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.]])