## 统计代写|主成分分析代写Principal Component Analysis代考|Incomplete PCA by Alternating Minimization

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|Incomplete PCA by Alternating Minimization

Although the convex-optimization-based approach can ensure correctness of the low-rank solution for the matrix completion problem, it requires solving a convex program of the same size as the matrix. When the data matrix $X$ is very large, parameterizing the low-rank solution $A$ and Lagrange multipliers $Z$ with two matrices of the same size as $X$ seems rather demanding, actually redundant. At least the low-rank solution $A$ could be parameterized more economically with its lowrank factors. Hence, if scalability of the algorithm is a serious concern, it makes sense to look for the low-rank factors of the solution matrix directly.

To this end, we introduce in this section an alternating minimization algorithm for solving the geometric PCA problem with missing data. The main idea behind this approach, which was probably first proposed in (Wiberg 1976), is to find $\mu, U$, and $Y$ that minimize the error $\left|X-\boldsymbol{\mu} \mathbf{1}^{\top}-U Y\right|_F^2$ considering only the known entries of $X$ in the set $\Omega=\left{(i, j): w_{i j}=1\right}$, i.e.,
\begin{aligned} \left|\mathcal{P}{\Omega}\left(X-\boldsymbol{\mu} \mathbf{1}^{\top}-U Y\right)\right|_F^2 & =\left|W \odot\left(X-\boldsymbol{\mu} \mathbf{1}^{\top}-U Y\right)\right|_F^2 \ & =\sum{i=1}^D \sum_{j=1}^N w_{i j}\left(x_{i j}-\mu_i-\boldsymbol{u}i^{\top} \boldsymbol{y}_j\right)^2, \end{aligned} where $x{i j}$ is the $(i, j)$ th entry of $X, \mu_i$ is the $i$ th entry of $\mu, \boldsymbol{u}i^{\top}$ is the $i$ th row of $U$, and $\boldsymbol{y}_j$ is the $j$ th column of $Y$. Notice that this cost function is the same as that in (3.10), except that the errors $\varepsilon{i j}=x_{i j}-\boldsymbol{u}i^{\top} \boldsymbol{y}_j$ associated with the missing entries $\left(w{i j}=0\right.$ ) are removed.In what follows, we will derive an alternating minimization algorithm for minimizing the cost function in (3.34). For the sake of simplicity, we will first derive the algorithm in the case of zero-mean and complete data. In this case, the problem in (3.34) reduces to a low-rank matrix approximation problem, which can be solved using the SVD, as described in Theorem 2.3. The alternating minimization algorithm to be derived provides an alternative to the SVD solution, which, however, can be more easily extended to the case of incomplete data, as we will see. Moreover, the algorithm can also be extended to the more challenging PCA problem with missing entries, as we will see.

## 统计代写|主成分分析代写Principal Component Analysis代考|PCA with Robustness to Corrupted Entries

In the previous section, we considered the PCA problem in the case that some entries of the data points are missing. In this section, we consider the PCA problem in the case that some of the entries of the data points have been corrupted by gross errors, known as intrasample outliers. The additional challenge is that we do not know which entries have been corrupted. Thus, the problem is to simultaneously detect which entries have been corrupted and replace them by their uncorrupted values. In some literature, this problem is referred to as the robust PCA problem (De la Torre and Black 2004; Candès et al. 2011).

