## Performance of a Markov Process

This applet calculates some simple performance measures for a Markov process.
The "TPM" tab can be used to set the transition probability matrix (TPM) governing the process.
Once values are input here and the "Accept" button is pressed, the tables on the
the other tabs can be viewed for information about the corresponding Markov process.

The "Evalues" tab will show the eigenvalues for the input TPM. One of them will be 1
(to within roundoff), and the others will have a magnitude less than 1 (perhaps will be
negative). The closer these other eigenvalues are to 1, the slower the convergence of
the Markov process to the limiting distribution. The "Evectors" tab shows the (left) eigenvectors
for the Markov process. Each column corresponds to an eigenvector, and the order of the columns
corresponds to the order of the eigenvalues in the Evalues table; the eigenvector corresponding
to the unit eigenvalue is the limiting distribution. You can check this limiting distribution
against that observed in the Markov chain applet, which actually
performs a Markov process for a given TPM.
The "Covariance" tab present a table of covariances for the Markov process. These values, divided
by the number of Markov steps performed, *M*, describes the reproducibility of a Markov
process of length *M*. That is, if the Markov process were run many, many times,
each run sampling *M* steps, then this matrix gives the variances and covariances
in the average state occupancies across these many runs. Smaller values correspond to a better
behaved (more reproducible) Markov sequence.

A few problems still exist with the interface (sigh). If the TPM disappears after you accept the values,
go to one of the other tabs and back, and it should reappear. If the tabs disappear when on the
TPM, hit the "Accept" button.

No matter what values you enter into the table, the applet will normalize the rows so they
sum to unity, so you can enter unnormalized probabilities if you wish.