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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B2. Istanbul 2004 
Evidence theory has been applied abroad in artificial 
intelligence field. Anand (1996) applied evidence theory to 
knowledge discovery by combination operator. More contents 
about evidence theory may refer to Shafer (1976). 
4.2 Fuzzy Evidence Theoretic Approaches for Spatial 
Knowledge Discovery 
Evidence theory can only process the uncertainty cased by 
randomness. In fact, spatial data and knowledge include both 
randomness and vagueness. When considering randomness and 
vagueness simultaneously of spatial data and knowledge, it 
may take account into combining fuzzy theory with probability 
theory. Fuzzy evidence theory can process the two kinds of 
uncertainty integrating uncertain reasoning. 
Herein, we consider spatial knowledge discovery as uncertain 
reasoning process based on fuzzy evidence theory, which 
include soft discretization of spatial data and uncertainty 
transformation between quantitative data and qualitative 
concept by applying Gaussian fuzzy function, uncertain 
knowledge discovery and representation by fuzzy D-S belief 
structure and uncertain reasoning. 
A fuzzy D-S belief structure is one of D-S belief structure that 
the focus element is the fuzzy sets. When apply combination 
operator to combine two fuzzy belief structures, only to apply 
fuzzy sets operation. For example, /m; and m, are the two 
fuzzy D-S belief structures in he frame of discemment, © . 
Thus, the new fuzzy belief structure is: 
m=m Um, (8) 
where the focus element is: F, = À, M B, , the membership 
function is max(4 aitx) > Bjr) ) and 
m(F,)=m, (A; )*m,(B; y. 
The fuzzy rule based on fuzzy D-S belief structure is as follows: 
R(r):1f(XyisA;) and (X,isA}) + <and (X ,isA”) 
then Y is m, (9) 
where m, is a fuzzy belief structure with focus element 
8, ein B,,-, B, IG -L-- p), itis a fuzzy partition of 
output space. m, (B,,)is the basic probability assignment of 
B, » which indicate that the belief degree ( probability) of the 
ria B,, Therefore, the output of rules is uncertain. This 
kind of rule form should take account into the propagation of 
evidences in knowledge integrating uncertain 
reasoning, 
discovery 
Suppose that X;=x;,i=1---n is a group of input values. 
Then the knowledge discovery and reasoning process based on 
D-$ belief structure is as follows: 
(1) Compute the activation degree of every rule 7, : 
7, — AA; (x;) or EL41(x;)] (10) 
1 I 
(2) Make certain the output of single rule according to 
activation degree and rule consequent: 
My. = OT, Mr) (11) 
where @ is containing operator; mi, 1s a fuzzy belief structure 
on J and its focus element is F,; F,is the fuzzy subset of 
the output space and its definition is: 
H gy Y T, ^ ug (Y) or up. (v)=r, * ugy (y) (12) 
B, is a focus element of m, ; the basic probability associated 
to. 7^, is: 
^ 
mr( Fm m, (By (13) 
* $* Output the combination rules, adopt no-null combination 
operator to combine fuzzy belief structure: 
M ^ 
m=@m, (14) 
pz] 
to every set. /, = 47 dod F js +}, where F jf is a focus 
r 
A 
element of 7m, , which lies in a focus element: 
rzl^,. 7 
When operator is average, E, may be defined as: 
1 M | 
Hg, Cy) 7 = jy £O) (16) 
r=l 
and its basic probability is: 
M 
m(Œ,)= I» Ga ) (17) 
=| 
So the output is a fuzzy D-S belief structure m with focus 
element £,(k=1,--- p" ). 
(4) Anti-fuzzy to fuzzy belief structure m : 
io qui. 
y = > Yi m(E, ) (18) 
k-l 
where y, is anti-fuzzy value of focus element £, : 
YEE, (v 
y= DE (o) ( 19) 
3 H E, (v) 
Here, we adopt Gaussian function as the membership function 
of fuzzy sets in input and output space. 
(x — cy 
exp (20) 
2-20 
Suppose that [/, 4] is discussion field of variable and /,u is 
minimum and maximum value respectively of every dimension