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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

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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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B. NASRI ET AL. 127

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.