# analysis - MSM analysis functions (msmtools.analysis)¶

This module contains functions to analyze a created Markov model, which is specified with a transition matrix T.

## Validation¶

 is_transition_matrix(T[, tol]) Check if the given matrix is a transition matrix. is_tmatrix(T[, tol]) Check if the given matrix is a transition matrix. is_rate_matrix(K[, tol]) Check if the given matrix is a rate matrix. is_connected(T[, directed]) Check connectivity of the given matrix. is_reversible(T[, mu, tol]) Check reversibility of the given transition matrix.

## Decomposition¶

Decomposition routines use the scipy LAPACK bindings for dense numpy-arrays and the ARPACK bindings for scipy sparse matrices.

 stationary_distribution(T) Compute stationary distribution of stochastic matrix T. statdist(T) Compute stationary distribution of stochastic matrix T. eigenvalues(T[, k, ncv, reversible, mu]) Find eigenvalues of the transition matrix. eigenvectors(T[, k, right, ncv, reversible, mu]) Compute eigenvectors of given transition matrix. rdl_decomposition(T[, k, norm, ncv, …]) Compute the decomposition into eigenvalues, left and right eigenvectors. timescales(T[, tau, k, ncv, reversible, mu]) Compute implied time scales of given transition matrix.

## Expected counts¶

 expected_counts(T, p0, N) Compute expected transition counts for Markov chain with n steps. expected_counts_stationary(T, N[, mu]) Expected transition counts for Markov chain in equilibrium.

## Passage times¶

 mfpt(T, target[, origin, tau, mu]) Mean first passage times (from a set of starting states - optional) to a set of target states.

## Committors and PCCA¶

 committor(T, A, B[, forward, mu]) Compute the committor between sets of microstates. pcca(T, m) Compute meta-stable sets using PCCA++ _ and return the membership of all states to these sets.

## Fingerprints¶

 fingerprint_correlation(T, obs1[, obs2, …]) Dynamical fingerprint for equilibrium correlation experiment. fingerprint_relaxation(T, p0, obs[, tau, k, ncv]) Dynamical fingerprint for relaxation experiment. expectation(T, a[, mu]) Equilibrium expectation value of a given observable. correlation(T, obs1[, obs2, times, maxtime, …]) Time-correlation for equilibrium experiment. relaxation(T, p0, obs[, times, k, ncv]) Relaxation experiment.

## Sensitivity analysis¶

 stationary_distribution_sensitivity(T, j) Sensitivity matrix of a stationary distribution element. eigenvalue_sensitivity(T, k) Sensitivity matrix of a specified eigenvalue. timescale_sensitivity(T, k) Sensitivity matrix of a specified time-scale. eigenvector_sensitivity(T, k, j[, right]) Sensitivity matrix of a selected eigenvector element. mfpt_sensitivity(T, target, i) Sensitivity matrix of the mean first-passage time from specified state. committor_sensitivity(T, A, B, i[, forward]) Sensitivity matrix of a specified committor entry. expectation_sensitivity(T, a) Sensitivity of expectation value of observable A=(a_i).