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.     

Appraisal of regional and index flood quantile estimators

Our results illustrate the performance of at-site and regional GEV/PWM flood quantile estimators in regions with different coefficients of variation, degrees of regional heterogeneity, record lengths, and number of sites. Analytic approximations of bias and variance are employed. For realistic GEV distributions and short records, the index-flood quantile estimator performs better than a 2-parameter GEV/PWM quantile estimator with a regional shape parameter, or a 3-parameter at-site GEV/PWM quantile estimator, in both humid and especially in arid regions, as long as the degree of regional heterogeneity is moderate. As regional heterogeneity or record lengths increases, 2-parameter estimators quickly dominate. Flood frequency models that assign probabilities larger than 2% to negative flows are unrealistic; experiments employing such distributions provide questionable results. This appraisal generally demonstrates the value of regionalizing estimators of the shape of a flood distribution, and sometimes the coefficient of variation.     

Appraisal of regional and index flood quantile estimators

Our results illustrate the performance of at-site and regional GEV/PWM flood quantile estimators in regions with different coefficients of variation, degrees of regional heterogeneity, record lengths, and number of sites. Analytic approximations of bias and variance are employed. For realistic GEV distributions and short records, the index-flood quantile estimator performs better than a 2-parameter GEV/PWM quantile estimator with a regional shape parameter, or a 3-parameter at-site GEV/PWM quantile estimator, in both humid and especially in arid regions, as long as the degree of regional heterogeneity is moderate. As regional heterogeneity or record lengths increases, 2-parameter estimators quickly dominate. Flood frequency models that assign probabilities larger than 2% to negative flows are unrealistic; experiments employing such distributions provide questionable results. This appraisal generally demonstrates the value of regionalizing estimators of the shape of a flood distribution, and sometimes the coefficient of variation.     

Simultaneous estimation of the parameters of the Hurst�CKolmogorov stochastic process

Various methods for estimating the self-similarity parameter (Hurst parameter, H) of a Hurst–Kolmogorov stochastic process (HKp) from a time series are available. Most of them rely on some asymptotic properties of processes with Hurst–Kolmogorov behaviour and only estimate the self-similarity parameter. Here we show that the estimation of the Hurst parameter affects the estimation of the standard deviation, a fact that was not given appropriate attention in the literature. We propose the least squares based on variance estimator, and we investigate numerically its performance, which we compare to the least squares based on standard deviation estimator, as well as the maximum likelihood estimator after appropriate streamlining of the latter. These three estimators rely on the structure of the HKp and estimate simultaneously its Hurst parameter and standard deviation. In addition, we test the performance of the three methods for a range of sample sizes and H values, through a simulation study and we compare it with other estimators of the literature.     

Updating estimates of low-streamflow statistics to account for possible trends

ABSTRACT

Accurate estimators of streamflow statistics are critical to the design, planning, and management of water resources. Given increasing evidence of trends in low-streamflow, new approaches to estimating low-streamflow statistics are needed. Here we investigate simple approaches to select a recent subset of the low-flow record to update the commonly used statistic of 7Q10, the annual minimum 7-day streamflow exceeded in 9 out of 10 years on average. Informed by low-streamflow records at 174 US Geological Survey streamgages, Monte Carlo simulation experiments evaluate competing approaches. We find that a strategy which estimates 7Q10 using the most recent 30 years of record when a trend is detected, reduces error and bias in 7Q10 estimators compared to use of the full record. This simple rule-based approach has potential as the basis for a framework for updating frequency-based statistics in the context of possible trends.     

A distribution function based bandwidth selection method for kernel quantile estimation

The key problem in nonparametric frequency analysis of flood and droughts is the estimation of the bandwidth parameter which defines the degree of smoothing. Most of the proposed bandwidth estimators have been based on the density function rather than the cumulative distribution function or the quantile that are the primary interest in frequency analysis. We propose a new bandwidth estimator derived from properties of quantile estimators. The estimator builds on work by Altman and Léger (1995). The estimator is compared to the well-known method of least squares cross-validation (LSCV) using synthetic data generated from various parametric distributions used in hydrologic frequency analysis. Simulations suggest that our estimator performs at least as well as, and in many cases better than, the method of LSCV. In particular, the use of the proposed plug-in estimator reduces bias in the estimation as compared to LSCV. When applied to data sets containing observations with identical values, typically the result of rounding or truncation, the LSCV and most other techniques generally underestimates the bandwidth. The proposed technique performs very well in such situations.     

