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The stationarity hypothesis is essential in hydrological frequency analysis and statistical inference. This assumption is often not fulfilled for large observed datasets, especially in the case of hydro-climatic variables. The Generalized Extreme Value distribution with covariates allows to model data in the presence of non-stationarity and/or dependence on covariates. Linear and non-linear dependence structures have been proposed with the corresponding fitting approach. The objective of the present study is to develop the GEV model with B-Spline in a Bayesian framework. A Markov Chain Monte Carlo (MCMC) algorithm has been developed to estimate quantiles and their posterior distributions. The methods are tested and illustrated using simulated data and applied to meteorological data. Results indicate the better performance of the proposed Bayesian method for rainfall quantile estimation according to BIAS and RMSE criteria especially for high return period events.
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Open Journal of Statistics, 2013, 3, 118-128
http://dx.doi.org/10.4236/ojs.2013.32013 Published Online April 2013 (http://www.scirp.org/journal/ojs)
Bayesian Estimation for GEV-B-Spline Model
Bouchra Nasri1*, Salaheddine El Adlouni2, Taha B. M. J. Ouarda1,3
1Center Eau Terre Environnement, Institut national de la Recherche Scientifique
Québec, Canada
2Département de Mathématique et de Statistique, Université de Moncton, Moncton, Canada
3Masdar Institue of Science and Technology, Abu Dhabi, UAE
Email: Bouchra.nasri@ete.inrs.ca, salah-eddine.el.adlouni@umoncton.ca, touarda@masdar.ac.ae
Received December 10, 2012; revised January 11, 2013; accepted January 25, 2013
Copyright © 2013 Bouchra Nasri et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
The stationarity hypothesis is essential in hydrological frequency analysis and statistical inference. This assumption is
often not fulfilled for large observed datasets, especially in the case of hydro-climatic variables. The Generalized Ex-
treme Value distribution with covariates allows to model data in the presence of non-stationarity and/or dependence on
covariates. Linear and non-linear dependence structures have been proposed with the corresponding fitting approach.
The objective of the present study is to develop the GEV model with B-Spline in a Bayesian framework. A Markov
Chain Monte Carlo (MCMC) algorithm has been developed to estimate quantiles and their posterior distributions. The
methods are tested and illustrated using simulated data and applied to meteorological data. Results indicate the better
performance of the proposed Bayesian method for rainfall quantile estimation according to BIAS and RMSE criteria
especially for high return period events.
Keywords: GEV; Bayesien; B-Spline; Nonlinearity; Covariate; Non-Stationarity
1. Introduction
Many fields of modern science and engineering have to
deal with rare events with significant consequences. Ex-
treme value theory (EVT) allows to providing the basis for
the statistical modeling of such extremes. The main result
of EVT shows that the maxima, of Independent and Iden-
tically Distributed (i.i.d.) events, are asymptotically Gen-
eralized Extreme Value (GEV) distributed [1]. In practice,
the hypotheses of the EVT are, generally, not fulfilled,
and a classical frequency analysis, of independent, ho-
mogeneous and stationary samples, is considered with a
large range of probability distributions to estimate the oc-
currence of extreme events. A number of methods have
been proposed to estimate GEV distribution’s parameters;
such as the method of moments (MM) [2,3], maximum
likelihood (ML) [4] and the method of probability wei-
ghted moments [5].
The Stationarity assumption is essential to carry out a
statistical frequency analysis. However, in many fields,
such as hydroclimatology, observed data series are not
stationary [6,7]. For hydrological datasets, two main types
of non-stationarity have been observed due to temporal
trends or cycles corresponding to the effect of other co-
variates. The second kind of non-stationarity, has been
largely studied during the last decade through the GEV
model with covariates [8-14] for local frequency analysis
and [15,16] for regional analysis.
Taking into account the effect of a covariate can be
considered in a polynomial form [9,11,17]. These poly-
nomial forms for estimating the GEV parameters were
developed by the introduction of covariates in a polyno-
mial form such as a linear or quadratic function. However,
the dependence between covariates and variables of in-
terest can take different structures.
[18] suggested the use of semi-parametric functions
such as smoothing splines to estimate the relationship
between the parameters and covariates. The smoothing
splines are based on the minimization of the penalized
sum of the squared errors and the choice of the smoothing
parameter [19]. The main disadvantages of this type of
function are that inference, often through the confidence
bands, is not straightforward and that a smoothing which
parameter needs to be specified at the beginning [20]. A
smoothing-based B-spline function resolves these prob-
lems and presents several others advantages.
B-spline functions are linear combinations of non
negative piecewise-polynomial real functions. A B-spline
function does not depend on the response variable, or the
*Corresponding author.
C
opyright © 2013 SciRes. OJS
B. NASRI ET AL. 119
variable of interested, but depend only on: 1) the support
of the covariates, 2) the number and position of knots and
3) the degree of B-Spline function [19]. The above ad-
vantages of B-Spline functions make it an appropriate
option to be used in the GEV model with covariates to
estimate the quantiles conditionally to given factors. The
GEV model with B-spline called mixed GEV-B-Spline
model, is rigorous and flexible and allows to fit a large
number of dependence structures [18,21]. [18] describes
smooth non-stationary generalized additive modeling for
sample extremes, in which spline smoothers are incorpo-
rated into models for exceedances over high thresholds
with the Generalized Pareto distribution. They developed
the maximum penalized likelihood estimation approach
with uncertainty assessed by using differences of devi-
ances and bootstrap simulation.
The main objective of the present study is to develop
the Generalized Extreme Value model with covariates
where the dependence structure is represented by B-spline
functions in a Bayesian framework. Prior distributions
have been proposed and the posterior distribution is
simulated through the Metropolis Hasting (MH) algorithm
based on the Monte Carlo Markov Chain (MCMC) me-
thod.
In the next section, the theoretical development of the
Bayesian method of parameter estimation is discussed. A
case study is then presented in Section 3. A comparison
between of the proposed Bayesian approach with classical
estimation methods such as the method of moments and
the maximum likelihood method is presented in the third
section. The last section corresponds to the conclusions
and recommendations for future work.
2. Bayesian GEV-B-Spline Model
2.1. GEV Distribution
The extreme value theory introduced by [22] shows that
the limiting distribution of the maximum is one of the
following distributions: Gumbel, Frechet or Weibull.
These three distributions can be grouped in a single Gen-
eralized Extreme Value (GEV) distribution:


