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

correlation, fingerprint_relaxation

References

[1]

Frank Noé, Sören Doose, Isabella Daidone, Marc Löllmann, Markus Sauer, John D Chodera, and Jeremy C Smith. Dynamical fingerprints for probing individual relaxation processes in biomolecular dynamics with simulations and kinetic experiments. Proceedings of the National Academy of Sciences, 108(12):4822–4827, 2011.

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])