I am reading a paper “Input variable identification- Fuzzy curves and fuzzy surfaces” by Yinghua Lin, George A. Cunningham , Stephen V. Coggeshall. In this paper authors proposed a method to reduce the number of input variables by applying fuzzy logic on the input variables. I am not able to understand the procedure clearly; I am writing the procedure and showing the graph also.

The steps are:

For each input variable we plot M data points(xik,yk) in (xi-y) space, i=1,2,...,n and k=1,2,...m.  For 3 input x1,x2,x3 graph is:

We create a fuzzy membership function ik(xi) and plot y ik(xi) at each data point xik.

           ik(xi) =exp  ,

                       k = 1,2,...,n

We defuzzify these fuzzy membership function by creating a fuzzy curve ci for each input variable xi,

ci(xi) =

We use the mean square error to rank the variables.

      MSE =

The error is calculated between each fuzzy curve and input variables.

We rank the input variables in ascending order of their rank and eliminate last 20 % of the variables.

Then fuzzy surface is created to remove related variables.

Fuzzy surface can be thought of as a “ two-dimensional fuzzy curve”. We define a fuzzy surface si,j  by:

Where xi and xj are input variables

 As with fuzzy curves, we compute a mean square error(MSE) for the fuzzy surfaces

Then we rank the variables in ascending order of MSE.

How they are getting second set of figures and third set of figures at step 2 and 3?