msmtools.analysis.fingerprint_relaxation

msmtools.analysis.fingerprint_relaxation(T, p0, obs, tau=1, k=None, ncv=None)

Dynamical fingerprint for relaxation experiment.

The dynamical fingerprint is given by the implied time-scale spectrum together with the corresponding amplitudes.

Parameters:
  • T ((M, M) ndarray or scipy.sparse matrix) – Transition matrix
  • obs1 ((M,) ndarray) – Observable, represented as vector on state space
  • obs2 ((M,) ndarray (optional)) – Second observable, for cross-correlations
  • k (int (optional)) – Number of time-scales and amplitudes to compute
  • tau (int (optional)) – Lag time of given transition matrix, for correct time-scales
  • ncv (int (optional)) – The number of Lanczos vectors generated, ncv must be greater than k; it is recommended that ncv > 2*k
Returns:

  • timescales ((N,) ndarray) – Time-scales of the transition matrix
  • amplitudes ((N,) ndarray) – Amplitudes for the relaxation experiment

See also

relaxation(), fingerprint_correlation()

References

[1]Noe, F, S Doose, I Daidone, M Loellmann, M Sauer, J D Chodera and J Smith. 2010. Dynamical fingerprints for probing individual relaxation processes in biomolecular dynamics with simulations and kinetic experiments. PNAS 108 (12): 4822-4827.

Notes

Fingerprints are a combination of time-scale and amplitude spectrum for a equilibrium correlation or a non-equilibrium relaxation experiment.

Relaxation

A relaxation experiment looks at the time dependent expectation value of an observable for a system out of equilibrium

\[\mathbb{E}_{w_{0}}[a(x, t)]=\sum_x w_0(x) a(x, t)=\sum_x w_0(x) \sum_y p^t(x, y) a(y).\]

The fingerprint amplitudes \(\gamma_i\) are given by

\[\gamma_i=\langle w_0, r_i\rangle \langle l_i, a \rangle.\]

And the fingerprint time scales \(t_i\) are given by

\[t_i=-\frac{\tau}{\log \lvert \lambda_i \rvert}.\]

Examples

>>> import numpy as np
>>> from msmtools.analysis import fingerprint_relaxation

>>> T = np.array([[0.9, 0.1, 0.0], [0.5, 0.0, 0.5], [0.0, 0.1, 0.9]])
>>> p0 = np.array([1.0, 0.0, 0.0])
>>> a = np.array([1.0, 0.0, 0.0])
>>> ts, amp = fingerprint_relaxation(T, p0, a)

>>> ts
array([        inf,  9.49122158,  0.43429448])

>>> amp
array([ 0.45454545,  0.5       ,  0.04545455])