Data, Problem, Method, and Result Invariance Theories in Fundamental Sciences of Estimation, Approximation, Data Modeling and Processing, and Solving General Problems

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

By solving problems [1], as well as data modeling, processing, approximation, and estimation, correspondence between data and method invariances by one-to-one transformations of coordinate systems is necessary to provide the objective sense of obtained results.

The essence of data and method invariances by coordinate system transformations is one-to-one correspondence between obtained results namely via the same one-to-one transformations.

In overmathematics [2-7] and fundamental sciences of estimation [8-13], approximation [14, 15], data modeling [16] and processing [17], as well as solving general problems [18], especially common and important transformation types are investigated. Among them are parallel translation invariance, rotation invariance, as well as tension-compression invariance.

Parallel translation invariance by any replacing the origin of a Cartesian coordinate system with the corresponding parallel translation of its axes is the most common.

Rotation invariance concerns any rotating a Cartesian or polar coordinate system about its origin by any angles. For a Cartesian coordinate system, combining any rotation of it about its origin by any angles with any parallel translation of its axes provides any rotation of it about any point. For rotation invariance, it is necessary that the physical dimensions (units) of the coordinates coincide and the lengths of the unit vectors of the coordinates are also equal to one another. But these conditions are not sufficient even if they are taken together. There are pure mathematical counterexamples, e.g. taking the three Euler angles on the axes of a three-dimensional Cartesian coordinate system or representing the polar angle and the distance (which is preliminarily divided by any length to obtain dimensionless distance) on the axes of a two-dimensional Cartesian coordinate system. The first of these two counterexamples shows that even adding the condition of coinciding the natures of the axes cannot lead to sufficiency. This invariance is typical for arbitrarily Cartesian representing purely geometric objects (points, lines, etc.) necessarily and (at least in the most practically important cases) sufficiently with coinciding the lengths of the unit vectors of the coordinates. Also note that coinciding the physical dimensions (units) of the coordinates is not sufficient for coinciding the natures of the axes. Cartesian representing the dependences either of the elongation on the length of a bar which is stretched by a force pair or of the Young modulus on a stress in a physically nonlinearly deformable solid.

Tension-compression invariance refers to any changing the lengths of the unit vectors of Cartesian coordinate axes with conserving their directions and the coordinate system origin. This invariance is typical for Cartesian representing physical data with generally different physical dimensions (units) of the coordinates, the lengths of the unit vectors of the coordinates being arbitrarily taken. It is also possible to consider any combinations of such one-to-one transformations of a Cartesian coordinate system with arbitrary parallel translations (this leads to any linear transformation of each coordinate separately, namely - unlike rotations - without using the remaining coordinates), and rotations of it.

Data invariance holds if, e.g., the given data are initially defined purely geometrically without any coordinate system. For example, a certain set of points belonging to a certain plane without any own coordinate system, any positioning this plane in a space, etc.

Result invariance holds if, e.g., the given data are initially defined purely geometrically without any coordinate system, too. Further consider the previous example in which the purely geometric straight line that is the nearest to the given set of points in a certain purely geometric sense is desired. Additionally, consider a certain set of straight lines belonging to a certain plane without any own coordinate system, any positioning this plane in a space, etc. Naturally, it is another example of data invariance. Here the purely geometric point that is the nearest to the given set of straight lines in a certain purely geometric sense is desired.

Problem invariance holds if and only if both data invariance and result invariance are valid, e.g. in any of the both previous examples.

Method invariance holds if and only if for any given data which can be processed by a ceriain method, preliminarily applying any coordinate system transformation of the corresponding type and further applying this method to these transformed data leads to a result which can be obtained via applying this transformation to the result of applying this method to the initial given data without their transformation. Equivalently, by subsequently applying both this method and any coordinate system transformation of the corresponding type to any given data which can be processed by this method, this method and this transformation commute with one another.

Nota bene: Methods are independent from problems and their data and results, and vice versa. The result of a problem can be unique whereas, in principle, distinct methods can be applied to this problem, and the results of these methods can be different.

Investigating invariance of some methods applicable to solving problems, e,g. in data modeling, processing, approximation, and estimation, leads to the following conclusions.

The classical least square method [1] by Legendre and Gauss in classical mathematics provides parallel translation invariance and tension-compression invariance without rotation invariance.

The least biquadratic method in fundamental sciences of estimation, approximation, and data processing [19] provides parallel translation invariance and tension-compression invariance without rotation invariance.

