A-FIS — A-Subsethood Fuzzy Inference System

A Python library implementing A-FIS (A-subsethood based Fuzzy Inference System), with rule learning, inference, regression, and visualization.

This library is the implementation accompanying the following conference paper:

A-Subsethood Fuzzy Inference System 2025 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE 2025) Reims, France, 06–10 July 2025 DOI: 10.1109/FUZZ62266.2025.11152043

The paper introduces an analogical reasoning scheme grounded in lattice theory that uses a novel subsethood measure to handle sparse fuzzy rule bases — directly activating the most relevant rules without requiring rule interpolation. An algorithmic strategy for regression and time series prediction is proposed, constructing a dedicated fuzzy inference system for each training point and combining their outputs via a distance-weighted approach. Experimental results demonstrate competitive performance with fewer rules compared to interpolation-based methods.

A preprint version of the paper is available in this repository: AFIS_FUZZ_IEEE_2025.pdf.

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Package Structure

afis/
├── core/
│   ├── afis_utils.py      # Membership functions, fuzzy rule structures, defuzzification
│   ├── A_FIS.py           # Main inference algorithm
│   ├── A_vee_B.py         # Supremum ν(A ∨ B) — analytical and numerical
│   └── wangmendel.py      # Wang-Mendel rule learning
├── visualization/
│   └── plotting.py        # Rule base plots, supremum visualization, diagnostics
├── regression/
│   ├── regressor.py       # AFISRegressor, k-fold evaluation, benchmarking
│   └── utils.py           # Correlation analysis, metrics, result plots
└── examples/
    ├── inference/          # 1D, ND, Gaussian, and supremum notebooks
    └── regression/         # Regression application notebook

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Installation

python3 -m venv venv
source venv/bin/activate          # Windows: venv\Scripts\activate
pip install numpy scipy scikit-learn pandas plotly tqdm jupyterlab

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Quick Start

Inference

from afis.core import FuzzySet, InferiorBorder, SuperiorBorder, Triangular, format_FN_N_Dim
from afis.visualization import create_rule_base, run_afis, plot_results

A1 = FuzzySet("A1", InferiorBorder(0, 5))
A2 = FuzzySet("A2", Triangular(0, 5, 10))
A3 = FuzzySet("A3", SuperiorBorder(5, 10))
Y1 = FuzzySet("Y1", InferiorBorder(0, 5))
Y2 = FuzzySet("Y2", Triangular(0, 5, 10))
Y3 = FuzzySet("Y3", SuperiorBorder(5, 10))

rule_base = create_rule_base(
    antecedents=[[A1, A2, A3]],
    consequents=[Y1, Y2, Y3],
    input_ranges=[(0, 10)],
    output_range=(0, 10),
    rules=[([0], 0), ([1], 1), ([2], 2)]
)

x = format_FN_N_Dim([3.0])
output, U, y_crisp, _ = run_afis(x, rule_base)
plot_results(rule_base, x, output, U, y_crisp)

Regression

import numpy as np
from afis.regression import AFISRegressor, generate_rule_base, evaluate_kfold

X, y = ...  # your data

config = {
    'agg_method': 'avg',        # 'avg' | 'min' | 'max' | 'product' | ['weighted_avg', weights]
    't_norm_type': 'product',   # 'product' | 'min' | 'luka' | ['hamacher', param]
    'imp_params': ['luka', 1],  # implication type and parameter
    'k_max': 10,                # neighbor search ceiling (or fixed k with 'k_fixed')
    'p_norm': 1,                # Minkowski distance order
}

rule_base = generate_rule_base(X, y, n_fuzzy_partitions=5, n_rules=30)

model = AFISRegressor(config)
model.fit(X, y, rule_base, X_val=X_val, y_val=y_val, optimize_k=True)
predictions = model.predict(X_test)
model.save('model.pkl')

K-Fold Cross-Validation

results = evaluate_kfold(
    X, y,
    rule_base_generator=lambda X, y: generate_rule_base(X, y, 5, 30),
    config=config,
    n_splits=5,
    n_repetitions=3,
)
print(f"RMSE: {results['mean_rmse']:.4f} ± {results['std_rmse']:.4f}")

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Membership Functions

Class	Parameters	Shape
Triangular	ini, top, end	Triangle
Trapezoidal	ini, top1, top2, end	Trapezoid
InferiorBorder	top, end	Left-saturated ramp
SuperiorBorder	ini, top	Right-saturated ramp
Gaussian	center, sigma	Gaussian bell

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Key API

afis.core

Symbol	Description
A_FIS(input, rule_base, ...)	Main inference function
format_FN_N_Dim(x, ftype)	Format inputs for A_FIS
fuzzy_imp(a, b, type, param)	Fuzzy implication I(a, b)
nu_A_vee_B_auto(A, B, U)	Supremum area ν(A ∨ B), auto-dispatch
wangmendel.generate_rule_base(...)	Generate rule base from data

afis.visualization

Symbol	Description
create_rule_base(...)	Build a FuzzyRuleBase from lists
run_afis(input, rule_base, ...)	Run inference and defuzzify
plot_results(...)	Plot antecedents, output, and defuzzified value
plot_supremum(...)	Visualize ν(X ∨ A) and SVI
show_svi_table(...)	Print SVI and S_A per rule
compute_activation_curves(...)	Sweep input and compute S_A curves

afis.regression

Symbol	Description
AFISRegressor	Full regression model: fit, predict, save, load
generate_rule_base(...)	Wang-Mendel rule generation
evaluate_kfold(...)	K-fold cross-validation
benchmark_method(...)	Repeated K-fold with held-out test sets
compute_metrics(y_true, y_pred)	Returns RMSE, MAE, R²

AFISRegressor config reference

Key	Default	Description
k_max	10	Upper bound for neighbor search; used as fixed k when optimize_k=False
k_fixed	—	If set, skips k optimization and uses this value directly
p_norm	1	Minkowski distance order for neighbor lookup
agg_method	'avg'	Aggregation across input dimensions
t_norm_type	'product'	T-norm for consequent scaling
imp_params	['luka', 1]	Implication type and parameter
param_range	(0.1, 50.0)	Search range for parametric implication optimization
param_step	0.5	Step size for implication parameter grid search
disc	100	Discretization points for numerical integration

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License

This project is licensed under the MIT License. You are free to use, modify, and distribute this code, provided that the original authors are credited.

If you use this library in academic work, please cite the reference paper above.

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Credits

The theoretical framework underlying this library was co-authored with Prof. Peter Sussner (IMECC — Institute of Mathematics, Statistics and Scientific Computing, University of Campinas, Brazil).

Fuzzy set classes and Wang-Mendel rule learning are based on the original work of Renato Lopes Moura, used here with modifications: https://github.com/renatolm/wang-mendel