2 hour seminar within the Geostatistics training course at WUR

1. Spatial Interpolation Comparison
Evaluation of spatial prediction methods
Tomislav Hengl
ISRIC — World Soil Information, Wageningen University
Geostatistics course, 25–29 October 2010, Wageningen

2. Based on
Hengl, T., MacMillan, R.A., 2011? Mapping efficiency and
information content. submitted to International Journal of
Applied Earth Observation and Geoinformation, special issue
Spatial Statistics Conference.
Geostatistics course, 25–29 October 2010, Wageningen

3. Topic
Geostatistics = a toolbox to generate maps from point data
i.e.to interpolate;
Geostatistics course, 25–29 October 2010, Wageningen

4. Topic
Geostatistics = a toolbox to generate maps from point data
i.e.to interpolate;
There are many possibilities;
Geostatistics course, 25–29 October 2010, Wageningen

5. Topic
Geostatistics = a toolbox to generate maps from point data
i.e.to interpolate;
There are many possibilities;
An inexperienced user will often be challenged by the amount
of techniques to run spatial interpolation;
Geostatistics course, 25–29 October 2010, Wageningen

6. Topic
Geostatistics = a toolbox to generate maps from point data
i.e.to interpolate;
There are many possibilities;
An inexperienced user will often be challenged by the amount
of techniques to run spatial interpolation;
. . .which method should we use?
Geostatistics course, 25–29 October 2010, Wageningen

7. Have you heard of SIC?
Geostatistics course, 25–29 October 2010, Wageningen

8. The spatial prediction game
Participants were invited to estimate values located at 1000
locations (right, crosses), using 200 observations (left, circles).
Geostatistics course, 25–29 October 2010, Wageningen

9. Lessons learned (from SIC)
Geostatistics course, 25–29 October 2010, Wageningen

10. Li and Heap (2008)
Geostatistics course, 25–29 October 2010, Wageningen

11. How many techniques are there?
Li and Heap (2008) list over 40 unique techniques.
1. Are all these equally valid?
2. How to objectively compare various methods (which criteria
to use)?
3. Which method to pick for your own case study?
Geostatistics course, 25–29 October 2010, Wageningen

12. There are not as many
There are roughly five main clusters of techniques:
1. splines (deterministic);
Geostatistics course, 25–29 October 2010, Wageningen

13. There are not as many
There are roughly five main clusters of techniques:
1. splines (deterministic);
2. kriging-based (plain geostatistics);
Geostatistics course, 25–29 October 2010, Wageningen

14. There are not as many
There are roughly five main clusters of techniques:
1. splines (deterministic);
2. kriging-based (plain geostatistics);
3. regression-based;
Geostatistics course, 25–29 October 2010, Wageningen

15. There are not as many
There are roughly five main clusters of techniques:
1. splines (deterministic);
2. kriging-based (plain geostatistics);
3. regression-based;
4. bayesian methods;
Geostatistics course, 25–29 October 2010, Wageningen

16. There are not as many
There are roughly five main clusters of techniques:
1. splines (deterministic);
2. kriging-based (plain geostatistics);
3. regression-based;
4. bayesian methods;
5. expert systems / machine learning;
Geostatistics course, 25–29 October 2010, Wageningen

17. The 5 criteria
1. the overall mapping accuracy, e.g.standardized RMSE at
control points — the amount of variation explained by the
predictor expressed in %;
Geostatistics course, 25–29 October 2010, Wageningen

18. The 5 criteria
1. the overall mapping accuracy, e.g.standardized RMSE at
control points — the amount of variation explained by the
predictor expressed in %;
2. the bias, e.g.mean error — the accuracy of estimating the
central population parameters;
Geostatistics course, 25–29 October 2010, Wageningen

19. The 5 criteria
1. the overall mapping accuracy, e.g.standardized RMSE at
control points — the amount of variation explained by the
predictor expressed in %;
2. the bias, e.g.mean error — the accuracy of estimating the
central population parameters;
3. the model robustness, also known as model sensitivity — in
how many situations would the algorithm completely fail /
how much artifacts does it produces?;
Geostatistics course, 25–29 October 2010, Wageningen

