Adept data processor implementing function similation with inverse deviation variation weighting

• US 6,181,976 B1
• Filed: 04/08/1998
• Issued: 01/30/2001
• Est. Priority Date: 04/18/1997
• Status: Expired due to Fees

• Current Assignee: Larry Stephen Chandler
• Original Assignee: Larry Stephen Chandler
• Inventors: Chandler, Larry Stephen
• Primary Examiner(s): Grant, William
• Assistant Examiner(s): MARC, MCDIEUNEL
• Application Number:
  US09/056,637
• Time in Patent Office: 1,028 Days
• Field of Search: 700/43, 700/49, 700/50, 700/51, 700/52, 700/53, 700/67, 700/91, 700/181, 700/246, 700/28, 700/29, 700/31, 700/48, 700/293, 700/32, 700/33, 702/27, 702/85
• US Class Current: 700/52
• CPC Class Codes: G06F 17/10 Complex mathematical operations function generation by table look-up G06F1/03; evaluation of elementary functions by calculation G06F7/544

Abstract

A coefficient generator, data processing system and method of data reduction in correspondence with multiple-term parametric relationships, and particularly provides for data reduction in correspondence with approximative descriptions which include multiple dependent-variable terms whose higher order derivatives do not vanish. Forms of inverse of deviation variations are implemented to provide weighting which will correct for nonlinear function variations. The forms of inverse deviation weighting are implemented for representing forms of residual weighting and forms of implied squared residual weighting. Represented phases of function similation establish adept equations and implement forms of data processing and representing. Adept data processors correspondingly provide for characterizing approximating relationships in correspondence with provided data.

Claims

1. A coefficient generator comprising means for implementing inverse deviation variation weighting for representing the weight of respective function related deviations;

said implementing means including;

means for representing information; and

means for generating a data representation of at least one weighting coefficient;

said implementing including;

rendering said at least one weighting coefficient in a form compatible to be included in representing a respective weighting factor of a corresponding addend for at least one application; and

representing at least one other weighting coefficient;

said weighting factor being respectively represented as one of a plurality of weighting factors of the same sign;

said at least one weighting coefficient and said at least one other weighting coefficient being included in representing said plurality of weighting factors;

said at least one weighting coefficient and said at least one other weighting coefficient being correspondingly rendered for implementing at least one form of said inverse deviation variation weighting;

said data representation of at least one weighting coefficient being generated in correspondence with at least one representative measure of respective proportion to at least one evaluation for at least one derivative;

said at least one derivative being a variable dependent derivative of a function comprising at least one isolated term function of a represented function deviation;

said variable dependent derivative being a function of at least one variable;

said function deviation being a function of a plurality of variables;

said data representation of at least one weighting coefficient being generated by;

establishing correspondence between said at least one weighting coefficient and said at least one application; and

effecting manipulative operations in which evaluated correspondence to said at least one derivative is included in representing said at least one weighting coefficient;

said at least one evaluation for said at least one derivative being determined in correspondence with represented measure for at least one of said plurality of variables;

said at least one weighting coefficient substantially corresponding in proportion to the absolute value of said at least one evaluation of said at least one derivative being raised to a negative power other than negative two;

said manipulative operations excluding subjecting representative measure for a derivative to absolute value rectification for generating a representation of at least one transformation weight factor in rendering implied squared residual weighting as exclusively comprising coordinate normalizing proportions, each included one of said coordinate normalizing proportions correspondingly representing weighting as implemented by a form of discriminate reduction data processing, said at least one transformation weight factor comprising at least one transformation weight factor coordinate normalizing proportion, said at least one transformation weight factor coordinate normalizing proportion being generated by converting represented numerical correspondence to represent at least one positive value in a manner peculiar to the prescribed operations of a final stage discriminate rectifier, said final stage discriminate rectifier being implemented for providing said at least one transformation weight factor coordinate normalizing proportion by subjecting representative measure for a derivative to operations including absolute value rectification whereby numerical correspondence to said derivative is effectively converted to represent at least one positive value, said positive value corresponding within processing accuracy to the absolute value of the inverse of said derivative being raised to a power other than two.

2. A coefficient generator as in claim 1, further comprising means for representing said weighting factor by a composite weighting coefficient;

said weighting factor comprising said composite weighting coefficient;

said composite weighting coefficient comprising a representation for said at least one weighting coefficient multiplied by a corresponding representation for the inverse of an effective variance;

said plurality of weighting factors being represented by a plurality of composite weighting coefficients in correspondence with measurements for at least one of said plurality of variables;

at least some of said measurements including corresponding measurement error.

