Directed Data 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), 22

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.

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 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).

Nota bene:

1. This method and all these theories are valid by coordinate system parallel translation invariance of the given data, which justifies further centralization transformation. Distance quadrat theories (DQT) and general theories of moments of inertia (GTMI) are valid by coordinate system rotation invariance of the given data. Power mean theories (PMT) naturally generalizing quadratic mean theories (QMT) are valid by coordinate system linear transformation invariance of the given data.

2. By great numbers of data points, increasing the power of theories brings many advantages and can became even necessary to provide adequate results. Power mean theories (PMT) naturally generalizing quadratic mean theories (QMT) simply provide using any positive power but are not invariant by coordinate system rotation of the given data. Distance quadrat theories (DQT) and general theories of moments of inertia (GTMI) are invariant by coordinate system rotation of the given data but are directly valid by the 2nd power only and admit their generalization for the 4th power with great additional complication. Now there are no obvious ways to provide their generalization for practical use by any other even natural-number powers, nothing to say about any positive power.

Therefore, it is very important to universalize power mean theories (PMT) naturally generalize quadratic mean theories (QMT) with providing their approximate invariance by coordinate system rotation invariance of the given data, which is principally possible in the case of coinciding the physical dimensions (units) of the both coordinate axes only.

Nota bene: Approximate invariance is a new very useful general concept.

Such universalization compensating the jumps of the sign function is the same by any positive power in power mean theories (PMT). Additionally, using namely the 2nd power provides not only clear simplification, but also direct comparison with distance quadrat theories (DQT) and general theories of moments of inertia (GTMI) all using namely the 2nd power, too.

Hence in this case, metatheories of directed numeric test and check systems may provide considering any rotation of already centralized two-dimensional data points whose linear approximation (linear bisector) is a priori clear due to the mirror symmetry of these data.

Nota bene: Preliminarily centralizing these data provides universalization of the ranges of the both coordinate axes by any rotation of them and brings clear simplification without any loss of generality.

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.

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 .

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

Then quadratic mean theories give

a = (Σ_j=1ⁿ y_j² / Σ_j=1ⁿ x_j²)^1/2 t^r 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 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)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 .

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) = (Σ_j=1ⁿ x_j² A² - 2Σ_j=1ⁿ x_jy_j A + Σ_j=1ⁿ y_j²)/(A² + 1)

by A = A_{1 , 2} providing ²S_min(A) and ²S_max(A) using values of slope A of the straight line obtained via this method and each of these theories, separately.

Determine ²S_min(A), ²S_max(A), and then

S = S_L = [²S_min(A) / ²S_max(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 = T_L = 1 - S = 1 - S_L = 1 - [²S_min(A) / ²S_max(A)]^1/2 .

Denote:

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

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

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

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

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

Nota bene: 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

S_DQT = S_GTMI ,

T_DQT = T_GTMI .

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.

Nota bene: To rotate these data about their center coinciding with the coordinate system origin in the same coordinate system Oxy by any angle β positive in the anticlockwise direction, also consider another coordinate system Ox'y' obtained from Oxy by its turning (rotating) about their common origin O namely by angle -β positive in the anticlockwise direction so that we have axis Ox' from Ox and axis Oy' from Oy . We obtain [1]

x' = x cos (-β) + y sin (-β) = x cos β - y sin β ,

y' = - x sin (-β) + y cos (-β) = x sin β + y cos β .

Now replace each point (x , y) with point (x' , y') and place this point (x' , y') in coordinate system Oxy but NOT in coordinate system Ox'y'.

Nota bene: Here x , y , x' , y' are real-number values without these designations. Therefore, there is no problem place this point (x' , y') in coordinate system Oxy but NOT in coordinate system Ox'y'. All the more, there is no necessity to show this auxiliary coordinate system Ox'y' used above to obtain the above transformation formulae for x' and y' via x and y .

Now we can rotate data about their center coinciding with the coordinate system origin in the same coordinate system Oxy 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.

Show suitable particular examples representing these both data types classified by their scatter and trend (directedness).

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.

Rotation-quasi-invariant 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

[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