Space-time downscaling under error in computer model output

Transcription

1 Space-time downscaling under error in computer model output University of Michigan Department of Biostatistics joint work with Alan E. Gelfand, David M. Holland, Peter Guttorp and Peter Craigmile

2 Introduction In many environmental disciplines data come from two sources: monitoring networks and numerical models Numerical models are deterministic mathematical models used to predict environmental spatio-temporal processes Describe the underlying physical and chemical processes via partial differential equations Equations solved via numerical methods by discretizing space and time Predictions are given in terms of averages over grid cells

3 Introduction Observed average temperature Year 7 Climate model ouput for average temperature Year Goteborg Borlange Stockholm Sparse locations Missing data Essentially, true value Large spatial domains No missingness Calibration concerns

4 Introduction Observed average temperature Year 7 Borlange Stockholm Climate model ouput for average temperature Year Sparse locations Missing data Essentially, true value Large spatial domains No missingness Calibration concerns

5 Our goal Fuse the two sources of data Obtain improved predictions at point level by downscaling the computer model output Explicitly address the difference in spatial scale Outputs from numerical models are given in terms of predictions over grid cells Observations from monitors are collected at points Calibrate the numerical model Correct outputs from numerical models

6 Previous approaches Approaches for downscaling include: Atmospheric sciences pproaches Algorithmic Model-based approaches Wikle and Berliner (5) Bayesian Melding of Fuentes and Raftery (5)

7 Downscaler: main idea Observed average temperature Year 7 Climate model ouput for average temperature Year Goteborg Borlange Stockholm

8 Downscaler: main idea Observed average temperature Year 7 Borlange Stockholm Climate model ouput for average temperature Year

9 Downscaler: main idea Observed average temperature Year 7 Borlange Stockholm Climate model ouput for average temperature Year To each point s in the domain S with observation Y (s) we associate the numerical model output at grid cell B, X (B), where B is such that s B: Y (s) X (B)

10 Downscaler Time t is fixed. Y (s) observation at point s, X (B) numerical model output at grid cell B. For s in B: Y (s) = β (s) + β 1 (s)x (B) + ε(s) ε(s) ind N(,τ ) with β i (s) = β i + β i (s), i=,1. β (s) and β 1 (s) correlated mean-zero GP.

11 Downscaler Time t is fixed. Y (s) observation at point s, X (B) numerical model output at grid cell B. For s in B: Y (s) = β (s) + β 1 (s)x (B) + ε(s) ε(s) ind N(,τ ) with β i (s) = β i + β i (s), i=,1. β (s) and β 1 (s) correlated mean-zero GP. Extension to space-time: For s in B and for each t Y (s,t) = β (s,t)+ β 1 (s,t)x (B,t)+ε(s,t) with β i (s,t) = β it + β i (s,t), i=,1. Temporal dependence in β it and β i (s,t): dynamic or independent ε(s,t) ind N(,τ )

12 Downscaler Very flexible and natural hierarchical Bayesian model specification Driven by true station data rather than uncalibrated model output Computationally feasible also for large spatial domains Allows local calibration of the numerical model output Endows the spatial process Y (s) with a non-stationary covariance structure Straightforward downscaling prediction at an unmonitored sites Better predictive performance than other methods (geostatistical and model-based)

13 Extending the downscaler Propose a neighbor-based extension of the downscaler model accounts for information in the numerical model output at neighboring grid cells accounts for uncertainty in the association Y (s) X (B) with s in B

14 Extending the downscaler Propose a neighbor-based extension of the downscaler model accounts for information in the numerical model output at neighboring grid cells accounts for uncertainty in the association Y (s) X (B) with s in B We call the new downscaler a smoothed downscaler using spatially-varying random weights

15 Static setting

16 Smoothed downscaler with spatially-varying weights Downscaler: Y (s) = β (s) + β 1 (s)x (B) + ε(s) s B

17 Smoothed downscaler with spatially-varying weights Downscaler: Y (s) = β (s) + β 1 (s)x (B) + ε(s) s B We don t use the numerical model X (B) output directly anymore. For each s, we introduce a new regressor X (s) and we assume that Y (s) = β (s) + β 1 X (s) + ε(s) ε(s) ind N(,τ ) where X (s) = g k=1 w k(s)x (B k ), and w k (s) random and spatially-varying.

