User:Guswen/Lukaszyk-Karmowski metric

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In mathematics, the Lukaszyk-Karmowski metric is a function defining a distance between two random variabless 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.

Continuous random variables[edit]

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

D(X, Y) = \int_{-\infty}^\infty \int_{-\infty}^\infty |x-y|f(x)g(y) \, dx\, dy

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:

D_{\delta\delta}(X, Y) = \int_{-\infty}^\infty \int_{-\infty}^\infty |x-y|\delta(x-\mu_x)\delta(y-\mu_y) \, dx\, dy = |\mu_x-\mu_y|

the Lukaszyk-Karmowski metric simply transforms into the metric between expected values \mu_x, \mu_y of the variables X and Y and obviously:

D_{\delta\delta}(X, X) = \int_{-\infty}^\infty \int_{-\infty}^\infty |x-x'|\delta(x-\mu_x)\delta(x'-\mu_x) \, dx\, dx' = |\mu_x-\mu_x| = 0.

For all the other cases however:

D\left(X, X\right) > 0.

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

\begin{align}
&{} D(X, Z) = \int_{-\infty}^\infty \int_{-\infty}^\infty |x-z|f(x)h(z) \, dx\, dz\ = \int_{-\infty}^\infty \int_{-\infty}^\infty |x-z|f(x)h(z) \, dx\, dz \int_{-\infty}^\infty g(y) dy\ = \\
&{} = \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty |(x-y)+(y-z)|f(x)g(y)h(z) \, dx\, dy\, dz\ \le \\
&{} \le \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty (|x-y|+|y-z|)f(x)g(y)h(z) \, dx\, dy\, dz\ = \\
&{} = \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty |x-y|f(x)g(y)h(z) \, dx\, dy\, dz\ + \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty |y-z|f(x)g(y)h(z) \, dx\, dy\, dz\ = \\
&{} = \int_{-\infty}^\infty \int_{-\infty}^\infty |x-y|f(x)g(y) \, dx\, dy\ + \int_{-\infty}^\infty \int_{-\infty}^\infty |y-z|g(y)h(z) \, dy\, dz\ = \\
&{} = D(X, Y) + D(Y, Z)
\end{align}

Therefore:


D(X, Z) \le D(X, Y)+D(Y, Z)

(\int_{-\infty}^\infty f(x) dx\ = \int_{-\infty}^\infty g(y) dy\ = \int_{-\infty}^\infty h(z) dz\ = 1)

L-K metric between two random variables X and Y having normal distributions and the same standard deviation \sigma = 0, \sigma = 0.2, \sigma = 0.4, \sigma = 0.6, \sigma = 0.8, \sigma = 1 (starting with the bottom curve). m_{xy} = |\mu_x-\mu_y| denotes a distance between means of X and Y.

In case if X and Y are dependent form each other, sharing a common joint probability distribution F(x, y), L-K metric has the following form:

\int_{-\infty}^\infty \int_{-\infty}^\infty |x-y|F(x, y) \, dx\, dy.

Example: two continuous random variables with normal distributions (NN)[edit]

If both probability distribution functions of random variables X and Y are normal distributions (N) having the same standard deviation σ, and moreover X and Y are independent, then evaluating D(XY) yields


D_{NN}(X, Y) = \mu_{xy} + \frac{2\sigma}{\sqrt\pi}\operatorname{exp}\left(-\frac{\mu_{xy}^2}{4\sigma^2}\right)-\mu_{xy} \operatorname{erfc} \left(\frac{\mu_{xy}}{2\sigma}\right)

where

\mu_{xy} = \left|\mu_x-\mu_y\right|,

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

In this case "zero value" of the function D_{NN}(X, Y) amounts:

\lim_{\mu_{xy}\to 0} D_{NN}(X, Y) = D_{NN}(X, X) = \frac{2\sigma}{\sqrt\pi}.

Example: two continuous random variables with uniform distributions (RR)[edit]

In case both random variables X and Y are characterized by uniform distributions (R) of the same standard deviation σ, integrating D(XY) yields:

D_{RR}(X, Y) = \begin{cases} \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{cases}

The minimal value of this kind of L-K metric amounts:

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

Discrete random variables[edit]

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

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:

D_{PP}(X, Y) = \sum_{x=0}^n\sum_{y=0}^n |x-y|\frac{{\lambda_x}^x{\lambda_y}^ye^{-(\lambda_x+\lambda_y)}}{x!y!}.

Random vectors[edit]

equidistant surface for Euclidean metric d^{2}(\mathbf{x},\mathbf{0}), \left(\mathbf{x,0}\right) \in \mathbb{R}^2
equidistant surface for Euclidean L-K metric D_{R\delta}^{2}(\mathbf{X},\mathbf{0}), \left(\mathbf{X,0}\right): \Omega \to \mathbb{R}^2

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

D(\mathbf{X}, \mathbf{Y}) =\int_{\Omega} \int_{\Omega} d(\mathbf{x}, \mathbf{y})F(\mathbf{x})G(\mathbf{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:

D(\mathbf{X}, \mathbf{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) \, dx_1\, dx_2\, dy_1\, dy_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:

D(\mathbf{X}, \mathbf{Y}) =\int_{\Omega} \int_{\Omega} d(\mathbf{x}, \mathbf{y})F(\mathbf{x}, \mathbf{y}) \, d\Omega_x \, d \Omega_y.