20. The 5 criteria
1. the overall mapping accuracy, e.g.standardized RMSE at
control points — the amount of variation explained by the
predictor expressed in %;
2. the bias, e.g.mean error — the accuracy of estimating the
central population parameters;
3. the model robustness, also known as model sensitivity — in
how many situations would the algorithm completely fail /
how much artifacts does it produces?;
4. the model reliability — how good is the model in estimating
the prediction error (how accurate is the prediction variance
considering the true mapping accuracy)?;
Geostatistics course, 25–29 October 2010, Wageningen

21. The 5 criteria
1. the overall mapping accuracy, e.g.standardized RMSE at
control points — the amount of variation explained by the
predictor expressed in %;
2. the bias, e.g.mean error — the accuracy of estimating the
central population parameters;
3. the model robustness, also known as model sensitivity — in
how many situations would the algorithm completely fail /
how much artifacts does it produces?;
4. the model reliability — how good is the model in estimating
the prediction error (how accurate is the prediction variance
considering the true mapping accuracy)?;
5. the computational burden — the time needed to complete
predictions;
Geostatistics course, 25–29 October 2010, Wageningen

22. Can we simplify this?
1. In theory, we could derive a single composite measure that
would then allow you to select ‘the optimal’ predictor for any
given data set (but this is not trivial!)
2. But how to assign weights to different criteria?
3. In many cases we simply finish using some na¨ıve predictor —
that is predictor that we know has a statistically more optimal
alternative, but this alternative is not feasible.
Geostatistics course, 25–29 October 2010, Wageningen

23. Automated mapping
In the intamap package1 decides which method to pick for you:
> meuse$value <- log(meuse$zinc)
> output <- interpolate(data=meuse, newdata=meuse.grid)
R 2009-11-11 17:09:14 interpolating 155 observations,
3103 prediction locations
[Time models loaded...]
[1] "estimated time for copula 133.479866956255"
Checking object ... OK
1
http://cran.r-project.org/web/packages/intamap/
Geostatistics course, 25–29 October 2010, Wageningen

24. Hypothesis
We need a single criteria to compare various prediction methods.
Geostatistics course, 25–29 October 2010, Wageningen

25. Mapping accuracy and survey costs
The cost of a soil survey is also a function of mapping scale,
roughly:
log(X) = b0 + b1 · log(SN) (1)
We can fit a linear model to the empirical table data from
e.g.Legros (2006; p.75), and hence we get:
X = exp (19.0825 − 1.6232 · log(SN)) (2)
where X is the minimum cost/ha in Euros (based on estimates in
2002). To map 1 ha of soil at 1:100,000 scale, for example, one
needs (at least) 1.5 Euros.
Geostatistics course, 25–29 October 2010, Wageningen

26. Survey costs and mapping scale
q
q
q
q
q
9.5 10.0 10.5 11.0 11.5 12.0 12.5
−10123
Scale number (log−scale)
MinimumsurveycostsinEUR/ha(log−scale)
Geostatistics course, 25–29 October 2010, Wageningen

27. Survey costs and mapping scale
Total costs of a soil survey can be estimated by using the size of
area and number of samples.
The effective scale number (SN) is:
SN = 4 ·
A
N
· 102
. . . SN =
A
N
· 102
(3)
where A is the surface of the study area in m2 and N is the total
number of observations.
Geostatistics course, 25–29 October 2010, Wageningen

28. Converges to:
X = exp 19.0825 − 1.6232 · log 0.0791 ·
A
N
· 102
(4)
Geostatistics course, 25–29 October 2010, Wageningen

29. Output map, from info perspective
The resulting (predictions) map is a sum of two signals:
Z∗
(s) = Z(s) + ε(s) (5)
where Z(s) is the true variation, and ε(s) is the error component.
The error component consists, in fact, of two parts: (1) the
unexplained part of soil variation, and (2) the noise (measurement
error). The unexplained part of soil variation is the variation we
somehow failed to explain because we are not using all relevant
covariates and/or due to the limited sampling intensity.
Geostatistics course, 25–29 October 2010, Wageningen

30. Prediction accuracy
In order to see how much of the global variation budget has been
explained by the model we can use:
RMSEr (%) =
RMSE
sz
· 100 (6)
where sz is the sampled variation of the target variable.
RMSEr (%) is a global estimate of the map accuracy, valid only
under the assumption that the validation points are spatially
independent from the calibration points, representative and large
enough ( 100).
Geostatistics course, 25–29 October 2010, Wageningen

