数学代写|模拟和蒙特卡洛方法作业代写simulation and monte carlo method代考|AEM4060

Doug I. Jones

Doug I. Jones

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  • Statistical Inference 统计推断
  • Statistical Computing 统计计算
  • Advanced Probability Theory 高等楖率论
  • 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
数学代写|模拟和蒙特卡洛方法作业代写simulation and monte carlo method代考|AEM4060

数学代写|模拟和蒙特卡洛方法作业代写simulation and monte carlo method代考|Duality

The aim of duality is to provide an alternative formulation of an optimization problem that is often more computationally efficient or has some theoretical significance (see [7], page 219). The original problem (1.66) is referred to as the primal problem, whereas the reformulated problem, based on Lagrange multipliers, is referred to as the dual problem. Duality theory is most relevant to convex optimization problems. It is well known that if the primal optimization problem is (strictly) convex, then the dual problem is (strictly) concave and has a (unique) solution from which the optimal (unique) primal solution can be deduced.

Definition 1.16.5 (Lagrange Dual Program) The Lagrange dual program of the primal program (1.66), is
$\begin{aligned} \max {\boldsymbol{\alpha}, \boldsymbol{\beta}} & \mathcal{L}^(\boldsymbol{\alpha}, \boldsymbol{\beta}) \ \text { subject to: } & \boldsymbol{\alpha} \geqslant 0, \end{aligned}$ where $\mathcal{L}^$ is the Lagrange dual function:
\mathcal{L}^(\boldsymbol{\alpha}, \boldsymbol{\beta})=\inf {\mathbf{x} \in \mathscr{X}} \mathcal{L}(\mathbf{x}, \boldsymbol{\alpha}, \boldsymbol{\beta}) .
It is not difficult to see that if $f^$ is the minimal value of the primal problem, then $\mathcal{L}^(\boldsymbol{\alpha}, \boldsymbol{\beta}) \leqslant f^$ for any $\boldsymbol{\alpha} \geqslant 0$ and any $\boldsymbol{\beta}$. This property is called weak duality. The Lagrangian dual program thus determines the best lower bound on $f^$. If $d^$ is the optimal value for the dual problem, then $d^$. The difference $f^-d^$ is called the duality gap.

The duality gap is extremely useful for providing lower bounds for the solutions of primal problems that may be impossible to solve directly. It is important to note that for linearly constrained problems, if the primal is infeasible (does not have a solution satisfying the constraints), then the dual is either infeasible or unbounded. Conversely, if the dual is infeasible, then the primal has no solution. Of crucial importance is the strong duality theorem, which states that for convex programs (1.66) with linear constrained functions $h_i$ and $g_i$ the duality gap is zero, and any $\mathrm{x}^$ and $\left(\boldsymbol{\alpha}^, \boldsymbol{\beta}^*\right)$ satisfying the KKT conditions are (global) solutions to the primal and dual programs, respectively. In particular, this holds for linear and convex quadratic programs (note that not all quadratic programs are convex).

For a convex primal program with $C^1$ objective and constraint functions, the Lagrangian dual function (1.70) can be obtained by simply setting the gradient (with respect to $\mathbf{x})$ of the Lagrangian $\mathcal{L}(\mathbf{x}, \boldsymbol{\alpha}, \boldsymbol{\beta})$ to zero. One can further simplify the dual program by substituting into the Lagrangian the relations between the variables thus obtained.

数学代写|模拟和蒙特卡洛方法作业代写simulation and monte carlo method代考|RANDOM NUMBER GENERATION

In the early days of simulation, randomness was generated by manual techniques, such as coin flipping, dice rolling, card shuffling, and roulette spinning. Later on, physical devices, such as noise diodes and Geiger counters, were attached to computers for the same purpose. ‘I’he prevailing belief held that only mechanical or electronic devices could produce truly random sequences. Although mechanical devices are still widely used in gambling and lotteries, these methods were abandoned by the computer-simulation community for several reasons: (1) mechanical methods were too slow for general use, (2) the generated sequences could not be reproduced, and (3) it was found that the generated numbers exhibit both bias and dependence. Although certain modern physical generation methods are fast and would pass most statistical tests for randomness (e.g., those based on the universal background radiation or the noise of a PC chip), their main drawback remains their lack of repeatability. Most of today’s random number generators are not based on physical devices but on simple algorithms that can be easily implemented on a computer. They are fast, require little storage space, and can readily reproduce a given sequence of random numbers. Importantly, a good random number generator captures all the important statistical properties of true random sequences, even though the sequence is generated by a deterministic algorithm. For this reason these generators are sometimes called pseudorandom.

