Spherical 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), 7

By data modeling, processing, estimation, and approximation [1], data scatter is relatively great in many cases and often allows no discriminating different analytic approximation expression types or forms, 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. In such a situation, 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.

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], 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].

Spherical theories in fundamental sciences of estimation, approximation, data modeling and processing are applicable to given three-dimensional data for which there exists a certain probable approximation law. In particular, it is also possible to combine these theories with other theories transforming the given data and to use not only their end data but also their intermediate data. In particular, preliminarily apply graph-analytic theories [20], principal graph types theories [21], invariance theories [22], groupwise centralization theories [23], bounds mean theories [24], linear two-dimensional [25] and three-dimensional [26] theories of moments of inertia, as well as general theories of moments of inertia [27] in fundamental sciences of estimation, approximation, data modeling and processing to the given data.

Spherical theories are completing (supplementing) additions to all these and other theories and consider all the given data points.

The ideas and essence of spherical theories are as follows:

1. Determining the least closed area containing all the data points.

2. Very roughly purely graphically determining a certain probable approximation law with the corresponding probable approximation graph which can be a planar, piecewise planar, or nonplanar surface in this three-dimensional case.

3. Dividing the area boundary graph into two subgraphs, namely above and below subgraphs (if this law graph is nonclosed) or outer and inner subgraphs (if this law graph is closed), with building the both extreme levels of the given data points.

4. Determining the type [21] of this law graph.

4.1. If this type is linear, namely a plane in this three-dimensional case, then by the fundamental principle of tolerable simplicity [2-7], first apply namely linear theories, e.g. least squared distance theories [28, 29], least biquadratic method [30], and quadratic mean theories for three dimensions [31] in fundamental sciences of estimation, approximation, data modeling and processing to the given data points. 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 plane whose limited parts can be complicated. For example, if limited parts of 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. Secondly, it is also possible to apply general central normalization theories [32] in fundamental sciences of estimation, approximation, data modeling and processing to the given data points in order to investigate whether theories nonlinearity can give essential advantages compared with linear theories.

4.2. If this type is piecewise linear, namely piecewise planar in this three-dimensional case, then by the fundamental principle of tolerable simplicity [2-7], first divide the given data into appropriate parts and consider them separately along with the corresponding parts of linear graphs.

4.3. If this type is quasilinear with equal curvature signs, namely a result of relatively slightly deforming (bending, twisting, distorting, or warping) parts of linear graphs (planes) to obtain surfaces without deflection (changing curvature signs), then by the fundamental principle of tolerable simplicity [2-7], along with Cartesian coordinate systems with their transformations equalizing the generally different mean curvatures, using spherical 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.

4.4. If this type is quasilinear with piecewise equal curvature signs, namely combining limited parts of quasilinear graphs with equal curvature signs, then by the fundamental principle of tolerable simplicity [2-7], first divide the given data into appropriate parts and consider them separately along with the corresponding parts of quasilinear graphs with piecewise equal curvature signs.

4.5. If this type is closed quasilinear with equal curvature signs which contains, e.g., spheres and ellipsoids in this three-dimensional case, then by the fundamental principle of tolerable simplicity [2-7], 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 spherical coordinate systems can bring additional advantages, too. To begin with, select the given data center as an origin.

Consider such an initially introduced spherical coordinate system.

By using a spherical coordinate system, it is also possible to additionally introduce the Cartesian interpretation of this spherical coordinate system, e.g. with an azimuth angle as an x-coordinate, an inclination angle as a y-coordinate, and a radial distance as a z-coordinate in this three-dimensional case.

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].

5. Introducing normalization transformation of all the points of the whole space in this three-dimensional case. The essence of this normalization transformation is dividing each space point distance from the selected origin by the distance of the nearest (to this point) intersection of the radial ray (straight half-line) with the probable approximation graph from the selected origin. Apply this normalization transformation both to all the given data points and to all the points of the probable approximation graph.

5.1. For each initial data point, consider the unique radial ray (straight half-line) containing this point and determine the set of all the intersections of this radial ray with the probable approximation graph.

5.1.1. If this set contains one element only, then consider the corresponding probable approximation graph point as the radial projection of this data point on this probable approximation graph. Further consider this data point and its radial projection on this line mutual one-to-one corresponding to one another.

5.1.2. Otherwise (if this set uniquantity [2-7] is greater than 1), among its elements, consider the set of the probable approximation graph points which are the nearest to this data point.

5.1.2.1. If this set clearly contains one element only, then consider the corresponding probable approximation graph point as the radial projection of this data point on this probable approximation graph. Further consider this data point and its radial projection on this line mutual one-to-one corresponding to one another.

