Computation of Confidence Limits for the Two Populations Extreme Value Type I Distribution

Computation of Confidence Limits for the Two Populations Extreme Value Type I Distribution

Jose A. Raynal-Villasenor

Civil and Environmental Engineering Department, Universidad de las Americas Puebla

72820 Cholula, Puebla, Mexico

M. Elena Raynal-Gutierrez and

Department of Civil Engineering, Monterrey Institute of Technology Campus Puebla

Via Atlixcayotl # 2301 72453 Puebla, Puebla, Mexico

Abstract

A methodology for obtaining the confidence limits for the two populations extreme value type I distribution is presented. The methodology is based on the application of the maximum likelihood method for estimating the parameters of the distribution and the confidence limits of the design events. The confidence limits are obtained by using of the variance-covariance matrix of the parameters and assuming a normality of the design events to compute them. Given the complexity of the likelihood function, its logarithmic form is used and a non-linear multivariable constrained optimization method is applied to maximize such

distribution of extreme values, TCEV, [3], [4], [5]

and [6], the mixed Gumbel, [7], [8], [9], and the

mixed general of extreme value, [10] and [11]. [12] used different mixtures of normal, gamma and Gumbel distributions to test the relevance of using mixture models, by computing the marginal likelihoods of single distribution models, and to verify the presence of a persistence in the time series by comparing independent and identically distributed and Markovian mixture models.

2. The Extreme Value Type I Distribution Function

The extreme value type I distribution function, for the maxima, is[13]:

x x

function to produce the maximum likelihood

F (x) exp exp

01

estimators of the parameters and its confidence limits of the distribution. An example of application

1

(1)

of the proposed methodology is contained in the paper. The results showed an improvement in the standard error of fit and confidence limits narrower than those produced by the one

population procedure.

where and x01 are the scale and location parameters, respectively.

The probability density function is given by, [13]:

1 x x01

f (x)

exp exp

1. Introduction

1

1

The method of maximum likelihood has been

exp x x01

recognized like one of the best methods of estimation of parameters of functions of probability distribution, the properties of its estimators like asymptotic unbiasness and sufficiency, as well as

1

(2)

the consistency and efficiency, have been briefed frequently in technical literature, [1], [2], etc. This method also has the virtue of being able to handle very complex likelihood functions with an amazing flexibility. The use of mixed functions of probability to fit samples coming from two or more populations has been proposed from time back, [1]. In the particular case of the functions of

where: – x and 0

1. Two Populations Extreme Value Type I Distribution Function

   Based in the general form for distribution function for two populations, [1] proposed:

   F(x)mix (1 p)F(x;1) pF(x;2 )

   (3)

   LnL(x; x01,1 , x02 , 2 , p)

   where p is the proportion of the second population

   1 p

   exp

   x x 01

   in the sample.

   The two populations extreme value type I

   N Ln 1

   1

   x x

   distribution function can be expressed as:

   i1

   e xp exp

   01

   (x x )

   1

   F (x)mix

   (1 p) exp exp 01

   p x x x x

   1

   exp 02 exp exp 02

   (x x

   2 2

   2

   p exp exp 01

   (8)

   2

   (4)

   In the proposed procedure, eq. (8) has been

   and the corresponding probability density function is:

   maximized directly by using the well-known non- linear mutlivariable constrained Rosenbrock optimization method, [14].

   f (x)

   1 pexp x x01

   mix

   1

   1

   5. Estimation of the Confidence Limits for

   x x

   the Two Populations Extreme Value Type

   exp exp

   p x x02

   01

   1

   x x02

   I Distribution

   The variance-covariance matrix for the two populations extreme value type I distributions, may

   be expressed as:

   exp exp exp

   2 2

   2

   Varx Covx ,

   Covx , x

   Covx , Covx , p

   01 01 1

   01 02

   01 2 01

   (5) Covx , Var Cov , x Cov , Cov , p

   01 1 1 1 02 1 2 1

2. The Method of Maximum Likelihood

   V Covx01, x02

   Covx ,

   Cov1, x02 Cov ,

   Varx02 Covx ,

   Covx02,2

   Var

   Covx02, p Cov , p

   The likelihood function for N independent and

   identically distributed X1, X2,…, Xn can be obtained as the joint probability density function,

