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

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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

When the distribution of the missing data indicator depends on the unobserved data, after conditioning on the observed data, i.e. when
$$\operatorname{Pr}\left(\mathbf{m}{i} \mid \mathbf{z}{i}, \mathbf{x}{i}\right) \neq \operatorname{Pr}\left(\mathbf{m}{i} \mid \mathbf{x}_{i}\right)$$

then the missing data are said to be missing not at random. This setting is the most general of all.

In the MNAR scenario the missingness indicator is assumed to be related with unmeasured predictors and/or the unobserved response, even conditionally on the observed data. MNAR data are also referred to as informative since the missing values contain information about the MNAR mechanism itself.

It should be stressed that, if the true mechanism is MNAR, simple CC analyses or naïve imputation methods inevitably produce biased results. The extent and direction of this bias is unpredictable, and even relatively small fractions of missing values might lead to a large bias.

There are different approaches to the treatment of this kind of missing data. Modelbased procedures are most commonly adopted. These aim at modeling the joint distribution of the measurement process and the dropout process, by specifying a missing data model (MDM). The MDM must take into account the residual dependence between the missingness indicator and the unobserved response. Below, we summarize the three main approaches.

• Pattern-mixture models. The joint distribution of $\mathbf{y}{i}$ and $\mathbf{m}{i}$ is factorized as
$$\operatorname{Pr}\left(\mathbf{y}{i}, \mathbf{m}{i}\right)=\operatorname{Pr}\left(\mathbf{y}{i} \mid \mathbf{m}{i}\right) \operatorname{Pr}\left(\mathbf{m}{i}\right)$$ This approach involves formulating separate submodels $\operatorname{Pr}\left(\mathbf{y}{i} \mid \mathbf{m}{i}\right)$ for each possible configuration of $\mathbf{m}{i}$, or, at least, for each observed configuration. This is appealing for studies where the main objective is to compare the response distribution in subgroups with possibly different missing value patterns. On the other hand, its specification can be cumbersome, while its interpretation at the population level may become difficult.
• Selection models. The joint distribution of $\mathbf{y}{i}$ and $\mathbf{m}{i}$ is factorized as
$$\operatorname{Pr}\left(\mathbf{y}{i}, \mathbf{m}{i}\right)=\operatorname{Pr}\left(\mathbf{m}{i} \mid \mathbf{y}{i}\right) \operatorname{Pr}\left(\mathbf{y}{i}\right)$$ This approach involves an explicit model to handle the distribution of the missing data process given the measurement mechanism. If correctly specified, the model for $\mathbf{y}{i}$ is estimated without bias and its interpretation is not compromised.

## 统计代写|主成分分析代写Principal Component Analysis代考|Methods to Handle Missing Data in Principal

In this section, we discuss selected missing data methods in PCA, some of which are relatively recent at the time of writing. PCA, originally introduced by Karl Pearson [34], is arguably one the most popular multivariate analysis techniques. It is often described as a tool for dimensionality reduction. Some authors consider PCA as a descriptive method, which needs not be based on distributional assumptions, whereas others provide probabilistic justifications in relation to sampling errors (see for example [20] or [21] for alternative interpretations of the PCA model). It is not our purpose to get embroiled in this discussion; here we take a probabilistic view as it is an essential framework for a statistical treatment of the missing data problem.
There are basically two main approaches where sampling comes into play: fixed and random effects PCA. (The random-effects approach can be formulated in either a frequentist or a Bayesian framework. We focus on the former, while more details on the latter can be found in [21]). In the fixed-effects approach, individuals are of direct interest. Therefore, individual-specific scores are parameters to be estimated. In symbols, the fixed-effects PCA model is given by
$$\mathbf{y}{i}=\mu+\mathbf{W b}{i}+\varepsilon_{i}, \quad i=1, \ldots, n$$
where $\mu=\left(\mu_{1}, \mu_{2}, \ldots, \mu_{p}\right)^{\mathrm{T}}$ is the vector with the mean of each variable, $\mathbf{b}{i}=$ $\left(b{i 1}, b_{i 2}, \ldots, b_{i q}\right)^{\mathrm{T}}$ is the $i$ th row vector of fixed scores and $\mathbf{W}$ is a $p \times q$ matrix of unknown loadings with elements $w_{j h}, j=1, \ldots, p, h=1, \ldots, q$, with $q \leq p$. The error is assumed to be zero-centered Gaussian with homoscedastic variance, $\varepsilon_{i} \sim N\left(\mathbf{0}, \psi \mathbf{I}{p}\right)$, where $\psi$ is a positive scalar. Model (6)’s parameter, $\boldsymbol{\theta}=\left(\boldsymbol{\mu}, \mathbf{W}, \psi, \mathbf{b}{1}, \mathbf{b}{2}, \ldots, \mathbf{b}{n}\right)$, can be estimated via maximum likelihood (or, equivalently, least squares) estimation (MLE). A downside of this approach is that the dimension of $\theta$ increases with the sample size.

