How to compute stationary distribution for martrix with more than 1 closed class

• sam48
In summary, the conversation discusses the presence of stationary distributions in a given matrix with an absorbing state. The speaker also asks for information on how to compute these distributions in a matrix with more than one closed class. The suggested method is to write the matrix in canonical form and use standard methods, such as solving the eigenvector equation.
sam48
Hi,

How any stationary distributions exists in below matrix and what are the value

[
.5 0 0 .5
.25 .5 .25 0
0 0 1 0
1/6 0 0 5/6
]

Any information regarding how to compute stationary distribution in a martrix with more than 1 closed class would be appreciated.
regards,

Your chain has an absorbing state (the third state, since the 3,3 entry is 1). Write this matrix in canonical form and analyze it with the standard methods.

One way is just to solve the eigenvector equation xA=x - since your example has 2 closed classes (the other being {1,4}), there will be two LI eigenvectors. Positivity and summation to 1 will tell you which linear combinations are valid.

What is a stationary distribution?

A stationary distribution is a probability distribution that represents the long-term behavior of a system. It describes the likelihood of a system being in a particular state over time, assuming that the system is in a steady state.

What is a closed class in matrix computation?

In matrix computation, a closed class refers to a set of states in a Markov chain that are not accessible from any other state. This means that once a system enters a closed class, it will never leave that class.

How do you compute the stationary distribution for a matrix with more than one closed class?

To compute the stationary distribution for a matrix with multiple closed classes, you can use the method of absorbing Markov chains. This involves rearranging the matrix into a specific form, solving a system of equations, and then normalizing the resulting vector to obtain the stationary distribution.

What factors can affect the computation of the stationary distribution?

The computation of the stationary distribution can be affected by the initial state of the system, the transition probabilities between states, and the number and size of closed classes in the matrix. Additionally, any errors in the computation or rounding of values can also impact the accuracy of the stationary distribution.

Why is computing the stationary distribution important in scientific research?

Computing the stationary distribution allows scientists to understand the long-term behavior of a system and make predictions about its future states. This is useful in a variety of fields, including biology, economics, and physics, as it can help researchers model and analyze complex systems and make informed decisions based on the results.

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