   01 2

   Covx01, p

   1 2 02 2 2 2

   Cov1, p Covx02, p Cov2 , p Varp

   (9)

   that is, [1]:

   N

   and its elements are:

   L(x, ) f (x )

   2 LL

   E 2

   2 2 2

   E LL

   E LL

   E LL

   E LL

   E LL

   E LL

   2 1

   E LL

   E LL

   i

   i1

   (6)

   x01

   2 LL

   x011 x01x02 x012

   2 LL 2 LL 2 LL

   x01p

   2 LL

   E x

   E 2

   E x

   E

   E p

   01 1 1

   1 02 1 2

   1

   2 LL 2 LL

   2 LL 2 LL

   2 LL

   where is the parameter vector and f (.) is the

   V E x x

   E x

   E x2

   E x

   E x p

   01 02

   1 02

   02

   02 2 02

   probability density function.

   2 LL

   2 LL

   2 LL

   2 LL 2 LL

   E x

   E

   E x

   E 2 E p

   The logarithmic version of the former equation is:

   01 2

   1 2

   02 2

   2

   2

   2 LL

   2 LL

   2 LL

   2 LL

   2 LL

   E x p

   E p

   E x p

   E p

   E p2

   N

   N

   LnL(x, )

   Lnf (x)

   01 1 02 2

   (10)

   i1

   (7)

   The second partial derivatives, which expected values must be obtained to evaluate the variance-

   Based in the statements of the previous section, the

   logarithmic likelihood function of the two populations exreme value type I distribution is:

   covariance matrix for the two populations extreme value type I distribution, are:

   x x01 x x02

   F x;1 exp

   F x; 2 exp

   x x

   1

   x x02

   2

   1 exp 01

   LnL

   1 p N

   1

   x x02

   x

   2

   DEN

   LnL

   1 p

   exp

   N

   N

   1

   01 1

   i1

   2

   x x

   (11)

   2

   2

   2 i1

   DEN

   F x;1 exp

   01

   (14)

   x x

   1

   LnL

   N f x;1 f x;2

   01

   x x

   p i1

   DEN

   (15)

   exp

   N

   N

   01 1

   LnL 1 p

   1

   where:

   1

   2

   1 i1

   DEN

   DEN = f(x)mix (16)

   x x

   F (x;1 ) exp exp 01

   1

   x x

   (17)

   (12) F (x;2 ) exp exp

   02

   x x02

   2

   (18)

   F x; 2 exp

   1 x x

   2

   f (x,1 )

   exp exp

   01

   x x

   1

   1

   1 exp 02

   LnL

   1 p N

   2

   x x01

   x02

   2

   2 i1

   DEN

   exp

   1

   1

   x x

   (19)

   f (x, 2 )

   exp exp 02

   (13)

   2

   exp

   x x

   2

   02

   2

   (20)

   1. Results and Discussion

      The gauging station Jaina in the state of Sinaloa, located in Northwestern Mexico, with period of record (1941-1991), has been chosen to show the procedure for obtaining the values of the confidence limits based in the two populations extreme value type I distribution. The initial values required by the procedure were estimated by using the computer code FLODRO 4.0 in [15]. The design values and their confidence limits for the one and two populations approaches are shown in Tables 1 and 2, respectively, and were obtained by using the computer code contained in [16].

      Table 1. One population design values and their confidence limits for gauging station Jaina, Sin

      (SE = 371.42)

      (1) (2) (3) (4) (5)

      Figure 1. Empirical and One Population Theoretical Probability Distribution Function

      and Confidence Limits for Gauging Station Jaina, Mexico

      The standard error of fitting (SE) has been computed as, [17]:

      N

      N

      5	821	1002	1184	363
      10	1044	1277	1509	465
      20	1256	1541	1825	569
      50	1528	1881	2235	707
      100	1731	2137	2543	812

      5	821	1002	1184	363
      10	1044	1277	1509	465
      20	1256	1541	1825	569
      50	1528	1881	2235	707
      100	1731	2137	2543	812

