### 统计代写|主成分分析代写Principal Component Analysis代考|MAST90085

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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 Data Mechanisms

The commonly adopted ontology of missing data [27] distinguishes among three cases: missing completely at random (MCAR), missing at random (MAR), and missing not at random (MNAR). These are called missing data mechanisms.

In our set up, let $\mathbf{Y}$ denote an $n$ by $p$ matrix of continuous measurements obtained from $n$ units on $p$ possibly correlated random variables $Y_{1}, Y_{2}, \ldots, Y_{p}$. Let also $\mathbf{M}$ denote an $n$ by $p$ matrix with row vectors $\mathrm{m}{i}, i=1,2, \ldots, n$, whose $j$ th entry is given by the binary indicator $$m{i j}= \begin{cases}1 & \text { if the } i j \text { th entry of } \mathbf{Y} \text { is missing, } \ 0 & \text { otherwise. }\end{cases}$$
The matrix M describes the pattern of missing data. In a likelihood framework, each element of $\mathbf{M}$ is assumed to be a random variable with marginal distribution given by a Bernoulli with probability $\pi_{i j}$, that is, $m_{i j} \sim \operatorname{Bin}\left(1, \pi_{i j}\right.$. All statements regarding how missing data are generated arise from assumptions on the joint distribution of $\mathbf{Y}$ and $\mathbf{M}$, and these assumptions, in turn, determine which methods are most appropriate to deal with the missing data. Before we discuss MCAR, MAR, and MNAR assumptions in detail, let us introduce some additional notation. Suppose that the $i$ th row of $\mathbf{Y}$ contains $s_{i} \geq 0$ missing values. Then, $\mathbf{y}{i}$ is partitioned into the $s{i} \times 1$ vector $\mathbf{z}{i}$ and the $\left(p-s{i}\right) \times 1$ vector $\mathbf{x}{i}$. That is, $\mathbf{x}{i}$ is the observed part of $\mathbf{y}{i}$ while $\mathbf{z}{i}$ is its unobserved part, which would have been recorded if at all possible. Finally, let $\mathbf{I}_{n}$ denote the identity matrix of order $n$.

The MCAR mechanism assumes that $\mathbf{y}{i}$ and $\mathrm{m}{i}$ are marginally independent. In other words, if in addition we assume independence among the $m_{i j}$ ‘s, this is equivalent to tossing a coin whose probability of heads equals $\pi_{i j}$, and to deleting the $j$ th entry of $\mathbf{y}_{i}$ if heads comes up. This is the strongest of the three assumptions, seldom tenable in practice.

## 统计代写|主成分分析代写Principal Component Analysis代考|Missing Completely at Random

When the distribution of the missing data indicator does not depend on either the observed or unobserved data, i.e. when
$$\operatorname{Pr}\left(\mathbf{m}{i} \mid \mathbf{z}{i}, \mathbf{x}{i}\right)=\operatorname{Pr}\left(\mathbf{m}{i}\right)$$
then the missing data are said to be missing completely at random.
Lack of measurement is therefore unpredictable (as if tossing a coin, so to speak) and, as such, not informative. One could therefore proceed with a complete case (CC) analysis without the risk of incurring into estimation bias. As mentioned in our introductory example, there are two possible choices for a $\mathrm{CC}$ analysis in the multivariate context: observations can be discarded listwise or component-wise. A listwise approach proceeds by discarding $\mathbf{y}{i}$ as soon as $s{i}>0$, that is, if any variable has not been measured, then the entire unit is discarded. A component-wise approach proceeds by using as much information contained in $\mathbf{x}{i}$ as possible (“nothing goes to waste”). For instance, if $m{i 1}=m_{i 2}=0$ and $m_{i 3}=1$, then $y_{i 1}$ and $y_{i 2}$ will contribute to the estimation of $s_{12}$ in a component-wise approach, but not in a listwise approach.

## 统计代写|主成分分析代写Principal Component Analysis代考|Missing at Random

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

then the missing data are said to be missing at random. This setting is more general than MCAR’s. Indeed, MCAR data are always MAR. However, the converse is false.
The event that a measurement is missing may depend on the measurement itself (e.g., wealthy people may be prone to refuse disclosing their income in a survey). However, conditional on the observed data, this dependence disappears (e.g., the propensity of people to disclose their income is completely explained by the knowledge of their assets).

On the one hand, in many cases a CC analysis under a MAR assumption is perfectly valid. As most statistical procedures are conditional on the observed data, $m$ cannot give any additional information on $z$. Hence, missing values can be simply ignored. On the other hand, it might be possible to incorporate in the analysis the uncertainty for not knowing $z$ by predicting $z$ from $x$. This procedure is known as imputation. There are several techniques that fall under this label.

A popular technique is hot deck single imputation. This involves selecting a number of donors, that is, units that are “similar” to the unit with missing values and then predicting the missing values through the average of the donors’ observed values. This method is simple and grounded on the fact that units that are similar with respect to the observed values should also be similar with respect to the unobserved ones (provided there is a strong association among variables). There are, however, some difficulties with this technique. Firstly, choosing the number of donors can be sometimes difficult. Secondly, hot deck imputation is not based on a statistical model, hence it lacks theoretical ground and it is difficult to adapt to specific problems. Finally, and most importantly, hot deck single imputation (as any other technique based on a single imputation) fails to take into account the uncertainty brought about by imputation. In other words, a predicted value replacing the missing value is effectively treated as a direct measurement.

To overcome the latter limitation, one can consider a multiple imputation approach which consists in repeatedly predicting the missing values. The results based on several predictions can be averaged to produce a final estimate, or they can be evaluated with respect to their sensitivity to specific imputed values.

Imputation (either single or multiple) can be made theoretically sound by drawing imputations from probability models. The latter are used to capture the generating mechanism of the missing values and can often formally take into account the imputation’s uncertainty.

## 统计代写|主成分分析代写Principal Component Analysis代考|Missing Data Mechanisms

MCAR 机制假设是一世和米一世是边缘独立的。换句话说，如果另外我们假设米一世j的，这相当于抛硬币，正面的概率等于圆周率一世j，并删除j的第是一世如果出现正面。这是三个假设中最强的一个，在实践中很少成立。

## 有限元方法代写

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## 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。