31. Kriging efficiency
Geostatistics course, 25–29 October 2010, Wageningen

32. Mapping efficiency
We propose two new measures of mapping success: (1) Mapping
efficiency, defined as the amount of money needed to map an area
of standard size and explain each one percent of variation in the
target variable:
θ =
X
A · RMSEr
[EUR · km−2
· %−1
] (7)
where X is the total costs of a survey, A is the size of area in
km−2, and RMSEr is the amount of variation explained by the
spatial prediction model.
Geostatistics course, 25–29 October 2010, Wageningen

33. Information production efficiency
(2) Equivalent measure of mapping efficiency is the information
production efficiency:
Υ =
X
gzip
[EUR · B−1
] (8)
where gzip is the size of data (in Bytes) left after compression and
after reformatting the values to match the effective precision
(based on Eq.10). This can be estimated as:
gzip = fc · (fE · M ) · cZ [B] (9)
where fc is the loss-less data compression factor that depends on
the compression algorithm, fE is the extrapolation adjustment
factor, cZ is the variable coding size, and M is the total number of
pixels.
Geostatistics course, 25–29 October 2010, Wageningen

34. Effective precision
Following the Nyquist frequency concept from signal processing,
which states that the original signal can be reconstructed if
sampling frequency is twice the maximum component frequency of
the signal, we can derive the effective precision — also known as
numerical resolution — of a produced prediction map as:
∆z =
RMSE
2
(10)
which means that there is no justification in saving the predictions
with better precision than half the average accuracy.
Geostatistics course, 25–29 October 2010, Wageningen

35. Nyquist frequency concept
q
q
qq
q
qq q
q
q
q
q q
qqq
q
q qq
q
qq q
q
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q
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q q
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q
q q qqq q q
q
q
qq
Figure: The Nyquist rate is the optimal rate that can be used to
compress a signal (it equals twice the maximum component frequency of
the signal) to allow perfect reconstruction of the signal from the samples.
Geostatistics course, 25–29 October 2010, Wageningen

36. Rounding numbers
Original data
2.25 4.08 6.25 4.23 2.56 1.21 0.98 0.98 0.85 0.4
4.24 4.69 7.17 4.37 2.08 1.4 1.44 0.96 0.89 0.31
3.62 5.39 5.27 3.11 2.04 1.57 1.67 1.43 0.61 0.28
2.72 8.75 7.77 4.63 2.88 2.34 2.93 1.49 0.57 0.25
2.83 10.55 14.45 5.79 3.13 2.95 2.85 0.89 0.34 0.22
2.87 5.45 10.34 5.01 2.42 1.88 1.5 0.61 0.3 0.23
1.19 2.69 3.76 3.63 1.86 0.97 1.24 0.64 0.37 0.26
0.86 1.22 1.39 2.71 2.17 1.61 2.37 1.56 0.66 0.47
0.67 1 1.23 1.53 2.04 3.12 5.74 3.71 1.53 0.92
1.18 1.48 1.35 2.13 2.11 3.64 7.56 6.92 2.97 1.96
Coded data
2 4 6 4 3 1 1 0
5 7 4 2 1 1 0
4 5 5 2 2 2 1 1
3 9 8 5 3 2 3 1 1
3 11 14 6 3 3 3
3 5 10 5 2 2 1
1 3 4 4 2 1 0
1 3 2 2 2
1 1 2 2 3 6 4 2 1
1 1 2 2 4 8 7 3 2
Geostatistics course, 25–29 October 2010, Wageningen

37. Exercise
To follow this exercise, obtain the DSM_examples.R script.
Download it to your machine and then run step-by-step.
Geostatistics course, 25–29 October 2010, Wageningen

38. Meuse data
> data(meuse)
> coordinates(meuse) <- ~x+y
> proj4string(meuse) <- CRS("+init=epsg:28992")
> sel <- !is.na(meuse$om)
> bubble(meuse[sel,], "om")
om
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1
5.3
6.9
9
17
Geostatistics course, 25–29 October 2010, Wageningen

39. Meuse
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0
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14
Geostatistics course, 25–29 October 2010, Wageningen

40. Eberg¨otzen (subset)
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Geostatistics course, 25–29 October 2010, Wageningen

41. Eberg¨otzen (subset)
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SAND.rk.3
0
10
20
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Geostatistics course, 25–29 October 2010, Wageningen

42. Eberg¨otzen (complete)
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SAND.rk.5
0
10
20
30
40
50
60
70
80
90
Geostatistics course, 25–29 October 2010, Wageningen