Partial L-moments for the analysis of censored flood samples / Utilisation des L-moments partiels pour l’analyse d’échantillons tronqués de crues

Abstract

Abstract A parameter estimation method is proposed for fitting the generalized extreme value (GEV) distribution to censored flood samples. Partial L-moments (PL-moments), which are variants of L-moments and analogous to ?partial probability weighted moments?, are defined for the analysis of such flood samples. Expressions are derived to calculate PL-moments directly from uncensored annual floods, and to fit the parameters of the GEV distribution using PL-moments. Results of Monte Carlo simulation study show that sampling properties of PL-moments, with censoring flood samples of up to 30% are similar to those of simple L-moments, and also that both PL-moment and LH-moments (higher-order L-moments) have similar sampling properties. Finally, simple L-moments, LH-moments, and PL-moments are used to fit the GEV distribution to 75 annual maximum flow series of Nepalese and Irish catchments, and it is found that, in some situations, both LH- and PL-moments can produce a better fit to the larger flow values than simple L-moments.     

Parameter and quantile estimation of the 2-parameter kappa distribution by maximum likelihood

Asymptotic properties of maximum likelihood parameter and quantile estimators of the 2-parameter kappa distribution are studied. Eight methods for obtaining large sample confidence intervals for the shape parameter and for quantiles of this distribution are proposed and compared by using Monte Carlo simulation. The best method is highlighted on the basis of the coverage probability of the confidence intervals that it produces for sample sizes commonly found in practice. For such sample sizes, confidence intervals for quantiles and for the shape parameter are shown to be more accurate if the quantile estimators are assumed to be log normally distributed rather than normally distributed (same for the shape parameter estimator). Also, confidence intervals based on the observed Fisher information matrix perform slightly better than those based on the expected value of this matrix. A hydrological example is provided in which the obtained theoretical results are applied.     

Comparative analysis of scalar upper tail indicators

ABSTRACT

Different upper tail indicators exist to characterize heavy tail phenomena, but no comparative study has been carried out so far. We evaluate the shape parameter (GEV), obesity index, Gini index and upper tail ratio (UTR) against a novel benchmark of tail heaviness – the surprise factor. Sensitivity analyses to sample size and changes in scale-to-location ratio are carried out in bootstrap experiments. The UTR replicates the surprise factor best but is most uncertain and only comparable between records of similar length. For samples with symmetric Lorenz curves, shape parameter, obesity and Gini indices provide consistent indications. For asymmetric Lorenz curves, however, the first two tend to overestimate, whereas Gini index tends to underestimate tail heaviness. We suggest the use of a combination of shape parameter, obesity and Gini index to characterize tail heaviness. These indicators should be supported with calculation of the Lorenz asymmetry coefficients and interpreted with caution.     

Analyse fréquentielle régionale des précipitations journalières maximales annuelles au Québec,Canada / Regional frequency analysis of annual maximum daily precipitation in Quebec,Canada

Abstract

Abstract A complete regional analysis of daily precipitations is carried out in the southern half of the province of Quebec, Canada. The first step of the regional estimation procedure consists of delineating the homogeneous regions within the area of study and testing for homogeneity within each region. The delineation of homogeneous regions is based on using L-moment ratios. A simulation-based testing of statistical homogeneity allows one to verify the inter-site variability. The second step of the procedure deals with the identification of the regional distribution and the estimation of its parameters. The General Extreme Value (GEV) distribution was identified as an appropriate parent distribution. This distribution has already been recommended by several previous research studies for regional frequency analysis of precipitation extremes. The parameters of the GEV distribution are estimated based on the computation of the regional L-CV, L-C_S and the mean of annual maximal daily precipitations. The third step consists of the estimation of precipitation quantiles corresponding to various return periods. The final procedure allows for the estimation of these quantiles at sites where no precipitation information is available. The use of a jack-knife resampling procedure with data from the province of Quebec allows one to demonstrate the robustness and efficiency of the regional estimation procedure. Values of the root mean square error were below 10% for a return period of 20 years, and 20% for a return period of 100 years.     

