The Gumbel logistic model for representing a multivariate stormevent

Sheng Yue *

Meteorological Services of Canada, Ontario Region, Environment Canada, 75 Farquhar Street, Guelph, Ontario, Canada N1H 3N4

Received 25 April 1999; received in revised form 5 May 2000; accepted 24 May 2000

Abstract

This study proposes the use of the Gumbel logistic model, the bivariate extreme value distribution with Gumbel marginals, to

analyze the joint distribution of annual maximum storm peaks (maximum rainfall intensities) and the corresponding storm amounts

which are mutually correlated. Parameters of the distribution are estimated using the method of moments (MM). On the basis of the

marginal distributions of these random variables, the joint distribution, the conditional probability distribution, and the associated

return periods can be readily deduced. The model is veri®ed using observed daily rainfall data from the Tokushima meteorological

station of Tokushima Prefecture, Japan. Results show that the model is suitable for representing the joint distribution of correlated

storm peaks and amounts that are Gumbel distributed. Ó 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Storm frequency analysis; Storm peak; Storm amount; Gumbel distribution; Bivariate extreme value distribution; Joint

probability distribution; Marginal distribution; Conditional distribution

1. Introduction

In recent years, it has been recognized that a hydro-logical extreme event such as a storm event appears tobe a multiple episodic event that might be characterizedby a few correlated variables such as storm maximumintensity (peak) and storm total amount. The severity ofthe storm event is a function of both storm peak andstorm total amount. However, storm frequency analysishas often centered on storm peak analysis only. Thissingle-variable storm frequency analysis provides alimited assessment of storm events. Many hydrologicalengineering planning, design, and management prob-lems require a more complete description of a stormevent, i.e., the joint distribution as well as the condi-tional distribution of these two correlated variables.Some meaningful works on the joint distribution withregard to storm events have been done by Hashino [14],Singh and Singh [18], Bacchi et al. [1], Yue [23], etc.Hashino [14] utilized the Freund bivariate exponentialdistribution [6] to represent the joint probability distri-bution of rainfall intensities and the correspondingmaximum storm surges at the Osaka Bay, Japan. Singh

and Singh [18] derived a bivariate probability densityfunction with exponential marginals to describe the jointdistribution of rainfall intensities and the correspondingdepths. Bacchi et al. [1] proposed another bivariate ex-ponential model with exponential marginals to analyzethe joint distribution of rainfall intensities and dura-tions. Yue [23] explored the use of the bivariate normalmodel for representing the joint statistical properties ofnon-normally distributed storm events that can be nor-malized by a transformation technique such as the Box±Cox approach [2].

In practice, extreme events such as maximum stormpeak and the corresponding storm amount might berepresented by the Gumbel distribution [3,8]. Thus, it isuseful to develop a bivariate extreme value distributionfor representing two correlated Gumbel distributedrandom variables. Theoretically, two bivariate extremevalue distributions with Gumbel marginals, namely theGumbel mixed model and the Gumbel logistic model,have been studied by Gumbel [9±11], Gumbel andMusta® [12], and Oliveira [16]. However, these modelshave been seldom applied to analyze joint statisticalproperties of correlated hydrological extreme events.The recent work by Yue et al. [22] investigated thesuitability of the Gumbel mixed model [9] for describinga multivariate ¯ood event. But the limitation of this

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Advances in Water Resources 24 (2001) 179±185

* Fax: +1-519-826-2083.

E-mail address: sheng.yue@ec.gc.ca (S. Yue).

0309-1708/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved.

PII: S 0 3 0 9 - 1 7 0 8 ( 0 0 ) 0 0 0 3 9 - 7

model is that the correlation coe�cient between tworandom variables must be in the range 06q6 2=3:

This study makes another attempt to use the Gumbellogistic model, the bivariate extreme value distributionmodel with Gumbel marginals to represent the jointprobability distribution of positively correlated stormpeaks and amounts. On the basis of the marginal dis-tributions of these random variables, the joint distribu-tion, the conditional probability distribution, and theassociated return periods can be readily derived. Section3 presents a practical application, the use of the modelfor representing the joint distribution of annual stormpeak and the corresponding storm amount at theTokushima meteorological station in Japan. The resultsindicate the usefulness of the model for describing jointstatistical properties of two positively correlated randomvariables with Gumbel marginals.

2. The Gumbel logistic model

The Gumbel logistic model with standard Gumbelmarginal distributions was originally proposed byGumbel [10,11] as follows:

F �x; y� � expnÿ �� ÿ lnF �x��m � � ÿ lnF �y��m�1=m

o�16m�; �1�

where F �x� and F �y� are, respectively, the marginaldistributions of X and Y, and are expressed as

F �x� � exp � ÿ exp� ÿ x��; �2a�

F �y� � exp � ÿ exp� ÿ y��; �2b�and where m�m P 1� is the parameter describing theassociation between the two random variables X and Y.The estimator of m is given by (see [12,15])

m � 1����������������1ÿ q�p �06 q6 1�; �3�

where q is the product±moment correlation coe�cientestimated by

q � E �X ÿ lX ��Y ÿ lY �� �rX rY

; �4�

where (lX ; rX ) and (lY ; rY ) are the means and standarddeviations of X and Y, respectively.

