On the use of partial probability weighted moments in the analysis of hydrological extremes

The use of partial probability weighted moments (PPWM) for estimating hydrological extremes is compared to that of probability weighted moments (PWM). Firstly, estimates from at‐site data are considered. Two Monte Carlo analyses, conducted using continuous and empirical parent distributions (of peak discharge and daily rainfall annual maxima) and applying four different distributions (Gumbel, Fréchet, GEV and generalized Pareto), show that the estimates obtained from PPWMs are better than those obtained from PWMs if the parent distribution is unknown, as happens in practice. Secondly, the use of partial L‐moments (obtained from PPWMs) as diagnostic tools is considered. The theoretical partial L‐diagrams are compared with the experimental data. Five different distributions (exponential, Pareto, Gumbel, GEV and generalized Pareto) and 297 samples of peak discharge annual maxima are considered. Finally, the use of PPWMs with regional data is investigated. Three different kinds of regional analyses are considered. The first kind is the regression of quantile estimates on basin area. The study is conducted applying the GEV distribution to peak discharge annual maxima. The regressions obtained with PPWMs are slightly better than those obtained with PWMs. The second kind of regional analysis is the parametric one, of which four different models are considered. The congruence between local and regional estimates is examined, using peak discharge annual maxima. The congruence degree is sometimes higher for PPWMs, sometimes for PWMs. The third kind of regional analysis uses the index flood method. The study, conducted applying the GEV distribution to synthetic data from a lognormal joint distribution, shows that better estimates are obtained sometimes from PPWMs, sometimes from PWMs. All the results seem to indicate that using PPWMs can constitute a valid tool, provided that the influence of ouliers, of course higher with censored samples, is kept under control. Copyright © 2007 John Wiley & Sons, Ltd.     

A review of methods of parameter estimation for the extreme value type-1 distribution

Four important methods of estimating the parameters of the extreme value type-1 (Gumbel) distribution, namely: (1) moments (MMM); (2) maximum likelihood (MML); (3) maximum entropy (MME); and (4) probability weighted moments (PWM), are considered, suitable solution procedures are recommended and related problems discussed. Approximate formulae for computing the variances-covariances of the estimators by each method are given in terms of the population value of the scale parameter, and these are shown to indicate very well the performance of the related method.

A simulation study to evaluate the performance of these methods in terms of commonly used criteria, i.e. the bias, root mean square error and goodness-of-fit statistic is described. It was found that the MMM is not as good as the remaining three methods, of which the PWM is best in terms of the bias, and the MML is best in terms of the root mean square error and efficiency. The MME, by all the criteria, ranks second best and follows the MML more closely than the PWM.     

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.     

Estimation of asymptotic variances of quantiles for the generalized logistic distribution

In this study, the parameter estimations for the 3-parameter generalized logistic (GL) distribution are presented based on the methods of moments (MOM), maximum likelihood (ML), and probability weighted moments (PWM). The asymptotic variances of the MOM, ML, and PWM quantile estimators for the GL distribution are expressed as functions of the sample size, return period, and parameters. A Monte Carlo simulation was performed to verify the derived expressions for variances and covariances between parameters and to evaluate the applicability of the derived asymptotic variances of quantiles for the MOM, ML and PWM methods. The simulation results generally show good agreement with the analytical results estimated from the asymptotic variances of parameters and quantiles when the shape parameter (β) of the GL distribution is between −0.10 and 0.10 for the MOM method and between −0.25 and 0.45 for the ML and PWM methods, respectively. In addition, the actual sample variances and the root mean square error (RMSE) of asymptotic variances of quantiles for various sample sizes, return periods, and shape parameters were presented. In order to evaluate the applicability of the estimation methods to real data and to compare the values of estimated parameter, quantiles, and confidence intervals based on each parameter estimation method, the GL distribution was fitted to the 24-h annual maximum rainfall data at Pohang, Korea.     

A new procedure to estimate the RUSLE EI30 index, based on monthly rainfall data and applied to the Algarve region, Portugal

One of the best indicators of the potential erosion risks is the rainfall–runoff erosivity factor (R) of the revised universal soil loss equation (RUSLE). Frequently, however, there is not enough data available to compute the R value, and other parameters, such as the modified Fournier index (F_mod), are used instead. But RUSLE is less effective if only the alternative procedures exist. One of the major discrepancies between R and the alternative parameters is time resolution: individual storms are used to calculate R while monthly averages over the year are used to calculate F_mod.

In this study, a multiple linear regression (r²=0.89) involving monthly EI₃₀, monthly rainfall for days with ≥10.0 mm and monthly number of days with rainfall ≥10.0 mm, for the Algarve region, is presented. Twenty-seven years of monthly rainfall erosivity values were computed for the 32 standard daily-read raingauge stations of the Algarve region.     

