Distance Quadrat 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)
10 (2010), 2
By estimation, approximation, and data processing, the least square method (LSM) [1] by Legendre and Gauss only usually applies to contradictory (e.g. overdetermined) problems, by methods of finite elements, points, etc. Overmathematics [2, 3] and fundamental sciences of estimation [4], approximation [5], and data processing [6] have discovered a lot of principal shortcomings [7] of the least square method. Additionally consider its simplest approach which is typical. Minimizing the sum of the squared differences of the alone preselected coordinates (e.g., ordinates in a two-dimensional problem) of the graph of the desired approximation function and of everyone among the given data depends on this preselection, ignores the remaining coordinates, and provides no coordinate system rotation invariance and hence no objective sense of the result. Moreover, the method is correct by constant approximation or no data scatter only and gives systematic errors increasing together with data scatter and the deviation (namely declination) of an approximation from a constant.
In fundamental sciences of estimation [4], approximation [5], and data processing [6], distance quadrat theories (DQT) are valid in the case of coordinate system rotation invariance. 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 least squared distance theories in the simplest but most important linear approximation in the two-dimensional case in which it is possible to explicitly algebraically provide the unique best quasisolution [2, 3] to the following problem setting.
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 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 ax + by = 0 containing origin O(0, 0). The distance between this line and the jth data point (xj , yj) and further the sum of the squared distances between this line and everyone of the n data points [j=1n (xj , yj)] are, respectively,
dj = |axj + byj|/(a2 + b2)1/2,
2S(a , b) = Σj=1n dj2 = Σj=1n(axj + byj)2/(a2 + b2).
This nonnegative differentiable function takes its minimum at a point with vanishing the both first order derivatives of this function by a and b:
2S'a = - 2a/(a2 + b2)2 Σj=1n (xj2a2 + 2xjyjab + yj2b2) + 2/(a2 + b2) Σj=1n (xj2a + xjyjb) = 0,
2S'b = - 2b/(a2 + b2)2 Σj=1n (xj2a2 + 2xjyjab + yj2b2) + 2/(a2 + b2) Σj=1n (xjyja + yj2b) = 0;
b[Σj=1n xjyj(a2 - b2) + (Σj=1n yj2 - Σj=1n xj2)ab] = 0,
a[Σj=1n xjyj(a2 - b2) + (Σj=1n yj2 - Σj=1n xj2)ab] = 0.
If b = 0, then by a = 0 ax + by = 0 becomes an identity and gives the whole plane x0y instead of a straight line, by a ≠ 0 vertical straight line x = 0. If b ≠ 0, then denote A = -a/b and obtain
Σj=1n xjyj A2 - (Σj=1n yj2 - Σj=1n xj2)A - Σj=1n xjyj = 0.
Now if Σj=1n xjyj = 0, then (Σj=1n yj2 - Σj=1n xj2)A = 0 and by Σj=1n xj2 = Σj=1n yj2 we obtain any real A and any proportional straight line y = Ax , whereas by Σj=1n xj2 ≠ Σj=1n yj2 , A = 0 and y = 0. If Σj=1n xjyj ≠ 0, then that quadratic equation gives two real solutions
A1 , 2 = {Σj=1n yj2 - Σj=1n xj2 ± [(Σj=1n yj2 - Σj=1n xj2)2 + 4(Σj=1n xjyj)2]1/2}/(2Σj=1n xjyj)
providing the required minimum of 2S by d2 2S / dA2 > 0 and its maximum by d2 2S / dA2 < 0, whereas the case d2 2S / dA2 = 0 needs further investigations. Here
2S(A) = Σj=1n dj2 = Σj=1n(Axj - yj)2/(A2 + 1) = Σj=1n(xj2A2 - 2xjyjA + yj2)/(A2 + 1) =
= (Σj=1n xj2 A2 - 2Σj=1n xjyj A + Σj=1n yj2)/(A2 + 1),
2S'A = - 2A/(A2 + 1)2 Σj=1n (xj2A2 - 2xjyjA + yj2) + 2/(A2 + 1) Σj=1n (xj2A - xjyj) =
2/(A2 + 1)2 [Σj=1n xjyj A2 - (Σj=1n yj2 - Σj=1n xj2)A - Σj=1n xjyj] = 0,
2S''AA = 2/(A2 + 1)3 [- 2Σj=1n xjyj A3 + 3(Σj=1n yj2 - Σj=1n xj2)A2 + 6Σj=1n xjyj A + Σj=1n xj2 - Σj=1n yj2].
Compare
2S(A) = (Σj=1n xj2 A2 - 2Σj=1n xjyj A + Σj=1n yj2)/(A2 + 1)
by A = A1 , 2 providing 2Smin(A) and 2Smax(A).
Determine 2Smin(A), 2Smax(A), and then
SL = [2Smin(A) / 2Smax(A)]1/2
as a measure of data scatter with respect to linear approximation.
This is an upper estimation of data scatter with respect to approximation at all because nonlinear approximation is also possible.
Denote a measure of data scatter with respect to approximation at all with S . Then SL ≥ S .
Also introduce a measure of data trend with respect to linear approximation
TL = 1 - SL = 1 - [2Smin(A) / 2Smax(A)]1/2
and a measure of data trend with respect to approximation at all
T = 1 - S .
Then, naturally, TL ≤ T .
Unlike the least square method, distance quadrat theories (DQT) provide best linear approximation to the given data by rotation invariance, e.g. in the following numeric tests, see Figures 1, 2 with replacing (x’, y’) via (x , y):
Figure 1. SL = 0.218. TL = 0.782
Figure 2. SL = 0.507. TL = 0.493
Nota bene: By linear approximation, the results of distance quadrat theories (DQT) and general theories of moments of inertia [4-6] coincide. By data symmetry axis (and the best linear approximation) y = ± x + C , the same also holds for quadratic mean theories [4-6]. Here y = x + 2. The LSM gives y = 0.909x + 2.364 (Figure 1) and even y = 0.591x + 3.636 (Figure 2) with the same data center (x , y) = (4, 6) and underestimating the modulus (absolute value) of the declination to the x-axis (which is typical) due to considering y-coordinate differences instead of distances with ignoring the declination of the approximation straight line to the x-axis.
Distance quadrat theories (DQT) are very efficient in data estimation, approximation, and processing by coordinate system rotation invariance.
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 Processing. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010
[7] 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