Let us first recall the PCA problem (see Section 2.1.2) in which we are given $N$ data points $\mathcal{X}=\left{x_j \in \mathbb{R}^D\right}_{j=1}^N$ drawn (approximately) from a $d$-dimensional affine subspace $S={\boldsymbol{x}=\boldsymbol{\mu}+U \boldsymbol{y}}$, where $\boldsymbol{\mu} \in \mathbb{R}^D$ is an arbitrary point in $S, U \in \mathbb{R}^{D \times d}$ is a basis for $S$, and $\left{y_j \in \mathbb{R}^d\right}_{j=1}^N$ are the principal components. In the robust PCA problem, we assume that the $i$ th entry $x_{i j}$ of a data point $\boldsymbol{x}j$ is obtained by corrupting the $i$ th entry $\ell{i j}$ of a point $\ell_j$ lying perfectly on the subspace $S$ by an error $e_{i j}$, i.e.,
$$x_{i j}=\ell_{i j}+e_{i j}, \quad \text { or } \quad \boldsymbol{x}_j=\boldsymbol{\ell}_j+\boldsymbol{e}_j, \quad \text { or } \quad X=L+E \text {, }$$ where $X, L, E \in \mathbb{R}^{D \times N}$ are matrices with entries $x_{i j}, \ell_{i j}$, and $e_{i j}$, respectively. Such errors can have a huge impact on the estimation of the subspace. Thus it is very important to be able to detect the locations of those errors,
$$\Omega=\left{(i, j): e_{i j} \neq 0\right}$$
as well as correct the erroneous entries before applying PCA to the given data. As discussed before, a key difference between the robust PCA problem and the incomplete PCA problem is that we do not know the location of the corrupted entries. This makes the robust PCA problem harder, since we need to simultaneously detect and correct the errors. Nonetheless, when the number of corrupted entries is a small enough fraction of the total number of entries, i.e., when $|\Omega|<\rho \cdot D N$ for some $\rho<1$, we may still hope to be able to detect and correct such errors. In the remainder of this section, we describe methods from robust statistics and convex optimization for addressing this problem.

# 主成分分析代考

## 统计代写|主成分分析代写Principal Component Analysis代考|Incomplete PCA by Alternating Minimization

$$\left|\mathcal{P} \Omega\left(X-\boldsymbol{\mu} \mathbf{1}^{\top}-U Y\right)\right|F^2=\left|W \odot\left(X-\boldsymbol{\mu} \mathbf{1}^{\top}-U Y\right)\right|_F^2 \quad=\sum i=1^D \sum{j=1}^N w_{i j}\left(x_{i j}-\mu_i\right.$$

## 统计代写|主成分分析代写Principal Component Analysis代考|PCA with Robustness to Corrupted Entries

Imathcal ${X}=\backslash l e f t\left{x _j \backslash\right.$ in $\left.\backslash m a t h b b{R}^{\wedge} D \backslash r i g h t\right} _{j=1}^{\wedge} N$ (大约) 从 $d$-维仿射子空间 $S=\boldsymbol{x}=\boldsymbol{\mu}+U \boldsymbol{y}$ ，在 分。在稳健的 PCA 问题中，我们假设 $i$ 第一个条目 $x_{i j}$ 一个数据点 $x j$ 是通过破坏获得的 $i$ 第一个条目 $\ell i j$ 的 一点 $\ell_j$ 完美地躺在子空间上 $S$ 由于错误 $e_{i j}$ ，那是，
$$x_{i j}=\ell_{i j}+e_{i j}, \quad \text { or } \quad \boldsymbol{x}j=\boldsymbol{\ell}_j+\boldsymbol{e}_j, \quad \text { or } \quad X=L+E,$$ 在哪里 $X, L, E \in \mathbb{R}^{D \times N}$ 是有条目的矩阵 $x{i j}, \ell_{i j}$ ，和 $e_{i j}$ ，分别。这种错误会对子空间的估计产生巨大 影响。因此，能够检测到这些错误的位置非常重要，
\Omega $=\backslash$ left ${(i, j):$ e_ ${i j} \backslash$ lneq o\right } }

## 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|Incomplete PPCA by Expectation Maximization

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|Incomplete PPCA by Expectation Maximization

