deeptime.markov.tools.estimation.girsanov_reweighted_count_matrix

deeptime.markov.tools.estimation.girsanov_reweighted_count_matrix(dtraj, lag, reweighting_factors, sliding=True, sparse_return=True, nstates=None)

Generate a Girsanov reweighted count matrix from given microstate trajectory. [1]

Parameters:

• dtraj (array_like or list of array_like) – Discretized trajectory or list of discretized trajectories.

• lag (int) – Lagtime in trajectory steps.

• reweighting (tuple) – Enforce a count-matrix with reweighting factors shape=(g,M). g is the likelihood ratio between probability measures with dim=(len(dtraj)-lag), if eta.shape==dtraj.shape. M is the likelihood ratio between the path probabilitiy densities with dim=(len(dtraj)-lag), if eta.shape==dtraj.shape.

• sliding (bool) – By default True, the sliding window approach is used for transition counting.

• sparse_return (bool, optional) – Whether to return a dense or a sparse matrix.

• nstates (int, optional) – Enforce a count-matrix with shape=(nstates, nstates).

Returns:

C – The Girsanov path reweighted count matrix at given lag in coordinate list format.

Return type:

scipy.sparse.coo_matrix

Notes

Transition counts can be obtained from microstate trajectory using the sliding approach. By sliding along the trajectory and counting all transitions sperated by the lagtime $\tau$.

Transition counts $c_{ij}(\tau)$ are reweighted according to

$$c_{ij}(\tau)=\lim_{m\rightarrow\infty}\sum_{\nu_k\in S_{\tau,m}} g([\mathbf{x}_0]_k)\mathbf{1}_{B_i}([\mathbf{x}_0]_k) \cdot M_{\mathbf{x},\tau}(\nu_k)\mathbf{1}_{B_j}([\mathbf{x}_n]_k).$$

Where $\mathbf{1}_{B_i}$ is the indicator function for state $B_i$.

References

[1]

Joana-Lysiane Schaefer and Bettina G Keller. Implementation of girsanov reweighting in openmm and deeptime. The Journal of Physical Chemistry B, 2024.

Example

>>> import numpy as np
>>> from deeptime.markov.tools.estimation import girsanov_reweighted_count_matrix

>>> dtraj = np.array([0, 0, 1, 0, 1, 1, 0, 0])
>>> tau = 2

In this example, the bias and target potential would be the same. Subtract the one to get the same length of the discrete trajectory len(eta+1) and random number array eta.

>>> g = [np.ones(len(dtraj)-1)]
>>> M = [np.zeros(len(dtraj)-1)]
>>> reweighting_factors = (g,M)

Use the reweighting approach as

>>> C_sliding = girsanov_reweighted_count_matrix(dtraj[:-1], tau, reweighting_factors)

The generated matrix is a sparse matrix in CSR-format. For convenient printing we convert it to a dense ndarray.

>>> C_sliding.toarray()
array([[1., 2.],
       [1., 1.]])