A non-stationary spatial weather generator for statistical ...
Transcript of A non-stationary spatial weather generator for statistical ...
A non-stationary spatial weather generator for
statistical modelling of daily precipitation
Pradeebane VAITTINADA AYAR and Juliette BLANCHET
International Workshop on Stochastic Weather Generators for HydrologicalApplications
Berlin, Germany β 20th September 2017
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Outline
1 Context
2 Overview of the model
3 Parameter estimation
4 First evaluations
5 Conclusions and Perspectives
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 2/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Outline
1 Context
2 Overview of the model
3 Parameter estimation
4 First evaluations
5 Conclusions and Perspectives
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 3/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Context
SCHADEX : method for flood determination and dam dimensioning at EDF
Need for a realistic daily rainfall simulator to provide inputs for SCHADEX
Currently available : Regional MEWP [cf. Evin et al., 2016] univariate model
Aim to build a generic rainfall simulator (working over different catchements) :
accounting for spatial dependencies (daily time scale and one km2 spatialresolution)well performing for high precipitation quantiles at catchement scale
First choice : to model rain occurrence and intensity at thesame time only from observations.
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 3/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Context
SCHADEX : method for flood determination and dam dimensioning at EDF
Need for a realistic daily rainfall simulator to provide inputs for SCHADEX
Currently available : Regional MEWP [cf. Evin et al., 2016] univariate model
Aim to build a generic rainfall simulator (working over different catchements) :
accounting for spatial dependencies (daily time scale and one km2 spatialresolution)well performing for high precipitation quantiles at catchement scale
First choice : to model rain occurrence and intensity at thesame time only from observations.
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 3/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Context
SCHADEX : method for flood determination and dam dimensioning at EDF
Need for a realistic daily rainfall simulator to provide inputs for SCHADEX
Currently available : Regional MEWP [cf. Evin et al., 2016] univariate model
Aim to build a generic rainfall simulator (working over different catchements) :
accounting for spatial dependencies (daily time scale and one km2 spatialresolution)well performing for high precipitation quantiles at catchement scale
First choice : to model rain occurrence and intensity at thesame time only from observations.
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 3/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
(a) Ardeche catchement at Sauze
Ardèche at Sauze basin
Longitude (km)
Latit
ude
(km
)
48 212 376 541 705 869 1033 1198
1620
1771
1922
2073
2224
2375
2526
2677
(b) Stations locations and altitudes
0
500
1000
1500
2000
Elevation (m)
β
ββ
β
β
ββ
β
β β
β
β
β
β
β
β
β
β
β
β
ββ
β
CHASSERADES
SAINTEβMARGUERITEβLAFIGERE
CUBIERES
LANGOGNE
BERZEME
SABLIERESALBAβLAβROMAINE
BOURGβSAINTβANDEOL
LOUBARESSEMIRABEL
PRIVAS
SAINTβETIENNEβDEβLUGDARES
SAINTβMONTAN
BAGNOLSβLESβBAINS
CHATEAUNEUFβDEβRANDON
FLORAC
LEβPONTβDEβMONTVERT
SAINTβSAUVEURβDEβGINESTOUX
VILLEFORT
LAPALUD
VALSβLESβBAINSCHAUDEYRAC
LEβBLEYMARD
CHASSERADES
SAINTEβMARGUERITEβLAFIGERE
CUBIERES
LANGOGNE
BERZEME
SABLIERESALBAβLAβROMAINE
BOURGβSAINTβANDEOL
LOUBARESSEMIRABEL
PRIVAS
SAINTβETIENNEβDEβLUGDARES
SAINTβMONTAN
BAGNOLSβLESβBAINS
CHATEAUNEUFβDEβRANDON
FLORAC
LEβPONTβDEβMONTVERT
SAINTβSAUVEURβDEβGINESTOUX
VILLEFORT
LAPALUD
VALSβLESβBAINSCHAUDEYRAC
LEβBLEYMARD
CHASSERADES
SAINTEβMARGUERITEβLAFIGERE
CUBIERES
LANGOGNE
BERZEME
SABLIERESALBAβLAβROMAINE
BOURGβSAINTβANDEOL
LOUBARESSEMIRABEL
PRIVAS
SAINTβETIENNEβDEβLUGDARES
SAINTβMONTAN
BAGNOLSβLESβBAINS
CHATEAUNEUFβDEβRANDON
FLORAC
LEβPONTβDEβMONTVERT
SAINTβSAUVEURβDEβGINESTOUX
VILLEFORT
LAPALUD
VALSβLESβBAINSCHAUDEYRAC
LEβBLEYMARD
CHASSERADES
SAINTEβMARGUERITEβLAFIGERE
CUBIERES
LANGOGNE
BERZEME
SABLIERESALBAβLAβROMAINE
BOURGβSAINTβANDEOL
LOUBARESSEMIRABEL
PRIVAS
SAINTβETIENNEβDEβLUGDARES
SAINTβMONTAN
BAGNOLSβLESβBAINS
CHATEAUNEUFβDEβRANDON
FLORAC
LEβPONTβDEβMONTVERT
SAINTβSAUVEURβDEβGINESTOUX
VILLEFORT
LAPALUD
VALSβLESβBAINSCHAUDEYRAC
LEβBLEYMARD
CHASSERADES
SAINTEβMARGUERITEβLAFIGERE
CUBIERES
LANGOGNE
BERZEME
SABLIERESALBAβLAβROMAINE
BOURGβSAINTβANDEOL
LOUBARESSEMIRABEL
PRIVAS
SAINTβETIENNEβDEβLUGDARES
SAINTβMONTAN
BAGNOLSβLESβBAINS
CHATEAUNEUFβDEβRANDON
FLORAC
LEβPONTβDEβMONTVERT
SAINTβSAUVEURβDEβGINESTOUX
VILLEFORT
LAPALUD
VALSβLESβBAINSCHAUDEYRAC
