A non-stationary spatial weather generator for statistical downscaling of precipitation. Pradeebane VAITTINADA AYAR and Mathieu VRAC.

Transcription

1 A non-stationary spatial weather generator for statistical downscaling of precipitation Pradeebane VAITTINADA AYAR and Mathieu VRAC StaRMIP Climate change impacts in the Mediterranean region Montpellier, France 17th October 2017

2 Outline P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15 1 Context 2 Overview of the model 3 Evaluations 4 Conclusions and Perspectives

3 Outline P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15 1 Context 2 Overview of the model 3 Evaluations 4 Conclusions and Perspectives

4 Context P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15 Realistic daily rainfall statistical simulator How to generate data at every location of a given domain? (not limited by observations location e.g. one km 2 spatial resolution) How to correctly model spatial (correlation) and temporal features (climate change)?

5 Context Realistic daily rainfall statistical simulator How to generate data at every location of a given domain? (not limited by observations location e.g. one km 2 spatial resolution) How to correctly model spatial (correlation) and temporal features (climate change)? Aim to build a generic rainfall simulator (working over different catchements) : accounting for spatial dependencies accounting for spatio-temporal non-stationnarities P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15

6 Context Realistic daily rainfall statistical simulator How to generate data at every location of a given domain? (not limited by observations location e.g. one km 2 spatial resolution) How to correctly model spatial (correlation) and temporal features (climate change)? Aim to build a generic rainfall simulator (working over different catchements) : accounting for spatial dependencies accounting for spatio-temporal non-stationnarities Spatial weather generator framework (flexibility, spatial structure). P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15

7 Context Overview of the model Evaluations Conclusions and Perspectives 46 N Mediterranean SQR Stations 69 Stations (from Limousin Auvergne 1500 RhôneAlpes 45 N Lozère Aveyron 44 N Latitude Tarn ARVIEUX HautesAlpes Ardèche Drôme Gard AlpesdeHauteProvence Vaucluse AlpesMaritimes 43 N Var 1st jan to 31st 500 Aude (Aigues-Mortes) to 1690m (Arvieux) Hérault AIGUESMORTES BouchesduRhône Altitudes : from MidiPyrénées Me te o France SQR database) dec PyrénéesOrientales 42 N 0 Elevation(m) 2 E 3 E 4 E 5 E 6 E 7 E 8 E Longitude P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15

8 Outline P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15 1 Context 2 Overview of the model 3 Evaluations 4 Conclusions and Perspectives

9 Model framework P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15 x D, Y (x, t) a r.v. characterising the precipitation over a given domain. For a given day, the rainfield is equal to a realisation of {Y (x, t)} x D

10 Model framework P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15 x D, Y (x, t) a r.v. characterising the precipitation over a given domain. For a given day, the rainfield is equal to a realisation of {Y (x, t)} x D What is {Y (x, t)} x D? {Y (x, t)} x D : based on a transformed and censored latent gaussian field to model rain occurrence and intensity at the same time [common in the literature e.g., Vischel et al., 2009; Rasmussen, 2013; Allard and Bourotte, 2015].

11 Model framework P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15 x D, Y (x, t) a r.v. characterising the precipitation over a given domain. For a given day, the rainfield is equal to a realisation of {Y (x, t)} x D What is {Y (x, t)} x D? {Y (x, t)} x D : based on a transformed and censored latent gaussian field to model rain occurrence and intensity at the same time [common in the literature e.g., Vischel et al., 2009; Rasmussen, 2013; Allard and Bourotte, 2015]. Gaussian field : fully defined by its mean vector and covariance structure.

12 Model framework P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15 x D, Y (x, t) a r.v. characterising the precipitation over a given domain. For a given day, the rainfield is equal to a realisation of {Y (x, t)} x D What is {Y (x, t)} x D? {Y (x, t)} x D : based on a transformed and censored latent gaussian field to model rain occurrence and intensity at the same time [common in the literature e.g., Vischel et al., 2009; Rasmussen, 2013; Allard and Bourotte, 2015]. Gaussian field : fully defined by its mean vector and covariance structure. Model steps Marginal model Spatial model

13 Marginal model P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15 F(Y ) = P 0 + (1 P 0 ) Ga (Y ) Ga a gamma CDF with shape α and rate β and P 0 the probability of zeros.