The random-effects specification of the probabilistic representation of PCA $[38$, $42]$ is given by the model
$$\mathbf{y}{i}=\mu+\mathbf{W u}{i}+\boldsymbol{\varepsilon}{i}, \quad i=1, \ldots, n$$ where $\mathbf{u}{i}=\left(u_{i 1}, u_{i 2}, \ldots, u_{i q}\right)^{\top}$ is the $i$ th row vector of latent scores and $\mathbf{W}$ is, as above, a matrix of unknown loadings. Furthermore, it is assumed that $\mathbf{u}$ is stochastically independent from $\varepsilon$. Conventionally, $\mathbf{u}{i} \sim N\left(0, \mathbf{I}{q}\right)$. If in addition the error is assumed to be zero-centered Gaussian with covariance matrix $\Psi, \varepsilon_{i} \sim N(0, \Psi)$, we obtain the multivariate normal distribution $\mathbf{y}{i} \sim N(\boldsymbol{\mu}, \mathbf{C}), \mathbf{C}=\mathbf{W} \mathbf{W}^{\top}+\boldsymbol{\Psi}$. We also assume that $\Psi=\psi \mathbf{I}{p}$, so that the elements of $\mathbf{y}{i}$ are conditionally independent, given $\mathbf{u}{i}$. The parameter $\mu=\left(\mu_{1}, \mu_{2}, \ldots, \mu_{p}\right)^{\top}$ allows for a location-shift fixed effect.

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

As mentioned before, single imputation methods treat imputed missing values as fixed (known). This means that the uncertainty related to the missing values is ignored, which generally leads to deflated standard errors. Josse et al. [23] proposed to deal with this issue by first performing a residual bootstrap procedure to obtain $B$ estimates of the PPCA parameters and then generate $B$ data matrices, each completed with samples from the predictive distribution of the missing values conditional on the observed values and the corresponding bootstrapped parameter set. More formally, one can proceed as follows:

1. obtain an initial estimate $\hat{\mu}, \hat{\mathbf{W}}, \hat{\mathbf{b}}_{i}, i=1, \ldots, n$, of the parameters in model (6) (e.g., via EM-PCA estimation). Reconstruct the data $\hat{\mathbf{Y}}$ with the first $q$ dimensions and calculate $\mathbf{R}=\mathbf{Y}-\hat{\mathbf{Y}}$, where the $n \times p$ matrix $\mathbf{R}$ of residuals has missing entries corresponding to those of $\mathbf{Y}$;
2. draw $B$ random samples from the non-missing rows of $\mathbf{R}$. Denote each replicate by $\mathbf{R}_{b}^{*}, b=1, \ldots, B$;
3. calculate $\mathbf{Y}{b}^{}=\hat{\mathbf{Y}}+\mathbf{R}{b}^{}, b=1, \ldots, B$ and estimate the PPCA parameters $\hat{\boldsymbol{\mu}}{b}^{}$, $\hat{\mathbf{W}}{b}^{}, \hat{\mathbf{b}}{b, i}^{}, i=1, \ldots, n$, from $\mathbf{Y}{b}^{}, b=1, \ldots, B ;$
4. for $\left{i: s_{i}>0\right}$, calculate $\mathbf{y}{b, i}^{}=\hat{\boldsymbol{\mu}}{b}^{}+\hat{\mathbf{W}}{b}^{} \hat{\mathbf{b}}{b, i}^{}+\mathbf{r}^{}$, where $\mathbf{r}^{}$ is a newly sampled residual from $\mathbf{R}$. Complete the vector $\mathbf{y}{i}$ with $\mathbf{y}{b, i}^{*}$ to obtain the $b$ th complete data matrix $\mathbf{Y}_{b}, b=1, \ldots, B$.

