∂在=∂吨r(在在)∂在=2在吨.
(8) 如果在,一种∈ℜ米×米和在是非奇异的，那么
∂吨r(一种在−1)∂在=−(在−1一种在−1)吨.

## 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。

## 机器学习代写|主成分分析作业代写PCA代考| Rayleigh Quotient and Its Characteristics

statistics-lab™ 为您的留学生涯保驾护航 在代写主成分分析PCA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。我们的专家在代写主成分分析PCA代写方面经验极为丰富，各种代写主成分分析PCA相关的作业也就用不着说。

• 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

If the negative direction of RQ gradient is regarded as the gradient flow of vector $\boldsymbol{x}$, e.g.’,
$$\dot{\boldsymbol{x}}=-[\boldsymbol{C}-r(\boldsymbol{x}) \boldsymbol{I}] \boldsymbol{x}$$
then vector $x$ can be computed iteratively by the following gradient algorithm:
$$\boldsymbol{x}(k+1)=\boldsymbol{x}(k)+\mu \dot{\boldsymbol{x}}=\boldsymbol{x}(k)-\mu[\boldsymbol{C}-r(\boldsymbol{x}) \boldsymbol{I}] \boldsymbol{x} .$$
It is worth noting that the gradient algorithm of RQ has faster convergence speed than the iterative algorithm of standard RQ.

In the following, the conjugate gradient algorithm for RQ will be introduced, where $\boldsymbol{A}$ in the RQ is a real symmetric matrix.

Starting from some initial vector, the conjugate gradient algorithm uses the iterative equation, e.g.,
$$\boldsymbol{x}{k+1}=\boldsymbol{x}{k}+\alpha_{k} \boldsymbol{P}{k}$$ to update and approach the eigenvector, associated with the minimal or maximal eigenvalue of a symmetric matrix. The real coefficient $\alpha{k}$ is
$$\alpha_{k}=\pm \frac{1}{2 D}\left(-B+\sqrt{B^{2}-4 C D}\right),$$
where ” $+$ ” is used in the updating of the eigenvector associated with the minimal eigenvalue, and “-” is used in the updating of the eigenvector associated with the maximal eigenvalue. The formulae for parameters D, B, C in the above equations are

$$\left{\begin{array}{c} D=P_{b}(k) P_{c}(k)-P_{a}(k) P_{d}(k) \ B=P_{b}(k)-\lambda_{k} P_{d}(k) \ C=P_{a}(k)-\lambda_{k} P_{c}(k) \ P_{a}(k)=\boldsymbol{P}{k}^{\mathrm{T}} \boldsymbol{A} \boldsymbol{x}{k} /\left(\boldsymbol{x}{k}^{\mathrm{T}} \boldsymbol{x}{k}\right) \ P_{b}(k)=\boldsymbol{p}{k}^{\mathrm{T}} \boldsymbol{A} \boldsymbol{p}{k} /\left(\boldsymbol{x}{k}^{\mathrm{T}} \boldsymbol{x}{k}\right) \ P_{c}(k)=\boldsymbol{p}{k}^{\mathrm{T}} \boldsymbol{x}{k} /\left(\boldsymbol{x}{k}^{\mathrm{T}} \boldsymbol{x}{k}\right) \ P_{d}(k)=\boldsymbol{p}{k}^{\mathrm{T}} \boldsymbol{p}{k} /\left(\boldsymbol{x}{k}^{\mathrm{T}} \boldsymbol{x}{k}\right) \ \lambda_{k}=r\left(\boldsymbol{x}{k}\right)=\boldsymbol{x}{k}^{\mathrm{T}} \boldsymbol{A} \boldsymbol{x}{k} /\left(\boldsymbol{x}{k}^{\mathrm{T}} \boldsymbol{x}{k}\right) . \end{array}\right.$$ At the $k+1$ th iteration, the search direction can be selected as $$\boldsymbol{p}{k+1}=\boldsymbol{r}{k+1}+b(k) \boldsymbol{p}{k}$$
where $b(-1)=0$ and $\boldsymbol{r}{k+1}$ is the residual vector at the $k+1$ th iteration. $\boldsymbol{r}{k+1}$ and $b(k)$ can be computed, respectively, as
$$\boldsymbol{r}{k+1}=-\frac{1}{2} \nabla{x} r\left(\boldsymbol{x}{k+1}\right)=\left(\lambda{k+1} \boldsymbol{x}{k+1}-\boldsymbol{A} \boldsymbol{x}{k+1}\right) /\left(\boldsymbol{x}{k=1}^{\mathrm{T}} \boldsymbol{x}{k+1}\right)$$
and
$$b(k)=-\frac{\boldsymbol{r}{k+1}^{\mathrm{T}} \boldsymbol{A} \boldsymbol{p}{k}+\left(\boldsymbol{r}{k+1}^{\mathrm{T}} \boldsymbol{r}{k+1}\right)\left(\boldsymbol{x}{k+1}^{\mathrm{T}} \boldsymbol{p}{k}\right)}{\boldsymbol{p}{k}^{\mathrm{T}}\left(\boldsymbol{A} \boldsymbol{p}{k}-\lambda_{k+1} \boldsymbol{I}\right) \boldsymbol{p}{k}}$$ Equations (2.5)-(2.9) constitute the conjugate gradient algorithm for RQ, which was proposed in [11]. If the updated $x{k}$ is normalized to one and “t” (or “-“) is selected in Eq. (2.6), the above algorithm will obtain the minimal (or maximal) eigenvalue of matrix $A$ and its associated eigenvectors.

## 机器学习代写|主成分分析作业代写PCA代考|Generalized Rayleigh Quotient

Definition $2.3$ Assume that $A \in \mathbb{C}^{n \times n}, \mathbf{B} \in \mathbb{C}^{n \times n}$ are both Hermitian matrices, and $\boldsymbol{B}$ is positive definite. The generalized RQ or generalized Rayleigh-Ritz of the matrix pencil $(\boldsymbol{A}, \boldsymbol{B})$ is a scalar function, e.g.,
$$r(\boldsymbol{x})=\frac{\boldsymbol{x}^{H} \boldsymbol{A} \boldsymbol{x}}{\boldsymbol{x}^{H} \boldsymbol{B} \boldsymbol{x}},$$
where $x$ is a quantity to be selected, and the objective is to maximize or minimize the generalized RQ.