1
, 0
, 0









,,, exp 1
,, , exp exp
y
Fy
y
Fy









(1)
The Gumbel distribution has two parameters defined
on R, the distribution function is obtained by tendering
0
.
The Fréchet distribution has three parameters defined
on the interval ,


 0
, obtained for
The Weibull distribution has three parameters defined
on the interval
.
,

 
 0
, obtained for
.
Considering a random variable following the GEV
distribution and t the time before the event T
Y.
Then is distributed according to a Geometric distribu-
tion with a parameter .
Y
y
t

pPY y
T
Let
12 n be i.i.d. random variables from the
GEV distribution. The probability that
,,,YY YΛ
0tkk



1
;1,2,3,
1
iT kT
ik
k
Pt k P Y y PY y k
pp

 



is
given by:
Ι

Λ
(2)
with:
1
TEt p

 
(3)
Since the variable Y follows the GEV distribution with
F as a repartition function. Equation (3) becomes:

11
1
TT
Et PY y F y


T
y

(4)
So the quantile of the GEV distribution is:
111
11log1
T
yF Et T

 

 

 

 


 
y
(5)
In the non-stationary case, the parameters of the GEV
are functions of time or other covariates. Consequently
the quantile T depends on these covariates. In the pre-
sent study, the parameters
an d
are supposed
constant. Let Y be a random variable that follows the
,,,
GEV , ,

, and
x
X
12 p
XX XΛ a vector of
covariates. The location parameter of the GEV model is a
function of covariates:

 

