Space-time regression-kriging using time series of predictors

1. Space-time regression-kriging
using time series of images
Tomislav Hengl
ISRIC World Soil Information, Wageningen University
R workshop, Mar 21th 2011

2. TAAC paper
R workshop, Mar 21th 2011

3. Space-time data
Universal kriging model for spatio-temporal data (Heuvelink
Grith, 2010):
T (s, t) = m(s, t) + ε(s, t) (1)
where m(s, t) is the deterministic part of the variation (i.e. a linear
function of the auxiliary variables), ε(s, t) is the residual for every
(s, t).
R workshop, Mar 21th 2011

4. Space-time cube
R workshop, Mar 21th 2011

5. Space-time semivariance
γ(si , ti ; sj , tj ) = 0.5 · E ( (si , ti ) − (sj , tj ))2 (2)
R workshop, Mar 21th 2011

6. Residuals
Residuals ( ) consist of three stationary and independent
components (Heuvelink Grith, 2010):
(s, t) = s (s) + t (t) + s,t (s, t) (3)
where s (s) is a purely spatial process (with constant realizations
over time), t (t) is a purely temporal process, and s,t (s, t) is a
space-time process for which distance in space is made comparable
to distance in time by introducing a space-time anisotropy ratio.
R workshop, Mar 21th 2011

7. Zonal anisotropies
The covariance structure can be represented by (Snepvangers et al.,
2003):
C(h, u) = Cs (h) + Ct (u) + Cs,t ( h2 + (α + u)2 ) (4)
where C(h, u) is the covariance at distance h in space, and
time-distance u, Cs (h) + Ct (u) allow the presence of zonal
anisotropies (dierent variogram sills in dierent directions), and
Cs,t ( h2 + (α + u)2 ) allows the presence of geometric anisotropy
represented with the ratio α.
R workshop, Mar 21th 2011

8. The data set
R workshop, Mar 21th 2011

9. In space-time cube
cdays
Y
X
R workshop, Mar 21th 2011

10. Variograms (separately)
365 days 159 stations
25
50
20
40
Semivariance
Semivariance
15
30
10
20
10
5
0
0
0 50000 100000 150000 200000 0 10 20 30 40 50 60
Distance (m) Distance (in days)
R workshop, Mar 21th 2011

11. Variograms (zonal anisotropy)
●
● ●
10 10 ●
● ●
● ●
●
●
8 8 ●
●
●
semivariance
semivariance
●
6 6
●
4 ● 4
● ● ●
● ●
● ● ●
● ●
●
● ●
●
2 ● 2
50000 100000 150000 200000 5 10 15
Distance (m) Distance (in days)
Marginal experimental variograms for residuals and tted models:
(left) space-domain only, (right) time-domain only.
R workshop, Mar 21th 2011

12. Final results
R workshop, Mar 21th 2011

13. Some experiences
By adding the time component we are better o.
R workshop, Mar 21th 2011

14. Some experiences
By adding the time component we are better o.
Automation of space-time regression-kriging (overlay,
regression modeling, variogram tting, predictions,
visualization in Google Earth) is anticipated.
R workshop, Mar 21th 2011

15. Some experiences
By adding the time component we are better o.
Automation of space-time regression-kriging (overlay,
regression modeling, variogram tting, predictions,
visualization in Google Earth) is anticipated.
Fitting and visualization of space-time variograms is a
bottle-neck!
R workshop, Mar 21th 2011

16. Some experiences
By adding the time component we are better o.
Automation of space-time regression-kriging (overlay,
regression modeling, variogram tting, predictions,
visualization in Google Earth) is anticipated.
Fitting and visualization of space-time variograms is a
bottle-neck!
Predictions need to be visualized as animations.
R workshop, Mar 21th 2011

17. Some experiences
By adding the time component we are better o.
Automation of space-time regression-kriging (overlay,
regression modeling, variogram tting, predictions,
visualization in Google Earth) is anticipated.
Fitting and visualization of space-time variograms is a
bottle-neck!
Predictions need to be visualized as animations.
We have ignored the one-way auto-correlation (time works
only one way)?
R workshop, Mar 21th 2011

18. Universal space-time reference
Each observation should have by default:
Longitude and latitude (WGS84) (or projected X, Y
coordinates + proj4 string);
Begin / end of the time interval in UTC (GMT) system;
Support size (in square meters);
Uncertainty or measurement error;
R workshop, Mar 21th 2011

19. Space-time algebra re-visited
Should we (re)dene and (re)implement
space-time (4D) algebra?
R workshop, Mar 21th 2011

20. What does this mean?
Distances always on a sphere (sphere geometry);
R workshop, Mar 21th 2011

21. What does this mean?
Distances always on a sphere (sphere geometry);
Always use information about uncertainty (weighted
regression);
R workshop, Mar 21th 2011

22. What does this mean?
Distances always on a sphere (sphere geometry);
Always use information about uncertainty (weighted
regression);
Always use information about the support size (nugget
estimation, cross-validation);
R workshop, Mar 21th 2011

23. What does this mean?
Distances always on a sphere (sphere geometry);
Always use information about uncertainty (weighted
regression);
Always use information about the support size (nugget
estimation, cross-validation);
Re-implement also any raster processing (geomorphometry,
resampling, ltering etc);
R workshop, Mar 21th 2011

24. What does this mean?
Distances always on a sphere (sphere geometry);
Always use information about uncertainty (weighted
regression);
Always use information about the support size (nugget
estimation, cross-validation);
Re-implement also any raster processing (geomorphometry,
resampling, ltering etc);
Use Google Earth to visualize any type of geographic data;
R workshop, Mar 21th 2011