LEβBLEYMARD
CHASSERADES
SAINTEβMARGUERITEβLAFIGERE
CUBIERES
LANGOGNE
BERZEME
SABLIERESALBAβLAβROMAINE
BOURGβSAINTβANDEOL
LOUBARESSEMIRABEL
PRIVAS
SAINTβETIENNEβDEβLUGDARES
SAINTβMONTAN
BAGNOLSβLESβBAINS
CHATEAUNEUFβDEβRANDON
FLORAC
LEβPONTβDEβMONTVERT
SAINTβSAUVEURβDEβGINESTOUX
VILLEFORT
LAPALUD
VALSβLESβBAINSCHAUDEYRAC
LEβBLEYMARD
CHASSERADES
SAINTEβMARGUERITEβLAFIGERE
CUBIERES
LANGOGNE
BERZEME
SABLIERESALBAβLAβROMAINE
BOURGβSAINTβANDEOL
LOUBARESSEMIRABEL
PRIVAS
SAINTβETIENNEβDEβLUGDARES
SAINTβMONTAN
BAGNOLSβLESβBAINS
CHATEAUNEUFβDEβRANDON
FLORAC
LEβPONTβDEβMONTVERT
SAINTβSAUVEURβDEβGINESTOUX
VILLEFORT
LAPALUD
VALSβLESβBAINSCHAUDEYRAC
LEβBLEYMARD
CHASSERADES
SAINTEβMARGUERITEβLAFIGERE
CUBIERES
LANGOGNE
BERZEME
SABLIERESALBAβLAβROMAINE
BOURGβSAINTβANDEOL
LOUBARESSEMIRABEL
PRIVAS
SAINTβETIENNEβDEβLUGDARES
SAINTβMONTAN
BAGNOLSβLESβBAINS
CHATEAUNEUFβDEβRANDON
FLORAC
LEβPONTβDEβMONTVERT
SAINTβSAUVEURβDEβGINESTOUX
VILLEFORT
LAPALUD
VALSβLESβBAINSCHAUDEYRAC
LEβBLEYMARD
CHASSERADES
SAINTEβMARGUERITEβLAFIGERE
CUBIERES
LANGOGNE
BERZEME
SABLIERESALBAβLAβROMAINE
BOURGβSAINTβANDEOL
LOUBARESSEMIRABEL
PRIVAS
SAINTβETIENNEβDEβLUGDARES
SAINTβMONTAN
BAGNOLSβLESβBAINS
CHATEAUNEUFβDEβRANDON
FLORAC
LEβPONTβDEβMONTVERT
SAINTβSAUVEURβDEβGINESTOUX
VILLEFORT
LAPALUD
VALSβLESβBAINSCHAUDEYRAC
LEβBLEYMARD
Ardèche at Sauze
X (km) β Lambert II extended
Y (
km)
β L
ambe
rt II
ext
ende
d
β
β
ββ
β
ββ
ββ
ββ
β
β
β
β
β
β
β
β
STβPIERREVILLE
MAYRESMONTPEZATβSOUBEYROL
ANTRAIGUESβAIZAC
AUBENAS
LABLACHΓRE
JOYEUSE
VALLONβPONTβD'ARC
MALONS
SENECHAS
STβMAURICEβDEβVENTALON
MASβDEβLAβBARQUE
BESSEGES
STEβEULALIE
USCALADESβETβRIEUTORD
MAZAN
LACβD'ISSARLΓS
ISSANLASβMEZEYRAC
MASMEJEAN/βBASTIDEβPUYLAURENT
STβPIERREVILLE
MAYRESMONTPEZATβSOUBEYROL
ANTRAIGUESβAIZAC
AUBENAS
LABLACHΓRE
JOYEUSE
VALLONβPONTβD'ARC
MALONS
SENECHAS
STβMAURICEβDEβVENTALON
MASβDEβLAβBARQUE
BESSEGES
STEβEULALIE
USCALADESβETβRIEUTORD
MAZAN
LACβD'ISSARLΓS
ISSANLASβMEZEYRAC
MASMEJEAN/βBASTIDEβPUYLAURENT
STβPIERREVILLE
MAYRESMONTPEZATβSOUBEYROL
ANTRAIGUESβAIZAC
AUBENAS
LABLACHΓRE
JOYEUSE
VALLONβPONTβD'ARC
MALONS
SENECHAS
STβMAURICEβDEβVENTALON
MASβDEβLAβBARQUE
BESSEGES
STEβEULALIE
USCALADESβETβRIEUTORD
MAZAN
LACβD'ISSARLΓS
ISSANLASβMEZEYRAC
MASMEJEAN/βBASTIDEβPUYLAURENT
STβPIERREVILLE
MAYRESMONTPEZATβSOUBEYROL
ANTRAIGUESβAIZAC
AUBENAS
LABLACHΓRE
JOYEUSE
VALLONβPONTβD'ARC
MALONS
SENECHAS
STβMAURICEβDEβVENTALON
MASβDEβLAβBARQUE
BESSEGES
STEβEULALIE
USCALADESβETβRIEUTORD
MAZAN
LACβD'ISSARLΓS
ISSANLASβMEZEYRAC
MASMEJEAN/βBASTIDEβPUYLAURENT
STβPIERREVILLE
MAYRESMONTPEZATβSOUBEYROL
ANTRAIGUESβAIZAC
AUBENAS
LABLACHΓRE
JOYEUSE
VALLONβPONTβD'ARC
MALONS
SENECHAS
STβMAURICEβDEβVENTALON
MASβDEβLAβBARQUE
BESSEGES
STEβEULALIE
USCALADESβETβRIEUTORD
MAZAN
LACβD'ISSARLΓS
ISSANLASβMEZEYRAC
MASMEJEAN/βBASTIDEβPUYLAURENT
STβPIERREVILLE
MAYRESMONTPEZATβSOUBEYROL
ANTRAIGUESβAIZAC
AUBENAS
LABLACHΓRE
JOYEUSE
VALLONβPONTβD'ARC
MALONS
SENECHAS
STβMAURICEβDEβVENTALON
MASβDEβLAβBARQUE
BESSEGES
STEβEULALIE
USCALADESβETβRIEUTORD
MAZAN
LACβD'ISSARLΓS
ISSANLASβMEZEYRAC
MASMEJEAN/βBASTIDEβPUYLAURENT
STβPIERREVILLE
MAYRESMONTPEZATβSOUBEYROL
ANTRAIGUESβAIZAC
AUBENAS
LABLACHΓRE
JOYEUSE
VALLONβPONTβD'ARC
MALONS
SENECHAS
STβMAURICEβDEβVENTALON
MASβDEβLAβBARQUE
BESSEGES
STEβEULALIE
USCALADESβETβRIEUTORD
MAZAN
LACβD'ISSARLΓS
ISSANLASβMEZEYRAC
MASMEJEAN/βBASTIDEβPUYLAURENT
STβPIERREVILLE
MAYRESMONTPEZATβSOUBEYROL
ANTRAIGUESβAIZAC
AUBENAS
LABLACHΓRE
JOYEUSE
VALLONβPONTβD'ARC
MALONS
SENECHAS
STβMAURICEβDEβVENTALON
MASβDEβLAβBARQUE
BESSEGES
STEβEULALIE
USCALADESβETβRIEUTORD
MAZAN
LACβD'ISSARLΓS
ISSANLASβMEZEYRAC
MASMEJEAN/βBASTIDEβPUYLAURENT
STβPIERREVILLE
MAYRESMONTPEZATβSOUBEYROL
ANTRAIGUESβAIZAC
AUBENAS
LABLACHΓRE
JOYEUSE
VALLONβPONTβD'ARC
MALONS
SENECHAS
STβMAURICEβDEβVENTALON
MASβDEβLAβBARQUE
BESSEGES
STEβEULALIE
USCALADESβETβRIEUTORD
MAZAN
LACβD'ISSARLΓS
ISSANLASβMEZEYRAC
MASMEJEAN/βBASTIDEβPUYLAURENT
695 709 722 736 749 763 776 790
1910
1921
1933
1944
1956
1967
1979
1990
FIGURE 1: Area : 2260 km2 β 42 Stations β Altitudes : from 47 to 1425 mAt least 20 years long time series
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 4/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Outline
1 Context
2 Overview of the model
3 Parameter estimation
4 First evaluations
5 Conclusions and Perspectives
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 5/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Model framework
βx β D,Y (x) a r.v. characterising the precipitation over a given domain. For a givenday, the rainfield is equal to a realisation of {Y (x)}xβD
What is {Y (x)}xβD ?The rainfield {Y (x)}xβD is obtained from a model based on a transformedlatent censored gaussian field quite widely used in the literature [e.g., Vischelet al., 2009; Rasmussen, 2013; Allard and Bourotte, 2015].