1
11 2 2
p
xii
i
p
p
fX
f
XfX fX
 
Λ
(6)
where i
f
represents the function that describes the rela-
tionship between the parameter and the covariate Xi.
In the classical GEV model with covariates, depend-
ence is represented through polynomial functions of lin-
ear or quadratic forms. In the following paragraph, we
present the GEV model with covariates where the de-
pendence structure is given by B-Splines. This model
will be called GEV-B-Spline.
2.2. The GEV-B-Spline Model
The function i
f
can be decomposed in the form of basic
spline functions:
Copyright © 2013 SciRes. OJS
B. NASRI ET AL.
120

ii
j

0,
1
m
jjdi
f
x Bx

(7)
where
 

1
r 0, 2
j
xx
Bx
j m


,,
1
1; 1
11
1
,0
for 0, 2
1, if fo
0, other wise
jd jd
jd j
jd
jd
jd j
jj
j
Bx B x
xx
xx
xx
jmd
xxx
Bx


 


(8)
,jd is a polynomial of degree d on each interval and
m is the number of control points.
Bx
Hence, the Equation (6) becomes

,0
11
xii i
ii

, ,
1
pp
m
ijjdi
j
f
xBx



 



1,0 1,1
,,1
1,
,,
pm
ppm
ppm
x

  
 
 
 
 
ΜΜ



 (9)
The matrix form of Equations (8) and (9) gives


0
,0 ,1
1,1 1 ,
,,
p
pm
Bx B





ββ
BΛ
(10)

0
1
x
j

β
B
β
(11)
where 1 is the unit vector of size p.
2.3. The GEV-B-Spline Model in Bayesian
Framework
The GEV-B-Spline is considered in a fully Bayesian
framework. For a given parameter prior distribution,

π
, the Bayes theorem allows to define the posterior
distribution:
 


π
fy
fy fy


0
,, ,,, where
(12)
where
t
  

 


ββ
is the vector of the parameters, and 0
β
and
β
are the
hyper-parameters of the location parameter.
[23] proposed the Beta
6, 9 distribution as a prior
distribution for the shape parameter of a stationary GEV
model, in order to avoid irrational estimations of the shape
parameter. In the present study, we considered an equi-
valent prior for the shape parameter; it is the normal dis-
tribution with mean 0.1 and variance 0.12.
[11] adopted this prior distribution for the GEV model
with covariates with polynomial dependence. Other stud-
ies have suggested adopting the normal distribution to
model the hyper parameters of the location parameter for
the GEV model with covariates and B-Spline dependence
[21,24].
0, NI
(13)
β
For the scale parameter, we used a non informative
prior distribution 1/σ
The posterior distribution of
is written as follows:




  

π
ππ*π
fy
fy fy
fy
fy



β
(14)
then






11
0
1
0
2
2
2
2
2
1*
11
1*
exp 1
1
2πdet exp 22π
0.1 1
exp 2*
j
j
k
yB
fy
yB





















































 


β
(15)
The posterior distribution

f
y
,,,,,.
is a function of
the hyperparameters 0

 
Considering a simple case of one covariate and m = 1
and d = 1, Equation (15) becomes:









11
1,0 1,1 1,1
1
1,0 1,1 1,1
22
1,0 1,1
2
2
2
2
*
11
*
exp 1
2πdet exp 2
0.1 1
exp *
2*
k
yBx
fy
yBx









































(16)
Copyright © 2013 SciRes. OJS
B. NASRI ET AL. 121
where
  
0,,
iii
x
, are computed using the inverse of
  
1
2, 0
32
*
xx B x
x
1
1,1 1,0
21
*
xx
Bx Bx
xx x


(17)
and
are the parameter set by the prior distribu-
tion. To estimate the above function, initial values of the
parameters ,,


then should be given in order to
simulate their joint posterior distribution by a MCMC al-
gorithm. The marginal distributions of the parameters
can be deduced by integrating Equation (15), with re-
spect to the rest of the parameter vector:

fy f

|*y f


N
0
N
(18)
The following section presents the details of the pro-
posed MCMC algorithm to estimate the GEV-B-Spline
parameter and quantile distributions.
2.4. MCMC Algorithm for the GEV-B-Spline
Model
The MCMC method constitutes an alternative to the nu-
merical methods, especially in Bayesian statistical analy-
sis. The basic idea of the MCMC method is, for each
parameter, to construct a Markov chain with the posterior
distribution being a stationary and ergodic distribution.
After running the Markov chain, of size , for a given
burn-in period , one obtains a sample from the poste-
rior distribution
π
x
15,000N
0
N
7000
. One popular method for con-
structing a Markov chain is via the Metropolis-Hastings
(MH) algorithm [25,26]. For the GML method, we si-
mulated realizations from the posterior distribution by
way of a single-component MH algorithm [27]. Each
parameter was updated using a random-walk Metropolis
algorithm with a Gaussian proposal density centered at the
current state of the chain. Some methods to assess the
convergence of the MCMC methods make it possible to
determine the length of the chain and the burn-in time
such as Raftery & Lewis and Subsampling methods [28].
In all cases, the convergence methods indicated that the
Markov chains converged within a few iterations. In this
study, we considered chains of size and a
burn-in period of runs. In every case, a sam-
ple of 0 values is collected from the pos-
terior of each of the elements of
8000
NN
. The GML corre-
sponds to the mode of the empirical posterior distribution
obtained from the values generated by the
MCMC algorithm.
0
NN
The MCMC algorithm produces also the conditional
quantile distribution for an observed value 0, of the co-
variate t. Indeed, for each iteration i of the MCMC
algorithm , the quantiles with non-exceedance
y
Y
1, ,iNΛ
1p

0
,
i
px
probability ,
vector
the cumulative distribution function of the GEV distribu-
tion:
x
corresponding to the parameter
 




00
,1log1
i
ii
px x i
yp
  (19)

0
i
y
where
is the position parameter conditional on the
particular value 0
x
of t
X
. Several statistical character-
istics of the conditional quantile distribution can be de-
termined from the values

00
,,,,
i
px
x
iN NΛ
u
, such as the
mean, the mode or the confidence interval. Principal step
of the MH algorithm can be summarized as follows [13]:
1) Choose a proposal distribution q;
from 2) Given the current state u, generate
qu
;
with probability
u
3) Accept
 



π
,min1,
π
uquu
uu uqu u

.

3. Case Study
3.1. Dataset
The proposed model is considered to model the maximum
annual rainfall (MAR) at Randsburg station (047253),
California for the period of 1938-2007. The Randsburg
station is located in the south east of the state of Califor-
nia
N,11735.37 .65 W
οο
. Figure 1 illustrates the geo-
graphic location of the Randsburg station. Figure 2
120.0
°
W
California
*047253
Figure 1. Geographic location of the Randsburg station.
Copyright © 2013 SciRes. OJS
B. NASRI ET AL.
Copyright © 2013 SciRes. OJS
122
1930 1940 1950 1960 1970 1980 1990 2000 2010
0
5
10
15
20
25
30
35
40
45
Years
Rainfall (mm)
Figure 2. Variation of maximum annual rainfall.
shows the 70-year variation of MAR at Randsburg Sta-
tion.
We consider the 70-year Southern Oscillation Index
(SOI) and Pacific Decadal Oscillation (PDO) time series
as covariates for MAR non-stationary quantile estimation.
The SOI and PDO describe the pressure and temperature
anomalies over the Pacific Ocean and have a clear impact
on water systems in North America [14,29]. By using SOI
and PDO as covariates in estimating the parameters of the
GEV-B-Spline model, we will take into account the effect
of multiannual climate fluctuations on extreme rainfall
events. We first apply the Mann Kendall test to examine
the existence of non-stationarity (Trend) in MAR time
series. The result shows that the MAR is not stationary at
1% significant level. The Spearman’s rho correlation
coefficient between the covariates and MAR is 0.52 and
0.51 for SOI and PDO respectively. These values are
significant at the 5% level. Figure 3 shows the variation
of maximum annual rainfall against SOI and PDO.
3.2. Model Development
For model development, the following function is first
fitted:
GEV-B-Spline
 