3. A coefficient generator as in claim 1, wherein said implementing includes representing said weighting factor by an alternately represented adept weighting coefficient.

4. A coefficient generator as in claim 3, wherein said at least one weighting coefficient is a coordinate related coefficient which is a function of at least one successive approximation for at least one approximating parameter.

5. A coefficient generator as in claim 1, wherein said implementing means includes means for accessing and processing information, whereby representations of at least two weighting coefficients are generated;

said at least two weighting coefficients comprising said at least one weighting coefficient and said at least one other weighting coefficient;

said representations of at least two weighting coefficients being generated in correspondence with at least two distinct evaluations for representing said at least one derivative;

said at least one derivative being a variable dependent partial derivative of said function deviation;

said at least two distinct evaluations corresponding to at least two respectively represented measures for said at least one variable;

said representations of at least two weighting coefficients being generated by;

accessing at least some information which establishes respective representative measure for each of said at least two distinct evaluations; and

said effecting manipulative operations including representing said at least two weighting coefficients in correspondence with said at least two distinct evaluations.

6. A coefficient generator as in claim 5, which is an even power rectifying coefficient generator.

7. A coefficient generator as in claim 5, which is a sign discriminate coefficient generator.

8. A coefficient generator as in claim 1, wherein said at least one weighting coefficient is a negative weighting coefficient.

9. A coefficient generator as in claim 8, wherein said implementing means includes a negafying exponential operator, and said manipulative operations include negafying operations.

10. A coefficient generator as in claim 1, wherein said data representation of at least one weighting coefficient is generated in correspondence with at least two evaluations of respective proportion to at least two derivatives;

said at least one derivative being included as one of said at least two derivatives;

said at least two derivatives being derivatives of different isolated term functions of said function deviation;

each of said at least two derivatives being taken with respect to the same variable;

said establishing including establishing respective representative measure in correspondence with each of said at least two derivatives; and

said manipulative operations including operations in which represented correspondence to said at least two evaluations is included in representing said at least one weighting coefficient as substantially corresponding in proportion to the absolute value of the product of said at least two evaluations being raised to a negative power other than negative two.

11. A coefficient generator as in claim 1, further comprising a control system configured for including said weighting factor to provide weighting of at least one residual term function as represented in a sum of weighted addends;

said at least one weighting coefficient being implemented in providing at least one form of residual weighting;

said weighting factor correspondingly representing said at least one form of residual weighting; and

said at least one application including said inverse deviation variation weighting in representing said at least one form of residual weighting.

12. A coefficient generator as in claim 1, wherein said implementing means includes means for accessing and processing information whereby said at least one weighting coefficient is rendered as a residual weighting coefficient by being included in representing at least one residuum;

said implementing including generating a representation for at least one sum comprising a plurality of evaluated addends;

each addend included in said plurality of addends being represented to include relative weighting of at least one corresponding weighting factor;

said effectuating including rendering a representation for at least one equation comprising said at least one sum;

each addend of said at least one sum being evaluated in correspondence with a respective measure for said at least one variable;

each said respective measure of said at least one variable representing a value for a corresponding measurement of said at least one variable;

said value being a respective value representing numerical correspondence to said measurement;

each said respective value being correspondingly represented as including at least one error associated with said measurement;

each said respective measure of said at least one variable being represented as including said at least one error;

each said plurality of evaluated addends respectively corresponding to a plurality of measures for at least one independent variable;

said representation for at least one sum of evaluated addends being generated by;

accessing some of said information which provides represented relative value for at least one fitting parameter; and

effectuating at least one form of equation residuum estimation;

said at least one application including said inverse deviation variation weighting in representing at least one form of residual weighting.

13. A coefficient generator as in claim 1, wherein said implementing means includes means for accessing and processing information whereby said at least one weighting coefficient is rendered for providing at least one form of implied squared residual weighting;

said at least one application including said inverse deviation variation weighting in representing said at least one form of implied squared residual weighting;

said implementing including;

accessing and processing at least some said information whereby said at least one representative measure is determined;

rendering at least one equation to include said plurality of weighting factors as represented to provide said at least one form of implied squared residual weighting; and

rendering means for representing a solution in compliance with said at least one equation as provided by a form of automated evaluating;

said solution being represented in correspondence with a plurality of measurements including measurements for said at least one variable;

said solution comprising a plurality of approximating parameters;

said plurality of approximating parameters including at least two approximating parameters which do not correspondingly represent constant terms of said function deviation;

said at least one weighting coefficient being a function of at least one of said at least two approximating parameters;

said manipulative operations excluding searching for and equating at least one approximating parameter to a value which corresponds to a minimum encountered value for a represented sum of weighted squared deviations, said minimum encountered value being considered with respect to alternate values for said plurality of approximating parameters, said at least one approximating parameter being represented and included in representing said at least one weighting coefficient.