14 Marginal model F(Y ) = P 0 + (1 P 0 ) Ga (Y ) Ga a gamma CDF with shape α and rate β and P 0 the probability of zeros. p(x, t) the probability of wet days : p(x, t) = 1 P 0 (x, t). P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15

15 Marginal model F(Y ) = P 0 + (1 P 0 ) Ga (Y ) Ga a gamma CDF with shape α and rate β and P 0 the probability of zeros. p(x, t) the probability of wet days : p(x, t) = 1 P 0 (x, t). Non-stationnarity introduced in the marginal model [VGLM Chandler and Wheater, 2002]. P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15

16 Marginal model P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15 F(Y ) = P 0 + (1 P 0 ) Ga (Y ) Ga a gamma CDF with shape α and rate β and P 0 the probability of zeros. p(x, t) the probability of wet days : p(x, t) = 1 P 0 (x, t). Non-stationnarity introduced in the marginal model [VGLM Chandler and Wheater, 2002]. α, β, p are functions of position and time varying predictors. α(x, t) = f α (X j (x, t)), β(x, t) = f β (X j (x, t)), p(x, t) = f p (X j (x, t)).

17 Marginal model P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15 F(Y ) = P 0 + (1 P 0 ) Ga (Y ) Ga a gamma CDF with shape α and rate β and P 0 the probability of zeros. p(x, t) the probability of wet days : p(x, t) = 1 P 0 (x, t). Non-stationnarity introduced in the marginal model [VGLM Chandler and Wheater, 2002]. α, β, p are functions of position and time varying predictors. α(x, t) = f α (X j (x, t)), β(x, t) = f β (X j (x, t)), p(x, t) = f p (X j (x, t)). Independent monthly estimation by maximum likelihood.

18 Spatial dependance Let the following transformation : H(x, t) = Φ 1 (F(Y (x, t))), where Φ is the CDF of the normal N(0, 1). Since Y (x, t) 0, H(x, t) T (x, t) Φ 1 (P 0 (x, t)) Then H(x, t) = max(t (x, t), G(x, t)), G(x, t) the latent Gaussian process GP(0, 1, C(h)) with marginal N(0, 1), C(h) the spatial covariance function. P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15

19 Spatial dependance Let the following transformation : H(x, t) = Φ 1 (F(Y (x, t))), where Φ is the CDF of the normal N(0, 1). Since Y (x, t) 0, H(x, t) T (x, t) Φ 1 (P 0 (x, t)) Then H(x, t) = max(t (x, t), G(x, t)), G(x, t) the latent Gaussian process GP(0, 1, C(h)) with marginal N(0, 1), C(h) the spatial covariance function. Procedure Estimate C(h) (monthly) from the H-transformed data by a censored maximum likelihood [Pesonen et al., 2015], Simulate G and deduce H, Retrieve Y by transforming back with Y (x) = F 1 Y (x,t)(φ(h(x, t))). P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15

20 Covariance function P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15 C S (h) = σ 2 exp ( ) h λ, the stationary exponential covariance (isotropic) function

21 Covariance function P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15 C S (h) = σ 2 exp ( ) h λ, the stationary exponential covariance (isotropic) function Geometrical anisotropy widely used in the literature with Σ a covariance matrix. h(x i, x j ) = (x i x j ) T Σ 1 (x i x j )

22 Covariance function C S (h) = σ 2 exp ( ) h λ, the stationary exponential covariance (isotropic) function Geometrical anisotropy widely used in the literature h(x i, x j ) = (x i x j ) T Σ 1 (x i x j ) with Σ a covariance matrix. How to include orographic and/or synoptic information in the covariance? P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15

23 Covariance function C S (h) = σ 2 exp ( ) h λ, the stationary exponential covariance (isotropic) function Geometrical anisotropy widely used in the literature h(x i, x j ) = (x i x j ) T Σ 1 (x i x j ) with Σ a covariance matrix. Non-stationarity introduced by Higdon et al. [1999] : kernel convolution in anisotropic function. P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15

