The Lab's Quarterly, 2008, n. 1

Page 31

Il Trimestrale. The Lab's Quarterly, 1, 2008

31

The key concept of these calibration systems is still the identification of some anchor points on the support variable that is to be transformed into a fuzzy set. The mechanism by which it is possible to switch from the quantitative support variable to the membership values for individual cases is explained in detail for both methods. In short, in the first method, called "direct": 

the anchors are determined for full membership, full nonmembership and for

the crossover point (membership 0.5), on the support variable; 

it is estimated the natural logarithm of the odd associated to the full member-

ship anchor (odd = membership degree / (1-membership degree)); 

is calculated the ratio between the log odd thus obtained and the deviation of

the value of full membership from the crossover point on the support variable; 

the ratiothus obtained serves as a multiplier that applies to deviations of the va-

lues of individual cases on support variable from the value of the crossover point, so returning the scores on the metric space of log odds; 

logs odds are converted into membership values through a mathematic tran-

sformation projecting them back in the 0-1 range. In particular, is used the inverse of the logit function, so: the e constant, base of natural logarithm, is exponentiated using the log odd as exponent, and the result is divided by the same value increased by 1. In formula: e(log odd) / (1 + e(log odd))

[1]

This is the procedure for values above the crossover point. For values below the crossover point a similar multiplier is used, obtained with the odd associated with the membership value of full non-membership in relation to the deviation of the value of full non-membership from the point of crossover on the support variable. The indirect method, instead, uses regression techniques (typically cubic, anyway non-linear) to rescale the values of the support variable, categorized into six subsets, on the metric space of log odds, to obtain scores that can be transformed into member-


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