## 统计代写|数据结构作业代写data structure代考|COS241

2023年2月6日

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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
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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
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## 统计代写|数据结构作业代写data structure代考|Test of TIDLE on a Two Clusters Case

To test our approach,t we consider simple synthetic datasets, each constituted of two clusters of different dimensionality generated by a multivariate distribution restricted to a linear subspace of the appropriate dimensionality in a 10 -dimensional ambient space. These clusters are randomly rotated [55], and translated along the diagonal $(1,1, \ldots, 1)$ of the unit hypercube so as to ensure a given distance between their centroids. An example of a similarly generated dataset in a three-dimensional space is shown in Fig. 2.10a. Figure $2.10$ presents the weights $w_k$ identified by TIDLE for the mixture as a function of the component index $k$. The variability of the method is accounted for by considering in each case 100 datasets randomly generated with the same characteristics. The distribution of each weight $w_k$ over the 100 runs is shown as a box plot. For all datasets, the clusters each contain 500 points and the neighbourhood size is set at $\kappa=100$.

Figure $2.10 \mathrm{~b}$ presents the results for the standard case, where the dimensionality of the two Gaussian distributed clusters is respectively three and seven. Those clusters (of unit variance) are well-separated with a distance of five hypercube diagonals, and subject to no noise. The three-dimensional component is detected in at least $95 \%$ of the runs and in these cases its associated weight is higher than $0.3$, as shown by the fifth percentile of the distribution of $w_3$. As for the seven-dimensional component, it is detected in at least $75 \%$ of cases, considering the first quartile of the distribution of $w_7$. Yet, for some datasets, several components of close dimensionality are detected for a same cluster. In particular, the seven-dimensional cluster is also partly explained by a six-dimensional component in at least $50 \%$ of cases, considering the median of the distribution of $w_6$.

## 统计代写|数据结构作业代写data structure代考|Link Between Distortions and Mapping Continuity

Distortions of neighbourhood relations may be interpreted in terms of mapping continuity. Intuitively, the continuity of a mapping $\widehat{\Phi}: \mathcal{D} \longrightarrow \mathcal{E}$ between metric spaces ensures that the image of a sufficiently small ball around a point $\xi_0$ by the mapping is comprised within a ball of given size around the point image $x_0=\widehat{\Phi}\left(\xi_0\right)$. More formally:

Definition $3.1$ A mapping $\widehat{\Phi}: \mathcal{D} \longrightarrow \mathcal{E}$, with $(\mathcal{D}, \Delta)$ and $(\mathcal{E}, D)$ metric spaces is said to be continuous in $\xi_0 \in \mathcal{D}$ if for all $\epsilon>0$, there exists $\omega>0$ so that for $\xi \in \mathcal{D}, \Delta\left(\xi_i, \xi\right)<\omega \Longrightarrow D\left(\widehat{\Phi}\left(\xi_0\right), \widehat{\Phi}(\xi)\right)<\epsilon$. This means that for any ball $\mathcal{B}\left(x_0, \epsilon\right)$ (centred at $x_i=\widehat{\Phi}\left(\xi_0\right)$ and of radius $\epsilon$ ) in the co-domain, there exists a radius $\omega$ such that the image by $\widehat{\Phi}$ of the ball $\mathcal{B}\left(\xi_0, \omega\right)$ is included in $\mathcal{B}\left(x_0, \epsilon\right)$ (see the illustration in Fig. 3.2).

In practice, most DR methods only define a discrete mapping $\Phi:\left{\xi_i\right} \longrightarrow\left{x_i\right}$. Thus, the formal concept of continuity may only be applied to an extension $\widehat{\Phi}: \mathcal{M} \longrightarrow \mathcal{E}$ of $\Phi$ to the entire data manifold $\mathcal{M}$.

A manifold tear or missed neighbourhood corresponds to a case where a neighbour $\xi$ of a data point $\xi_0$ (which means a point that would be in “any” ball centred at $\xi_0$ ) is not mapped within a ball around the image $x_0=\widehat{\Phi}\left(\xi_0\right)$ of $\xi_0$. Hence, this type of distortion suggests a breach of continuity of the mapping $\widehat{\Phi}$.
Conversely, a manifold gluing or false neighbourhood implies a breach of continuity for the mapping inverse $\widehat{\Phi}^{-1}$. Indeed, it means that a neighbour $x$ of an embedded point $x_0$, which is a point that would be in “any” ball centred at $x_0$, is not mapped within a ball around the image $\xi=\widehat{\Phi}^{-1}(x)$ of $x$. Note that this relies on the assumption that $\widehat{\Phi}$ admits an inverse.

In that regard,t an ideal mapping, subject to no distortions would be a homeomorphism or bi-continuous function, namely an invertible function that is continuous and whose inverse is continuous. Distortions indicators described in Sect. 3.2.4 assess the breach of continuity for the theoretical mapping $\widehat{\Phi}$ (and its inverse) based on the available information for the discrete mapping $\Phi$, which is its restriction to the sample points $\left{\xi_i\right}$. For rank-based indicators, this is done by considering the preservation of $\kappa$-neighbourhoods, which are balls centred at the points and whose radii are defined by the distance to the $\kappa^{\text {th }}$ nearest neighbour of each point.

# 数据结构代考

## 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。