Methods in least squared distance theory in fundamental science of solving general problems [20] provide parallel translation invariance and rotation invariance without tension-compression invariance.

Methods in least squared distance theories in fundamental sciences of estimation, approximation, and data processing [21] provide parallel translation invariance and rotation invariance without tension-compression invariance.

Methods in linear two-dimensional theories of moments of inertia in fundamental sciences of estimation, approximation, and data processing [22] provide parallel translation invariance and rotation invariance without tension-compression invariance.

Methods in linear three-dimensional theories of moments of inertia in fundamental sciences of estimation, approximation, and data processing [23] provide parallel translation invariance and rotation invariance without tension-compression invariance.

Methods in general theories of moments of inertia in fundamental sciences of estimation, approximation, and data processing [24] provide parallel translation invariance and rotation invariance without tension-compression invariance.

Methods in quadratic mean theories for two dimensions in fundamental sciences of approximation and data processing [25] provides parallel translation invariance and tension-compression invariance without rotation invariance.

Methods in quadratic mean theories for three dimensions in fundamental sciences of approximation and data processing [26] provides parallel translation invariance and tension-compression invariance without rotation invariance.

Methods in groupwise centralization replacement theories in fundamental sciences of estimation, approximation, and data processing [27] provide parallel translation invariance and rotation invariance without tension-compression invariance.

Methods in outliers processing theories in fundamental sciences of estimation, approximation, and data processing [28] provide parallel translation invariance and rotation invariance without tension-compression invariance.

Methods in two-dimensional principal central normalization theories in fundamental sciences of estimation, approximation, and data processing [29] provide parallel translation invariance and rotation invariance without tension-compression invariance.

Methods in three-dimensional principal central normalization theories in fundamental sciences of estimation, approximation, and data processing [30] provide parallel translation invariance and rotation invariance without tension-compression invariance.

Methods in best circumferential approximation theories in fundamental sciences of estimation, approximation, and data processing [31] provide parallel translation invariance and rotation invariance without tension-compression invariance.

Methods in best spherical approximation theories in fundamental sciences of estimation, approximation, and data processing [32] provide parallel translation invariance and rotation invariance without tension-compression invariance.

Methods in iterative polar approximation theories in fundamental sciences of estimation, approximation, and data processing [33] provide parallel translation invariance and rotation invariance without tension-compression invariance.

Nota bene: Both applicability and invariance of a method do not provide its adequacy. For example, the least square method [1] by Legendre and Gauss only usually applies to contradictory (e.g., overdetermined) problems by estimation, approximation, and data processing, by methods of finite elements, points, etc. This method also provides parallel translation invariance and tension-compression invariance without rotation invariance.But overmathematics [2-7] and fundamental sciences of estimation [8-13], approximation [14, 15], data modeling [16] and processing [17], as well as solving general problems [18], have discovered a lot of principal shortcomings of the least square method. Additionally consider its simplest approach which is typical. Minimizing the sum of the squared differences of the alone preselected coordinates (e.g., ordinates in a two-dimensional problem) of the graph of the desired approximation function and of everyone among the given data depends on this preselection, ignores the remaining coordinates, and provides no objective sense of the result. Moreover, the method is correct in the unique case of a constant approximation only and gives systematic errors increasing together with the deviation (namely declination) of an approximation from a constant.

Acknowledgements to Anatolij Gelimson for our constructive discussions on coordinate system transformation invariances and his valuable remarks.

References

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[25] Lev Gelimson. Quadratic Mean Theories for Two Dimensions in Fundamental Sciences of Approximation and Data Processing (Essential). Mathematical Journal of the “Collegium” All World Academy of Sciences, Munich (Germany), 10 (2010), 4

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[28] Lev Gelimson. Outliers Processing Theories in Fundamental Sciences of Estimation, Approximation, and Data Processing (Essential). Mathematical Journal of the “Collegium” All World Academy of Sciences, Munich (Germany), 11 (2011), 5

[29] Lev Gelimson. Two-Dimensional Principal Central Normalization 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

[30] Lev Gelimson. Three-Dimensional Principal Central Normalization 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), 2

[31] Lev Gelimson. Best Circumferential Approximation 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), 5

[32] Lev Gelimson. Best Spherical Approximation 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), 6

[33] Lev Gelimson. Iterative Polar Approximation 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), 7