Estimating distribution parameters using optimization techniques

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1 Hydrological Sciences -Journal- des Sciences Hydrologiques,39,4, August Estimating distribution parameters using optimization techniques INTRODUCTION FANG XIN YU & BABAK NAGHAVI Louisiana Transportation Research Center, 4101 Gourrier Ave., Baton Rouge, LA 70808, USA Abstract An improved parameter estimation procedure has been developed by using optimization techniques and applied to estimate the parameters of the log-pearson type 3 (LP3) distribution. As a result, an improved estimation method was found. The new methods estimates the mean and the standard deviation of the log-transformed data by the method of moments and estimates the coefficient of skewness by minimizing both the relative root average square error (RRASE) and the relative average bias (RAB). Monte Carlo simulation was conducted for four selected LP3 populations. As compared with the method of moments, larger reductions in standard root mean square error (SRMSE) and standard bias (SBIAS) for quantile prediction can be achieved by the new method for small sample sizes and large return periods of quantités. In addition, the new method can always fit the observed data better than the method of moments. Utilisation de techniques d'optimisation pour l'estimation des paramètres d'une distribution Résumé Une procédure d'estimation des paramètres d'une distribution utilisant des techniques d'optimisation a été développée et appliquée à l'estimation des paramètres de la loi log-pearson III (LP3). Une méthode d'estimation améliorée a ainsi été définie. La nouvelle méthode estime la moyenne et l'écart-type des transformées logarithmiques des données par la méthode des moments et estime leur coefficient d'asymétrie en minimisant à la fois la racine de l'erreur quadratique moyenne relative et le biais moyen relatif. Des simulations de Monte Carlo ont été réalisées sur quatre populations suivant une loi log-pearson III. Comparée à la méthode des moments, la nouvelle méthode permet d'obtenir une importante réduction de la racine de l'erreur quadratique moyenne normalisée et du biais normalisé des quantiles estimés pour les échantillons de petite taille et les longs temps de retour. De plus, cette nouvelle méthode permet toujours d'obtenir un meilleur ajustement aux données empiriques que la méthode des moments. Accurate estimation of flood quantiles is needed for the cost-effective design of hydraulic structures. By conventional flood frequency analysis, one may evaluate the performances of several frequency distributions and parameter estimation methods by using the available data and select the best combination of distribution and estimation method for quantile prediction. The conventional procedure, however, does not warrant the best quantile prediction. As an Open for discussion until 1 February 1995

2 392 F. X. Yu& B. Naghavi example, Yu et al. (1993) have shown that the combination of the method of moments () and least squares can yield the best fit to observed data but predicts flood quantiles rather poorly as compared with three other popular methods. The US Water Resources Council (1967) recommended the log- Pearson type 3 (LP3) distribution along with the method of moments for parameter estimation for at-site frequency analysis. Many studies have been carried out to test whether is superior to alternative methods for estimating the parameters of the LP3 distribution (Bobee & Robitaille, 1977; Kuczera, 1982; Arora & Singh, 1989; Naghavi et al., 1991). No general consensus on the performance of a specific estimation method has been reached to date. An examination of past studies on parameter estimation indicated that if a method is found to perform well for a specific distribution by using Monte Carlo simulation, it may perform poorly by using observed data sets, and vice versa (Arora & Singh, 1989; Jain & Singh, 1987). The reasons for these seemingly contradictory results are: (a) the underlying population distribution is unknown for observed data; (b) an estimator that performs better for one distribution may not necessarily do so for another distribution; (c) an estimator may not perform uniformly better for different shapes of the (d) same distribution; and the performance indices for observed data and for Monte Carlo simulated data are usually not the same and may mislead the performance evaluation. The objectives of this study were to develop an improved parameter estimation procedure and to apply the proposed estimation procedure to at-site flood frequency analysis. DATA ANALYSIS Annual maximum flood data from 94 Louisiana stream gauges were obtained from the US Geological Survey. Stations either having less than years of records or having been regulated were excluded from the 94 gauging stations. Four gauging stations having drainage areas of less than 10 square miles and having record lengths of less than 30 years were eliminated from the data sets because observations from very small drainage areas with short periods of records are subject to large errors. The locations of the remaining 90 stations are shown in Fig. 1. The average record length for the 90 data sets was 36 years. The coefficient of variation of the original data varied from 0.29 to 0.71, and the coefficient of skewness from to In order to evaluate the performance of a parameter estimation method by Monte Carlo simulation, four LP3 populations were selected based on Louisiana stream records. The LP3 population parameters (Arora & Singh, 1989) and corresponding observation stations in Louisiana are listed in Table 1.

