Summary In many cases, a Markov chain exhibits a long-term limiting behavior. The chain settles down to an equilibrium distribution, which is independent of its initial state. The long-term behavior of a Markov chain is related to how often states are visited. This chapter addresses the ...
Long-Term Frequency Interpretations 从频率学的角度来说,当试验次数 n 很大的时候,事件发生的频率同样可以表示事件发生的概率,那么 \pi_j=\lim_{n \rightarrow \infty}{\frac{v_{ij}(n)}{n}}, v_{ij}(n) 状态i 转移到到 j 的总次数。 记q_{kj}(n) 是状态 k\rightarrow j 的次数,那么有 ...
In this section, we review Markov chains and discuss some key results. 9.1.1 Overview A Markov chain is a model of the random motion of an object in a discrete set of possible locations. Two versions of this model are of interest to us: discrete time and continuous time. In discrete ti...
[Section 1] DISCRETE-TIME MARKOV CHAINS(系统每一步的状态可能不同)(系统的下一个状态只和当前状态有关) The state changes at certain discrete time instants, indexed by an integer variable n . State space(每一步的状态可能不同) At each time step n , the state of the chain is denoted by ...
A continuous-time Markov chain is time reversible if the process in forward time is indistinguishable from the process in reversed time. A consequence is that for all states i and j, the long-term forward transition rate from i to j is equal to the long-term backward rate from j to i....
Markov chains have many applications as statistical models of real-world processes.Introduction Russian mathematician Andrey Markov.A Markov chain is a stochastic process with the Markov property. The term "Markov chain" refers to the sequence of random variables such a process moves through, with ...
Markov Chains are a powerful tool in probability theory and statistics. They represent a sequence of random events where the probability of an event depends only on the outcome of the previous event. The chain consists of a set of states and a set of transition probabilities between states. Ma...
This means in the long term we areequallyas likely to be in any of the three states! Note: This wasn’t a dense deep derivation of the stationary distribution as I didn’t want it turn into a textbook! However, there are many in-depth examples online abouteigenvalue decomposition. ...
不过,对于 periodic Markov chain,其稳态分布不再表示 Markov chain 在无穷多步后处于状态 i 的极限概率,而只对应访问状态 i 的long-term frequency. 任意的 finite Markov chain 中都存在一个 recurrent 的 communicating class,其对应的 subchain 就会是 irreducible 并且 recurrent 的,如果其还是 aperiodic 的,...
404 MARKOV CHAINSSynergy between these two approaches to marine conservation and management arises from use of marine reserves as tools in the marine ecosystem–based manage-ment portfolio. Within reserves and reserve networks, conservation objectives may be achieved as resident target species and ...