In order to solve for the generalized RQ, define a new vector $\tilde{\boldsymbol{x}}=\boldsymbol{B}^{1 / 2} \boldsymbol{x}$, where $\boldsymbol{B}^{1 / 2}$ is the square root of the positive definite $\boldsymbol{B}$. Replace $\boldsymbol{x}$ by $\boldsymbol{B}^{-1 / 2} \tilde{\boldsymbol{x}}$ in (2.43). Then it holds that
$$r(\tilde{\boldsymbol{x}})=\frac{\tilde{\boldsymbol{x}}^{H}\left(\boldsymbol{B}^{-1 / 2}\right)^{H} \boldsymbol{A}\left(\boldsymbol{B}^{-1 / 2}\right)^{H} \tilde{\boldsymbol{x}}}{\tilde{\boldsymbol{x}}^{H} \tilde{\boldsymbol{x}}},$$
which shows that the generalized RQ of matrix pencil $(\boldsymbol{A}, \boldsymbol{B})$ is equivalent to the RQ of matrix product $\left(\boldsymbol{B}^{-1 / 2}\right)^{H} \boldsymbol{A}\left(\boldsymbol{B}^{-1 / 2}\right)^{H}$. From the Rayleigh-Ritz theorem, it is clear that when vector $\tilde{x}$ is the eigenvector associated with the smallest eigenvalue $\lambda_{\min }$ of matrix product $\left(\boldsymbol{B}^{-1 / 2}\right)^{H} \boldsymbol{A}\left(\boldsymbol{B}^{-1 / 2}\right)^{H}$, the generalized RQ obtains $\lambda_{\min }$. And if vector $\tilde{\boldsymbol{x}}$ is the eigenvector associated with the largest eigenvalue $\lambda_{\max }$ of matrix product $\left(\boldsymbol{B}^{-1 / 2}\right)^{H} \boldsymbol{A}\left(\boldsymbol{B}^{-1 / 2}\right)^{H}$, the generalized RQ obtains $\lambda_{\max }$.

In the following, we review the eigen decomposition of matrix product $\left(\boldsymbol{B}^{-1 / 2}\right)^{H} \boldsymbol{A}\left(\boldsymbol{B}^{-1 / 2}\right)^{H}$, e.g.,
$$\left(\boldsymbol{B}^{-1 / 2}\right)^{H} \boldsymbol{A}\left(\boldsymbol{B}^{-1 / 2}\right)^{H} \tilde{\boldsymbol{x}}=i \tilde{\boldsymbol{x}} .$$
If $\mathbf{B}=\sum_{i=1}^{n} \beta_{i} v_{i} v_{i}^{H}$ is an eigen decomposition of matrix $\boldsymbol{B}$, then
$$\mathbf{B}^{1 / 2}=\sum_{i=1}^{n} \sqrt{\beta_{i}} v_{i} v_{i}^{H}$$
and $\boldsymbol{B}^{1 / 2} \boldsymbol{B}^{1 / 2}=\boldsymbol{B}$. Since matrix $\boldsymbol{B}^{1 / 2}$ and $\boldsymbol{B}^{-1 / 2}$ have the same eigenvectors and their eigenvalues are reciprocals to each other, then it follows that
$$\mathbf{B}^{-1 / 2}=\sum_{i=1}^{n} \frac{1}{\sqrt{\beta_{i}}} v_{i} v_{i}^{H},$$
which shows that $\boldsymbol{B}^{-1 / 2}$ is also an Hermitian matrix, e.g., $\left(\boldsymbol{B}^{-1 / 2}\right)^{H}=\boldsymbol{B}^{-1 / 2}$.

## 机器学习代写|主成分分析作业代写PCA代考|Differential and Integral of Matrix with Respect to Scalar

If $\boldsymbol{A}(t)=\left{a_{i j}(t)\right}_{m \times n}$ is a real matrix function of scalar $t$, then its differential and integral are, respectively, defined as
\left{\begin{aligned} \frac{\mathrm{d}}{\mathrm{d} t} \boldsymbol{A}(t) &=\left{\frac{\mathrm{d}}{\mathrm{d}{t}} a{i j}(t)\right}_{m \times n} \ \int A(t) \mathrm{d} t &=\left{\int a_{i j}(t) \mathrm{d} t\right}_{m \times n} \end{aligned}\right.
If $\boldsymbol{A}(\mathrm{t})$ and $\boldsymbol{B}(\mathrm{t})$ are, respectively, $m \times n$ and $n \times r$ matrices, then
$$\frac{\mathrm{d}}{\mathrm{d} t}[\boldsymbol{A}(t) \boldsymbol{B}(t)]=\left[\frac{\mathrm{d} \boldsymbol{A}(t)}{\mathrm{d} t}\right] \boldsymbol{B}(t)+\boldsymbol{A}(t)\left[\frac{\mathrm{d} \boldsymbol{B}(t)}{\mathrm{d} t}\right] .$$
If $\boldsymbol{A}(\mathrm{t})$ and $\boldsymbol{B}(\mathrm{t})$ are both $m \times n$ matrices, then
$$\frac{\mathrm{d}}{\mathrm{d} t}[\boldsymbol{A}(t)+\boldsymbol{B}(t)]=\frac{\mathrm{d} \boldsymbol{A}(t)}{\mathrm{d} t}+\frac{\mathrm{d} \boldsymbol{B}(t)}{\mathrm{d} t}$$
If $A(\mathrm{t})$ is a rank- $n$ invertible square matrix, then
$$\frac{\mathrm{d} \boldsymbol{A}^{-1}(t)}{\mathrm{d} t}=-\boldsymbol{A}^{-1}(t) \frac{\mathrm{d} \boldsymbol{A}(t)}{\mathrm{d} t} \boldsymbol{A}^{-1}(t) .$$

## 主成分分析代写

X˙=−[C−r(X)一世]X

X(ķ+1)=X(ķ)+μX˙=X(ķ)−μ[C−r(X)一世]X.