Random vectors - the Euclidean form[edit]

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:

D_{**}^{(p)}(\mathbf{X}, \mathbf{Y}) = \left( {\sum_i{D_{**}(X_i, Y_i)}^p} \right)^{\frac1p}

where:

D_{**}(X_i, Y_i)\,

is a particular form of L-K metric of random variables chosen in dependence of the distributions of particular coefficients X_i and 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:
\forall{\mathbf{X}, \mathbf{Y}}\ D_{**}^{(p)}(\mathbf{X}, \mathbf{Y}) = 0 \ \nLeftrightarrow \ \mathbf{X} = \mathbf{Y} \,
since:
D_{**}^{(p)}(\mathbf{X}, \mathbf{X}) = 0 \Leftrightarrow \ \forall{i} \ D_{**}(X_i, X_i) = 0
but from the properties of L-K metric for random variables it follows that:
\exists\ X_i\ D_{**}(X_i, X_i) > 0
  • It is non-negative and symmetric since the particular coefficients are also non-negative and symmetric:
\forall\ i \ D_{**}(X_i, Y_i) > 0 \,
\forall\ i \ D_{**}(X_i, Y_i) = D_{**}(Y_i, X_i)
  • It satisfies the triangle inequality:
\forall\ \mathbf{X}, \mathbf{Y}, \mathbf{Z} \ D_{**}^{(p)}(\mathbf{X}, \mathbf{Z}) \le D_{**}^{(p)}(\mathbf{X}, \mathbf{Y}) + D_{**}^{(p)}(\mathbf{Y}, \mathbf{Z})
since (cf. Minkowski inequality):
\begin{align}
&{} \left( {\sum_i{D_{**}(X_i, Y_i)}^p} \right)^{\frac1p} + \left( {\sum_i{D_{**}(Y_i, Z_i)}^p} \right)^{\frac1p}\ \ge \\
&{} \ge \left( {\sum_i{D_{**}(X_i, Y_i) + D_{**}(Y_i, Z_i)}^p} \right)^{\frac1p} \ge \\
&{} \ge \left( {\sum_i{D_{**}(X_i, Z_i)}^p} \right)^{\frac1p}
\end{align}

Physical interpretation[edit]

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:

dP = |\psi(x, y, z)|^2 dV\,.

A quantum particle in a box[edit]

L-Kmetric between a quantum particle in one dimensional box of length L and a given point ξ of the box (0 \le \xi \le L).

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

\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 \xi \in (0, L)\, of the box amounts:

\begin{align}
&{} D(X, \xi) = \int\limits_{0}^L |x-\xi||\psi_m(x)|^2dx = \\
&{} = \frac{\xi^2}{L} - \xi +L\left(\frac{1}{2}-\frac{\sin^2(\frac{m\pi\xi}{L})}{m^2\pi^2}\right).
\end{align}

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:

d(0,0.2L) + D(0.2L, X) \approx 0.2L + 0.3171L = 0.517L \neq 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.

Two quantum particles in a box[edit]

Probability metric D(Y, Y) between two particles X, Y in a potential well for the first ten energy values m, n of these particles.

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

\psi_m(x) = \sqrt{\frac{2}{L}} \sin{\left(\frac{m \pi x}{L} \right)}, \,
\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:

\begin{align}
&{} D(X, Y) = \int\limits_{0}^L \int\limits_{0}^L |x-y||\psi_m(x)|^2|\psi_n(y)|^2 \, dx\, dy \\
&{} = L\left(\frac{1}{3}(m+n)^2 - \frac{m^4 + n^4 + 2m^3n + 2mn^3 + 2m^2n^2}{2m^2n^2\pi^2} \right).
\end{align}

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

\min(D(X, Y)) = L\left(\frac{4}{3}-\frac{4}{\pi^2} \right) \approx 0.93L \,.

According to properties of this function, the minimum distance is nonzero. In fact it is close to the length L of the potential well. For other energy levels it is even greater than the length of the well.

Popular explanation[edit]

Normal distributions of two random variables X and Y of the same variance for three locations of their means μx, μy

Suppose than we have to measure the distance between point μx and point μy, 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 μx and each surveyor of the second group will measure distance between 0 and μy.

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

Calculating the arithmetic mean for all pairs |xi - yj| we should then obtain the value of L-K metric DNN(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].

Practical applications[edit]

Lukaszyk-Karmowski metric may be used instead of a metric operator (commonly 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 map.

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

See also[edit]

References[edit]

  1. ^ Metryka Pomiarowa, przykłady zastosowań aproksymacyjnych w mechanice doświadczalnej (Measurement metric, examples of approximation applications in experimental mechanics), PhD thesis, Szymon Łukaszyk (author), Wojciech Karmowski (supervisor), Tadeusz Kościuszko Cracow University of Technology, submitted December 31, 2001, completed March 31, 2004
  2. ^ A new concept of probability metric and its applications in approximation of scattered data sets, Łukaszyk Szymon, Computational Mechanics Volume 33, Number 4, 299-304, Springer-Verlag 2003 doi: 10.1007/s00466-003-0532-2
  3. ^ The Number Sense: How the Mind Creates Mathematics, Stanislas Dehaene, Oxford University Press US, 1999, ISBN 0195132408, p. 73-75
  4. ^ Radial Basis Function, Lambert M. Surhone, Miriam T. Timpledon, Susan F. Marseken, VDM Publishing House 2010, ISBN 9786131215087
  5. ^ Classification of Arabic Documents by a Model of Fuzzy Proximity with a Radial Basis Function, Taher Zaki, Driss Mammass, Abdellatif Ennaji, Fathallah Nouboud, International Journal of Future Generation Communication and Networking Vol. 3, No. 4, December, 2010
  6. ^ Small-scale health-related indicator acquisition using secondary data spatial interpolation, Gang Meng, Jane Law, Mary E. Thompson, International Journal of Health Geographics 2010, , 9:50 doi:10.1186/1476-072X-9-50
  7. ^ Social and Spatial Determinants of Adverse Birth Outcome Inequalities in Socially Advanced Societies. A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Planning, Gang Meng, University of Waterloo, Canada, 2010