18 Smoothed downscaler with spatially-varying weights Downscaler: Y (s) = β (s) + β 1 (s)x (B) + ε(s) s B We don t use the numerical model X (B) output directly anymore. For each s, we introduce a new regressor X (s) and we assume that Y (s) = β (s) + β 1 X (s) + ε(s) ε(s) ind N(,τ ) where X (s) = g k=1 w k(s)x (B k ), and w k (s) random and spatially-varying. Let r k, k = 1,...,g be the centroids of numerical model grid cells B k, k = 1,...,g, then w k (s) := K (s r k;ψ) exp(q(r k )) g l=1 K (s r l;ψ) exp(q(r l ))

19 Smoothed downscaler with spatially-varying weights Downscaler: Y (s) = β (s) + β 1 (s)x (B) + ε(s) s B We don t use the numerical model X (B) output directly anymore. For each s, we introduce a new regressor X (s) and we assume that Y (s) = β (s) + β 1 X (s) + ε(s) ε(s) ind N(,τ ) where X (s) = g k=1 w k(s)x (B k ), and w k (s) random and spatially-varying. Let r k, k = 1,...,g be the centroids of numerical model grid cells B k, k = 1,...,g, then w k (s) := K (s r k;ψ) exp(q(r k )) g l=1 K (s r l;ψ) exp(q(r l )) Q( ) mean-zero GP with variance σ Q, exponential covariance function and decay parameter φ Q. K (s r k ;ψ) = exp( ψ s r k ).

20 Smoothed downscaler with spatially-varying random weights Observed average temperature Year 7 Climate model ouput for average temperature Year Goteborg Borlange Stockholm

21 Smoothed downscaler with spatially-varying random weights Observed average temperature Year 7 Borlange Stockholm Climate model ouput for average temperature Year

22 Smoothed downscaler with spatially-varying random weights Observed average temperature Year 7 Borlange Stockholm Climate model ouput for average temperature Year

23 Smoothed downscaler with spatially-varying random weights Observed average temperature Year 7 Borlange Stockholm Climate model ouput for average temperature Year

24 w_(s) w_(s) w_1(s) w_3(s) w_5(s) Smoothed downscaler with spatially-varying random weights Observed average temperature Year 7 Borlange Stockholm Climate model ouput for average temperature Year To each point s in the domain S with observation Y (s) we associate X (s), where X (s)= g k=1 w k(s)x(b k ) Y (s) X (s)

25 Spatio-temporal setting

26 Smoothed downscaler with spatially-varying weights Y (s,t) observations at s and time t, X (B,t) numerical model output at grid cell B and time t: Y (s,t) = β (s,t) + β 1,t X (s,t) + ε(s,t) ε(s,t) ind N(,τ ) with β (s,t) = β t + β (s,t), X (s,t) = g k=1 w k(s,t)x (B k,t), and w k (s,t) random and varying both in space and time.

27 Smoothed downscaler with spatially-varying weights Y (s,t) observations at s and time t, X (B,t) numerical model output at grid cell B and time t: Y (s,t) = β (s,t) + β 1,t X (s,t) + ε(s,t) ε(s,t) ind N(,τ ) with β (s,t) = β t + β (s,t), X (s,t) = g k=1 w k(s,t)x (B k,t), and w k (s,t) random and varying both in space and time. w k (s,t) := K (s r k;ψ) exp(q(r k,t)) g l=1 K (s r l;ψ) exp(q(r l,t))

28 Smoothed downscaler with spatially-varying weights Y (s,t) observations at s and time t, X (B,t) numerical model output at grid cell B and time t: Y (s,t) = β (s,t) + β 1,t X (s,t) + ε(s,t) ε(s,t) ind N(,τ ) with β (s,t) = β t + β (s,t), X (s,t) = g k=1 w k(s,t)x (B k,t), and w k (s,t) random and varying both in space and time. w k (s,t) := K (s r k;ψ) exp(q(r k,t)) g l=1 K (s r l;ψ) exp(q(r l,t)) Temporal dependence in β it, i =,1, β (s,t) and Q(r k,t): either (i) independent in time, or (ii) dynamic.

