Defuzzification

A defuzzifier collapses the aggregated output set into a crisp value. Select by name in a Mamdani system (fz.Mamdani(defuzz=...)) or call the functions directly from fuzzytool.defuzz.

Name	Method
centroid	center of gravity (default)
bisector	splits the area into two equal halves
mom	mean of maxima
som	smallest of maxima
lom	largest of maxima
wtaver / height	weighted average Σ x·y / Σ y
coa	center of the largest contiguous area

Two of these are worth knowing about specifically.

wtaver treats the samples as discrete weighted points instead of integrating the area. It is cheaper than centroid and insensitive to uneven sampling of the universe.

coa exists for a failure mode centroid has: when aggregation leaves two separated humps, the centroid lands in the valley between them — a value no rule ever supported. coa keeps only the widest connected region of positive membership and takes its centroid, so the answer is always something a rule actually argued for.

fz.Mamdani(defuzz="coa")     # never answers "between" two disjoint humps

from fuzzytool import defuzz
import numpy as np

x = np.linspace(0, 30, 501)
y = np.maximum(0, 1 - np.abs(x - 20) / 10)
defuzz.centroid(x, y)   # ~20

A custom defuzzifier is any callable (x, y) -> float; pass it directly:

fz.Mamdani(defuzz=lambda x, y: float(x[y.argmax()]))

When no rule fires the output set is empty; the crisp value is then chosen by the Mamdani system's on_no_rule policy rather than by the defuzzifier.