The use of the Akaike Information Criterion in the identification of an optimum flood frequency model

Abstract

For a long time now, the hydrologist has been faced with the problem of finding which of the many possible probability distribution functions can be used most effectively in flood frequency analyses. This problem has been mainly due to the insufficiency of the conventional goodness-of-fit procedures when used with the typically skewed flood probability distributions. In this study, the Akaike Information Criterion (AIC) goodness-of-fit test is used to identify more objectively the optimum model for flood frequency analysis in Kenya from a class of competing models. The class is comprised of (a) seven three-parameter density functions, namely, log-normal, Pearson, log-Pearson, Fisher-Tippet, log-Fisher-Tippet, Walter Boughton and log-Walter Boughton; and (b) two five-parameter density functions, namely, Wakeby and log-Wakeby. The AIC is also used in this study as a method of testing for the existence of outlier peak-flow values in the peak annual data used. A modified version of the chi-square goodness-of-fit test is also used, but only for the sake of comparison with the AIC.     

Estimation of confidence intervals of quantiles for the Weibull distribution

 Estimation of confidence limits and intervals for the two- and three-parameter Weibull distributions are presented based on the methods of moment (MOM), probability weighted moments (PWM), and maximum likelihood (ML). The asymptotic variances of the MOM, PWM, and ML quantile estimators are derived as a function of the sample size, return period, and parameters. Such variances can be used for estimating the confidence limits and confidence intervals of the population quantiles. Except for the two-parameter Weibull model, the formulas obtained do not have simple forms but can be evaluated numerically. Simulation experiments were performed to verify the applicability of the derived confidence intervals of quantiles. The results show that overall, the ML method for estimating the confidence limits performs better than the other two methods in terms of bias and mean square error. This is specially so for γ≥0.5 even for small sample sizes (e.g. N=10). However, the drawback of the ML method for determining the confidence limits is that it requires that the shape parameter be bigger than 2. The Weibull model based on the MOM, ML, and PWM estimation methods was applied to fit the distribution of annual 7-day low flows and 6-h maximum annual rainfall data. The results showed that the differences in the estimated quantiles based on the three methods are not large, generally are less than 10%. However, the differences between the confidence limits and confidence intervals obtained by the three estimation methods may be more significant. For instance, for the 7-day low flows the ratio between the estimated confidence interval to the estimated quantile based on ML is about 17% for T≥2 while it is about 30% for estimation based on MOM and PWM methods. In addition, the analysis of the rainfall data using the three-parameter Weibull showed that while ML parameters can be estimated, the corresponding confidence limits and intervals could not be found because the shape parameter was smaller than 2.     

中国海域及邻区地震时间分布特征研究

地震时间分布特征研究是进行地震预测和地震危险性分析的重要基础.以中国海域统一地震目录为基础资料,以指数分布模型、伽马分布模型、威布尔分布模型、对数正态分布模型以及布朗过程时间分布(BPT)模型为目标模型,采用极大似然法估算模型参数.根据赤池信息准则(AIC)、贝叶斯信息准则(BIC)以及K-S检验结果确定能够描述海域地...     

Using regional regression within index flood procedures and an empirical Bayesian estimator

Studies have illustrated the performance of at-site and regional flood quantile estimators. For realistic generalized extreme value (GEV) distributions and short records, a simple index-flood quantile estimator performs better than two-parameter (2P) GEV quantile estimators with probability weighted moment (PWM) estimation using a regional shape parameter and at-site mean and L-coefficient of variation (L-CV), and full three-parameter at-site GEV/PWM quantile estimators. However, as regional heterogeneity or record lengths increase, the 2P-estimator quickly dominates. This paper generalizes the index flood procedure by employing regression with physiographic information to refine a normalized T-year flood estimator. A linear empirical Bayes estimator uses the normalized quantile regression estimator to define a prior distribution which is employed with the normalized 2P-quantile estimator. Monte Carlo simulations indicate that this empirical Bayes estimator does essentially as well as or better than the simpler normalized quantile regression estimator at sites with short records, and performs as well as or better than the 2P-estimator at sites with longer records or smaller L-CV.     

