Method and apparatus for rapid approximation of system model

Abstract

Systems, methods, apparatus and mechanisms that iteratively generate one or more equations (conforming to one or more functions or functions types) of increasing complexity (e.g., degree or order) to provide one or more corresponding approximations of a System Under Study (SUS), where each approximation or “fit” provides additional information about the SUS and where the various equations may be summed to provide a final “fit” having an adequate level of accuracy.

What is claimed is: 
     
         1 . An apparatus for modeling a System Under Study (SUS), the SUS exhibiting at least one output adapted in response to a plurality of input parameters, the apparatus comprising a processor configured to:
 generate an improved approximation model for said SUS comprising a summation of an initial approximation model, a constant term, and a plurality of polynomials, each polynomial being associated with a respective N-tuplet of input parameters for greater than zero integer values of N, wherein polynomials included within said improved approximation model are generated by iteratively performing the following steps until a sufficiently accurate fit is achieved:
 rank ordering the N-tuplets of input parameters; 
 selecting a highest ranking N-tuplet of input parameters from among the remaining N-tuplets of input parameters to be processed; 
 fitting a polynomial involving input parameters in the selected N-tuplet of input parameters; 
 updating the current approximation model by adding the fitted polynomial; 
 adjusting the constant term as needed to match the evaluations of the SUS and the current approximation model at a point; and 
 evaluating fit of the said current approximation model to SUS; 
   wherein data representative of the improved approximation model of said SUS is stored in memory.   
     
     
         2 . The apparatus in  claim 1 , wherein fitting a polynomial involving input parameters in a selected N-tuplet of input parameters comprises:
 iteratively performing the following steps until a sufficiently accurate fit is achieved:
 selecting a plurality of model generation points in the N-dimensional space defined by axis parallel lines of the selected N-tuplet of input parameters, where model generation points are incremented by one or more for subsequent iterations in the said N-dimensional space; 
 evaluating the SUS at the model generation points; 
 fitting a polynomial involving input parameters from the said N-tuplet of input parameters to the SUS evaluation; and 
 evaluating fit of said polynomial to SUS; 
   and removing polynomial terms that do not involve all input parameters from the said N-tuplet of input parameters.   
     
     
         3 . The apparatus of  claim 1 , wherein rank ordering is determined according to a difference between evaluations of SUS and the current approximation model at ranking stress points generated using at least one of orthogonal array design, fractional factorial design, full factorial design, design of experiments and space filling designs. 
     
     
         4 . The apparatus in  claim 1 , wherein rank ordering is determined according to at least one of importance, presumed importance, random order, past data, alphabetical order and order of entry. 
     
     
         5 . The apparatus of  claim 4 , wherein importance of input parameters is determined by main effects analysis of the difference between SUS evaluations and current approximation model evaluations at ranking stress points. 
     
     
         6 . The apparatus of  claim 4 , wherein importance of N-tuplets of input parameters is determined by N-parameter interaction effects analysis of the difference between evaluations of SUS and the current approximation model at ranking stress points. 
     
     
         7 . The apparatus of  claim 1 , wherein fitting a polynomial involving input parameters in the selected N-tuplet of input parameters is performed using at least one of orthogonal polynomials, matrix inversion, Lagrange polynomials and solving simultaneous equations. 
     
     
         8 . The apparatus of  claim 1 , wherein sufficiency of accuracy is determined by goodness of fit criteria comprising at least one of a maximum absolute difference, a sum of absolute differences, a mean square of selected top absolute differences, and a mean square error. 
     
     
         9 . The apparatus of  claim 1 , wherein said sufficiently accurate fit is achieved in response to reaching at least one of a dimensional limit, a cost constraint, a time constraint and a resource constraint. 
     
     
         10 . The apparatus of  claim 1 , wherein the initial approximation model is equal to one of the following: zero, a user specified model, a system specified model and a previously improved approximation model. 
     
     
         11 . The apparatus of  claim 1 , wherein the fit is determined by generating one or more stress points and evaluating a difference between evaluations of the current approximation model and the SUS at the said stress points according to at least one goodness of fit criteria comprising an absolute difference, a sum of absolute differences, a mean square of selected top absolute differences, and a mean square error. 
     