Xķ+1=Xķ+一种ķ磷ķ更新和逼近与对称矩阵的最小或最大特征值相关的特征向量。实际系数一种ķ是

$$\左{D=磷b(ķ)磷C(ķ)−磷一种(ķ)磷d(ķ) 乙=磷b(ķ)−λķ磷d(ķ) C=磷一种(ķ)−λķ磷C(ķ) 磷一种(ķ)=磷ķ吨一种Xķ/(Xķ吨Xķ) 磷b(ķ)=pķ吨一种pķ/(Xķ吨Xķ) 磷C(ķ)=pķ吨Xķ/(Xķ吨Xķ) 磷d(ķ)=pķ吨pķ/(Xķ吨Xķ) λķ=r(Xķ)=Xķ吨一种Xķ/(Xķ吨Xķ).\对。一种吨吨H和ķ+1吨H一世吨和r一种吨一世这n,吨H和s和一种rCHd一世r和C吨一世这nC一种nb和s和l和C吨和d一种s\boldsymbol{p}{k+1}=\boldsymbol{r}{k+1}+b(k) \boldsymbol{p}{k} 在H和r和b(−1)=0一种ndrķ+1一世s吨H和r和s一世d在一种l在和C吨这r一种吨吨H和ķ+1吨H一世吨和r一种吨一世这n.rķ+1一种ndb(ķ)C一种nb和C这米p在吨和d,r和sp和C吨一世在和l是,一种s \boldsymbol{r}{k+1}=-\frac{1}{2} \nabla{x} r\left(\boldsymbol{x}{k+1}\right)=\left(\lambda{k +1} \boldsymbol{x}{k+1}-\boldsymbol{A} \boldsymbol{x}{k+1}\right) /\left(\boldsymbol{x}{k=1}^{\mathrm {T}} \boldsymbol{x}{k+1}\right) 一种nd b(k)=-\frac{\boldsymbol{r}{k+1}^{\mathrm{T}} \boldsymbol{A} \boldsymbol{p}{k}+\left(\boldsymbol{r}{ k+1}^{\mathrm{T}} \boldsymbol{r}{k+1}\right)\left(\boldsymbol{x}{k+1}^{\mathrm{T}} \boldsymbol{p }{k}\right)}{\boldsymbol{p}{k}^{\mathrm{T}}\left(\boldsymbol{A} \boldsymbol{p}{k}-\lambda_{k+1} \ boldsymbol{I}\right) \boldsymbol{p}{k}}$$ 方程 (2.5)-(2.9) 构成了 RQ 的共轭梯度算法，该算法在 [11] 中提出。如果更新Xķ被归一化为 1 并且在方程式中选择“t”（或“-”）。(2.6)，上述算法将得到矩阵的最小（或最大）特征值一种及其相关的特征向量。

## 机器学习代写|主成分分析作业代写PCA代考|Generalized Rayleigh Quotient

r(X)=XH一种XXH乙X,

r(X~)=X~H(乙−1/2)H一种(乙−1/2)HX~X~HX~,

(乙−1/2)H一种(乙−1/2)HX~=一世X~.

## 机器学习代写|主成分分析作业代写PCA代考|Differential and Integral of Matrix with Respect to Scalar

\left{\begin{aligned} \frac{\mathrm{d}}{\mathrm{d} t} \boldsymbol{A}(t) &=\左{\frac{\mathrm{d}}{\mathrm{d} {t}} a {ij}(t)\right}_{m \times n} \ \int A(t) \mathrm{d} t &=\left{\int a_{ij}(t) \mathrm{d} t\right}_{m \times n} \end{aligned}\right. 一世F一种(吨)一种nd乙(吨)一种r和,r和sp和C吨一世在和l是,米×n一种ndn×r米一种吨r一世C和s,吨H和n \frac{\mathrm{d}}{\mathrm{d} t}[\boldsymbol{A}(t) \boldsymbol{B}(t)]=\left[\frac{\mathrm{d} \boldsymbol{ A}(t)}{\mathrm{d} t}\right] \boldsymbol{B}(t)+\boldsymbol{A}(t)\left[\frac{\mathrm{d} \boldsymbol{B} (t)}{\mathrm{d} t}\right] 。 一世F一种(吨)一种nd乙(吨)一种r和b这吨H米×n米一种吨r一世C和s,吨H和n \frac{\mathrm{d}}{\mathrm{d} t}[\boldsymbol{A}(t)+\boldsymbol{B}(t)]=\frac{\mathrm{d} \boldsymbol{A} (t)}{\mathrm{d} t}+\frac{\mathrm{d} \boldsymbol{B}(t)}{\mathrm{d} t} 一世F一种(吨)一世s一种r一种nķ−n一世n在和r吨一世bl和sq在一种r和米一种吨r一世X,吨H和n \frac{\mathrm{d} \boldsymbol{A}^{-1}(t)}{\mathrm{d} t}=-\boldsymbol{A}^{-1}(t) \frac{\mathrm {d} \boldsymbol{A}(t)}{\mathrm{d} t} \boldsymbol{A}^{-1}(t) 。

## 有限元方法代写

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## MATLAB代写

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## 机器学习代写|主成分分析作业代写PCA代考| Eigenvalue Decomposition

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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 机器学习代写|主成分分析作业代写PCA代考|Eigenvalue Decomposition of Hermitian Matrix

All the discussions on eigenvalues and eigenvectors in the above hold for general matrices, and they do not require the matrices to be real symmetric or complex conjugate symmetric. However, in the statistical and information science, one usually encounter real symmetric or Hermitian (complex conjugate symmetric) matrices. For example, the autocorrelation matrix of a real measurement data vector $\boldsymbol{R}=E\left{\boldsymbol{x}(t) \boldsymbol{x}^{T}(t)\right}$ is real symmetric, while the autocorrelation matrix of a complex measurement data vector $\boldsymbol{R}=E\left{\boldsymbol{x}(t) \boldsymbol{x}^{H}(t)\right}$ is Hermitian. On the other hand, since a real symmetric matrix is a special case of Hermitian matrix and the eigenvalues and eigenvectors of a Hermitian matrix have a series of important properties, and it is necessary to discuss individually the eigen analysis of Hermitian matrix.

