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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

## 统计代写|随机控制代写Stochastic Control代考|Standard Particle Swarm Optimization

Dr. Kennedy and Dr. Eberhart proposed the PSO algorithm in 1995 (Kennedy, \& Eberhart, 1995), which derived from the behavior research of flock foraging, and the research found out that the PSO theory can be applied to the function optimization, then it was developed into a universal optimization algorithm gradually. As the concept of PSO is simple and easy to implement, at the same time, it has profound intelligence background, the PSO algorithm attracted extensive attention when it was first proposed and has become a hot topic of research. The search of PSO spreads all over the solution space, so the global optimal solution can be easily got, what is more, the PSO requires neither continuity nor differentiability for the target function, even doesn’t require the format of explicit function, the only requirement is that the problem should be computable. In order to realize the PSO algorithm, a swarm of random particles should be initialized at first, and then get the optimal solution through iteration calculation. For each iteration calculation, the particles found out their individual optimal value of pbest through tracking themselves and the global optimal value of gbest through tracking the whole swarm. The following formula is used to update the velocity and position.
$$\begin{gathered} v_{u d}^{k+1}=w \cdot v_{u d}^k+c_1 \cdot \text { vond } \cdot\left(p_{u d}-x_{d d}^k\right)+c_2 \cdot \operatorname{nnd} \cdot\left(p_{g i}-x_{d d}^k\right) \ x_{i d}^{k+1}=x_{i d}^k+v_{i d}^{k+1} \end{gathered}$$
In the formula (1) and (2), $i=1,2, \ldots, m, m$ refers to the total number of the particles in the swarm; $d=1,2, \ldots, \mathrm{n}, d$ refers to the dimension of the particle; $v_{i d}^k$ is the No. $d$ dimension component of the flight velocity vector of iteration particle $i$ of the No. $k$ times. $x_{i d}^k$ is the No.

## 统计代写|随机控制代写Stochastic Control代考|Improved Particle Swarm Optimization Algorithm

The PSO algorithm is simple, but research shows that, when the particle swarm is over concentrated, the global search capability of particle swarm will decline and the algorithm is easy to fall into local minimum. If the aggregation degree of the particle swarm can be controlled effectively, the capability of the particle swarm optimizing to the global minimum will be improved. According to the formula (1), the velocity $v$ of the particle will become smaller gradually as the particles move together in the direction of the global optimal location gbest. Supposed that both the social and cognitive parts of the velocity become smaller, the velocity of the particles will not become larger, when both of them are close to zero, as $w<1$, the velocity will be rapidly reduced to 0 , which leads to the loss of the space exploration ability. When the initial velocity of the particle is not equal to zero, the particles will move away from the global optimal location of gbest by inertial movement. When the velocity is close to zero, all the particles will move closer to the location of gbest and stop movement. Actually, the PSO algorithm does not guarantee convergence to the global optimal location, but to the optimal location gbest of the swarm(LU Zhensu \& HOU Zhirong, 2004). Furthermore, as shown in the formula (2), the value of the particle velocity also represents the distance of particle relative to the optimal location gbest. When the particles become farther from the gbest, the particle velocity will be greater, on the contrary, when the particles become closer to the gbest, the velocity will be smaller gradually. Therefore, as shown in the formula (1), by means of the extreme variation of the swarm individual, the velocity of the particles can be controlled in order to prevent the particles from gathering at the location gbest quickly, which can control the swarm diversity effectively. Known from the formula (1), when the variability measures are taken, both the particle activity and increases the global search capability of particle swarm to a large extent. The improved PSO(MPSO) is carried out on the basis of standard PSO, which increases the The improved $\mathrm{PSO}$ (MPSO) is carried out on the basis of standard PSO, which increases the variation operation of optimal location for the swarm individual. The method includes the following steps:
(1) Initializing the position and velocity of particle swarm at random;
(2) The value pbest of the particle is set as the current value, and the gbest for the optimal particle location of the initial swarm ;
(3) Determining whether to meet the convergence criteria or not, if satisfied, turn to step 6; Otherwise, turn to step 4;
(4) In accordance with the formula (1) and (2), updating the location and velocity of the particles, and determining the current location of pbest and gbest :
(5) Determining whether to meet the convergence criteria or not, if satisfied, turn to step 6; Otherwise, carrying out the optimal location variation operation of swarm individuals according to the formula (3), then turn to step 4 ;

