### 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
• (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.

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## MATLAB代写

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