Gaussian field : fully defined by its mean vector and covariance structure.
Model stepsMarginal model
Spatial model
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 5/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Model framework
βx β D,Y (x) a r.v. characterising the precipitation over a given domain. For a givenday, the rainfield is equal to a realisation of {Y (x)}xβD
What is {Y (x)}xβD ?The rainfield {Y (x)}xβD is obtained from a model based on a transformedlatent censored gaussian field quite widely used in the literature [e.g., Vischelet al., 2009; Rasmussen, 2013; Allard and Bourotte, 2015].
Gaussian field : fully defined by its mean vector and covariance structure.
Model stepsMarginal model
Spatial model
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 5/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Model framework
βx β D,Y (x) a r.v. characterising the precipitation over a given domain. For a givenday, the rainfield is equal to a realisation of {Y (x)}xβD
What is {Y (x)}xβD ?The rainfield {Y (x)}xβD is obtained from a model based on a transformedlatent censored gaussian field quite widely used in the literature [e.g., Vischelet al., 2009; Rasmussen, 2013; Allard and Bourotte, 2015].Gaussian field : fully defined by its mean vector and covariance structure.
Model stepsMarginal model
Spatial model
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 5/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Model framework
βx β D,Y (x) a r.v. characterising the precipitation over a given domain. For a givenday, the rainfield is equal to a realisation of {Y (x)}xβD
What is {Y (x)}xβD ?The rainfield {Y (x)}xβD is obtained from a model based on a transformedlatent censored gaussian field quite widely used in the literature [e.g., Vischelet al., 2009; Rasmussen, 2013; Allard and Bourotte, 2015].Gaussian field : fully defined by its mean vector and covariance structure.
Model stepsMarginal model
Spatial model
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 5/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Marginal model
Extended Generalized Pareto of Type II (EGPII)Γ Model the full range of rainfall intensities without any threshold selection.Naveau et al. [2016] : P(Y 6 y) = G
(HΞΎ( yΟ
))
where
HΞΎ, the Generalized Pareto Distribution
G(v) = vΞΊ
Courtesy : Naveau et al. [2016] ΞΎ = 0.5
0 1 2 3 4 5 6 7
0.0
0.2
0.4
0.6
0.8
1.0
x
Den
sity
f(x)
ΞΎ=0.5
ΞΊ = 1ΞΊ = 2ΞΊ = 5Gamma
0 2 4 6 8 10
050
100
150
200
U = βln(βln(F))
prec
ipita
tion
(mm
/day
)
ΞΊ = 1ΞΊ = 2ΞΊ = 5Gamma
ΞΎ=0.510 20 50 100 1000 10000
Temps de retour T (annees)
0 1 2 3 4 5 6 7
0.0
0.2
0.4
0.6
0.8
1.0
x
Den
sity
f(x)
ΞΎ=0.5
ΞΊ = 1ΞΊ = 2ΞΊ = 5Gamma
0 2 4 6 8 10
050
100
150
200
U = βln(βln(F))
prec
ipita
tion
(mm
/day
)
ΞΊ = 1ΞΊ = 2ΞΊ = 5Gamma
ΞΎ=0.510 20 50 100 1000 10000
Temps de retour T (annees)
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 6/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Marginal model
Extended Generalized Pareto of Type II (EGPII)Γ Model the full range of rainfall intensities without any threshold selection.Naveau et al. [2016] : P(Y 6 y) = G
(HΞΎ( yΟ
))where
HΞΎ, the Generalized Pareto Distribution
G(v) = vΞΊ
Courtesy : Naveau et al. [2016] ΞΎ = 0.5
0 1 2 3 4 5 6 7
0.0
0.2
0.4
0.6
0.8
1.0
x
Den
sity
f(x)
ΞΎ=0.5
ΞΊ = 1ΞΊ = 2ΞΊ = 5Gamma
0 2 4 6 8 10
050
100
150
200
U = βln(βln(F))
prec
ipita
tion
(mm
/day
)
ΞΊ = 1ΞΊ = 2ΞΊ = 5Gamma
ΞΎ=0.510 20 50 100 1000 10000
Temps de retour T (annees)
0 1 2 3 4 5 6 7
0.0
0.2
0.4
0.6
0.8
1.0
x
Den
sity
f(x)
ΞΎ=0.5
ΞΊ = 1ΞΊ = 2ΞΊ = 5Gamma
0 2 4 6 8 10
050
100
150
200
U = βln(βln(F))pr
ecip
itatio
n (m
m/d
ay)
ΞΊ = 1ΞΊ = 2ΞΊ = 5Gamma
ΞΎ=0.510 20 50 100 1000 10000
Temps de retour T (annees)
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 6/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Marginal model
Extended Generalized Pareto of Type II (EGPII)Γ Model the full range of rainfall intensities without any threshold selection.Naveau et al. [2016] : P(Y 6 y) = G
(HΞΎ( yΟ
))where
HΞΎ, the Generalized Pareto Distribution
G(v) = vΞΊ
Constraints
Low values driven by ΞΊ, large values driven by ΞΎ
Weilbull type lower tail behavior (bounded orshort-tailed)
Frechet type upper tail behavior (unbouded orheavy-tailed)
Courtesy : Naveau et al. [2016] ΞΎ = 0.5
0 1 2 3 4 5 6 7
0.0
0.2
0.4
0.6
0.8
1.0
x
Den
sity
f(x)
ΞΎ=0.5
ΞΊ = 1ΞΊ = 2ΞΊ = 5Gamma
0 2 4 6 8 10
050
100
150
200
U = βln(βln(F))
prec
ipita
tion
(mm
/day
)
ΞΊ = 1ΞΊ = 2ΞΊ = 5Gamma
ΞΎ=0.510 20 50 100 1000 10000
Temps de retour T (annees)
0 1 2 3 4 5 6 7
0.0
0.2
0.4
0.6
0.8
1.0
x
Den
sity
f(x)
ΞΎ=0.5
ΞΊ = 1ΞΊ = 2ΞΊ = 5Gamma
0 2 4 6 8 10
050
100
150
200
U = βln(βln(F))pr
ecip
itatio
n (m
m/d
ay)
ΞΊ = 1ΞΊ = 2ΞΊ = 5Gamma
ΞΎ=0.510 20 50 100 1000 10000
Temps de retour T (annees)
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 6/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Marginal model
Extended GP-II : without threshold
F (y ;Ο, ΞΎ) =
(
1 β(
1 + ΞΎyΟ
)β1/ΞΎ)ΞΊ
if ΞΎ 6= 0,(1 β exp
{βyΟ
})ΞΊ, if ΞΎ = 0.