14. A data processing system comprising:

a control system;

means for accessing and processing information; and

means for implementing at least one form of inverse deviation variation weighting for representing the weight of respective function related deviations;

said data processing system being configured for providing said implementing;

said implementing including;

representing at least one weighting coefficient;

rendering said at least one weighting coefficient in a form compatible to be included in representing a respective weighting factor of a corresponding addend;

representing at least one equation; and

representing said at least one weighting factor to implement said at least one form of inverse deviation variation weighting in representing said at least one equation;

said at least one equation being rendered in a form compatible to be included in representing a set of independent equations as rendered for solution by said data processing system;

said at least one weighting coefficient being included in representing said at least one equation;

representation for said at least one weighting coefficient being generated in correspondence with representative measure of respective proportion to at least one evaluation for at least one derivative;

said at least one derivative being a variable dependent derivative of a function comprising at least one isolated term function of a represented function deviation;

said variable dependent derivative being a function of at least one variable;

said function deviation being a function of a plurality of variables;

said representative measure of said at least one derivative being determined in correspondence with represented measure of at least one of said plurality of variables;

said at least one weighting coefficient being represented as substantially corresponding in proportion to the absolute value of said representative measure of said at least one derivative being raised to a negative power other than negative two;

said representation for at least one weighting coefficient being generated by;

establishing correspondence between said representative measure and at least some of said information; and

effecting manipulative operations in which evaluated correspondence to said at least one derivative is included in representing said at least one weighting coefficient;

said control system being configured for including said at least one equation in said set of independent equations as rendered for providing said solution;

said data processing system constituting means for including correspondence to said at least one equation in representing said set of independent equations as formulated for said solution;

said control system being configured for evaluating a plurality of approximating parameters in compliance with said set of independent equations;

said data processing system constituting means for evaluating said plurality of approximating parameters in compliance with said set of independent equations, said plurality of approximating parameters including a least two approximating parameters which do not constitute corresponding constant terms of said function deviation;

said plurality of approximating parameters being evaluated by means which do not establish a minimum value for a represented parametric expression as considered with respect to alternate values for said plurality of approximating parameters;

said set of independent equations not constituting a viable form for evaluating said alternate values which would minimize said represented parametric expression.

15. A data processing system as in claim 14, wherein said processing information includes generating a data representation of at least one fitting parameter in correspondence with a represented variation;

said data representation being generated by accessing some of said information which characterizes said represented variation;

said represented variation being characterized in correspondence with at least two variables;

said implementing including representing correspondence between some of said information and at least one sum of weighted addends;

said at least one equation comprising said at least one sum;

each addend of said at least one sum being evaluated in correspondence with a respective measure of at least one of said plurality of variables;

each said respective measure representing a value for a corresponding measurement of said at least one of said plurality of variables;

said value being a respective value representing numerical correspondence to said measurement;

said respective value for at least one of said plurality of variables being correspondingly represented as including error;

said plurality of evaluated addends respectively corresponding to a plurality of measures for at least one independent variable; and

said effectuating including evaluating said at least one fitting parameter in correspondence with a plurality of respective measures for characterizing said represented variation.

16. A data processing system as in claim 15, further comprising a memory and means for storing information whereby said data representation is generated and stored in said memory;

said means for accessing including means for accessing said data representation.

17. A data processing system as in claim 14, wherein at least one equation of said set of independent equations relates parametric expression for a sum of weighted residuals to a relative estimate for a residuum;

said accessing including accessing information which establishes an approximated value for said relative estimate; and

said data processing system further comprising a memory and means for representing said approximated value in formulating said at least one equation.

18. A data processing system as in claim 14, further comprising means for training channeled convergence in correspondence with reduction restraints.

19. A data processing system as in claim 18, further comprising means for effecting zeta parameter iterations.

20. A data processing system as in claim 14, wherein said implementing includes representing said weighting factor by an alternately represented adept weighting coefficient.

