数学代写|随机过程统计代写Stochastic process statistics代考|STAT4061

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随机过程 用于表示在时间上发展的统计现象以及在处理这些现象时出现的理论模型,由于这些现象在许多领域都会遇到,因此这篇文章具有广泛的实际意义。

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我们提供的随机过程统计Stochastic process statistics及其相关学科的代写,服务范围广, 其中包括但不限于:

  • Statistical Inference 统计推断
  • Statistical Computing 统计计算
  • Advanced Probability Theory 高等概率论
  • 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|随机过程统计代写Stochastic process statistics代考|STAT4061

数学代写|随机过程统计代写Stochastic process statistics代考|Notion of Stochastic Processes

Loosely speaking, the mathematical description of a random phenomenon as it changes in time is a stochastic process. Since the last century there has been greater realisation that stochastic (or non-deterministic) models are more realistic than deterministic models in many situations. Observations taken at different time points rather than those taken at a fixed period of time began to draw the attention of scientists. The physicists and communication engineers played a leading role in the development of dynamic indeterminism. Many a phenomenon occurring in physical and life sciences are studied not only as a random phenomenon but also as one changing with time or space. Similar considerations are also made in other areas such as social sciences, economics and management sciences, and so on. The scope of applications of stochastic processes which are functions of time or space or both is ever increasing.

A stochastic process is a family of random variables $\left{X_{t}\right}$, where $t$ takes values in the index set $T$ (sometimes called a parameter set or a time set).
The values of $X_{t}$ are called the state space and will be denoted by $S$.
If $T$ is countable then the stochastic process is called a stochastic sequence (or discrete parameter stochastic process). If $S$ is countable then the stochastic process is called a discrete state (space) process.

If $S$ is a subset of the real line the stochastic process is called a real valued process.
If $T$ takes continuously uncountable number of values like $(0, \infty)$ or $(-\infty, \infty)$ the stochastic process is called a continuous time process. To emphasize its dependence on $t$ and sample point $w$, we shall denote the stochastic process by $X(t, w), t \in T, w \in \Omega$ i.e. for each $w \in \Omega, X_{t}=X(t$,
$w)$ is a function of $t$.
This graph is known as the “typical sample function” or “realization of the stochastic process” $X(t, w)$.

数学代写|随机过程统计代写Stochastic process statistics代考|Different Types of Stochastic Processes

Following are the most important types of stochastic processes we come across:

  1. Independent stochastic sequence (Discrete time process)
    $T={1,2,3, \ldots}$ and $\left{X_{t}, t \in T\right}$ are independent random variables.
  2. Renewal process (Discrete time process)
    Here $T={0,1,2,3, \ldots], S=[0, \infty]$.
    If $X_{n}$ are i.i.d. non-negative random variables and $S_{n}=X_{1}+\ldots+X_{n}$ then $\left{S_{n}\right}$ forms a discrete time (renewal process).
  3. Independent increment process (Continuous time process)
    $T=\left{t_{0}, \infty\right}$, where $t_{0}$ be any real number (+ or $-$ ). For every
    t_{0}<t_{1}<\ldots<t_{n}, t_{i} \in T, i=1,2, \ldots, n
    if $X_{t_{0}}, X_{t_{1}}-X_{t_{0}}, X_{t_{2}}-X_{t_{1}}, \ldots, X_{t_{n}}-X_{t_{n-1}}$ are independent for all possible choices of $(1.1)$, then the stochastic process $\left{X_{r}, t \in T\right}$ is called independent increment stochastic process.
  4. Markov process
    \text { If } \begin{aligned}
    P\left[X_{i_{n+1}} \in A \mid X_{i_{n}}=a_{n}\right.&\left., X_{t_{n-1}}=a_{n-1}, \ldots, X_{t_{0}}=a_{0}\right] \
    &=P\left[X_{t_{n+1}} \in A \mid X_{t_{n}}=a_{n}\right] \text { holds for all choices of } \
    t_{0}<t_{1}<t_{2} &<\ldots<t_{n+1}, t_{i} \in T \cdot i=0,1,2, \ldots, n+1
    and $A \in B$, the Borel field of the state space $S$, then $\left{X_{t}, t \in T\right}$ is called a Markov process.
  5. Martingale or fair game process
    E\left[X_{t_{n+1}} \mid X_{t_{n}}=a_{n}, X_{t_{n-1}}=a_{n-1}, \ldots, X_{t_{0}}=a_{0}\right]=a_{n}
    i.e. $E\left[X_{t_{n+1}} \mid X_{t_{n}}, \ldots, X_{t_{0}}\right]=X_{t_{n}}$ a.s. for all choices of the partition (1.1), then $\left{X_{t}, t \in T\right}$ is called a Martingale process.
  6. Stationary process
    If the joint distribution of $\left(X_{t_{1}+t_{h}}, \ldots, X_{t_{n}+h}\right)$ are the same for all $h>0$ and
    t_{1}<t_{2}<\ldots<t_{n}, t_{i} \in T, t_{i}+h \in T
    then $\left{X_{t}, t \in T\right}$ is called a stationary process (strictly stationary process).