1. Eigenvalue and Eigenvector of Hermitian matrix.
Some important properties of eigenvalues and eigenvectors of Hermitian matrices can be summarized as follows:
(1) The eigenvalues of an Hermitian matrix $A$ must be a real number.
(2) Let $(\lambda, \boldsymbol{u})$ be an eigen pair of an Hermitian matrix $\boldsymbol{A}$. If $\boldsymbol{A}$ is invertible, then $(1 / \lambda, u)$ is an eigen pair of matrix $A^{-1}$.
(3) If $\lambda_{k}$ is a multiple eigenvalue of Hermitian matrix $A^{H}=A$, and its multiplicity is $m_{k}$, then $\operatorname{rank}\left(\boldsymbol{A}-\lambda_{k} \boldsymbol{I}\right)=n-m_{k}$.
(4) Any Hermitian matrix $A$ is diagonalizable, namely $U^{-1} \boldsymbol{A} U=\Sigma$.
(5) All the eigenvectors of an Hermitian matrix are linearly independent, and they are mutual orthogonal, namely the eigen matrix $\boldsymbol{U}=\left[\boldsymbol{u}{1}, \boldsymbol{u}{2}, \ldots, \boldsymbol{u}{n}\right]$ is a unitary matrix and it meets $\boldsymbol{U}^{-1}=\boldsymbol{U}^{H}$. (6) From property (5), it holds that $\boldsymbol{U}^{H} \boldsymbol{A} \boldsymbol{U}=\Sigma=\operatorname{diag}\left(\lambda{1}, \lambda_{2}, \ldots, \lambda_{n}\right)$ or $\boldsymbol{A}=\boldsymbol{U} \Sigma \boldsymbol{U}^{H}$, which can be rewritten as: $\boldsymbol{A}=\sum_{i=1}^{n} \lambda_{i} \boldsymbol{u}{i} \boldsymbol{u}{i}^{H}$. This is called the spectral decomposition of a Hermitian matrix.
(7) The spread formula of the inverse of an Hermitian matrix $A$ is
$$\boldsymbol{A}^{-1}=\sum_{i=1}^{n} \frac{\mathrm{I}}{\lambda_{i}} \boldsymbol{u}{i} \boldsymbol{u}{i}^{H}$$
Thus, if one know the eigen decomposition of an Hermitian matrix $\boldsymbol{A}$, then one can directly obtain the inverse matrix $A^{-1}$ using the above formula.
(8) For two $n \times n$ Hermitian matrices $\boldsymbol{A}$ and $\boldsymbol{B}$, there exists a unitary matrix so that $\boldsymbol{P}^{H} \boldsymbol{A} \boldsymbol{P}$ and $\boldsymbol{P}^{H} \boldsymbol{B P}$ are both diagonal if and only if $\boldsymbol{A B}=\boldsymbol{B A}$.
(9) For two $n \times n$ non-negative definite Hermitian matrices $\boldsymbol{A}$ and $\boldsymbol{B}$, there exists a nonsingular matrix $P$ so that $P^{H} A P$ and $P^{H} B P$ are both diagonal.

## 机器学习代写|主成分分析作业代写PCA代考|Generalized Eigenvalue Decomposition

Let $A$ and $B$ both be $n \times n$ square matrices, and they constitute a matrix pencil or matrix pair, written as $(\boldsymbol{A}, \boldsymbol{B})$. Now we consider the following generalized eigenvalue problem. That is, to compute all scalar $\lambda$ such that
$$A u=\lambda B u$$
has nonzero solution $\boldsymbol{u} \neq 0$, where the scalar $\lambda$ and the nonzero vector $\boldsymbol{u}$ are called the generalized eigenvalue and the generalized eigenvector of matrix pencil $(\boldsymbol{A}, \boldsymbol{B})$, respectively. A generalized eigenvalue and its associated generalized eigenvector are called generalized eigen pair, written as $(\lambda, \boldsymbol{u})$. Equation (2.35) is also called the generalized eigen equation. It is obvious that the eigenvalue problem is a special case when the matrix pencil is chosen as $(\boldsymbol{A}, \boldsymbol{I})$.

Theorem 2.6 $\lambda \in \mathbb{C}$ and $\mathbf{u} \in \mathbb{C}^{n}$ are respectively the generalized eigenvalue and the associated generalized eigenvector of matrix pencil $(\boldsymbol{A}, \boldsymbol{B})_{n \times n}$ if and only if:
(1) $\operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{B})=0$.
(2) $\boldsymbol{u} \in \operatorname{Null}(\boldsymbol{A}-\lambda \boldsymbol{B})$, and $\boldsymbol{u} \neq 0$.
In the natural science, sometimes it is necessary to discuss the eigenvalue problem of the generalized matrix pencil.

Suppose that $n \times n$ square matrices $\boldsymbol{A}$ and $\boldsymbol{B}$ are both Hermitian, and $B$ is positive definite. Then $(\boldsymbol{A}, \boldsymbol{B})$ is called the regularized matrix pencil.

The eigenvalue problem of regularized matrix pencil is similar to the one of Hermitian matrix.