5.1.2.2. Otherwise (if this set is fuzzy, namely there are two probable approximation graph points for which it is unclear which of them is nearer to this data point than the remaining probable approximation graph point), consider the set of these both probable approximation graph points which both are the nearest to this data point. The set uniquantity q = 2. Divide 1 (the data point quantity) by this uniquantity q = 2 and consider this data point as the quantiset [2-7] of q = 2 quantielements, each of them coinciding with this data point and has quantity 1/q = 1/2 . Then for each of these both radial projection points on this probable approximation graph, select precisely one data point quantielement and consider it and this projection mutual one-to-one corresponding to one another.

5.2. For each initial data point quantielement, divide its distance from the selected origin by the distance of the corresponding radial projection point from the same selected origin to obtain the normalized radial distance of the corresponding normalized data point quantielement.

5.3. Apply the same normalization transformation to the area boundary graph consisting of two subgraphs, namely above and below subgraphs (if this law graph is nonclosed) or outer and inner subgraphs (if this law graph is closed), with building the both extreme levels of the given data points, to obtain the corresponding normalized area boundary graph and the both normalized area boundary subgraphs.

5.4. Apply the same normalization transformation to this probable approximation graph itself to naturally obtain the unit sphere in this three-dimensional case. Its center coincides with the same selected origin. The radius of this unit sphere is 1. This unit sphere is the corresponding normalized probable approximation graph.

5.5. Apply the same normalization transformation to the initially introduced spherical coordinate system to obtain the corresponding normalized spherical coordinate system.

6. Interpreting both all the normalized data points quantielements and this normalized probable approximation graph in this normalized spherical coordinate system.

7. Applying spherical theories themselves both to all the normalized data points quantielements and to this normalized probable approximation graph in this normalized spherical coordinate system.

8. Determining a normalized graph which can be considered a mean (middle) normalized graph in this normalized spherical coordinate system between the both normalized area boundary subgraphs in a certain reasonable sense with taking all the normalized data and valid data invariance type into account. By rotation invariance, use general theories of moments of inertia [25] in fundamental sciences of estimation, approximation, data modeling and processing.

9. Directly considering this mean (middle) normalized graph to be namely a normalized graph of this desired approximation law.

10. Applying the inversion of this normalization transformation to this mean (middle) normalized graph in this normalized spherical coordinate system for determining a graph which can be considered a mean (middle) graph in the initially introduced spherical coordinate system between the both area boundary subgraphs in a certain reasonable sense with taking all the normalized data and valid data invariance type into account. By rotation invariance, use general theories of moments of inertia [25] in fundamental sciences of estimation, approximation, data modeling and processing.

11. Directly considering this mean (middle) graph in the initially introduced spherical coordinate system to be namely a graph of this desired approximation law.

Given n (n ∈ N⁺ = {1, 2, ...}, n > 3) points [_j=1ⁿ (x_j , y_j , z_j)] = {(x₁ , y₁ , z₁), (x₂ , y₂ , z₂), ... , (x_n , y_n , z_n)] with any real coordinates in the initial Cartesian three-dimensional coordinate system Oxyz .

Determine the radial distance r_j , the azimuth angle φ_j , and the inclination angle θ_j of the jth data point in the initial spherical three-dimensional coordinate system with Oφθr .

Apply the normalization transformation and obtain the radial distance r°_j , the azimuth angle φ°_j , and the inclination angle θ°_j of the jth normalized data point in the normalized spherical three-dimensional coordinate system O°φ°θ°r° .

Now determine the normalized radial distance

r° = Σ_j=1ⁿ r°_j / n

of a normalized sphere.

Directly consider this normalized sphere as a mean (middle) normalized graph of this desired approximation law.

Apply the inversion of this normalization transformation to this mean (middle) normalized graph in this normalized spherical coordinate system for determining a graph which can be considered a mean (middle) graph in the initially introduced spherical coordinate system.

Directly consider this mean (middle) graph in the initially introduced spherical coordinate system to be namely a graph of this desired approximation law.

Spherical theories consider all the given data points and provide relatively simply approximating all the given data.

To improve data modeling, processing, estimation, and approximation, it is also possible to preliminarily locally represent each data point group with its center whose quantity equals the number of the points in this group and then applying both graphical and analytic approaches to the already groupwise centralized data, namely to a quantiset [2-7] of their local groupwise centers.

Consider a group of data points quantielements, generally their quantiset. Denote the ith coordinate with x_i (i = 1, 2, ... , m , m ∈ N⁺ = {1, 2, ...}, e.g. by m = 3, x₁ for x-coordinate, x₂ for y-coordinate, and x₃ for z-coordinate), the ith coordinate of the jth data point quantielement [2-7] (j = 1, 2, ... , n , n ∈ N⁺ = {1, 2, ...}) of a group of data points quantielements with x_ij , and the quantity of the element in this quantielement with q_j . Replace this group with its weighted central data point quantielement. The ith coordinate of the element of this quantielement is

x_i = Σ_j=1ⁿ q_jx_jj / Σ_j=1ⁿ q_j ,

and the quantity of the element in this quantielement is

q = Σ_j=1ⁿ q_j .

The variety of spherical theories and their variability provide their algorithms flexibility.

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

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

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