21. A data processing system as in claim 14, wherein said means for accessing and processing information comprises:

an adept alternate coefficient generator; and

means for generating representations of at least two coordinate related coefficients;

said weighting factor respectively comprising one of said at least two coordinate related coefficients;

said representations being generated in correspondence with at least two distinct evaluations for representing said at least one derivative;

said at least two distinct evaluations corresponding to at least two respectively represented measures for said at least one variable;

said at least one derivative being a variable dependent partial derivative of said function deviation;

said representations being generated by;

accessing at least some information which establishes respective representative measure for each of said at least two distinct evaluations; and

said effecting manipulative operations including operations in which represented correspondence to said at least two distinct evaluations is respectively included in representing said at least two coordinate related coefficients as having the same sign;

said manipulative operations excluding absolute value rectification being used for rendering a weighting factor as a transformation weight factor which is included in implementing at least one form of discriminate reduction data processing.

22. A data processing system as in claim 14, wherein said representation of at least one weighting coefficient is generated in correspondence with at least two evaluations of respective proportion to at least two derivatives;

said at least one derivative being included as one of said at least two derivatives;

said at least two derivatives being derivatives of different isolated term functions of said function deviation;

each of said at least two derivatives being taken with respect to the same variable;

said establishing including establishing respective representative measure in correspondence with each of said at least two derivatives; and

said manipulative operations including operations in which represented correspondence to said at least two evaluations is included in representing said at least one weighting coefficient as substantially corresponding in proportion to the absolute value of the product of said at least two evaluations being raised to a negative power other than negative two.

24. A data processing system as in claim 14, wherein said weighting factor is implemented for providing at least one form of implied weighting;

said at least one weighting coefficient being represented as a function of at least one successive approximation for at least one of said at least two approximating parameters;

said weighting factor correspondingly rendering said at least one form for said implied weighting;

said manipulative operations excluding absolute value rectification being used for rendering a weighting factor as a transformation weight factor which is included in implementing at least one form of discriminate reduction data processing.

25. A data processing system as in claim 14, further comprising means for effecting characteristic form iterations.

26. A data processing system as in claim 14, wherein said implementing includes representing at least one estimated sum of weighted addends;

said effectuating including;

accessing some of said information which provides represented relative value for at least one fitting parameter; and

effectuating at least one form of equation residuum estimation;

said control system being configured for providing said effectuating.

27. A data processing system as in claim 14, further comprising analog circuitry for providing at least some said manipulative operations;

said manipulative operations including generating an analog representation of at least one term function, wherein at least some said information is a real time analog signal being subjected to analog preprocessing whereby a corresponding analog representation of at least one integrated sum is generated;

said integrated sum representing a plurality of infinitesimal addends;

said at least one equation comprising said integrated sum;

said integrated sum being determined in correspondence with at least one respective measure for at least one of said plurality of variables;

each said respective measure representing a value for a corresponding measurement of said at least one of said plurality of variables;

said value being a respective value representing numerical correspondence to said measurement, and being correspondingly represented as including error;

each of said plurality of infinitesimal addends respectively corresponding to a plurality of measures for at least one independent variable;

said analog representation of said at least one term function being generated by;

generating at least one analog signal for providing weighting;

formulating at least one analog product signal which is an analog product of said at least one analog signal for providing weighting and said real time analog signal; and

formulating said analog representation of said at least one term function by integrating said analog product signal, over at least one interval of time.

28. A data processing system as in claim 14, wherein said weighting factor is a composite weighting coefficient included in providing at least one form of implied squared residual weighting;

said composite weighting coefficient comprising a representation of said at least one weighting coefficient multiplied by a corresponding representation for the inverse of an effective variance;

at least some measurements for at least one of said plurality of variables being considered as including corresponding measurement error.

29. A data processing system as in claim 14, wherein said weighting factor is a composite weighting coefficient included in providing at least one form of residual weighting;

said composite weighting coefficient comprising a representation of said at least one weighting coefficient multiplied by a corresponding representation for the inverse of the square root of an effective variance;

at least some measurements for at least one of said plurality of variables being considered as including corresponding measurement error.

23. A data processing system as in 14, claim wherein said control system is configured for representing said weighting factor as a coefficient for providing weighting of at least one residual term function as included in a sum of weighted addends;

said at least one weighting coefficient correspondingly rendering at least one form of residual weighting.