3 Estimating distribution parameters 393

4 394 F. X. Yu & B. Naghavi For each LP3 population, four sample sizes of,, and were considered. 0 samples for each sample size were drawn for each LP3 population. Table 1 Parameters for four selected LP3 populations Set no. Station no. My 7, CRITERIA FOR PERFORMANCE EVALUATION In terms of quantile prediction, the performance indices for Monte Carlo simulated data are usually the standard root mean square error (SRMSE) and the standard bias (SBIAS) (Arora & Singh, 1989): SRMSE = 1 mti m r x c (i) -x X 1 2 _ (1) SBIAS m 1 -E x(i) -x (2) where m is the number of samples of the same sample size n, x is the population quantile generated by use of the population parameters, and x c (i) is the quantile computed by using the parameters estimated from sample i of size n. The indices for observed data are usually the relative root average square error (RRASE) and the relative average bias (RAB) (Bobee & Robitaille, 1977): RRASE 1 2 (3) 1 " RAB = -J] «TIT x c (i) -x o (i) (4) where x 0 (i) is the ith largest annual maximum flood value at a gauging station.

5 Estimating distribution parameters 395 PROPOSED PARAMETER ESTIMATION PROCEDURE The conventional frequency analysis (CFA) procedure may be improved if optimization techniques are employed. The proposed estimation procedure builds on the CFA procedure and can be described as follows. First, the conventional frequency analysis procedure is applied to obtain the best combination of distribution and estimation method for the data used. Second, some or all parameters of the selected distribution are optimized by minimizing the selected objective function. Finally, flood quantiles are computed by using the set of optimal parameters. The procedure described above should be tested through Monte Carlo simulation before any actual application. Since the performance indices for the observed data sets are usually the RRASE and RAB, the objective function may logically be selected to minimize both the RRASE and the RAB: MINz = RRASE +1 RAB (5) The new parameter estimation procedure assumes that there exists a variety of parameter estimation methods for the same distribution and that the CFA procedure can yield a relatively good estimate of the distribution parameters. Figure 2 shows the scheme of the proposed estimation procedure and Fig. 3 illustrates the idea of the new estimation procedure. It is seen from Fig. 3 that ÂLYS0S TOO!» COLLECT & COMPILE DATA CONVENTIONAL ANALYSIS (Evaluate the performances of some popular combinations of distribution and method and select the best one to determine the initial parameters) PARAMETER OPTIMIZATION (Optimize parameters by minimizing selected performance indices) PREDICT FLOOD QUANTILES Fig. 2 Scheme of the new estimation procedure.

6 396 F. X. Yu & B. Naghavi some local minimum points of the objective function may not be physically acceptable. For example, if parameters p x and p 2 are restricted to be positive, points A and B in Fig. 3 are not acceptable. On the other hand, if none of the conventional methods can estimate the distribution parameters closely enough to the overall minimum point (say point C in Fig. 3), the new estimation method may not locate that overall minimum point either. However, the new estimation method can improve the parameter estimation from the best conventional method. For instance, if the method of moments is selected as the best estimation method for the selected distribution by the conventional frequency analysis procedure, the new estimation method can improve the parameter estimation via the by minimizing the selected performance indices (referring to point D in Fig. 3). Fig. 3 Illustration of the new estimation procedure ( = point estimated by the method of moments; OPT = point estimated by the optimization method).

