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# 统计代写|非参数统计代写Nonparametric Statistics代考|STAT6610 The Markov Chain Approach

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## 统计代写|非参数统计代写Nonparametric Statistics代考|The Markov Chain Approach

Some theory and concepts in the area of what is called a finite homogeneous MC have been applied to derive the run-length distribution and various characteristics of the run-length distribution of some time-weighted charts, such as the CUSUM and the EWMA. The reader can look up, for example, Fu and Lou (2003) and Balakrishnan and Koutras (2002) for a more detailed treatment of the topics; however, most existing SPC books don’t seem to treat this important topic. In this section, some of the basic terminologies, definitions, results, and theorems are provided that give the necessary background for calculating the run-length distribution of some control charts via the $\mathrm{MC}$ approach. These theorems and results are critical to the following chapters; illustrations are provided.

The $\mathrm{MC}$ approach entails that the charting statistic is viewed as following a MC (as a stochastic process), with a state space $S$ and a transition probability matrix $M$. The state space consists of two types of states:

1. one absorbing state (i.e., this state is entered when the chart signals, i.e., when the charting statistic is greater than or equal to the $U C L$, or less than or equal to the $L C L$ for a two-sided chart); and
2. $\nu$ transient or non-absorbing states, so that there are $\nu+1$ states in total.

The $(\nu+1) \times(\nu+1)$ transition probability matrix, $M$, is often written in a partitioned form
where the sub-matrix $Q_{\nu \times \nu}$ contains all the probabilities of going from one transient state to another and is called the essential transition probability matrix. The column vector $\underline{p}{\nu \times 1}$ contains all the probabilities of going from each transient state to the absorbing state; $\underline{0}{1 \times \nu}^{\prime}$ is a row vector of zeros, consisting of the probabilities of going from the absorbing state to each transient state (which are all zero), and the scalar value 1 is the probability of staying in the absorbing state once it has been entered. Note that the key component in using the MC approach is to obtain the essential transition probability matrix $Q_{\nu \times \nu}$.

## 统计代写|非参数统计代写Nonparametric Statistics代考|The Integral Equation Approach

The integral equation approach utilizes mathematics and combinatorics to find a closed form expression of the run-length distribution. This approach is sometimes challenging in that the expression obtained is typically complex or difficult to evaluate numerically. Often, the exact expression of the run-length distribution can be found after a considerable amount of work, but here a simulation is done instead, since it is much easier.

## 统计代写|非参数统计代写Nonparametric Statistics代考|The Markov Chain Approach

1. 一种吸收状态（即，当图表发出信号时，即当图表统计大于或等于在C大号，或小于或等于大号C大号对于双面图表）；和
2. n瞬态或非吸收状态，因此有n+1共州。

，其中子矩阵问n×n包含从一种瞬态到另一种瞬态的所有概率，称为基本转移概率矩阵。列向量p―n×1包含从每个瞬态到吸收状态的所有概率；0―1×n′是一个由零组成的行向量，由从吸收状态到每个瞬态（全为零）的概率组成，标量值 1 是进入吸收状态后保持在吸收状态的概率。请注意，使用 MC 方法的关键部分是获得基本转移概率矩阵问n×n.

## 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。