From the combination of a Kriged output map containing the estimates and its output error map, you can create confidence interval maps by using some MapCalc statements.
When the errors in the error map have a normal distribution, you can find critical values of confidence levels in a probability distribution table of a standard normal curve.
In general, , where
is the Kriging estimation and
is the estimated error (standard deviation). The multiplication factors c (critical value) for the estimated errors (
) in the error map for different one-sided confidence levels are:
Confidence level: |
90% |
95% |
97.5% |
99% |
99.5% |
Critical value c: |
1.282 |
1.645 |
1.960 |
2.326 |
2.576 |
Example 1:
When for example you want to test whether for a metal concentration level of e.g. 100 mg/m3 (reference level), Kriged output values are outside the 98% confidence interval (after Pebesma 1996):
Level100 |
= |
IFF(Estim> 100 + 2*Error, "Upper", IFF(Estim< 100 - 2*Error, "Lower", "Uncertain")) |
where Estim is the output Kriging map and Error is the output error map.
In the Raster Map Definition dialog box, select your class domain Confi98.
Map Level100 contains 3 classes:
You can edit the representation as you wish.
In the same way, you can choose another reference level, or another critical value for another confidence interval.
Example 2:
Instead of creating for instance four different maps for four number of reference levels (e.g. for metal concentrations of 25, 50, 100 and 200 mg/m3), you can also make a combined confidence interval map which shows the actual values, i.e. the upper and lower boundaries, of a certain confidence interval.
Groups of pixels in the output confidence interval map will be shown as 2 'triangles' as if a group of pixels is divided along one of the diagonals of the groups. One half of the pixels in a group show the value of the lower boundary of the confidence interval and the other half show the value of the upper boundary of that confidence interval. In this example, each group of pixels consists of 4 lines and 3 columns. This is achieved by using the MOD (i.e. the modulus) function on the line numbers (%L) and column numbers (%C) of the map (see below).
Suppose you interested in the 98% confidence interval. You will thus use a multiplication factor (c) of 2 as above.
Upper = Estim + 2 * Error
Lower = Estim - 2 * Error
where Estim is the output Kriging map and Error is the output error map.
In the Raster Map Definition dialog box, select for both maps your value domain.
Conf98 = iff((%L mod 4) - (%C mod 3) <= 0, Upper, Lower)
Confidence interval map Conf98 then shows 'triangles' of values for the upper and lower boundaries of the confidence interval.
To obtain a confidence interval map with larger 'triangles', you can use larger values for the modulus functions in the last MapCalc statement.
Reference:
See also:
Anisotropic Kriging : functionality