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CONTEXT 2
ees |
CONTEXT 0
POSITION
THEMATIC
TEMPORAL
UN-
CERTAINTY
SENSITIVITY
ANALYSIS
PROPAGATION
OF UNCERTAINTY]
CONTEXT O
SPECIFIC REQUIREMENTS FOR IMPROVED DATA
Figure 2: A generalized model for handling uncertainty and
fitness for use in GIS.
This model is intended for implementation with
existing data structures in GIS software. For
vendors it represents evolution rather than
revolution.
AN EXAMPLE OF USING THE MODEL
Flexibility of the model can best be illustrated
through a simple example. A propagation metric is
currently being refined that will permit the
integration of positional, thematic and temporal
uncertainty using fuzzy numbers and fuzzy sets
(Zadeh, 1965; Kaufmann & Gupta, 1985). The current
example however will be limited to a consideration
of thematic or characterization uncertainty.
The propagation metric is summarized in Table 2.
These are normal convex fuzzy sets which have been
derived through rounding memberships ( ui [xi ) from
triangular fuzzy sets. The simplified fuzzy sets as
shown have the important properties that they
spread towards the mid-values (i.e. become more
fuzzy as one is more uncertain about the data and
therefore conform with linguistic use of fuzzy
sets) and are a constant orthogonal distance from
each other. Thus more complex fuzzy sets resulting
from propagation can be unambiguously matched with
a simplified fuzzy set. Direct interpretation or
constructing representations of verbalizations from
fuzzy sets is not always easy. A further feature of
these simplified fuzzy sets is that as a series
they normalize (i.e. where uwi=1 ) from 0 to 1 at an
interval of 0.1. Each fuzzy set can therefore be
referred to as a number akin to a probability
statement (with which there is generally a high
level of familiarity) and which to avoid confusion
with the literature is called a ‘fuzzy
expectation’ (®E). Singly or in combination it is
possible to map verbalizations about data
uncertainty or fitness-for-use in and out of the
system.
Figure 3 shows sample categorical data sets, Layer
1 and Layer 2. Each has four categories of data.
The observer who collected the data (say, by aerial
photographic interpretation) has identified four
levels of uncertainty and has assigned each polygon
to one of them. Uncertainty that a polygon contains
category A may arise either because there is
insufficient evidence in the photograph or because
the unit contains a level of heterogeneity. The
four linguistic classes of uncertainty have been
mapped as =~E by the observer (see Table 3). A
linguistic class may be cover a range of RE
combined with a Boolean OR, or overlap with the
adjacent class. Within the GIS data structure,
pointers can be made to a lookup table of «E which
is a small overhead compared with the 3,628,000
possible fuzzy sets between O and 1. In Figure 4
the two layers have been overlayed to form map L1L2
(the numbers in some of the polygons are for
reference below). The following Boolean selection
is carried out with the convential binary output
from a GIS given in Figure 5:
(CC AOR W ) AND #C ) AND #Y )
where # indicates NOT. The uncertainties can be
propagated using the same operators. In order to
see the rsults of this, two user's assessment
Schemes are used (see Table 3) where Scheme B is
more stringent in terms of its requirements for
fitness-for-use. Figure 6 shows the distribution of
X ;
01.2.3 .4 .5.6.71.8.9 1
fe 0 | 1
ux [1|9 1/9
zp |.2]. 1.9
ze |s 4 1.8 .4
yc |4 11.7|.8
(t T5 1,7 8
a [s 3 tris
t 17 4.81 8.4
i .8 65 9 1 9 5
.9 9 1.9
n |1 F
n
Table 2: Fuzzy expectation, «E, as underlying
simplified fuzzy sets.
