Coarse-graining with PCCA+¶

Reduce a large MSM to a small one over metastable basins, two function calls:

from gpvolve import pcca_plus, coarse_grain

chi = pcca_plus(msm.transition_matrix, n_clusters=3)
P_coarse = coarse_grain(msm.transition_matrix, chi)

chi is the (n, k) membership matrix (row-stochastic, non-negative). P_coarse is the (k, k) row-stochastic Galerkin projection.

Inspecting the basins¶

from gpvolve import metastable_sets

sets = metastable_sets(chi)
for j, indices in enumerate(sets):
    print(f"cluster {j}: {len(indices)} states")

metastable_sets is the hard-argmax assignment. Combine it with find_peaks to identify which basin each fitness peak ends up in.

Picking n_clusters¶

The classic heuristic is the implied-timescales gap: compute timescales tau_k for the first several modes, look for a clear separation between tau_{k-1} and tau_k. The k for which the gap is largest is a reasonable choice.

from gpvolve import timescales

ts = timescales(msm.transition_matrix, k=10)
print(ts)

For small maps with obvious topology (two-peak landscape, single fitness ridge) the right k is often pickable by eye.