Directed Data Rotation Test System Metatheories 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), 23

Separate numeric tests and checks for particular methods and theories are well-known in classical mathematics [1] and natural sciences.s.

Metatheories of directed numeric test and check systems in overmathematics [2, 3] and fundamental sciences of estimation [4], approximation [5], data modeling [6] and processing [7] make it possible, not only to provide systematic tools of comprehensively testing and checking particular theories and methods, but also to develop them and even to create many principally new theories and methods.

Show the essence of metatheories of directed numeric test and check systems by linear approximation to (a linear bisector of) two-dimensional data via the least square method (LSM) [1, 8] by Legendre and Gauss, distance quadrat theories (DQT), general theories of moments of inertia (GTMI), as well as possibly rotation-quasi-invariant power mean theories (PMT) naturally generalizing quadratic mean theories (QMT).

Use a data set with clear trend as follows.

To provide clearly best linear approximation (bisectors), take the given data to be central and additionally mirror-symmetric with respect to the x-axis and rotate these data about their center coinciding with the coordinate system origin by any angle β positive in the anticlockwise direction.

It is especially important to determine some finite set of the most characteristic angles β in half-open (and half-closed) interval [0, π) including endpoint 0 but excluding endpoint π because 2π is a period of functions sin β and cos β at all, and the centrality and additionally mirror-symmetricity of the given data with respect to the x-axis now provides the periodicity of rotating such data about their center coinciding with the coordinate system origin with period π .

This set should contain:

0 and π/2 as the discontinuity points of quadratic and other power mean theories due to the jumps of the function sign z at z = 0 both to the left and to the right;

some points in half-open (and half-closed) interval [0, π) which are very close (plot-coinciding) to the both above discontinuity points 0 (from the right only) and π/2 (both from the left and from the right), as well as π (from the left only), e.g. π/180, π/2 - π/180, π/2 + π/180, and π - π/180;

π/4 and 3π/4 as the angles of precisely coinciding linear approximation (bisectors) via distance quadrat theories (DQT), general theories of moments of inertia (GTMI), as well as both usual and rotation-quasi-invariant power mean theories (PMT) naturally generalizing quadratic mean theories (QMT). The same also holds for angles 0 and π/2 already taken;

π/8, 3π/8, 5π/8, and 7π/8 as the angles of bisectors of the angles between 0 and π/4, π/4 and π/2, π/2 and 3π/4, as well as 3π/4 and π, respectively, because we can expect that namely at these bisector angles, the differences between linear approximation (bisectors) via distance quadrat theories (DQT) and general theories of moments of inertia (GTMI), from the one side, as well as both usual and rotation-quasi-invariant power mean theories (PMT) naturally generalizing quadratic meanc mean theories (QMT) from the other side, could be near to the greatest values of these differences;

possibly some additional characteristic angles playing important roles in trigonometry, e.g. π/6, π/3, 2π/3, and 5π/6, as well as some additional intermediate angles, e.g. π/18, π/2 - π/18, π/2 + π/18, and π - π/18, to reduce the greatest angle intervals between the remaining angles.

It is reasonable to investigate linear approximation to (bisectors of) both relatively well-directed and weakly (hardly) directed (very scattered, diffuse) data by such rotations.

Nota bene: To provide direct comparativity of linear approximation to (the bisectors of) the given data via the least square method (LSM) [1, 8] by Legendre and Gauss, distance quadrat theories (DQT), general theories of moments of inertia (GTMI), as well as power mean theories (PMT) and rotation-quasi-invariant power mean theories (RQIPMT) naturally generalizing quadratic mean theories (QMT) and rotation-quasi-invariant quadratic mean theories (RQIQMT), respectively, use the same universal criterion given by distance quadrat theories (DQT).

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) using values of slope A of the straight line obtained via this method and each of these theories, separately.

Determine 2Smin(A), 2Smax(A), and then

S = SL = [2Smin(A) / 2Smax(A)]1/2

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

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

T = TL = 1 - S = 1 - SL = 1 - [2Smin(A) / 2Smax(A)]1/2 .

Denote:

S and T obtained via the least square method (LSM) with SLSM and TLSM = 1 - SLSM , respectively;

S and T obtained via distance quadrat theories (DQT) with SDQT and TDQT = 1 - SDQT , respectively;

S and T obtained via general theories of moments of inertia (GTMI) with SGTMI and TGTMI = 1 - SGTMI , respectively;

S and T obtained via quadratic mean theories (QMT) with SQMT and TQMT = 1 - SQMT , respectively;

S and T obtained via rotation-quasi-invariant quadratic mean theories (RQIQMT) with SRQIQMT and TRQIQMT = 1 - SRQIQMT , respectively.

Nota bene:

1. Distance quadrat theories (DQT) and general theories of moments of inertia (GTMI) have different forms and approaches but the same essence and hence always coinciding results. Therefore, we always obtain

SDQT = SGTMI ,

TDQT = TGTMI .

2. Distance quadrat theories (DQT) and general theories of moments of inertia (GTMI) are invariant by any rotation of the given data and always provide best linear approximation (bisectors).

3. In the present realization of metatheories of directed numeric test and check systems, coordinate system rotation invariance of the given data takes place. Therefore, power mean theories (PMT) and rotation-quasi-invariant power mean theories (RQIPMT) naturally generalizing quadratic mean theories (QMT) and rotation-quasi-invariant quadratic mean theories (RQIQMT), respectively, provide best linear approximation (bisectors) if and only if these theories give the same linear approximation (bisectors) as distance quadrat theories (DQT) and general theories of moments of inertia (GTMI) do, see Figures 1-20.

