In mathematics, the Lukaszyk–Karmowski metric is a function defining a distance between two random variables or two random vectors.^[1][2] This function is not a metric as it does not satisfy the identity of indiscernibles condition of the metric, that is for two identical arguments its value is greater than zero.

The Lukaszyk–Karmowski metric D between two continuous independent random variables X and Y is defined as:

${\displaystyle D(X,Y)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|x-y|f(x)g(y)dxdy}$

where f(x) and g(y) are the probability density functions of X and Y respectively.

One may easily show that such metrics above do not satisfy the identity of indiscernibles condition required to be satisfied by the metric of the metric space. In fact they satisfy this condition if and only if both arguments X, Y are certain events described by Dirac delta density probability distribution functions. In such a case:

${\displaystyle D_{\delta \delta }(X,Y)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|x-y|\delta (x-{\mu }_x)\delta (y-{\mu }_y)dxdy=|{\mu }_x-{\mu }_y|}$

the Lukaszyk–Karmowski metric simply transforms into the metric between expected values ${\displaystyle {\mu }_x}$, ${\displaystyle {\mu }_y}$ of the variables X and Y and obviously:

${\displaystyle D_{\delta \delta }(X,X)=|{\mu }_x-{\mu }_x|=0.}$

For all the other cases however:

${\displaystyle D(X,X)>0.}$

The Lukaszyk–Karmowski metric satisfies the remaining non-negativity and symmetry conditions of metric directly from its definition (symmetry of modulus), as well as subadditivity/triangle inequality condition:

${\displaystyle \begin{array}{cc}& D(X,Z)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|x-z|f(x)h(z)dxdz={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|x-z|f(x)h(z)dxdz{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}g(y)dy\\ & ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|(x-y)+(y-z)|f(x)g(y)h(z)dxdydz\\ & \le {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(|x-y|+|y-z|)f(x)g(y)h(z)dxdydz\\ & ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|x-y|f(x)g(y)h(z)dxdydz+{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|y-z|f(x)g(y)h(z)dxdydz\\ & ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|x-y|f(x)g(y)dxdy+{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|y-z|g(y)h(z)dydz\\ & =D(X,Y)+D(Y,Z)\end{array}}$

Thus

${\displaystyle D(X,Z)\le D(X,Y)+D(Y,Z).}$

In the case where X and Y are dependent on each other, having a joint probability density function f(x, y), the L-K metric has the following form:

${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|x-y|f(x,y)dxdy.}$

If both random variables X and Y have normal distributions with the same standard deviation σ, and if moreover X and Y are independent, then D(X, Y) is given by

${\displaystyle D_{NN}(X,Y)={\mu }_{xy}+\frac{2\sigma }{\sqrt{\pi }}\exp (-\frac{{\mu }_{xy}^2}{4{\sigma }^2})-{\mu }_{xy}\mathrm{erfc}\left(\frac{{\mu }_{xy}}{2\sigma }\right),}$

where

${\displaystyle {\mu }_{xy}=|{\mu }_x-{\mu }_y|,}$

where erfc(x) is the complementary error function and where the subscripts NN indicate the type of the L-K metric.

In this case, the lowest possible value of the function ${\displaystyle D_{NN}(X,Y)}$ is given by

${\displaystyle \underset{{\mu }_{xy}\to 0}{lim}{D}_{NN}(X,Y)=D_{NN}(X,X)=\frac{2\sigma }{\sqrt{\pi }}.}$

When both random variables X and Y have uniform distributions (R) of the same standard deviation σ, D(X, Y) is given by

${\displaystyle D_{RR}(X,Y)=\{\begin{array}{cc}\frac{24\sqrt{3}{\sigma }^3-{\mu }_{xy}^3+6\sqrt{3}\sigma {\mu }_{xy}^2}{36{\sigma }^2},& {\mu }_{xy}<2\sqrt{3}\sigma ,\\ {\mu }_{xy},& {\mu }_{xy}\ge 2\sqrt{3}\sigma .\end{array}}$

The minimal value of this kind of L–K metric is

${\displaystyle D_{RR}(X,X)=\frac{2\sigma }{\sqrt{3}}.}$

In case the random variables X and Y are characterized by discrete probability distribution the Lukaszyk–Karmowski metric D is defined as:

${\displaystyle D(X,Y)=\sum _i\sum _j|x_i-y_j|P(X=x_i)P(Y=y_j).}$

For example for two discrete Poisson-distributed random variables X and Y the equation above transforms into:

${\displaystyle D_{PP}(X,Y)=\sum _{x=0}^n\sum _{y=0}^n|x-y|\frac{{{\lambda }_x}^x{{\lambda }_y}^y{e}^{-({\lambda }_x+{\lambda }_y)}}{x!y!}.}$

The Lukaszyk–Karmowski metric of random variables may be easily extended into metric D(X, Y) of random vectors X, Y by substituting ${\displaystyle |x-y|}$ with any metric operator d(x,y):

${\displaystyle D(𝐗,𝐘)={\int }_{\Omega }{\int }_{\Omega }d(𝐱,𝐲)F(𝐱)G(𝐲)d{\Omega }_{x}d{\Omega }_y.}$

For example substituting d(x,y) with an Euclidean metric and assuming two-dimensionality of random vectors X, Y would yield:

${\displaystyle D(𝐗,𝐘)={\int }_{\Omega }{\int }_{\Omega }\sqrt{\sum _{i=1}^2|x_i-y_i{|}^2}F(x_1,x_2)G(y_1,y_2)d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2.}$

This form of L–K metric is also greater than zero for the same vectors being measured (with the exception of two vectors having Dirac delta coefficients) and satisfies non-negativity and symmetry conditions of metric. The proofs are analogous to the ones provided for the L–K metric of random variables discussed above.

In case random vectors X and Y are dependent on each other, sharing common joint probability distribution F(X, Y) the L–K metric has the form:

${\displaystyle D(𝐗,𝐘)={\int }_{\Omega }{\int }_{\Omega }d(𝐱,𝐲)F(𝐱,𝐲)d{\Omega }_{x}d{\Omega }_y.}$

If the random vectors X and Y are not also only mutually independent but also all components of each vector are mutually independent, the Lukaszyk–Karmowski metric for random vectors is defined as:

${\displaystyle D_{\ast \ast }^{(p)}(𝐗,𝐘)={\left(\sum _i{D_{\ast \ast }(X_i,Y_i)}^p\right)}^{\frac{1}{p}}}$

where:

${\displaystyle D_{\ast \ast }(X_i,Y_i)}$

is a particular form of L–K metric of random variables chosen in dependence of the distributions of particular coefficients ${\displaystyle X_i}$ and ${\displaystyle Y_i}$ of vectors X, Y .

Such a form of L–K metric also shares the common properties of all L–K metrics.

• It does not satisfy the identity of indiscernibles condition:

${\displaystyle \mathrm{\forall }𝐗,𝐘D_{\ast \ast }^{(p)}(𝐗,𝐘)=0\nLeftrightarrow 𝐗=𝐘}$

since:

${\displaystyle D_{\ast \ast }^{(p)}(𝐗,𝐗)=0\iff \mathrm{\forall }i{D}_{\ast \ast }(X_i,X_i)=0}$

but from the properties of L–K metric for random variables it follows that:

${\displaystyle \mathrm{\exists }{X}_i{D}_{\ast \ast }(X_i,X_i)>0}$

• It is non-negative and symmetric since the particular coefficients are also non-negative and symmetric:

${\displaystyle \mathrm{\forall }i{D}_{\ast \ast }(X_i,Y_i)>0}$

${\displaystyle \mathrm{\forall }i{D}_{\ast \ast }(X_i,Y_i)=D_{\ast \ast }(Y_i,X_i)}$

• It satisfies the triangle inequality:

${\displaystyle \mathrm{\forall }𝐗,𝐘,𝐙D_{\ast \ast }^{(p)}(𝐗,𝐙)\le D_{\ast \ast }^{(p)}(𝐗,𝐘)+D_{\ast \ast }^{(p)}(𝐘,𝐙)}$

since (cf. Minkowski inequality):

${\displaystyle \begin{array}{cc}& {\left(\sum _i{D_{\ast \ast }(X_i,Y_i)}^p\right)}^{\frac{1}{p}}+{\left(\sum _i{D_{\ast \ast }(Y_i,Z_i)}^p\right)}^{\frac{1}{p}}\ge \\ & \ge {\left(\sum _i{D_{\ast \ast }(X_i,Y_i)+D_{\ast \ast }(Y_i,Z_i)}^p\right)}^{\frac{1}{p}}\ge \\ & \ge {\left(\sum _i{D_{\ast \ast }(X_i,Z_i)}^p\right)}^{\frac{1}{p}}\end{array}}$

The Lukaszyk–Karmowski metric may be considered as a distance between quantum mechanics particles described by wavefunctions ψ, where the probability dP that given particle is present in given volume of space dV amounts:

${\displaystyle dP=|\psi (x,y,z){|}^{2}dV.}$

For example the wavefunction of a quantum particle (X) in a box of length L has the form:

${\displaystyle {\psi }_m(x)=\sqrt{\frac{2}{L}}\sin \left(\frac{m\pi x}{L}\right),}$

In this case the L–K metric between this particle and any point ${\displaystyle \xi \in (0,L)}$ of the box amounts:

${\displaystyle \begin{array}{cc}& D(X,\xi )=\underset{0}{\overset{L}{\int }}|x-\xi ||{\psi }_m(x){|}^{2}dx=\\ & =\frac{{\xi }^2}{L}-\xi +L(\frac{1}{2}-\frac{{\sin }^2(\frac{m\pi \xi }{L})}{m^2{\pi }^2}).\end{array}}$

From the properties of the L–K metric it follows that the sum of distances between the edge of the box (ξ = 0 or ξ= L) and any given point and the L–K metric between this point and the particle X is greater than L–K metric between the edge of the box and the particle. E.g. for a quantum particle X at an energy level m = 2 and point ξ = 0.2:

${\displaystyle d(0,0.2L)+D(0.2L,X)\approx 0.2L+0.3171L=0.517L\ne D(0,X)=0.5L=d(0,0.5L).}$

Obviously the L–K metric between the particle and the edge of the box (D(0, X) or D(L, X)) amounts 0.5L and is independent on the particle's energy level.

A distance between two particles bouncing in one dimensional box of length L having time-independent wavefunctions:

${\displaystyle {\psi }_m(x)=\sqrt{\frac{2}{L}}\sin \left(\frac{m\pi x}{L}\right),}$

${\displaystyle {\psi }_n(y)=\sqrt{\frac{2}{L}}\sin \left(\frac{n\pi y}{L}\right),}$

may be defined in terms of Lukaszyk–Karmowski metric of independent random variables as:

${\displaystyle \begin{array}{cc}& D(X,Y)=\underset{0}{\overset{L}{\int }}\underset{0}{\overset{L}{\int }}|x-y||{\psi }_m(x){|}^2|{\psi }_n(y){|}^{2}dxdy\\ & =\{\begin{array}{cc}L\left(\frac{4{\pi }^2{m}^2-15}{12{\pi }^2{m}^2}\right)& m=n,\\ L\left(\frac{2{\pi }^2{m}^2{n}^2-3m^2-3n^2}{6{\pi }^2{m}^2{n}^2}\right)& m\ne n\end{array}\end{array}}$

The distance between particles X and Y is minimal for m = 1 i n = 1, that is for the minimum energy levels of these particles and amounts:

${\displaystyle min(D(X,Y))=L\left(\frac{4{\pi }^2-15}{12{\pi }^2}\right)\approx 0.2067L.}$

According to properties of this function, the minimum distance is nonzero. For greater energy levels m, n it approaches to L/3.

Suppose than we have to measure the distance between point µ_x and point µ_y, which are collinear with some point 0. Suppose further that we instructed this task to two independent and large groups of surveyors equipped with tape measures, wherein each surveyor of the first group will measure distance between 0 and µ_x and each surveyor of the second group will measure distance between 0 and µ_y.

Under the following assumptions we may consider the two sets of received observations x_i, y_j as random variables X and Y having normal distribution of the same variance σ ² and distributed over "factual locations" of points µ_x, µ_y.

Calculating the arithmetic mean for all pairs |x_i − y_j| we should then obtain the value of L–K metric D_NN(X, Y). Its characteristic curvilinearity arises from the symmetry of modulus and overlapping of distributions f(x), g(y) when their means approach each other.

An interesting experiment the results of which coincide with the properties of L-K metric was performed in 1967 by Robert Moyer and Thomas Landauer who measured the precise time an adult took to decide which of two Arabic digits was the largest. When the two digits were numerically distanced such as 2 and 9. subjects responded quickly and accurately. But their response time slowed by more than 100 milliseconds when they were closer such as 5 and 6, and subjects then erred as often as Once in every ten trials. The distance effect was present both among highly intelligent persons, as well as those who were trained to escape it.^[3]

A Lukaszyk–Karmowski metric may be used instead of a metric operator (commonly the Euclidean distance) in various numerical methods, and in particular in approximation algorithms such us radial basis function networks ,^[4][5] inverse distance weighting or Kohonen self-organizing maps.

This approach is physically based, allowing the real uncertainty in the location of the sample points to be considered. ^[6][7]

• Probabilistic metric space
• Statistical distance

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