24 Covariance function C S (h) = σ 2 exp ( ) h λ, the stationary exponential covariance (isotropic) function Geometrical anisotropy widely used in the literature h(x i, x j ) = (x i x j ) T Σ 1 (x i x j ) with Σ a covariance matrix. Non-stationarity introduced by Higdon et al. [1999] : kernel convolution in anisotropic function. Generalized by Paciorek and Schervish [2006] P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15

25 Spatio-temporal non-stationary covariance function P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15 One solution for spatial non-stationarity : C NS (x i, x j ) = Σ i 1 4 Σj 1 4 Σ i + Σ j exp (x i x j ) T ( Σi + Σ j 2 ) 1 (x i x j ) Σ i, Σ j : 2 2 covariance matrix (local kernels)

26 Spatio-temporal non-stationary covariance function P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15 One solution for spatial non-stationarity : C NS (x i, x j ) = Σ i 1 4 Σj 1 4 Σ i + Σ j exp (x i x j ) T ( Σi + Σ j 2 ) 1 (x i x j ) Σ i, Σ j : 2 2 covariance matrix (local kernels) x i D, Σ i = Σ(x i, t), following Huser and Genton [2016] (Locally isotropic case) : Σ i = ( ω 2 (x i, t) 0 0 ω 2 (x i, t) ω(x i, t) = f ω (X j (x i, t)) function of position and time varying predictors. )

27 Outline P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15 1 Context 2 Overview of the model 3 Evaluations 4 Conclusions and Perspectives

28 Context Overview of the model Evaluations Conclusions and Perspectives Stations selection by Latitude 46 N Covariates Limousin geographical : LON, LAT, ALT Auvergne RhôneAlpes 45 N Lozère Aveyron 44 N Latitude temporal : only the ERA-I grid-cell Tarn AlpesdeHauteProvence Vaucluse AlpesMaritimes Hérault BouchesduRhône Var 43 N Z850) Gard containing the station (D2, SLP, HautesAlpes Ardèche Drôme MidiPyrénées Aude PyrénéesOrientales 42 N 2 E 3 E 5 E 4 E 6 E 7 E 8 E Longitude P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15

29 Context Overview of the model Evaluations Conclusions and Perspectives Stations selection by Latitude 46 N Covariates Limousin geographical : LON, LAT, ALT Auvergne RhôneAlpes 45 N Lozère Aveyron 44 N Latitude temporal : only the ERA-I grid-cell Tarn AlpesdeHauteProvence Vaucluse AlpesMaritimes Hérault BouchesduRhône 43 N Var Scheme Z850) Gard containing the station (D2, SLP, HautesAlpes Ardèche Drôme MidiPyrénées Calibration (monthly) 1/1/ Aude 31/12/2004 (red stations) PyrénéesOrientales Validation 1/1/ /12/ N (blue stations) - Evaluation set 2 E 3 E 5 E 4 E 6 E 7 E 8 E Longitude P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15

30 Context Overview of the model Evaluations Conclusions and Perspectives Stations selection by Latitude 46 N Covariates Limousin geographical : LON, LAT, ALT Auvergne RhôneAlpes 45 N Lozère Aveyron 44 N Latitude temporal : only the ERA-I grid-cell Tarn AlpesdeHauteProvence Vaucluse AlpesMaritimes Hérault BouchesduRhône 43 N Var Scheme Z850) Gard containing the station (D2, SLP, HautesAlpes Ardèche Drôme MidiPyrénées Calibration (monthly) 1/1/ Aude 31/12/2004 (red stations) PyrénéesOrientales Validation 1/1/ /12/ N (blue stations) - Evaluation set 2 E 3 E 5 E 4 E 6 E 7 E 8 E Longitude 6 set-up (100 simulations over the whole period) Covariates All only geographical none Marginal MARST MARS Covariance COVST COVS COVISO P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15