## 机器学习代写|主成分分析作业代写PCA代考|Rayleigh Quotient

Definition 2.1 The Rayleigh quotient (RQ) of an Hermitian matrix $C \in \mathbb{C}^{n \times n}$ is a scalar, defined as
$$r(\boldsymbol{u})=r(\boldsymbol{u}, \boldsymbol{C})=\frac{\boldsymbol{u}^{H} \boldsymbol{C} \boldsymbol{u}}{\boldsymbol{u}^{H} \boldsymbol{u}},$$
where $u$ is a quantity to be selected. The objective is to maximize or minimize the Rayleigh quotient.
The most relevant properties of the $R Q$ are can be summarized as follows:
(1) Homogeneity: $r(\alpha \boldsymbol{u}, \beta \boldsymbol{u})=\beta r(\boldsymbol{u}, \boldsymbol{C}) \quad \forall \alpha, \beta \neq 0$.
(2) Translation invariance: $\boldsymbol{r}(\boldsymbol{u}, \boldsymbol{C}-\alpha \boldsymbol{I})=\boldsymbol{r}(\boldsymbol{u}, \boldsymbol{C})-\alpha$.
(3) Boundedness: Since $\boldsymbol{u}$ ranges over all nonzero vectors, $r(\boldsymbol{u})$ fills a region in the complex plane which is called the field of values of $\boldsymbol{C}$. This region is closed, bounded, and convex. If $\boldsymbol{C}=\boldsymbol{C}^{*}$ (selfadjoint matrix), the field of values is the real interval bounded by the extreme eigenvalues.
(4) Orthogonality: $\boldsymbol{u} \perp(\boldsymbol{C}-r(\boldsymbol{u}) \boldsymbol{I}) \boldsymbol{u}$.
(5) Minimal residual: $\forall \boldsymbol{u} \neq 0 \wedge \forall$ scalar $\mu,|(\boldsymbol{C}-r(\boldsymbol{u}) \boldsymbol{I}) \boldsymbol{u}| \leq|(\boldsymbol{C}-\mu \boldsymbol{I}) \boldsymbol{u}|$.
Proposition $2.1$ (Stationarity) Let $C$ be a real symmetric n-dimensional matrix with eigenvalues $\lambda_{n} \leq \lambda_{n-1} \leq \cdots \lambda_{1}$ and associated unit eigenvectors $z_{1}, z_{2}, \ldots, z_{n}$. Then it holds that $\lambda_{1}=\max r(\boldsymbol{u}, \boldsymbol{C}), \lambda_{n}=\min r(\boldsymbol{u}, \boldsymbol{C})$. More generally, the critical points and critical values of $r(\boldsymbol{u}, \boldsymbol{C})$ are the eigenvectors and eigenvalues of $\boldsymbol{C}$.

Proposition $2.2$ (Degeneracy): The $R Q$ critical points are degenerate because at these points the Hessian matrix is not invertible. Then the RQ is not a Morse function in every open subspace of the domain containing a critical point.

Furthermore, the following important theorems also holds for RQ.
Courant-Fischer Theorem: Let $C \in \mathbb{C}^{n \times n}$ be an Hermitian matrix, and its eigenvalues are $\lambda_{1} \geq \lambda_{2} \geq \cdots \leq \lambda_{n}$, then it holds that for $\lambda_{k}(1 \leq k \leq u)$ :
$$\lambda_{k}=\min {S, \operatorname{dim}(S)=\boldsymbol{n}-k+1} \max {\boldsymbol{u} \in S, \boldsymbol{u} \neq 0}\left(\frac{\boldsymbol{u}^{H} \boldsymbol{C} \boldsymbol{u}}{\boldsymbol{u}^{H} \boldsymbol{u}}\right)$$
The Courant-Fischer Theorem can also written as
$$\lambda_{k}=\min {S, \operatorname{dim}(S)=k} \max {\boldsymbol{u} \in S, \boldsymbol{u} \neq 0}\left(\frac{\boldsymbol{u}^{H} \boldsymbol{C} \boldsymbol{u}}{\boldsymbol{u}^{H} \boldsymbol{u}}\right)$$

## 机器学习代写|主成分分析作业代写PCA代考|Eigenvalue Decomposition of Hermitian Matrix

1. Hermitian 矩阵的特征值和特征向量。
Hermitian 矩阵的特征值和特征向量的一些重要性质可以概括如下：
(1) Hermitian 矩阵的特征值一种必须是实数。
(2) 让(λ,在)是 Hermitian 矩阵的特征对一种. 如果一种是可逆的，那么(1/λ,在)是矩阵的特征对一种−1.
(3) 如果λķ是 Hermitian 矩阵的多重特征值一种H=一种，其多重性为米ķ， 然后秩⁡(一种−λķ一世)=n−米ķ.
(4) 任何 Hermitian 矩阵一种是可对角化的，即在−1一种在=Σ.
(5) Hermitian矩阵的所有特征向量都是线性独立的，并且相互正交，即特征矩阵在=[在1,在2,…,在n]是酉矩阵并且满足在−1=在H. (6) 根据性质 (5)，它认为在H一种在=Σ=诊断⁡(λ1,λ2,…,λn)或者一种=在Σ在H，可以改写为：一种=∑一世=1nλ一世在一世在一世H. 这称为 Hermitian 矩阵的谱分解。
(7) Hermitian 矩阵的逆矩阵的展开公式一种是
$$\boldsymbol{A}^{-1}=\sum_{i=1}^{n} \frac{\mathrm{I}}{\lambda_{i}} \boldsymbol{u} {i} \ boldsymbol{u} {i}^{H}$$
因此，如果知道 Hermitian 矩阵的特征分解一种，则可以直接得到逆矩阵一种−1使用上面的公式。
(8) 两人份n×n厄米矩阵一种和乙，存在一个酉矩阵，使得磷H一种磷和磷H乙磷都是对角线当且仅当一种乙=乙一种.
(9) 两人份n×n非负定 Hermitian 矩阵一种和乙, 存在一个非奇异矩阵磷以便磷H一种磷和磷H乙磷都是对角线。

## 机器学习代写|主成分分析作业代写PCA代考|Generalized Eigenvalue Decomposition

(1)这⁡(一种−λ乙)=0.
(2) 在∈空值⁡(一种−λ乙)， 和在≠0.

