Geoscience Reference
In-Depth Information
Then the probability of that object being in the
intersect
of sets
A
and
B
is given by Equation 12.1,
in the
union
by Equation 12.2 and in the
inverse
by Equation 12.3:
(12.1)
px
()
∩
=
px px
() ()
⋅
AB
A
B
in which
A
∩
B
symbolises objects that belong to set
A
and set
B
(12.2)
px
()
=
px px px
() () ()
+
−
AB
∪
A
B
AB
∩
in which
A
∪
B
symbolises objects that belong to set
A
or set
B
(12.3)
px
()
′
=−
1
px
()
A
A
in which
p
(
x
)
A
′
indicates not belong to set
A
.
Examples of crisp sets are the set of people allowed to practise medicine as a doctor or the set of
people with degrees. Crisp geographical sets are the census division of, at least, European and North
American countries, where a specific region is distinguished, each household being assigned to one
and only one region and people are then counted as being within and/or belonging to that region for
the census; there is a simple one-to-many relationship between regions and people, and within the
hierarchy of regions, the census tracts or output areas are defined by
iat
boundaries (Smith, 2001).
In temporal analysis, an obvious crisp set is the year. Large crowds of people celebrate the change
from 1 year to the next, and many time series analyses divide information into arbitrary, crisp divi-
sions of time (e.g. years and months).
The crisp set is the basis of most conventional set and statistical analysis. For example, to estab-
lish the equality of the means of two groups of values, we use the
t
-test; here, a null hypothesis
is set up that the means are equal and an alternative hypothesis that they are not equal. We then
determine the value of the test statistic (
t
) and compare it with the threshold value which comes
from tables or a calculation of the distribution by a computer. The value calculated from the obser-
vations is then compared with the threshold value, and a decision is made whether to accept or
reject the null hypothesis (Ebdon, 1985; Harris and Jarvis, 2011). In a test with a particular level
of confidence, and with a particular data set, only two outcomes are valid: to accept or to reject
the null hypothesis. No matter how close the calculated test statistic is to the threshold, no other
outcome is acceptable. More advanced interpretations admit to a continuous probability of the null
hypothesis being correct or false, or else investigators conveniently select another value for the
confidence interval so that the test does confirm their preconceptions. In short, although hypoth-
esis testing is a matter of clear-cut crisp decisions, few investigators adopt a slavish adherence to
the acceptance or rejection of the null hypothesis, admitting some vagueness as to the meaning
of a threshold.
12.4 FUZZY SET THEORY
Zadeh (1965) first put forward fuzzy set theory as a direct response to the shortcomings of crisp sets.
To illustrate his proposal, Zadeh used the set of tall people. Most humans have an idea about this,
but any two humans are unlikely to agree on how high someone has to be in order to be a member
of the set, and whether a particular person matches one individual's conception of the set might
change. Furthermore, someone may not be prepared to commit to the idea that a particular person
is either tall or not tall. Rather, if questioned, a person might say they were nearly tall, or pretty tall,
using a linguistic hedge or qualifier to express a degree to which the person in question is within
their concept of tall people.
A similar concept to tallness is the idea of baldness (Burrough, 1992) where people may be pre-
pared to concede that someone with no hair is definitely bald, as is a person with a 100 hairs, but
