Interval type-2 (IT2)

A type-1 fuzzy set assigns each point a single membership degree. An interval type-2 set assigns an interval [lower, upper]: it is bounded by a lower membership function (LMF) and an upper membership function (UMF), and the gap between them is the footprint of uncertainty (FOU). This models uncertainty about the membership function itself — useful when "good" or "high" has no sharp, agreed-upon definition.

IT2 rules use the same operator syntax as type-1 rules; only the membership functions and the engine change.

Building IT2 sets

import fuzzytool as fz

# A Gaussian whose mean is uncertain across [630, 680]:
fair = fz.it2_gauss_uncertain_mean(630, 680, sigma=60)

# A Gaussian with uncertain spread:
mid = fz.it2_gauss_uncertain_std(5.0, sigma_lo=0.8, sigma_hi=1.4)

# Height uncertainty: LMF is a scaled-down copy of a type-1 MF:
hi = fz.it2_scale(fz.gauss(8, 1.5), scale=0.6)

# Or supply the lower and upper MFs explicitly:
custom = fz.it2(lower=fz.tri(2, 5, 8), upper=fz.tri(0, 5, 10))

Membership is an interval — fair(650) returns (lower, upper); the engines also read fair.lower(x) and fair.upper(x).

IT2 inference

IT2Mamdani uses center-of-sets type reduction; IT2TSK type-reduces crisp consequents directly. Both return the midpoint of the type-reduced interval [y_l, y_r].

score   = fz.Variable("score", (300, 850))
score["good"] = fz.it2_gauss_uncertain_mean(740, 780, 50)
score["poor"] = fz.it2_gauss_uncertain_mean(420, 480, 70)
premium = fz.Variable("premium", (0, 12))
premium["low"]  = fz.it2_gauss_uncertain_mean(1.5, 2.5, 1.5)
premium["high"] = fz.it2_gauss_uncertain_mean(9.5, 10.5, 1.5)

sys = fz.IT2Mamdani()
sys.rule(score["good"], premium["low"])
sys.rule(score["poor"], premium["high"])
sys(score=780)    # -> 2.29  (midpoint of the type-reduced interval)

A ready-made example is fuzzytool.datasets.credit_risk_it2.

Type reduction (Karnik-Mendel)

Type reduction collapses the interval-valued output into a crisp interval [y_l, y_r]. fuzzytool implements the Karnik-Mendel algorithm as a single reusable primitive (fuzzytool.type2.reduction): km_endpoint finds one endpoint, karnik_mendel returns both, and centroid_it2 applies it to compute an IT2 set's centroid interval.

import numpy as np
from fuzzytool.type2.reduction import centroid_it2

good = fz.it2_gauss_uncertain_mean(740, 780, 50)
universe = np.linspace(300, 850, 400)
centroid_it2(good, universe)   # -> (738.05, 772.07), the [y_l, y_r] centroid interval

Visualization

import matplotlib.pyplot as plt
from fuzzytool import viz

viz.plot_it2_variable(score)   # draws each term's LMF/UMF with a shaded FOU
plt.show()