Space-time data workshop at IfGI
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This document discusses space-time regression-kriging using time series of images. It presents a universal kriging model for spatio-temporal data that treats observations as having both spatial and temporal components. Variograms are fitted separately for the spatial and temporal dimensions as well as for their combined zonal anisotropy. The data set examined consists of daily soil moisture observations over multiple years at various locations. Automation of the full space-time regression-kriging process and improved visualization of predictions over time are areas for future work.
Space-time data workshop at IfGI
- 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