12
MAR GEV SOI PDO , ,ff


12
,
f
f are independent spline functions, for which the
degree and the number of nodes should be determined. In
this application the number of nodes and the degree of the
function are both chosen to take the value 3.
Table 1 shows the GEV-Spline parameters fitted to
SOI and PDO time series using a Bayesian method. Fig-
ures 4 and 5 show the estimated 2, 20 and 50-year return
period maximum rainfall quantiles as function of the
covariates (SOI and PDO). It can be seen that, generally,
the SOI has a negative correlation with precipitation,
while PDO is positively correlated with precipitation.
The negative values of SOI (e.g. El Nino phase) and
positive values of PDO (Warm Phase of PDO) coincide
with the relatively high MAR observations. MAR quan-
tiles increase slowly with increasing PDO values and
then increase exponentially for PDO values greater than
1. On the other hand, different inflexion points, in the
relationship between SOI and MAR are observed (for
example at SOI = 1.5, SOI = 0 and SOI = 1.5), indicat-
ing a more complex relationship between SOI and MAR
than between PDO and MAR.
4. Parameter Estimation Comparison
In this section, we propose a comparison of the Bay-
esian parameter estimation method for the GEV-B-Spline
model (BAYES) and other estimation methods such as
the conventional method of moments (MM) and the me-
thod of maximum likelihood (ML). The theoretical back-
grounds of these two methods for the GEV-B-Spline
model are presented in Appendices 1 and 2, respectively.
The comparison of these methods is carried out based on
a simulation of MAR-SOI relationship only. The quantile
with a non-exceedance probability 1 p is computed for
the maximum SOI using the parameters given by the
Bayesian method (Table 1). The objective is to compare
the quantile estimation methods for the quantiles esti-
mated from 1000 samples of size generated
from each estimation method. The parameter values cho-
sen for simulation are
70n
01234
44.8, 2.6, 81.6, 53.6, 54.4 ;
7.2; 0.17 .


  

