Power Mean Theories for Three Dimensions 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), 16

To solving contradictory (e.g., overdetermined) problems in approximation and data processing, the least square method (LSM) [1] by Legendre and Gauss only usually applies.

In overmathematics [2, 3] and fundamental sciences of estimation [4], approximation [5], data modeling [6] and processing [7], power mean theories (PMT) naturally generalize quadratic mean theories (QMT) and are valid by coordinate system linear transformation invariance of the given data, too. Additionally to the 2D case, show the essence of power mean theories by linear approximation in the 3D case.

Given n (n ∈ N⁺ = {1, 2, ...}, n > 3) points [_j=1ⁿ (x'_j , y'_j , z'_j)] = {(x'₁ , y'₁ , z'₁), (x'₂ , y'₂ , z'₂), ... , (x'_n , y'_n , z'_n)] with any real coordinates. Use clearly invariant centralization transformation x = x' - Σ_j=1ⁿ x'_j / n , y = y' - Σ_j=1ⁿ y'_j / n , z = z' - Σ_j=1ⁿ z'_j / n to provide coordinate system Oxyz central for the given data and further work in this system with points [_j=1ⁿ (x_j , y_j , z_j)] to be approximated with a plane ax + by + cz = 0 containing origin O(0, 0, 0).

Quadratic mean theories use the signed geometric mean similarly to the case of two dimensions [2-7] pairwise using the obtained values of a/b and b/a , b/c and c/b , as well as a/c and c/a :

a/b = sign(Σ_j=1ⁿ x_jz_j Σ_j=1ⁿ y_jz_j - Σ_j=1ⁿ x_jy_j Σ_j=1ⁿ z_j²) {[Σ_j=1ⁿ y_j² Σ_j=1ⁿ z_j² - (Σ_j=1ⁿ y_jz_j)²]/[Σ_j=1ⁿ x_j² Σ_j=1ⁿ z_j² - (Σ_j=1ⁿ x_jz_j)²]}^1/2,

b/c = sign(Σ_j=1ⁿ x_jy_j Σ_j=1ⁿ x_jz_j - Σ_j=1ⁿ x_j² Σ_j=1ⁿ y_jz_j) {[Σ_j=1ⁿ x_j² Σ_j=1ⁿ z_j² - (Σ_j=1ⁿ x_jz_j)²]/[Σ_j=1ⁿ x_j² Σ_j=1ⁿ y_j² - (Σ_j=1ⁿ x_jy_j)²]}^1/2,

a/c = sign(Σ_j=1ⁿ x_jy_j Σ_j=1ⁿ y_jz_j - Σ_j=1ⁿ x_jz_j Σ_j=1ⁿ y_j²) {[Σ_j=1ⁿ y_j² Σ_j=1ⁿ z_j² - (Σ_j=1ⁿ y_jz_j)²]/[Σ_j=1ⁿ x_j² Σ_j=1ⁿ y_j² - (Σ_j=1ⁿ x_jy_j)²]}^1/2.

If

sign(Σ_j=1ⁿ x_jy_j Σ_j=1ⁿ y_jz_j - Σ_j=1ⁿ x_jz_j Σ_j=1ⁿ y_j²) = sign(Σ_j=1ⁿ x_jz_j Σ_j=1ⁿ y_jz_j - Σ_j=1ⁿ x_jy_j Σ_j=1ⁿ z_j²) sign(Σ_j=1ⁿ x_jy_j Σ_j=1ⁿ x_jz_j - Σ_j=1ⁿ x_j² Σ_j=1ⁿ y_jz_j),

then necessary relation a/c = a/b b/c is provided. Finally, e.g., equation z = - a/c x - b/c y gives

z = sign(Σ_j=1ⁿ x_jz_j Σ_j=1ⁿ y_j² - Σ_j=1ⁿ x_jy_j Σ_j=1ⁿ y_jz_j) {[Σ_j=1ⁿ y_j² Σ_j=1ⁿ z_j² - (Σ_j=1ⁿ y_jz_j)²]/[Σ_j=1ⁿ x_j² Σ_j=1ⁿ y_j² - (Σ_j=1ⁿ x_jy_j)²]}^1/2 x +

sign(Σ_j=1ⁿ x_j² Σ_j=1ⁿ y_jz_j - Σ_j=1ⁿ x_jy_j Σ_j=1ⁿ x_jz_j) {[Σ_j=1ⁿ x_j² Σ_j=1ⁿ z_j² - (Σ_j=1ⁿ x_jz_j)²]/[Σ_j=1ⁿ x_j² Σ_j=1ⁿ y_j² - (Σ_j=1ⁿ x_jy_j)²]}^1/2 y .