THEME 1B ESTIMATION DES PARAMETRES DE L'EROSION ET DES QUANTITES DE SEDIMENTS DANS LES BASSINS FLUVIAUX OU LES DONNEES SONT INSUFFISANTES-RELATIONS ENTRE L'EROSION ET LES QUANTITES DE SEDIMENTS ET LEURS FACTEURS CLIMATIQUES ET LES CARACTERISTIQUES DU BASSIN VERSANT ET DU LIT LUI-MEME

ABSTRACT

The extreme value type III distribution was derived by using the principle of maximum entropy. The derivation required only two constraints to be determined from data, and yielded a procedure for estimation of distribution parameters. This method of parameter estimation was comparable to the methods of moments (MOM) and maximum likelihood estimation (MLE) for the low flow data used.     

Parameter estimation for 2-parameter generalized pareto distribution by POME

The principle of maximum entropy (POME) was employed to derive a new method of parameter estimation for the 2-parameter generalized Pareto (GP2) distribution. Monte Carlo simulated data were used to evaluate this method and compare it with the methods of moments (MOM), probability weighted moments (PWM), and maximum likelihood estimation (MLE). The parameter estimates yielded by POME were comparable or better within certain ranges of sample size and coefficient of variation.     

中国大陆地震空间分布模型检验

本文以中国大陆地震目录为基础资料,以泊松模型为零假设模型,并将Neyman-Scott空间丛集过程的各子模型设为检验模型,采用K-function点过程分析法和最大似然估计法计算各模型参数,并以AIC准侧判定模型的拟合优度,来检验中国大陆地震空间分布模型.检验结果表明:泊松模型的拟合优度最差,说明地震在空间的分布不是完全随机的;广义Thomas模型的拟合优度最好,说明地震的空间分布是丛集的,可用由两个高斯核组成的广义Thomas模型较好地描述.研究结果还表明,同一研究区内,采用不同时段具有不同最小完整起始震级的地震目录计算得到的地震空间分布的丛集尺度几乎不变,这意味着地震空间丛集尺度不受小地震的控制,且可能与研究区的断层规模有关.     

Estimation of quantiles and confidence intervals for the log-Gumbel distribution

The log-Gumbel distribution is one of the extreme value distributions which has been widely used in flood frequency analysis. This distribution has been examined in this paper regarding quantile estimation and confidence intervals of quantiles. Specific estimation algorithms based on the methods of moments (MOM), probability weighted moments (PWM) and maximum likelihood (ML) are presented. The applicability of the estimation procedures and comparison among the methods have been illustrated based on an application example considering the flood data of the St. Mary's River.     

Impact of Three-Parameter Weibull Models in Probabilistic Assessment of Earthquake Hazards

This paper investigates the suitability of a three-parameter (scale, shape, and location) Weibull distribution in probabilistic assessment of earthquake hazards. The performance is also compared with two other popular models from same Weibull family, namely the two-parameter Weibull model and the inverse Weibull model. A complete and homogeneous earthquake catalog (Yadav et al. in Pure Appl Geophys 167:1331–1342, 2010) of 20 events (M ≥ 7.0), spanning the period 1846 to 1995 from north–east India and its surrounding region (20°–32°N and 87°–100°E), is used to perform this study. The model parameters are initially estimated from graphical plots and later confirmed from statistical estimations such as maximum likelihood estimation (MLE) and method of moments (MoM). The asymptotic variance–covariance matrix for the MLE estimated parameters is further calculated on the basis of the Fisher information matrix (FIM). The model suitability is appraised using different statistical goodness-of-fit tests. For the study area, the estimated conditional probability for an earthquake within a decade comes out to be very high (≥0.90) for an elapsed time of 18 years (i.e., 2013). The study also reveals that the use of location parameter provides more flexibility to the three-parameter Weibull model in comparison to the two-parameter Weibull model. Therefore, it is suggested that three-parameter Weibull model has high importance in empirical modeling of earthquake recurrence and seismic hazard assessment.     