## 机器学习代写|主成分分析作业代写PCA代考|Rayleigh Quotient

r(在)=r(在,C)=在HC在在H在,

(1) 同质性：r(一种在,b在)=br(在,C)∀一种,b≠0.
(2)平移不变性：r(在,C−一种一世)=r(在,C)−一种.
(3) 有界性：自在范围在所有非零向量上，r(在)填充复平面中的一个区域，该区域称为值域C. 这个区域是封闭的、有界的和凸的。如果C=C∗（自伴随矩阵），值域是由极值特征值界定的实区间。
(4) 正交性：在⊥(C−r(在)一世)在.
(5) 最小残差：∀在≠0∧∀标量μ,|(C−r(在)一世)在|≤|(C−μ一世)在|.

Courant-Fischer 定理：让C∈Cn×n是 Hermitian 矩阵，其特征值为λ1≥λ2≥⋯≤λn, 那么它认为对于λķ(1≤ķ≤在) :
λķ=分钟小号,暗淡⁡(小号)=n−ķ+1最大限度在∈小号,在≠0(在HC在在H在)
Courant-Fischer 定理也可以写成
λķ=分钟小号,暗淡⁡(小号)=ķ最大限度在∈小号,在≠0(在HC在在H在)

## 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。

## 机器学习代写|主成分分析作业代写PCA代考| Properties of SVD

statistics-lab™ 为您的留学生涯保驾护航 在代写主成分分析PCA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。我们的专家在代写主成分分析PCA代写方面经验极为丰富，各种代写主成分分析PCA相关的作业也就用不着说。

• 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

## 机器学习代写|主成分分析作业代写PCA代考|Properties of SVD

Assume $\boldsymbol{A} \in \Re^{m \times n}, \quad \boldsymbol{B} \in \Re^{m \times n}$, and $r_{A}=\operatorname{rank}(A), \quad p=\min {m, n}$. The singular values of matrix $A$ can be arranged as follows: $\sigma_{\max }=\sigma_{1} \geq \sigma_{2} \geq \cdots$ $\geq \sigma_{p-1} \geq \sigma_{p}=\sigma_{\min } \geq 0$, and denote by $\sigma_{i}(\boldsymbol{B})$ the $i$ th largest singular value of matrix B. A few properties of SVD can summarized as follows [6]:
(1) The relationship between the singular values of a matrix and the ones of its submatrix.

Theorem $2.3$ (interlacing theorem for singular values). Assume $A \in \Re^{m \times n}$, and its singular values satisfy $\sigma_{1} \geq \sigma_{2} \geq \cdots \geq \sigma_{r}$, where $r=\min {m, n} .$ If $\boldsymbol{B} \in \mathbb{X}^{p \times q}$ is a submatrix of $\boldsymbol{A}$, and its singular values satisfy $\gamma_{1} \geq \gamma_{2} \geq \cdots \geq \gamma_{\min {p, q}}$, then it holds that
$$\sigma_{i} \geq \gamma_{i}, \quad i=1,2, \ldots, \min {p, q}$$
and
$$\gamma_{i} \geq \sigma_{i+(m-p)+(n-q)}, \quad i \leq \min {p+q-m, p+q-n} .$$
From Theorem 2.3, it holds that: If $\boldsymbol{B} \in \Re^{m \times(n-1)}$ is a submatrix of $\mathbf{A} \in \Re^{m \times n}$ by deleting any column of matrix $\boldsymbol{A}$, and their singular values are arranged in non-decreasing order, then it holds that
$$\sigma_{1}(\boldsymbol{A}) \geq \sigma_{1}(\boldsymbol{B}) \geq \sigma_{2}(\boldsymbol{A}) \geq \sigma_{2}(\boldsymbol{B}) \geq \cdots \geq \sigma_{h}(\boldsymbol{A}) \geq \sigma_{h}(\boldsymbol{B}) \geq 0$$
where $h=\min {m, n-1}$.
If $\boldsymbol{B} \in \Re^{\Re^{(m-1) \times n}}$ is a submatrix of $\boldsymbol{A} \in \Re^{m \times n}$ by deleting any row of matrix $\boldsymbol{A}$, and their singular values are arranged as non-decreasing order, then it holds that
$$\sigma_{1}(\boldsymbol{A}) \geq \sigma_{1}(\boldsymbol{B}) \geq \sigma_{2}(\boldsymbol{A}) \geq \sigma_{2}(\boldsymbol{B}) \geq \cdots \sigma_{h}(\boldsymbol{A}) \geq \sigma_{h}(\boldsymbol{B}) \geq 0$$
(2) The relationship between the singular values of a matrix and its norms. The spectral norm of a matrix $\boldsymbol{A}$ is equal to its largest singular value, namely,

According to the SVD theorem of matrix and the unitary invariability property of Frobenius norm $|\boldsymbol{A}|_{F}$ of matrix $\boldsymbol{A}$, namely $\left|\boldsymbol{U}^{H} \boldsymbol{A} \boldsymbol{V}\right|_{F}=|\boldsymbol{A}|_{F}$, it holds that
$$|\boldsymbol{A}|_{F}=\left[\sum_{i=1}^{m} \sum_{j=1}^{n}\left|a_{i j}\right|^{2}\right]^{1 / 2}=\left|\boldsymbol{U}^{H} \boldsymbol{A} \boldsymbol{V}\right|_{F}=|\Sigma|_{F}=\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}+\cdots+\sigma_{F}^{2}}$$
That is to say, the Frobenius norm of any matrix is equal to the square root of the sum of the squares of all nonzero singular values of this matrix. Consider the rank- $k$ approximation of matrix $A$ and denote it as $\boldsymbol{A}{k}$, in which $k{k}$ is defined as follows:
$$A_{k}=\sum_{i=1}^{k} \sigma_{i} \boldsymbol{u}{i} v{i}^{H}, k<r$$

