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# 计算机代写|流形学习代写Manifold learning代考|Math214

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## 计算机代写|流形学习代写Manifold learning代考|Multidimensional Scaling

Multidimensional scaling (MDS) is a family of algorithms each member of which seeks to identify an underlying manifold consistent with a given set of data. A useful motivation for MDS can be viewed in the following way. Imagine a map of a particular geographical region, which includes several cities and towns. Such a map is usually accompanied by a two-way table of distances between selected pairs of those towns and cities. The number in each cell of that table gives the degree of “closeness” (or proximity) of the row city to the column city that identifies that cell. The general problem of MDS reverses that relationship between the map and table of proximities. With MDS, one is given only the table of proximities, and the problem is to reconstruct the map as closely as possible. There is one more wrinkle: the number of dimensions of the map is unknown, and so we have to determine the dimensionality of the underlying (linear) manifold that is consistent with the given table of proximities.
Proximity Matrices
Proximities do not have to be distances, but can be a more complicated concept. We can talk about the proximity of any two entities to each other, where by “entity” we might mean an object, a brand-name product, a nation, a stimulus, and so on. The proximity of a pair of such entities could be a measure of association (e.g., the absolute value of a correlation coefficient), a confusion frequency (i.e., to what extent one entity is confused with another in an identification exercise), or some other measure of how alike (or how different) one perceives the entities to be. A proximity can be a continuous measure of how physically close one entity is to another or it could be a subjective judgment recorded on an ordinal scale, but where the scale is sufficiently well-calibrated as to be considered continuous. In other scenarios, especially in studies of perception, a proximity will not be quantitative, but will be a subjective rating of “similarity” (how close a pair of entities are to each other) or “dissimilarity” (how unalike are the pair of entities). The only thing that really matters in MDS is that there should be a monotonic relationship (either increasing or decreasing) between the “closeness” of two entities and the corresponding similarity or dissimilarity value.

## 计算机代写|流形学习代写Manifold learning代考|Classical Scaling

Although there are several different versions of MDS, we describe here only the classical scaling method. Other methods are described in Izenman (2008, Chapter 13).

So, suppose we are given $n$ points $\mathbf{X}1, \ldots, \mathbf{X}_n \in \Re^r$ from which we compute an $(n \times n)$ matrix $\boldsymbol{\Delta}=\left(\delta{i j}\right)$ of dissimilarities, where
$$\delta_{i j}=\left|\mathbf{X}i-\mathbf{X}_j\right|=\left{\sum{k=1}^r\left(X_{i k}-X_{j k}\right)^2\right}^{1 / 2}$$
is the dissimilarity between $\mathbf{X}i=\left(X{i 1}, \cdots, X_{i r}\right)^\tau$ and $\mathbf{X}j=\left(X{j 1}, \cdots, X_{j r}\right)^\tau, i, j=$ $1,2, \ldots, n$; these dissimilarities are the Euclidean distances between all $m=n(n-1) / 2$ pairs of points in that space. Squaring both sides of (1.23) and expanding the right-hand side yields
$$\delta_{i j}^2=\left|\mathbf{X}i\right|^2+\left|\mathbf{X}_j\right|^2-2 \mathbf{X}_i^\tau \mathbf{X}_j$$ Note that $\delta{i 0}^2=\left|\mathbf{X}i\right|^2$ is the squared distance from the point $\mathbf{X}_i$ to the origin. Let $$b{i j}=\mathbf{X}i^\tau \mathbf{X}_j=-\frac{1}{2}\left(\delta{i j}^2-\delta_{i 0}^2-\delta_{j 0}^2\right)$$

Summing (1.24) over $i$ and over $j$ gives the following identities:
\begin{aligned} n^{-1} \sum_i \delta_{i j}^2 & =n^{-1} \sum_i \delta_{i 0}^2+\delta_{j 0}^2 \ n^{-1} \sum_j \delta_{i j}^2 & =\delta_{i 0}^2+n^{-1} \sum_j \delta_{j 0}^2 \ n^{-2} \sum_i \sum_j \delta_{i j}^2 & =2 n^{-1} \sum_i \delta_{i 0}^2 . \end{aligned}
Let $a_{i j}=-\frac{1}{2} \delta_{i j}^2$. Using the usual “dot” notation, we define $a_i .=n^{-1} \sum_j a_{i j}, a_{. j}=$ $n^{-1} \sum_i a_{i j}$, and $a . .=n^{-2} \sum_i \sum_j a_{i j}$. Substituting (1.26)-(1.28) into (1.25) and then simplifying, we get
$$b_{i j}=a_{i j}-a_{i \cdot}-a_{. j}+a_{\ldots}$$

## 计算机代写|流形学习代写Manifold learning代考|Classical Scaling

$$b_{i j}=a_{i j}-a_{i \cdot}-a_{. j}+a_{\ldots}$$

## 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。