deeptime.markov.tools.analysis.fingerprint_correlation¶
- deeptime.markov.tools.analysis.fingerprint_correlation(T, obs1, obs2=None, tau=1, k=None, ncv=None)¶
Dynamical fingerprint for equilibrium correlation experiment. [1].
- 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 correlation experiment
See also
References
Notes
Fingerprints are a combination of time-scale and amplitude spectrum for a equilibrium correlation or a non-equilibrium relaxation experiment.
Auto-correlation
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
\[t_i=-\frac{\tau}{\log \lvert \lambda_i \rvert}. \]Cross-correlation
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
\[\gamma_i=\langle a, r_i\rangle_{\mu} \langle l_i, b \rangle. \]Examples
>>> import numpy as np >>> from deeptime.markov.tools.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])