Flood frequency analysis of Turkey using L‐moments method

The index flood procedure coupled with the L‐moments method is applied to the annual flood peaks data taken at all stream‐gauging stations in Turkey having at least 15‐year‐long records. First, screening of the data is done based on the discordancy measure (D_i) in terms of the L‐moments. Homogeneity of the total geographical area of Turkey is tested using the L‐moments based heterogeneity measure, H, computed on 500 simulations generated using the four parameter Kappa distribution. The L‐moments analysis of the recorded annual flood peaks data at 543 gauged sites indicates that Turkey as a whole is hydrologically heterogeneous, and 45 of 543 gauged sites are discordant which are discarded from further analyses. The catchment areas of these 543 sites vary from 9·9 to 75121 km² and their mean annual peak floods vary from 1·72 to 3739·5 m³ s^?1. The probability distributions used in the analyses, whose parameters are computed by the L‐moments method are the general extreme values (GEV), generalized logistic (GLO), generalized normal (GNO), Pearson type III (PE3), generalized Pareto (GPA), and five‐parameter Wakeby (WAK). Based on the L‐moment ratio diagrams and the |Z_dist|‐statistic criteria, the GEV distribution is identified as the robust distribution for the study area (498 gauged sites). Hence, for estimation of flood magnitudes of various return periods in Turkey, a regional flood frequency relationship is developed using the GEV distribution. Next, the quantiles computed at all of 543 gauged sites by the GEV and the Wakeby distributions are compared with the observed values of the same probability based on two criteria, mean absolute relative error and determination coefficient. Results of these comparisons indicate that both distributions of GEV and Wakeby, whose parameters are computed by the L‐moments method, are adequate in predicting quantile estimates. Copyright © 2011 John Wiley & Sons, Ltd.     

Linear combinations of order statistics to estimate the quantiles of generalized pareto and extreme values distributions

 Ad hoc techniques for estimating the quantiles of the Generalized Pareto (GP) and the Generalized Extreme Values (GEV) distributions are introduced. The estimators proposed are based on new estimators of the position and the scale parameters recently introduced in the Literature. They provide valuable estimates of the quantiles of interest both when the shape parameter is known and when it is unknown (this latter case being of great relevance in practical applications). In addition, weakly-consistent estimators are introduced, whose calculation does not require the knowledge of any parameter. The procedures are tested on simulated data, and comparisons with other techniques are shown. The research was partially supported by Contract n. ENV4-CT97-0529 within the project “FRAMEWORK” of the European Community – D.G. XII. Grants by “Progetto Giovani Ricercatori” are also acknowledged.     

Unbiased estimation of flood risk with the GEV distribution

Conventional flood frequency analysis is concerned with providing an unbiased estimate of the magnitude of the design flow exceeded with the probabilityp, but sampling uncertainties imply that such estimates will, on average, be exceeded more frequently. An alternative approach is therefore, to derive an estimator which gives an unbiased estimate of flow risk: the difference between the two magnitudes reflects uncertainties in parameter estimation. An empirical procedure has been developed to estimate the mean true exceedance probabilities of conventional estimates made using a GEV distribution fitted by probability weighted moments, and adjustment factors have been determined to enable the estimation of flood magnitudes exceeded with, on average, the desired probability.     

Weighted estimate of extreme quantile: an application to the estimation of high flood return periods

Parametric models are commonly used in frequency analysis of extreme hydrological events. To estimate extreme quantiles associated to high return periods, these models are not always appropriate. Therefore, estimators based on extreme value theory (EVT) are proposed in the literature. The Weissman estimator is one of the popular EVT-based semi-parametric estimators of extreme quantiles. In the present paper we propose a new family of EVT-based semi-parametric estimators of extreme quantiles. To built this new family of estimators, the basic idea consists in assigning the weights to the k observations being used. Numerical experiments on simulated data are performed and a case study is presented. Results show that the proposed estimators are smooth, stable, less sensitive, and less biased than Weissman estimator.     

Bias,variance and computational properties of Kijko’s estimators of the upper limit of magnitude distribution,M<Subscript>max</Subscript>

It is often assumed in probabilistic seismic hazard analysis that the magnitude distribution has an upper limit M _max, which indicates a limitation on event size in specific seismogeneic conditions. Accurate estimation of M _max from an earthquake catalog is a matter of utmost importance. We compare bias, dispersion and computational properties of four popular M _max estimators, introduced by Kijko and others (e.g., Kijko and Sellevoll 1989, Kijko and Graham 1998, Kijko 2004) and we recommend the ones which can be the most fruitful in practical applications. We provide nomograms for evaluation of bias and standard deviation of the recommended estimators for combinations of sample sizes and distribution parameters. We suggest to use the bias nomograms to correct the M _max estimates. The nomograms of standard deviation can be used to determine minimum sample size for a required accuracy of M _max.     

