Triangular Fuzzy Rescaling Distance
Proposes Triangular Fuzzy Rescaling Distance (d_TR), a novel metric that integrates Linear Rescaling directly into TFN distance calculations, eliminating the need for separate normalization stages Formally proves d_TR satisfies all metric properties: non-negativity, identity, symmetry, and triangle inequality Demonstrates d_TR is bounded, scale-invariant, and origin-invariant, making it robust for heterogeneous fuzzy data Enables dimension weighting via a weighting vector for prioritizing attrib
Analysis
TL;DR
- Proposes Triangular Fuzzy Rescaling Distance (d_TR), a novel metric that integrates Linear Rescaling directly into TFN distance calculations, eliminating the need for separate normalization stages
- Formally proves d_TR satisfies all metric properties: non-negativity, identity, symmetry, and triangle inequality
- Demonstrates d_TR is bounded, scale-invariant, and origin-invariant, making it robust for heterogeneous fuzzy data
- Enables dimension weighting via a weighting vector for prioritizing attributes in multi-criteria contexts
- Applicable to synthetic indicator construction, distance-based machine learning, and multicriteria decision aiding
Why It Matters
This work addresses a fundamental limitation in fuzzy decision-making systems: the inability to compare Triangular Fuzzy Numbers across heterogeneous scales without ad-hoc preprocessing. By embedding rescaling directly into the distance metric, it simplifies pipelines and reduces error propagation from separate normalization steps, which is critical for practitioners building fuzzy-based ML systems or decision-support tools.
Technical Details
- Core innovation: d_TR combines Linear Rescaling (LRE) with triangular fuzzy number distance computation in a single unified operation, rather than treating normalization as a pre-processing step
- Mathematical properties: Rigorously proven to satisfy the four axioms of a metric space (non-negativity, identity of indiscernibles, symmetry, triangle inequality), plus boundedness, scale-invariance, and origin-invariance
- Weighting mechanism: Incorporates a weighting vector to assign relative importance to different dimensions, enabling prioritized comparison across attributes with different units or scales
- Target domain: Triangular Fuzzy Numbers (TFNs), the most widely used representation of fuzzy uncertainty in practical applications
- Application areas: Synthetic indicator construction, distance-based machine learning algorithms (e.g., fuzzy k-NN, clustering), and multicriteria decision aiding (MCDA) frameworks
Industry Insight
- Organizations relying on fuzzy logic for decision support systems can adopt d_TR to streamline data preprocessing pipelines, reducing engineering complexity and potential normalization-induced distortions
- The scale-invariance and origin-invariance properties make d_TR particularly valuable for cross-domain benchmarking and federated learning scenarios where data comes from heterogeneous sources with different measurement conventions
- As fuzzy methods gain traction in regulated industries (finance, healthcare) where explainable uncertainty quantification is required, metrics with formal mathematical guarantees like d_TR will see increased adoption in production ML systems
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