31 PR Context Overview of the model Evaluations Conclusions and Perspectives Q5Q95 Q5Q95 PROJ RMSE = 0.36 mm Seasonal Cycle PROJ (monthly COR = mean) PR (mm/day) PR (mm/day) PR (mm/day) PR (mm/day) PR (mm/day) PR (mm/day) PR (mm/day) MARST SpaWGEN SQR COV.SPA.TPS PROJ Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Months Q5Q95 SpaWGEN SQR COV.SPA PROJ RMSE = 0.36 mm PROJ COR = Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Q5Q95 Months PROJ RMSE = 0.36 mm SpaWGEN SQR COV.SPA PROJ COR = Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Q5Q95 Months PROJ RMSE = 0.37 mm PROJ COR = SpaWGEN SQR COV.SPA PROJ Feb Mar Apr May Jun Jul Aug Sep Oct Jan Nov Dec Months Q5Q95 SpaWGEN SQR STATIO PROJ RMSE = 0.37 mm PROJ COR = Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Q5Q95 Months PROJ RMSE = 0.37 mm SpaWGEN SQR STATIO PROJ COR = Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Q5Q95 Months PROJ RMSE = 0.38 mm PROJ COR = SpaWGEN SQR STATIO PROJ Feb Mar Apr May Jun Jul Aug Sep Oct Jan Nov Dec PR PR (mm/day) PR (mm/day) PR (mm/day) PR (mm/day) PR (mm/day) PR (mm/day) PR (mm/day) MARS PROJ RMSE = 0.59 mm PROJ COR = SpaWGENs SQR COV.SPA.TPS PROJ Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Months Q5Q95 SpaWGENs SQR COV.SPA PROJ PROJ RMSE = 0.59 mm PROJ COR = Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Q5Q95 Months PROJ RMSE = 0.59 mm SpaWGENs SQR COV.SPA PROJ PROJ COR = Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Q5Q95 Months PROJ RMSE = 0.6 mm PROJ COR = SpaWGENs SQR COV.SPA PROJ Feb Mar Apr May Jun Jul Aug Sep Oct Jan Nov Dec Months Q5Q95 SpaWGENs SQR STATIO PROJ RMSE = 0.6 mm PROJ COR = Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Q5Q95 Months PROJ RMSE = 0.6 mm SpaWGENs SQR STATIO PROJ COR = Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Q5Q95 Months PROJ RMSE = 0.58 mm PROJ COR = SpaWGENs SQR STATIO PROJ Feb Mar Apr May Jun Jul Aug Sep Oct Jan Nov Dec COVST COVS COVISO Months Months Q5Q95 Q5Q95 P. Vaittinada Ayar & M. Vrac PROJ RMSE = 0.38 mm cc-mistrals Montpellier, Oct. PROJ RMSE 17, = mm 11/15

