INTRODUCTION 



Mathematical description of asymmetric curve forms expected in a regression 

 relation can be difficult when descriptor components must apply over the entire range 

 of each independent variable involved. Under such constraint, failure of the analyst 

 to find suitably accurate forms (Bartlett [1947] and Draper and Hunter [1969] 

 recognized the necessity for allowing the analyst to establish his own acceptance 

 criteria) among his available alternatives might prompt adoption of a segmented descrip- 

 tor system. For our purposes, a segmented descriptor system is one wherein the 

 relation to be described is divided into two or more segments covering the ranges of 

 one or more independent variables. A descriptor is developed for the portion of the 

 response curve within each segment and applies there exclusively. The contiguous 

 array of these segments then portrays the entire relation. The segmented approach 

 might be regarded as less elegant, perhaps, from the standpoint of mathematical and 

 statistical manipulability, but this may be a necessary trade-off for descriptor 

 accuracy . 



Segmented descriptors are developed for several three-dimensional relations 

 using systems outlined in Matchacurves- 1 and -2 (Jensen and Homeyer 1970, 1971) and in 

 Matchacurve-3 (Jensen 1973) . The first example involves a two-segment descriptor. At 

 the time of analysis, the general form of the relationship was known. A bell-shaped 

 curve, possibly asymmetric, was expected over the range of one independent variable, 

 while a sigmoidal curve was expected over the second. Strong interaction was likely 

 to occur between the independent variables. 



Specification of a viable mathematical hypothesis for this potentially complex 

 form, from prior knowledge alone, was considered to be impractical. And, rather than 

 dedicate the data to the statistical evaluation of poorly specified hypotheses and 

 suffer a potential loss of information, the analyst elected to exhaust the data 

 graphically. The graphed model was then described mathematically. The resulting 

 function was refitted to response values at 36 control points on the graph, this by 

 least squares in the simple model.... 



Y = 6 (model transform) + e 



Example #2, a three-segment descriptor, was completely specified in graphic form 

 from prior knowledge. The objective here was simply to describe the graphed model 

 mathematically, using the Matchacurve system. Data were not involved. 



Both the graphic and general mathematical forms of these models are presented in 

 the text. Familiarity with the Matchacurve system will enhance understanding of the 

 segmentation adopted. Specific mathematical forms and associated explanatory material 

 are given in the appendix to reinforce the reader's knowledge, as needed, of mathema- 

 tical component development by means of Matchacurve. 



1 



