Interactive Rough Sets Visualiser

Simulator Controls

Adjust parameters below to inspect changes in approximation boundaries.

Grid Granularity 40 Controls partition granulation size (equivalence classes).

Point in Target (X)

Point Outside Target

Lower Approx. (Certainty)

Upper Approx. (Possibility)

Initialising simulator...

Crisp Rough Set Approximation

Uses an equivalence grid (granulation) to compute sharp approximations.

Interactive HTML5 Canvas (500 × 500 px) Click or drag sliders to recalculate

Mathematical Background: Crisp vs Fuzzy Rough Sets

In Pawlak's classical rough set theory, approximations are built upon crisp equivalence relations that divide the universe into disjoint granules (elementary sets). The Lower Approximation contains granules completely contained within the target set (guaranteed membership), while the Upper Approximation contains granules having non-empty overlap with the target (possible membership). The difference defines the Boundary Region.

When attributes are real-valued, discretization into rigid crisp cells can produce artificial boundaries. Fuzzy-rough set theory replaces crisp partitions with continuous fuzzy similarity relations, employing fuzzy implicators (e.g. Kleene-Dienes, Gödel, Łukasiewicz) to calculate continuous degrees of lower and upper membership for every point in the feature space.

Approximation Formulation

Similarity Relation

Implicator Operator

Active Target Class

Similarity Parameter (σ)

2D Membership Map

Grid Sampling Step

Data Reduction Algorithm

2D Training Points Space Click canvas to add points

3D Membership Approximation Surface Drag to rotate / scroll to zoom

Click "Render 3D Lower Surface" or "Render 3D Upper Surface" above to generate the interactive 3D model.