32 ) PR ( ) Context Overview of the model Evaluations Conclusions and Perspectives Q5Q95 PROJ RMSE = mm Q5Q95 CALIB COR = 0.72 PROJ COR = 0.69 Interannual variability (Fall : SON) PR (mm/year) PR (mm/year) PR (mm/year) PR (mm/year) PR (mm/year) PR (mm/year) PR (mm/year) MARST 1979 SpaWGEN SQR 1994COV.SPA.TPS Fall Q5Q95 CALIB RMSE = mm Years CALIB RMSE = mm Q5Q95 PROJ RMSE = mm CALIB COR = 0.72 PROJ COR = 0.69 SpaWGEN SQR COV.SPA Fall CALIB RMSE 2009 = mm Q5Q95 PROJ RMSE = mm CALIB COR = 0.71 Years PROJ COR = 0.68 SpaWGEN SQR COV.SPA Fall CALIB RMSE 2009 = mm Q5Q95 PROJ RMSE = mm Years CALIB COR = 0.71 PROJ COR = SpaWGEN SQR COV.SPA Fall Years CALIB RMSE = mm Q5Q95 PROJ RMSE = mm CALIB COR = 0.71 PROJ COR = 0.68 SpaWGEN SQR STATIO Fall CALIB RMSE 2009 = mm Q5Q95 PROJ RMSE = mm CALIB COR = 0.71 Years PROJ COR = 0.68 SpaWGEN SQR STATIO Fall CALIB RMSE 2009 = mm Q5Q95 PROJ RMSE = mm Years CALIB COR = 0.71 PROJ COR = SpaWGEN SQR 1999 STATIO 2004 Fall Years CALIB RMSE = 88.3 mm PROJ RMSE = mm CALIB COR = 0.71 PROJ COR = 0.68 PR ( PR (mm/year) PR (mm/year) PR (mm/year) PR (mm/year) PR (mm/year) PR (mm/year) PR (mm/year) MARS 1979 SpaWGENs SQR 1994 COV.SPA.TPS Fall 2014 Q5Q95 CALIB RMSE = mm PROJ RMSE = mm CALIB COR = 0.01 PROJ COR = 0.01 Years CALIB RMSE = mm Q5Q95 PROJ RMSE = mm CALIB COR = 0.01 PROJ COR = 0.01 SpaWGENs SQR COV.SPA Fall CALIB 2004RMSE 2009 = mm Q5Q95 PROJ RMSE = mm CALIB COR = 0.01 Years PROJ COR = 0.01 SpaWGENs SQR COV.SPA Fall CALIB 2004RMSE 2009 = mm Q5Q95 PROJ RMSE = mm Years CALIB COR = 0.01 PROJ COR = SpaWGENs SQR 1999 COV.SPA 2004 Fall Years CALIB RMSE = mm Q5Q95 PROJ RMSE = mm CALIB COR = 0.01 PROJ COR = 0.01 SpaWGENs SQR STATIO Fall CALIB 2004RMSE 2009 = mm Q5Q95 PROJ RMSE = mm CALIB COR = 0 Years PROJ COR = 0.01 SpaWGENs SQR STATIO Fall CALIB 2004RMSE 2009 = mm Q5Q95 PROJ RMSE = mm Years CALIB COR = 0 PROJ COR = SpaWGENs SQR 1999 STATIO 2004Fall Years CALIB RMSE = mm PROJ RMSE = mm CALIB COR = 0 PROJ COR = 0.01 COVST COVS COVISO P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15

33 Log-odds ratio P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15 The log-odds ratio can be seen as analogous to spatial correlation of continuous variable for binary variables [Charles et al., 1999] TABLE 1: Contingency table for two stations x i, x j NR(x i ) R(x i ) NR(x j ) n 00 n 01 R(x j ) n 10 n 11 The log-odds ratio is given by : ( ) n00 n 11 LOR = log n 01 n 10

34 SpaWGEN SQR COV.SPA.TPS PROJ Fall SpaWGENs SQR COV.SPA.TPS PROJ Fall Log-odds ratio (Fall : SON) SpaWGEN SQR COV.SPA.TPS MARST PROJ Fall SpaWGENs SQR COV.SPA.TPS PROJ Fall MARS Observation logodds ratio COVST Observation logodds ratio SpaWGEN SQR COV.SPA PROJ Fall SpaWGENs SQR COV.SPA PROJ Fall Simulation logodds Simulation ratio logodds Simulation ratio logodds ratio Simulation logodds Simulation ratio logodds Simulation ratio logodds ratio Simulation logodds Simulation ratio logodds Simulation ratio logodds ratio Observation logodds ratio SpaWGEN SQR COV.SPA PROJ Fall SpaWGEN SQR COV.SPA PROJ Fall Observation logodds ratio SpaWGEN SQR STATIO PROJ Fall Observation logodds ratio Observation logodds ratio SpaWGEN SQR STATIO PROJ Fall Simulation logodds Simulation ratio logodds Simulation ratio logodds ratio Simulation logodds Simulation ratio logodds Simulation ratio logodds ratio Simulation logodds Simulation ratio logodds Simulation ratio logodds ratio Observation logodds ratio SpaWGENs SQR COV.SPA PROJ Fall Observation logodds ratio SpaWGENs SQR STATIO PROJ Fall Observation logodds ratio Observation logodds ratio SpaWGENs SQR COV.SPA PROJ Fall SpaWGENs SQR STATIO PROJ Fall COVS Observation logodds ratio Observation logodds ratio SpaWGEN SQR STATIO PROJ Fall SpaWGENs SQR STATIO PROJ Fall COVISO Observation logodds ratio Observation logodds ratio P. Vaittinada 0 Ayar 1 & 2M. Vrac cc-mistrals Montpellier, 3 4 5Oct. 6 17, /15