7 Estimating distribution parameters 397 To minimize the objective function of equation (5), conjugate gradient optimization (CGO) was employed. A one-dimensional minimization subroutine developed by Yu & Singh (1993) was used to implement the linear search required by the CGO algorithm. The CGO algorithm has been described in many textbooks (Himmelblau, 1972; Press et al., 1986) and will not be discussed here. AN APPLICATION TO AT-SITE FLOOD FREQUENCY ANALYSIS Naghavi & Yu (1992) applied the conventional flood frequency analysis procedure to the 90 sets of Louisiana data by evaluating five frequently used distributions and three parameter estimation methods. The five distributions were: (1) two-parameter log-normal (LN02); (2) three-parameter log-normal (LN03); (3) Pearson type 3 (PT3); (4) log-pearson type 3 (LP3); and (5) extreme value type 1 (EV1). The three estimation methods were: (1) ; (2) maximum likelihood (MLE); and (3) method of maximum entropy (MME). The average RRASE and average RAB values for the 90 stations are listed in Tables 2 and 3, respectively. Table 2 shows that the LP3 distribution is the most suitable distribution (i.e. resulting in the smallest RRASE) for Louisiana flood data regardless of which method is used, and that the yields the smallest average RRASE for the LP3 distribution. From Table 3, however, the EV1 distribution with MME gives the smallest RAB. In this situation, the LP3/ would be selected as the best combination of distribution and estimation method for the observed data because the RRASE is normally considered to be a more preferred index than the RAB, provided that the computed RAB is not excessively large compared with other alternative combinations. In conclusion, the LP3/ is the best choice for predicting flood quantiles for the 90 gauging stations in Louisiana based on the CFA procedure. Therefore, the proposed parameter estimation procedure proceeds from this LP3/ combination. Table 2 Average RRASE for five distributions and three estimation methods for 90 sets of Louisiana flood data Distribution LN02 LN03 PT3 LP3 EV1 MLE MME Max Mm Max Mm Max Mm

8 398 F. X, Yu & B, Naghavi Table 3 Average RAB for five distributions and three estimation methods for 90 sets of Louisianaflooddata Distribution LN02 LN03 PT3 LP3 EV1 MLE MME Max Mm Max Mm Max Mm Past experience of Monte Carlo simulation showed that, for a medium sample size (say 30 or larger) parameter estimation by the is very accurate for n y, reasonably accurate for a y, and inaccurate for y y. By viewing the merit and weakness of the optimization scheme shown in Fig. 3, it may be concluded that if a parameter can be estimated accurately enough by an alternative method, that parameter may preferably not be included in the parameter set to be optimized. To illustrate this point, four possible combination methods were investigated in this at-site application. Table 4 lists the four combinations, in which y, S y and G y are the estimated values of the log-transformed mean n y, standard deviation a y, and coefficient of skewness y y, respectively; and â, b and c are the estimated values of the LP3 distribution parameters a, b and c by the. The four combination methods were initially tested by using 90 sets of Monte Carlo simulated data generated using the 5th set of LP3 population parameters listed in Table 1. As a result, for seven selected quantiles at return periods of 2, 5, 10, 25, 50, and 0 years and six sample sizes (15, 25, 30,,, 80, and 500) with 10 samples for each sample size, the l (shown in Table 4) which estimates the n y and a y by the and y y by minimizing the objective function of equation (5), gave the smallest SRMSE and SBIAS. Therefore, this method (l) was selected as the best method for the LP3 distribution and is hereafter referred to as the method. Table 4 Four alternative combination methods Method Parameter estimated by Starting point Optimization l MM02 MM03 MM04 (My, w "y) (7,) ("v. 7,) (Hy, ay, 7V.) (a, b, c) (Q (S y, G v ) (y, s v, G v ) (a, b, c)

9 Estimating distribution parameters 399 EVALUATION OF THE METHOD BY MONTE CARLO SIMULATION To test the method, 0 samples for each of the four sample sizes of,, and were drawn for each of the four LP3 populations listed in Table 1. Performance indices of SRMSE and SBIAS for the and were computed and compared. Six selected quantités corresponding to return periods of 10, 25, 50,, 0 and 500 years were used for the comparison. Table 5 shows the computed SRMSE values. In general, the larger the return period, the larger reduction in SRMSE can be achieved by the. Also, the smaller the sample size, the larger reduction in SRMSE can be made by the as compared with the. Table 6 lists the computed SBIAS for six selected quantiles and four sample sizes. Although both methods yielded small values of bias, the relatively improved the SBIAS substantially. Figures 4 and 5 show the average values of SRMSE and SBIAS for the six selected quantiles for the two methods. Clearly, the significantly improves the Table 5 SRMSE for six quantiles for four LP3 populations using 0 samples for each sample size Method Size Ô10 Ô25 e» QM) Q0 Qswi LP3 Population 1: LP3 Population 2: LP3 Population 3: LP3 Population 4:

