### 数学代写|理论计算机代写theoretical computer science代考|Margin-Based Semi-supervised Learning

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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

## 数学代写|理论计算机代写theoretical computer science代考|Review of Concepts Related to the Density Peaks and the Apollonius Circle

In this section, the concepts of the Apollonius circle are discussed. We propose the neighborhood structure of the Apollonius circle to label the unlabeled data.
Apollonius circle is one of the most famous problems in geometry [18]. The goal is to find accurate neighborhoods. In a set of points, there is no information about the relationship between the points and some databases may not even contain topological information. It is more accurate than the Gabriel graph [25] and neighborhood graph in the neighborhood construction. It evaluates important neighborhood entirely. In this method, the first step is to find high density points and the second step is to build neighborhood groups with the Apollonius circle. The third step is analyzing points outside the radius of the Apollonian circles or points within the region.

## 数学代写|理论计算机代写theoretical computer science代考|Finding High Density Points

Rodriguez and Laio [20] presented the algorithm to find high density points (DPC). The high density points are found by this method and then stored in an array.

The points are shown by the vector $\mathbf{M}=\left(M_{1 i}, M_{2 i}, \ldots, M_{m i}\right)$ where $m$ is the number of attributes, also $N_{M_{i}}$ shows $k$ nearest neighbors of $M_{i} \cdot d\left(M_{i}, M_{j}\right)$ is the Euclidean distance between $M_{i}$ and $M_{j}$. The percent of the neighborhood is shown by $p$. The number of neighbors is obtained by $r=p \times n$, where $n$ is the number of data points. The local density $\rho_{i}$ is then defined as:
$$\begin{gathered} \rho_{i}=\exp \left(-\left(\frac{1}{r} \sum_{M_{j} \in N\left(M_{i}\right)} d\left(M_{i}, M_{J}\right)^{2}\right)\right) \ d\left(M_{i}, M_{j}\right)=\left|M_{i}-M_{j}\right| \end{gathered}$$
where $\delta_{i}$ is the minimum distance between $M_{i}$ and any other sample with higher density than $p_{i}$, which is define as below:
$$\delta_{i}= \begin{cases}\min {\rho{i}<\rho_{j}}\left{d\left(M_{i}, M_{j}\right)\right}, & \text { if } \exists j \quad \rho_{i}<\rho_{j} \ \max {j}\left{d\left(M{i}, M_{j}\right)\right}, & \text { otherwise }\end{cases}$$
Peaks (high density points) are obtained using the score function. The points that have the highest score are considered as peak point.
$$\operatorname{score}\left(M_{i}\right)=\delta_{i} \times \rho_{i}$$
In this article, number of peaks are selected based on the number of classes. Peaks are selected from the labeled set. We assign the label of peaks to the unlabeled data based on neighborhood radius of peaks. Neighboring groups are found by the Apollonius circle.

## 数学代写|理论计算机代写theoretical computer science代考|Neighborhood Groups with the Apollonius Circle

The Apollonius circle is the geometric location of the points on the Euclidean plane which have a specified ratio of distances to two fixed points $\mathrm{A}$ and $\mathrm{B}$, this ratio is called $K$ [17]. Apollonius circle can be seen in Fig. 2 .
$$K=d_{1} / d_{2} .$$
The Apollonius circle based on $A$ and $B$ is defined as:
$$C_{A B}=\left{\begin{array}{lll} C_{A} & \text { if } & K<1 \\ C_{B} & \text { if } & K>1 \ C_{\text {inf }} & \text { if } & K=1 \end{array}\right.$$
After finding the high density points, we sort these points. The peak points are indicated by $P=\left(P_{1}, P_{2}, \ldots, P_{m}\right)$ which are arranged in pairs $\left(P_{t}, P_{t+1}\right), t \in{1,2, \ldots, m-1}$, the data points are denoted by $M=$ $\left{M_{i} \mid i \in{1,2, \ldots, n-m}, M_{i} \notin P\right}$. In the next step, data points far from the peak points are determined by the formula 7 . Finally the distance of the furthest point from the peak points is calculated by the formula 8 .
$$\begin{gathered} F d_{P_{t}}=\max \left{d\left(P_{t}, M_{i}\right) \mid M_{i} \in M \text { and } d\left(P_{t}, M_{i}\right)<d\left(P_{t}, P_{t+1}\right)\right. \ \text { and } \left.d\left(P_{t}, M_{i}\right)<\min {l=1, l \in P} d\left(P{l}, M_{i}\right) \text { s.t. } t \neq l\right} \ F P_{t}=\left{M_{i} \mid d\left(P_{t}, M_{i}\right)=F d_{t}\right} . \end{gathered}$$
The points between the peak point and the far point are inside the Apollonius circle, circle [18].

