Power Mean Coordinate Theories in Fundamental Sciences of Estimation, Approximation, Data Modeling and Processing

by

© Ph. D. & Dr. Sc. Lev Gelimson

Academic Institute for Creating Fundamental Sciences (Munich, Germany)

Mathematical Journal

of the "Collegium" All World Academy of Sciences

Munich (Germany)

11 (2011), 13

By data modeling, processing, estimation, and approximation in classical mathematics [1], namely the well-known Euclidean distance as the quadratic mean of coordinate differences is typically used. Generally, the m-dimensional Euclidean space R^m is the set of all the points (x₁ , x₂ , ... , x_m)} with real coordinates and the Euclidean norm and distance between such points. The Minkowski distance of order q , or a q-norm distance, between points (x₁ , x₂ , ... , x_m) and (y₁ , y₂ , ... , y_m) is

[Σ_j=1^m |x_j - y_j|^q]^1/q

for any real number q ≥ 1 to provide the triangle inequality [1].

The 1-norm, especially 2-norm (Euclidean), and infinity-norm

max[_j=1^m |x_j - y_j|]

distances are common. All the remainingor q-norm distances are very rarely used.

In overmathematics [2, 3] and fundamental sciences of estimation [4], approximation [5], data modeling and processing [6], general power mean coordinate theories introduce power mean of coordinate differences for any unireal coordinates and any positive unireal power q which can also be less than 1 because the triangle inequality [1] is in overmathematics [2, 3] desired but not obligatory. This gives many additional possibilities.

Nota bene: Especially by more than 4 data points, the 2nd power (using of which is simple and bequem) can be insufficient and inadequate. Using other powers can bring complication in searching for a desired solution to a specific problem but is always possible and brings many advantages in estimation.

In the simplest 2D case, for any points M and N with coordinates pairs (x_M , y_M) and (x_N , y_N), respectively, in a rectangular Cartesian coordinate system Oxy , the well-known distance between these points as the quadratic mean of their corresponding coordinate differences is

d[M(x_M , y_M), N(x_N , y_N)] = [(x_M - x_N)² + (y_M - y_N)²]^1/2.

A natural generalization of such a distance is a power mean of coordinate differences for any power q > 0:

^qd[M(x_M , y_M), N(x_N , y_N)] = [|x_M - x_N|^q + |y_M - y_N|^q]^1/q.

Using moduli (absolute values) provides possibility to consider any power q > 0.

It is clear that d = ²d .

Like a usual distance, a power mean of coordinate differences for any power q > 0 is invariant by any translation transformation of a rectangular Cartesian coordinate system but cannot be invariant by even homogeneous multiplication of coordinates and, therefore, by linear transformation of such a coordinate system.

A usual distance is invariant by any rotation transformation of a rectangular Cartesian coordinate system.

Let us investigate the behavior of a power mean of coordinate differences for any power q > 0 by any rotation transformation of a rectangular Cartesian coordinate system. Rotate such a coordinate system Oxy about its origin O by any angle α positive in the anticlockwise direction so that we have axis Ox' from Ox and axis Oy' from Oy . Now

x' = x cos α + y sin α ,

y' = - x sin α + y cos α ,

^qd[M(x'_M , y'_M), N(x'_N , y'_N)] = [|x'_M - x'_N|^q + |y'_M - y'_N|^q]^1/q =

[|(x_M - x_N)cos α + (y_M - y_N)sin α|^q + |- (x_M - x_N)sin α + (y_M - y_N)cos α|^q]^1/q.

By q = 2 only, we have

²d[M(x'_M , y'_M), N(x'_N , y'_N)] = [(x_M - x_N)² + (y_M - y_N)²]^1/2 =

{[(x_M - x_N)cos α + (y_M - y_N)sin α]² + [- (x_M - x_N)sin α + (y_M - y_N)cos α]²}^1/2 =

[(x_M - x_N)² + (y_M - y_N)²]^1/2 = d[M(x_M , y_M), N(x_N , y_N)].

Due to the specific features of a power function and trigonometric functions sin α and cos α by q = 2 only, it is clear that by any q ≠ 2, generally, there is no rotation invariance. Therefore, if there is data rotation invariance and we want to keep (conserve) it, then we have to use a common distance only. This fully holds, e.g., for least squared and biquadratic distance theories, as well as theories of moments of inertia in fundamental sciences of estimation, approximation, data modeling and processing. This partially holds, e.g., for groupwise centralization theories, bounds and levels mean theories, circumferential and spherical theories, principal bisector partition theories, general center and bisector theories in these sciences.

Otherwise, we can use a power mean of coordinate differences by any power q > 0. This holds, e.g., for the least biquadratic method, general central normalization theories, quadratic and other power mean theories, iterative polar theories, and coordinate partition theories in these sciences.

Nota bene: Any positive power q in a power mean of coordinate differences is valid for this natural generalization of a distance itself and plays an internal role for a distance and its generalization. Any positive power p in least squared and biquadratic distance theories, theories of moments of inertia, the least biquadratic method, general central normalization theories, as well as quadratic and other power mean theories in these sciences has another meaning, e.g. raising a distance into such a power with an external role for a distance and its generalization. Therefore, we can independently choose such powers p and q .

These theories are very efficient in data estimation, approximation, and processing.

Acknowledgements to Anatolij Gelimson for our constructive discussions on coordinate system transformation invariances and his very useful remarks.

References

[1] Encyclopaedia of Mathematics. Ed. M. Hazewinkel. Volumes 1 to 10. Kluwer Academic Publ., Dordrecht, 1988-1994

[2] Lev Gelimson. Providing Helicopter Fatigue Strength: Flight Conditions [Overmathematics and Other Fundamental Mathematical Sciences]. In: Structural Integrity of Advanced Aircraft and Life Extension for Current Fleets – Lessons Learned in 50 Years After the Comet Accidents, Proceedings of the 23rd ICAF Symposium, Dalle Donne, C. (Ed.), 2005, Hamburg, Vol. II, 405-416

[3] Lev Gelimson. Overmathematics: Fundamental Principles, Theories, Methods, and Laws of Science. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010

[4] Lev Gelimson. Fundamental Science of Estimation. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010

[5] Lev Gelimson. Fundamental Science of Approximation. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010

[6] Lev Gelimson. Fundamental Science of Data Modeling and Processing. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010

[7] Lev Gelimson. General Data Direction, as well as Scatter and Trend Measure and Estimation Theories (Essential). Mathematical Journal of the “Collegium” All World Academy of Sciences, Munich (Germany), 11 (2011), 10

[8] Lev Gelimson. Corrections and Generalizations of the Least Square Method. In: Review of Aeronautical Fatigue Investigations in Germany during the Period May 2007 to April 2009, Ed. Dr. Claudio Dalle Donne, Pascal Vermeer, CTO/IW/MS-2009-076 Technical Report, International Committee on Aeronautical Fatigue, ICAF 2009, EADS Innovation Works Germany, 2009, 59-60

[9] Lev Gelimson. Least Biquadratic Distance Theories in Fundamental Sciences of Estimation, Approximation, Data Modeling and Processing (Essential). Mathematical Journal of the “Collegium” All World Academy of Sciences, Munich (Germany), 11 (2011), 11