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# 统计代写|时间序列分析作业代写time series analysis代考|Stationary Stochastic Processes

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## 统计代写|时间序列分析作业代写time series analysis代考|Stochastic Processes

Consider the following experiment (see Figure 22): we hook up a resistor to an oscilloscope in such a way that we can examine the voltage variations across the resistor as a function of time. Every time we press a “reset” button on the oscilloscope, it displays the voltage variations for the one-second interval following the reset. Since the voltage variations are presumably caused by such factors as small temperature variations in the resistor, each time we press the reset button, we will observe a different display on the oscilloscope. Owing to the complexity of the factors that influence the display, there is no way that we can use the laws of physics to predict what will appear on the oscilloscope. However, if we repeat this experiment over and over, we soon see that, although we view a different display each time we press the reset button, the displays resemble each other: there is a characteristic “bumpiness” shared by all the displays.

We can model this experiment by considering a large box in which we have placed pictures of all the oscilloscope displays that we could possibly observe. Pushing the reset button corresponds to reaching into the box and choosing “at random” one of the pictures. Loosely speaking, we call the box of all possible pictures together with the mechanism by which we select the pictures a stochastic process. The one particular picture we actually draw out at a given time is called a realization of the stochastic process. The collection of all possible realizations is called the ensemble.

## 统计代写|时间序列分析作业代写time series analysis代考|Notation

Since it is important to be clear about the type of stochastic process we are discussing, we shall use the following notational conventions throughout this book:
1 $\left{X_{t}\right}$ refers to a real-valued discrete parameter stochastic process whose t th component is $X_{t}$, while
2 ${X(t)}$ refers to a real-valued continuous parameter stochastic process whose component at time t is $X(t)$.
3 When the index set for a stochastic process is not explicitly stated as is the case for both $\left{X_{t}\right}$ and ${X(t)}$, we shall assume that it is the set of all integers Z = {. . . , −2, −1, 0,1, 2, . . .}for a discrete parameter process and the entire real axis R = {t : −∞ < t <∞}for a continuous parameter process. Note that ” t ” is being used in two different ways here: the $t$ in $X_{t}$ is a unitless index referring to the $t$ th element of the process $\left{X_{t}\right}$ , whereas the t in $X(t)$ has physically meaningful units such as seconds or days (hence $X(t)$ is the element occurring at time t of the process ${X(t)}$.
4 On occasion we will need to discuss more than one stochastic process at a time. To distinguish among them, we will either introduce another symbol besides $X$ (such as in $\left{Y_{t}\right}$ or add another index before the time index. For example, {Xj,t} and {Xk,t} refer to the jth and kth discrete parameter processes, while {Xj (t)} and {Xk(t)} refer to two continuous parameter processes. Another way to handle multiple processes is to define a vector whose elements are stochastic processes. This approach leads to what are known as vector-valued stochastic processes or multivariate stochastic processes, which we do not deal with in this book.
5 We reserve the symbol Z for a complex-valued RV whose real and imaginary components are real-valued RVs. With an index added, {Zt} is a complex-valued discrete parameter stochastic process with a tth component formed from, say, the real-valued RVs
X0,t and X1,t; i.e., Zt = X0,t + iX1,t, where idef =√−1 (i.e., i is “equal by definition”to √−1). Likewise, {Z(t)} is a complex-valued continuous parameter stochastic process with a tth component formed from two real-valued RVs, say, X0(t) and X1(t); i.e.,Z(t) = X0(t) + iX1(t).