When m � 1, the bivariate distribution splits into theproduct of the two marginal distributions, and becomes

F �x; y� � F �x�F �y�: �5�This represents the independent case and the prod-

uct±moment correlation coe�cient q is equal to zero.Let the marginal distributions of the two random

variables take the follow general forms:

F �x� � exp

�ÿ exp

�ÿ xÿ ux

ax

��; �6a�

F �y� � exp

�ÿ exp

�ÿ y ÿ uy

ay

��; �6b�

where (ux; ax) and (uy ; ay) are the location and scaleparameters of the Gumbel distributions of X and Y,respectively.

The joint probability density function (p.d.f.) is de-rived using Eq. (1) and is expressed as follows:

f �x; y� � o2F �x; y�oxoy

� F �x; y�axay

� eÿm�xÿux�

ax

�� e

ÿm�yÿuy �ay

��1ÿ2m�=m

� eÿm�xÿux�

ax

�"� e

ÿm�yÿuy �ay

�1=m

� mÿ 1

#

� eÿm xÿuxax�yÿuy

ay

� �: �7�

The joint cumulative distribution function (cdf) of therandom variables X and Y takes the same form asEq. (1) in which the marginal distributions are given byEqs. (6a) and (6b).

The conditional cdf FX jY �xjy� � Pr �X 6 xjY 6 y� of Xgiven Y 6 y is presented by

FX jY �x yj � � F �x; y�FY �y� � exp exp

ÿnÿ �y ÿ uy�=ay

�ÿ exp�� ÿ m�xÿ ux�=ax�

� expÿÿ m�y ÿ uy�=ay

��1m

o�8�

Similarly, the conditional probability distribution func-tion of Y given X 6 x can be expressed by an equivalentformula.

The return periods exceeding certain values of thevariables X and Y are represented as follows:

TX � 1

1ÿ F �x� �F �x� � Pr �X 6 x��; �9a�

TY � 1

1ÿ F �y� �F �y� � Pr �Y 6 y��: �9b�

On the basis of the same principle, the joint return pe-riod T �x; y� of X and Y associated with the event (X > x,Y > y, or X > x and Y > y, i.e., at least one value of xand y is exceeded) can be represented by

T �x; y� � 1

1ÿ F �x; y� �F �x; y� � Pr �X 6 x; Y 6 y��:

�9c�

180 S. Yue / Advances in Water Resources 24 (2001) 179±185

The conditional return period of X given Y 6 y andthe conditional return period of Y given X 6 x are, re-spectively, presented as follows:

TX jY � 1

1ÿ F �xjy� �F �xjy� � Pr �X 6 xjY 6 y��; �9d�

TY Xj � 1

1ÿ F �yjx� �F �yjx� � Pr �Y 6 yjX 6 x��: �9e�

3. Application

In order to demonstrate the applicability of theGumbel logistic model, 102-year daily rainfall data from1892 to 1996 (except the years 1896, 1965, and 1976)were made use of, which were observed at the Toku-shima meteorological observation station (TMOA) inJapan. The data from 1892 to 1990 were recorded in thebook by TMOA [19]. The data after 1990 were obtainedin the Monthly Meteorological Report by TMOA. Thecorrections concerning the changes in the station loca-tion, the measurement equipment, and the measurementprocedure have been made by TMOA [19].

3.1. De®nition of a storm sequence

A storm time series can be constructed using theannual maximum series (AMS) approach, or the partialduration series (PDS) (or peak over threshold (POT))approach. An AMS is constructed by selecting the an-nual maximum value of each year, i.e., only one eventper year is retained. This naturally leads to events thatare generally independently and identically distributed.The PDS consists of all values that exceed a certainthreshold. The main advantage of the PDS approach isthat it is not con®ned to only one event per year and itallows for more extreme events to be considered. Thekey unresolved problem of the PDS is how to selectappropriate thresholds. This issue has been addressed inthe work of Cunnane [4], Valadares Tavares and Ev-aristo Da Silva [20], Wang [21], and Rasmussen et al.[17]. Wang [21] has documented that the AMS and thePDS are similar for a long-term time series. In thepresent study, theAMS approach is employed toconstructa storm peak time series and the corresponding stormamount time series using the observed daily rainfalldata.