TstRt000.gif

Figure 1. β = 0. Sbest = SDQT = SGTMI = SLSM = SQMT = SRQIQMT ≈ 0.31623.

TstRt001.gif

Figure 2. β = π/180. Sbest = SDQT = SGTMI ≈ 0.316228. SLSM ≈ 0.316233. SQMT ≈ 0.432334. SRQIQMT ≈ 0.316232.

TstRt010.gif

Figure 3. β = π/18. Sbest = SDQT = SGTMI ≈ 0.3162. SLSM ≈ 0.3167. SQMT ≈ 0.3604. SRQIQMT ≈ 0.3171.

TstRt023.gif

Figure 4. β = π/8. Sbest = SDQT = SGTMI ≈ 0.3162. SLSM ≈ 0.3189. SQMT ≈ 0.3272. SRQIQMT ≈ 0.3170.

TstRt030.gif

Figure 5. β = π/6. Sbest = SDQT = SGTMI ≈ 0.3162. SLSM ≈ 0.3214. SQMT ≈ 0.3203. SRQIQMT ≈ 0.3166.

TstRt045.gif

Figure 6. β = π/4. Sbest = SDQT = SGTMI = SQMT = SRQIQMT ≈ 0.684. SLSM ≈ 0.669.

TstRt060.gif

Figure 7. β = π/3. Sbest = SDQT = SGTMI ≈ 0.3162. SLSM ≈ 0.3600. SQMT ≈ 0.3203. SRQIQMT ≈ 0.3166.

TstRt068.gif

Figure 8. β = 3π/8. Sbest = SDQT = SGTMI ≈ 0.3163. SLSM ≈ 0.3967. SQMT ≈ 0.3272. SRQIQMT ≈ 0.3170.

TstRt080.gif

Figure 9. β = 4π/9. Sbest = SDQT = SGTMI ≈ 0.3163. SLSM ≈ 0.6391. SQMT ≈ 0.3604. SRQIQMT ≈ 0.3171.

TstRt089.gif

Figure 10. β = 89π/180. Sbest = SDQT = SGTMI ≈ 0.316228. SLSM ≈ 2.772737. SQMT ≈ 0.432334. SRQIQMT ≈ 0.316232.

TstRt090.gif

Figure 11. β = π/2. Sbest = SDQT = SGTMI = SRQIQMT ≈ 0.316. SLSM ≈ 3.162. SQMT ≈ 0.445.

TstRt091.gif

Figure 12. β = 91π/180. Sbest = SDQT = SGTMI ≈ 0.316228. SLSM ≈ 2.772737. SQMT ≈ 0.432334. SRQIQMT ≈ 0.316232.

TstRt100.gif

Figure 13. β = 5π/9. Sbest = SDQT = SGTMI ≈ 0.3162. SLSM ≈ 0.6391. SQMT ≈ 0.3604. SRQIQMT ≈ 0.3171.

TstRt113.gif

Figure 14. β = 5π/8. Sbest = SDQT = SGTMI ≈ 0.3163. SLSM ≈ 0.3967. SQMT ≈ 0.3272. SRQIQMT ≈ 0.3170.

TstRt120.gif

Figure 15. β = 2π/3. Sbest = SDQT = SGTMI ≈ 0.3162. SLSM ≈ 0.3600. SQMT ≈ 0.3203. SRQIQMT ≈ 0.3166.

TstRt135.gif

Figure 16. β = 3π/4. Sbest = SDQT = SGTMI = SQMT = SRQIQMT ≈ 0.316. SLSM ≈ 0.331.

TstRt150.gif

Figure 17. β = 5π/6. Sbest = SDQT = SGTMI ≈ 0.3162. SLSM ≈ 0.3214. SQMT ≈ 0.3203. SRQIQMT ≈ 0.3166.

TstRt158.gif

Figure 18. β = 7π/8. Sbest = SDQT = SGTMI ≈ 0.3162. SLSM ≈ 0.3189. SQMT ≈ 0.3272. SRQIQMT ≈ 0.3170.

TstRt170.gif

Figure 19. β = 17π/18. Sbest = SDQT = SGTMI ≈ 0.3162. SLSM ≈ 0.3167. SQMT ≈ 0.3604. SRQIQMT ≈ 0.3171.

TstRt179.gif

Figure 20. β = 179π/180. Sbest = SDQT = SGTMI ≈ 0.316228. SLSM ≈ 0.316233. SQMT ≈ 0.432334. SRQIQMT ≈ 0.316232.

Nota bene:

1. Invert impossible values SLSM > 1 to obtain correct values of SLSM also providing correct values of TLSM . Such impossible values show paradoxical mutual replacement of the maximum and minimum values of the sums of the above squared distances. The cause of this replacement is that the least square method paradoxically gives (quasi)horizontal linear approximation to 2D data with (quasi)vertical best linear approximation.

2. Adequate distance quadrat theories (DQT), general theories of moments of inertia (GTMI), quadratic mean theories (QMT) and other power mean theories (PMT), as well as rotation-quasi-invariant quadratic mean theories (RQIQMT) and other power mean theories (RQIPMT) are very efficient in data estimation, approximation, and processing and reliable even by great data scatter and give, e.g., best linear approximation to any 2D data.

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

[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