Chapter 1 . Appllcations of Fuzzy Logic
9
1.4
Fuzzy Regionalization: A Fuzzy Kriging Approach
Kriging belongs to the most popular methods of spatial interpolation, but its
application is often restricted owing to an insufficient amount of data. If the
number of available measurements is too low for conventional kriging methods,
the data set can be supplemented using additional imprecise data subjectively
estimated by an expert. Fuzzy kriging utilizes exact (crisp) measurement data as
weIl as imprecise estimates obtained from an expert (Bardossy et al. 1989; 1990;
Diamond 1989; Kacewicz 1994). The imprecision and uncertainty of these
estimates can be handled with the fuzzy sets. The logical structure of this fuzzy
kriging procedure with both crisp and fuzzy data and a crisp theoretical variogram
is shown in Figure 1.2. The zigzag line marks stages with fuzzy data input in form
of fuzzy numbers. At two stages fuzziness is introduced into the calculation. First,
fuzziness in the input values causes fuzziness in the experimental variogram. An
expert takes the experimental variogram and its fuzziness into account when
fitting the crisp theoretical variogram. Second, the fuzzy input values are used at
the final step of kriging, namely at the calculation of the interpolated values.
Therefore, if the input data set contains at least one fuzzy number the kriging
results have the form of fuzzy numbers, too.
As an example of an application of fuzzy kriging in spatial interpolation of
geological data a fuzzy kriging interpolation of hydraulic-conductivity values
from an aquifer in northwestem Germany could be mentioned (Piotrowski et a1.
1996). Because of a high spatial variability of data and irregular distribution of
data points the modification of the original data set was necessary. After
supplementing the original data set (557 boreholes) with 30 imprecise (fuzzy)
points the kriging variance has been significantly reduced. The Fuzzy Evaluation
and Kriging System FUZZEKS developed at the University in Kiel (Barteis 1997)
has been used as a tool for spatial interpolation. The authors of this application
consider the fuzzy kriging approach in interpolation hydrogeological parameters
as an important tool with a potential of quantifying vast pieces of information
available as expert knowledge.
1.5
Fuzzy Knowledge-Based Modelling
As mentioned above, fuzzy knowledge-based modelling can be particularly useful
in cases where the relations between the components of an ecosystem are not
exactly known or where we do not have any analytical model for these relations,
or where we have an insufficient amount of data for statistical analysis. Ecologists
often use vague and ill-defined natural language to describe their knowledge
(Salski 1992; 1999). Therefore this knowledge can be represented by a set of
linguistic "IF -THEN" roles, which can be interpreted as a linguistic description of
9
1.4
Fuzzy Regionalization: A Fuzzy Kriging Approach
Kriging belongs to the most popular methods of spatial interpolation, but its
application is often restricted owing to an insufficient amount of data. If the
number of available measurements is too low for conventional kriging methods,
the data set can be supplemented using additional imprecise data subjectively
estimated by an expert. Fuzzy kriging utilizes exact (crisp) measurement data as
weIl as imprecise estimates obtained from an expert (Bardossy et al. 1989; 1990;
Diamond 1989; Kacewicz 1994). The imprecision and uncertainty of these
estimates can be handled with the fuzzy sets. The logical structure of this fuzzy
kriging procedure with both crisp and fuzzy data and a crisp theoretical variogram
is shown in Figure 1.2. The zigzag line marks stages with fuzzy data input in form
of fuzzy numbers. At two stages fuzziness is introduced into the calculation. First,
fuzziness in the input values causes fuzziness in the experimental variogram. An
expert takes the experimental variogram and its fuzziness into account when
fitting the crisp theoretical variogram. Second, the fuzzy input values are used at
the final step of kriging, namely at the calculation of the interpolated values.
Therefore, if the input data set contains at least one fuzzy number the kriging
results have the form of fuzzy numbers, too.
As an example of an application of fuzzy kriging in spatial interpolation of
geological data a fuzzy kriging interpolation of hydraulic-conductivity values
from an aquifer in northwestem Germany could be mentioned (Piotrowski et a1.
1996). Because of a high spatial variability of data and irregular distribution of
data points the modification of the original data set was necessary. After
supplementing the original data set (557 boreholes) with 30 imprecise (fuzzy)
points the kriging variance has been significantly reduced. The Fuzzy Evaluation
and Kriging System FUZZEKS developed at the University in Kiel (Barteis 1997)
has been used as a tool for spatial interpolation. The authors of this application
consider the fuzzy kriging approach in interpolation hydrogeological parameters
as an important tool with a potential of quantifying vast pieces of information
available as expert knowledge.
1.5
Fuzzy Knowledge-Based Modelling
As mentioned above, fuzzy knowledge-based modelling can be particularly useful
in cases where the relations between the components of an ecosystem are not
exactly known or where we do not have any analytical model for these relations,
or where we have an insufficient amount of data for statistical analysis. Ecologists
often use vague and ill-defined natural language to describe their knowledge
(Salski 1992; 1999). Therefore this knowledge can be represented by a set of
linguistic "IF -THEN" roles, which can be interpreted as a linguistic description of