Graph-Analytic 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), 1

To data approximation also in their modeling, processing, and estimation [1], pure analytic approach alone is blind and often leads to false results. Without graphically interpreting the given data, it is almost impossible to discover relations between them and their laws to provide adequate data processing. For reasonably analytically approximating the given data, it is necessary and very useful to create conditions for efficiently applying analytic methods. For example, graphically interpreting the given data usually immediately leads to simple hypotheses on analytic approximation character, e.g. linear, piecewise linear, parabolic, hyperbolic, circumferential, elliptic, sinusoidal, etc. by two-dimensional data or linear, piecewise linear, paraboloidal, hyperboloidal, spherical, ellipsoidal, etc. by three-dimensional data.

On the other hand, purely graphically interpreting the given data alone cannot provide sufficient possibilities not only for data processing, but also for discovering relations between them and their laws to provide adequate data processing.

As ever, the fundamental principle of tolerable simplicity [2-7] plays a key role.

In overmathematics [2-7] and fundamental sciences of estimation [8-13], approximation [14, 15], as well as data modeling [16] and processing [17], graph-analytic theories provide reasonable interaction of the both (graphical and analytic) approaches to the given data and their modeling, processing, estimation, and approximation.

To clearly graphically interpret the given three-dimensional data, it is very useful to provide their two-dimensional modeling via suitable data transformation if possible. For example, this is the case by strength data due to fundamental science of strength data unification, modeling, analysis, processing, approximation, and estimation [18, 19].

In some cases, the initial coordinate system is very suitable for discovering the relations between the given data. Otherwise, namely by coordinate system translation invariance of the given data, compare graphically interpreting them by two approaches. The first of them is straightforward and directly uses the given data themselves. The second approach to them preliminarily provides their centralization via appropriate centralization transformations which are clearly invariant. Namely, centralize the given data by subtracting every coordinate of the data center from the corresponding coordinate of every data point.

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 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)].

In the three-dimensional case, given n (n ∈ N⁺ = {1, 2, ...}, n > 3) points [_j=1ⁿ (x'_j , y'_j , z'_j)] = {(x'₁ , y'₁ , z'₁), (x'₂ , y'₂ , z'₂), ... , (x'_n , y'_n , z'_n)] with any real coordinates. Use centralization transformation x = x' - Σ_j=1ⁿ x'_j / n , y = y' - Σ_j=1ⁿ y'_j / n , z = z' - Σ_j=1ⁿ z'_j / n to provide coordinate system Oxyz central for the given data and further work in this system with points [_j=1ⁿ (x_j , y_j , z_j)].

In many cases, centralizing the given data brings clear advantages due to simplifying analytic expressions with initially unknown desired parameters (e.g. factors) in appropriate general forms expressing the essence of a hypothesis to be investigated.

The same holds for further rotating already centralized coordinate systems about their origins by appropriate angles to provide principal central coordinate systems with vanishing the nondiagonal elements of the tensors of moments of inertia due to introducing the corresponding coordinate system rotation transformations.

Further, it is possible to equalize the mean weights of all the coordinates via dividing them by the square roots of the sums of the second powers of the corresponding coordinates of the given data.

Along with Cartesian coordinate systems, using polar coordinate systems can bring additional advantages. By potential approximation optimization, there are two possibilities for selecting poles as origins of polar coordinate systems, namely either predefined (fixed) or variable poles. Poles variability providing much more optimization freedom brings not only expressions complication as pure technical difficulties, but also the advance of a pole to the center of the given data. Such an advance is true if a pole position is internal for a probable closed approximation graph (line, surface, etc.). Otherwise, such an advance is false and leads to illusion. In this case, use predefined (fixed) poles. For a probable closed approximation graph (line, surface, etc.), to begin with, use the center of the given data as such a pole. Generally, divide this graph into parts without deflection and approximate each part separately via selecting its probable center of curvature. It is also possible to use iterations either with changing the position of a pole from one iteration to another or always fix this position. In the last case, approximation graph variability can compensate fixing a pole. To search for a desired approximation, additionally introduce the Cartesian interpretation of a polar coordinate system, e.g. with a polar angle as an abscissa and a distance as an ordinate.

These theories are very efficient in estimation, approximation, and data processing.

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. Basic New Mathematics. Monograph. Drukar Publishers, Sumy, 1995

[3] Lev Gelimson. General Analytic Methods. Abhandlungen der WIGB (Wissenschaftlichen Gesellschaft zu Berlin), 3 (2003), Berlin

[4] Lev Gelimson. Elastic Mathematics. Abhandlungen der WIGB (Wissenschaftlichen Gesellschaft zu Berlin), 3 (2003), Berlin

[5] Lev Gelimson. Elastic Mathematics. General Strength Theory. Mathematical, Mechanical, Strength, Physical, and Engineering Monograph. The “Collegium” All World Academy of Sciences Publishers, Munich (Germany), 2004

[6] 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

[7] Lev Gelimson. Overmathematics: Fundamental Principles, Theories, Methods, and Laws of Science. Mathematical Monograph. The “Collegium” All World Academy of Sciences Publishers, Munich (Germany), 2009

[8] Lev Gelimson. General estimation theory. Transactions of the Ukraine Glass Institute, 1 (1994), 214-221

[9] Lev Gelimson. General Estimation Theory. Mathematical Monograph. The “Collegium” All World Academy of Sciences Publishers, Munich (Germany), 2001

[10] Lev Gelimson. General Estimation Theory Fundamentals. Mathematical Journal of the “Collegium” All World Academy of Sciences, Munich (Germany), 1 (2001), 3

[11] Lev Gelimson. General Estimation Theory Fundamentals (along with its line by line translation into Japanese). Mathematical Journal of the “Collegium” All World Academy of Sciences, Munich (Germany), 9 (2009), 1

[12] Lev Gelimson. General Estimation Theory (along with its line by line translation into Japanese). Mathematical Monograph. The “Collegium” All World Academy of Sciences Publishers, Munich (Germany), 2011

[13] Lev Gelimson. Fundamental Science of Estimation. Mathematical Monograph. The “Collegium” All World Academy of Sciences Publishers, Munich (Germany), 2011

[14] Lev Gelimson. General Problem Theory. Abhandlungen der WIGB (Wissenschaftlichen Gesellschaft zu Berlin), 3 (2003), Berlin

[15] Lev Gelimson. Fundamental Science of Approximation. Mathematical Monograph. The “Collegium” All World Academy of Sciences Publishers, Munich (Germany), 2011

[16] Lev Gelimson. Fundamental Science of Data Modeling. Mathematical Monograph. The “Collegium” All World Academy of Sciences Publishers, Munich (Germany), 2011

[17] Lev Gelimson. Fundamental Science of Data Processing. Mathematical Monograph. The “Collegium” All World Academy of Sciences Publishers, Munich (Germany), 2011

[18] Lev Gelimson. Fundamental Science of Strength Data Unification, Modeling, Analysis, Processing, Approximation, and Estimation (Essential). Strength and Engineering Journal of the “Collegium” All World Academy of Sciences, Munich (Germany), 10 (2010), 3

[19] Lev Gelimson. Fundamental Science of Strength Data Unification, Modeling, Analysis, Processing, Approximation, and Estimation (Fundamentals). Strength Monograph. The “Collegium” All World Academy of Sciences Publishers, Munich (Germany), 2010