     
         12 . The apparatus of  claim 11 , wherein stress points are generated using at least one of orthogonal arrays, fractional factorial designs, full factorial designs, design of experiments and space filling designs. 
     
     
         13 . The apparatus of  claim 11 , wherein one or more stress points generated in one iteration are used in at least one subsequent iteration. 
     
     
         14 . The apparatus of  claim 11 , wherein one or more stress points are generated for each iteration. 
     
     
         15 . The apparatus of  claim 1 , further comprising:
 generating summary statistics associated with the improved approximation model of said SUS; and   storing data representative of generated summary statistics in memory.   
     
     
         16 . The apparatus of  claim 15 , wherein said summary statistics comprise at least one of mean, standard deviation, maxima, minima, inflection points, confidence intervals, probability percentiles, zeros of an equation, integrals associated with one or more input parameters and derivatives associated with one or more input parameters. 
     
     
         17 . The apparatus of  claim 1 , further comprising:
 post-processing said improved approximation model of said SUS according to at least one of an equation, a distribution and a region of interest of input parameters to generate thereby decision rules; and   storing data representative of the decision rules in memory.   
     
     
         18 . The apparatus of  claim 1 , wherein rank ordering of input parameters performed only in a first iteration. 
     
     
         19 . The apparatus of  claim 1 , wherein a sufficiency of accuracy parameter is adapted so that an iteration is performed until polynomials for all of a specified set of N-tuplets of input parameters have been included in the current approximation model. 
     
     
         20 . The apparatus of  claim 19 , wherein evaluation of fit of the current approximation model to the SUS occurs only after polynomials for all of a specified set of N-tuplets of input parameters have been included in the current approximation model. 
     
     
         21 . The apparatus of  claim 1 , wherein fitting a polynomial involving input parameters in the selected N-tuplet of input parameters comprises constructing a polynomial with an origin proximate the center point. 
     
     
         22 . The apparatus of  claim 1 , wherein fitting a polynomial involving input parameters in the selected N-tuplet of input parameters is accomplished by performing the steps of:
 selecting a plurality of model generation points in the N-dimensional space defined by axis parallel lines of the selected N-tuplet of input parameters;   evaluating the SUS at the model generation points;   fitting a polynomial involving input parameters from the said N-tuplet of input parameters to the SUS evaluation; and   removing polynomial terms that do not involve all input parameters from the said N-tuplet of input parameters.   
     
     
         23 . The apparatus of  claim 2 , wherein the fit is determined by:
 generating one or more stress points in the N-dimensional space defined by the axis parallel lines of the input parameters in the N-tuplet of input parameters that are distinct from the model generation points; and   evaluating a difference between evaluations of the said polynomial and the SUS at the said stress points according to at least one goodness of fit criteria comprising absolute difference, sum of absolute differences, mean square of selected top absolute differences and mean square error.   
     
     
         24 . The apparatus of  claim 23 , wherein stress points are generated using at least one of orthogonal arrays, fractional factorial designs, full factorial designs, design of experiments and space filling designs. 
     
     
         25 . The apparatus in  claim 1 , wherein fitting a polynomial involving input parameters in the selected N-tuplet of input parameters comprises:
 iteratively by performing the following steps until a sufficiently accurate fit is achieved:
 selecting a plurality of model generation points in or close to the N-dimensional space defined by axis parallel lines of the selected N-tuplet of input parameters, where model generation points are incremented by at least one subsequent iteration within approximately the said N-dimensional space; 
 evaluating the SUS at the model generation points; 
 fitting a polynomial involving input parameters from the said N-tuplet of input parameters to the SUS evaluation; and 
 evaluating fit of said polynomial to SUS; 
   and removing polynomial terms that do not involve all parameters from the said N-tuplet of input parameters.   
     
     
         26 . The apparatus in  claim 1 , wherein fitting a polynomial involving input parameters in the selected N-tuplet of input parameters comprises iteratively performing the following steps until a sufficiently accurate fit is achieved:
 selecting a plurality of model generation points in the N-dimensional space defined by axis parallel lines of the selected N-tuplet of input parameters, where model generation points are incremented by at least one for subsequent iterations in the said N-dimensional space;   evaluating the SUS at the model generation points;   fitting a polynomial involving input parameters from the said N-tuplet of input parameters to the SUS evaluation; and   evaluating fit of said polynomial to SUS.   
     
