Rotation-Quasi-Invariant Quadratic and Other Power Mean 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), 20

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. Overmathematics [2, 3] and fundamental sciences of estimation [4], approximation [5], data modeling [6] and processing [7] have discovered a lot of principal shortcomings [2-8] of this method. Additionally, 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 objective sense of the result. Moreover, the method is correct in the unique case of a constant approximation only and gives systematic errors increasing together with the declination of an approximation function.

In 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. Show the essence of power mean theories by a linear approximation (bisector) 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 clearly invariant centralization transformation x = x' - Σ_j=1ⁿ x'_j / n , y = y' - Σ_j=1ⁿ y'_j / n to provide coordinate system xOy central for the given data and further work in this system with points [_j=1ⁿ (x_j , y_j)] to be approximated with a straight line y = ax containing origin O(0, 0).

Quadratic mean theories give

a = (Σ_j=1ⁿ y_j² / Σ_j=1ⁿ x_j²)^1/2 sign Σ_j=1ⁿ x_jy_j ,

y = sign Σ_j=1ⁿ x_jy_j (Σ_j=1ⁿ y_j² / Σ_j=1ⁿ x_j²)^1/2 x

for the transformed centralized data, whereas for the initial noncentralized data we obtain

y' = sign Σ_j=1ⁿ (x'_j - Σ_j=1ⁿ x'_j / n)(y'_j - Σ_j=1ⁿ y'_j / n) [Σ_j=1ⁿ (y'_j - Σ_j=1ⁿ y'_j / n)²/ Σ_j=1ⁿ (x'_j - Σ_j=1ⁿ x'_j / n)²]^1/2(x'_j - Σ_j=1ⁿ x'_j / n) + Σ_j=1ⁿ y'_j / n .

Power mean theories naturally give by any power p > 0

a = (Σ_j=1ⁿ |y_j|^p / Σ_j=1ⁿ |x_j|^p)^1/p sign Σ_j=1ⁿ x_jy_j ,

y = sign Σ_j=1ⁿ x_jy_j (Σ_j=1ⁿ |y_j|^p / Σ_j=1ⁿ |x_j|^p)^1/p x

for the transformed centralized data, whereas for the initial noncentralized data we obtain

y' = sign Σ_j=1ⁿ (x'_j - Σ_j=1ⁿ x'_j / n)(y'_j - Σ_j=1ⁿ y'_j / n) [Σ_j=1ⁿ |y'_j - Σ_j=1ⁿ y'_j / n|^p/ Σ_j=1ⁿ |x'_j - Σ_j=1ⁿ x'_j / n|^p]^1/p(x'_j - Σ_j=1ⁿ x'_j / n) + Σ_j=1ⁿ y'_j / n .

Nota bene:

1. Using moduli (absolute values) provides possibility to consider any power p > 0. The above sign remains due to coordinate system linear transformation invariance of the given data.

2. We can keep here the 2nd power because we consider here that sign only. The above sign remains due to coordinate system linear transformation invariance of the given data.

By nonzero but relatively very small absolute values of Σ_j=1ⁿ x_jy_j , namely by |Σ_j=1ⁿ x_jy_j| << (Σ_j=1ⁿ x_j² Σ_j=1ⁿ y_j²)^1/2, the used sign can become oversensitive to small data variations. In such a case, use either horizontal y = 0 (y' = Σ_j=1ⁿ y'_j / n) by ²_yS = Σ_j=1ⁿy_j² - (Σ_j=1ⁿy_j)²/n < ²_xS = Σ_j=1ⁿx_j² - (Σ_j=1ⁿx_j)²/n or vertical x = 0 (x' = Σ_j=1ⁿ x'_j / n) by ²_yS > ²_xS straight line approximation. The last line cannot be obtained by general equation y = ax and has to be considered separately.

After centralization, additionally introduce normalization transformation for any power p > 0

X = x/(Σ_j=1ⁿ |x_j|^p)^1/p ,

Y = y/(Σ_j=1ⁿ |y_j|^p)^1/p

to provide coordinate system XOY central normalized for the given data and further work in this system with points [_j=1ⁿ (X_j , Y_j)] to be approximated with a straight line Y = AX containing origin O(0, 0). Note that y = ax gives

Y(Σ_j=1ⁿ |y_j|^p)^1/p = a(Σ_j=1ⁿ |x_j|^p)^1/p X

and hence

A = (Σ_j=1ⁿ |x_j|^p / Σ_j=1ⁿ |y_j|^p)^1/p a ,

a = (Σ_j=1ⁿ |y_j|^p / Σ_j=1ⁿ |x_j|^p)^1/p A .