The above concepts are used to label the unlabeled samples confidently. In our proposed algorithm, label of the peak points are assigned to the unlabeled example which are inside the Apollonius circle. The steps are shown in Fig. 3 .

## 数学代写|理论计算机代写theoretical computer science代考|Finding High Density Points

Rodriguez 和 Laio [20] 提出了寻找高密度点 (DPC) 的算法。通过这种方法找到高密度点，然后将其存储在数组中。

ρ一世=经验⁡(−(1r∑米j∈ñ(米一世)d(米一世,米Ĵ)2)) d(米一世,米j)=|米一世−米j|

\delta_{i}= \begin{cases}\min {\rho{i}<\rho_{j}}\left{d\left(M_{i}, M_{j}\right)\right}, & \text { if } \exists j \quad \rho_{i}<\rho_{j} \ \max {j}\left{d\left(M{i}, M_{j}\right)\right}， & \text { 否则 }\end{cases}\delta_{i}= \begin{cases}\min {\rho{i}<\rho_{j}}\left{d\left(M_{i}, M_{j}\right)\right}, & \text { if } \exists j \quad \rho_{i}<\rho_{j} \ \max {j}\left{d\left(M{i}, M_{j}\right)\right}， & \text { 否则 }\end{cases}

## 数学代写|理论计算机代写theoretical computer science代考|Neighborhood Groups with the Apollonius Circle

ķ=d1/d2.

$$C_{AB}=\left{C一种 如果 ķ<1C乙 如果 ķ>1 C信息 如果 ķ=1\对。 找到高密度点后，我们对这些点进行排序。峰值点由 P=\left(P_{1}, P_{2}, \ldots, P_{m}\right) 表示，它们成对排列 \left(P_{t}, P_{t +1}\right), t \in{1,2, \ldots, m-1}，数据点记为M= \left{M_{i} \mid i \in{1, 2, \ldots, nm}, M_{i} \notin P\right}。在下一步中，远离峰值点的数据点由公式7确定。最后通过公式8计算最远点到峰值点的距离。找到高密度点后，我们对这些点进行排序。峰值点由 P=\left(P_{1}, P_{2}, \ldots, P_{m}\right) 表示，它们成对排列 \left(P_{t}, P_{t +1}\right), t \in{1,2, \ldots, m-1}，数据点记为M= \left{M_{i} \mid i \in{1, 2, \ldots, nm}, M_{i} \notin P\right}。在下一步中，远离峰值点的数据点由公式7确定。最后通过公式8计算最远点到峰值点的距离。 \begin{gathered} F d_{P_{t}}=\max \left{d\left(P_{t}, M_{i}\right) \mid M_{i} \in M \text { and } d \left(P_{t}, M_{i}\right)<d\left(P_{t}, P_{t+1}\right)\right。\ \text { 和 } \left.d\left(P_{t}, M_{i}\right)<\min {l=1, l \in P} d\left(P{l}, M_{i }\right) \text { st } t \neq l\right} \F P_{t}=\left{M_{i} \mid d\left(P_{t}, M_{i}\right)=F d_{t}\right} 。\结束{聚集}\begin{gathered} F d_{P_{t}}=\max \left{d\left(P_{t}, M_{i}\right) \mid M_{i} \in M \text { and } d \left(P_{t}, M_{i}\right)<d\left(P_{t}, P_{t+1}\right)\right。\ \text { 和 } \left.d\left(P_{t}, M_{i}\right)<\min {l=1, l \in P} d\left(P{l}, M_{i }\right) \text { st } t \neq l\right} \F P_{t}=\left{M_{i} \mid d\left(P_{t}, M_{i}\right)=F d_{t}\right} 。\结束{聚集}$$

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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。