FUZZEKS [to Index]
General overview

The Fuzzy Evaluation and Kriging System FUZZEKS can be used for Both procedures utilize both measurements and fuzzy input data.
The regionalization part can be used without the aggregation part.

Spatial interpolation

The spatial interpolation is done by fuzzy kriging, a modified kriging method that allows to use fuzzy numbers as input data.
Kriging (non-fuzzy) is a common interpolation method, which bases on a statistical analysis of input data, precisely on experimental variograms, which can be handled by FUZZEKS.

An advantage of kriging is that an estimation variance can be calculated for the kriging results. It can help in the interpretation of the kriging results.

If fuzzy input data are used, the fuzzyness of the kriging result also helps in its interpretation (as discussed below in the chapter about applications).

Aggregation of different parameters

If more parameters should be taken into account for regionalization, they can be aggregated before kriging.

In order to aggregate these parameters they have to be transformed to a common scale (by the so-called membership functions) and then be combined by different operators.

Example: In order to get membership values of a given area for some quality such as "suitable as waste disposal site" some soil parameters (such as clay content, Cl-concentration, the thickness of some layer, etc.) must be transformed by approriate membership functions. Then they are on a common scale and can be combined using functions like "weighted sum" (and others).

These aggregation facilities also allow fuzzy numbers as input.

Display and output of results

FUZZEKS offers many display-options to view the results in 2 spatial dimensions (such as isolines, greyshades and colors) and FUZZEKS also allows to view the results for a part of the area only (along a cut/section and for a point).
ASCII-representations are available too.


Land evaluation (e.g. soil quality of a given area with respect to suitability for growing maize) is one possible application field [Burrough, 1989] (References are at the bottom of this page).
FUZZEKS has alredy been used in the field of Geology (to perform fuzzy kriging of hydrogeological data [Piotrowski et al., 1994, 1995, 1996]).
Other applications are planned.

The simplified example demonstrates an examination of suitability as waste disposal site.

There are many advantages in using fuzzy input data. An example is the possibility to incorporate expert knowledge (can be used for the definition of fuzzy numbers) in places where exact measurements are rare in order to reduce the kriging variance. As the kriging variance decreases, the fuzziness appears in the result. At the first glance this may look like no advantage in the end. But the result now presents more information, because the vague information is taken into account that could not be used by conventional methods.

Another point is that data often incorporate fuzziness quite naturally, as e.g. measurement tolerances (which can be expressed as fuzzy numbers). It can be of great advantage to know about the result's tolerances when the results have to be judged. If the fuzziness is not taken into account, results can not be judged as accurately.

Tips on how to read this documentation and
description of window types

In order to get a first impression of FUZZEKS and its facilities, examination of the simplified example - which demonstrates most of the facilities - is strongly recommended.

If you like to see quickly how FUZZEKS can be used in order to do fuzzy kriging only, look at "Quick start: Fuzzy kriging".

Once you are used to the window concept of FUZZEKS, the "Quick reference / Window description" section of the table of contents can be employed to find a topic in this help text in a very fast way: First select the window type you are interested in. You'll get a page with a picture of a typical window of that type. Then you can simply select the item you want information on with the mouse. In order to simplify the use of this feature (and also to help understanding the simplified example), an overview of FUZZEKS window-types follows.

How to get help on a specific topic describes all facilities to find an appropriate help page.
The simplest ways to do it is