ΞΎ is set to 0 Γ powered exponential used as marginaldistribution.
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 7/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Marginal model
Extended GP-II : without threshold
F (y ;Ο, ΞΎ) =
(
1 β(
1 + ΞΎyΟ
)β1/ΞΎ)ΞΊ
if ΞΎ 6= 0,(1 β exp
{βyΟ
})ΞΊ, if ΞΎ = 0.
ΞΎ > 0 Γ too large return levels for long return periods e.g. 1000 to 10000 years(standard return period in dam construction)
ΞΎ is set to 0 Γ powered exponential used as marginaldistribution.
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 7/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Marginal model
Extended GP-II : without threshold
F (y ;Ο, ΞΎ) =
(
1 β(
1 + ΞΎyΟ
)β1/ΞΎ)ΞΊ
if ΞΎ 6= 0,(1 β exp
{βyΟ
})ΞΊ, if ΞΎ = 0.
ΞΎ > 0 Γ too large return levels for long return periods e.g. 1000 to 10000 years(standard return period in dam construction)
ΞΎ is set to 0 Γ powered exponential used as marginaldistribution.
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 7/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Spatial dependance
Let the following transformation :H(x) = Ξ¦β1(F (Y (x))), where Ξ¦ is the CDF of the normal N(0, 1).
Γ Since Y (x) > 0, H(x) > T (x) β‘ Ξ¦β1(FY(x)(0))
Γ Then H(x) = max(T (x),G(x)),
G(x) the latent Gaussian process GP(0, 1,C(h))with marginal N(0, 1), C(h) the spatial covariance function.
ProcedureEstimate C(h) from the H-transformed data,
Simulate G and deduce H,
Retrieve Y by transforming back with Y (x) = Fβ1Y(x)(Ξ¦(H(x))).
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 8/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Spatial dependance
Let the following transformation :H(x) = Ξ¦β1(F (Y (x))), where Ξ¦ is the CDF of the normal N(0, 1).
Γ Since Y (x) > 0, H(x) > T (x) β‘ Ξ¦β1(FY(x)(0))
Γ Then H(x) = max(T (x),G(x)),
G(x) the latent Gaussian process GP(0, 1,C(h))with marginal N(0, 1), C(h) the spatial covariance function.
ProcedureEstimate C(h) from the H-transformed data,
Simulate G and deduce H,
Retrieve Y by transforming back with Y (x) = Fβ1Y(x)(Ξ¦(H(x))).
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 8/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Covariance function
CS(h) = Ο2exp(βhΞ»
), the stationary exponential covariance (isotropic) function
Geometrical anisotropy widely used in the literature
h(xi , xj) =β
(xi β xj)TΞ£β1(xi β xj)
with Ξ£ a covariance matrix, Ξ£β1 = MT M :
M =
(cosΟ sinΟ
βb sinΟ b cosΟ
)
with b the elongation coefficient, Ο β[βΟ2 ,
Ο2
]
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 9/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Covariance function
CS(h) = Ο2exp(βhΞ»
), the stationary exponential covariance (isotropic) function
Geometrical anisotropy widely used in the literature
h(xi , xj) =β
(xi β xj)TΞ£β1(xi β xj)
with Ξ£ a covariance matrix, Ξ£β1 = MT M :
M =
(cosΟ sinΟ
βb sinΟ b cosΟ
)
with b the elongation coefficient, Ο β[βΟ2 ,
Ο2
]
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 9/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Covariance function
CS(h) = Ο2exp(βhΞ»
), the stationary exponential covariance (isotropic) function
Geometrical anisotropy widely used in the literature
h(xi , xj) =β
(xi β xj)TΞ£β1(xi β xj)
with Ξ£ a covariance matrix, Ξ£β1 = MT M :
M =
(cosΟ sinΟ
βb sinΟ b cosΟ
)
with b the elongation coefficient, Ο β[βΟ2 ,
Ο2
]How to include topographical information in the
covariance ?
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 9/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Covariance function
CS(h) = Ο2exp(βhΞ»
), the stationary exponential covariance (isotropic) function
Geometrical anisotropy widely used in the literature
h(xi , xj) =β
(xi β xj)TΞ£β1(xi β xj)
with Ξ£ a covariance matrix, Ξ£β1 = MT M :
M =
(cosΟ sinΟ
βb sinΟ b cosΟ
)
with b the elongation coefficient, Ο β[βΟ2 ,
Ο2
]Non-stationarity introduced by Higdon et al. [1999] : kernel
convolution in anisotropic function.