      (xi yi) 2

      1 / 2

      SE i 1

      1. Return Period (years)

      2. Lower Limit (m3/s)

      3. Design Value (m3/s)

      4. Upper Limit (m3/s)

      5. Interval Width between Confidence Limits (m3/s)

         Table 2. Two populations design values and their confidence limits for gauging station Jaina, Sin (SE = 276.15)

         (N mj)

         (21)

         (1)	(2)	(3)	(4)	(5)
         5	886	1007	1129	243
         10	1345	1498	1651	306
         20	1756	1938	2120	364
         50	2261	2479	2698	437

         100 2631 2876 3121 490

         1. Return Period (years)

         2. Lower Limit (m3/s)

         3. Design Value (m3/s)

         4. Upper Limit (m3/s)

         5. Interval Width between Confidence Limits (m3/s)

         A graphical representation of these results can be observed in figures 1 and 2, for the one population approach and for the two populations model, respectively.

         Figure 2. Empirical and Two Populations Theoretical Probability Distribution Function

         and Confidence Limits for Gauging Station Jaina, Mexico

         The Gumbels reduced variate, required to produce the abscissa axis in graphical displays of flood data, models applied and its confidence limits, is obtained as follows:

         y = – Ln(-Ln(1-1/Tr)) (22)

         where Tr is the return period in years.

         It is observed that the two populations model fits the flood sample much better (SE = 276.15) compared with the one population model (SE= 371.42). The two populations model produced a narrower confidence limits, too.

         The application of the proposed approach is restricted to the fact that the computer code for the Rosenbrocks constrained multivariable method

         must be available, given that performing the required computations for such method without a computer code is just out of the question.

   2. Conclusions

      A procedure for the obtaining of the confidence limits for the two populations extreme value type I distribution has been described here, based on the method of maximum likelihood. The procedure has given good results so far with the samples of data analyzed until now, one of which was used as an example of application of the proposed methodology. It can be observed that in this example of application, the standard error of fitting has been reduced significantly and the width of the confidence limits was reduced, too. Based on this arguments, the authors recommend the procedure here shown as an effective tool for annual flood frequency analysis when two populations are detected within a sample of flood data.

   3. Acknowledgements

      The authors wishes thank to the Universidad de las Americas Puebla for the support granted to make this publication possible.

   4. References

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2. C. T. Haan, Statistical methods in hydrology, The Iowa State University Press, Ames, Iowa. 1977

3. E. J. Gumbel, Statistics of extremes, Columbia University Press, New York, N Y. 1958

4. P. Todorovic and J. Rousselle Some problems of flood analysis, Wat. Resour. Res., 7(5):1144-1150, 1971

5. R. V. Canfield, The Distribution of the extremes of a mixture of random variable with applications to hydrology, in Input for Risk Analysis in Water Systems, E A McBean, K W Hipel T E Unny, eds. , 77-84, Water Resources Publications, Littleton, Colorado. 1979

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   D.F., Mexico (In Spanish). 1970

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    parameters of the mixed general extreme value distribution, Proc. IX National Congress on Hydraulics,

    Queretaro, Qro., Mexico, 79-90. (In Spanish). 1986

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12. G. Evin, J. Merleau, and L. Perreault, Two components mixtures of normal, gamma and Gumbel distributions for hydrological applications, Wat. Resour. Res., 47, 2011, Art. No: W08525, DOI: 10.1029/2010WR010266.

13. Natural Environment Research Council (NERC)

    Flood studies report, vol. 1 Hydrological studies,

    Whitefriars LDT. London. UK. 1975

14. J. L.Kuester and J. H. Mize, Optimization with FORTRAN, 386-398, McGraw-Hill Book Co., New York. 1973

15. J. A. Raynal-Villasenor, Frequency Analysis of Hydrologic Extremes, Lulu.com. 2010

16. M. E. Raynal-Gutierrez, An interactive computer package for the estimation of the confidence limits for the two populations Gumbel distribution, B. Sc. Thesis, Department of Civil Engineering, Universidad de las Americas Puebla, Puebla, Mexico. (In Spanish). 2001

17. G. W. Kite, Frequency and risk analyses in hydrology, Water Resources Publications, Littleton, Colorado. 1988