Distancea

Distancea is a parametric distance measure used to compare two points in a common feature space. It generalizes the L_p norm and is defined for a set X in R^n and points x = (x1, ..., xn) and y = (y1, ..., yn) by d_a(x, y) = (sum_{i=1}^n w_i |x_i − y_i|^a)^{1/a}, with a > 0 and weights w_i > 0. When all weights are equal to 1, this reduces to the standard L_p distance.

Special cases and properties: a = 1 yields the Manhattan distance; a = 2 yields the Euclidean distance; as a → ∞, d_a approaches the Chebyshev distance max_i |x_i − y_i|. For a ≥ 1, d_a satisfies the metric axioms: non-negativity, identity of indiscernibles, symmetry, and the triangle inequality. For 0 < a < 1, the function is nonnegative and symmetric but generally violates the triangle inequality, so it is not a metric. The weights w_i allow anisotropic scaling across coordinates, enabling the distance to reflect feature importance or measurement units.

Computation and usage: distancea is computed in O(n) time per pair. It is used in clustering, nearest-neighbor search, and dimensionality reduction to define similarity between data points. The choice of a and the weights shape sensitivity to coordinate differences, influencing algorithm performance and results.

History and context: Distancea is part of the broader family of L_p distances introduced in functional analysis and popularized in data analysis as p-norm or Minkowski distances, with applications across machine learning and statistics. See also L_p norm, Minkowski distance, and related metrics.