## 统计代写|数据结构作业代写data structure代考|RU101

2023年2月6日

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
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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|Correlation Dimension

The correlationt dimension $[29,82]$ relies on the assumption that a manifold of intrinsic dimensionality $\partial$ locally behaves as a Euclidean space of that dimensionality. In such a Euclidean space, the volume of a ball of radius $\sigma$ is proportional to $\sigma^{\partial}$. Hence, considering that the density of points in the manifold is uniform, the number of neighbours in the neighbourhood of size of any given point is proportional to that volume for sufficiently small values of $\sigma$. As a result, for a set of $N$ points uniformly sampled from that manifold, the proportion of point pairs within a distance $\sigma$ from each other, called $C(\sigma)$, follows for $\sigma \rightarrow 0$, the scaling law:
$$C(\sigma) \propto \sigma^{\partial} \Longleftrightarrow \log C(\sigma)=\partial \log \sigma+\text { const. }$$
In practice, the correlation dimension is obtained as the slope of the linear part of the graph of $\log C(\sigma)$ against $\log \sigma$ [82], as illustrated in Fig. 2.6a.

However, this approach is subject to a conflicting requirement for the definition of the linear part. The scales $\sigma$ must be small enough to satisfy the scaling law of Eq. (2.2), while the largest of those scales must be sufficiently high to encompass a statistically significant number of pairwise distances [59]. Moreover, due to the curse of dimensionality, the proportion of small distances decreases with the dimensionality of the manifold, so that very big datasets are necessary to reach the same quality of estimation. As such, for high dimensional manifolds with a reasonable number of samples the correlation dimension leads to a systematic under-estimation of the dimension. This is confirmed by Fig. 2.6b, showing that the estimated slope decrease with the number of samples.

To tackle this issue, several solutions have been proposed. An empirical method [29] consists in estimating the connection between the estimated dimensionality and the true dimensionality based on synthetic datasets of known dimensionality and containing the same number of samples as the dataset of interest. This relation may then be inverted to obtain a corrected estimation of the intrinsic dimensionality from the biased estimate of the dimensionality. Another possibility is to consider the geodesic distances (approximated with shortest path distances on a nearestneighbours graph as detailed in [175]), instead of the ambient space distances [81]. As such, the manifold curvature is accounted for, allowing to extend the range of values $\sigma$ satisfying the scaling law to higher scales. This also enables to define a scale dependent dimensionality by considering the slope at each point of the curve $\log C(\sigma)$ against $\log \sigma$

## 统计代写|数据结构作业代写data structure代考|Nearest Neighbours Dimension

The two-NN estimator [62] considers points drawn from a locally uniform distribution and that the probability to find a point in a region of space is proportional to its volume. With those assumptions it may be shown that the ratio $\eta$ between the distance to the second and first neighbours of a point follows the Pareto distribution with parameter $\partial$. Compared with the correlation dimension, this approach allows to drop the hypothesis of a uniform density over the entire dataset.

The Paretot distribution corresponds to the probability density function $f$ and the associated cumulative density function $F$ (illustrated Fig. 2.7a):
$$f(\eta)=\partial \eta^{-\partial-1} 1_{[1,+\infty[}(\eta) \quad \text { and } \quad F(\eta)=\left(1-\eta^{-\partial}\right) 1_{[1,+\infty[}(\eta)$$

where $1_{[1,+\infty}$ is the indicator function of the set $[1,+\infty[$ and the ratio of distances is formally defined for each point $i$ as:
$$\eta_i \triangleq \frac{\Delta_i \tilde{v}_i(2)}{\Delta_i \tilde{v}_i(1)}$$
with $\tilde{v}_i$ the neighbourhood permutation for point $i$ (as defined in Sect.1.1.2). Considering the expression of the Cumulative Distribution Function (CDF) provided by Eq. (2.3), the parameter $\partial$ of that distribution may be estimated as the slope of $-\log (1-\widehat{F}(\eta))$ against $\log \eta$, where $\widehat{F}(\eta)$ is the empirical estimation of the CDF. To obtain this slope, the two-NN method fits a line to that curve by linear regression removing the $10 \%$ of the points with highest ratio $\eta$. This filtering allows to alleviate the effect of outliers which imply high variations of local density, thus violating the hypothesis. Figure $2.7 \mathrm{~b}$ illustrates this method for a dataset with strongly varying density.

# 数据结构代考

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

$$C(\sigma) \propto \sigma^{\partial} \Longleftrightarrow \log C(\sigma)=\partial \log \sigma+\text { const. }$$

## 统计代写|数据结构作业代写data structure代考|Nearest Neighbours Dimension

two-NN 估计器 [62] 考虑从局部均匀分布中提取的点，

(如图 2.7a 所示)：
$f(\eta)=\partial \eta^{-\partial-1} 1_{[1,+\infty[}(\eta) \quad$ and $\quad F(\eta)=\left(1-\eta^{-\partial}\right)$

$$\eta_i \triangleq \frac{\Delta_i \tilde{v}_i(2)}{\Delta_i \tilde{v}_i(1)}$$

## 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。