2D Data Testing Distance Quadrat Theories, Inertia Moment Theories, and Quadratic 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), 9

To contradictory (e.g. overdetermined) problems consisting of sets of given relations with some unknowns, there are no precise solutions. It is necessary to relatively simply find such values of these unknowns that all these relations are approximately satisfied with possibly small deviations in certain reasonable sense. Such values are called quasisolutions to the corresponding problems.

In particular, by data modeling, processing, estimation, and approximation [1], data scatter is relatively great in many cases.

Let us compare the results of linearly fitting 2D Data of some characteristic types via distance quadrat theories (DQT) (by rotation invariance), general theories of moments of inertia (GTMI) (by rotation invariance), and quadratic mean theories (QMT) (by linear transformation invariance) in overmathematics [2-7] and fundamental sciences of estimation [8-13], approximation [14, 15], data modeling [16] and processing [17]. Note that 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 [19, 20]. Apply graph-analytic theories [21], principal graph types theories [22], and groupwise centralization theories [23] in fundamental sciences of estimation, approximation, data modeling and processing to the given data.

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

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)] to be approximated with a straight line ax + by = 0 containing origin O(0, 0).

In the following numeric tests, along with initial data points and linear approximations to them, or their bisectors, data scatter measures S_L with respect to the corresponding linear bisectors are also shown, see Figures 1-8 with replacing (x’, y’) via (x , y):.

Figure 1. S_L = 0.218 (DQT & GTMI, QMT)

Figure 2. S_L = 0.507 (DQT & GTMI, QMT)

Figure 3. S_L = 0.366 (DQT & GTMI), 0.507 (QMT)

Figure 4. S_L = 0.366 (DQT & GTMI), 0.507 (QMT)

Figure 5. S_L = 0.218 (DQT & GTMI, QMT)

Figure 6. S_L = 0.507 (DQT & GTMI, QMT)

Figure 7. S_L = 0.444 (DQT & GTMI, QMT)

Figure 8. S_L = 0.525 (DQT & GTMI), 0.562 (QMT)

Nota bene: By linear approximation, the results of distance quadrat theories (DQT) and general theories of moments of inertia (GTMI) coincide. By Σ_j=1ⁿ y_j² = Σ_j=1ⁿ x_j² (and the best linear approximation y = ± x + C), the same also holds for quadratic mean theories (QMT) (Figures 1, 2, 5, 6, 7). Here y = ± x + 2 (Figures 1, 2, 5, 6). By Σ_j=1ⁿ y_j² ≠ Σ_j=1ⁿ x_j² , QMT give other results than DQT and GTMI (Figure 8). But QMT are valid by another invariance type than DQT and GTMI. DQT & GTMI and QMT 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. 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 [Overmathematics and Other Fundamental Mathematical Sciences]. 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 Solving General Problems. Mathematical Monograph. The “Collegium” All World Academy of Sciences Publishers, Munich (Germany), 2011

[19] 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

[20] 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

[21] Lev Gelimson. Graph-Analytic Theories in Fundamental Sciences of Estimation, Approximation, Data Modeling and Processing (Essential). Mathematical Journal of the “Collegium” All World Academy of Sciences, Munich (Germany), 11 (2011), 2

[22] Lev Gelimson). Principal Graph Types Theories in Fundamental Sciences of Estimation, Approximation, Data Modeling and Processing (Essential). Mathematical Journal of the “Collegium” All World Academy of Sciences, Munich (Germany), 11 (2011), 3

[23] Lev Gelimson. Groupwise Centralization Theories in Fundamental Sciences of Estimation, Approximation, Data Modeling and Processing (Essential). Mechanical and Physical Journal of the “Collegium” All World Academy of Sciences, Munich (Germany), 11 (2011), 1

[24] Lev Gelimson. Data, Problem, Method, and Result Invariance Theories in Fundamental Sciences of Estimation, Approximation, Data Modeling and Processing, and Solving General Problems (Essential). Mathematical Journal of the “Collegium” All World Academy of Sciences, Munich (Germany), 11 (2011), 1