29 Application 5 5 Borlange Stockholm Goteborg Yearly average temperature: observations and regional climate model output for years Climate model output for southern Sweden: the Rossby Centre Regional Climate model RCA3 at 1-km grid resolution (g =,). 17 monitoring sites used for fitting the model and sites used for validation: Borlange and Goteborg Missing data:.5% Time-varying parameters: independent in time

30 Observations and climate model output Observations Regional climate model output Scatterplot of observed yearly average temperature at the 17 training sites versus the climate model output at the corresponding grid cells. Correlation coefficient: R =.

31 Observations, climate model output and predictions Borlange Goteborg We will look at observations and predictions at the 17 training sites by the climate model output and the downscaler model with spatially-varying random weights.

32 Observations, climate model output and predictions Location 1 Location Location 3 Temperature Temperature Temperature Year Year Year Location Location 5 Location Temperature Temperature Temperature Year Year Year Location 7 Location Location 9 Temperature Temperature Temperature Year Year Year Observed yearly average temperature at the 17 training sites (black line), climate model output (blue line), downscaler predictions (red line)

33 Observations, climate model output and predictions Location 1 Location 11 Location 1 Temperature Temperature Temperature Year Year Year Location 13 Location 1 Location 15 Temperature Temperature Temperature Year Year Year Location 1 Location 17 Temperature Temperature Year Year Observed yearly average temperature at the 17 training sites (black line), climate model output (blue line), downscaler predictions (red line)

34 Mean Absolute Error Mean Absolute Error 1 3 Downscaler Regional climate model Year Mean absolute error at the 17 training sites: climate model output (blue line), downscaler (red line)

35 Results Predictive performance at the 17 training sites, averaged over space and time. Results are in degree Celsius. Coverage Avg. length Method MSPE MAPE 95% PI 95% PI Regional climate model 1.9. Smoothed downscaler with spatially-varying weights % 7.

36 Observations, climate model output and predictions Goteborg Average yearly temperature (in degree Celsius) Year Observed yearly average temperature at Goteborg (black line), climate model output (blue line), downscaler predictions (red line). The shaded grey area indicates the 95% prediction interval.

37 Observations, climate model output and predictions Borlange Average yearly temperature (in degree Celsius) Year Observed yearly average temperature at Borlange (black line), climate model output (blue line), downscaler predictions (red line). The shaded grey area indicates the 95% prediction interval.

38 Results Predictive performance at Borlange and Goteborg averaged over years Results are in degree Celsius. Coverage Avg. length City Method MSPE MAPE 95% PI 95% PI Regional climate model Goteborg Smoothed downscaler with spatially-varying weights % 5.7 Borlange Regional climate model Smoothed downscaler with spatially-varying weights % 5.9

39 Predictions: year 199 Climate model ouput Year 199 Downscaler Year Goteborg Borlange Stockholm 5 5 Goteborg Borlange Stockholm

40 Predictions: year 199 Downscaler Climate model output Year Borlange Stockholm Goteborg

41 Predictions: year 7 Climate model ouput Year 7 Downscaler Year Goteborg Borlange Stockholm 5 5 Goteborg Borlange Stockholm

42 Predictions: year 7 Downscaler Climate model ouput Year Borlange Stockholm Goteborg

43 Conclusions Presented a method to downscale the output from numerical models that accounts for uncertainty in the association of a site to its putatively associated grid cells In our application, predictions are less biased than the predictions from a regional climate model. Currently, we are comparing the merits of this approach versus those of an upscaling approach