### 数学代写|随机过程统计代写Stochastic process statistics代考|STAT3061

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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|随机过程统计代写Stochastic process statistics代考|Genetic Algorithms

Genetic algorithms are among the first developed stochastic optimization methods. They were proposed by Holland (1975) and are a subtype of the so-called evolutionary algorithms. They emulate the evolution of a species, where the more capable individuals in a given population have higher chances to pass their genes to further generations. In mathematical terms, a solution $\bar{x}$ is an individual in the $\mathrm{GA}$, and a given set of solutions forms a population. One of the particularities of the GAs is that the solutions are represented in terms of chromosomes, i.e., chains containing the genetic information of each individual. That codified representation is known as genotype, whereas the “manifestation” of the genotype, i.e., the physical/mathematical system, is known as phenotype. Some examples of chromosomes are shown in Figure 3.1, where binary, integer, and alphabetic representations can be observed. The type of representation to be used depends on the problem to be solved. Each locus (position on the chain) can show different values (alleles), and each combination of locus and values represents different solutions for the objective function $\mathrm{f}(\bar{x})$. For engineering applications, codification with real numbers is more advisable because of the similarity between the genotype and the phenotype spaces (Gen and Cheng, 2000).

Once the problem has been codified, an initial solution is required to start the algorithm. The GAs function with a set of solutions in a simultaneous way, thus the initial point is indeed a population of solutions, each one with particular characteristics (i.e., different genetic information) that differentiate it from the others. The initial population is generated randomly. Then, it is necessary to evaluate the individuals on that first generation of solutions to determine which of them are good individuals and which are bad individuals. This classification is given by the so-called fitness function, which is strongly related to the objective function. Thus,for minimization, “good individuals” are those with a low value of $f(\bar{x})$. Then, a selection procedure is started. In this step, some of the individuals in the generation are selected to reproduce and give birth to the next generation, which is expected to have better characteristics than those of the previous generation. In general, the best individuals of the generation (i.e., those with the better values of $f(\bar{x})$ have higher probabilities of being selected for reproduction). Nevertheless, other individuals can also be selected to give genetic variability in the following generation, ensuring that a wider space in the feasible region is analyzed.

## 数学代写|随机过程统计代写Stochastic process statistics代考|Differential Evolution

Differential evolution is an evolutionary method. It shares some characteristics with the GAs. It was first proposed by Storn and Price (1997) as a strategy to solve the Chebyshev polynomial fitting problem (Shaoqiang et al., 2010). Similar to the GAs, the DE method functions with generations, where each generation comprises a number of parameter vectors representing a set of solutions. It is important to recall that DE uses a real-number representation at the parameter vectors. To start with the algorithm, an initial solution is

randomly generated. The solutions in the generation are evaluated through the fitness function, which is related to the objective function. To produce the next generation, an individual $\bar{x}{\text {SEL }}$ is randomly selected as candidate to be substituted through the crossover operation. Here, three other individuals $\left(\bar{x}{p 1}, \bar{x}{p 2}, \bar{x}{p 3}\right)$ are selected as parents, where one of the individuals will act as the main parent. Then, a fraction of the difference between the values of each variable in the other two parents is computed. Those values are added to the value of the respective variable in the main parent, which can be expressed as follows:
$$\left(\bar{x}{\mathrm{NEW}}\right)^{T}=\left(\bar{x}{p 1}\right)^{T}+F \times\left[\left(\bar{x}{p 2}\right)^{T}-\left(\bar{x}{p 3}\right)^{T}\right]$$
where $F$ is a randomly generated number and $F \in(0,1)$ (Abbass et al., 2001). The new individual $\bar{x}{\mathrm{NEW}}$ is then compared with the selected individual. If $\mathrm{f}\left(\bar{x}{\text {NEW }}\right)$ is better than $\mathrm{f}\left(\bar{x}{\text {SEL }}\right)$, then $\bar{x}{\text {SEL }}$ is replaced by $\bar{x}{\text {NEW }}$ in the population. Otherwise, $\bar{x}{\text {SEL. }}$ remains as an individual for the next generation. This operation takes place until the new generation has been completed. The procedure continues until the CC has been reached, which may imply a maximum number of generations. Figure $3.4$ shows a graphical representation of the DE method.

## 数学代写|随机过程统计代写Stochastic process statistics代考|Tabu Search

Tabu search is an optimization method proposed by Glover (1977, 1989). The method was originally developed to solve combinatorial problems related with scheduling and covering (Glover, 1989). One of the most important concepts, from which the method takes its name, is the so-called tabu list. The tabu list consists of a set containing some of the solutions that have been already proved. In general, the method starts with a single initial solution $\bar{x}$, setting the tabu list as empty. The initial solution is then perturbed several times to generate a number of new solutions $\bar{x}^{\prime}$, which is known as the neighborhood of $\bar{x}, N(\bar{x})$. This neighborhood can be obtained by applying a modification $m$ to the initial solution, i.e., $\bar{x}^{\prime}=\bar{x}^{\prime} \pm m$ (Fiechter, 1994). At the first steps of the algorithm, it is possible to move to a solution $f\left(\bar{x}^{\prime}\right)$, no matter if it is better than $\mathrm{f}(\bar{x})$ or not. A given number of the last obtained solutions is then added to the tabu list. The solutions in the previous iteration are then compared and the best one is selected as the new suboptimal. The new solution is then perturbed to generate another set of alternative solutions. For a next iteration, if a new solution is contained in the tabu list, it must be rejected and an alternative solution is proposed. As the iterations advance, the oldest components of the tabu list are deleted. The method continues until the stop criterion is reached, which may imply a maximum number of iterations. Figure $3.5$ presents a graphical representation of the tabu search method.

## 数学代写|随机过程统计代写Stochastic process statistics代考|Differential Evolution

(X¯ñ和在)吨=(X¯p1)吨+F×[(X¯p2)吨−(X¯p3)吨]

## 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。