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 9/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Spatial non-stationary covariance function
Generalized by Paciorek and Schervish [2006]
One solution for spatial non-stationarity :
CNS(xi , xj) = |Ξ£i |14 |Ξ£j |
14
β£β£β£β£Ξ£i + Ξ£j
2
β£β£β£β£β 12
exp
β
β(xi β xj)T
(Ξ£i + Ξ£j
2
)β1
(xi β xj)
Ξ£i ,Ξ£j : 2Γ 2 covariance matrix (local kernels, βi, j Ξ£i = Ξ£j ) Γ classical anisotropyβxi β D, Ξ£β1
i = MTi Mi :
Mi =1Ξ»i
(cosΟi sinΟi
βbi sinΟi bi cosΟi
)=
(Ai cosΟi Ai sinΟi
βBi sinΟi Bi cosΟi
)
with Ai , Bi (> 0) et Οi as functions of the covariates (lon, lat, elev).
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 10/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Spatial non-stationary covariance function
Generalized by Paciorek and Schervish [2006]
One solution for spatial non-stationarity :
CNS(xi , xj) = |Ξ£i |14 |Ξ£j |
14
β£β£β£β£Ξ£i + Ξ£j
2
β£β£β£β£β 12
exp
β
β(xi β xj)T
(Ξ£i + Ξ£j
2
)β1
(xi β xj)
Ξ£i ,Ξ£j : 2Γ 2 covariance matrix (local kernels, βi, j Ξ£i = Ξ£j ) Γ classical anisotropy
βxi β D, Ξ£β1i = MT
i Mi :
Mi =1Ξ»i
(cosΟi sinΟi
βbi sinΟi bi cosΟi
)=
(Ai cosΟi Ai sinΟi
βBi sinΟi Bi cosΟi
)
with Ai , Bi (> 0) et Οi as functions of the covariates (lon, lat, elev).
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 10/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Spatial non-stationary covariance function
Generalized by Paciorek and Schervish [2006]
One solution for spatial non-stationarity :
CNS(xi , xj) = |Ξ£i |14 |Ξ£j |
14
β£β£β£β£Ξ£i + Ξ£j
2
β£β£β£β£β 12
exp
β
β(xi β xj)T
(Ξ£i + Ξ£j
2
)β1
(xi β xj)
Ξ£i ,Ξ£j : 2Γ 2 covariance matrix (local kernels, βi, j Ξ£i = Ξ£j ) Γ classical anisotropyβxi β D, Ξ£β1
i = MTi Mi :
Mi =1Ξ»i
(cosΟi sinΟi
βbi sinΟi bi cosΟi
)=
(Ai cosΟi Ai sinΟi
βBi sinΟi Bi cosΟi
)
with Ai , Bi (> 0) et Οi as functions of the covariates (lon, lat, elev).
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 10/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Outline
1 Context
2 Overview of the model
3 Parameter estimation
4 First evaluations
5 Conclusions and Perspectives
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 11/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
The estimation procedure is performed over six sub-set based on :
a rainy days classification in K = 3 weather types [WT,Garavaglia et al., 2010],
S = 2 seasons defined as low risk (LO dec. to aug.) and high risk (HI sep. to nov.)seasons.
Marginal distribution parameters : maximum likelihood,
Covariance function parameters : censored maximum likelihood [Pesonenet al., 2015].
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 11/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
The estimation procedure is performed over six sub-set based on :
a rainy days classification in K = 3 weather types [WT,Garavaglia et al., 2010],
S = 2 seasons defined as low risk (LO dec. to aug.) and high risk (HI sep. to nov.)seasons.
Marginal distribution parameters : maximum likelihood,
Covariance function parameters : censored maximum likelihood [Pesonenet al., 2015].
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 11/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Covariance estimation and censored likelihood
gd(xi) not observed : only gd(xi) > Td(xi).
Let Zd = {xi |gd(xi)is censored} (i.e. stations without rainfall the day d),
Let Od = {xi |gd(xi)is observed}
Ξ² the parameters of C(h)
The censored likelihood given the {gd(xi)}xiβZd is Lc(Ξ²) =β
d
Lcd(Ξ²) with :
1 Lcd(Ξ²) = fGP(gd(x1), . . . , gd(xN);Ξ²), if it rains at all stations
2 Lcd(Ξ²) = fGP({gd(xi)}xiβOd ;Ξ²)
ΓP({Gd(xi) 6 Td(xi)}xiβZd |{Gd(xi) = gd(xi)}xiβOd ;Ξ²),if not
fGP are GP densities with covariance function C
P({Gd(xi) 6 Td(xi)}xiβZd |{Gd(xi) = gd ,m(xi)}xiβOd ;Ξ²) is a conditional GP CDF
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 12/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Covariance estimation and censored likelihood
gd(xi) not observed : only gd(xi) > Td(xi).
Let Zd = {xi |gd(xi)is censored} (i.e. stations without rainfall the day d),
Let Od = {xi |gd(xi)is observed}
Ξ² the parameters of C(h)
The censored likelihood given the {gd(xi)}xiβZd is Lc(Ξ²) =β
d
Lcd(Ξ²) with :
1 Lcd(Ξ²) = fGP(gd(x1), . . . , gd(xN);Ξ²), if it rains at all stations
2 Lcd(Ξ²) = fGP({gd(xi)}xiβOd ;Ξ²)
ΓP({Gd(xi) 6 Td(xi)}xiβZd |{Gd(xi) = gd(xi)}xiβOd ;Ξ²),if not
fGP are GP densities with covariance function C
P({Gd(xi) 6 Td(xi)}xiβZd |{Gd(xi) = gd ,m(xi)}xiβOd ;Ξ²) is a conditional GP CDF
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 12/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Covariance estimation and censored likelihood
gd(xi) not observed : only gd(xi) > Td(xi).
Let Zd = {xi |gd(xi)is censored} (i.e. stations without rainfall the day d),
Let Od = {xi |gd(xi)is observed}
Ξ² the parameters of C(h)
The censored likelihood given the {gd(xi)}xiβZd is Lc(Ξ²) =β
d
Lcd(Ξ²) with :
1 Lcd(Ξ²) = fGP(gd(x1), . . . , gd(xN);Ξ²), if it rains at all stations
2 Lcd(Ξ²) = fGP({gd(xi)}xiβOd ;Ξ²)
ΓP({Gd(xi) 6 Td(xi)}xiβZd |{Gd(xi) = gd(xi)}xiβOd ;Ξ²),if not
fGP are GP densities with covariance function C
P({Gd(xi) 6 Td(xi)}xiβZd |{Gd(xi) = gd ,m(xi)}xiβOd ;Ξ²) is a conditional GP CDF
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 12/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Covariance estimation and censored likelihood
gd(xi) not observed : only gd(xi) > Td(xi).