30. A produced data representation of processed information generated by a data processing system, said data processing system comprising:

means for accessing, processing, and representing information;

means for implementing inverse deviation variation weighting for representing the weight of respective function related deviations; and

means for rendering said data representation in substance;

said produced data representation comprising said data representation as rendered in substance;

said implementing including;

representing at least one weighting coefficient;

rendering said at least one weighting coefficient in a form compatible to be included in representing a respective weighting factor of a corresponding addend; and

representing correspondence between at least some information, at least one equation, and said data representation;

representing said at least one equation; and

including the representation of said at least one weighting coefficient to implement said inverse deviation variation weighting in representing said at least one equation;

said at least one equation being rendered in a form compatible to be included in representing a set comprising at least one independent equation as rendered for solution by said data processing system;

said at least one weighting coefficient being included in representing said at least one equation;

representation for said at least one weighting coefficient being generated in correspondence with a representative measure of respective proportion to at least one evaluation for at least one derivative;

said at least one derivative being a variable dependent derivative of a function comprising at least one isolated term function of a represented function deviation;

said function deviation being a function of a plurality of variables;

said representative measure of said at least one derivative being determined in correspondence with a represented measure of at least one of said plurality of variables;

said represented measure of at least one of said plurality of variables being represented as including error;

said at least one weighting coefficient being represented as substantially corresponding in proportion to the absolute value of said representative measure of said at least one derivative being raised to a negative power other than negative two;

said representation of at least one weighting coefficient being generated by;

establishing correspondence between said representative measure and said at least some information; and

effecting manipulative operations in which evaluated correspondence to said at least one derivative is included in representing said at least one weighting coefficient;

said means for rendering said data representation in substance including;

means for generating said data representation; and

means for transferring said data representation for utilization;

said data representation being generated by;

accessing some information which characterizes a represented variation as a function of at least one measure;

establishing correspondence between said represented variation and said function deviation;

representing said weight of respective function related deviations as implemented in correspondence with said function deviation and said represented variation;

effectuating said implementing; and

effecting said rendering;

said effectuating including;

activating means for processing and representing information whereby said data representation is generated;

representing correspondence between said data representation, said at least one equation, and at least some of said information which characterizes said represented variation;

rendering a representative form for said at least one independent equation; and

generating said data representation in correspondence with said at least one equation and at least some said information which characterizes said represented variation;

said manipulative operations excluding subjecting representative measure for a derivative to absolute value rectification for generating a representation of at least one transformation weight factor in rendering implied squared residual weighting as exclusively comprising coordinate normalizing proportions, each included one of said coordinate normalizing proportions correspondingly representing weighting as implemented by a form of discriminate reduction data processing, said at least one transformation weight factor comprising at least one transformation weight factor coordinate normalizing proportion, said at least one transformation weight factor coordinate normalizing proportion being generated by converting represented numerical correspondence to represent at least one positive value in a manner peculiar to the prescribed operations of a final stage discriminate rectifier, said final stage discriminate rectifier being implemented for providing said at least one transformation weight factor coordinate normalizing proportion by subjecting representative measure for a derivative to operations including absolute value rectification whereby numerical correspondence to said derivative is effectively converted to represent at least one positive value, said positive value corresponding within processing accuracy to the absolute value of the inverse of said derivative being raised to a power other than two.

31. A produced data representation as in claim 30, wherein said implementing includes rendering a representation of said represented variation in substance;

said data representation comprising a representation of said represented variation as rendered in substance;

said data representation being generated by;

accessing at least some of said information which characterizes said represented variation; and

generating said solution;

said solution comprising a plurality of approximating parameters being evaluated in correspondence with represented measure of at least two variables;

said plurality of approximating parameters including a least two approximating parameters which do not correspondingly represent constant terms of said function deviation;

said manipulative operations including evaluating said weighting factor as a function of at least one of said plurality of approximating parameters;

said plurality of approximating parameters representing a convergent solution to a set of independent equations;

said set of independent equations comprising said at least one independent equation;

said convergent solution being generated by means other than searching for and equating approximating parameters to values which correspond to a minimum encountered value for a represented sum of weighted squared deviations.

32. A produced data representation as in claim 31, wherein said means for transferring includes a memory for storing data for access by an application program being executed on a processing system, said processed information being stored in said memory, and said application program correspondingly rendering said data representation in substance.

33. A produced data representation as in claim 30, wherein said data representation comprises means for designating the generating of functional representation of a group of equations;

said group of equations comprising at least one set of independent equations for evaluating at least one fitting parameter; and

said group of equations including said at least one equation.

