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Łukaszyk, S. Łukaszyk-Karmowski Metric. Encyclopedia. Available online: https://encyclopedia.pub/entry/56660 (accessed on 16 June 2024).

Łukaszyk S. Łukaszyk-Karmowski Metric. Encyclopedia. Available at: https://encyclopedia.pub/entry/56660. Accessed June 16, 2024.

Łukaszyk, Szymon. "Łukaszyk-Karmowski Metric" *Encyclopedia*, https://encyclopedia.pub/entry/56660 (accessed June 16, 2024).

Łukaszyk, S. (2024, May 19). Łukaszyk-Karmowski Metric. In *Encyclopedia*. https://encyclopedia.pub/entry/56660

Łukaszyk, Szymon. "Łukaszyk-Karmowski Metric." *Encyclopedia*. Web. 19 May, 2024.

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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

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 μ_{x}, μ_{y}, 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 μ_{xy}=|μ_{x}-μ_{y}|. 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

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 LK-metric^{[4]}. Remarkably, the identity of indiscernibles ontological axiom, introduced to philosophy by Gottfried Wilhelm Leibniz around 1686, is also invalidated by the ugly duckling theorem^{[5]} stated in 1969 and asserting that every two objects one perceives are equally similar (or equally dissimilar). Consequently, the identity of indiscernibles is neither a logical nor empirical principle.

This characteristic non-zero distance effect built in LK-metric allows to avoid ill-conditioning problems in radial basis function interpolation^{[6]}^{[7]} and inverse distance weighting^{[8]}^{[9]}^{[10]}^{[11]}, where the interpolation accuracy can be improved by choosing the type of distance metric^{[12]}^{[11]} and leads to a smooth interpolation function^{[13]}. By preventing zero distances based on parameter uncertainty, LK-metric can, furthermore, be used in analysis of nondeterministic dynamical systems with competing attractors^{[14]}. Since 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^{[15]}. The actual value of an uncertain object is modeled by a probability density function^{[16]}. LK-metric has been successfully applied in various fields of science and technology^{[17]}^{[18]}^{[19]}^{[20]}^{[21]}^{[22]}^{[23]}^{[24]}^{[13]}^{[25]}^{[26]}^{[27]}^{[28]}^{[29]}^{[30]}^{[7]}^{[31]}^{[32]}^{[33]}^{[34]}^{[35]}^{[10]}^{[36]}^{[37]}^{[38]}^{[39]}^{[40]}^{[41]}^{[42]}^{[43]}^{[44]}^{[45]}^{[11]}^{[46]}^{[47]}^{[48]}^{[49]}^{[14]}^{[50]}^{[51]}.

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