Distance Biquadrat 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), 11
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], data modeling and 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. And the second power can be not sufficient, especially by great data point numbers.
In fundamental sciences of estimation [4], approximation [5], data modeling and processing [6], distance biquadrat theories (DBQT), as well as 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 biquadratic 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 y = ax (a is any real constant) containing origin O(0, 0). Case x = 0 (a vertical straight line) should be considered separately if necessary. The distance between this line and the jth data point (xj , yj) and further the sum of the biquadratic distances between this line and everyone of the n data points [j=1n (xj , yj)] are, respectively,
dj = |axj - yj|/(a2 + 1)1/2,
4S(a) = Σj=1n dj4 = Σj=1n(axj - yj)4/(a2 + 1)2 =
[Σj=1n xj4 a4 - 4Σj=1n xj3yj a3 + 6Σj=1n xj2yj2 a2 - 4Σj=1n xjyj3 a + Σj=1n yj4]/(a2 + 1)2 .
This nonnegative differentiable function takes its least value at a point with vanishing the first order derivative of this function by a :
4S'a = - 4a/(a2 + 1)3 Σj=1n (axj - yj)4 + 4/(a2 + 1)2 Σj=1n (axj - yj)3xj = 0,
Σj=1n xj3yj a4 + (Σj=1n xj4 - 3Σj=1n xj2yj2) a3 + 3(Σj=1n xjyj3 - Σj=1n xj3yj) a2 + (3Σj=1n xj2yj2 - Σj=1n yj4) a- Σj=1n xjyj3 = 0,
a4 + (Σj=1n xj4 - 3Σj=1n xj2yj2)/Σj=1n xj3yj a3 + 3(Σj=1n xjyj3/Σj=1n xj3yj - 1) a2 + (3Σj=1n xj2yj2 - Σj=1n yj4)/Σj=1n xj3yj a- Σj=1n xjyj3/Σj=1n xj3yj = 0,
a4 + A3 a3 + A2 a2 + A1 a + A0 = 0
where
A3 = (Σj=1n xj4 - 3Σj=1n xj2yj2)/Σj=1n xj3yj ,
A2 = 3(Σj=1n xjyj3/Σj=1n xj3yj - 1),
A1 = (3Σj=1n xj2yj2 - Σj=1n yj4)/Σj=1n xj3yj ,
A0 = - Σj=1n xjyj3/Σj=1n xj3yj ,
To provide vanishing the factor by a3 in this quartic equation in a , following the Ferrari method [1], introduce such h that
a = h - A3 / 4
and obtain equation (in h)
h4 + (A2 - 3/8 A32) h2 + (A1 - 1/2 A2A3+ 1/8 A33) h + A0 - 1/4 A1A3+ 1/16 A2A32 - 3/256 A34 = 0,
h4 + Ph2 + Qh + R = 0
where
P = A2 - 3/8 A32 ,
Q = A1 - 1/2 A2A3+ 1/8 A33 ,
R = A0 - 1/4 A1A3+ 1/16 A2A32 - 3/256 A34 .
Now introduce real-number parameter A , consider
h4 + Ph2 + Qh + R = (h2 + 1/2 P + A)2 - S
where
S = 2Ah2 - Qh + A2 + PA + 1/4 P2 - R ,
and find such value A that S is the square of some linear polynomial in h . Then and only then the discriminant of S vanishes:
Q2 - 8A(A2 + PA + 1/4 P2 - R) = 0 ,
A3 + PA2 + (1/4 P2 - R)A - 1/8 Q2 = 0 .
Introducing such B that
A = B - 1/3 P ,
we obtain a reduced cubic equation [1]
B3 + pB + q = 0,
namely
B3 + (- 1/12 P2 - R)B + (- 1/108 P3 + 1/3 PR - 1/8 Q2) = 0
with
p = - 1/12 P2 - R ,
q = - 1/108 P3 + 1/3 PR - 1/8 Q2.