Let Zd = {xi |gd(xi)is censored} (i.e. stations without rainfall the day d),
Let Od = {xi |gd(xi)is observed}
Ξ² the parameters of C(h)
The censored likelihood given the {gd(xi)}xiβZd is Lc(Ξ²) =β
d
Lcd(Ξ²) with :
1 Lcd(Ξ²) = fGP(gd(x1), . . . , gd(xN);Ξ²), if it rains at all stations
2 Lcd(Ξ²) = fGP({gd(xi)}xiβOd ;Ξ²)
ΓP({Gd(xi) 6 Td(xi)}xiβZd |{Gd(xi) = gd(xi)}xiβOd ;Ξ²),if not
fGP are GP densities with covariance function C
P({Gd(xi) 6 Td(xi)}xiβZd |{Gd(xi) = gd ,m(xi)}xiβOd ;Ξ²) is a conditional GP CDF
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 12/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Covariance estimation and censored likelihood
gd(xi) not observed : only gd(xi) > Td(xi).
Let Zd = {xi |gd(xi)is censored} (i.e. stations without rainfall the day d),
Let Od = {xi |gd(xi)is observed}
Ξ² the parameters of C(h)
The censored likelihood given the {gd(xi)}xiβZd is Lc(Ξ²) =β
d
Lcd(Ξ²) with :
1 Lcd(Ξ²) = fGP(gd(x1), . . . , gd(xN);Ξ²), if it rains at all stations
2 Lcd(Ξ²) = fGP({gd(xi)}xiβOd ;Ξ²)
ΓP({Gd(xi) 6 Td(xi)}xiβZd |{Gd(xi) = gd(xi)}xiβOd ;Ξ²),if not
fGP are GP densities with covariance function C
P({Gd(xi) 6 Td(xi)}xiβZd |{Gd(xi) = gd ,m(xi)}xiβOd ;Ξ²) is a conditional GP CDF
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 12/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Covariance estimation and censored likelihood
gd(xi) not observed : only gd(xi) > Td(xi).
Let Zd = {xi |gd(xi)is censored} (i.e. stations without rainfall the day d),
Let Od = {xi |gd(xi)is observed}
Ξ² the parameters of C(h)
The censored likelihood given the {gd(xi)}xiβZd is Lc(Ξ²) =β
d
Lcd(Ξ²) with :
1 Lcd(Ξ²) = fGP(gd(x1), . . . , gd(xN);Ξ²), if it rains at all stations
2 Lcd(Ξ²) = fGP({gd(xi)}xiβOd ;Ξ²)
ΓP({Gd(xi) 6 Td(xi)}xiβZd |{Gd(xi) = gd(xi)}xiβOd ;Ξ²),if not
fGP are GP densities with covariance function C
P({Gd(xi) 6 Td(xi)}xiβZd |{Gd(xi) = gd ,m(xi)}xiβOd ;Ξ²) is a conditional GP CDF
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 12/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Outline
1 Context
2 Overview of the model
3 Parameter estimation
4 First evaluations
5 Conclusions and Perspectives
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 13/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
First evaluations
Calibration over the whole period with available data (1948-2013),
100 simulations over the period,
The simulations are only conditioned by the season and weather type the daybelongs to.
Evaluation of the marginal and spatial dependance properties. In particular, theadded-value of NS-covariance is explored. To this end three different types ofexponential covariance function are tested :
isotropic (ISO)anisotropic (AN)non-stationary (NSAN)
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 13/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
First evaluations
Calibration over the whole period with available data (1948-2013),
100 simulations over the period,
The simulations are only conditioned by the season and weather type the daybelongs to.
Evaluation of the marginal and spatial dependance properties. In particular, theadded-value of NS-covariance is explored. To this end three different types ofexponential covariance function are tested :
isotropic (ISO)anisotropic (AN)non-stationary (NSAN)
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 13/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Mean rainfall & occurrence frequency (LO WT2)
ββ
β
β
ββ
β
β
βββ
β
β
β ββ
ββ
ββ β
β
ββ
β
βββ β
β
β β
ββ
β β
β
β
4 6 8 10 12
Mean
NSANβ β
β
β
βββ
β
β
ββ
βββ
β
β β
β
ββ
β ββ
10 15 20 25 30
Mean>0
ββ β
β
β β
β
β
βββ β
β β
ββ
ββ
βββ
ββ
β
ββ
β
β
ββ
β ββ
β β
ββ
β
β
β
ββ
β
ββ
β
β
ββ
30 35 40 45 50 55
P1
0
200
400
600
800
1000
1200
1400
Elevation (m)
β
β
ββ
β β
βββ
β β
β
4 6 8 10 12
ANββ β
β
β
β
β
β β
β
β
ββ
10 15 20 25 30
ββ
ββ
β ββ
β
β
β
β
β
β ββββ β
ββ
ββ
β
β β
ββ β
β ββ
ββ
β
β
β
ββ β β
ββ
β
30 35 40 45 50 55
ββββ
β
β
ββ
β
β
β
4 6 8 10 12
ISOPR (mm/day)
β
β
β
β
β
β β
β
β ββ
ββ β
β β
β
βββ
ββ
β
10 15 20 25 30
PR (mm/day)β
β
β
β β
ββ β
β
β
β
ββ
ββ
β
ββ ββ
β β
β
β ββ
β
β
β
β
β β βββ
β
30 35 40 45 50 55
%
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 14/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Interannual variabilityLO WT2
020
040
060
080
010
00
Years
PR
(m
m/y
ear)
1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010
LO GT2 cov mod: NSAN
MeanQ5βQ95OBS
RMSE = 146.66 mmCOR = 0.49
020
040
060
080
0
Years
PR
(m
m/y
ear)
1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010
LO GT2 cov mod: AN
MeanQ5βQ95OBS
RMSE = 147.44 mmCOR = 0.47
020
040
060
080
0
Years
PR
(m
m/y
ear)
1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010
LO GT2 cov mod: ISO
MeanQ5βQ95OBS
RMSE = 146.08 mmCOR = 0.48
020
040
060
080
010
00
Years
PR
(m
m/y
ear)
1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010
LO GT2 cov mod: NSAN
MeanQ5βQ95OBS
RMSE = 146.66 mmCOR = 0.49
020
040
060
080
0
Years
PR
(m
m/y
ear)
1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010
LO GT2 cov mod: AN
MeanQ5βQ95OBS
RMSE = 147.44 mmCOR = 0.47
020
040
060
080
0
Years