ARIMA forecasting of ambient air pollutants (O3, NO, NO2 and CO)

In the present study, a stationary stochastic ARMA/ARIMA [Autoregressive Moving (Integrated) Average] modelling approach has been adapted to forecast daily mean ambient air pollutants (O₃, CO, NO and NO₂) concentration at an urban traffic site (ITO) of Delhi, India. Suitable variance stabilizing transformation has been applied to each time series in order to make them covariance stationary in a consistent way. A combination of different information-criterions, namely, AIC (Akaike Information Criterion), HIC (Hannon–Quinn Information Criterion), BIC (Bayesian Information criterion), and FPE (Final Prediction Error) in addition to ACF (autocorrelation function) and PACF (partial autocorrelation function) inspection, has been tried out to obtain suitable orders of autoregressive (p) and moving average (q) parameters for the ARMA(p,q)/ARIMA(p,d,q) models. Forecasting performance of the selected ARMA(p,q)/ARIMA(p,d,q) models has been evaluated on the basis of MAPE (mean absolute percentage error), MAE (mean absolute error) and RMSE (root mean square error) indicators. For 20 out of sample forecasts, one step (i.e., one day) ahead MAPE for CO, NO₂, NO and O₃, have been found to be 13.6, 12.1, 21.8 and 24.1%, respectively. Given the stochastic nature of air pollutants data and in the light of earlier reported studies regarding air pollutants forecasts, the forecasting performance of the present approach is satisfactory and the suggested forecasting procedure can be effectively utilized for short term air quality forewarning purposes.     

Parameter estimation for 3-parameter log-logistic distribution (LLD3) by Pome

The principle of maximum entropy (POME) was employed to derive a new method of parameter estimation for the 3-parameter log-logistic distribution (LLD3). Monte Carlo simulated data were used to evaluate this method and compare it with the methods of moments (MOM), probability weighted moments (PWM), and maximum likelihood estimation (MLE). Simulation results showed that POME's performance was superior in predicting quantiles of large recurrence intervals when population skew was greater than or equal to 2.0. In all other cases, POME's performance was comparable to other methods.     

Relationships between three dispersion measures used in flood frequency analysis

Three dispersion measures of a random variable, i.e., the standard deviation, the mean deviation (MD) about the mean and the second L-moment, are analyzed in terms of their properties and mutual relationships. Emphasis is placed on the MD, as it is less recognized than two other dispersion measures. The relationships between the dispersion measures are derived for distributions commonly applied in flood frequency analysis (FFA). For distributions that are unbounded, there is a distribution-dependent constant value of the ratio of dispersion measures, or equivalently of respective coefficients of variation. For two-parameter distributions that are lower-bounded, the relationship between the coefficients of variation is also distribution dependent and is not linear. For lower-bounded three-parameter distributions, the dispersion measure ratios, or equivalently the ratios of coefficients of variation, depend on the coefficient of skewness and show a strong distributional dependence. For selected distributions, the three dispersion measures are compared both in terms of the robustness to the largest samples element and the accuracy of upper quantile estimation. The MD statistics may be highly competitive to the two other dispersion measure statistics if applied in FFA for parameters estimation.     

Parameter estimation for 3-parameter log-logistic distribution (LLD3) by Pome

The principle of maximum entropy (POME) was employed to derive a new method of parameter estimation for the 3-parameter log-logistic distribution (LLD3). Monte Carlo simulated data were used to evaluate this method and compare it with the methods of moments (MOM), probability weighted moments (PWM), and maximum likelihood estimation (MLE). Simulation results showed that POME's performance was superior in predicting quantiles of large recurrence intervals when population skew was greater than or equal to 2.0. In all other cases, POME's performance was comparable to other methods.     