35 Co Context Overview of the model Evaluations Conclusions and Perspectives PROJ COR = 0.99 Q5Q95 Q5Q95 Correlogram (zero included - Fall : SON) Correlation Correlation Correlation Correlation Correlation Correlation Correlation MARST PROJ RMSE = SpaWGEN SQR 157COV.SPA.TPS PROJ Fall 419 Distance (km) PROJ RMSE = 0.07 PROJ COR = 0.99 Q5Q95 SpaWGEN SQR COV.SPA PROJ Fall PROJ RMSE 376 = PROJ COR = 0.99 Distance (km) Q5Q95 SpaWGEN SQR COV.SPA PROJ Fall PROJ RMSE 376 = PROJ COR = 0.99 Distance (km) Q5Q SpaWGEN SQR 201 COV.SPA PROJ 332Fall Distance (km) PROJ RMSE = 0.07 PROJ COR = 0.99 Q5Q95 SpaWGEN SQR STATIO PROJ Fall PROJ RMSE 376 = PROJ COR = 0.99 Distance (km) Q5Q95 SpaWGEN SQR STATIO PROJ Fall PROJ RMSE 376 = PROJ COR = 0.99 Distance (km) Q5Q SpaWGEN SQR 201 STATIO PROJ 332 Fall Distance (km) PROJ RMSE = 0.08 PROJ COR = 0.99 Co Correlation Correlation Correlation Correlation Correlation Correlation Correlation MARS PROJ RMSE = 0.05 PROJ COR = SpaWGENs SQR 157 COV.SPA.TPS PROJ Fall419 Distance (km) PROJ RMSE = 0.05 PROJ COR = 0.99 Q5Q95 SpaWGENs SQR COV.SPA PROJ Fall PROJ RMSE 376 = PROJ COR = 0.99 Distance (km) Q5Q95 pawgens SQR COV.SPA PROJ Fall PROJ RMSE 376 = PROJ COR = 0.99 Distance (km) Q5Q SpaWGENs SQR 201COV.SPA PROJ 332 Fall Distance (km) PROJ RMSE = 0.06 PROJ COR = 0.99 Q5Q95 SpaWGENs SQR STATIO PROJ Fall PROJ RMSE 376 = PROJ COR = 0.99 Distance (km) Q5Q95 SpaWGENs SQR STATIO PROJ Fall PROJ RMSE 376 = PROJ COR = 0.99 Distance (km) Q5Q SpaWGENs SQR 201STATIO PROJ 332Fall Distance (km) PROJ RMSE = 0.07 PROJ COR = 0.99 COVST COVS COVISO P. Vaittinada Q5Q95 Ayar & M. Vrac cc-mistrals Q5Q95 Montpellier, Oct. 17, /15

36 Outline P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15 1 Context 2 Overview of the model 3 Evaluations 4 Conclusions and Perspectives

37 Conclusions & Perspectives Good agreement in terms of marginal statistics Underestimation of the spatial dependency NS-covariance does not really improve the results in terms of marginal statistics and spatial dependency (Is it worth it?) P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15

38 Conclusions & Perspectives Good agreement in terms of marginal statistics Underestimation of the spatial dependency NS-covariance does not really improve the results in terms of marginal statistics and spatial dependency (Is it worth it?) YES : loglikelihood significantly increased (varying acc. month) Another way to characterize spatial dependency : odd ratios and correlograms too general P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15

39 Conclusions & Perspectives Good agreement in terms of marginal statistics Underestimation of the spatial dependency NS-covariance does not really improve the results in terms of marginal statistics and spatial dependency (Is it worth it?) YES : loglikelihood significantly increased (varying acc. month) Another way to characterize spatial dependency : odd ratios and correlograms too general Application : forced by climate scenarios. P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15

