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

2022年7月20日

couryes-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
couryes™为您提供可以保分的包课服务

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

Recall from Section $2.1 .2$ that in the PCA problem, we are given $N$ data points $\mathcal{X} \doteq\left{x_{j} \in \mathbb{R}^{D}\right}_{j=1}^{N}$ drawn (approximately) from a $d$-dimensional affine subspace $S \doteq{x=\mu+U y}$, where $\mu \in \mathbb{R}^{D}$ is an arbitrary point in $S, U \in \mathbb{R}^{D \times d}$ is a basis for $S$, and $\mathcal{Y}=\left{y_{j} \in \mathbb{R}^{d}\right}_{j=1}^{N}$ are the principal components.

In this section, we consider the PCA problem in the case that some of the given data points are incomplete. A data point $\boldsymbol{x}=\left[x_{1}, x_{2}, \ldots, x_{D}\right]^{\top}$ is said to be incomplete when some of its entries are missing or unspecified. For instance, if the $i$ th entry $x_{i}$, of $x$ is missing, then $x$ is known only up to a line in $\mathbb{R}^{D}$, i.e.,
\begin{aligned} \boldsymbol{x} \in L & \doteq\left{\left[x_{1}, \ldots, x_{i-1}, x_{i}, x_{i+1}, \ldots, x_{D}\right]^{\top}, x_{i} \in \mathbb{R}\right} \ &=\left{x_{-i}+x_{i} e_{i}, x_{i} \in \mathbb{R}\right} \end{aligned}
where $\boldsymbol{x}{-i}=\left[x{1}, \ldots, x_{i-1}, 0, x_{i+1}, \ldots, x_{D}\right]^{\top} \in \mathbb{R}^{D}$ is the vector $\boldsymbol{x}$ with its ith entry zeroed out and $e_{i}=[0, \ldots, 0,1,0, \ldots, 0]^{\top} \in \mathbb{R}^{D}$ is the ith basis vector. More generally, if the point $x$ has $M$ missing entries, without loss of generality we can partition it as $\left[\begin{array}{l}\boldsymbol{x}{U} \ \boldsymbol{x}{O}\end{array}\right]$, where $\boldsymbol{x}{U} \in \mathbb{R}^{M}$ denotes the unobserved entries and $x{O} \in \mathbb{R}^{D-M}$ denotes the observed entries. Thus, $x$ is known only up to the following $M$-dimensional affine subspace:
$$x \in L \doteq\left{\left[\begin{array}{c} 0 \ x_{O} \end{array}\right]+\left[\begin{array}{c} I_{M} \ 0 \end{array}\right] x_{U}, x_{U} \in \mathbb{R}^{M}\right}$$
Incomplete PCA When the Subspace Is Known
Let us first consider the simplest case, in which the subspace $S$ is known. Then we know that the point $\boldsymbol{x}$ belongs to both $L$ and $S$. Therefore, given the parameters $\mu$ and $U$ of the subspace $S$, we can compute the principal components $y$ and the missing entries $\boldsymbol{x}_{U}$ by intersecting $L$ and $S$. In the case of one missing entry (illustrated in Figure 3.1), the intersection point can be computed from a $\boldsymbol{x}=\boldsymbol{x}{-i}+x{i} \boldsymbol{e}{i}=\boldsymbol{\mu}+U \boldsymbol{y} \Longrightarrow\left[U-\boldsymbol{e}{i}\right]\left[\begin{array}{l}\boldsymbol{y} \ x_{i}\end{array}\right]=\boldsymbol{x}_{-i}-\boldsymbol{\mu} .$

## 统计代写|主成分分析代写Principal Component Analysis代考|Incomplete PCA by Mean and Covariance Completion

Recall from Section 2.1.2 that the optimization problem associated with geometric PCA is
$$\min {\mu, U,\left{y{j}\right}} \sum_{j=1}^{N}\left|x_{j}-\mu-U y_{j}\right|^{2} \text { s.t. } U^{\top} U=I_{d} \text { and } \sum_{j=1}^{N} y_{j}=\mathbf{0} .$$
We already know that the solution to this problem can be obtained from the mean and covariance of the data points,
$$\hat{\mu}{N}=\frac{1}{N} \sum{j=1}^{N} x_{j} \quad \text { and } \quad \hat{\Sigma}{N}=\frac{1}{N} \sum{j=1}^{N}\left(x_{j}-\hat{\mu}{N}\right)\left(x{j}-\hat{\mu}{N}\right)^{\top}$$ respectively. Specifically, $\boldsymbol{\mu}$ is given by the sample mean $\hat{\mu}{N}, U$ is given by the top $d$ eigenvectors of the covariance matrix $\hat{\Sigma}{N}$, and $y{j}=U^{\top}\left(x_{j}-\mu\right)$. Alternatively, an optimal solution can be found from the rank- $d$ SVD of the mean-subtracted data matrix $\left[x_{1}-\hat{\mu}{N}, \ldots, x{N}-\hat{\mu}_{N}\right]$, as shown in Theorem $2.3$.