In this section, we derive an EM algorithm (see Appendix B.2.1) for solving the PPCA problem with missing data. Recall from Section 2.2 that in the PPCA model, each data point is drawn as $x \sim \mathcal{N}\left(\mu_x, \Sigma_x\right)$, where $\mu_x=\mu$ and $\Sigma_x=B B^{\top}+\sigma^2 I_D$, where $\mu \in \mathbb{R}^D, B \in \mathbb{R}^{D \times d}$, and $\sigma>0$. Recall also from (2.56) that the log-likelihood of the PPCA model is given by
$$\mathscr{L}=-\frac{N D}{2} \log (2 \pi)-\frac{N}{2} \log \operatorname{det}\left(\Sigma_x\right)-\frac{1}{2} \sum_{j=1}^N \operatorname{trace}\left(\Sigma_x^{-1}\left(x_j-\mu\right)\left(x_j-\mu\right)^{\top}\right)$$
where $\left{x_j\right}_{j=1}^N$ are $N$ i.i.d. samples of $\boldsymbol{x}$. Since the samples are incomplete, we can partition each point $x$ and the parameters $\mu_x$ and $\Sigma_x$ as
$$\left[\begin{array}{l} x_U \ x_O \end{array}\right]=P x, \quad\left[\begin{array}{l} \mu_U \ \mu_O \end{array}\right]=P \mu, \text { and }\left[\begin{array}{cc} \Sigma_{U U} & \Sigma_{U O} \ \Sigma_{O U} & \Sigma_{O O} \end{array}\right]=P \Sigma_x P^{\top} .$$
Here $\boldsymbol{x}O$ is the observed part of $\boldsymbol{x}, \boldsymbol{x}_U$ is the unobserved part of $\boldsymbol{x}$, and $P$ is any permutation matrix that reorders the entries of $x$ so that the unobserved entries appear first. Notice that $P$ is not unique, but we can use any such $P$. Notice also that the above partition of $x, \mu_x$, and $\Sigma_x$ could be different for each data point, because the missing entries could be different for different data points. When strictly necessary, we will use $\boldsymbol{x}{j U}$ and $\boldsymbol{x}_{j O}$ to denote the unobserved and observed parts of point $\boldsymbol{x}_j$, respectively, and $P_j$ to denote the permutation matrix. Otherwise, we will avoid using the index $j$ in referring to a generic point.

## 统计代写|主成分分析代写Principal Component Analysis代考|Matrix Completion by Convex Optimization

The EM-based approaches to incomplete PPCA discussed in the previous section rely on (a) explicit parameterizations of the low-rank factors and (b) minimization of a nonconvex cost function in an alternating minimization fashion. Specifically, such approaches alternate between completing the missing entries given the parameters of a PPCA model for the data and estimating the parameters of the model from complete data. While simple and intuitive, such approaches suffer from two important disadvantages. First, the desired rank of the matrix needs to be known in advance. Second, due to the greedy nature of the EM algorithm, it is difficult to ensure convergence to the globally optimal solution. Therefore, a good initialization of the EM-based algorithm is critical for converging to a good solution.

In this section, we introduce an alternative approach that solves the low-rank matrix completion problem via a convex relaxation. As we will see, this approach allows us to complete a low-rank matrix by minimizing a convex objective function, which is guaranteed to have a globally optimal minimizer. Moreover, under rather benign conditions on the missing entries, the global minimizer is guaranteed to be the correct low-rank matrix, even without knowing the rank of the matrix in advance.
A rigorous justification for the correctness of the convex relaxation approach requires a deep knowledge of high-dimensional statistics and geometry that is beyond the scope of this book. However, this does not prevent us from introducing and summarizing here the main ideas and results, as well as the basic algorithms offered by this approach. Practitioners can apply the useful algorithm to their data and problems, whereas researchers who are more interested in the advanced theory behind the algorithm may find further details in (Cai et al. 2008; Candès and Recht 2009; Candès and Tao 2010; Gross 2011; Keshavan et al. 2010a; Zhou et al. 2010a).
Compressive Sensing of Low-Rank Matrices
The matrix completion problem can be considered a special case of the more general class of problems of recovering a high-dimensional low-rank matrix $X$ from highly compressive linear measurements $B=\mathcal{P}(X)$, where $\mathcal{P}$ is a linear operator that returns a set of linear measurements $B$ of the matrix $X$. It is known from highdimensional statistics that if the linear operator $\mathcal{P}$ satisfies certain conditions, then the rank minimization problem
$$\min _A \operatorname{rank}(A) \quad \text { s.t. } \quad \mathcal{P}(A)=B$$
is well defined, and its solution is unique (Candès and Recht 2009). However, it is also known that under general conditions, the task of finding such a minimal-rank solution is in general an NP-hard problem.

