### 统计代写|主成分分析代写Principal Component Analysis代考|MATH5855

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• 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|Missing Not at Random

A number of statistical approaches have been developed to cope with nonignorable missing mechanisms and such approaches have been applied in different analytic frameworks. However, the approaches used in PCA have typically focused on assumptions of ignorability $[19,21,42]$. Recently, Geraci and Farcomeni [11] extended Tipping and Bishop’s [42] EM approach to the case in which the vector $\mathbf{y}$ is partially observed and the missing data mechanism is nonignorable. Specifically, they proposed an adaptation of Ibrahim et al.’s [17] methods for missing responses in random-effects models with non-monotone patterns of missing data.

Suppose that $\mathbf{y}{i}$ contains $s{i}, s_{i}<p$, missing values. The $i$ th contribution to the complete data density of $\left(\mathbf{y}{i}, \mathbf{u}{i}, \mathbf{m}{i}\right)$ is given by $$f\left(\mathbf{y}{i}, \mathbf{u}{i}, \mathbf{m}{i} \mid \boldsymbol{\theta}, \eta\right)=f\left(\mathbf{y}{i} \mid \mathbf{u}{i}, \boldsymbol{\theta}\right) f\left(\mathbf{u}{i}\right) f\left(\mathbf{m}{i} \mid \mathbf{y}{i}, \eta\right), \quad i=1, \ldots, n,$$ where the additional factor $f\left(\mathbf{m}{i} \mid \mathbf{y}{i}, \eta\right)$, indexed by the parameter $\eta$, is the MDM, which we assume to be independent from $\mathbf{u}{i}$. This assumption simplifies the subsequent steps of the estimation algorithm, although it can be relaxed at the cost of increased computational time (see $[17,18]$ for a discussion).

Estimation of $\boldsymbol{\theta}$ would in general require marginalizing the log-likelihood based on (10) over the unobserved data, which however leads to a rather intractable integral of dimension $s_{i}+q$. Instead, the EM algorithm can be applied. The E-step at the $(t+1)$ th iteration is defined as follows:
$$Q\left(\lambda \mid \lambda^{(t)}\right)=\mathrm{E}{\mathbf{z}, \mathbf{u} \mid \mathbf{x}, \mathbf{m}, \lambda^{(m}}{l(\lambda ; \mathbf{Y}, \mathbf{U}, \mathbf{M})}$$ with $\lambda=(\boldsymbol{\theta}, \eta) \quad$ and $\quad l(\lambda ; \mathbf{Y}, \mathbf{U}, \mathbf{M})=\sum{i}^{n} \log f\left(\mathbf{y}{i} \mid \mathbf{u}{i}, \boldsymbol{\theta}\right)+\log f\left(\mathbf{u}{i}\right)+$ $\log f\left(\mathbf{m}{i} \mid \mathbf{y}{i}, \eta\right)$, and where the expectation is taken with respect to the conditional distribution of $\mathbf{z}{i}$ and $\mathbf{u}_{i}$, given the observed data, evaluated at $\lambda^{(t)}$.

The E-step (11), however, does not yet offer a computational advantage since it is not easy to solve analytically. Therefore, Geraci and Farcomeni [11] applied a Monte Carlo E-step [17]. They considered an adaptive rejection Metropolis sampling (ARMS) algorithm [12] and specified the following MDM

$$f\left(\mathbf{m}{i} \mid \mathbf{y}{i}, \eta\right)=\prod_{j=1}^{p} \pi_{i j}^{m_{i j}}\left(1-\pi_{i j}\right)^{1-m_{i j}}$$
where $\pi_{i j}$ is the probability that $y_{i j}$ is missing, conditional on the response $\mathbf{y}_{i}$. The ARMS algorithm is convenient as, basically, no tuning is needed. In addition, it is run in parallel for each row of the data matrix, which therefore greatly speeds up the computation. Moreover, the PPCA Gaussian model provides scope for further reductions in computational time during the calculation of the E-step. All the technical details are given in appendix.

