1. Advanced Tutorial on : Global offset and residual covariance ENVR 468 Prahlad Jat and Marc Serre

2. Agenda Why use a global offset? How is the global offset calculated ? Remove the global offset from data Effect of global offset on covariance

3. Why use a Global Offset? We may be interested in mapping a global trend (global warming). To model short range variability more accurately. The Trend Analysis can help to identify a global trend in the user dataset if it exists.

4. Variability =f (short range, long range variability) Short range variability can in some cases be modeled in the global offset in the data. However, there is a real danger of over fitting the data when using the global offset and leaving too little variation in the residuals to properly account for the uncertainty in the prediction. What is our dilemma ?

5. Desirable: I. Low residual variability (for global offset with small range variability) II. Long autocorrelation range in covariance model (very flat global offset) A global offset with small range variability is very informative and therefore leaves little autocorrelation in the residuals. A flat global offset leaves too much variability in the residuals. A tradeoff between residual variability and autocorrelation range is needed: One should choose a mean trend which captures some variability and leaves reasonable autocorrelation in the residuals What we want to achieve ?

6. Model the Global Offset Temporal plot of Z versus time t for Monitoring Station 1 and 2 There is a temporal trend of increasing values with time

7. Model the Global Offset Spatial plot of Z versus monitoring event 1 and 2 There is a spatial trend of increasing values from left to right

8. Model the Global Offset Residual data plots There is no trend in residual

9. Model the Global Offset We model the S/TRF Z(s,t) as the sum of a global offset m z (s,t) and residual S/TRF X(s,t) Z(s,t) = m z (s,t) + x(s,t)

10. Model the Global Offset BMEGUI assumes that the global offset is a space/time additive separable function i.e. space/time mean trend Where : m s (s) is the spatial component and m t (t) is the temporal component m z (s,t) = m s (s) + m t (t)

11. Model the Global Offset Temporal plot of log PM2.5 (ug/m3) versus time (days)

12. Model the Global Offset Global Offset

13. Model the Global Offset Temporal plot of log PM2.5 (ug/m3) versus time (days) Take the sum of all observations at time and divide it by number of observations; apply exponential filter for smoothness in the trend

14. T radius S radius Smoothen the Global Offset

15. Model the Global Offset Global Offset for MS=4

16. Model the Global Offset Time series of observed log(PM2.5) at MS 4. We want to model the global offset at this MS Apply exponential filter to smoothen the global offset Remove the global offset and obtain residuals for covariance modeling

17. Model the Mean Trend Time series of observed log(PM2.5) at MS 4. Plot mt, the temporal component of the global offset Shift the temporal global offset to zero (i.e. calculate mt–mean(mt)) Add the spatial component of the global offset, i.e. add ms to mt– mean(mt)

18. Model the Global Trend Add spatial trend to this final temporal trend spatial trend + [temporal Global trend – mean of temporal global trend] m z (s,t) = m s (s) + m t (t)

19. Removing the mean trend from data Remove mean trend (i.e. global offset) and obtain residuals Use residual data for covariance modeling x(s,t) = Z(s,t) - m z (s,t) m z (s,t) = m s (s) + m t (t)

20. Mean Trend in BMEGUI Case1 : Flat mean trend Case 2: Informative mean trend

21. Mean Trend in BMEGUI Case1 : Flat mean trend Case 2: Informative mean trend

22. Covariance Models in BMEGUI Case1 : Flat mean trend Case 2: Informative mean trend Flat Mean trend Structure 1 Structure 2 SpatialTemporalSpatialTemporal Sill0.2 0.19 Modelexp Range4710075 Very smoothened Mean trend Structure 1 Structure 2 SpatialTemporalSpatialTemporal Sill0.05 0.0619 Modelexp Range1.55325

23. Covariance Models in BMEGUI Flat Mean trend Structure 1 Structure 2 SpatialTemporalSpatialTemporal Sill0.2 0.19 Modelexp Range4710075 Very smoothened Mean trend Structure 1 Structure 2 SpatialTemporalSpatialTemporal Sill0.05 0.0619 Modelexp Range1.55325 Case1 : Flat mean trend Case 2: Informative mean trend

24. Fitted Covariance Models Spatial ComponentTemporal Component case Search radius (deg.) Smoothing range (deg.) Search radius (days) Smoothing range (days) 115 1000 31160 40.2 10 50.1 55 20.001 0.1 Changes in the smoothness in the mean trend we observe changes in the experimental covariance. An extremely smoothed (i.e. flat) mean trend results in higher residual variance and larger spatial and temporal autocorrelation ranges. On the other hand, very informative mean trend results in smaller residual variance but shorter spatial and temporal autocorrelation ranges.

25. Temporal Dist. Est. in BMEGUI Case1 : Flat mean trend Case 2: Informative mean trend

26. Mean Trend Conclusions Each mean trend model represents a tradeoff between residual variance and autocorrelation range. Very flat mean trend: the highest residual variance but longer autocorrelation Very informative mean trend: low residual variance but short autocorrelation range The optimal level is the breakpoint where further decrease in smoothness results a drastic decreases in autocorrelation range. (green circle)