TSK Lab NoteType-1 vs Interval Type-2

Reproducible fuzzy inference study seed 2026

When does uncertainty earn its place in a fuzzy model?

A transparent benchmark comparing Type-1 and Interval Type-2 TSK inference as noisy inputs disturb a learned signal.

Gaussian antecedentsNie–Tan outputTraining-only scaling
Explore the evidence
Synthetic target 01 / 06
y = sin(x)
−π0π
A controlled testbed for separating model behavior from dataset complexity.
Type-1 test RMSELoadingSingle split · seed 2026
IT2 test RMSELoadingδ = 0.2 · same split
Held-out samplesLoading168 train / 72 test
Automated checks25Full pytest suite

The inference stack

Models & mathematics

02 / 06
Type-1 TSK

One membership value per rule and feature

Each rule measures how closely an input sits to its Gaussian center. The rule’s linear consequent is blended with the other rules according to normalized firing strength.

μkj(xj) = exp(−(xj − mkj)² / (2σkj²))
fk = ∏j μkj(xj)
Interval Type-2

A bounded width, not a new consequent

IT2 keeps the same Gaussian means but expresses width uncertainty through lower and upper membership functions. The implementation uses the specified Nie–Tan effective strength.

σlo = σ(1 − δ)   σhi = σ(1 + δ)
flo,k = ∏ μlo,kj   fhi,k = ∏ μhi,kj
TSK consequent
gk(x) = ak0 + ∑j akjxj
Type-1 output
y = ∑k fkgk(x)∑k fk
Nie–Tan output
y = ∑k(flo,k + fhi,k)gk(x)∑k(flo,k + fhi,k)
Why δ = 0 matters.When lower and upper widths coincide, flo,k = fhi,k = fk. The shared factor of two cancels during normalization, so matched parameters reproduce the Type-1 prediction.

Measured, not inferred

Results

03 / 06
Primary comparison

Same held-out split, two fuzzy interpretations

Seed 2026 · 168 training points · 72 test points · six rules · δ = 0.2

IT2 − Type-1 RMSELoading
Single-split comparison loaded from results/t1_it2_comparison_metrics.csv
ModelRMSEMAESeedDelta
Loading verified CSV…
Exploratory

Delta sensitivity

Same test split
Loaded from it2_delta_sensitivity_metrics.csv
δRMSEMAE
Loading…
10 seeds × 3 levels

Robustness snapshot

Mean ± SD
Loaded from t1_it2_robustness_aggregate.csv
Noise αModelRMSEMAE
Loading…

Interpretation boundary: these are descriptive results on one synthetic function. No Wilcoxon test or real-world claim is made here.

See the uncertainty

Visualizations

04 / 06
Six panels showing Type-1 Gaussian membership curves and IT2 lower and upper membership bounds
Membership geometryEach panel is one learned rule. The orange footprint shows the interval created by δ = 0.2; the blue curve is the Type-1 reference.Open full image ↗
Held-out sine targets with Type-1 and IT2 predictions
Held-out predictionsThe same seed-2026 test inputs are evaluated by both models against the true sine target.Open full image ↗
Robustness plot comparing RMSE and MAE across noise levels
Noise robustnessError bars show standard deviation across ten fixed seeds for the implemented noise levels.Open full image ↗

Protocol, made explicit

Experimental method

05 / 06
01

Generate

Use 240 evenly spaced inputs over [−π, π] with target y = sin(x).

02

Split

Seeded permutations create 168 training and 72 held-out test points per condition.

03

Fit

Standardization, k-means centers, spreads, and consequent fitting use training data only.

04

Stress

Robustness noise is Normal(0, (α · std(Xtrain,clean))²), sampled independently for train and test.

Specification note.SPEC.md describes a future five-level study (α = 0, 0.1, 0.2, 0.3, 0.5) and Wilcoxon comparisons. The current implemented robustness benchmark uses α = 0, 0.1, 0.3 and reports no statistical significance test; this mismatch is intentionally disclosed.

Open the work

Reproducibility & files

06 / 06
Windows / PowerShell

Run from the project root

python -m http.server 8000
# open http://localhost:8000

python -m pytest -v
python experiments\it2_comparison.py
python experiments\robustness_experiment.py
python experiments\plot_model_visualizations.py

The page loads CSV metrics with JavaScript. A local server avoids browser restrictions that can block CSV fetches from file://.