There are several ways to obtain bootstrapped residuals. One approach is to draw a sample with replacement from the entries of the matrix $\mathbf{R}$. Another approach, recommended by [21], is to sample the residuals from a zero-centered Gaussian with variance estimated from the non-missing entries of $\mathbf{R}$. Improved results might be obtained with corrected residuals (e.g., leave-one-out residuals).

Once $B$ complete data matrices have been generated, the simplest analytic approach is to carry out a PPCA on each $\mathbf{Y}_{b}$, and then calculate the average of the $B$ sets of parameters. A multiple imputation PPCA of the Orange data set is given in R code 3.4. Our example is based on $B=100$ replicates, with $q=2$. The individuals and variables maps obtained from average scores and loadings are plotted in Fig. 2. As compared to the complete case PCA and the single imputation EM-PCA (Table 1), the multiple imputation PCA produced noticeably different estimates of $\hat{\mathbf{W}}$, especially for the second principal axis (R code $3.4$ ). The uncertainty due to the missing values is also shown in Fig. 2. For example, individual scores 1, 9, and 10 showed more total variability, as given by the area of the ellipses, and more variability along the second axis, as reflected in the eccentricity of the ellipses. The uncertainty was greater for sweetness, color intensity, and bitterness, and, as in the case of the individual scores, it was more prominent in relation to the second axis.

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

• 模式混合模型。联合分布是一世和米一世被分解为
公关⁡(是一世,米一世)=公关⁡(是一世∣米一世)公关⁡(米一世)这种方法涉及制定单独的子模型公关⁡(是一世∣米一世)对于每个可能的配置米一世，或者至少对于每个观察到的配置。这对于主要目标是比较可能具有不同缺失值模式的子组中的响应分布的研究很有吸引力。另一方面，它的规范可能很麻烦，而在人口水平上的解释可能会变得困难。
• 选择模型。联合分布是一世和米一世被分解为
公关⁡(是一世,米一世)=公关⁡(米一世∣是一世)公关⁡(是一世)这种方法涉及一个显式模型来处理给定测量机制的缺失数据过程的分布。如果正确指定，模型为是一世估计没有偏见，它的解释没有妥协。

## 统计代写|主成分分析代写Principal Component Analysis代考|Methods to Handle Missing Data in Principal

PCA概率表示的随机效应规范[38, 42]由模型给出

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

1. 获得初步估计μ^,在^,b^一世,一世=1,…,n，模型（6）中的参数（例如，通过EM-PCA估计）。重构数据是^与第一个q尺寸和计算R=是−是^, 其中n×p矩阵R残差的缺失条目对应于是;
2. 画乙非缺失行中的随机样本R. 用Rb∗,b=1,…,乙;
3. 计算是b=是^+Rb,b=1,…,乙并估计 PPCA 参数μ^b, 在^b,b^b,一世,一世=1,…,n， 从是b,b=1,…,乙;
4. 为了\left{i: s_{i}>0\right}\left{i: s_{i}>0\right}， 计算是b,一世=μ^b+在^bb^b,一世+r， 在哪里r是新采样的残差R. 完成向量是一世和是b,一世∗获得b完整的数据矩阵是b,b=1,…,乙.

## 有限元方法代写

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## MATLAB代写

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