B. NASRI ET AL. 123
-2.5 -2 -1 .5 -1 -0.5 00.5 11.5 22.5
0
10
20
30
40
50
SOI
-3 -2 -1 0 1 2 3
0
10
20
30
40
50
PDO
Rainfall (mm)
Figure 3. Annual maximum rainfall against SOI and PDO index.
Table 1. Bayesian estimation of the parameters of the mo-
del.
Climate Index
Parameter
SOI PDO
0
114.728 11.355
1
48.112 44.616
2
147.695 0
3
116.138 4.556
4
158.933 7.932
5
0 30.110
5.678 7.566
0.124 0.145
The comparison is carried out using the bias and the
root mean square error (RMSE) of quantile estimations at
non-exceedance probabilities, 1 p = 0.5, 0.8, 0.9, 0.99
corresponding to return periods of 2, 5, 10, 100. The re-
sults are given in Table 2.
Results show that the Bayesian estimation for the
GEV-B-Spline model in all cases represents the best re-
sults. However, this estimation method requires large
time-consuming numerical calculations and does not meet
a convergence point easily. For our case, the MCMC me-
thod details, such as the choice of numerical method
burning period and number of iterations are the key points
to the convergence of the MCMC algorithm. On the other
hand, even if the method of moments is the easiest method
to implement, the corresponding results are largely un-
satisfactory. The method of maximum likelihood, how-
ever, is a compromise between the other two methods. It is
interesting to note that for the case of low return periods,
i.e. 2, 5TT10 and T
years, the maximum like-
lihood method gives almost comparable results with the
Bayesian estimation. However, the error of the ML me-
thod increases rapidly with the increase in the return pe-
riod and the method becomes increasingly less effective.
Therefore, the Bayesian method leads to a superior per-
formance for the estimation of the extreme rainfall quan-
tiles for all return periods. The Bayesian method offers
also a general framework to combine observed and sub-
jective information and the possibility to estimate the
entire predictive distribution of the parameters and quan-
tiles.
5. Conclusions and Recommendations
Statistical risk assessment is of great importance in hy-
drology and many other fields of applied statistics. The
last two decades have witnessed the development of a
number of statistical modeling approaches for extreme
values in the presence of non-stationarity or dependence
on covariates. The GEV-B-Spline model which takes
into account the non-stationarity and nonlinearity offers a
great flexibility and takes into account the heavy tailed
character of the extreme distribution. The present study
roposes a Bayesian estimation framework of the GEV- p
Copyright © 2013 SciRes. OJS
B. NASRI ET AL.
124
-2 .5 -2 -1 .5 -1 -0 . 5 00.5 11.5 22.5
0
5
10
15
20
25
30
35
40
45
SOI
Data
T=2 years
T=50 years
T=20 years
Rainfall (mm)
Figure 4. GEV-B-Spline estimators of the 2, 20 and 50-year return period quantiles conditional upon SOI.
-3 -2 -1 0 1 2 3
0
10
20
30
40
50
60
PDO
Data
T=2 ye ar s
T=50 ye ar s
T=20 ye ar s
Rainfall (mm)
Figure 5. GEV-B-Spline estimators of the 2, 20 and 50-year return period quantiles conditional upon PDO.
Table 2. Comparison of estimation methods.
Probability BIAS RMSE
BAYES MM ML BAYES MM ML
0.5 0.020 0.075 0.052 0.403 0.715 0.435
0.8 0.060 0.094 0.090 0.418 0.901 0.514
0.9 0.148 0.249 0.177 0.450 1.847 1.525
0.99 0.182 0.655 0.448 0.826 3.128 2.879
B-spline model for hydro-meteorological variables. The
Bayesian approach is general, flexible and connected
with the decision theory. It combines observed and prior
information, estimates the entire posterior distribution of
the parameters and quantiles and thus allows to estimate
the credibility intervals.
Results of the simulated data show the advantage of
the proposed method for quantile estimation of an ex-
treme variable such as maximum rainfall especially for
high return period.
Copyright © 2013 SciRes. OJS
B. NASRI ET AL. 125
The evaluation for the quantile uncertainty using BIAS
and RMSE criteria also indicated the superiority of the
proposed method in comparison to other estimation me-
thods, especially for high return period quantile estima-
tion. However, the uncertainty of quantile estimation of
low return periods does not show a significant difference
between the bayesian and the maximum likelihood me-
thod. On the other hand, one can see that the numerical
calculation is the main disadvantage of these types of
models when the number of covariates increases which
may lead to divergence problem. The quantile regression
model can be a good alternative to overcome this prob-
lem [30,31]. Therefore, future work can focus on the
comparison of extreme value models with regression
quantiles in order to use different covariates in quantile
estimations.
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Appendix 1: GEV-B-Spline Moment
Let Y be a random variable that follows a GEV distribu-
tion therefore:
GEV , ,
x
Y

(A1)
With
x
f
X

01
nn
ii
ii
is a parameter that depends on a
covariate X.
 
0ii
f
xBx


Bx


0ii
Bx
(A2)
B is a spline basis function.
Where

1
n
i
fx


(A3)
And thus

0xii
Bx
1
n
i
 


0
V , ,
(A4)
Then

1
GE
n
ii
i
ZY Bx

 
(A5)
The following equations are used to estimate the pa-
rameters 0,,
:
2
2.9554
cc


ˆ7.8590
(A6)
2
ˆ
ˆ
ˆ
1
l
ˆ
12

(A7)

01
ˆ
ˆ
l
ˆ
11

  (A8)
With







3
3
12
10
1
2
log 2
2,
3log3
,
12
6
12
1,
21
1
nii
i
i
n
j
j
ct
y
cy n
tl
ii
cnn n
lb y
n
j
ly
nn






21
1
6
1
1
nn
ii
jj
i
nn
y
n


(A9)
The other i
Appendix 2: GEV-B-Spline Maximum
Likelihood
Let Y be a random variable that follows a GEV distribu-
tion therefore:
values are estimated using the linear
regression between Y and the basis matrix B of the
B-spline corresponding to the covariates.