## 统计代写|随机控制代写Stochastic Control代考|标准粒子群优化

Dr。Kennedy和Eberhart博士在1995年提出了PSO算法(Kennedy， ＆Eberhart, 1995)，该算法起源于对羊群觅食行为的研究，研究发现粒子群优化理论可以应用于函数优化，并逐渐发展为一种通用的优化算法。由于粒子群算法的概念简单、易于实现，同时又具有深厚的智能背景，因此它一提出就受到了广泛的关注，成为了研究的热点。粒子群算法的搜索分布在整个解空间中，因此可以很容易地得到全局最优解，而且粒子群算法对目标函数既不要求连续性，也不要求可微性，甚至不要求显式函数的格式，只要求问题是可计算的。为了实现粒子群算法，首先要初始化一群随机粒子，然后通过迭代计算得到最优解。在每次迭代计算中，粒子通过对自身的跟踪得到各自的pbest最优值，通过对整个群的跟踪得到gbest的全局最优值。下面的公式用于更新速度和位置。
$$\begin{gathered} v_{u d}^{k+1}=w \cdot v_{u d}^k+c_1 \cdot \text { vond } \cdot\left(p_{u d}-x_{d d}^k\right)+c_2 \cdot \operatorname{nnd} \cdot\left(p_{g i}-x_{d d}^k\right) \ x_{i d}^{k+1}=x_{i d}^k+v_{i d}^{k+1} \end{gathered}$$

## 统计代写|随机控制代写Stochastic Control代考|改进的粒子群优化算法

PSO算法简单，但研究表明，当粒子群过于集中时，粒子群的全局搜索能力会下降，算法容易陷入局部极小值。如果能有效地控制粒子群的聚集程度，将提高粒子群优化到全局最小的能力。根据式(1)，速度 $v$ 当粒子向全局最优位置gbest方向运动时，粒子的值会逐渐变小。假设速度的社会和认知部分都变小了，当它们都接近零时，粒子的速度不会变大，如 $w<1$，速度将迅速降至0，从而导致空间探索能力的丧失。当粒子的初速度不等于零时，粒子会通过惯性运动远离全局最优位置gbest。当速度接近于零时，所有的粒子都将向gbest的位置靠近并停止运动。实际上，PSO算法并不能保证收敛到全局最优位置，而是收敛到群的最优位置gbest (LU Zhensu ＆侯志荣，2004)。此外，如式(2)所示，粒子速度的值也表示粒子相对于最佳位置gbest的距离。当粒子离最佳点越远时，粒子速度越大，反之，当粒子离最佳点越近时，粒子速度逐渐变小。因此，如式(1)所示，通过群体个体的极值变化，可以控制粒子的速度，以防止粒子快速聚集在最佳位置，从而有效地控制群体多样性。由式(1)可知，当采取变率措施时，粒子的活跃性和粒子群的全局搜索能力都有很大的提高。改进的粒子群算法是在标准粒子群算法的基础上进行的，提高了改进的粒子群算法 $\mathrm{PSO}$ (MPSO)是在标准粒子群算法的基础上进行的，增加了群体个体最优位置的变异操作。该方法包括以下步骤:
(1)随机初始化粒子群的位置和速度;
(2)将粒子的值pbest设为当前值，gbest为初始粒子群的最优粒子位置;
(3)确定是否满足收敛条件，如果满足则转到步骤6;
(4)根据公式(1)和(2)，更新粒子的位置和速度，确定pbest和gbest的当前位置:
(5)确定是否满足收敛条件，如果满足则转到步骤6;否则，根据式(3)进行群体个体最优位置变异操作，则转到步骤4;

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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。

assignmentutor™您的专属作业导师
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