Hello List (and Jorge),



I’m looking for advice on back-transforming kriged values.  I know this has
previously been discussed on the list but haven’t found a satisfactory
solution for my application.  Apologies if I’ve missed it somewhere.



I’m trying to estimate the density of two species at unsampled locations,
based on sampled locations.  The sample distribution for species 1 is
zero-inflated and not right-skewed enough to be truly lognormal with a
maximum of ~85,000.  The distribution for species 2 is very-well
approximated by the lognormal distribution with no zero values and a range
of 100 - 80,000. For both species, there is evidently a high sampling error
or finer-scale variability as nuggets from semivariograms are ~50% of the
sill value.



I initially tried a log10 transformation and ordinary kriging (gstat). For
species 1, I’m added 1 to deal with the zeros and co-kriged with data from
adjacent years while species 2 is just ordinary kriging. From this, I’ve
tried direct (biased) back-transformation the standard bias-correction by
adding half the krig variance, and transverse-gaussian kriging (lambda=0).
In all cases, I used a leave-one-out method to attempt to recreate the
original data and none perform very well.



For species 1, the biased back-transformation does the best job, with
similar ranges and means, a heavy left tail as a result of the zero values
but a shift in the median from 3.1 to 2.5.  The bias-corrected method
produces much higher estimates with an increase in the mean and upper end of
the range.  The transverse-gaussian method truncates the distribution on
both ends.



Log+1  distribution of Species 1:

Raw Data

 Biased

 Bias-Corrected

 T_Gauss

  Min.

0.0

Min.

-0.1

Min.

1.0

Min.

2.2

1st Qu.

0.0

1st Qu.

1.3

1st Qu.

2.5

1st Qu.

2.5

Median

3.1

Median

2.5

Median

3.6

Median

2.8

Mean

2.5

Mean

2.5

Mean

3.7

Mean

3.1

3rd Qu.

4.3

3rd Qu.

3.8

3rd Qu.

5.0

3rd Qu.

3.7

Max.

4.9

Max.

4.8

Max.

6.2

Max.

4.6



Histograms available at:

https://docs.google.com/leaf?id=0B_LclJI7HZSJZTU0ZjRjZmUtNWY0Yy00MjgxLWFjNDQtZDhmNTUzZTgzNDE4&hl=en&authkey=CJnZgbAP




For species 2, all kriging methods reasonably re-create the central tendency
of the distribution but severely truncate both tails.



Log distribution of Species 2:

Raw Data

 Biased

 Bias-Corrected

 T_Gauss

  Min.

2.0

Min.

2.7

Min.

2.8

Min.

2.8

1st Qu.

2.9

1st Qu.

3.1

1st Qu.

3.2

1st Qu.

3.2

Median

3.2

Median

3.2

Median

3.3

Median

3.2

Mean

3.2

Mean

3.2

Mean

3.3

Mean

3.2

3rd Qu.

3.5

3rd Qu.

3.3

3rd Qu.

3.4

3rd Qu.

3.3

Max.

4.7

Max.

3.6

Max.

3.7

Max.

3.6



Histograms available at:

https://docs.google.com/leaf?id=0B_LclJI7HZSJZDM3NmUyMmQtMmFjZi00YzA4LTg0NTgtOThlYjRhODY3NTk3&hl=en&authkey=CL2_4NIO



At this point, I generally prefer the results from the biased backtransform
and feel that I’m dealing more with smoothing effects than actual bias.



Does anyone have suggestions for alternatives I haven’t tried?



It looks like the method outlined in Yamamoto 2007 (Computational
Geosciences 11:219), could be useful but I don’t know if anyone has been
implemented it in an R function or if it is applicable to co-kriging
methods.



More pragmatic than scientific, do people ever just rescale to the raw
distribution based on rank, similar to an omnibus normalizing procedure?



Suggestions are appreciated.



Thanks, Burton

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