34. A method of implementing the inverse of represented deviation variation as rendered to establish the respective weight of function related deviations for characterizing an approximating relationship, said method comprising:

activating means for processing and representing information whereby at least one set of independent equations is represented in correspondence with at least some of said information; and

effectuating at least some data processing;

said effectuating including;

rendering a representation for at least one equation to be included in said at least one set of independent equations;

evaluating approximative correspondence for representing said at least one set of independent equations as related to at least some of said information;

rendering a representation of said at least one set of independent equations for providing a solution;

establishing and implementing said respective weight of function related deviations in said rendering; and

correspondingly characterizing said approximating relationship by said evaluating and said rendering;

said means for processing and representing information comprising;

a control system;

means for accessing information; and

said control system being configured for providing said representing, said implementing, and said effectuating;

said representing and said implementing including;

representing at least one weighting coefficient;

representing said at least one equation;

rendering said at least one weighting coefficient in a form compatible to be included in representing a respective weighting factor of a corresponding addend; and

including the representation of said at least one weighting coefficient to implement said respective weight of function related deviations in representing said at least one equation;

said at least one equation being rendered in a form compatible to be included in representing said at least one set of independent equations as rendered for providing said solution;

representation for said at least one weighting coefficient being generated in correspondence with a representative measure of respective proportion to at least one evaluation for at least one derivative;

said at least one derivative being a variable dependent derivative of a function comprising at least one isolated term function of a represented function deviation;

said variable dependent derivative being a function of at least one variable;

said function deviation being a function of a plurality of variables;

said representative measure of said at least one derivative being determined in correspondence with a represented measure of at least one of said plurality of variables;

said at least one weighting coefficient being represented as substantially corresponding in proportion to the absolute value of said representative measure of said at least one derivative being raised to a negative power other than negative two;

said representation for at least one weighting coefficient being generated by;

establishing correspondence between said representative measure and at least some of said information; and

effecting manipulative operations in which evaluated correspondence to said at least one derivative is included in representing said at least one weighting coefficient;

said at least one set of independent equations comprising a plurality of equations;

said plurality of equations rendering means for evaluating a plurality of approximating parameters in correspondence with said approximating relationship;

said control system being rendered for evaluating said plurality of approximating parameters in compliance with said at least one set of independent equations;

said plurality of approximating parameters including at least two approximating parameters which do not correspondingly represent constant terms of said function deviation;

said at least one weighting coefficient being a function of at least one of said at least two approximating parameters;

said manipulative operations excluding rendering said evaluated correspondence by searching for and equating at least one approximating parameter to a value which corresponds to a minimum encountered value for a represented sum of weighted squared deviations, said minimum encountered value being considered with respect to alternate values for said plurality of approximating parameters, said at least one approximating parameter being represented and included in representing said at least one weighting coefficient.

35. A method as in claim 34, wherein said manipulative operations exclude subjecting representative measure for a derivative to absolute value rectification for generating a representation of at least one transformation weight factor in rendering implied squared residual weighting as exclusively comprising coordinate normalizing proportions, each included one of said coordinate normalizing proportions correspondingly representing weighting as implemented by a form of discriminate reduction data processing, said at least one transformation weight factor comprising at least one transformation weight factor coordinate normalizing proportion, said at least one transformation weight factor coordinate normalizing proportion being generated by converting represented numerical correspondence to represent at least one positive value in a manner peculiar to the prescribed operations of a final stage discriminate rectifier, said final stage discriminate rectifier being implemented for providing said at least one transformation weight factor coordinate normalizing proportion by subjecting representative measure for a derivative to operations including absolute value rectification whereby numerical correspondence to said derivative is effectively converted to represent at least one positive value, said positive value corresponding within processing accuracy to the absolute value of the inverse of said derivative being raised to a power other than two.

36. A method as in claim 35, wherein said effectuating includes accessing and processing information, whereby representations of at least two weighting coefficients are generated;

said at least one weighting coefficient being one of said at least two weighting coefficients;

said representations of at least two weighting coefficients being generated in correspondence with at least two distinct evaluations for representing said at least one derivative;

said at least one derivative being a variable dependent partial derivative of said function deviation;

said at least two distinct evaluations corresponding to at least two respectively represented measures for said at least one variable;

said representations of at least two weighting coefficients being generated by;

accessing at least some information which establishes respective representative measure for each of said at least two distinct evaluations; and

said effecting manipulative operations including representing said at least two weighting coefficients in correspondence with said at least two distinct evaluations.

37. A method as in claim 34, whereby a data representation of at least one represented variation is generated in correspondence with at least some of said information and said approximating relationship;

said data representation being generated by evaluating said plurality of approximating parameters in compliance with said set of independent equations.