The

- spatial (2-dimensional) interpolation (also called regionalization) and
- aggregation of different parameters (before the interpolation)

The regionalization part can be used without the aggregation part.

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).

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.

ASCII-representations are available too.

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.

description of window types

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.

- The background window is used for general purposes such as: defining where data should be stored, loading input data, and exiting the program. After defining where data should be stored the user does not need to save explicitily, because the program updates the files automatically (when changes are detected).
- The management and composition window
is used as central management component. If you want to open or find
any other window of FUZZEKS, simply press
**F6**in order to switch to the management window. Clicking at the appropriate item opens or activates the corresponding window.

The tools to define the aggregation of the parameters are also located in this window. - Membership function windows are used to define a transformation that is needed as preparation to aggregate different parameters.
- Kriging windows are used to deal with variograms and the display of fuzzy kriging results.

How to get help on a specific topic describes all
facilities to find an appropriate help page.

The simplest ways to do it is

- to seek through the "Detailed system description index" section in the table of contents or
- to search for a keyword by selecting the appropriate button at the top of the Windows-help window.

- A. Bárdossy, I. Bogardi, and W.E.Kelly:

"Geostatistics utilizing imprecise (fuzzy) information",

Fuzzy Sets and Systems, vol.31, pp.311-327, 1989

Fuzzy kriging (using fuzzy and crisp variograms) - P. A. Burrough:

"Fuzzy mathematical methods for soil survey and land evaluation",

Journal of Soil Science, 1989, 40, pp.477-492

Composition of (crisp) soil parameters using membership function ("Semantic Import Model") as preparation and weighted sum as composition - L. Zadeh:

"Fuzzy Sets",

Information and Control 8, 1965; pp.338-353

Often referenced in papers about fuzzy sets, Extension Principle is introduced - J. A. Piotrowski, F. Bartels, A. Salski, G. Schmidt (1994):

"Fuzzy Kriging of Imprecise Hydrogeological Data",

International Association for Mathematical Geology, Annual Conference Mont Tremblant, Quebec, Canada, October 1-5, 1994; Proceedings: pp.282-288

First application of the fuzzy kriging part of FUZZEKS - J. A. Piotrowski, F. Bartels, A. Salski, G. Schmidt (1995):

"Fuzzy logic in hydrogeology - closer to nature?",

9 Int. Conf. on the state of the Art of Ecological Modelling (ISEM'95), 11-15 August 1995, Beijing, China; Abstracts: S.91. - J. A. Piotrowski, F. Bartels, A. Salski, G. Schmidt (1995):

"Geostatistische Regionalisierung hydrogeologischer Parameter mit Fuzzy Kriging",

62. Tagung der Arbeitsgem. Nordwestdeutscher Geologen, Hamburg-Bergedorf, Tagungsband, 12-19 - J. A. Piotrowski, F. Bartels, A. Salski, G. Schmidt (1996):

"Fuzzy-Kriging-Regionalisierung hydrogeologischer Parameter",

In: Merkel, B., Dietrich, P.G., Struckmeier, W. & L Löhnert, E.P. (Hrsg.): Grundwasser und Rohstoffgewinnung. GeoCongress 2, Verlag Sven von Loga, Köln, 400-405. - J. A. Piotrowski, F. Bartels, A. Salski, G. Schmidt (1996):

"Geostatistical regionalization of glacial aquitard thickness in northwestern Germany, based on fuzzy kriging",

Mathematical Geology 28(4): 437-452. - J. A. Piotrowski, F. Bartels, A. Salski, G. Schmidt (1996):

"Estimation of hydrogeological parameters for groundwater modelling with fuzzy geostatistics: closer to nature?",

In: Kovar, K. & van der Heijde, P. (eds.) Calibration and Reliability in Groundwater modelling; Int. Conf. ModelCARE'96, Golden (Colorado), USA, 24.-26. Sept. 1996; Proceedings, 511-520. - F. Bartels:

"Entwicklung eines Fuzzy-Auswertungs- und Krigingsystems für raumbezogene Daten",

M.Sc. thesis, Institute of Informatics, University of Kiel (in German)

The most detailed and complete theoretical documentation of FUZZEKS