Note that, due to normalization,

Σ_j=1ⁿ |X_j|^p = Σ_j=1ⁿ [|x_j| / (Σ_j=1ⁿ |x_j|^p)^1/p]^p = 1,

Σ_j=1ⁿ |Y_j|^p = Σ_j=1ⁿ [|y_j| / (Σ_j=1ⁿ |y_j|^p)^1/p]^p = 1.

In quadratic mean theories (QMT) we have obtained two solutions only

A₁ = 1,

A₂ = - 1

which both are fully independent of a power and even of the specific data in their initial and transformed forms. Therefore, we can conside these solutions in our natural generalization of quadratic mean theories (QMT), too.

Consequently determine

^pS(A₁) = Σ_j=1ⁿ |Y_j - X_j|^p ,

^pS(A₂) = Σ_j=1ⁿ |Y_j + X_j|^p ,

^pS_min(A) = min{^pS(A₁), ^pS(A₂)},

^pS_max(A) = max{^pS(A₁), ^pS(A₂)},

Note that theoretically by

Σ_j=1ⁿ X_jY_j = 0,

practically by

|Σ_j=1ⁿ X_jY_j| << 1,

we have to investigate the pair of straight lines Y = 0 and X = 0.

Otherwise, we have Y = X and Y = - X obtained above by a ≠ 0 and A ≠ 0.

Check whether namely

A = sign(Σ_j=1ⁿ X_jY_j)

provides

^pS_min(A) = Σ_j=1ⁿ |Y_j - sign(Σ_j=1ⁿ X_jY_j) X_j|^p ,

whereas

A = - sign(Σ_j=1ⁿ X_jY_j)

provides

^pS_max(A) = Σ_j=1ⁿ |Y_j + sign(Σ_j=1ⁿ X_jY_j) X_j|^p.

Then define and determine

^1|pS = [^pS_min(A) / ^pS_max(A)]^1/p

as a measure of data scatter with respect to linear approximation and power p .

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|pS ≥ S .

Also introduce a measure of data trend with respect to linear approximation and power p

^1|pT = 1 - ^1|pS = 1 - [^pS_min(A) / ^pS_max(A)]^1/p

and a measure of data trend with respect to approximation at all

T = 1 - S .

Then, naturally, ^1|pT ≤ T .

But this approach is not universal enough because such a dependence of a and y on Σ_j=1ⁿ x_jy_j is not continuous due to the jumps of the function sign z at z = 0 both to the left and to the right.

It is possible to give still more universal (but much more complicated) formulae for a and y . Namely, denote

t = |Σ_j=1ⁿ x_jy_j|/(Σ_j=1ⁿ x_j² Σ_j=1ⁿ y_j²)^1/2

to provide continuity due to

sign Σ_j=1ⁿ x_jy_j = sign [Σ_j=1ⁿ x_jy_j /(Σ_j=1ⁿ x_j² Σ_j=1ⁿ y_j²)^1/2],

Σ_j=1ⁿ x_jy_j = |Σ_j=1ⁿ x_jy_j| sign Σ_j=1ⁿ x_jy_j = sign [Σ_j=1ⁿ x_jy_j /(Σ_j=1ⁿ x_j² Σ_j=1ⁿ y_j²)^1/2] |Σ_j=1ⁿ x_jy_j|

via multiplying the above values of a and y with t^r whose power r(t) is a suitable function of t .

Nota bene: 0 ≤ t ≤ 1 [1].

A deep analysis of obtaining the formulae for a and y in quadratic mean theories, namely including expression Σ_j=1ⁿ x_jy_j at power 1 by small values of ratio Σ_j=1ⁿ y_j² / Σ_j=1ⁿ x_j² , at power 0 by Σ_j=1ⁿ y_j² / Σ_j=1ⁿ x_j² = 1, and at power -1 by great values of this ratio Σ_j=1ⁿ y_j² / Σ_j=1ⁿ x_j² leads to idea

u = 1 - 4/π arctan[(Σ_j=1ⁿ y_j² / Σ_j=1ⁿ x_j²)^1/2],

r = 1 - 4/π arctan[(Σ_j=1ⁿ y_j² / Σ_j=1ⁿ x_j²)^1/2 t^u]

with still better results by coinciding the physical dimensions (units) of x and y .

Nota bene: 0 ≤ t ≤ 1 [1] but, generally, variable t cannot take value 1. A deep analysis shows that the greatest value of t is

t_max = |Σ_j=1ⁿ x_j²(0) - Σ_j=1ⁿ y_j²(0)|/[Σ_j=1ⁿ x_j²(0) + Σ_j=1ⁿ y_j²(0)]

where data rotation angle 0 relates to the initial data (before rotation) in the principal central coordinate system with

Σ_j=1ⁿ x_j(0)y_j(0) = 0.

