Least Biquadratic Method 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), 3
The least square method (LSM) [1] by Legendre and Gauss only usually applies to solving contradictory (e.g., overdetermined) problems, which is always the case in data processing. Overmathematics [2, 3] and fundamental sciences of estimation [4-6], approximation [7], and data processing [8] have discovered a lot of principal shortcomings [9] of this method. Additionally, by more than 4 data points, the second power can paradoxically give smaller errors of better approximations and can be increased due to the least biquadratic method (LBQM) in fundamental sciences of estimation [4-6], approximation [7], and data processing [8].
By coordinate system translation invariance of the given data, centralize them by subtracting every coordinate of the data center from the corresponding coordinate of every data point. Show the essence of the method in the simplest but most important linear approximation in the two-dimensional case.
Given n (n ∈ N+ = {1, 2, ...}, n > 2) points [j=1n (x'j , y'j )] = {(x'1 , y'1), (x'2 , y'2), ... , (x'n , y'n)] with any real coordinates. Use centralization transformation x = x' - Σj=1n x'j / n , y = y' - Σj=1n y'j / n clearly invariant to provide coordinate system xOy central for the given data and further work in this system with points [j=1n (xj , yj)] to be approximated with a straight line y = ax containing origin O(0, 0) to minimize the sum of the 4th powers of the ordinate differences between this line and everyone of the n data points [j=1n (xj , yj)]:
4S(a) = Σj=1n (axj - yj)4,
4S'a = 4Σj=1n (axj - yj)3xj = 0,
Σj=1n xj4 a3 - 3Σj=1n xj3yj a2 + 3Σj=1n xj2yj2 a - Σj=1n xjyj3 = 0.
Introducing a' = a - Σj=1n xj3yj / Σj=1n xj4 , we obtain a reduced cubic equation [1] a'3 + pa' + q = 0, namely
a'3 + 3[Σj=1n xj2yj2 / Σj=1n xj4 - (Σj=1n xj3yj)2/(Σj=1n xj4)2]a' - Σj=1n xjyj3 / Σj=1n xj4 + 3Σj=1n xj2yj2 Σj=1n xj3yj / (Σj=1n xj4)2 - 2(Σj=1n xj3yj)3/ (Σj=1n xj4)3 = 0
with
p = 3[Σj=1n xj2yj2 / Σj=1n xj4 - (Σj=1n xj3yj)2/(Σj=1n xj4)2],
q = - Σj=1n xjyj3 / Σj=1n xj4 + 3Σj=1n xj2yj2 Σj=1n xj3yj / (Σj=1n xj4)2 - 2(Σj=1n xj3yj)3/ (Σj=1n xj4)3.
Due to the Cardano formulae [1], we obtain by Q = (p/3)3 + (q/2)2 > 0 one real solution and two conjugated imaginary solutions, by Q = 0 one real solution and another doubled real solution (a triple real solution by p = q = 0), by Q < 0 three different real solutions:
a'1 = A + B ,
a'2 , 3 = - (A + B)/2 ± 31/2/2 i (A - B) where i2 = -1,
A = (- q/2 + Q1/2)1/3,
B = (- q/2 - Q1/2)1/3,
for each value A , take value B with AB = - p/3; for real equations (which is here the case), take real values of A and B . We consider exclusively real values of a' and a . Namely the minimum of 4S(a) is provided due to
4S''aa = 12Σj=1n (axj - yj)2xj2 > 0
in any nontrivial case.
Increasing the power improves the LSM results via the LBQM, especially by not too great data scatter. In numeric tests [Figures 1, 2 with replacing (x’, y’) via (x , y)], the LSM gives y = 0.909x + 2.364 and even y = 0.591x + 3.636, whereas the LBQM brings y = 0.968x + 2.127 and y = 0.698x + 3.206. Data symmetry straight line y = x + 2 is the best linear approximation given, e.g., by least squared distance theories.
Figure 1
Figure 2
The least biquadratic method (LBQM) is efficient in data 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. 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. General estimation theory. Transactions of the Ukraine Glass Institute, 1 (1994), 214-221. The same with its translation into Japanese. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2001
[5] Lev Gelimson. General Estimation Theory. The “Collegium” All-World Academy of Sciences Publishers, Munich (Germany), 2004. The same with its translation into Japanese. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010
[6] Lev Gelimson. Fundamental Science of Estimation. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010
[7] Lev Gelimson. Fundamental Science of Approximation. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010
[8] Lev Gelimson. Fundamental Science of Data Processing. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010
[9] 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