PR
(m
m/y
ear)
1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010
LO GT2 cov mod: ISO
MeanQ5βQ95OBS
RMSE = 146.08 mmCOR = 0.48
020
040
060
080
010
00
Years
PR
(m
m/y
ear)
1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010
LO GT2 cov mod: NSAN
MeanQ5βQ95OBS
RMSE = 146.66 mmCOR = 0.49
020
040
060
080
0
Years
PR
(m
m/y
ear)
1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010
LO GT2 cov mod: AN
MeanQ5βQ95OBS
RMSE = 147.44 mmCOR = 0.47
020
040
060
080
0
Years
PR
(m
m/y
ear)
1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010
LO GT2 cov mod: ISO
MeanQ5βQ95OBS
RMSE = 146.08 mmCOR = 0.48
NSAN
AN
ISO
HI WT2
020
040
060
080
0
Years
PR
(m
m/y
ear)
1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010
HI GT2 cov mod: NSAN
MeanQ5βQ95OBS
RMSE = 138.21 mmCOR = 0.45
020
040
060
080
0Years
PR
(m
m/y
ear)
1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010
HI GT2 cov mod: AN
MeanQ5βQ95OBS
RMSE = 137.45 mmCOR = 0.45
020
040
060
080
0
Years
PR
(m
m/y
ear)
1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010
HI GT2 cov mod: ISO
MeanQ5βQ95OBS
RMSE = 137.51 mmCOR = 0.45
020
040
060
080
0
Years
PR
(m
m/y
ear)
1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010
HI GT2 cov mod: NSAN
MeanQ5βQ95OBS
RMSE = 138.21 mmCOR = 0.45
020
040
060
080
0
Years
PR
(m
m/y
ear)
1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010
HI GT2 cov mod: AN
MeanQ5βQ95OBS
RMSE = 137.45 mmCOR = 0.45
020
040
060
080
0
Years
PR
(m
m/y
ear)
1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010
HI GT2 cov mod: ISO
MeanQ5βQ95OBS
RMSE = 137.51 mmCOR = 0.45
020
040
060
080
0
Years
PR
(m
m/y
ear)
1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010
HI GT2 cov mod: NSAN
MeanQ5βQ95OBS
RMSE = 138.21 mmCOR = 0.45
020
040
060
080
0
Years
PR
(m
m/y
ear)
1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010
HI GT2 cov mod: AN
MeanQ5βQ95OBS
RMSE = 137.45 mmCOR = 0.45
020
040
060
080
0
Years
PR
(m
m/y
ear)
1950 1956 1962 1968 1974 1980 1986 1992 1998 2004 2010
HI GT2 cov mod: ISO
MeanQ5βQ95OBS
RMSE = 137.51 mmCOR = 0.45
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 15/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Log-odds ratio
The log-odds ratio can be seen as analogous to spatial correlation of continuousvariable for binary variables [Charles et al., 1999]
TABLE 1: Contingency table for two stations xi, xj
NR(xi) R(xi)
NR(xj) n00 n01
R(xj) n10 n11
The log-odds ratio is given by :
LOR = log(
n00n11
n01n10
)
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 16/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Log-odds ratioLO WT2
β
β
β
ββ
β β
β
ββ
β
β
βββββ
β
ββββ
ββ
βββββββββββββ βββ
β
βββ
β
β β
β
βββ
βββ β
β
β
β
βββ
ββββββββ
β
ββ ββββββββ
βββ
ββ β
βββ
β
β
βββ
β
ββ
ββ
β
β
ββββ β ββ βββββ
β βββββββ
β β
βββ
β
β
ββββ ββ
ββ
ββ
ββββββββ
βββββ
ββ βββ
ββ
ββ
β
ββ
ββ
βββ βββ β
β
ββ
βββββ
ββββββββββ β
β
βββ
βββββ
β
β
ββ
βββββββ
ββββ
ββββββ
β
ββ
β
β
ββββββ
ββ
βββββ β ββ ββ
βββββ
ββββββ βββ
β
β
ββββββ
β
βββ
βββββββ
ββββββ
ββββββββ
β
ββββ ββ
β
βββββ
ββββββ
ββ
ββββ
βββ
ββββ
ββ
ββββββ β
β
βββ
ββββ
ββββββββ
ββββββββ
ββββββ β
β
ββ
ββββββββββββ ββββββββ
ββββββ
ββββ
β
βββββ ββββββ
βββ
βββ
β
ββββββ
β
βββ
ββββββββ
βββββ
ββββββ
βββββ
ββ
ββ
ββββββββ
βββββββ
βββββββ
βββ
ββ
ββββββββ
ββββ
βββββββ
β
ββ
βββ
β
ββββββ ββ
ββββββββββ
ββ
β
βββ
β
ββββββββββ
ββ
βββββββ
βββ
ββ
βββββββββββββ
βββββββ
βββ
βββββ ββ
β
ββ
ββ
ββ βββββ
βββ
ββββ ββββ
βββββ
ββββββ ββ
βββββ
βββββββ βββββββ
β
ββββββββ
ββββ
βββββββ
ββββββββ
βββββββββββ
βββ
ββββββββ βββββββ
ββββ β
ββββ βββ
βββββ
ββββ
βββββ ββββ
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46
8LO GT2 cov mod: NSAN
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0 2 4 6 8
02
46
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0 2 4 6 8
02
46
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0 2 4 6 8
02
46
8
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0 2 4 6 8
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46
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46
8
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0 2 4 6 8
02
46
8
LO GT2 cov mod: AN
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0 2 4 6 8
02
46
8
LO GT2 cov mod: ISO
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0 2 4 6 8
02
46
8
HI GT2 cov mod: NSAN
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0 2 4 6 8
02
46
8
HI GT2 cov mod: AN
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0 2 4 6 8
02
46
8
HI GT2 cov mod: ISO
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0 2 4 6 8
02
46
8
HI GT2 cov mod: NSAN
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0 2 4 6 8
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46
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HI GT2 cov mod: AN
Observation logβodds ratio
Sim
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0 2 4 6 8
02
46
8
HI GT2 cov mod: ISO
Observation logβodds ratio
Sim
ulat
ion
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odds
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βββββββββββ
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ββββββ β
