Principal Graph Types 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), 3

To data modeling, processing, estimation, and approximation, graph-analytic approaches [1] are well-known.

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 along with fundamental science of strength data unification, modeling, analysis, processing, approximation, and estimation [18, 19] provide reasonable interaction [20] of the both (graphical and analytic) approaches to the given data and their modeling, processing, estimation, and approximation. Pure analytic approach to data 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 interpretating 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. To clearly graphically interprete the given three-dimensional data, it is very useful to provide their two-dimensional modeling via suitable data trnsformation 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].

To efficiently apply graphical approach to the given data and their modeling, processing, estimation, and approximation, it is very useful to predefine the most important principal types of graphs for graphically interpreting the given data.

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

The simplest and most straightforward types of graphs are linear, namely a straight line in the two-dimensional case and a plane in the three-dimensional case. If there are bounds and limitations which allow using certain predefined parts only, it leads to clear complications. This holds even in the simplest case of a straight line whose limited parts can be its intervals, half-closed intervals, and closed intervals, or segments, as well as more complicated parts. All the more, bounds and limitations for a plane in the three-dimensional case lead to much more complications of higher levels. For example, if limited parts of a straight line or a plane do not contain the base of the perpendicular from a given data point onto a graph, then the graph point which is the nearest to a given data point can be not alone but multiple and it is also possible that there are no such nearest points at all, e.g. in the case when in any arbitrarily small neighborhood (vicinity) of the base of that perpendicular, there are points in admissible parts of graphs. The same can be valid not only by this linear graph type.

Combining limited parts of linear graphs naturally leads to the piecewise linear graph type. In this case, divide the given data into appropriate parts and consider them separately along with the corresponding parts of linear graphs.

Graphs nonlinearity naturally leads to much more complications of higher levels.

To begin with, consider quasilinear case of relatively slightly deforming (bending, twisting, distorting, or warping) parts of linear graphs (straight lines, usual two-dimensional and multidimensional planes) to obtain arcs and surfaces without deflection (changing curvature signs). By this quasilinear graph type with equal curvature signs, along with Cartesian coordinate systems with their transformations equalizing the generally different mean curvatures, using polar coordinate systems with either predefined (fixed) or variable poles can bring additional advantages. To select such poles, preliminarily consider probable centers of curvature and ranges of their variability.

Combining limited parts of quasilinear graphs with equal curvature signsnaturally leads to the quasilinear graph type with piecewise equal curvature signs. In this case, divide the given data into appropriate parts and consider them separately along with the corresponding parts of quasilinear graphs with piecewise equal curvature signs.

For the closed quasilinear graph type with equal curvature signs which contains, e.g., spheres and ellipsoids, along with Cartesian coordinate systems with their transformations equalizing the sums of the second powers of the homonymous coordinates of the given data points, using polar coordinate systems can bring additional advantages, too. To begin with, select the given data center as a pole.

By using polar coordinate systems, 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.

In many practically important cases, these simplest graph types suffice for data modeling, processing, estimation, and approximation. Otherwise, additionally introduce more complicated graph types, e.g. quantigraph types containing quantigraphs belonging to the quantisets building quantialgebras in quantianalysis in overmathematics [2-7].

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

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