## 机器学习代写|主成分分析作业代写PCA代考|Eigenvalue Problem and Eigen Equation

The basic problem of the eigenvalue can be stated as follows. Given an $n \times n$ matrix $\boldsymbol{A}$, determine a scalar $\lambda$ such that the following algebra equation
$$A u=\lambda u, \quad u \neq 0$$
has an $n \times 1$ nonzero solution. The scalar $\lambda$ is called as an eigenvalue of matrix $A$, and the vector $\boldsymbol{u}$ is called as the eigenvector associated with $\lambda$. Since the eigenvalue
$\lambda$ and eigenvector $\boldsymbol{u}$ appear in couples, $(\lambda, \boldsymbol{u})$ is usually called as an eigen pair of matrix $\boldsymbol{A}$. Although the eigenvalues can be zeros, the eigenvectors cannot be zero. In order to determine a nonzero vector $\boldsymbol{u}, \mathrm{Eq} .$ (2.17) can be modified as
$$(A-\lambda I) u=0$$
The above equation should come into existence for any vector $\boldsymbol{u}$, so the unique condition under which Eq. $(2.18)$ has a nonzero solution $\boldsymbol{u}=0$ is that the determinant of matrix $\boldsymbol{A}-\lambda \boldsymbol{I}$ is equal to zero, namely
$$\operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{I})=0 .$$
Thus, the solution of the eigenvalue problem consists of the following two steps:
(1) Solve all scalar $\lambda$ (eigenvalues) which make the matrix $\boldsymbol{A}-\lambda \boldsymbol{I}$ singular.
(2) Given an eigenvalue $\lambda$ which makes $\boldsymbol{A}=\lambda \boldsymbol{I}$ singular, and to solve all nonzero vectors which meets $(\boldsymbol{A}-\lambda \boldsymbol{I}) \boldsymbol{x}=\boldsymbol{0}$, i.e., the eigenvectors corresponding to $\lambda$.
According to the relationship between the singular values of a matrix and its determinant, a matrix is singular if and only if $\operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{I})=\boldsymbol{0}{,}$, namely $$(\boldsymbol{A}-\lambda \boldsymbol{I}) x \text { singular } \Leftrightarrow \operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{I})=\boldsymbol{0}$$ The matrix $(\boldsymbol{A}-\lambda \boldsymbol{I})$ is called as the eigen matrix of $\boldsymbol{A}$. When $\boldsymbol{A}$ is an $n \times n$ matrix, spreading the left side determinant of Eq. (2.20) can obtain a polynomial equation (power- $n$ ), namely $$\alpha{0}+\alpha_{1} \lambda+\cdots+\alpha_{n-1} \lambda^{n-1}+(-1)^{n} \lambda^{n}=0,$$
which is called as the eigen equation of matrix $\boldsymbol{A}$. The polynomial $\operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{I})$ is called as the eigen polynomial.

## 机器学习代写|主成分分析作业代写PCA代考|Eigenvalue and Eigenvector

In the following, we list some major properties about the eigenvalues and eigenvector of a matrix $A$.
Several important terms about the eigenvalues and eigenvectors [6]:
(1) The eigenvalue $\lambda$ of a matrix $A$ is called as having algebraic multiplicity $\mu$, if $\lambda$ is a $\mu$-repeated root of the eigen equation $\operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{I})=\boldsymbol{0}$.
(2) If the algebraic multiplicity of eigenvalue $\lambda$ is equal to one, the eigenvalue is called as single eigenvalue. Non-single eigenvalues are called as multiple eigenvalues.
(3) The eigenvalue $\lambda$ of a matrix $\boldsymbol{A}$ is called as having geometric multiplicity $\gamma$, if the number of linear independent eigenvectors associated with $\lambda$ is equal to $\gamma$.
(4) An eigenvalue is called half-single eigenvalue if its algebraic multiplicity is equal to geometric multiplicity. Not half-single eigenvalues are called as wane eigenvalues.
(5) If matrix $\boldsymbol{A}{n \times n}$ is a general complex matrix and $\lambda$ is its eigenvalue, the vector $v$ which meets $A v=\lambda v$ is called as the right eigenvector associated with the eigenvalue $\lambda$, and the eigenvector $\boldsymbol{u}$ which meets $\boldsymbol{u}^{H} \boldsymbol{A}=\lambda \boldsymbol{u}^{H}$ is called as the left eigenvector associated with the eigenvalue $\lambda$. If $A$ is Hermitian matrix and all its eigenvalues are real number, then it holds that $\boldsymbol{v}=\boldsymbol{u}$, that is to say, the left and right eigenvectors of a Hermitian matrix are the same. Some important properties can be summarized as follows: (1) Matrix $A\left(\in \Re^{n \times n}\right)$ has $n$ eigenvalues, of which the multiple eigenvalues are computed according to their multiplicity. (2) If $\boldsymbol{A}$ is a real symmetrical matrix or Hermitian matrix, all its eigenvalues are real numbers. (3) If $\boldsymbol{A}=\operatorname{diag}\left(a{11}, a_{22}, \ldots, a_{\mathrm{nn}}\right)$, its eigenvalues are $a_{11}, a_{22}, \ldots, a_{\mathrm{nn}}$; If $\boldsymbol{A}$ is a trigonal matrix, its diagonal elements are all its eigenvalues.
(4) For $\boldsymbol{A}\left(\in \Re^{n \times n}\right)$, if $\lambda$ is the eigenvalue of matrix $\boldsymbol{A}, \lambda$ is also the eigenvalue of matrix $A^{\mathrm{T}}$. If $\lambda$ is the eigenvalue of matrix $A, \lambda^{*}$ is the eigenvalue of matrix $A^{H}$. If $\lambda$ is the eigenvalue of matrix $A, \lambda+\sigma^{2}$ is the eigenvalue of matrix $\boldsymbol{A}+\sigma^{2} \boldsymbol{I}$. If $\lambda$ is the eigenvalue of matrix $\boldsymbol{A}, 1 / \lambda$ is the eigenvalue of matrix $A^{-1}$.
(5) All eigenvalues of matrix $A^{2}=A$ are either 0 or 1 .
(6) If $A$ is a real orthogonal matrix, all its eigenvalues are on the unit circle.
(7) If a matrix is singular, at least one of its eigenvalues is equal to zero.
(8) The sum of all the eigenvalues is equal to its trace, namely $\sum_{i=1}^{n} \lambda_{i}=\operatorname{tr}(\boldsymbol{A})$.
(9) The nonzero eigenvectors $\boldsymbol{u}{1}, \boldsymbol{u}{2}, \ldots, \boldsymbol{u}{n}$ associated with different eigenvalues $\lambda{1}, \lambda_{2}, \ldots \lambda_{n}$ are linearly independent.
(10) If matrix $A\left(\in \mathcal{H}^{\mathrm{n} \times n}\right)$ has $r$ nonzero eigenvalues, then it holds that $\operatorname{rank}(\boldsymbol{A}) \geq r$; If zero is a non-multiple eigenvalue, then $\operatorname{rank}(\boldsymbol{A}) \geq n-1$; If $\operatorname{rank}(\boldsymbol{A}-\lambda \boldsymbol{I}) \geq n-1$, then $\lambda$ is an eigenvalue of matrix $\boldsymbol{A}$.
(11) The product of all eigenvalues of matrix $A$ is equal to the determinant of matrix $\boldsymbol{A}$, namely $\prod_{i=1}^{n} \lambda_{i}=\operatorname{det}(\boldsymbol{A})=|\boldsymbol{A}|$.
(12) A Hermitian matrix $\boldsymbol{A}$ is positive definite (or positive semi-definite), if and only if all its eigenvalues are positive (or non-negative).

