数学代写|matlab代写|Introduction of Probability Theory

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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|matlab代写|Introduction of Probability Theory

数学代写|matlab代写|Beginnings of Probability Theory

Beginnings of Probability Theory. Probabilities represent the state of knowledge about physical phenomena by providing something more useful than “I don’t know” to questions involving uncertainty. One of the mysteries in the history of science is why it took so long for mathematicians to formalize a subject of such practical importance. The Romans were selling insurance and annuities long before expectancy and risk were concepts of serious mathematical interest. Much later, the Italians were issuing insurance policies against business risks in the early Renaissance, and the first known attempts at a theory of probabilities-for games of chance-occurred in that period. The Italian Girolamo Cardano ${ }^{5}$ (1501-1576) performed an accurate analysis of probabilities for games involving dice. He assumed that successive tosses of the dice were statistically independent events. He and the contemporary Indian writer Brahmagupta stated without proof that the accuracies of empirical statistics tend to improve with the number of trials. This would later be formalized as a law of large numbers.

More general treatments of probabilities were developed by Blaise Pascal (16231662), Pierre de Fermat (1601-1655), and Christiaan Huygens (1629-1695). Fermat’s work on combinations was taken up by Jakob (or James) Bernoulli (1654-1705), who is considered by some historians to be the founder of probability theory. He gave the first rigorous proof of the law of large numbers for repeated independent trials (now called Bernoulli trials). Thomas Bayes (1702-1761) derived his famous rule for statistical inference sometime after Bernoulli. Abraham de Moivre (1667-1754), Pierre Simon Marquis de Laplace (1749-1827), Adrien Marie Legendre (1752-1833), and Carl Friedrich Gauss (1777-1855) continued this development into the nineteenth century.

Between the early nineteenth century and the mid-twentieth century, the probabilities themselves began to take on more meaning as physically significant attributes. The idea that the laws of nature embrace random phenomena, and that these are treatable by probabilistic models began to emerge in the nineteenth century. The development and application of probabilistic models for the physical world expanded rapidly in that period. It even became an important part of sociology. The work of James Clerk Maxwell (1831-1879) in statistical mechanics established the probabilistic treatment of natural phenomena as a scientific (and successful) discipline.

An important figure in probability theory and the theory of random processes in the twentieth century was the Russian academician Andrei Nikolaeovich Kolmogorov (1903-1987). Starting around 1925, working with H. Ya. Khinchin and others, he reestablished the foundations of probability theory on measurement theory, which became the accepted mathematical basis of probability and random processes. Along with Norbert Wiener (1894-1964), he is credited with founding much of the theory of prediction, smoothing and filtering of Markov processes, and the general theory of ergodic processes. His was the first formal theory of optimal estimation for systems involving random processes.

数学代写|matlab代写|Wiener Filter

Norbert Wiener $(1894-1964)$ is one of the more famous prodigies of the early twentieth century. He was taught by his father until the age of 9 , when he entered high school. He finished high school at the age of 11 and completed his undergraduate degree in mathematics in three years at Tufts University. He then entered graduate school at Harvard University at the age of 14 and completed his doctorate degree in the philosophy of mathematics when he was 18 . He studied abroad and tried his hand at several jobs for six more years. Then, in 1919 , he obtained a teaching appointment at the Massachusetts Institute of Technology (MIT). He remained on the faculty at MIT for the rest of his life.

In the popular scientific press, Wiener is probably more famous for naming and promoting cybernetics than for developing the Wiener filter. Some of his greatest mathematical achievements were in generalized harmonic analysis, in which he extended the Fourier transform to functions of finite power. Previous results were restricted to functions of finite energy, which is an unreasonable constraint for signals on the real line. Another of his many achievements involving the generalized Fourier transform was proving that the transform of white noise is also white noise. ${ }^{6}$

数学代写|matlab代写|Kalman Filter

Rudolf Emil Kalman was born on May 19,1930 , in Budapest, the son of Otto and Ursula Kalman. The family emigrated from Hungary to the United States during World War II. In 1943, when the war in the Mediterranean was essentially over, they traveled through Turkey and Africa on an exodus that eventually brought them to Youngstown, Ohio, in 1944 . Rudolf attended Youngstown College there for three years before entering MIT.

Kalman received his bachelor’s and master’s degrees in electrical engineering at MIT in 1953 and 1954 , respectively. His graduate advisor was Ernst Adolph Guillemin, and his thesis topic was the behavior of solutions of second-order difference equations [114]. When he undertook the investigation, it was suspected that second-order difference equations might be modeled by something analogous to the describing functions used for second-order differential equations. Kalman discovered that their solutions were not at all like the solutions of differential equations. In fact, they were found to exhibit chaotic behavior.

In the fall of 1955 , after a year building a large analog control system for the E. I. DuPont Company, Kalman obtained an appointment as lecturer and graduate student at Columbia University. At that time, Columbia was well known for the work in control theory by John R. Ragazzini, Lotfi A. Zadeh, ${ }^{7}$ and others. Kalman taught at Columbia until he completed the Doctor of Science degree there in $1957 .$

For the next year, Kalman worked at the research laboratory of the International Business Machines Corporation in Poughkeepsie and for six years after that at the research center of the Glenn L. Martin company in Baltimore, the Research Institute for Advanced Studies (RIAS).

