Quadratic 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)
10 (2010), 5
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], quadratic mean theories are valid by coordinate system linear transformation invariance of the given data. Additionally to the 2D case, show the essence of these 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' - Σj=1n x'j / n , y = y' - Σj=1n y'j / n , z = z' - Σj=1n z'j / n to provide coordinate system Oxyz central for the given data and further work in this system with points [j=1n (xj , yj , zj)] to be approximated with a plane ax + by + cz = 0 containing origin O(0, 0, 0).
First, use the least square method [1] by its common approach to minimizing the sum of the squared x-coordinate differences between this plane (with its equation x = - b/a y - c/a z explicit for x) and everyone of the n data points [j=1n (xj , yj , zj)]:
2xS(b/a , c/a) = Σj=1n (xj + b/a yj + c/a zj)2,
2xS'b/a = 2Σj=1n (xj + b/a yj + c/a zj)yj = 0,
2xS'c/a = 2Σj=1n (xj + b/a yj + c/a zj)zj = 0,
Σj=1n yj2 b/a + Σj=1n yjzj c/a = - Σj=1n xjyj ,
Σj=1n yjzj b/a + Σj=1n zj2 c/a = - Σj=1n xjzj ,
b/a = (Σj=1n xjzj Σj=1n yjzj - Σj=1n xjyj Σj=1n zj2)/[Σj=1n yj2 Σj=1n zj2 - (Σj=1n yjzj)2],
c/a = (Σj=1n xjyj Σj=1n yjzj - Σj=1n xjzj Σj=1n yj2)/[Σj=1n yj2 Σj=1n zj2 - (Σj=1n yjzj)2]
providing namely the minimum of 2xS(b/a , c/a) at these values of b/a and c/a .
Secondly, use the least square method [1] by its common approach to minimizing the sum of the squared y-coordinate differences between this plane (with its equation y = - a/b x - c/b z explicit for y) and everyone of the n data points [j=1n (xj , yj , zj)]:
2yS(a/b , c/b) = Σj=1n (a/b xj + yj + c/b zj)2,
2yS'a/b = 2Σj=1n (a/b xj + yj + c/b zj)xj = 0,
2yS'c/b = 2Σj=1n (a/b xj + yj + c/b zj)zj = 0,
Σj=1n xj2 a/b + Σj=1n xjzj c/b = - Σj=1n xjyj ,
Σj=1n xjzj a/b + Σj=1n zj2 c/b = - Σj=1n yjzj ,
a/b = (Σj=1n xjzj Σj=1n yjzj - Σj=1n xjyj Σj=1n zj2)/[Σj=1n xj2 Σj=1n zj2 - (Σj=1n xjzj)2],
c/b = (Σj=1n xjyj Σj=1n xjzj - Σj=1n xj2 Σj=1n yjzj)/[Σj=1n xj2 Σj=1n zj2 - (Σj=1n xjzj)2]
providing namely the minimum of 2yS(a/b , c/b) at these values of a/b and c/b .
Thirdly, use the least square method [1] by its common approach to minimizing the sum of the squared z-coordinate differences between this plane (with its equation z = - a/c x - b/c y explicit for z) and everyone of the n data points [j=1n (xj , yj , zj)]:
2zS(a/c , b/c) = Σj=1n (a/c xj + b/c yj + zj)2,
2zS'a/c = 2Σj=1n (a/c xj + b/c yj + zj)xj = 0,
2zS'b/c = 2Σj=1n (a/c xj + b/c yj + zj)yj = 0,
Σj=1n xj2 a/c + Σj=1n xjyj b/c = - Σj=1n xjzj ,
Σj=1n xjyj a/c + Σj=1n yj2 b/c = - Σj=1n yjzj ,
a/c = (Σj=1n xjyj Σj=1n yjzj - Σj=1n xjzj Σj=1n yj2)/[Σj=1n xj2 Σj=1n yj2 - (Σj=1n xjyj)2],
b/c = (Σj=1n xjyj Σj=1n xjzj - Σj=1n xj2 Σj=1n yjzj)/[Σj=1n xj2 Σj=1n yj2 - (Σj=1n xjyj)2]
providing namely the minimum of 2zS(a/c , b/c) at these values of a/c and b/c .
Now 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(Σj=1n xjzj Σj=1n yjzj - Σj=1n xjyj Σj=1n zj2) {[Σj=1n yj2 Σj=1n zj2 - (Σj=1n yjzj)2]/[Σj=1n xj2 Σj=1n zj2 - (Σj=1n xjzj)2]}1/2,
b/c = sign(Σj=1n xjyj Σj=1n xjzj - Σj=1n xj2 Σj=1n yjzj) {[Σj=1n xj2 Σj=1n zj2 - (Σj=1n xjzj)2]/[Σj=1n xj2 Σj=1n yj2 - (Σj=1n xjyj)2]}1/2,
a/c = sign(Σj=1n xjyj Σj=1n yjzj - Σj=1n xjzj Σj=1n yj2) {[Σj=1n yj2 Σj=1n zj2 - (Σj=1n yjzj)2]/[Σj=1n xj2 Σj=1n yj2 - (Σj=1n xjyj)2]}1/2.
If
sign(Σj=1n xjyj Σj=1n yjzj - Σj=1n xjzj Σj=1n yj2) = sign(Σj=1n xjzj Σj=1n yjzj - Σj=1n xjyj Σj=1n zj2) sign(Σj=1n xjyj Σj=1n xjzj - Σj=1n xj2 Σj=1n yjzj),
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(Σj=1n xjzj Σj=1n yj2 - Σj=1n xjyj Σj=1n yjzj) {[Σj=1n yj2 Σj=1n zj2 - (Σj=1n yjzj)2]/[Σj=1n xj2 Σj=1n yj2 - (Σj=1n xjyj)2]}1/2 x +
sign(Σj=1n xj2 Σj=1n yjzj - Σj=1n xjyj Σj=1n xjzj) {[Σj=1n xj2 Σj=1n zj2 - (Σj=1n xjzj)2]/[Σj=1n xj2 Σj=1n yj2 - (Σj=1n xjyj)2]}1/2 y .
Quadratic 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