When some entries of each $\boldsymbol{x}{j}$ are missing, we cannot directly compute $\hat{\mu}{N}$ or $\hat{\Sigma}_{N}$ as in (3.11). A straightforward method for dealing with missing entries was introduced in (Jolliffe 2002). It basically proposes to compute the sample mean and covariance from the known entries of $X$. Specifically, the entries of the incomplete mean and covariance can be computed as
$$\hat{\mu}{i}=\frac{\sum{j=1}^{N} w_{i j} x_{i j}}{\sum_{j=1}^{N} w_{i j}} \text { and } \hat{\sigma}{i k}=\frac{\sum{j=1}^{N} w_{i j} w_{k j}\left(x_{i j}-\hat{\mu}{i}\right)\left(x{k j}-\hat{\mu}{k}\right)}{\sum{j=1}^{N} w_{i j} w_{k j}}$$
where $i, k=1, \ldots, D$. However, as discussed in (Jolliffe 2002), this simple approach has several disadvantages. First, the estimated covariance matrix need not be positive semidefinite. Second, these estimates are not obtained by optimizing any statistically or geometrically meaningful objective function (least squares, maximum likelihood, etc.) Nonetheless, estimates $\hat{\mu}{N}$ and $\hat{\Sigma}{N}$ obtained from the naive approach in (3.12) may be used to initialize the methods discussed in the next two sections, which are iterative in nature. For example, we may initialize the columns of $U$ as the eigenvectors of $\hat{\Sigma}{N}$ associated with its $d$ largest eigenvalues. Then given $\hat{\mu}{N}$ and $\hat{U}$, we can complete each missing entry as described in (3.6).

# 主成分分析代考

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

Imathcal ${\mathrm{X}} \backslash$ doteq left $\left{\mathrm{x}{-}{\mathrm{j}} \backslash\right.$ in $\backslash$ math bb $\left.{\mathrm{R}} \wedge \mathrm{D}\right} \backslash$ right $}{-}{\mathrm{j}=1} \wedge{\mathrm{N}}$ (大约) 从一个 $d$ 维仿射子 空间 $S \doteq x=\mu+U y$ ， 在哪里 $\mu \in \mathbb{R}^{D}$ 是一个任意点 $S, U \in \mathbb{R}^{D \times d}$ 是一个基础 $S$ ，和

$\boldsymbol{x}=\left[x_{1}, x_{2}, \ldots, x_{D}\right]^{\top}$ 当它的某些条目丢失或末指定时，称为不完整。例如，如果 $i$ 第 条目 $x_{i}$ ，的 $x$ 不见了，那么 $x$ 只知道最茤一行 $\mathbb{R}^{D}$ ，那是，

$S$. 因此，给定参数 $\mu$ 和 $U$ 子空间的 $S$ ，我们可以计算主成分 $y$ 和丢失的条目 $x_{U}$ 通过相交 $L$

$\boldsymbol{x}=\boldsymbol{x}-i+x i \boldsymbol{e i}=\boldsymbol{\mu}+U \boldsymbol{y} \Longrightarrow[U-\boldsymbol{e i}]\left[\boldsymbol{y} x_{i}\right]=\boldsymbol{x}_{-i}-\boldsymbol{\mu} .$

## 统计代写|主成分分析代写Principal Component Analysis代考|Incomplete PCA by Mean and Covariance Completion

$$\hat{\mu} N=\frac{1}{N} \sum j=1^{N} x_{j} \quad \text { and } \quad \hat{\Sigma} N=\frac{1}{N} \sum j=1^{N}\left(x_{j}-\hat{\mu} N\right)(x j-\hat{\mu} N)$$

$$\hat{\mu} i=\frac{\sum j=1^{N} w_{i j} x_{i j}}{\sum_{j=1}^{N} w_{i j}} \text { and } \hat{\sigma} i k=\frac{\sum j=1^{N} w_{i j} w_{k j}\left(x_{i j}-\hat{\mu} i\right)(x k j-\hat{\mu} k)}{\sum j=1^{N} w_{i j} w_{k j}}$$

## 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。