During the monsoon season in Japan, standing rainyfronts and hurricanes or typhoons causes the heavieststorm both in storm peak (maximum rainfall intensity)and storm amount. As this type of rain continues to fallover a few days, it is considered to be a storm event.Because recorded historical rainfall data is often in theform of averages over a period of one day, one storm isde®ned as continuous daily rainfalls, as shown in Fig. 1.

Let the storm peak I (mm/day), be the maximum dailyrainfall in a year selected by Eq. (10a) and the corre-sponding total rainfall amount A (mm) be given byEq. (10b).

In�maxfuni; i � 1; 2; . . . ; 365�366�g �n � 1; 2; . . . ; 102�;�10a�

An �XDn

s�1

uns �n � 1; 2; . . . ; 102�; �10b�

where uni is the ith daily rainfall amount (mm/day) forthe nth year, uns the sth daily rainfall amount of thestorm n corresponding to the storm peak In, and Dn isthe storm duration (day) of the storm n.

The maximum daily rainfall (storm peak In) for eachyear is ®rst selected using Eq. (10a) from the observed102-year daily rainfall data at the Tokushima meteoro-logical observation station; then the correspondingstorm duration Dn is counted and the storm amount An

is computed using Eq. (10b).

3.2. Marginal distributions of storm peaks and amounts

3.2.1. Empirical probabilitiesThe non-exceedance probability is estimated using

the Gringorton formula [5,7,13]

Pk � k ÿ 0:44

N � 0:12; �11�

where Pk is the cumulative frequency, the probabilitythat a given value is less than the kth smallest observa-tion in the data set of N observations.

3.2.2. Parameter estimationIn a single-variable frequency analysis domain, the

location and scale parameters of the Gumbel distribu-tion can be estimated using a few approaches such as themaximum likelihood method (ML) and the method ofmoments (MM). While for the analysis of the jointdistribution of two correlated Gumbel distributed

Fig. 1. Characteristic values of a storm event.

S. Yue / Advances in Water Resources 24 (2001) 179±185 181

random variables, if the ML can provide a reliable es-timate of the association parameter between the vari-ables remains unknown. This paper presents thesimplest way to derive the modelÕs parameters, i.e., tocompletely identify the modelÕs parameters through themarginal distributions of the variables. Gumbel andMusta® [12] proposed to estimate the association pa-rameter via the MM. This study follows their idea thatthe associated parameter is derived using Eqs. (3) and(4). The correlation coe�cient (q) between storm peaksand amounts is estimated by Eq. (4), and takes 0.731.The association parameter m between storm peaks andamounts is computed using Eq. (3), and is equal to1.928. In order to keep the consistency of the method-ology, the location and scale parameters are also esti-mated using the MM as follows:

a ����6p

pS; �12a�

u � M ÿ 0:577a; �12b�where M and S are the mean and standard deviation ofthe sample data, respectively. The estimated mean andstandard deviation of storm peaks and amounts fromthe sample data are listed in Table 1. The estimatedparameters of the Gumbel distribution are also pre-sented in Table 1.

The v2 test is executed to test the goodness of ®t of theGumbel distribution. The v2 test statistics are 7.65 forthe storm peak and 8.82 for the storm amount. Thecritical value of the v2 test v2

0:05�9� is 16.92. Thus, the nullhypothesis H0 that the underlying distributions of boththe storm peaks and amounts are Gumbel distributed isaccepted at the signi®cant level 0.05. The empiricalprobabilities computed by Eq. (11) and the theoreticalprobabilities calculated using Eqs. (6a) and (6b) for theobserved storm peaks and amounts are respectively il-lustrated in Figs. 2(a) and (b).

3.3. Statistics of the joint distribution of storm peaks (I)and amounts (A)

3.3.1. Validity of the proposed modelThe empirical non-exceedance joint probabilities of

storm peaks (I) and amounts (A) are estimated using anequivalent form of Eq. (11) in order to keep consistentwith the marginal case and is given by Yue et al. [22]

Pkj � Pr �I 6 ik;A6 aj� �Pk

m�1

Pjl�1 nml ÿ 0:44

N � 0:12; �13�

where N is the total number of observations �N � 102�,and nml is the number of occurrences of the combina-tions of im and al.

Theoretical joint probabilities of the real occurrencecombinations of ik and aj are estimated using Eq. (1).The empirical and theoretical joint probabilities ofstorm peaks and amounts are plotted in Fig. 3 in which

Table 1

Statistics of the storm peak (I) and amount (A)

Variables Statistics Parameters of the Gumbel distribution

M S u a

I (mm/day) 146.657 59.201 120.014 46.159

A (mm) 217.294 101.157 171.769 78.872

Fig. 2. (a) Distribution of storm peaks; (b) distribution of storm

amounts.

