The ergodic property can be written Stated in another way, it says that, at the limit, the early behaviour of the trajectory becomes negligible and only the long run stationary behaviour really matter when computing the temporal mean.
For a given page, all the allowed links have then equal chance to be clicked. So if the initial distribution q is a stationary distribution then it will stay the same for all future time steps.
Usually the term "Markov chain" is reserved for a process with a discrete set of times, that is, a discrete-time Markov chain DTMC   but a few authors use the term "Markov process" to refer to a continuous-time Markov chain CTMC without explicit mention. First, we denote So we want to compute here m R,R.
However, thanks to the Markov property, the dynamic of a Markov chain is pretty easy to define. Additionally, the transition matrix must be a stochastic matrix, a matrix whose entries in each row must add up to exactly 1. The random variables at different instant of time can be independent to each other coin flipping example or dependent in some way stock price example as well as they can have continuous or discrete state space space of possible outcomes at each instant of time. Assume that we have a tiny website with 7 pages labeled from 1 to 7 and with links between the pages as represented in the following graph. Indeed, the probability of any realisation of the process can then be computed in a recurrent way. A series of independent events for example, a series of coin flips satisfies the formal definition of a Markov chain. So if the initial distribution q is a stationary distribution then it will stay the same for all future time steps.
Thanks for reading! For example, flipping a coin every day defines a discrete time random process whereas the price of a stock market option varying continuously defines a continuous time random process.
From a theoretical point of view, it is interesting to notice that one common interpretation of the PageRank algorithm relies on the simple but fundamental mathematical notion of Markov chains.
This typically leaves them unable to successfully produce sequences in which some underlying trend would be expected to occur.
Markov chains linear algebra
In the second section, we will discuss the special case of finite state space Markov chains. The chain on the right one edge has been added is irreducible: each state can be reached from any other state. Therefore, every day in our simulation will have a fifty percent chance of rain. A visualization of the weather example The Model Formally, a Markov chain is a probabilistic automaton. History[ edit ] Andrey Markov studied Markov chains in the early 20th century. The second sequence seems to jump around, while the first one the real data seems to have a "stickyness". So, we want to compute the probability Here, we use the law of total probability stating that the probability of having s0, s1, s2 is equal to the probability of having first s0, multiplied by the probability of having s1 given we had s0 before, multiplied by the probability of having finally s2 given that we had, in order, s0 and s1 before. We can then define a random process also called stochastic process as a collection of random variables indexed by a set T that often represent different instants of time we will assume that in the following. For example, the algorithm Google uses to determine the order of search results, called PageRank , is a type of Markov chain.
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