Similarity functions

frsutils constructs fuzzy relations by comparing samples feature by feature and aggregating the resulting similarities with a T-norm. A feature-level similarity maps two values to the unit interval, where larger values represent greater similarity.

Implemented functions

For feature values \(x\) and \(y\), let \(d = x-y\).

Name	Formula	Parameters	Registered aliases
Linear	(\max(0, 1-	d	))
Gaussian	\(\exp[-d^2/(2\sigma^2)]\)	\(\sigma>0\)	gaussian, gauss

The linear function is most interpretable when feature differences are on a meaningful scale, commonly after normalization. The Gaussian function provides a smooth similarity whose decay is controlled by sigma.

For a data matrix with multiple features, build_similarity_matrix computes a similarity matrix for each feature and combines the feature-level values using the selected similarity T-norm. The diagonal is set to one.

Public API example

import numpy as np

from frsutils import build_similarity_matrix

X = np.array([[0.1, 0.2], [0.2, 0.1]], dtype=float)

similarity_matrix = build_similarity_matrix(
    X,
    similarity="gaussian",
    similarity_sigma=0.5,
    similarity_tnorm="minimum",
)

Use package-root imports in user code. The classes in frsutils.core are implementation details and may change independently of the public API.

References

1. Dubois, D., & Prade, H. (1990). Rough fuzzy sets and fuzzy rough sets. International Journal of General Systems, 17(2–3), 191–209. https://doi.org/10.1080/03081079008935107
2. Radzikowska, A. M., & Kerre, E. E. (2002). A comparative study of fuzzy rough sets. Fuzzy Sets and Systems, 126(2), 137–155. https://doi.org/10.1016/S0165-0114(01)00032-X