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# Technique for Multidimensional Nonlinear Curve Fitting

IP.com Disclosure Number: IPCOM000078151D
Original Publication Date: 1972-Nov-01
Included in the Prior Art Database: 2005-Feb-25
Document File: 3 page(s) / 79K

IBM

## Related People

Lawlor, FD: AUTHOR [+2]

## Abstract

This technique is for the generation of a model of a nonlinear relationship involving multiple variables, by examining a first relationship of one independent variable to the dependent variable under the situation where other independent variables are relatively constant, modelling this first relationship, determining a second model for the relationship of each coefficient in the first relationship to the second independent variable, and continuing the generation process in an iterative manner until all independent variables are included in the relationship.

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Technique for Multidimensional Nonlinear Curve Fitting

This technique is for the generation of a model of a nonlinear relationship involving multiple variables, by examining a first relationship of one independent variable to the dependent variable under the situation where other independent variables are relatively constant, modelling this first relationship, determining a second model for the relationship of each coefficient in the first relationship to the second independent variable, and continuing the generation process in an iterative manner until all independent variables are included in the relationship.

The following is specifically referred to the case in which there are three independent variables, for which data has been gathered against the dependent variable.

In step 1, the various variables and constants are placed into the computer memory and the default values, if any, are associated with them. In particular, those functions to which the data is to be fit are selected and placed into memory, together with standard regression routines for calculating their coefficients and the variable NCOEFS(f) standing for number of coefficients in each function.

In step 2, the accumulated data are read into the memory. This step includes the formatting of the data in core and the setting of pointers to indicate the boundaries of each data set.

Step 3 involves the determination of the relationship between the dependent variable y and the first independent variable x1. This determination involves a curve fitting of functions to the data for each of the functional forms available, to determine which curve best suits the data. This process is carried out for each of the level 1 data sets and is detailed in steps 30 to 34. At 33 a call is issued for the FITCALL routine which provides the best fit for each of the various functions available to the data in the first data set, and at its conclusion, returns to block 34 where the query - is the counter I equal to or greater than the number of data sets at level 1 (KURVDSN) - is asked. If the answer is a NO, the process continues until all of the data sets at level 1 have been fitted to a best function and then a return is made to block 4.

In step 4, one functional form of all those fitted to the data is selected over the others. The primary consideration for selection is which curve provides the best fit for representing the functional relationship of y to x1. While the criteria for selection of the best functional form is arbitrary, the following algorithm is used. The Error Sum of Squares (ESS) for all of the data sets in level 1 for a particular function is totalled and that process is repeated for all of the functional forms, after which the function with the smallest total value is selected as the best functional form for the data provided in level 1.

Then in step 5, a test is made to determine whether all curve fitting steps are finished. If there is a second independent variable,...