## 统计代写|主成分分析代写Principal Component Analysis代考|Appendix – EM Algorithm for PPCA with MNAR Values

In this appendix, we provide additional details on the Monte Carlo EM algorithm introduced in Sect. $3.2$ and we derive a simplified E-step where the random effects are integrated out from the complete data log-likelihood.

The Monte Carlo E-step requires sampling from $f\left(\mathbf{z}{i}, \mathbf{u}{i} \mid \mathbf{x}{i}, \mathbf{m}{i}, \lambda^{(t)}\right)$. This task can be carried out efficiently via ARMS [12] using the full conditionals
\begin{aligned} &f\left(\mathbf{z}{i} \mid \mathbf{x}{i}, \mathbf{u}{i}, \mathbf{m}{i}, \lambda^{(t)}\right) \propto f\left(\mathbf{y}{i} \mid \mathbf{u}{i}, \lambda^{(t)}\right) f\left(\mathbf{m}{i} \mid \mathbf{y}{i}, \lambda^{(t)}\right) \ &f\left(\mathbf{u}{i} \mid \mathbf{x}{i}, \mathbf{z}{i}, \mathbf{m}{i}, \lambda^{(t)}\right) \propto f\left(\mathbf{y}{i} \mid \mathbf{u}{i}, \lambda^{(t)}\right) f\left(\mathbf{u}{i}\right) \end{aligned} An implementation of ARMS is available in the R package HI [35]. A sample $\xi{i 1}, \ldots, \xi_{i K}$ for $i=1, \ldots, n$ is obtained at each EM iteration $t$, where the $\left(s_{i}+q\right) \times 1$ vector $\xi_{i k}=\left(\overline{\mathbf{z}}{i k}, \overline{\mathbf{u}}{i k}\right), k=1, \ldots, K$, contains ‘imputed’ values for $\mathbf{z}{i}$ and $\mathbf{u}{i}$ (with the understanding that $\boldsymbol{\xi}{i k}=\overline{\mathbf{u}}{i k}$ if $s_{i}=0$ ). Here the Monte Carlo sample size $K$ is kept constant throughout. Alternative strategies with varying $K^{(t)}$ that may increase the speed or the accuracy of the EM algorithm can be considered $[2,17]$. The E-step (11) is approximated by
$$Q\left(\lambda \mid \lambda^{(t)}\right)=\frac{1}{K} \sum_{i=1}^{n} \sum_{k=1}^{K} l\left(\lambda ; \xi_{i k}, \mathbf{x}{i}, \mathbf{m}{i}\right)$$
The maximization of (15) with respect to $\lambda$ is straightforward. Define $\tilde{\mathbf{y}}{i k}=\left(\tilde{\mathbf{z}}{i k}, \mathbf{x}{i}\right)$ if $s{i}>0$ or $\tilde{\mathbf{y}}{i k}=\mathbf{y}{i}$ if $s_{i}=0, i=1, \ldots, n, k=1, \ldots, K$. The maximum likelihood solution of the M-step at the $(t+1)$ th iteration is given by
\begin{aligned} \hat{\boldsymbol{\mu}}^{(t+1)} &=\frac{1}{n K} \sum_{i=1}^{n} \sum_{k=1}^{K}\left(\tilde{\mathbf{y}}{i k}-\hat{\mathbf{W}}^{(t)} \tilde{\mathbf{u}}{i k}\right) \ \hat{\mathbf{W}}^{(t+1)} &=\left{\sum_{i=1}^{n} \sum_{k=1}^{K}\left(\tilde{\mathbf{y}}{i k}-\hat{\boldsymbol{\mu}}^{(t+1)}\right) \tilde{\mathbf{u}}{i k}^{\top}\right}\left(\sum_{i=1}^{n} \sum_{k=1}^{K} \tilde{\mathbf{u}}{i k} \tilde{\mathbf{u}}{i k}^{T}\right)^{-1} \ \hat{\psi}^{(t+1)} &=\frac{1}{n K p} \sum_{i=1}^{n} \sum_{k=1}^{K}\left|\tilde{\mathbf{y}}{i k}-\hat{\boldsymbol{\mu}}^{(t+1)}-\hat{\mathbf{W}}^{(t+1)} \tilde{\mathbf{u}}{i k}\right|_{2}^{2} \end{aligned}
Analogously, the MLE of $\eta$ can be easily obtained using standard results for generalized linear models.

