### 统计代写|应用随机过程代写Stochastic process代考|Discrete time Markov chains and extensions

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• Statistical Inference 统计推断
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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|应用随机过程代写Stochastic process代考|Discrete time Markov chains and extensions

As we mentioned in Chapter 1, Markov chains are one of the simplest stochastic processes to study and are characterized by a lack of memory property, so that future observations depend only on the current state and not on the whole of the past history of the process. Despite their simplicity, Markov chains can be and have been applied to many real problems in areas as diverse as web-browsing behavior, language modeling, and persistence of surnames over generations. Furthermore, as illustrated in Chapter 2, with the development of Markov chain Monte Carlo (MCMC) methods, Markov chains have become a basic tool for Bayesian analysis.

In this chapter, we shall study the Bayesian analysis of discrete time Markov chains, focusing on homogeneous chains with a finite state space. We shall also analyze many important subclasses and extensions of this basic model such as reversible chains, branching processes, higher order Markov chains, and discrete time Markov processes with continuous state spaces. The properties of the basic Markov chain model and these variants are outlined from a probabilistic viewpoint in Section 3.2.

In Section 3.3, inference for time homogeneous, discrete state space, first-order chains is considered. Then, Section $3.4$ provides inference for various extensions and particular classes of chains. A case study on the analysis of wind directions is presented in Section $3.5$ and Markov decision processes are studied in Section 3.6. The chapter concludes with a brief discussion.

## 统计代写|应用随机过程代写Stochastic process代考|Higher order chains and mixtures

Generalizing from Definition 1.6, a discrete time stochastic process, $\left{X_{n}\right}$ is a Markov chain of order $r$ if $P\left(X_{n}=x_{n} \mid X_{0}=x_{0}, \ldots, X_{n-1}=x_{n-1}\right)=P\left(X_{n}=x_{n} \mid X_{n-r}=\right.$ $x_{n-r}, \ldots, X_{n-1}=x_{n-1}$ ) so that the state of the chain is determined by the previous $r$ states. It is possible to represent such a chain as first-order chain by simply combining states.

Example 3.1: Consider a second-order, homogeneous Markov chain $\left{X_{n}\right}$ with two possible states (1 and 2) and write $p_{i j l}=P\left(X_{n}=l \mid X_{n-1}=j, X_{n-2}=i\right)$ for $i, j$, $l=1,2$. Then the first-order transition matrix is

The disadvantage of modeling higher order Markov chain models in such a way is that the number of states necessary to reduce such models to a first-order Markov chain is large. For example, if $X_{n}$ can take values in ${1, \ldots, K}$, then $K^{r}$ states are needed to define an $r$ th order chain. Therefore, various alternative approaches to modeling $r$ th order dependence have been suggested. One of the most popular ones is the mixture transition distribution (MTD) model of Raftery (1985). In this case, it is assumed that
$$P\left(X_{n}=x_{n} \mid X_{n-1}=x_{n-1}, \ldots, X_{n-r}=x_{n-r}\right)=\sum_{i=1}^{r} w_{i} p_{x_{n-i} x_{n}}$$
where $\sum_{i=1}^{r} w_{i}=1$ and $\boldsymbol{P}=\left(p_{i j}\right)$ is a transition matrix. This approach leads to more parsimonious modeling than through the full $r$ th order chain. In particular, in Example 3.1, four free parameters are necessary to model the full second-order chain, whereas using the MTD model only three free parameters are necessary. Inference for higher order Markov chains and for the MTD model is examined in Section 3.4.2.

## 统计代写|应用随机过程代写Stochastic process代考|Discrete time Markov processes with continuous state space

As noted in Chapter 1, Markov processes can be defined with both discrete and continuous state spaces. We have seen that for a Markov chain with discrete state space, the condition for the chain to have an equilibrium distribution is that the chain is aperiodic and that all states are positive recurrent. Although the condition of positive recurrence cannot be sensibly applied to chains with continuous state space, a similar condition known as Harris recurrence applies to chains with continuous state space, which essentially means that the chain can get close to any point in the future. It is known that Harris recurrent, aperiodic chains also possess an equilibrium distribution, so that if the conditional probability distribution of the chain is $P\left(X_{n} \mid X_{n-1}\right)$, then the equilibrium density $\pi$ satisfies
$$\pi(x)=\int P(x \mid y) \pi(y) \mathrm{d} y .$$
As with Markov chains with discrete state space, a sufficient condition for a process to possess an equilibrium distribution is to be reversible.

Example 3.2: Simple examples of continuous space Markov chain models are the autoregressive (AR) models. The first-order AR process was outlined in Example 1.1. Higher order dependence can also be incorporated. An $\mathrm{AR}(k)$ model is defined by
$$X_{n}=\phi_{0}+\sum_{i=1}^{k} \phi_{i} X_{n-i}+\epsilon_{n}$$

The condition for this process to be (weakly) stationary is the well-known unit roots condition that all roots of the polynomial
$$\phi_{0} z^{k}-\sum_{i=1}^{k} \phi_{i} z^{k-i}$$
must lie within the unit circle, that is, each root $z_{i}$ must satisfy $\left|z_{i}\right|<1$. $\triangle$
Inference for AR processes and other continuous state space processes is briefly reviewed in Section 3.4.3.

## 统计代写|应用随机过程代写Stochastic process代考|Discrete time Markov processes with continuous state space

Xn=φ0+∑一世=1ķφ一世Xn−一世+εn

φ0和ķ−∑一世=1ķφ一世和ķ−一世

3.4.3 节简要回顾了 AR 过程和其他连续状态空间过程的推理。

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