     
         27 . The apparatus of  claim 2 , wherein fitting a polynomial involving input parameters from the said N-tuplet of input parameters to the SUS evaluation is performed using at least one of orthogonal polynomials, matrix inversion, Lagrange polynomials and solving simultaneous equations. 
     
     
         28 . The apparatus of  claim 2 , wherein sufficiency of accuracy is determined by goodness of fit criteria comprising at least one goodness of fit criterion comprising an absolute difference, a sum of absolute differences, a mean square of selected top absolute differences and a mean square error. 
     
     
         29 . An apparatus for modeling a System Under Study (SUS), the SUS exhibiting at least one output adapted in response to a plurality of input parameters, the apparatus comprising a processor configured to:
 generate an improved approximation model for said SUS comprising a summation of an initial approximation model, a constant term, and a plurality of complex function elements, each complex function element being associated with a respective N-tuplet of input parameters for greater than zero integer values of N, wherein complex function elements included within said improved approximation model are generated by iteratively performing the following steps until a sufficiently accurate fit is achieved:
 rank ordering the N-tuplets of input parameters; 
 selecting a highest ranking N-tuplet of input parameters from among the remaining N-tuplets of input parameters to be processed; 
 fitting a complex function element involving input parameters in the selected N-tuplet of input parameters; 
 updating the current approximation model by adding the fitted complex function element; 
 adjusting the constant term as needed to match the evaluations of the SUS and the current approximation model at a point; and 
 evaluating fit of the said current approximation model to SUS; 
   wherein data representative of the improved approximation model of said SUS is stored in memory.   
     
     
         30 . The apparatus of  claim 29 , wherein the complex function elements are associated with one of a polynomial function, a wavelet function, a Fourier series and a Walsh function. 
     
     
         31 . A tangible and non-transient computer readable storage medium storing instructions which, when executed by a computer, adapt the operation of the computer to provide a method of modeling a System Under Study (SUS), the SUS exhibiting at least one output adapted in response to a plurality of input parameters, the method comprising:
 generating an improved approximation model for said SUS comprising a summation of an initial approximation model, a constant term, and a plurality of polynomials, each polynomial being associated with a respective N-tuplet of input parameters for greater than zero integer values of N, wherein polynomials included within said improved approximation model are generated by iteratively performing the following steps until a sufficiently accurate fit is achieved:
 rank ordering the N-tuplets of input parameters; 
 selecting a highest ranking N-tuplet of input parameters from among the remaining N-tuplets of input parameters to be processed; 
 fitting a polynomial involving input parameters in the selected N-tuplet of input parameters; 
 updating the current approximation model by adding the fitted polynomial; 
 adjusting the constant term as needed to match the evaluations of the SUS and the current approximation model at a point; and 
 evaluating fit of the said current approximation model to SUS; 
   wherein data representative of the improved approximation model of said SUS is stored in memory.   
     
     
         32 . A computer program product wherein computer instructions, when executed by a processor in a computing device, adapt the operation of the computing device to provide a method of modeling a System Under Study (SUS), the SUS exhibiting at least one output adapted in response to a plurality of input parameters, the method comprising:
 generating an improved approximation model for said SUS comprising a summation of an initial approximation model, a constant term, and a plurality of polynomials, each polynomial being associated with a respective N-tuplet of input parameters for greater than zero integer values of N, wherein polynomials included within said improved approximation model are generated by iteratively performing the following steps until a sufficiently accurate fit is achieved:
 rank ordering the N-tuplets of input parameters; 
 selecting a highest ranking N-tuplet of input parameters from among the remaining N-tuplets of input parameters to be processed; 
 fitting a polynomial involving input parameters in the selected N-tuplet of input parameters; 
 updating the current approximation model by adding the fitted polynomial; 
 adjusting the constant term as needed to match the evaluations of the SUS and the current approximation model at a point; and 
 evaluating fit of the said current approximation model to SUS; 
   wherein data representative of the improved approximation model of said SUS is stored in memory.