## 机器学习代写|主成分分析作业代写PCA代考|Properties of SVD

(1)矩阵的奇异值与其子矩阵的奇异值之间的关系。

σ一世≥C一世,一世=1,2,…,分钟p,q

C一世≥σ一世+(米−p)+(n−q),一世≤分钟p+q−米,p+q−n.

σ1(一种)≥σ1(乙)≥σ2(一种)≥σ2(乙)≥⋯≥σH(一种)≥σH(乙)≥0

σ1(一种)≥σ1(乙)≥σ2(一种)≥σ2(乙)≥⋯σH(一种)≥σH(乙)≥0
(2) 矩阵的奇异值与其范数之间的关系。矩阵的谱范数一种等于它的最大奇异值，即

|一种|F=[∑一世=1米∑j=1n|一种一世j|2]1/2=|在H一种在|F=|Σ|F=σ12+σ22+⋯+σF2

## 机器学习代写|主成分分析作业代写PCA代考|Eigenvalue Problem and Eigen Equation

λ和特征向量在出现在情侣中，(λ,在)通常称为矩阵的特征对一种. 虽然特征值可以为零，但特征向量不能为零。为了确定一个非零向量在,和q.(2.17) 可以修改为
(一种−λ一世)在=0

(1) 求解所有标量λ（特征值）构成矩阵一种−λ一世单数。
(2) 给定一个特征值λ这使得一种=λ一世奇异的，并求解所有满足的非零向量(一种−λ一世)X=0，即对应的特征向量λ.

## 机器学习代写|主成分分析作业代写PCA代考|Eigenvalue and Eigenvector

1）特征值λ矩阵的一种被称为具有代数多重性μ， 如果λ是一个μ-特征方程的重复根这⁡(一种−λ一世)=0.
(2) 若特征值的代数重数λ等于一，则该特征值称为单一特征值。非单一特征值称为多重特征值。
(3) 特征值λ矩阵的一种被称为具有几何多重性C，如果与相关的线性独立特征向量的数量λ等于C.
(4) 如果一个特征值的代数重数等于几何重数，则称其为半单特征值。非半单特征值称为衰减特征值。
(5) 如果矩阵一种n×n是一个一般的复矩阵，并且λ是它的特征值，向量在满足一种在=λ在被称为与特征值相关的右特征向量λ, 和特征向量在满足在H一种=λ在H被称为与特征值相关的左特征向量λ. 如果一种是 Hermitian 矩阵并且它的所有特征值都是实数，那么它认为在=在，也就是说，一个厄密矩阵的左右特征向量是相同的。一些重要的性质可以概括如下： (1) 矩阵一种(∈ℜn×n)拥有n特征值，其中的多个特征值是根据它们的多重性计算的。(2) 如果一种是一个实对称矩阵或 Hermitian 矩阵，它的所有特征值都是实数。(3) 如果一种=诊断⁡(一种11,一种22,…,一种nn)，其特征值为一种11,一种22,…,一种nn; 如果一种是一个三角矩阵，它的对角元素都是它的特征值。
(4) 为一种(∈ℜn×n)， 如果λ是矩阵的特征值一种,λ也是矩阵的特征值一种吨. 如果λ是矩阵的特征值一种,λ∗是矩阵的特征值一种H. 如果λ是矩阵的特征值一种,λ+σ2是矩阵的特征值一种+σ2一世. 如果λ是矩阵的特征值一种,1/λ是矩阵的特征值一种−1.
(5) 矩阵的所有特征值一种2=一种是 0 或 1 。
(6) 如果一种是一个实正交矩阵，它的所有特征值都在单位圆上。
(7) 如果一个矩阵是奇异的，至少它的一个特征值等于 0。
(8) 所有特征值之和等于它的迹，即∑一世=1nλ一世=tr⁡(一种).
(9) 非零特征向量在1,在2,…,在n与不同的特征值相关联λ1,λ2,…λn是线性独立的。
(10) 如果矩阵一种(∈Hn×n)拥有r非零特征值，那么它认为秩⁡(一种)≥r; 如果零是非多重特征值，则秩⁡(一种)≥n−1; 如果秩⁡(一种−λ一世)≥n−1， 然后λ是矩阵的特征值一种.
(11) 矩阵所有特征值的乘积一种等于矩阵的行列式一种，即∏一世=1nλ一世=这⁡(一种)=|一种|.
(12) Hermitian 矩阵一种是正定的（或半正定的），当且仅当它的所有特征值都是正的（或非负的）。

## 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。