Due to the Cardano formulae [1], we obtain by H = (p/3)3 + (q/2)2 > 0 one real solution and two conjugated imaginary solutions, by H = 0 one real solution and another doubled real solution (a triple real solution by p = q = 0), by H < 0 three different real solutions:
B1 = F + G ,
B2 , 3 = - (F + G)/2 ± 31/2/2 i (F - G) where i2 = -1,
F = (- q/2 + H1/2)1/3,
G = (- q/2 - H1/2)1/3,
for each value F , take value G with FG = - p/3; for real equations (which is here the case), take real values of F and G . We consider exclusively real values of B and A .
By such value A , we have
S = 2Ah2 - Qh + A2 + PA + 1/4 P2 - R = 2A[h - Q/(4A)]2 ,
h4 + Ph2 + Qh + R = (h2 + 1/2 P + A)2 - S = (h2 + 1/2 P + A)2 - 2A[h - Q/(4A)]2 ,
(h2 + 1/2 P + A)2 - 2A[h - Q/(4A)]2 = 0.
Check whether A ≥ 0. If it is the case, then we obtain two quadratic equations in h :
{h2 + (2A)1/2h - Q/[2(2A)1/2]+ 1/2 P + A}{h2 - (2A)1/2h + Q/[2(2A)1/2]+ 1/2 P + A} = 0;
h2 + (2A)1/2h - Q/[2(2A)1/2]+ 1/2 P + A = 0,
h1 , 2 = - (A/2)1/2 ± {A/2 + Q/[2(2A)1/2]- 1/2 P - A}1/2 ;
h2 - (2A)1/2h + Q/[2(2A)1/2]+ 1/2 P + A = 0,
h3 , 4 = (A/2)1/2 ± {A/2 - Q/[2(2A)1/2]- 1/2 P - A}1/2 .
By each solution h , determine the corresponding solution
a = h - A3 / 4
and
4S''aa = d2 4S(a) / dA2 = (20a2 - 4)/(a2 + 1)4 Σj=1n(axj - yj)4 - 32a/(a2 + 1)3 Σj=1n (axj - yj)3xj + 12/(a2 + 1)2 Σj=1n (axj - yj)2xj2 ,
or
4S''aa = 1/(a2 + 1)4 [Σj=1n xj4 (- 12a4 + 12a2) + Σj=1n xj3yj (- 8a5 + 64a3 - 24a) + Σj=1n xj2yj2 (36a4 - 96a2 + 12) + Σj=1n xyj3 (- 48a3 + 48a) + Σj=1n yj4 (20a2 - 4)]
providing the required minimum of 4S(a) by d2 4S / dA2 > 0 and its maximum by d2 4S / dA2 < 0, whereas the case d2 4S / dA2 = 0 needs further investigations.
Compare
4S(a) = [Σj=1n xj4 a4 - 4Σj=1n xj3yj a3 + 6Σj=1n xj2yj2 a2 - 4Σj=1n xjyj3 a + Σj=1n yj4]/(a2 + 1)2
by a = a1 , 2, 3, 4 providing 4Smin(a) and 4Smax(a).
Determine 4Smin(a), 4Smax(a), and then
1|4S = [4Smin(a) / 4Smax(a)]1/4
as a measure of data scatter with respect to linear (1st power) approximation and the 4th power of a distance.
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 1|4S ≥ S .
Also introduce a measure of data trend with respect to linear approximation and the 4th power of a distance
1|4T = 1 - 1|4S = 1 - [4Smin(a) / 4Smax(a)]1/4
and a measure of data trend with respect to approximation at all
T = 1 - S .
Then, naturally, 1|4T ≤ T .
Unlike the least square method, least biquadratic distance theories 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. 1|4S = 0.189. 1|4T = 0.811
Figure 2. 1|4S = 0.487. 1|4T = 0.513
Nota bene: By linear approximation, as well as data symmetry axis (and the best linear approximation) y = ± x + C , the results of least biquadratic distance theories and biquadratic mean theories [4-6] coincide. 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 biquadrat theories (DBQT) 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
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[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
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[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. 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