Any fast way to compute the fixed vector of a Markov chain transistion matrix?

In summary, the fixed vector of a Markov chain transition matrix can be computed by finding the eigenvector associated with the eigenvalue of 1. This can be done using methods such as power iteration, inverse iteration, or the QR algorithm. Faster methods such as the Arnoldi iteration or the Lanczos iteration can also be used. The fixed vector is a stable equilibrium point and will not change unless the transition matrix is altered. It represents the long-term steady-state distribution of the Markov chain and can provide insights into the behavior and stability of the system. However, there are assumptions and limitations, such as the Markov chain must be irreducible and aperiodic, and the transition matrix must be square with a finite number of states
  • #1
samuelandjw
22
0
* I have already posted this in the General Math, but I guess the problem is more like a linear algebra problem.

Currently I am using a rather simple way, to solve vector w from (M-I)w=0 (replace one equation by w1+w2+...wn=1). Is there any faster way to do this? Thank you.
 
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  • #2
I would calculate the Transition matrix's eigenvector corresponding to [itex]\lambda[/itex]=1.
 
  • #3
SprucerMoose said:
I would calculate the Transition matrix's eigenvector corresponding to [itex]\lambda[/itex]=1.

basically the same as mine.
 

1. How do you compute the fixed vector of a Markov chain transition matrix?

The fixed vector of a Markov chain transition matrix can be computed by finding the eigenvector associated with the eigenvalue of 1. This can be done using various methods such as power iteration, inverse iteration, or the QR algorithm.

2. Is there a faster way to compute the fixed vector?

Yes, there are faster methods such as the Arnoldi iteration or the Lanczos iteration which can converge to the fixed vector more quickly than traditional methods.

3. Can the fixed vector of a Markov chain transition matrix change?

No, the fixed vector is a stable equilibrium point and will not change unless the transition matrix is altered.

4. What is the significance of the fixed vector in a Markov chain?

The fixed vector represents the long-term steady-state distribution of the Markov chain. It can provide insights into the behavior and stability of the system.

5. Are there any assumptions or limitations when computing the fixed vector of a Markov chain transition matrix?

Yes, the Markov chain must be irreducible and aperiodic in order for the fixed vector to exist. Additionally, the transition matrix must be square and have a finite number of states.

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