Power mean theories naturally give by any power p > 0:(Σ_j=1ⁿ |y_j|^p / Σ_j=1ⁿ |x_j|^p)^1/p s

a/b = sign(Σ_j=1ⁿ x_jz_j Σ_j=1ⁿ y_jz_j - Σ_j=1ⁿ x_jy_j Σ_j=1ⁿ z_j²) {[Σ_j=1ⁿ |y_j|^p Σ_j=1ⁿ |z_j|^p - |Σ_j=1ⁿ y_jz_j|^p]/[Σ_j=1ⁿ |x_j|^p Σ_j=1ⁿ |z_j|^p - |Σ_j=1ⁿ x_jz_j|^p]}^1/p,

b/c = sign(Σ_j=1ⁿ x_jy_j Σ_j=1ⁿ x_jz_j - Σ_j=1ⁿ x_j² Σ_j=1ⁿ y_jz_j) {[Σ_j=1ⁿ |x_j|^p Σ_j=1ⁿ |z_j|^p - |Σ_j=1ⁿ x_jz_j|^p]/[Σ_j=1ⁿ |x_j|^p Σ_j=1ⁿ |y_j|^p - |Σ_j=1ⁿ x_jy_j|^p]}^1/p,

a/c = sign(Σ_j=1ⁿ x_jy_j Σ_j=1ⁿ y_jz_j - Σ_j=1ⁿ x_jz_j Σ_j=1ⁿ y_j²) {[Σ_j=1ⁿ |y_j|^p Σ_j=1ⁿ |z_j|^p - |Σ_j=1ⁿ y_jz_j|^p]/[Σ_j=1ⁿ |x_j|^p Σ_j=1ⁿ |y_j|^p - |Σ_j=1ⁿ x_jy_j|^p]}^1/p.

If

sign(Σ_j=1ⁿ x_jy_j Σ_j=1ⁿ y_jz_j - Σ_j=1ⁿ x_jz_j Σ_j=1ⁿ y_j²) = sign(Σ_j=1ⁿ x_jz_j Σ_j=1ⁿ y_jz_j - Σ_j=1ⁿ x_jy_j Σ_j=1ⁿ z_j²) sign(Σ_j=1ⁿ x_jy_j Σ_j=1ⁿ x_jz_j - Σ_j=1ⁿ x_j² Σ_j=1ⁿ y_jz_j),

then necessary relation a/c = a/b b/c is provided. Finally, e.g., equation z = - a/c x - b/c y gives

z = sign(Σ_j=1ⁿ x_jz_j Σ_j=1ⁿ y_j² - Σ_j=1ⁿ x_jy_j Σ_j=1ⁿ y_jz_j) {[Σ_j=1ⁿ |y_j|^p Σ_j=1ⁿ |z_j|^p - |Σ_j=1ⁿ y_jz_j|^p]/[Σ_j=1ⁿ |x_j|^p Σ_j=1ⁿ |y_j|^p - |Σ_j=1ⁿ x_jy_j|^p]}^1/p x +

sign(Σ_j=1ⁿ x_j² Σ_j=1ⁿ y_jz_j - Σ_j=1ⁿ x_jy_j Σ_j=1ⁿ x_jz_j) {[Σ_j=1ⁿ |x_j|^p Σ_j=1ⁿ |z_j|^p - |Σ_j=1ⁿ x_jz_j|^p]/[Σ_j=1ⁿ |x_j|^p Σ_j=1ⁿ |y_j|^p - |Σ_j=1ⁿ x_jy_j|^p]}^1/p y .

Power mean theories are very efficient in data estimation, approximation, and processing and reliable even by great data scatter.

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. 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. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010

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