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
• 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
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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

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