Most computer languages already contain a built-in pseudorandom number generator. The user is typically requested only to input the initial seed, $X_0$, and upon invocation the random number generator produces a sequence of independent, uniform $(0,1)$ random variables. We therefore assume in this book the availability of such a “black box” that is capable of producing a stream of pseudorandom numbers. In Matlab, for example, this is provided by the rand function. The “seed” of the random number generator, which can be set by the rng function, determines which random stream is used, and this is very useful for testing purposes.

数学代写|模拟和蒙特卡洛方法作业代写simulation and monte carlo method代考|AEM4060




$\begin{aligned} \max {\boldsymbol{\alpha}, \boldsymbol{\beta}} & \mathcal{L}^(\boldsymbol{\alpha}, \boldsymbol{\beta}) \ \text { subject to: } & \boldsymbol{\alpha} \geqslant 0, \end{aligned}$,其中$\mathcal{L}^$是拉格朗日对偶函数:
\mathcal{L}^(\boldsymbol{\alpha}, \boldsymbol{\beta})=\inf {\mathbf{x} \in \mathscr{X}} \mathcal{L}(\mathbf{x}, \boldsymbol{\alpha}, \boldsymbol{\beta}) .
不难看出,如果$f^$是原问题的最小值,那么$\mathcal{L}^(\boldsymbol{\alpha}, \boldsymbol{\beta}) \leqslant f^$对于任意$\boldsymbol{\alpha} \geqslant 0$和任意$\boldsymbol{\beta}$。这个性质叫做弱对偶性。拉格朗日对偶程序确定了$f^$的最佳下界。如果$d^$是对偶问题的最优值,则$d^$。差异$f^-d^$被称为对偶差。

对偶间隙对于为原始问题的解提供下界是非常有用的,这些问题可能无法直接求解。值得注意的是,对于线性约束问题,如果原矩阵是不可行的(没有满足约束条件的解),那么对偶矩阵要么是不可行的,要么是无界的。相反,如果对偶不可行,则原子无解。重要的是强对偶定理,它表明对于凸规划(1.66)具有线性约束函数$h_i$和$g_i$的对偶差距为零,并且任何满足KKT条件的$\mathrm{x}^$和$\left(\boldsymbol{\alpha}^, \boldsymbol{\beta}^*\right)$分别是原规划和对偶规划的(全局)解。特别地,这对线性和凸二次规划都成立(注意不是所有的二次规划都是凸的)

对于具有$C^1$目标和约束函数的凸原程序,只需将拉格朗日函数$\mathcal{L}(\mathbf{x}, \boldsymbol{\alpha}, \boldsymbol{\beta})$的梯度(相对于$\mathbf{x})$的梯度)设为零,即可得到拉格朗日对偶函数(1.70)。我们可以将由此得到的变量之间的关系代入拉格朗日量,从而进一步简化对偶程序

数学代写|模拟和蒙特卡洛方法作业代写模拟和蒙特卡罗方法代考|RANDOM NUMBER GENERATION




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术语 广义线性模型(GLM)通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归,以及方差分析和方差分析(仅含固定效应)。



有限元是一种通用的数值方法,用于解决两个或三个空间变量的偏微分方程(即一些边界值问题)。为了解决一个问题,有限元将一个大系统细分为更小、更简单的部分,称为有限元。这是通过在空间维度上的特定空间离散化来实现的,它是通过构建对象的网格来实现的:用于求解的数值域,它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统,以模拟整个问题。然后,有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。





随机过程,是依赖于参数的一组随机变量的全体,参数通常是时间。 随机变量是随机现象的数量表现,其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值(如1秒,5分钟,12小时,7天,1年),因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中,往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录,以得到其自身发展的规律。


多元回归分析渐进(Multiple Regression Analysis Asymptotics)属于计量经济学领域,主要是一种数学上的统计分析方法,可以分析复杂情况下各影响因素的数学关系,在自然科学、社会和经济学等多个领域内应用广泛。


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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