Minkowski Distance - generalised Lp norm (p ≥ 1)

Overview

d_p ≥ 0. Monotone in p: d_1 ≥ d_2 ≥ … ≥ d_∞ for the same pair. p controls the sensitivity to large differences.

Output

distance, lower is better

Data

Crisp, complete numeric matrix

Size

2+ alternatives, 3-12 criteria works best

Used for

Similarity assessment, clustering

How it works

1. 1

   Compute Minkowski distance with parameter p: d_p = (Σ_j |a_j−b_j|^p)^{1/p}. Special cases: p=1→Manhattan; p=2→Euclidean; p→∞→Chebyshev.

   $$d_p(\mathbf{a},\mathbf{b})={\left(\sum _{j=1}^n|a_j-b_j{|}^p\right)}^{1/p},p\ge 1$$

   Minkowski 1910 (Lp norm; pending PDF page verification)

Edge cases and pitfalls

• Large p amplifies the maximum gap; small p spreads weight over all gaps equally.

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