[Example home page]

A first glance at the result

Choosing a display type for this application

Kriging Basics / Theoretical concepts explains about the fuzzy kriging method in detail.

This page roughly describes the use of fuzzy kriging with FUZZEKS. If you are not familiar to kriging methods it may be easier to understand these topics in full detail when reading about Kriging Basics / Theoretical concepts first or, if you want further information about some special topic, have a look at the table of contents (quite at the end of the table of contents are the corresponding topics).

The kriging window contains two items:

On the left side there is a display that contains both the experimental and the theoretical variogram. The experimental variogram is shown by the crosses and serves as statistical basis to find a theoretical variogram. The theoretical variogram is displayed as red curve and serves as basis of the kriging process. Its parameters, like type of the curve and range/sill/nugget-effect must be given by the user.

On the right side there is the kriging result as 2-dimensional graphics.

The variogram

• The match with the experimental variogram, which is calculated by using the input data.
• The kriging result should not contradict the user's possible assumptions on how the isolines are supposed to run.
• A cross-validation can be done with other programs (using crisp kriging, but the same theoretical variogram). This procedure leaves out sequentially any single point of the input and kriges a value at this location in order to compare real values with estimated ones.
  This option is not part of FUZZEKS at this point of time.
• The kriging variance (which also can be calculated by FUZZEKS) should be not too high in any area of interest.
  Of course, the user should not look at the absolute values with respect to the kriging variance.
  It also must be considered that the calculation of the kriging variance bases on the theoretical variogram as well as the kriging result itself (i.e. if the theoretical variogram is wrong the kriging variance can not be trusted).

This is described in more detail in Theoretical variogram fitting (Function types and parameters)

The result

The above picture shows the result of the parameter aggregation as shown on the previous page, but the above picture demonstrates another type of display:
The places where values are estimated to be clearly higher than 0.8 (which means that the minimum possible value is greater than 0.8) are colored green and the places where values are estimated to be clearly lower than 0.8 (which means that the maximum possible value is smaller than 0.8) are colored magenta.
The black contour marks the line where the most possible border of the region with values higher than 0.8 is.
The white space between magenta and green indicates the area where it is not clear wether the value is higher, equal or lower than 0.8 (i.e. the minimum possible value is smaller or equal than 0.8 and the maximum possible value is larger or equal than 0.8).

This is not the only possible way to display the results.
FUZZEKS offers various options:

• The type of display. You can select this out of a list of available types:
  
  The element to be displayed is stated left of the colon (":"), the method to do it on the right.
  • vari. means "estimated kriging variance".
    value means "by kriging estimated most possible values".
    value(lo) means "by kriging estimated minimal values".
    fuzzy means "the fuzziness" (i.e. possible deviations to both sides (the size of the range of the values with possibility degree larger than 0) ).
    fuzzy(lo) means "the possible deviations to the bottom".
  • iso means "using isolines".
    darkness means "the darker the region is, the higher is the value".
    mag./cyan means "possible deviations to lower/higher values are displayed by the density of magenta/cyan points".
    For magenta-lo,cyan-hi see the example above. brightness means "brighter regions indicate larger values".
  
  The next page discusses how to find an appropriate type of display for the example.
  Following are settings, which can be influenced via options dialog windows:
• Isolines: The distance (in value space) and the lettering.
• The resolution of the result (higher resolutions need longer calculation times but deliver more exact results).
• Whether the variance should be calculated or not (this also uses more calculation time).
• Result values can also be shown for a single point or a section. The fuzzy values can be seen in more detail then.
  These facilities are shown on two of the next pages.
• An ASCII-output for a given grid is possible too (shown as 3rd next page).

Choosing a display type for this application

Kriging Basics / Theoretical concepts explains about the fuzzy kriging method in detail.
Theoretical variogram fitting (Function types and parameters)