## 统计代写|主成分分析代写Principal Component Analysis代考|PCA and Robust PCAs

Let us consider a training set of $N n$-dimensional samples $\left{\mathbf{x}{i}\right}{i=1^{*}}^{N}$ Assuming that the samples have zero-mean, PCA is to find an orthonormal projection matrix $\mathbf{W} \in$ $\mathbb{R}^{n \times m}\left(m \ll n\right.$ ) by which the projected samples $\left{\mathbf{y}{i}=\mathbf{W}^{T} \mathbf{x}{i}\right}_{i=1}^{N}$ have the maximum variance in the reduce space. It is formulated as the following:
$$\mathbf{W}{P C A}=\underset{\mathbf{W}}{\arg \max } \operatorname{tr}\left(\mathbf{W}^{T} \mathbf{S W}\right)$$ where $\mathbf{S}=\frac{1}{N} \sum{i=1}^{N} \mathbf{x}{i} \mathbf{x}{i}^{T}$ is a sample covariance matrix and $\operatorname{tr}(\mathbf{A})$ is the trace of a square matrix $\mathbf{A}$. The projection matrix $\mathbf{W}{P C A}$ can be also found from the viewpoint of projection errors, i.e., it minimizes the average of the squared projection errors or reconstruction errors. Mathematically, it is represented as the optimization problem minimizing the following cost function: $$J{L_{2}}(\mathbf{W})=\frac{1}{N} \sum_{i=1}^{N}\left|\mathbf{x}{i}-\mathbf{W} \mathbf{W}^{T} \mathbf{x}{i}\right|_{2}^{2}$$
where $|\mathbf{x}|_{2}$ is the $L_{2}$-norm of a vector $\mathbf{x}$. The two optimization problems are equivalent and easily solved by obtaining the $m$ eigenvectors associated with the $m$ largest eigenvalues of $\mathbf{S}$. Although PCA is simple and powerful, it is prone to outliers [8, 13] because $J_{L_{2}}(\mathbf{W})$ is based on the mean squared reconstruction error. To learn a subspace robust to outliers, Ke and Kanade [13] proposed to minimize an $L_{1}$-norm based objective function as follows:
$$J_{L_{1}}(\mathbf{W})=\frac{1}{N} \sum_{i=1}^{N}\left|\mathbf{x}{i}-\mathbf{W} \mathbf{W}^{T} \mathbf{x}{i}\right|_{1}$$
where $|\mathbf{x}|_{1}$ is the $L_{1}$-norm of a vector $\mathbf{x}$. They also present an iterative method to obtain the solution for minimizing $J_{L_{1}}(\mathbf{W})$.

## 统计代写|主成分分析代写Principal Component Analysis代考|Missing Not at Random

F(是一世,在一世,米一世∣θ,这)=F(是一世∣在一世,θ)F(在一世)F(米一世∣是一世,这),一世=1,…,n,其中附加因子F(米一世∣是一世,这), 由参数索引这, 是 MDM，我们假设它独立于在一世. 这个假设简化了估计算法的后续步骤，尽管它可以以增加计算时间为代价来放宽（参见[17,18]讨论）。