## 统计代写|时间序列分析作业代写time series analysis代考|Basic Theory for Stochastic Processes

Let us first consider the real-valued discrete parameter stochastic process $\left{X_{t}\right}$. Since, for $t$ fixed, $X_{t}$ is an RV, it has an associated cumulative probability distribution function (CPDF) given by
$$F_{t}(a)=\mathbf{P}\left[X_{t} \leq a\right],$$
where the notation $\mathbf{P}[A]$ indicates the probability that the event $A$ will occur. Because $\left{X_{t}\right}$ is a stochastic process, our primary interest is in the relationships amongst the various RVs that are part of it. These are expressed by various higher-order CPDFs. For example, for any $t_{0}$ and $t_{1}$ in the index set $T$,
$$F_{t_{0}, t_{1}}\left(a_{0}, a_{1}\right)=\mathbf{P}\left[X_{t_{0}} \leq a_{0}, X_{t_{1}} \leq a_{1}\right]$$
gives the bivariate CPDF for $X_{t_{0}}$ and $X_{t_{1}}$, where the notation $\mathbf{P}\left[A_{0}, A_{1}\right]$ refers to the probability of the intersection of the events $A_{0}$ and $A_{1}$. More generally, for any integer $N \geq 1$ and any $t_{0}, t_{1}, \ldots, t_{N-1}$ in the index set, we can define the $N$-dimensional CPDF by
$$F_{t_{0}, t_{1}, \ldots, t_{N-1}}\left(a_{0}, a_{1}, \ldots, a_{N-1}\right)=\mathbf{P}\left[X_{t_{0}} \leq a_{0}, X_{t_{1}} \leq a_{1}, \ldots, X_{t_{N-1}} \leq a_{N-1}\right]$$

## 统计代写|时间序列分析作业代写TIME SERIES ANALYSIS代考|NOTATION

1\left{X_{t}\right}\left{X_{t}\right}指的是一个实值离散参数随机过程，其第 t 个分量是X吨, 而
2X(吨)指实值连续参数随机过程，其在时间 t 的分量为X(吨).
3 当随机过程的指数集没有像两者一样明确说明时\left{X_{t}\right}\left{X_{t}\right}和X(吨)，我们假设它是所有整数 Z = { 的集合。. . , -2, -1, 0,1, 2, . . .}对于离散参数过程，整个实轴 R = {t : −∞ < t <∞}对于连续参数过程。请注意，“t”在这里以两种不同的方式使用：吨在X吨是一个无单位索引，指的是吨过程的要素\left{X_{t}\right}\left{X_{t}\right}，而 t 在X(吨)具有物理意义的单位，例如秒或天H和nC和$X(吨一世s吨H和和l和米和n吨这CC在rr一世nG一种吨吨一世米和吨这F吨H和pr这C和ss{X吨}.4这n这CC一种s一世这n在和在一世lln和和d吨这d一世sC在ss米这r和吨H一种n这n和s吨这CH一种s吨一世Cpr这C和ss一种吨一种吨一世米和.吨这d一世s吨一世nG在一世sH一种米这nG吨H和米,在和在一世ll和一世吨H和r一世n吨r这d在C和一种n这吨H和rs是米b这lb和s一世d和sX(s在CH一种s一世n\left{Y_{t}\right}$ 或在时间索引之前添加另一个索引。例如，{Xj,t} 和 {Xk,t} 指的是第 j 个和第 k 个离散参数过程，而 {Xj吨} 和 {Xk吨} 指的是两个连续的参数过程。处理多个过程的另一种方法是定义一个向量，其元素是随机过程。这种方法导致了所谓的向量值随机过程或多元随机过程，我们在本书中不涉及。
5 我们为复值 RV 保留符号 Z，其实部和虚部都是实值 RV。添加索引后，{Zt} 是复值离散参数随机过程，其第 t 个分量由实值 RV
X0,t 和 X1,t 形成；即，Zt = X0,t + iX1,t，其中 idef =√−1一世.和.,一世一世s“和q在一种lb是d和F一世n一世吨一世这n”吨这√−1. 同样，{Z吨} 是一个复值连续参数随机过程，其第 t 个分量由两个实值 RV 组成，例如 X0吨和 X1吨; 即，Z吨 = X0吨+ IX1吨.

## 统计代写|时间序列分析作业代写TIME SERIES ANALYSIS代考|BASIC THEORY FOR STOCHASTIC PROCESSES

F吨(一种)=磷[X吨≤一种],

F吨0,吨1(一种0,一种1)=磷[X吨0≤一种0,X吨1≤一种1]

F吨0,吨1,…,吨ñ−1(一种0,一种1,…,一种ñ−1)=磷[X吨0≤一种0,X吨1≤一种1,…,X吨ñ−1≤一种ñ−1]

## 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。