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

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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.

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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。