Estimation of water cloud model vegetation parameters using a genetic algorithm

Abstract

The water cloud model is used to account for the effect of vegetation water content on radar backscatter data. The model generally comprises two parameters that characterize the vegetated terrain, A and B, and two bare soil parameters, C and D. In the present study, parameters A and B were estimated using a genetic algorithm (GA) optimization technique and compared with estimates obtained by the sequential unconstrained minimization technique (SUMT) from measured backscatter data. The parameter estimation was formulated as a least squares optimization problem by minimizing the deviations between the backscatter coefficients retrieved from the ENVISAT ASAR image and those predicted by the water cloud model. The bias induced by three different objective functions was statistically analysed by generating synthetic backscatter data. It was observed that, when the backscatter coefficient data contain no errors, the objective functions do not induce any bias in the parameter estimation and the true parameters are uniquely identified. However, in the presence of noise, these objective functions induce bias in the parameter estimates. For the cases considered, the objective function based on the sum of squares of normalized deviations with respect to the computed backscatter coefficient resulted in the best possible estimates. A comparison of the GA technique with the SUMT was undertaken in estimating the water cloud model parameters. For the case considered, the GA technique performed better than the SUMT in parameter estimation, where the root mean squared error obtained from the GA was about half of that obtained by the SUMT.

Editor D. Koutsoyiannis; Associate editor L. See

Citation Kumar, K., Hari Prasad, K.S. and Arora, M.K., 2012. Estimation of water cloud model vegetation parameters using a genetic algorithm. Hydrological Sciences Journal, 57 (4), 776–789.     

Probability Estimates of Extreme Temperature Events: Stochastic Modelling Approach vs. Extreme Value Distributions

The paper deals with the probability estimates of temperature extremes (annual temperature maxima and heat waves) in the Czech Republic. Two statistical methods of probability estimations are compared; one based on the stochastic modelling of time series of the daily maximum temperature (TMAX) using the first-order autoregressive (AR(1)) model, the other consisting in fitting the extreme value distribution to the sample of annual temperature peaks.The AR(1) model is able to reproduce the main characteristics of heat waves, though the estimated probabilities should be treated as upper limits because of deficiencies in simulating the temperature variability inherent to the AR(1) model. Theoretical extreme value distributions do not yield good results when applied to maximum annual lengths of heat waves and periods of tropical days (TMAX 30°C), but it is the best method for estimating the probability and recurrence time of annual one-day temperature extremes. However, there are some difficulties in the application: the use of the two-parameter Gumbel distribution and the three-parameter generalized extreme value (GEV) distribution may lead to different results, particularly for long return periods. The resulting values also depend on the chosen procedure of parameter estimation. Based on our findings, the shape parameter testing for the GEV distribution and the L moments technique for parameter estimation may be recommended.The application of the appropriate statistical tools indicates that the heat wave and particularly the long period of consecutive tropical days in 1994 were probably a more rare event than the record-breaking temperatures in July 1983 exceeding 40°C. An improvement of the probability estimate of the 1994 heat wave may be expected from a more sophisticated model of the temperature series.     

Probability density estimation in stochastic environmental models using reverse representations

The estimation of probability densities of variables described by stochastic differential equations has long been done using forward time estimators, which rely on the generation of forward in time realizations of the model. Recently, an estimator based on the combination of forward and reverse time estimators has been developed. This estimator has a higher order of convergence than the classical one. In this article, we explore the new estimator and compare the forward and forward–reverse estimators by applying them to a biochemical oxygen demand model. Finally, we show that the computational efficiency of the forward–reverse estimator is superior to the classical one, and discuss the algorithmic aspects of the estimator.     

Techniques for statistical analysis and prediction of geophysical fluid systems

Abstract

Application of statistical estimators to analysis and prediction is examined from the point of view of geophysical fluid dynamics. The fundamental difficulty is that estimators constructed from observational records of limited length (the usual case in GFD) are sensitive to sampling errors in the statistics upon which they are based. To achieve meaningful results, the number of data, or input, parameters must be limited. The relationship between statistical and dynamical models (particularly clear for linear systems) coupled with certain statistical methods are explored with respect to the problem of input parameter selection, both for linear and nonlinear systems. Methods of assessing the effects of sampling errors in hindcasts are discussed and techniques for minimizing these effects in forecasts are evaluated. A method of efficiently condensing statistical models to a few input parameters and transfer functions is given. Finally the steps of hindcast analysis and forecaster construction are discussed from the practical point of view.