Therefore, for any data rotation angle β , we can also take

t(β) = |Σ_j=1ⁿ x_j(β)y_j(β)|/[Σ_j=1ⁿ x_j²(β) Σ_j=1ⁿ y_j²(β)]^1/2 [Σ_j=1ⁿ x_j²(0) + Σ_j=1ⁿ y_j²(0)]/|Σ_j=1ⁿ x_j²(0) - Σ_j=1ⁿ y_j²(0)|,

r(β) = sign[Σ_j=1ⁿ x_j²(β) - Σ_j=1ⁿ y_j²(β)] 2/π arctan[1/t(β) - 1],

or

r(β) = sign[Σ_j=1ⁿ x_j²(β) - Σ_j=1ⁿ y_j²(β)] 2/π arctan{[Σ_j=1ⁿ x_j²(β) Σ_j=1ⁿ y_j²(β)]^1/2/|Σ_j=1ⁿ x_j(β)y_j(β)| |Σ_j=1ⁿ x_j²(0) - Σ_j=1ⁿ y_j²(0)|/[Σ_j=1ⁿ x_j²(0) + Σ_j=1ⁿ y_j²(0)] - 1}.

Finally, multiply the above values of a and y with t^r.

Then quadratic mean theories give

a(β) = [Σ_j=1ⁿ y_j²(β) / Σ_j=1ⁿ x_j²(β)]^1/2 t(β)^r(β) sign Σ_j=1ⁿ x_j(β)y_j(β) ,

y(β) = sign Σ_j=1ⁿ x_j(β)y_j(β) [Σ_j=1ⁿ y_j²(β) / Σ_j=1ⁿ x_j²(β)]^1/2 t(β)^r(β) x

for the transformed centralized data, whereas for the initial noncentralized data we obtain

y' = sign[Σ_j=1ⁿ (x'_j - Σ_j=1ⁿ x'_j / n)(y'_j - Σ_j=1ⁿ y'_j / n)] [Σ_j=1ⁿ (y'_j - Σ_j=1ⁿ y'_j / n)²/ Σ_j=1ⁿ (x'_j - Σ_j=1ⁿ x'_j / n)²]^1/2 t^r (x'_j - Σ_j=1ⁿ x'_j / n) + Σ_j=1ⁿ y'_j / n .

Power mean theories naturally give by any power p > 0

a = (Σ_j=1ⁿ |y_j|^p / Σ_j=1ⁿ |x_j|^p)^1/p t^r sign Σ_j=1ⁿ x_jy_j ,

y = sign Σ_j=1ⁿ x_jy_j (Σ_j=1ⁿ |y_j|^p / Σ_j=1ⁿ |x_j|^p)^1/p t^r x

for the transformed centralized data, whereas for the initial noncentralized data we obtain

y' = sign Σ_j=1ⁿ (x'_j - Σ_j=1ⁿ x'_j / n)(y'_j - Σ_j=1ⁿ y'_j / n) [Σ_j=1ⁿ |y'_j - Σ_j=1ⁿ y'_j / n|^p/ Σ_j=1ⁿ |x'_j - Σ_j=1ⁿ x'_j / n|^p]^1/p t^r (x'_j - Σ_j=1ⁿ x'_j / n) + Σ_j=1ⁿ y'_j / n .

Unlike the LSM, PMT provide best linear approximation to the given data, e.g. in numeric tests, see Figures 1, 2 with replacing (x’, y’) via (x , y):

Figure 1. ^1|1S = 0.25, ^1|1T = 0.75. ^1|2S = 0.218, ^1|2T = 0.782. ^1|4S = 0.189, ^1|4T = 0.811.

Figure 2. ^1|1S = 0.541, ^1|1T = 0.459. ^1|2S = 0.507, ^1|2T = 0.493. ^1|4S = 0.487, ^1|4T = 0.513.

Nota bene: By linear approximation, the results of distance quadrat theories (DQT) and general theories of moments of inertia (GTMI) [4, 5] coincide. By Σ_j=1ⁿ |y_j|^p = Σ_j=1ⁿ |x_j|^p (and the best linear approximation y = ± x + C), the same also holds for PMT. Here y = x + 2 (Figures 1, 2). By Σ_j=1ⁿ |y_j|^p ≠ Σ_j=1ⁿ |x_j|^p , PMT give other results than DQT and GTMI. But PMT are valid by another invariance type than DQT and GTMI. The data symmetry straight line y = x + 2 is the best linear approximation in the both above tests. The LSM gives y = 0.909x + 2.364 (Figure 1) and even y = 0.591x + 3.636 (Figure 2) with the same data center (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.

Universalized quadratic and other 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

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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

[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

[8] 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