Kriging belongs to the most popular methods of spatial interpolation of data. The

Imprecise data are represented in FUZZEKS by **fuzzy numbers**.

A **fuzzy set of** a set * X* is defined by a membership function

It is similar to a subset of

See also references at bottom of Overview/References.

A fuzzy set of * R* (the real numbers) is called a

(Convex means that the function decreases monotonically to both sides of the point where the value 1 is taken.)

Have a look at
"Load Input File / FDF-inputfile-format"
for further details on how to formulate fuzzy input data for FUZZEKS.

**Example**: Definition of a fuzzy number 4. If values of the parameter x can not be lower than 2 and not greater than 5 then a membership function may look like:

(In this example:*m(2)=0*,*m(3)=0.5*,*m(4)=1*, etc.)

For FUZZEKS one could define this fuzzy number in the form

**4 / 0: 2-5**

(first the point where the membership function takes the value 1, then the range of values with membership values grater than 0).**Remark:**The conventional crisp real numbers can be embedded in the set of fuzzy numbers, so that crisp numbers are a special case of fuzzy numbers. Their membership function is 1 for the given real number and 0 for all other real numbers. This is because a crisp real number always expresses only this number (all other numbers are impossible, so the membership value is 0).

The output of fuzzy kriging is an

Fuzzy kriging is based on a **theoretical variogram** which is result of
a statistical analysis of the parameter.
It tells about the similarity of values with respect to their distance.

The **experimental variogram** (calculated utilizing the input data) is used
as a statistical tool to find the theoretical variogram.
FUZZEKS can calculate it taking the fuzziness into account (which yields
a fuzzy experimental variogram),
although the theoretical variogram can only be crisp.
The reason is that the user can take the fuzziness into account when fitting
a theoretical variogram curve.

**Remark**: The experimental (semi)variogram can be formulated as

where the*xi*are the input coordinates, the*Z(xi)*are the input values which can take a form of fuzzy numbers mentioned above, and*h*is the distance-vector between points. In practice the direction of the distance-vector is not taken into account (in this case the vector*xi-xj*is replaced by the distance of*xi*and*xj*).

To facilitate the kriging procedure FUZZEKS offers

The following description is valid for the crisp case. If fuzzy values are
used, the Extension Principle (well known in fuzzy theory,
see also references at bottom of Overview/References,
especially [Zadeh, 1965]) is used to convert
the main kriging equation (3 headlines below) into a fuzzy function.

(If you want to have a rough description of the application of the
extension principle, have a look at the
Overview of membership function window.)
Below (at the main kriging equation) is an additional remark regarding
the fuzzy version of the main kriging equation.

Z(x): Input parameter value at the location x

Z(x): Estimated value at the location x

xi: Locations where an input parameter value exists (i=1..n)

(x1,x2)=(x1-x2): The semivariogram (=VAR(Z(x1)-Z(x2))/2), which is the same as the earlier definition of the experimental variogram because of the following assumptions

: Kriging variance at the location x ( E((Z(x)-Z(x))^2) )

The expected parameter value exists and does not depend on x.

(2) VAR(Z(x+h)-Z(x))/2=(h)

The difference Z(x+h)-Z(x) has finite variance

and does not depend on x.

(Remark: For a fixed x the coefficients are variables of real numbers only.)

**Remark for the case of fuzzy input values Z(x):**
The are crisp values although the input values are fuzzy
(in the case of a crisp theoretical variogram, which is true for FUZZEKS
as described above);
they can be obtained by the crisp solution (which is described below).

E(Z(x)-Z(x)) = 0

lead to two equations:

(1)

(2)

The are chosen in a way that a minimal estimation variance (2) will be obtained (observing (1)).

Remark: Minimizing (2) gives n equations (the n differentials equal zero) and (1) is one more equation.

Remark: Using the Lagrange principle, one can solve these equations (unique solution). A unique solution can always been found because of the selection of theoretical variograms that is allowed by FUZZEKS.

FUZZEKS allows to use kriging by offering

- Variogram display and tools to fit the theoretical variogram
- Facilities to display the kriging result