182 S. Yue / Advances in Water Resources 24 (2001) 179±185

the solid line represents the theoretical probabilities(arranged in ascending order), and the correspondingempirical probabilities are indicated by the plus sign.The x-axis is the corresponding order number of acombination of ik and aj. It is evident that no signi®cant

di�erence can be detected. Thus, it is concluded that themodel is suitable for representing the joint distributionof the correlated Gumbel distributed storm peaks andamounts.

3.3.2. Contours of the joint cdf F �i; a� and joint returnperiod T �i; a� of I and A

The joint cdf and joint return period correspondingto given I � �0; 10; 20; . . . ; 500� and A � �0; 20; 40; . . . ;800� are computed using Eqs. (1) and (9c) and are dis-played in Figs. 4(a) and (b), respectively. The corre-sponding contours of the joint cdf and the joint returnperiod of I and A are plotted in Figs. 4(c) and (d), re-spectively. These contours indicate that the proposedmethod can contribute meaningfully in solving severalproblems of hydrological engineering design and man-agement, for which single variable storm frequencyanalysis cannot provide answers. For example, given anoccurrence probability or a return period of a stormevent, we can obtain various occurrence combinationsof storm peaks and amounts, and vice versa. These re-sults are especially useful in the case that both stormpeak and amount must be considered for hydrologicalengineering design and management.

Fig. 4. (a) Joint cumulative distribution function of storm peaks and amounts; (b) joint return periods of storm peaks and amounts; (c) contours of

the joint cumulative distribution function of storm peaks and amounts; (d) contours of the joint return period of storm peaks and amounts.

Fig. 3. Comparison of empirical and theoretical probabilities of storm

peaks and amounts.

S. Yue / Advances in Water Resources 24 (2001) 179±185 183

3.3.3. Conditional return periodsThe conditional return period TI jA of storm peak I

given storm amount A is estimated based on Eq. (8) and(9d), and is shown in Fig. 5(a). Similarly, the conditionalreturn period TAjI of A given I is plotted in Fig. 5(b). Itcan be seen that the proposed method also allows one toobtain information concerning the occurrence returnperiods of the storm peak under the condition that agiven storm amount occurs, and vice versa.

For the purpose of comparison, the return period TI

of the storm peak I is computed using Eq. (9a) and isdisplayed by the dashed line in Fig. 5(a). In fact, it is theconditional return period TI jA of the storm peak I giventhe storm amount A when A!1. It can be seen thatgiven a return period (for example, 100 years), the cor-responding storm peak value obtained by single-variablefrequency analysis is greater than those obtained by thejoint distribution. This implies that if we ignore the closecorrelation between the storm peak and amount and

execute single-variable frequency analysis on storm peakonly, then the severity of the storm event will be over-estimated in the study case. If hydrological engineeringplanning, design, and management are based on thisstorm peak, then this overestimation will lead to in-creased cost.

Similarly, the return period TA of storm amount A iscomputed using Eq. (9b) and is presented by the dashedline in Fig. 5(b). We can obtain the same inference asthat of the storm peak case.

4. Conclusions

This study presents a methodology for using theGumbel logistic model to analyze the joint distributionsof two positively correlated extreme random variablesthat are Gumbel distributed. The model is used to de-velop the joint distribution of combinations of stormpeaks and amounts. Based on this model, if the marginaldistributions of two random variables can be repre-sented by the Gumbel distribution, one can readily ob-tain the joint probability distribution, the conditionaldistributions, and the associated return periods of thesevariables. The parameters of the model are easily esti-mated from the sample data based on the marginaldistributions of the random variables.

This approach is veri®ed using observed daily rainfalldata from the Tokushima meteorological station ofTokushima prefecture, Japan. A good agreement isobserved between the theoretical and empirical distri-butions. The results point out that the proposed methodprovides additional information which cannot be ob-tained by single variable storm frequency analysis, suchas the joint return periods, and the conditional returnperiods of these variables. These results also indicatethat the proposed model can contribute meaningfully insolving several problems of hydrological engineeringdesign and management. For example, given a stormevent return period, it is possible to obtain various oc-currence combinations of storm peaks and amounts,and vice versa. These various scenarios can be of greatusefulness in the analysis and assessment of the riskassociated with several hydrological problems.

Acknowledgements

The work had been accomplished when the authorwas in INRS-EAU, University of Quebec, Canada. Theauthor gratefully acknowledges the Natural Sciencesand Engineering Research Council (NSERC) of Can-ada, Hydro-Quebec, and INRS-EAU for partially sup-porting this study. The author also thanks Prof. M.Hashino of University of Tokushima in Japan for hiskind assistance concerning this research. The construc-

Fig. 5. (a) Conditional return period of storm peaks given storm

amount A6 a; (b) conditional return period of storm amounts given

storm peak I 6 i.

184 S. Yue / Advances in Water Resources 24 (2001) 179±185

tive comments of anonymous reviewers signi®cantlyimproved the quality of this paper.

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