Abstract
This analysis introduces a generalization of the basic statistical concepts of expectation values and variance for non-Euclidean metrics induced by Lp-norms. The non-Euclidean Lp means are defined by exploiting the fundamental property of minimizing the Lp deviations that compose the Lp variance. These Lp expectation values embody a generic formal scheme of means characterization. Having the p-norm as a free parameter, both the Lp-normed expectation values and their variance are flexible to analyze new phenomena that cannot be described under the notions of classical statistics based on Euclidean norms. The new statistical approach provides insights into regression theory and Statistical Physics. Several illuminating examples are examined.
Highlights
2500 years after the Pythagorean discipline and the “self-imposed” intense study on the arithmetic, geometric and harmonic means, the power-means of the elements {yi }N i=1, yi ∈ Dy ⊆ R, ∀ i = 1, . . . , N, ∑ N p 1/p, were introduced [1] as suitable generalization of the former given by Mp ({yi }N i=1 ) = ( i=1 yi /N )Pythagorean ones, for p = 1, 0, −1, respectively
In [7] a novel generalized characterization of means was introduced, namely, the non-Euclidean means, based on metrics induced by Lp norms, wherein the median is included as special case for p = 1 (L1 ), while the non-Euclidean φ-means can be defined. (See the work of [19], where the general clustering approaches is investigated using, among others, similar well-defined means based on non-Euclidean optimization.) In this way, the Lp expectation value of a given energy spectrum k=1 is defined, representing the non-Euclidean adaptation of internal energy Up
In the following three examples from Statistical Mechanics, we examine the systems of (1) gas in thermal equilibrium, (2) space plasmas out of thermal equilibrium, and (3) multi-dimensional quantum harmonic oscillator at thermal equilibrium
Summary
As soon as the nonlinear function φ is appropriately chosen, φ-means-based filters can efficiently reduce noise at a preferred scale of signal values. In [7] a novel generalized characterization of means was introduced, namely, the non-Euclidean means, based on metrics induced by Lp norms, wherein the median is included as special case for p = 1 (L1 ), while the non-Euclidean φ-means can be defined. (See the work of [19], where the general clustering approaches is investigated using, among others, similar well-defined means based on non-Euclidean optimization.) In this way, the Lp expectation value of a given energy spectrum.
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