43. OK vs RK
2.0 2.5 3.0 3.5
0.00.20.40.60.81.0
Sampling intensity (log)
Amountofvariationexplained
Ordinary kriging Regression−kriging
Geostatistics course, 25–29 October 2010, Wageningen

44. Prediction accuracy and survey costs
Geostatistics course, 25–29 October 2010, Wageningen

45. Summary results
For the two case studies there is a gain of 7% for mapping
organic matter (Meuse), and 13% and for mapping sand
content (Eberg¨otzen) using regression-kriging vs ordinary
kriging.
Geostatistics course, 25–29 October 2010, Wageningen

46. Summary results
For the two case studies there is a gain of 7% for mapping
organic matter (Meuse), and 13% and for mapping sand
content (Eberg¨otzen) using regression-kriging vs ordinary
kriging.
to map organic carbon for the Meuse case study, one would
need to spend 13.1 EUR km−2 %−1 (1.13 EUR B−1); to map
sand content for the Eberg¨otzen case study would costs
11.1 EUR km−2 %−1 (5.88 EUR B−1).
Geostatistics course, 25–29 October 2010, Wageningen

47. Summary results
For the two case studies there is a gain of 7% for mapping
organic matter (Meuse), and 13% and for mapping sand
content (Eberg¨otzen) using regression-kriging vs ordinary
kriging.
to map organic carbon for the Meuse case study, one would
need to spend 13.1 EUR km−2 %−1 (1.13 EUR B−1); to map
sand content for the Eberg¨otzen case study would costs
11.1 EUR km−2 %−1 (5.88 EUR B−1).
Information production efficiency is possibly a more robust
measure of mapping quality than mapping efficiency because
it is scale-independent and because it accounts for
extrapolation effects.
Geostatistics course, 25–29 October 2010, Wageningen

48. Conclusions
Mapping efficiency (cost / area / percent of variance
explained) is a possible universal criteria to compare prediction
methods.
Geostatistics course, 25–29 October 2010, Wageningen

49. Conclusions
Mapping efficiency (cost / area / percent of variance
explained) is a possible universal criteria to compare prediction
methods.
Maps are not what they seem.
Geostatistics course, 25–29 October 2010, Wageningen

50. Conclusions
Mapping efficiency (cost / area / percent of variance
explained) is a possible universal criteria to compare prediction
methods.
Maps are not what they seem.
Geostatistics really outperforms non-statistical methods (but
this is area/data dependent).
Geostatistics course, 25–29 October 2010, Wageningen

51. Conclusions
Mapping efficiency (cost / area / percent of variance
explained) is a possible universal criteria to compare prediction
methods.
Maps are not what they seem.
Geostatistics really outperforms non-statistical methods (but
this is area/data dependent).
It’s not about the making beautiful maps, it’s about
understanding what they mean.
Geostatistics course, 25–29 October 2010, Wageningen

52. Conclusions
Mapping efficiency (cost / area / percent of variance
explained) is a possible universal criteria to compare prediction
methods.
Maps are not what they seem.
Geostatistics really outperforms non-statistical methods (but
this is area/data dependent).
It’s not about the making beautiful maps, it’s about
understanding what they mean.
If you deal with several equally valid (independent) methods,
maybe you should consider combining them?
Geostatistics course, 25–29 October 2010, Wageningen

53. Comparing methods
Geostatistics course, 25–29 October 2010, Wageningen

54. Literature
Dubois, G. (Ed.), 2005. Automatic mapping algorithms for routine
and emergency monitoring data. Report on the Spatial Interpolation
Comparison (SIC2004) exercise. EUR 21595 EN. Office for Official
Publications of the European Communities, Luxembourg, p. 150.
Hengl, T., 2009. A Practical Guide to Geostatistical Mapping, 2nd
edition. University of Amsterdam, 291 p. ISBN 978-90-9024981-0.
Li, J., Heap, A., 2008. A review of spatial interpolation methods for
environmental scientists. Record 2008/23. Geoscience Australia,
Canberra, p. 137.
Pebesma, E., Cornford, D., Dubois, D., Heuvelink, G.B.M.,
Hristopoulos, D., Pilz, J., Stohlker, U., Morin, G., Skoien, J.O.,
2010. INTAMAP: The design and implementation of an
interoperable automated interpolation web service. Computers &
Geosciences, In Press, Corrected Proof.
Geostatistics course, 25–29 October 2010, Wageningen