数学代写|随机过程统计代写Stochastic process statistics代考|An Introduction to Stationary Processes

A stochastic process $\left{X_{t}, t \in T\right}$ with $E X_{t}^{2}<\infty$ for all $t \in T$ is called covariance stationary or stationary in the wide-sense or weakly stationary if its covariance

function $C_{s, t}=E\left(X_{1} X_{s}\right)$ depends only on the difference $|t-s|$ for all $t, s \in T$. Nute that in our definition we have taken a zero mean stochastic process.
Examples of stationary processes
(a) Electrical pulses in communication theory are often postulated to describe a stationary process. Of course, in any physical system there is a transient period at the beginning of a signal. Since typically this has a short duration compared to the signal length, a stationary model may be appropriate. In electrical communication theory, often both the electrical potential and the current are represented as complex variables. Here we may encounter complex-valued stationary processes.
(b) The spatial and/or planar distributions of stars of galaxies, plants and animals, are often stationary. Time parameter set $T$ might be Euclidean space, the surface of a sphere or the plane.

A stationary distribution may be postulated for the height of a wave and $T$ is taken to be a set of longitudes and latitudes, again two dimensional.
(c) Economic time series, such as unemployment, gross national product, national income etc., are often assumed to correspond to a stationary process, at least after some correction for long-term growth has been made.

数学代写|随机过程统计代写Stochastic process statistics代考|STAT4061


数学代写|随机过程统计代写Stochastic process statistics代考|Notion of Stochastic Processes


随机过程是一系列随机变量\left{X_{t}\right}\left{X_{t}\right}, 在哪里吨取索引集中的值吨(有时称为参数集或时间集)。


数学代写|随机过程统计代写Stochastic process statistics代考|Different Types of Stochastic Processes


  1. 独立随机序列(离散时间过程)
    吨=1,2,3,…和\left{X_{t}, t \in T\right}\left{X_{t}, t \in T\right}是独立的随机变量。
  2. 更新过程(离散时间过程)
    这里 $T={0,1,2,3, \ldots], S=[0, \infty].我FX_{n}一个r和一世.一世.d.n○n−n和G一个吨一世在和r一个nd○米在一个r一世一个bl和s一个ndS_{n}=X_{1}+\ldots+X_{n}吨H和n\left{S_{n}\right}$ 形成一个离散时间(更新过程)。
  3. 独立增量过程(Continuous time process)
    T=\left{t_{0}, \infty\right}T=\left{t_{0}, \infty\right}, 在哪里吨0是任何实数(+ 或−)。对于每一个
    如果X吨0,X吨1−X吨0,X吨2−X吨1,…,X吨n−X吨n−1对于所有可能的选择都是独立的(1.1),然后是随机过程\left{X_{r}, t \in T\right}\left{X_{r}, t \in T\right}称为独立增量随机过程。
  4. 马尔科夫过程
     如果 磷[X一世n+1∈一个∣X一世n=一个n,X吨n−1=一个n−1,…,X吨0=一个0] =磷[X吨n+1∈一个∣X吨n=一个n] 适用于所有选择  吨0<吨1<吨2<…<吨n+1,吨一世∈吨⋅一世=0,1,2,…,n+1
    和一个∈乙, 状态空间的 Borel 场小号, 然后\left{X_{t}, t \in T\right}\left{X_{t}, t \in T\right}称为马尔科夫过程。
  5. 鞅或公平博弈过程
    IE和[X吨n+1∣X吨n,…,X吨0]=X吨n至于分区(1.1)的所有选择,那么\left{X_{t}, t \in T\right}\left{X_{t}, t \in T\right}称为鞅过程。
  6. 平稳过程
    然后\left{X_{t}, t \in T\right}\left{X_{t}, t \in T\right}称为平稳过程(strictly平稳过程)。

数学代写|随机过程统计代写Stochastic process statistics代考|An Introduction to Stationary Processes

随机过程\left{X_{t}, t \in T\right}\left{X_{t}, t \in T\right}和和X吨2<∞对所有人吨∈吨称为协方差平稳或广义上的平稳或弱平稳,如果它的协方差

功能Cs,吨=和(X1Xs)仅取决于差异|吨−s|对所有人吨,s∈吨. 请注意,在我们的定义中,我们采用了零均值随机过程。
(a) 通信理论中的电脉冲通常被假定为描述静止过程。当然,在任何物理系统中,信号开始时都有一个瞬态周期。由于与信号长度相比,这通常具有较短的持续时间,因此固定模型可能是合适的。在电通信理论中,通常电势和电流都表示为复变量。在这里,我们可能会遇到复值平稳过程。
(b) 星系、植物和动物的恒星的空间和/或平面分布通常是静止的。时间参数集吨可能是欧几里得空间、球面或平面。

(c) 经济时间序列,例如失业、国民生产总值、国民收入等,通常被假定为对应于一个平稳的过程,至少在对长期增长进行了一些修正之后是这样。

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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。