A note on the applicability of log-Gumbel and log-logistic probability distributions in hydrological analyses: II. Assumed pdf

Abstract

Applicability of log-Gumbel (LG) and log-logistic (LL) probability distributions in hydrological studies is critically examined under real conditions, where the assumed distribution differs from the true one. The set of alternative distributions consists of five two-parameter distributions with zero lower bound, including LG and LL as well as lognormal (LN), linear diffusion analogy (LD) and gamma (Ga) distributions. The log-Gumbel distribution is considered as both a false and a true distribution. The model error of upper quantiles and of the first two moments is analytically derived for three estimation methods: the method of moments (MOM), the linear moments method (LMM) and the maximum likelihood method (MLM). These estimation methods are used as methods of approximation of one distribution by another distribution. As recommended in the first of this two-part series of papers, MLM turns out to be the worst method, if the assumed LG or LL distribution is not the true one. It produces a huge bias of upper quantiles, which is at least one order higher than that of the other two methods. However, the reverse case, i.e. acceptance of LN, LD or Ga as a hypothetical distribution, while the LG or LL distribution is the true one, gives the MLM bias of reasonable magnitude in upper quantiles. Therefore, one should avoid choosing the LG and LL distributions in flood frequency analysis, especially if MLM is to be applied.     

用于地震预警的P波震相到时自动拾取

P波震相的自动拾取可用于地震预警中地震事件判别和地震定位,是实现基于地震台网地震预警的首要条件.针对地震预警中P波震相拾取的特点,本文发展了一套基于长短时平均(STA/LTA)和池赤准则(AIC)算法的多步骤P波自动拾取技术,应用Delaunay三角剖分提出了一种非几何相关的干扰信号剔除方法,并应用福建省数字地震台网记录对方法进行了验证,目前方法已经用到了福建省地震预警试验系统中.     

Entropy-based parameter estimation for extended Burr XII distribution

Two entropy-based methods, called ordinary entropy (ENT) method and parameter space expansion method (PSEM), both based on the principle of maximum entropy, are applied for estimating parameters of the extended Burr XII distribution. With the parameters so estimated, the Burr XII distribution is applied to six peak flow datasets and quantiles (discharges) corresponding to different return periods are computed. These two entropy methods are compared with the methods of moments (MOM), probability weighted moments (PWM) and maximum likelihood estimation (MLE). It is shown that PSEM yields the same quantiles as does MLE for discrete cases, while ENT is found comparable to the MOM and PWM. For shorter return periods (<10–30 years), quantiles (discharges) estimated by these four methods are in close agreement, but the differences amongst them grow as the return period increases. The error in quantiles computed using the four methods becomes larger for return periods greater than 10–30 years.     

Application of linear stochastic models for drought forecasting in the Büyük Menderes river basin, western Turkey

In the present study, a seasonal and non-seasonal prediction of the Standardized Precipitation Index (SPI) time series is addressed by means of linear stochastic models. The methodology presented here is to develop adequate linear stochastic models known as autoregressive integrated moving average (ARIMA) and multiplicative seasonal autoregressive integrated moving average (SARIMA) to predict drought in the Büyük Menderes river basin using SPI as drought index. Temporal characteristics of droughts based on SPI as an indicator of drought severity indicate that the basin is affected by severe and more or less prolonged periods of drought from 1975 to 2006. Therefore, drought prediction plays an important role for water resources management. ARIMA modeling approach involves the following three steps: model identification, parameter estimation, diagnostic checking. In model identification step, considering the autocorrelation function (ACF) and partial autocorrelation function (PACF) results of the SPI series, different ARIMA models are identified. The model gives the minimum Akaike Information Criterion (AIC) and Schwarz Bayesian Criterion (SBC) is selected as the best fit model. Parameter estimation step indicates that the estimated model parameters are significantly different from zero. Diagnostic check step is applied to the residuals of the selected ARIMA models and the results indicated that the residuals are independent, normally distributed and homoscedastic. For the model validation purposes, the predicted results using the best ARIMA models are compared to the observed data. The predicted data show reasonably good agreement with the actual data. The ARIMA models developed to predict drought found to give acceptable results up to 2 months ahead. The stochastic models developed for the Büyük Menderes river basin can be employed to predict droughts up to 2 months of lead time with reasonably accuracy.     