10 0 F. X. Yu & B. Naghavi quantile prediction of the for smaller sample sizes. As a special case, the estimated the six selected quantiles very poorly for population 4 for sample sizes and. In contrast, the yielded excellent estimates for the same population and same sample sizes. Table 6 SBIASfor six quantiles for four LP3 populations using 0 samples for each sample size Method Size Ô10 e«ô50 OlOO S0 Qsoo LP3 Population 1 LP3 Population 2; ' LP3 Population 3: LP3 Population 4: ' EVALUATION OF THE METHOD BY THE OBSERVED FLOOD DATA The method was further evaluated by using the 90 sets of observed annual maximum flood data from Louisiana. Table 7 shows the maximum, average and minimum values of RRASE and RAB for the, MLE, MME and. On average, the RRASE values for the observed data sets for the, MLE, MME and were , 0.83, 0.80 and respectively. The was found to be the best method and the MLE the worst. The reduced RRASE by 14% as compared with the, by 30% compared with the MLE and by 29% compared with the MME. The

11 Estimating distribution parameters 1 A Pop. i = Population i «1.5 c to O Pop. 3. Pop. 3 WIMO co or CO 0) en Pop Sample Size Fig. 4 Average SRMSE for six selected quantités to O x CO tr 0.1 CD CO 03 > < Sample Size Fig. 5 Average SBIASfor six selected quantiles. 1

12 2 F. X. Yu & B, Naghavi average RAB values for the, MLE, MME and were , , and respectively, for the 90 observed data sets. The method reduced RAB by 46% compared with the, by 38% compared with the MLE and by 42% compared with the MME. The better fitting capability of the is expected because the minimizes both RRASE and RAB. Table 7 Average RRASE and RAB for 90 sets of flood data for four estimation methods MLE MME RRASE: Maximum Average Minimum RAB: Maximum Average Minimum CONCLUSION An improved parameter estimation procedure has been developed for flood frequency analysis. To improve the accuracy of at-site quantile prediction for LP3 distribution, the method has been developed by applying a proposed parameter optimization procedure. Tests showed that the method performed better than the both in quantile prediction and in fitting observed flood data. Large reduction of SRMSE and SBIAS by the are expected for larger return periods (larger than or equal to 50 years) and smaller sample sizes (smaller than or equal to ). Acknowledgments This project was funded by the Federal Highway Administration through the Louisiana Transportation Research Center under LTRC project number 92-lGT(B). REFERENCES Arora, K. & Singh, V. P. (1989) A comparative evaluation of the estimators of the log- Pearson type 3 distribution./. Hydrol. 105, Bobee, B. B. & Robitaille, R. (1977) The use of Pearson type 3 and log-pearson type 3 distributions revised. Wat. Resour. Res. 13(2), Himmelblau, D. M. (1972) Applied Nonlinear Programming. McGraw-Hill, New York, USA. Jain, D. & Singh, V. P. (1987) Comparison of some flood frequency analysis distributions using empirical data. In: Hydrologie Frequency Modelling, ed. V. P. Singh, D. Reidel Publishing. Kuczera, G. (1982) Robust flood frequency models. Wat. Resour. Res. 18(2),

13 Estimating distribution parameters 3 Naghavi, B., Singh, V. P. & Yu, F. X. (1991) LADOTD 24-hour rainfall frequency maps and I-D-F curves. Louisiana Transportation Research Center, LTRC Report No. 236, Baton Rouge, LA, USA. Naghavi, B. & Yu, F. X. (1992) Flood frequency analysis using optimization techniques. Louisiana Transportation Research center, LTRC Report No. 249, Baton Rouge, LA, USA. Press, W. H., Flannery, B. P., Teukolsky, S. A. & Velterling, W. T. (1986) Numerical Recipes Cambridge University Press, London, UK. Yu, F. X. & Singh, V. P. (1993) An efficient and derivative-free algorithm for finding the minimum of a 1-D user-defined function. Adv. Engng. Software. 16, Yu, F. X., Wei-Qing Li, Singh, V. P. & Naghavi, B. (1993) Estimating LP3 parameters using a combination of the method of moments and the least squares. J. Environ. Sci. 1(2), Received 24 November 1992; accepted 11 April 1994

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