FuzzyMethods: Adaptive Fuzzy Systems and Neuro-Fuzzy Modeling 📊

This repository contains implementations and experiments with fuzzy logic-based systems focused on nonlinear function approximation, regression, and predictive modeling.

The main goal of this project is to investigate and validate methods for:

• Automatic fuzzy rule extraction;
• Neuro-fuzzy learning systems;
• Nonlinear function approximation;
• Dynamic system modeling;
• Prediction of complex and chaotic behaviors.

The project explores two main approaches:

• Wang-Mendel fuzzy rule extraction, focused on interpretability and symbolic knowledge generation;
• ANFIS (Adaptive Neuro-Fuzzy Inference System), focused on parameter optimization through neural learning.

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🛠 Technologies

• Python
• NumPy
• TensorFlow / Keras
• Fuzzy Logic Systems
• ANFIS
• Wang-Mendel Algorithm
• Neuro-Fuzzy Networks
• Scientific Computing
• Data Analysis

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🧠 Implemented Models

1. Wang-Mendel Fuzzy Rule Extraction (WMModel)

The classical Wang-Mendel (WM) algorithm is implemented to generate an interpretable fuzzy knowledge base from numerical data.

The model maps input-output relationships by extracting fuzzy rules directly from observations.

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Dynamic Fuzzy Partitioning

The input space is partitioned dynamically using five triangular membership functions based on statistical properties of the dataset:

min, q25, q50, q75, max

This approach allows the model to automatically adapt the fuzzy universe of discourse according to the distribution of the input data.

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Antecedent Combination

The fuzzy antecedents are combined using the product T-norm:

$$ \mu_A(x)=\prod_i \mu_i(x_i) $$

where each membership value contributes to the final rule activation degree.

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Defuzzification

The output prediction is obtained using a weighted center-of-area approach:

$$ y=\frac{\sum_i w_i c_i}{\sum_i w_i} $$

where:

• $w_i$ represents the rule activation degree;
• $c_i$ represents the consequent center.

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Adaptive Geometric Extrapolation

One of the main contributions of this implementation is an adaptive extrapolation mechanism.

When a combination of fuzzy antecedents has no corresponding training samples, the model performs a geometric neighborhood search.

The algorithm searches for rules sharing exactly:

$$ N-1 $$

common antecedents.

The final prediction is estimated using:

• Euclidean distance between membership-function centers;
• Degree of certainty (DOC) weighting.

This allows the model to generalize beyond observed regions of the input space.

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2. ANFIS (Adaptive Neuro-Fuzzy Inference System)

The project also implements a neuro-fuzzy model based on the Takagi-Sugeno architecture.

Unlike Wang-Mendel, which generates static fuzzy rules, ANFIS represents the fuzzy inference system as a trainable neural architecture.

The model optimizes:

• Membership function parameters (antecedent layer);
• Linear consequent coefficients.

Optimization is performed through learning algorithms such as:

• Gradient descent;
• Hybrid optimization strategies.

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ANFIS Architecture

The model follows the classical five-layer structure:

1. Fuzzification layer

   • Computes membership values.

2. Rule layer

   • Calculates rule activation strengths.

3. Normalization layer

   • Normalizes firing strengths.

4. Consequent layer

   • Computes Takagi-Sugeno functions.

5. Output layer

   • Aggregates the final prediction.

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🔬 Experiments

XOR Logical Problem

The classical XOR (Exclusive OR) problem was used to evaluate the ability of fuzzy systems to represent nonlinear decision boundaries.

Since XOR is not linearly separable, it is a useful benchmark for testing whether fuzzy rules and overlapping membership functions can generate complex decision surfaces.

The experiment evaluates:

• Rule granularity;
• Membership overlap;
• Nonlinear separation capability.

Both models were evaluated:

• Wang-Mendel;
• ANFIS.

Although both achieved satisfactory results, ANFIS obtained better performance due to its adaptive parameter optimization capability.

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📈 Dynamic Benchmark

Narendra-Li Chaotic System Approximation

To evaluate the capability of fuzzy models to approximate nonlinear and chaotic dynamics, the classical Narendra-Li benchmark was used.

The dataset is transformed into an autoregressive temporal representation using a lag window.

The model receives five input variables.

Inputs

$$ X = {y_t, y_{t-1}, y_{t-2}, u_t, u_{t-1}} $$

Target

The target variable represents the continuous nonlinear dynamic signal:

$$ y $$

The benchmark evaluates the ability of fuzzy systems to model temporal dependencies and nonlinear behavior.

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📊 Results

XOR Benchmark

Modelo Wang-Mendel

Modelo ANFIS

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Narendra-Li Benchmark

Model	RMSE	MAE
Wang-Mendel	0.1376	0.2974
ANFIS	0.0005	0.0168

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📁 Project Structure

├── fuzzy_model.py          # WMModel implementation and rule extrapolation logic
├── anfis_model.py          # ANFIS neural fuzzy architecture
├── utils.py                # Membership functions, inference utilities, and helpers
├── teste_narendra_li.py    # Narendra-Li benchmark training and evaluation
├── validacao_xor.py        # XOR experiment and validation
├── pyproject.toml          # Project configuration and dependencies (uv)
└── logs/                   # Training logs, JSON metrics, and CSV results

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🚀 Future Improvements

Possible future extensions:

• Add additional fuzzy partitioning strategies;
• Compare different membership functions (Gaussian, trapezoidal, triangular);
• Implement alternative T-norm operators;
• Add GPU acceleration for ANFIS training;
• Compare against traditional neural networks;
• Create visualization dashboards for fuzzy rules;
• Evaluate additional nonlinear dynamic benchmarks.

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📚 References

• Wang, L. X., & Mendel, J. M. (1992). Generating fuzzy rules by learning from examples.
• Jang, J. S. R. (1993). ANFIS: Adaptive-Network-Based Fuzzy Inference System.
• Narendra, K. S., & Li, C. (1996). Identification and control of dynamical systems using neural networks.