### 机器学习代写|流形学习代写manifold data learning代考|Data on Manifolds

statistics-lab™ 为您的留学生涯保驾护航 在代写流形学习manifold data learning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。我们的专家在代写流形学习manifold data learning代写方面经验极为丰富，各种代写流形学习manifold data learning相关的作业也就用不着说。

• 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

## 机器学习代写|流形学习代写manifold data learning代考|Data on Manifolds

All the manifold-learning algorithms that we will describe in this chapter assume that finitely many data points, $\left{\mathbf{y}{i}\right}$, are randomly drawn from a smooth $t$-dimensional manifold $\mathcal{M}$ with a metric given by geodesic distance $d^{\mathcal{M}}$. These data points are (linearly or nonlinearly) embedded by a smooth map $\psi$ into high-dimensional input space $\mathcal{X}=\Re^{r}$, where $t \ll r$, with Euclidean metric $|$. $|{\mathcal{X}}$. This embedding results in the input data points $\left{\mathbf{x}{i}\right}$. In other words, the embedding map is $\psi: \mathcal{M} \rightarrow \mathcal{X}$, and a point on the manifold, $\mathbf{y}{i} \in \mathcal{M}$, can be expressed as $\mathbf{y}=\phi(\mathbf{x}), \mathbf{x} \in \mathcal{X}$, where $\phi=\psi^{-1}$.

The goal is to recover $\mathcal{M}$ and find an explicit representation of the map $\psi$ (and recover the ${\mathbf{y}}$ ), given either the input data points $\left{\mathbf{x}{i}\right}$ in $\mathcal{X}$, or the proximity matrix of distances between all pairs of those points. When we apply these algorithms, we obtain estimates $\left{\widehat{\mathbf{y}}{i}\right} \subset \Re^{t^{\prime}}$ that provide reconstructions of the manifold data $\left{\mathbf{y}_{i}\right} \subset \Re^{t}$, for some $t^{\prime}$. Clearly, if $t^{\prime}=t$, we have been successful. In general, we expect $t^{\prime}>3$, and so the results will be impractical for visualization purposes. To overcome this difficulty, while still providing a low-dimensional representation, we take only the points of the first two or three of the coordinate vectors of the reconstruction and plot them in a two-or three-dimensional space.

## 机器学习代写|流形学习代写manifold data learning代考|Linear Manifold Learning

Most statistical theory and applications that deal with the problem of dimensionality reduction are focused on linear dimensionality reduction and, by extension, linear manifold learning. A linear manifold can be visualized as a line, a plane, or a hyperplane, depending upon the number of dimensions involved. Data are observed in some high-dimensional space and it is usually assumed that a lower-dimensional linear manifold would be the most appropriate summary of the relationship between the variables. Although data tend not to live on a linear manifold, we view the problem as having two kinds of motivations. The first such motivation is to assume that the data live close to a linear manifold, the distance off the manifold determined by a random error (or noise) component. A second way of thinking about linear manifold learning is that a linear manifold is really a simple linear approximation to a more complicated type of nonlinear manifold that would probably be a better fit to the data. In both scenarios, the intrinsic dimensionality of the linear manifold is taken to be much smaller than the dimensionality of the data.

Identifying a linear manifold embedded in a higher-dimensional space is closely related to the classical statistics problem of linear dimensionality reduction. The recommended way of accomplishing linear dimensionality reduction is to create a reduced set of linear transformations of the input variables. Linear transformations are projection methods, and so the problem is to derive a sequence of low-dimensional projections of the input data that possess some type of optimal properties. There are many techniques that can be used for either linear dimensionality reduction or linear manifold learning. In this chapter, we describe only two linear methods, namely, principal component analysis and multidimensional scaling. The earliest projection method was principal component analysis (dating back to 1933), and this technique has become the most popular dimensionality-reducing technique in use today. A related method is that of multidimensional scaling (dating back to 1952), which has a very different motivation. An adaptation of multidimensional scaling provided the core element of the IsOMAP algorithm for nonlinear manifold learning.

