8.6.9. A Proposed Procedure for Use of Fuzzy Set Theory
in Integrated Analysis of Spatial Data

 

The problem,

Given spatial data E = {e1, e2, ... , em} from m different sources S1, S2, ... , Sm, one wishes to decide which hypothesis among n of them H = {H1, H2, ... , Hn} is most likely to happen. Or in a classification problem, one wishes to decide which class among n classes {C1, C2, ... , Cn} is the most appropriate one into which E to be classified. Formally stated, one wishes to find out a projection F such that

F : S1 x S2 x ... x Sm Æ H

which satisfies

(1) 0 ² FHj(E) ² 1 for j = 1, 2, ... , n

(2)  FHj(E) = 1 .

It requires relatively deep mathematical knowledge to determine a projection from the Cartesian product space S1 x S2 x ... x Sm to H, interested reader may find Kruse et al. (1991) a starting point. This may be relaxed by finding a projection between each source Si to H.

Therefore,

One may follow the steps listed below to solve the problem posed.

Step 1. Consider each element in H fuzzy set  j, j = 1, 2, ... , n. Determine the fuzzy membership function on each source Si, i = 1, 2, ... , m for each Hj, j = 1, 2, ... , n. Thus a total of m x n membership functions need to be found. Usually, expert knowledge or fuzzy sets.

Step 2. Combine evidences from different sources to validate hypotheses or to conduct classification. Fuzzy set operations including union, intersection, complement and algebraic operation can be used for such purposes.

Step 3. Compare combined degree of membership for each hypothesis (class), confirm the hypothesis with the highest degree of membership.

Gong (1993) and a fuzzy classifier in a forest ecological classification research (Crain et al., 1993) are all following this procedure. It needs to be further validated. The assumption here is obviously each hypothesis is independent to the other.