统计代写|数据结构作业代写data structure代考|CS166

Doug I. Jones

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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|Reasons of Distortions Ubiquity

In the process of mapping high dimensional data to a low-dimensional space, distortions are present in most cases. First, the best possible map according to the stress function of a given DR technique (i.e. the global optimum of that stress) may not always be reached. Indeed, most of them have non-convex stress functions, so that optimization algorithms may remain stuck at local optima and never converge towards that global optimum (see Sect. 7.1).

Furthermore, some datasets (associated to an hypothetical underlying manifold) may be non-mappable in a given embedding space. This means that even the best map may not preserve all relevant information. Assuming that a non-distorted mapping is an homeomorphism (see Sect. 3.2.2), datasets are not mappable in an embedding space, if they live on a manifold that is not homeomorphic to this

embedding space. However, in practice, the true underlying manifold associated to a discrete dataset is unknown. Thus, a non-smooth mappable manifold could technically be fitted to any discrete dataset.

If the intrinsic dimensionality of the manifold is strictly higher than that of the embedding space, there exists no homeomorphism between the two spaces. Thus, such a discrepancy is a sufficient condition for the non-mappability of the dataset. This is rather frequent in visual exploration for which the embedding dimensionality is constrained to be very low (at most three for visualization with a single scatter plot). Yet, this condition is not necessary, since some manifolds may not be mapped to embedding space of adequate dimensionality due to topological incompatibility. This is for example the case of a intrinsically two-dimensional sphere, which despite its dimensional compatibility with the two-dimensional plane, is not homeomorphic to it. Note that by puncturing it in a point (introducing a tear), the sphere becomes homeomorphic to a plane (e.g., the Riemman mapping).

统计代写|数据结构作业代写data structure代考|Distance-Based Indicators

The quality of an embedding may be assessed by comparing the distances $\Delta_{i j}$ and $D_{i j}$ between the two spaces. Most indicators from that framework are derived from the normalized “stress” [21], which is both a stress function for DR and used to assess map quality:
$$\mathscr{S} \triangleq \frac{\sum_i \sum_j\left(\Delta_{i j}-D_{i j}\right)^2}{\sum_i \sum_j \Delta_{i j}^2}$$

This indicator is however sensitive to scaling of the embedding space distances. Thus, other map evaluation and interpretation criteria tend to independently normalize the distances in both spaces. Yet, a normalization dependent on the embedding distances may differ from one map to another. Hence, it induces additional variability in the evaluation scores.

The projection precision score [160] uses root mean square of distances to the $\kappa$ nearest neighbours for normalization, in line with the focus of many DR methods on the preservation of small distances:
$$\mathscr{S}i^{\mathrm{pps}}(\kappa) \triangleq \sqrt{\sum{k=1}^\kappa\left(\frac{\Delta_{i \tilde{v}i(k)}}{\sqrt{\sum{l=1}^\kappa \Delta_{i \tilde{v}i(l)}}}-\frac{D{i \tilde{n}i(k)}}{\sqrt{\sum{l=1}^\kappa D_{i \tilde{n}_i(l)}^2}}\right)^2},$$
where $\tilde{v}_i$ and $\tilde{n}_i$ are the neighbourhood permutations defined in Sect. 1.1.2. The use of a normalizing value aggregating distances to several points seems to lead to a rather robust indicator. We may note that this normalizing value is also defined pointwise, scaling differently each neighbourhood.

Martins et al. [122] use the following point-wise aggregated error, using absolute values rather than squares:
$$\mathscr{E}i \triangleq \sum{j \neq i}\left|\mathscr{E}{i j}\right| \quad \text { with } \quad \mathscr{E}{i j} \triangleq \frac{D_{i j}}{D_{\max }}-\frac{\Delta_{i j}}{\Delta_{\max }}$$

数据结构代考

统计代写|数据结构作业代写data structure代考|Distance-Based Indicators

$$\mathscr{S} \triangleq \frac{\sum_i \sum_j\left(\Delta_{i j}-D_{i j}\right)^2}{\sum_i \sum_j \Delta_{i j}^2}$$

$$\mathscr{S} i^{\mathrm{pps}}(\kappa) \triangleq \sqrt{\sum k=1^\kappa\left(\frac{\Delta_{i \tilde{v} i(k)}}{\sqrt{\sum l=1^\kappa \Delta_{i \tilde{v} i(l)}}}\right.}$$

有限元方法代写

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MATLAB代写

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