msmtools.analysis.fingerprint_correlation(T, obs1, obs2=None, tau=1, k=None, ncv=None)

Dynamical fingerprint for equilibrium correlation experiment.

  • 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

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


[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.


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


The auto-correlation of an observable \(a(x)\) for a system in equilibrium is

\[\mathbb{E}_{\mu}[a(x,0)a(x,t)]=\sum_x \mu(x) a(x, 0) a(x, t)\]

\(a(x,0)=a(x)\) is the observable at time \(t=0\). It can be propagated forward in time using the t-step transition matrix \(p^{t}(x, y)\).

The propagated observable at time \(t\) is \(a(x, t)=\sum_y p^t(x, y)a(y, 0)\).

Using the eigenvlaues and eigenvectors of the transition matrix the autocorrelation can be written as

\[\mathbb{E}_{\mu}[a(x,0)a(x,t)]=\sum_i \lambda_i^t \langle a, r_i\rangle_{\mu} \langle l_i, a \rangle.\]

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

\[\gamma_i=\langle a, r_i\rangle_{\mu} \langle l_i, a \rangle.\]

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

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


The cross-correlation of two observables \(a(x)\), \(b(x)\) is similarly given

\[\mathbb{E}_{\mu}[a(x,0)b(x,t)]=\sum_x \mu(x) a(x, 0) b(x, t)\]

The fingerprint amplitudes \(\gamma_i\) are similarly given in terms of the eigenvectors

\[\gamma_i=\langle a, r_i\rangle_{\mu} \langle l_i, b \rangle.\]


>>> import numpy as np
>>> from msmtools.analysis import fingerprint_correlation
>>> T = np.array([[0.9, 0.1, 0.0], [0.5, 0.0, 0.5], [0.0, 0.1, 0.9]])
>>> a = np.array([1.0, 0.0, 0.0])
>>> ts, amp = fingerprint_correlation(T, a)
>>> ts
array([        inf,  9.49122158,  0.43429448])
>>> amp
array([ 0.20661157,  0.22727273,  0.02066116])