GEV , ,
x
Y

(A10)
f
With xX
  
0
01
nn
ii ii
ii
is a parameter that depends on a
covariate X.
xBx Bx




f
(A11)
B is a spline basis function.
The maximum likelihood (ML) function is written as
1
1
1
1
1
1
1
,,
1exp 1
1
1exp exp exp
nx
n
tt
t
tt
n
tt tt
tn
Ly
y
y
yy



 

























  
 


 

 
 

 

n0
(A12)
1 is the number of observations when
0
.
, the log-likelihood function is: In the case of

1
1
1
;,,
log 1
1
1log1
nx
n
tt
t
n
tt
t
ly
y
n
y





  



 
 






1, 1mp
(A13)
The ML estimators are the solution of an equation
system formed by setting to zero the partial derivates of
(A13) with respect to each parameter.
In the case of one covariate and , we have
4 parameters to estimate
01
,,,






1
01,1 1
1
01,1 1
1
;,,
log 1
1
1log1
nx
ntt
t
ntt
t
ly
yBx
n
yBx






 

 


 


 
 
 

01
,,,
(A14)
The ML estimators of the parameter

are
the solution of the following system:
Copyright © 2013 SciRes. OJS
B. NASRI ET AL.
Copyright © 2013 SciRes. OJS
128
 




1
1
01,1
1
1
01,1 1
1
1
1
0
1
1,1
1
1
0ln1
1
00
1
00
1
0
ntt
nt
tt
tt
ntt
nt
tt
n
nt
tt
nt
t
t
yB
lw
ww w
yBx
lw
nw
lw
w
lw
Bx w




 

  





 



 




 

