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Łukaszyk-Karmowski Metric: History
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Contributor: Szymon Łukaszyk

The Łukaszyk–Karmowski metric (LK-metric) defines a distance between two random variables or vectors. LK-metric is not a metric as it does not satisfy the identity of indiscernibles axiom of the metric; for the same random variables, its value is greater than zero, providing they are not both degenerated.

  • distance functions
  • identity of indiscernibles
  • ugly duckling theorem

1. Introduction

The LK-metric[1] between two continuous random variables X and Y having a joint probability density function (PDF) F(x,y) is defined as

If X and Y are independent, then

where f(x) and g(y) are PDFs of X and Y, and subscripts denote their types. For example, if X and Y have normal PDFs having the same standard deviation σ but different means μx, μy, then

LK-metric between two random variables having normal PDFs and the same standard deviations σ = {0, 0.2, 0.4, 0.6, 0.8, 1}.

where μxy=|μx-μy|. For discrete X and Y, LK-metric has a form

and for random vectors X and Y, LK-metric becomes

where d(x,y) is a metric function, such as the Euclidean metric. In case, X and Y are mutually and internally independent, a simplified form of LK-metric can also be defined as

If X and Y are degenerated, almost sure variables having the Dirac delta (or one-point, in the discrete case) PDFs, then LK-metric becomes the metric between their mean values.

and obviously

However, in any other case

LK-metric satisfies all the remaining axioms of the metric. It is symmetric by definition, and it satisfies the triangle inequality

Thus

since

The LK-metric is not the only distance function that does not satisfy the identity of indiscernibles axiom[2]. For example, the partial metric[3] also allows each object not necessarily to have zero distance from itself. However, the partial metric satisfies two additional axioms of small self-distances and modified triangle inequality, which are not satisfied by the LK-metric[4]. Remarkably, the identity of indiscernibles ontological axiom, introduced to philosophy by Gottfried Wilhelm Leibniz around 1686 for distinguishing between monads[5], is also refuted for first-quantized bosons, fermions, and paraparticles[6] and independently faces Black’s two-spheres counterexample[7] and the trivialization dilemma. Finally, the ugly duckling theorem[8] asserts that every two objects one perceives are equally similar (or equally dissimilar). Consequently, the identity of indiscernibles is neither a logical nor an empirical principle.

This characteristic non-zero distance effect built in the LK-metric allows for avoiding ill-conditioning problems in radial basis function interpolation[9][10] and inverse distance weighting[11][12][13][14], where the interpolation accuracy can be improved by choosing the type of distance metric[15][14] and leads to a smooth interpolation function[16]. By preventing zero distances due to parameter uncertainty, the LK-metric can, furthermore, be used to analyze nondeterministic dynamical systems with competing attractors [17]. Since the LK-metric represents the mean of distances between all the outcomes of the two uncertain objects, it can also be used in uncertain nearest neighbor classification[18]. The actual value of an uncertain object is modeled by a probability density function[19]. LK-metric has been successfully applied in various fields of science and technology[20][21][22][23][24][25][26][27][16][28][29][30][31][32][33][10][34][35][36][37][38][13][39][40][41][42][43][44][45][46][47][48][14][49][50][51][52][17][53][54].

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