Computes soft overlap matrices quantifying how the basin partitions of two functions intersect. The overlap accounts for basin multiplicity through soft membership weights.
Value
A list containing:
- O_pp
Matrix of y-max with z-max overlaps.
- O_mm
Matrix of y-min with z-min overlaps.
- O_pm
Matrix of y-max with z-min overlaps.
- O_mp
Matrix of y-min with z-max overlaps.
- total_mass
Sum of vertex masses.
Details
For two functions y and z with basin structures, the overlap matrix entry \(O_{ij}^{++}\) measures the effective intersection between y-maximum basin i and z-maximum basin j:
$$O_{ij}^{++} = \sum_v m_0(v)\,\mu_i^{y,+}(v)\,\mu_j^{z,+}(v)$$
where \(m_0(v)\) is vertex mass and \(\mu\) are normalized membership weights.
Four overlap matrices are computed: O_pp: y-max with z-max (positive-positive association regions) O_mm: y-min with z-min (negative-negative association regions) O_pm: y-max with z-min (opposite association regions) O_mp: y-min with z-max (opposite association regions)
Large O_pp entries indicate regions where both functions tend toward high values. Large O_pm entries indicate regions where y is high but z is low.