 
1
0
n
t
10
x












(A15)
where
1
1 1
t
x
01
,
1t
t
yB
w








 (A16)
Numerical methods must be used to solve the system
(A15). In the present study we used the Newton Raphson
method.
... L'introduction des covariables peut être effectuée au niveau de n'importe quel paramètre ou même à deux ou à trois paramètres à la fois dans la loi de probabilité choisie (dans notre cas c'est la loi GEV ou GPD). L'effet d'une covariable sur la variable d'intérêt peut être pris en compte dans une forme polynomiale (linéaire ou quadratique) [Coles, 2001;Ouarda & Adlouni, 2011;Cannon, 2010] ou dans d'autres formes non paramétriques (Splines, etc) [Chavez-Demoulin & Davison, 2005;Nasri et al., 2013]. ...
... The link between these parameters and the covariates can be linear [e.g., Coles, 2001;Cannon, 2010] or nonlinear [e.g., Chavez-Demoulin & Davison, 2005;Neville et al., 2011;Nasri et al., 2013Nasri et al., , 2016. ...
... where α X can be a linear [e.g., Coles, 2001;Cannon, 2010] or a nonlinear [e.g., Chavez-Demoulin & Davison, 2005;Neville et al., 2011;Nasri et al., 2013Nasri et al., , 2016 function of the covariables X. ...
... Several tests are used to detect non-stationarity in time series including the KPSS test (Kwiatkowski et al., 1992), the Leybourne-McCabe test (Leybourne and McCabe, 1994) and the Mann Kendall test (Mann, 1945). The last one is the most commonly used one in hydro-climatological studies (e.g., Dry and Wood, 2005;Cunderlik and Burn, 2002;Cunderlik and Ouarda, 2009;Nasri et al., 2013;Fiala et al., 2010;Khaliq et al., 2009). The frequency analysis of a non-stationary series calls for a different understanding than the conventional approach involving stationarity. ...
... Introducing covariates in one of these distributions can be done through any parameter. The effect of a covariate can be modeled by making one or more parameter linearly (e.g., Coles, 2001;Cannon, 2010) or nonlinearly (e.g., Chavez-Demoulin and Davison, 2005;Nasri et al., 2013;Neville et al., 2011) dependent on the covariate. The covariate method is largely developed and has been used in the literature to understand the variation in hydrological time series. ...
... Also can provides, in some cases, some convergence problem (ii) the interactions of the predictors make the model much more complicated because it requires multivariate function modelling. For this reason, some recent studies (e.g., Nasri et al., 2013Nasri et al., , 2016 suggest to use the quantile regression method, which was introduced by (Koenker and Bassett, 1987). Quantile regression provides the conditional quantiles of the response variable for a fixed value of covariates rather than only the conditional mean. ...
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... In nonstationary models, distribution parameters are made conditional on time-dependent covariates. The relations between distribution parameters and covariates can take the form of simple linear combinations (El Adlouni et al. 2007; El Adlouni and Ouarda 2009) or more complex models such as B-splines (Nasri et al. 2013;Thiombiano et al. 2017). When the POI-GP model is adopted, usually, the scale parameter (σ) of the GP is made conditional on covariates, the shape parameter (κ) of the GP is kept constant, and the rate parameter (λ) of the POI is made conditional on covariates (Kyselý et al. 2010;Mondal and Mujumdar 2015;Thiombiano et al. 2018). ...
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... It is a parametric curve derived from a linear Nonstationary frequency analysis of extreme daily precipitation combination of polynomial pieces, also called basis splines of degree d with k knots. B-spline functions have the advantages to be more rigorous and flexible and allow representing nonlinear dependence structures (De Boor 2001;Nasri et al. 2013). In the present study, the GPD model with B-splines is considered to model the effect of covariates on precipitation extremes. ...
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... In nonstationary models, distribution parameters are made conditional on time-dependent covariates. The relations between distribution parameters and covariates can take the form of simple linear combinations (El Adlouni et al. 2007; El Adlouni and Ouarda 2009) or more complex models such as B-splines (Nasri et al. 2013;Thiombiano et al. 2017). When the POI-GP model is adopted, usually, the scale parameter (σ) of the GP is made conditional on covariates, the shape parameter (κ) of the GP is kept constant, and the rate parameter (λ) of the POI is made conditional on covariates (Kyselý et al. 2010;Mondal and Mujumdar 2015;Thiombiano et al. 2018). ...
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The Middle East can experience extended wintertime spells of exceptionally hot weather, which can result in prolonged droughts and have major impacts on the already scarce water resources of the region. Recent observational studies point at increasing trends in mean and extreme temperatures in the Middle East, while climate projections seem to indicate that, in a warming weather scenario, the frequency, intensity and duration of warm spells will increase. The nonstationary warm spell frequency analysis approach proposed herein allows considering both climate variability through global climatic oscillations and climate change signals. In this study, statistical distributions with parameters conditional on covariates representing time, to account for temporal trend, and climate indices are used to predict the frequency, duration and intensity of wintertime warm spells in the Middle East. Such models could find a large applicability in various fields of climate research, and in particular in the seasonal prediction of warm spell severity. Based on previous studies linking atmospheric circulation patterns in the Atlantic to extreme temperatures in the Middle East, we use as covariates two classic modes of ‘fast’ and ‘slow’ climatic variability in the Atlantic Ocean (i.e., the Northern Atlantic Oscillation and the Atlantic Multidecadal Oscillation respectively). Results indicate that the use of covariates improves the goodness-of-fit of models for all warm spell characteristics.
... It is a parametric curve derived from a linear combination of polynomial pieces, also called basis splines of degree d with k knots. B-spline functions have the advantages to be more rigorous and flexible and allow representing nonlinear dependence structures (De Boor 2001;Nasri et al. 2013). In the present study, the GPD model with B-splines is considered to model the effect of covariates on precipitation extremes. ...
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... Log of raw moments (log E Y h λ and log E Y h 1 ) are computed for the given durations (λ) and different order of moment (h) The scaling exponent (θ) is estimated using the computed log of raw moments by the ordinary least square method. The GEV distribution is widely employed in the modeling of extremes [54][55][56][57][58][59][60][61][62][63][64]. Additionally, the GEV distribution has been used for modeling extreme rainfall events in many studies [65][66][67][68]. ...
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... See, e.g., Bouyé and Salmon [5] when d = 1. This relation between the conditional distribution of Y given X = x and the associated copula was used recently to propose conditional quantile estimators, as alternative to the quantile regression methods [11] or the parameter approach [6,15,16]. ...
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