Manual Markov Chains

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What is a Markov Chain?

In order for it to be an absorbing Markov chain, all other transient states must be able to reach the absorbing state with a probability of 1. Absorbing Markov chains have specific unique properties that differentiate them from the normal time-homogeneous Markov chains. One of these properties is the way in which the transition matrix can be written. With a chain with t transient states and r absorbing states, the transition matrix P can be written in canonical form as follows:.

Where Q is a t x t matrix, R is a t x r matrix, 0 is a r x t zero matrix, and I r is a r x r identity matrix.

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In particular, the decomposition of the transition matrix into the fundamental matrix allows for certain calculations such as the expected number of steps until absorption from each state. The fundamental matrix N is calculated as follows:. Where I t is a t x t identity matrix.

The expected number of steps is based on the concept of linearity of expectation, and it is calculated as follows:. Where 1 is a column vector of the same length as the number of transient states with all entries being 1. Furthermore, we can calculate the probability of being absorbed by a specific absorbing state when starting from any given transient state.

Introduction to Markov Chains

This probability is calculated as follows:. Markov chains are widely used in many fields such as finance, game theory, and genetics. However, the basis of this tutorial is how to use them to model the length of a company's sales process since this could be a Markov process. This was in fact validated by testing if sequences are detailing the steps that a deal went through before successfully closing complied with the Markov property.

On Learning Markov Chains

This analysis carried the assumption that the probabilities of a given deal moving forward in our sales process was constant from month to month for a given industry in order to use time-homogenous Markov chains. That is a Markov chain in which the transition probabilities between states stayed constant as time went on the number of steps k increased. This analysis was conducted using the R programming language.

R has a handy package called a Markov Chain that can handle a vast array of Markov chain types. To begin with, the first thing we did was to check if our sales sequences followed the Markov property.

Introduction to Markov Chains

To that end, the Markov Chain package carries a handy function called verifyMarkovProperty that tests if a given sequence of events follows the Markov property by performing Chi-square tests on a series of contingency tables derived from the sequence of events. Large p-values indicate that the null hypothesis of a sequence following the Markov property should not be rejected. Read more about accessing full-text.

The notion of steadily attracting state is new.

Artificial Intelligence - foundations of computational agents -- Markov Chains

We additionally derive practical conditions by showing that the rank condition on the controllability matrix needs to be verified only at a globally attracting state resp. We illustrate that the conditions are easy to verify on a non-trivial and non-artificial example of Markov chain arising in the context of adaptive stochastic search algorithms to optimize continuous functions in a black-box scenario.

Lecture 32: Markov Chains Continued - Statistics 110

Source Bernoulli , Volume 25, Number 1 , Zentralblatt MATH identifier Chotard, Alexandre; Auger, Anne. Verifiable conditions for the irreducibility and aperiodicity of Markov chains by analyzing underlying deterministic models. In a previous page, we studied the movement between the city and suburbs.

Let us discuss another example on population dynamics. Example: Age Distribution of Trees in a Forest Trees in a forest are assumed in this simple model to fall into four age groups: b k denotes the number of baby trees in the forest age group years at a given time period k ; similarly y k , m k and o k denote the number of young trees years of age , middle-aged trees age , and old trees older than 45 years of age , respectively.

The length of one time period is 15 years.

How does the age distribution change from one time period to the next? The model makes the following three assumptions: A certain percentage of trees in each age group dies. Surviving trees enter into the next age group; old trees remain old. Lost trees are replaced by baby trees.

Note that the total tree population does not change over time.