40 Conclusions & Perspectives P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15

41 Thank you for your attention

42 Reference Allard, Denis and Bourotte, Marc [2015]. Disaggregating daily precipitations into hourly values with a transformed censored latent Gaussian process. In: Stochastic Environmental Research and Risk Assessment. Vol. 29. no. 2, pp Chandler, R. E. and Wheater, H. S. [2002]. Analysis of rainfall variability using generalized linear models : A case study from the west of Ireland. In: Water Resour. Res.. Vol. 38(10), p Charles, Stephen P., Bates, Bryson C., and Hughes, James P. [1999]. A spatiotemporal model for downscaling precipitation occurrence and amounts. In: Journal of Geophysical Research : Atmospheres. Vol no. D24, pp Fealy, Rowan and Sweeney, John [2007]. Statistical downscaling of precipitation for a selection of sites in Ireland employing a generalised linear modelling approach. In: International Journal of Climatology. Vol. 27. no. 15, pp Higdon, Dave, Swall, J, and Kern, J [1999]. Non-stationary spatial modeling. In: Bayesian statistics. Vol. 6. no. 1, pp Huser, Raphaël and Genton, Marc G. [2016]. Non-Stationary Dependence Structures for Spatial Extremes. In: Journal of Agricultural, Biological, and Environmental Statistics. Vol. 21. no. 3, pp Paciorek, Christopher J. and Schervish, Mark J. [2006]. Spatial modelling using a new class of nonstationary covariance functions. In: Environmetrics. Vol. 17. no. 5, pp Pesonen, Maiju, Pesonen, Henri, and Nevalainen, Jaakko [2015]. Covariance matrix estimation for left-censored data. In: Computational Statistics & Data Analysis. Vol. 92, pp Rasmussen, P. F. [2013]. Multisite precipitation generation using a latent autoregressive model. In: Water Resources Research. Vol. 49. no. 4, pp Vischel, Théo, Lebel, Thierry, Massuel, Sylvain, and Cappelaere, Bernard [2009]. Conditional simulation schemes of rain fields and their application to rainfall runoff modeling studies in the Sahel. In: Journal of Hydrology. Vol no Surface processes and water cycle in West Africa, studied from the AMMA-CATCH observing system, pp

43 Marginal model P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15 F(Y ) = P 0 + (1 P 0 ) Ga (Y ) Ga a gamma CDF with shape α and rate β. Non-stationnarity introduced in the marginal model [VGLM Chandler and Wheater, 2002] : α, β, P 0. log(α(x, t)) = α 0 + log(β(x, t)) = β 0 + N α j X j (x, t), j=1 N β j X j (x, t), p(x, t) the frequency of wet days : p(x, t) = 1 P 0 (x, t), p(x, t) estimated by a j=1 logistic regression [e.g.fealy and Sweeney, 2007] : log ( ) p(x, t) = 1 p(x, t) =S { }} { N p 0 + p j X j (x, t) p(x, t) = j=1 exp(s) 1 + exp(s).

44 Spatio-temporal non-stationary covariance function P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15 N log(ω(x i, t)) = ω 0 + ω j X j (x i, t) j=1

45 Covariance estimation and censored likelihood g d (x i ) not observed : only g d (x i ) > T d (x i ). Let Z d = {x i g d (x i )is censored} (i.e. stations without rainfall the day d), Let O d = {x i g d (x i )is observed} β the parameters of C(h) The censored likelihood given the {g d (x i )} xi Z d is L c (β) = d L c d(β) with : 1 L c d (β) = f GP(g d (x 1 ),..., g d (x N ); β), if it rains at all stations 2 L c d (β) = f GP({g d (x i )} xi O d ; β) P({G d (x i ) T d (x i )} xi Z d {G d (x i ) = g d (x i )} xi O d ; β),if not f GP are GP densities with covariance function C P({G d (x i ) T d (x i )} xi Z d {G d (x i ) = g d (x i )} xi O d ; β) is a conditional GP CDF P. Vaittinada Ayar & M. Vrac cc-mistrals Montpellier, Oct. 17, /15