Posted on Categories:Stochastic Porcesses, 数学代写, 随机过程

# 数学代写|随机过程Stochastic Porcesses代考|MA546 Basic setup and results

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|随机过程代写Stochastic Porcesses代考|Basic setup and results

In this section, we shall outline the most important probabilistic results for CTMCs. We shall assume that $\left{X_t\right}_{t \in T}$ is a continuous time stochastic process that evolves within a finite state space, say $E={1,2, \ldots, K}$. When the process enters into state $i$, it remains there for an exponentially distributed time period with mean $1 / v_i$. At the end of this time period, the process will move to a different state $j \neq i$ with probability $p_{i j}$, such that $\sum_{j=1}^K p_{i j}=1, \forall i$, and $p_{i i}=0$. Clearly, for physical or logical reasons, some additional $p_{i j}$ could also be zero. As in Chapter 3 , the transition probability matrix is defined to be $\mathbf{P}=\left(p_{i j}\right)$. This defines an embedded (discrete time) Markov chain. The process $\left{X_t\right}$ will be designated a CTMC with parameters $\mathbf{P}$ and $\boldsymbol{v}=\left(v_1, \ldots, v_K\right)^T$.

One important class of CTMCs, which will be analyzed in detail in later chapters are birth-death processes.

## 数学代写|随机过程代写Stochastic Porcesses代考|Inference and prediction for CTMCs

Here, we study inference and prediction for CTMCs. We first consider inference for chain parameters and then examine the forecasting of both the short- and long-term behavior of a CTMC. We will suppose throughout the most general case where the transition matrix, $\mathbf{P}$, and the transition rates, $v$, are unknown and unrelated, that is, that the elements of $\mathbf{P}$ are not known functions of $\boldsymbol{v}$.

Assume that we observe the initial state of the chain, say $x_0$ and the times, $t_i$, and states, $x_i$, for $i=1, \ldots, n$, of the first $n$ transitions of the chain. Then, the likelihood function can be written as
$$l(\mathrm{P}, \boldsymbol{v} \mid \text { data })=\prod_{i=1}^n v_{x_{i-1}} \exp \left(-v_{x_{i-1}}\left(t_i-t_{i-1}\right)\right) p_{x_{i-1} x_i} \propto \prod_{i=1}^K v_i^{n_i} \exp \left(-v_i T_i\right) \prod_{j=1}^K p_{i j}^{n_{i j}},$$
where $n_{i j}$ is the number of observed transitions from $i$ to $j, T_i$ is the total time spent in state $i$ and $n_i=\sum_{j=1}^K n_{i j}$ is the total number of transitions out of state $i$, for $i, j \in{1, \ldots, K}$. Given the lack of memory property of the exponential distribution, many alternative experiments have likelihood functions of the same form.

## 数学代写|随机过程代写Stochastic Porcesses代考|Inference and prediction for CTMCs

$$l(\mathrm{P}, \boldsymbol{v} \mid \text { data })=\prod_{i=1}^n v_{x_{i-1}} \exp \left(-v_{x_{i-1}}\left(t_i-t_{i-1}\right)\right) p_{x_i-1} x_i \propto \prod_{i=1}^K v_i^{n_i} \exp \left(-v_i T_i\right) \prod_{j=1}^K p_{i j}^{n_{i j}},$$
.鉴于指数分布缺乏记杍特性，许多替代实验都具有相同形江的似然函数。

## 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。