Complexity vs. Simplicity: Groundwater Model Ranking Using Information Criteria

A groundwater model characterized by a lack of field data about hydraulic model parameters and boundary conditions combined with many observation data sets for calibration purpose was investigated concerning model uncertainty. Seven different conceptual models with a stepwise increase from 0 to 30 adjustable parameters were calibrated using PEST. Residuals, sensitivities, the Akaike information criterion (AIC and AICc), Bayesian information criterion (BIC), and Kashyap's information criterion (KIC) were calculated for a set of seven inverse calibrated models with increasing complexity. Finally, the likelihood of each model was computed. Comparing only residuals of the different conceptual models leads to an overparameterization and certainty loss in the conceptual model approach. The model employing only uncalibrated hydraulic parameters, estimated from sedimentological information, obtained the worst AIC, BIC, and KIC values. Using only sedimentological data to derive hydraulic parameters introduces a systematic error into the simulation results and cannot be recommended for generating a valuable model. For numerical investigations with high numbers of calibration data the BIC and KIC select as optimal a simpler model than the AIC. The model with 15 adjusted parameters was evaluated by AIC as the best option and obtained a likelihood of 98%. The AIC disregards the potential model structure error and the selection of the KIC is, therefore, more appropriate. Sensitivities to piezometric heads were highest for the model with only five adjustable parameters and sensitivity coefficients were directly influenced by the changes in extracted groundwater volumes.     

Models for extremes using the extended three-parameter Burr XII system with application to flood frequency analysis / Modèles d’extrêmes utilisant le système Burr XII étendu à trois paramètres et application à l’analyse fréquentielle des crues

Abstract

Abstract A new theoretically-based distribution in frequency analysis is proposed. The extended three-parameter Burr XII distribution includes the generalized Pareto distribution, which is used to model the exceedences over threshold; log-logistic distribution, which is also advocated in flood frequency analysis; and Weibull distribution, which is a part of the generalized extreme value distribution used to model annual maxima as special cases. The extended Burr distribution is flexible to approximate extreme value distribution. Note that both the generalized Pareto and generalized extreme value distributions are limiting results in modelling the exceedences over threshold and block extremes, respectively. From a modelling perspective, generalization might be necessary in order to obtain a better fit. The extended three-parameter Burr XII distribution is therefore a meaningful candidate distribution in the frequency analysis. Maximum likelihood estimation for this distribution is investigated in the paper. The use of the extended three-parameter Burr XII distribution is demonstrated using data from China.     

Identification of uncertainty in low flow frequency analysis using Bayesian MCMC method

This study employs the Bayesian Markov Chain Monte Carlo (MCMC) method with the Metropolis–Hastings algorithm and maximum likelihood estimation (MLE) using a quadratic approximation of the likelihood function for the evaluation of uncertainties in low flow frequency analysis using a two‐parameter Weibull distribution. The two types of prior distributions, a non‐data‐based distribution and a data‐based distribution using regional information collected from neighbouring stations, are used to establish a posterior distribution. Eight case studies using the synthetic data with a sample size of 100, generated from two‐parameter Weibull distribution, are performed to compare with results of analysis using MLE and Bayesian MCMC. Also, Bayesian MCMC and MLE are applied to 36 years of gauged data to validate the efficiency of the developed scheme. These examples illustrate the advantages of Bayesian MCMC and the limitations of MLE based on a quadratic approximation. From the point of view of uncertainty analysis, Bayesian MCMC is more effective than MLE using a quadratic approximation when the sample size is small. In particular, Bayesian MCMC method is more attractive than MLE based on a quadratic approximation because the sample size of low flow at the site of interest is mostly not enough to perform the low flow frequency analysis. Copyright © 2007 John Wiley & Sons, Ltd.