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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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

## 机器学习代写|流形学习代写manifold data learning代考|LEM Results

Figures 2.5-2.7 show the results after using LEM for different values of $k$. As the value of $k$ increases from 1 to higher values we notice the spreading of the embedded data. The bottom subplot shows the nearest neighbor graph with $k=1$ as shown in Figure 2.7. The right plot shows the embedding of the graph. It is interesting to observe how the embedded data loses its local neighborhood information. The embedding practically happens along the second principal eigenvector (the first being Zero Vector). As the value of $k$ is increased to 2 , we observe that embedding happens along the second and third principal axes. See Figure 2.7. For $k=1$ the graph is highly disconnected and for $k=2$ the graphs has much less isolated pieces of graphs. One interesting thing to observe is that as the connectivity of the graph increases the low-dimensional representation begins to preserve the local information.

The graph with $k=2$ and its embedding is shown in Figure $2.8$. Increasing the neighborhood information to 2 neighbors is still not able to represent the continuity of the original manifold. Figure $2.7$ shows the graph with $k=3$ and its embedding. Increasing the neighborhood information to 3 neighbors better represents the continuity of the original manifold. Figure $2.5$ shows the graph with $k=5$ and its embedding. Increasing the neighborhood information to 5 neighbors better represents the continuity of the original manifold. Similar results are obtained by increasing the the number of neighbors, however, it should be noted that when the number of neighbors is very high then the graph starts to get influenced by ambient neigbhors.

We see similar results for the face images. The three plots in Figure $2.6$ show the embedding results obtained using LEM when the neighborhood graphs are created using $k=1, k=2$, and $k=5$. The top and the middle plot validate the limitation of LEM for $k=1$ and $k=2$. As expected, for $k=5$ there is continuity of facial images in the embedded space.

## 机器学习代写|流形学习代写manifold data learning代考|Bibliographical and Historical Remarks

Dimensionality reduction is an important research area in data analysis with an extensive research literature. Both linear and non-linear methods exist, and each category has both supervised and unsupervised versions. In this section we will briefly mention some of the salient works that have been proposed in the area of locally preserving manifold learning: see $[8]$ for a broader survey.

Lee and Seung [12] showed that many high dimensional data such as a series of related images, video frames, etc. lie on a much lower-dimensional manifold instead of being scattered throughout the feature space. This particular observation has motivated researchers to develop dimension reduction algorithms that try to learn an embedded manifold in a high-dimensional space.

ISOMAP [14] learns the manifold by exploring geodesic distances. In fact the algorithm tries to preserve the geometry of the data on the manifold by noting the points in the neighborhood of each point. The algorithm is defined as such:

1. Form a neighborhood graph $\mathrm{G}$ for the dataset, based, for instance, on the $K$ nearest neighbors of each point $x_{i}$.
2. For every pair of nodes in the graph, compute the shortest path, using Dijkstras algorithm, as an estimate of intrinsic distance on the data manifold. The weights of edges of the graphs are computed based on the Euclidean distance measure.
3. Classical Multi-Dimensional Scaling algorithm is computed using these pairwise distances to find a lower dimensional embedding $y_{i}$.

Bernstein et al. [22] have described the convergence properties of the estimation procedure for the intrinsic distances. For large and dense data sets, computation of pairwise distances is time consuming, and moreover the calculation of eigenvalues can be computationally intensive for large data sets. Such constraints have motivated researchers to find simpler variations of the Isomap algorithm. One such algorithm uses subsampled data called landmarks. Firstly, it calculates Isomap for random points called landmarks and between those landmarks a simple triangulation algorithm is applied.

Locally Linear Embedding (LLE) is an unsupervised learning method based on global and local optimization [11]. It is is similar to Isomap in the sense that it generates a graphical representation of the data set. However, it is different from Isomap as it only attempts to preserve local structures of the data. Because of the locality property used in LLE, the algorithm allows for successful embedding of nonconvex manifolds. An important point to be noted is that LLE creates the local properties of a manifold using the linear combinations of $k$ nearest neighbors of the data $x_{i}$. LLE attempts to create a local regression like model and thereby tries to fit a hyperplane through the data point $x_{i}$. This appears to be reasonable for smooth manifolds where the nearest neighbors align themselves well in a linear space. For very non-smooth or noisy data sets, LLE does not perform well. It has been noted that LLE preserves the reconstruction weights in the space of lower dimensionality, as the reconstruction weights of a data point are invariant to linear transformational operations like translation, rotation, etc.

# 流形学习代写

## 机器学习代写|流形学习代写manifold data learning代考|Bibliographical and Historical Remarks

Lee 和 Seung [12] 表明，许多高维数据（例如一系列相关图像、视频帧等）位于低得多的流形上，而不是分散在整个特征空间中。这一特殊观察促使研究人员开发降维算法，试图在高维空间中学习嵌入式流形。

ISOMAP [14] 通过探索测地距离来学习流形。事实上，该算法试图通过注意每个点附近的点来保留流形上数据的几何形状。算法定义如下：

1. 形成邻域图G对于数据集，例如，基于ķ每个点的最近邻X一世.
2. 对于图中的每一对节点，使用 Dijkstras 算法计算最短路径，作为数据流形上内在距离的估计。图的边权重是根据欧几里得距离度量计算的。
3. 使用这些成对距离计算经典的多维缩放算法以找到较低维的嵌入是一世.

## 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。

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