F(米一世∣是一世,这)=∏j=1p圆周率一世j米一世j(1−圆周率一世j)1−米一世j

## 统计代写|主成分分析代写Principal Component Analysis代考|Appendix – EM Algorithm for PPCA with MNAR Values

Monte Carlo E-step 需要从F(和一世,在一世∣X一世,米一世,λ(吨)). 这个任务可以通过 ARMS [12] 使用完整的条件有效地执行

F(和一世∣X一世,在一世,米一世,λ(吨))∝F(是一世∣在一世,λ(吨))F(米一世∣是一世,λ(吨)) F(在一世∣X一世,和一世,米一世,λ(吨))∝F(是一世∣在一世,λ(吨))F(在一世)R 包 HI [35] 中提供了 ARMS 的实现。一个样品X一世1,…,X一世ķ为了一世=1,…,n在每次 EM 迭代中获得吨, 其中(s一世+q)×1向量X一世ķ=(和¯一世ķ,在¯一世ķ),ķ=1,…,ķ, 包含“估算”值和一世和在一世（理解为X一世ķ=在¯一世ķ如果s一世=0）。这里是蒙特卡洛样本量ķ始终保持不变。具有不同的替代策略ķ(吨)可以考虑提高 EM 算法的速度或准确性[2,17]. E-step (11) 近似为

(15) 的最大化关于λ很简单。定义是~一世ķ=(和~一世ķ,X一世)如果s一世>0或者是~一世ķ=是一世如果s一世=0,一世=1,…,n,ķ=1,…,ķ. M 步的最大似然解(吨+1)第一次迭代由下式给出

\begin{aligned} \hat{\boldsymbol{\mu}}^{(t+1)} &=\frac{1}{n K} \sum_{i=1}^{n} \sum_{k= 1}^{K}\left(\tilde{\mathbf{y}}{i k}-\hat{\mathbf{W}}^{(t)} \tilde{\mathbf{u}}{i k}\右) \ \hat{\mathbf{W}}^{(t+1)} &=\left{\sum_{i=1}^{n} \sum_{k=1}^{K}\left( \tilde{\mathbf{y}}{i k}-\hat{\boldsymbol{\mu}}^{(t+1)}\right) \tilde{\mathbf{u}}{i k}^{\top }\right}\left(\sum_{i=1}^{n} \sum_{k=1}^{K} \tilde{\mathbf{u}}{i k} \tilde{\mathbf{u}} {i k}^{T}\right)^{-1} \ \hat{\psi}^{(t+1)} &=\frac{1}{n K p} \sum_{i=1}^ {n} \sum_{k=1}^{K}\left|\tilde{\mathbf{y}}{i k}-\hat{\boldsymbol{\mu}}^{(t+1)}-\ hat{\mathbf{W}}^{(t+1)} \tilde{\mathbf{u}}{i k}\right|_{2}^{2} \end{aligned}\begin{aligned} \hat{\boldsymbol{\mu}}^{(t+1)} &=\frac{1}{n K} \sum_{i=1}^{n} \sum_{k= 1}^{K}\left(\tilde{\mathbf{y}}{i k}-\hat{\mathbf{W}}^{(t)} \tilde{\mathbf{u}}{i k}\右) \ \hat{\mathbf{W}}^{(t+1)} &=\left{\sum_{i=1}^{n} \sum_{k=1}^{K}\left( \tilde{\mathbf{y}}{i k}-\hat{\boldsymbol{\mu}}^{(t+1)}\right) \tilde{\mathbf{u}}{i k}^{\top }\right}\left(\sum_{i=1}^{n} \sum_{k=1}^{K} \tilde{\mathbf{u}}{i k} \tilde{\mathbf{u}} {i k}^{T}\right)^{-1} \ \hat{\psi}^{(t+1)} &=\frac{1}{n K p} \sum_{i=1}^ {n} \sum_{k=1}^{K}\left|\tilde{\mathbf{y}}{i k}-\hat{\boldsymbol{\mu}}^{(t+1)}-\ hat{\mathbf{W}}^{(t+1)} \tilde{\mathbf{u}}{i k}\right|_{2}^{2} \end{aligned}

## 统计代写|主成分分析代写Principal Component Analysis代考|PCA and Robust PCAs

Ĵ大号2(在)=1ñ∑一世=1ñ|X一世−在在吨X一世|22

Ĵ大号1(在)=1ñ∑一世=1ñ|X一世−在在吨X一世|1

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