数学代写|随机过程统计代写Stochastic process statistics代考|STAT7004

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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|Simulated Annealing

Kirkpatrick et al. (1983) established that there is a similarity between the behavior of a system reaching thermal equilibrium and the performance of an optimization procedure. This was considered the basis for the creation of the simulated annealing approach, which emulates the phenomena of annealing in solids. In the annealing procedure, a solid is first heated at high temperature. At this stage, the energy of the system is quite high, and the atoms in the solid are randomly distributed. Then, the temperature is slowly reduced until a new equilibrium is reached. This procedure continues until the atoms are ordered in a crystalline structure, where the energy of the system is at its minimum. The solid treated with annealing is quite resistant. However, if the temperature was not gradually reduced, a thermal shock is induced, and the solid becomes fragile. In the simulated annealing method, a simulated temperature is used to control the algorithm. As in the physical phenomena, this temperature must be high to cause a random behavior. Once the initial simulated temperature is selected, an initial solution $\bar{x}$ is proposed, and the value of the objective function $f(\bar{x})$ is computed. Then, a second solution $\bar{x}^{\prime}$ is proposed, computing the value of $f\left(\bar{x}^{\prime}\right)$. The two solutions are compared and, if $f\left(\bar{x}^{\prime}\right)$ is better than $\mathrm{f}(\bar{x})$ (i.e., if, for a minimization, $\mathrm{f}\left(\bar{x}^{\prime}\right) \geq \mathrm{f}(\bar{x})$ ), then $\bar{x}^{\prime}$ is selected as the new solution. If $\mathrm{f}\left(\bar{x}^{\prime}\right)$ is worse than $\mathrm{f}(\bar{x})$, it is not immediately discarded. Instead, a probability of selection is computed using the Metropolis formula (Metropolis et al., 1953):
$$P(\Delta)=\exp \left(-\frac{\Delta}{T}\right)$$
where
$$\Delta=\mathrm{f}\left(\bar{x}^{\prime}\right)-\mathrm{f}(\bar{x})$$
If the probability of selection is higher than a random number $a$, then $\bar{x}^{\prime}$ is selected as the new solution. Otherwise, the method returns to the previous proposal, $\bar{x}$. This implies that, if a given solution is “bad,” it still has probabilities of being selected as a new solution. This is helpful to perform a search on all feasible regions, avoiding local optimum. The proposal and selection of new solutions continue until a stationary point is reached. Then, the temperature is decreased, and a new set of proposals is established. As the temperature decreases, the probability of selection is lower. Thus, when the algorithm advances, the “bad” solutions have less chances to be selected because the method is expected to be converging to the global optimum. The algorithm stops when the freezing temperature $\left(T_{\text {freeze }}\right)$ is reached, where the solution is stable and the same solution is selected among a certain number of proposals. Figure $3.6$ shows a graphical representation of the simulated annealing.

数学代写|随机过程统计代写Stochastic process statistics代考|Ant Colony Optimization

The ant colony optimization method was developed by Dorigo and Gambardella (1997) and is inspired by the behavior of ant colonies. In general, the ants indicate the way from their nest to a food source by depositing pheromones on the ground. When various ants are taking different paths, the path representing the shortest distance will have a higher concentration of pheromones, thus most of the ants will be attracted to follow that route and will increase the concentration of pheromones even more. In the end, all the ants nearby will follow the shortest path. The ant colony optimization method emulates this behavior. Two parameters are of importance to the algorithm: the pheromone value and the age of a solution. The algorithm starts with a randomly generated set of solutions. Then, a local search procedure starts, where a simulated ant selects a solution, in terms of the pheromone values. Then, in terms of the age of the current solution, a new solution is selected. If the fitness function of the new solution is better than that of the previous solution, the new solution is selected. Otherwise, the previous solution remains. In both cases, the values of age and pheromone are modified. Then, a global search takes place and the pheromone values are updated to consider the phenomena of pheromone evaporation. The procedure continues until the CC is achieved (Jayaraman et al., 2010). This method has been used for applications such as the optimization of project scheduling (Merkle et al., 2002), the optimization of water distribution systems (Maier et al., 2003), and the scheduling of batch pro-cesses (Jayaraman et al., 2010). Nevertheless, to the best of the authors’ knowlchemical processes.

数学代写|随机过程统计代写Stochastic process statistics代考|Particle Swarm Optimization

This optimization method is based on the social behavior of animal species, particularly that of human beings. The method was proposed by Kennedy and Eberhart (1995), and it is related to both artificial life and evolutionary programming. It has been developed from the observations and simulations of Reynolds (1987) not only about the movement patterns of flying animals (birds), land animals, and water animals (fishes), but also considering the abstractness that characterizes the human decision-making. In general, the algorithm functions with a set of entities known as particles. An initial population of particles is first generated, and a position and a velocity are assigned to each particle, where the position is given by a solution for the optimization problem. Then, the fitness function is evaluated for each position, and the best value is selected. Then, the position and the velocity of each particle are updated, which implies a movement of the particle in the direction of the best previous solution. The movements continue until a stop criterion is reached (Jarboui et al., 2010). This approach has been applied to the dynamic analysis of reactive systems (Ourique et al., 2002), the parameter estimation of a polypropylene reactor (Martinez Prata et al., 2009), and the optimal design of heat exchangers (Patel and Rao, 2010).

Δ=F(X¯′)−F(X¯)

有限元方法代写

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MATLAB代写

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