## 机器学习代写|流形学习代写manifold data learning代考|Principal Component Analysis

PRINCIPAL COMPONENT ANALYSIS (PCA) (Hotelling, 1933) was introduced as a technique for deriving a reduced set of orthogonal linear projections of a single collection of correlated variables, $\mathbf{X}=\left(X_{1}, \cdots, X_{r}\right)^{\tau}$, where the projections are ordered by decreasing variances. The amount of information in a random variable can be measured by its variance, which is a second-order property. PCA has also been referred to as a method for “decorrelating” $\mathbf{X}$, and so several researchers in different fields have independently discovered this technique. For example, PCA is also called the Karhunen-Loève transform in communications theory and empirical orthogonal functions in atmospheric science. As a technique for dimensionality reduction, PCA has been used in lossy data compression, pattern recognition, and image analysis. In chemometrics, PCA is used as a preliminary step for constructing derived variables in biased regression situations, leading to principal component regression.

PCA is also used as a means of discovering unusual facets of a data set. This can be accomplished by plotting the top few pairs of principal component scores (those having largest variances) in a scatterplot. Such a scatterplot can identify whether $\mathbf{X}$ actually lives on a low-dimensional linear manifold of $\Re^{r}$, as well as provide help identifying multivariate outliers, distributional peculiarities, and clusters of points. If the bottom set of principal components each have near-zero variance, then this implies that those principal components are essentially constant and, hence, can be used to identify the presence of collinearity and possibly outliers that might distort the intrinsic dimensionality of the vector $\mathbf{X}$.
Population Principal Components
Suppose that the input variables are the components of a random $r$-vector,
$$\mathbf{X}=\left(X_{1}, \cdots, X_{r}\right)^{\top}$$
where $\mathbf{A}^{\tau}$ denotes the transpose of the matrix $\mathbf{A}$. In this chapter, all vectors will be column vectors. Further, assume that $\mathbf{X}$ has mean vector $\mathrm{E}{\mathbf{X}}=\boldsymbol{\mu}{X}$ and $(r \times r)$ covariance matrix $\mathrm{E}\left{\left(\mathbf{X}-\boldsymbol{\mu}{X}\right)\left(\mathbf{X}-\boldsymbol{\mu}{X}\right)^{\tau}\right}=\mathbf{\Sigma}{X X}$. PCA replaces the input variables $X_{1}, X_{2}, \ldots, X_{r}$ by a new set of derived variables, $\xi_{1}, \xi_{2}, \ldots, \xi_{t}, t \leq r$, where
$$\xi_{j}=\mathbf{b}{j}^{\tau} \mathbf{X}=b{j 1} X_{1}+\cdots+b_{j r} X_{r}, \quad j=1,2, \ldots, r .$$
The derived variables are constructed so as to be uncorrelated with each other and ordered by the decreasing values of their variances. To obtain the vectors $\mathbf{b}{j}, j=1,2, \ldots, r$, which define the principal components, we minimize the loss of information due to replacement. In $\mathrm{PCA}$, “information” is interpreted as the “total variation” of the original input variables, $$\sum{j=1}^{r} \operatorname{var}\left(X_{j}\right)=\operatorname{tr}\left(\Sigma_{X X}\right)$$

## 机器学习代写|流形学习代写manifold data learning代考|Principal Component Analysis

PCA 也被用作发现数据集异常方面的一种手段。这可以通过在散点图中绘制前几对主成分分数（具有最大方差的分数）来完成。这样的散点图可以识别是否X实际上生活在一个低维线性流形上ℜr，并提供帮助识别多元异常值、分布特性和点簇。如果底部的主成分集每个都具有接近零的方差，则这意味着这些主成分基本上是恒定的，因此可用于识别共线性的存在以及可能扭曲向量固有维度的可能异常值X.

X=(X1,⋯,Xr)⊤

Xj=bjτX=bj1X1+⋯+bjrXr,j=1,2,…,r.

## 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。