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0 2 4 6 8
02
46
8
HI GT2 cov mod: NSAN
Observation logβodds ratio
Sim
ulat
ion
logβ
odds
rat
io
ββ
β
β
β
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0 2 4 6 8
02
46
8
HI GT2 cov mod: AN
Observation logβodds ratio
Sim
ulat
ion
logβ
odds
rat
io
ββ
β
β
β
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0 2 4 6 8
02
46
8
HI GT2 cov mod: ISO
Observation logβodds ratio
Sim
ulat
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odds
rat
io
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 16/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Correlogram (zero included)LO WT2
0.0
0.2
0.4
0.6
0.8
1.0
Distance (km)
Cor
rela
tion
0 6 12 19 27 35 42 50 58 65 73 81 88
LO GT2 cov mod: NSAN
MeanQ5βQ95OBS
GT2 RMSE = 0.1GT2 COR = 0.98
0.0
0.2
0.4
0.6
0.8
1.0
Distance (km)
Cor
rela
tion
0 6 12 19 27 35 42 50 58 65 73 81 88
LO GT2 cov mod: AN
MeanQ5βQ95OBS
GT2 RMSE = 0.11GT2 COR = 0.99
0.0
0.2
0.4
0.6
0.8
1.0
Distance (km)
Cor
rela
tion
0 6 12 19 27 35 42 50 58 65 73 81 88
LO GT2 cov mod: ISO
MeanQ5βQ95OBS
GT2 RMSE = 0.09GT2 COR = 0.99
0.0
0.2
0.4
0.6
0.8
1.0
Distance (km)
Cor
rela
tion
0 6 12 19 27 35 42 50 58 65 73 81 88
LO GT2 cov mod: NSAN
MeanQ5βQ95OBS
GT2 RMSE = 0.1GT2 COR = 0.98
0.0
0.2
0.4
0.6
0.8
1.0
Distance (km)
Cor
rela
tion
0 6 12 19 27 35 42 50 58 65 73 81 88
LO GT2 cov mod: AN
MeanQ5βQ95OBS
GT2 RMSE = 0.11GT2 COR = 0.99
0.0
0.2
0.4
0.6
0.8
1.0
Distance (km)
Cor
rela
tion
0 6 12 19 27 35 42 50 58 65 73 81 88
LO GT2 cov mod: ISO
MeanQ5βQ95OBS
GT2 RMSE = 0.09GT2 COR = 0.99
0.0
0.2
0.4
0.6
0.8
1.0
Distance (km)
Cor
rela
tion
0 6 12 19 27 35 42 50 58 65 73 81 88
LO GT2 cov mod: NSAN
MeanQ5βQ95OBS
GT2 RMSE = 0.1GT2 COR = 0.98
0.0
0.2
0.4
0.6
0.8
1.0
Distance (km)
Cor
rela
tion
0 6 12 19 27 35 42 50 58 65 73 81 88
LO GT2 cov mod: AN
MeanQ5βQ95OBS
GT2 RMSE = 0.11GT2 COR = 0.99
0.0
0.2
0.4
0.6
0.8
1.0
Distance (km)
Cor
rela
tion
0 6 12 19 27 35 42 50 58 65 73 81 88
LO GT2 cov mod: ISO
MeanQ5βQ95OBS
GT2 RMSE = 0.09GT2 COR = 0.99
NSAN
AN
ISO
HI WT2
0.0
0.2
0.4
0.6
0.8
1.0
Distance (km)
Cor
rela
tion
0 6 12 19 27 35 42 50 58 65 73 81 88
HI GT2 cov mod: NSAN
MeanQ5βQ95OBS
GT2 RMSE = 0.04GT2 COR = 0.98
0.0
0.2
0.4
0.6
0.8
1.0
Distance (km)
Cor
rela
tion
0 6 12 19 27 35 42 50 58 65 73 81 88
HI GT2 cov mod: AN
MeanQ5βQ95OBS
GT2 RMSE = 0.04GT2 COR = 0.99
0.0
0.2
0.4
0.6
0.8
1.0
Distance (km)
Cor
rela
tion
0 6 12 19 27 35 42 50 58 65 73 81 88
HI GT2 cov mod: ISO
MeanQ5βQ95OBS
GT2 RMSE = 0.04GT2 COR = 0.99
0.0
0.2
0.4
0.6
0.8
1.0
Distance (km)
Cor
rela
tion
0 6 12 19 27 35 42 50 58 65 73 81 88
HI GT2 cov mod: NSAN
MeanQ5βQ95OBS
GT2 RMSE = 0.04GT2 COR = 0.98
0.0
0.2
0.4
0.6
0.8
1.0
Distance (km)
Cor
rela
tion
0 6 12 19 27 35 42 50 58 65 73 81 88
HI GT2 cov mod: AN
MeanQ5βQ95OBS
GT2 RMSE = 0.04GT2 COR = 0.99
0.0
0.2
0.4
0.6
0.8
1.0
Distance (km)
Cor
rela
tion
0 6 12 19 27 35 42 50 58 65 73 81 88
HI GT2 cov mod: ISO
MeanQ5βQ95OBS
GT2 RMSE = 0.04GT2 COR = 0.99
0.0
0.2
0.4
0.6
0.8
1.0
Distance (km)
Cor
rela
tion
0 6 12 19 27 35 42 50 58 65 73 81 88
HI GT2 cov mod: NSAN
MeanQ5βQ95OBS
GT2 RMSE = 0.04GT2 COR = 0.98
0.0
0.2
0.4
0.6
0.8
1.0
Distance (km)
Cor
rela
tion
0 6 12 19 27 35 42 50 58 65 73 81 88
HI GT2 cov mod: AN
MeanQ5βQ95OBS
GT2 RMSE = 0.04GT2 COR = 0.99
0.0
0.2
0.4
0.6
0.8
1.0
Distance (km)
Cor
rela
tion
0 6 12 19 27 35 42 50 58 65 73 81 88
HI GT2 cov mod: ISO
MeanQ5βQ95OBS
GT2 RMSE = 0.04GT2 COR = 0.99
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 17/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Outline
1 Context
2 Overview of the model
3 Parameter estimation
4 First evaluations
5 Conclusions and Perspectives
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 18/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Conclusions & Perspectives
Good agreement in terms of marginal statistics with an underestimation of therain occurrence probability at stations
Underestimation of the spatial dependency
NS-covariance does not really improve the results in terms of marginal statisticsand spatial dependency (Is it worth it ?)
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 18/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Conclusions & Perspectives
Good agreement in terms of marginal statistics with an underestimation of therain occurrence probability at stations
Underestimation of the spatial dependency
NS-covariance does not really improve the results in terms of marginal statisticsand spatial dependency (Is it worth it ?)
Another way to characterize spatial dependency,
Add cross-validation procedure,
Remove days with too many missing values in the dataset,
3-days cumulated amounts are really important for flood determination : Γadd temporal correlation (AR process)
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 18/18
Context Overview of the model Parameter estimation First evaluations Conclusions and Perspectives
Conclusions & Perspectives
LO WT2
HI WT2
P. Vaittinada Ayar & J. Blanchet SWGEN-Hydro β Berlin, Sept. 20, 2017 18/18
Thank you for yourattention
Reference
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