数学代写|matlab代写|Introduction of Probability Theory


数学代写|matlab代写|Beginnings of Probability Theory

概率论的开端。概率通过为涉及不确定性的问题提供比“我不知道”更有用的东西来代表关于物理现象的知识状态。科学史上的一个谜团是为什么数学家花了这么长时间才正式确定一个具有如此实际重要性的主题。早在期望和风险成为具有严肃数学意义的概念之前,罗马人就开始销售保险和年金。很久以后,意大利人在文艺复兴早期为商业风险发行了保险单,并且在那个时期发生了第一次已知的关于概率论的尝试——机会博弈。意大利吉罗拉莫·卡尔达诺5(1501-1576)对涉及骰子的游戏的概率进行了准确的分析。他假设连续掷骰子是统计上独立的事件。他和当代印度作家 Brahmagupta 在没有证据的情况下表示,经验统计的准确性往往会随着试验次数的增加而提高。这后来被形式化为大数定律。

Blaise Pascal (16231662)、Pierre de Fermat (1601-1655) 和 Christiaan Huygens (1629-1695) 开发了更一般的概率处理方法。费马关于组合的工​​作被雅各布(或詹姆斯)伯努利(1654-1705)接手,他被一些历史学家认为是概率论的创始人。他为重复独立试验(现在称为伯努利试验)给出了大数定律的第一个严格证明。托马斯·贝叶斯(Thomas Bayes,1702-1761)在伯努利之后的某个时间得出了他著名的统计推断规则。Abraham de Moivre (1667-1754)、Pierre Simon Marquis de Laplace (1749-1827)、Adrien Marie Legendre (1752-1833) 和 Carl Friedrich Gauss (1777-1855) 将这一发展延续到了 19 世纪。

在 19 世纪早期和 20 世纪中叶之间,概率本身开始具有更多意义,作为物理上重要的属性。自然规律包含随机现象,并且这些可以通过概率模型处理的想法在 19 世纪开始出现。在那个时期,物理世界的概率模型的开发和应用迅速扩大。它甚至成为社会学的重要组成部分。James Clerk Maxwell (1831-1879) 在统计力学方面的工作将自然现象的概率处理确立为一门科学(且成功的)学科。

二十世纪概率论和随机过程理论的重要人物是俄罗斯院士安德烈·尼古拉奥维奇·科尔莫哥洛夫(1903-1987)。从 1925 年左右开始,与 H. Ya 合作。Khinchin 等人,他在测量理论上重新建立了概率论的基础,这成为概率和随机过程公认的数学基础。与 Norbert Wiener (1894-1964) 一起,他被认为建立了马尔可夫过程的预测、平滑和滤波理论以及遍历过程的一般理论。他是第一个关于涉及随机过程的系统的最优估计的正式理论。

数学代写|matlab代写|Wiener Filter

诺伯特·维纳(1894−1964)是二十世纪初最著名的神童之一。直到 9 岁,他才进入高中,由父亲教他。他在 11 岁时读完高中,并在塔夫茨大学用三年时间完成了数学本科学位。14岁考入哈佛大学研究生院,18岁完成数学哲学博士学位。他在国外学习并尝试了六年多的工作。然后,在 1919 年,他获得了麻省理工学院 (MIT) 的任教职位。他的余生一直在麻省理工学院任教。


数学代写|matlab代写|Kalman Filter

鲁道夫·埃米尔·卡尔曼于 1930 年 5 月 19 日出生于布达佩斯,是奥托和乌苏拉·卡尔曼的儿子。二战期间,全家从匈牙利移民到美国。1943 年,当地中海战争基本结束时,他们穿越土耳其和非洲出走,最终于 1944 年将他们带到了俄亥俄州的扬斯敦。在进入麻省理工学院之前,鲁道夫在那里就读了三年的扬斯敦学院。

卡尔曼分别于 1953 年和 1954 年在麻省理工学院获得电气工程学士学位和硕士学位。他的研究生导师是 Ernst Adolph Guillemin,他的论文题目是二阶差分方程解的行为[114]。当他进行调查时,有人怀疑二阶微分方程可能由类似于用于二阶微分方程的描述函数的东西建模。卡尔曼发现他们的解根本不像微分方程的解。事实上,他们被发现表现出混乱的行为。

1955 年秋天,在为 EI DuPont 公司构建了一个大型模拟控制系统一年后,Kalman 获得了哥伦比亚大学讲师和研究生的任命。当时,哥伦比亚因 John R. Ragazzini、Lotfi A. Zadeh 在控制理论方面的工作而闻名,7和别的。卡尔曼在哥伦比亚大学任教,直到他在那里完成了理学博士学位1957.

接下来的一年,卡尔曼在位于波基普西的国际商业机器公司的研究实验室工作了六年,之后又在位于巴尔的摩的 Glenn L. Martin 公司的研究中心高级研究所 (RIAS) 工作了六年。

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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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