# 主成分分析代考

## 统计代写|主成分分析代写Principal Component Analysis代考|Incomplete PPCA by Expectation Maximization

$$\mathscr{L}=-\frac{N D}{2} \log (2 \pi)-\frac{N}{2} \log \operatorname{det}\left(\Sigma_x\right)-\frac{1}{2} \sum_{j=1}^N \operatorname{trace}\left(\Sigma_x^{-1}\left(x_j-\mu\right)\left(x_j-\mu\right)^{\top}\right)$$

## 统计代写|主成分分析代写Principal Component Analysis代考|Matrix Completion by Convex Optimization

$$\min _A \operatorname{rank}(A) \quad \text { s.t. } \quad \mathcal{P}(A)=B$$

## 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。

## MTH9893 Principal component analysis课程简介

This course covers univariate and multivariate time series analysis, conditional heteroscedastic models, principal component analysis, and factor models. Students will learn about implementing univariate and multivariate volatility models. Note: Students cannot receive credit for both MTH 9867 and MTH 9893.

## PREREQUISITES

This course covers univariate and multivariate time series analysis, conditional heteroscedastic models, principal component analysis, and factor models. Students will learn about implementing univariate and multivariate volatility models. Note: Students cannot receive credit for both MTH 9867 and MTH 9893.

## MTH9893 Principal component analysis HELP（EXAM HELP， ONLINE TUTOR）

Exercise 5.1 (Clustering Points in a Plane). Describe how Algorithm 5.1 can also be applied to a set of points in the plane $\left{x_j \in \mathbb{R}^2\right}_{j=1}^N$ that are distributed around a collection of cluster centers $\left{\boldsymbol{\mu}i \in \mathbb{R}^2\right}{i=1}^n$ by interpreting the data points as complex numbers: ${z \doteq x+y \sqrt{-1} \in \mathbb{C}}$. In particular, discuss what happens to the coefficients and roots of the fitting polynomial $p_n(z)$.

Exercise 5.3 (Level Sets and Normal Vectors). Let $f(x): \mathbb{R}^D \rightarrow \mathbb{R}$ be a smooth function. For a constant $c \in \mathbb{R}$, the set $S_c \doteq\left{x \in \mathbb{R}^D \mid f(x)=c\right}$ is called a level set of the function $f ; S_c$ is in general a $(D-1)$-dimensional submanifold. Show that if $|\nabla f(x)|$ is nonzero at a point $x_0 \in S_c$, then the gradient $\nabla f\left(x_0\right) \in \mathbb{R}^D$ at $x_0$ is orthogonal to all tangent vectors of the level set $S_c$.

Exercise 5.7 (Two Subspaces in General Position). Consider two linear subspaces of dimension $d_1$ and $d_2$ respectively in $\mathbb{R}^D$. We say that they are in general position if an arbitrarily small perturbation of the position of the subspaces does not change the dimension of their intersection. Show that two subspaces are in general position if and only if
$$\operatorname{dim}\left(S_1 \cap S_2\right)=\min \left{d_1+d_2-D ; 0\right} .$$

Exercise 5.8. Implement the basic algebraic subspace clustering algorithm, Algorithm 5.4 , and test the algorithm for different subspace arrangements with different levels of noise.

Exercise 5.12 (Robust Estimation of Fitting Polynomials). We know that samples from an arrangement of $n$ subspaces, their Veronese lifting, all lie on a single subspace $\operatorname{span}\left(V_n(D)\right)$. The coefficients of the fitting polynomials are simply the null space of $\boldsymbol{V}_n(D)$. If there is noise, the lifted samples approximately span a subspace, and the coefficients of the fitting polynomials are eigenvectors associated with the small eigenvalues of $\boldsymbol{V}_n(D)^{\top} \boldsymbol{V}_n(D)$. However, if there are outliers, the lifted samples together no longer span a subspace. Notice that this is the same situation that robust statistical techniques such as multivariate trimming (MVT) are designed to deal with. See Appendix B.5 for more details. In this exercise, show how to combine MVT with ASC so that the resulting algorithm will be robust to outliers. Implement your scheme and find out the highest percentage of outliers that the algorithm can handle (for various subspace arrangements).

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

Statistics-lab™可以为您提供cuny.edu MTH9893 Principal component analysis主成分分析课程的代写代考辅导服务！ 请认准Statistics-lab™. Statistics-lab™为您的留学生涯保驾护航。

## 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。