Simple scaling properties of Canadian annual average streamflow

The spatial scaling properties of Canadian annual average streamflow (abbreviated as AASF) are assessed using both the product moments (PMs) and the probability weighted moments (PWMs) of AASF across the entire country and in its sub-climatic regions. By the PMs, the log relationship between the kth moments of AASF and the drainage area can be almost represented by a perfect straight line across the entire country and in its sub-climatic regions, whose regression parameters are a linear function of the moment order. By the PWMs, the logarithm of the kth PWM is a linear function of the logarithm of drainage area for the entire country and its sub-climatic regions, where its slope (or scale exponent) in a region is constant and is independent of the order. These results indicate that Canadian AASF exhibits simple scaling and drainage area alone may describe most of the variability in the moments of AASF. The third approach, based on the log linearity between quantiles and drainage area, is applied to Region 2, also demonstrate simple scaling of AASF in that region, as concluded from using PMs and PWMs methods, which indicates that all three methods are consistent. The simple scaling results provide a basis for using the index flood method to conduct regional frequency analysis of AASF in Canada.     

Frequency analyses of annual extreme rainfall series from 5 min to 24 h

The parameter estimation methods of (1) moments, (2) maximum‐likelihood, (3) probability‐weighted moments (PWM) and (4) self‐determined PWM are applied to the probability distributions of Gumbel, general extreme values, three‐parameter log‐normal (LN3), Pearson‐3 and log‐Pearson‐3. The special method of computing parameters so as to make the sample skewness coefficient zero is also applied to LN3, and hence, altogether 21 candidate distributions resulted. The parameters of these distributions are computed first by original sample series of 14 successive‐duration annual extreme rainfalls recorded at a rain‐gauging station. Next, the parameters are scaled by first‐degree semi‐log or log‐log polynomial regressions versus rainfall durations from 5 to 1440 min (24 h). Those distributions satisfying the divergence criterion for frequency curves are selected as potential distributions, whose better‐fit ones are determined by a conjunctive evaluation of three goodness‐of‐fit tests. Frequency tables, frequency curves and intensity–duration–frequency curves are the outcome. Copyright © 2010 John Wiley & Sons, Ltd.     

On the informative value of the largest sample element of log-Gumbel distribution

Extremes of stream flow and precipitation are commonly modeled by heavytailed distributions. While scrutinizing annual flow maxima or the peaks over threshold, the largest sample elements are quite often suspected to be low quality data, outliers or values corresponding to much longer return periods than the observation period. Since the interest is primarily in the estimation of the right tail (in the case of floods or heavy rainfalls), sensitivity of upper quantiles to largest elements of a series constitutes a problem of special concern. This study investigated the sensitivity problem using the log-Gumbel distribution by generating samples of different sizes (n) and different values of the coefficient of variation by Monte Carlo experiments. Parameters of the log-Gumbel distribution were estimated by the probability weighted moments (PWMs) method, method of moments (MOMs) and maximum likelihood method (MLM), both for complete samples and the samples deprived of their largest elements. In the latter case, the distribution censored by the non-exceedance probability threshold, F _T , was considered. Using F _T instead of the censored threshold T creates possibility of controlling estimator property. The effect of the F _T value on the performance of the quantile estimates was then examined. It is shown that right censoring of data need not reduce an accuracy of large quantile estimates if the method of PWMs or MOMs is employed. Moreover allowing bias of estimates one can get the gain in variance and in mean square error of large quantiles even if ML method is used.     

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.     

Dipole moment scaling for convection-driven planetary dynamos

Scaling laws are derived for the time-average magnetic dipole moment in rotating convection-driven numerical dynamo models. Results from 145 dynamo models with a variety of boundary conditions and heating modes, covering a wide section of parameter space, show that the time-average dipole moment depends on the convective buoyancy flux F. Two distinct regimes are found above the critical magnetic Reynolds number for onset of dynamo action. In the first regime the external magnetic field is dipole-dominant, whereas for larger buoyancy flux or slower rotation the external field is dominated by higher multipoles and the dipole moment is reduced by a factor of 10 or more relative to the dipolar regime. For dynamos driven by basal heating, the dipole moment M increases like M  F^1/3 in the dipolar regime. Reversing dipolar dynamos tend to cluster near the multipolar transition, which is shown to depend on a local Rossby number parameter. The geodynamo lies close to this transition, suggesting an explanation for polarity reversals and the possibility of a weaker dipole earlier in Earth history. Internally heated dynamos generate smaller dipole moments overall and show a gradual transition from dipolar to multipolar states. Our scaling yields order of magnitude agreement with the dipole moments of Earth, Jupiter, Saturn, Uranus, Neptune, and Ganymede, and predicts a multipolar-type dynamo for Mercury.     

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.     

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.     

On the strength of electric currents and zonal magnetic fields at the top of the Earth''s core: Methodology and preliminary estimates

A new method is described for estimating: (a) the meridional electric current density, j_θ, (b) the vertical growth rate of the zonal magnetic field, ∂B_φ/∂r, or its scale-height, B_φ/∂B_φ/∂r) and (c) the vertical growth rate of the vertical current density, ∂j_r/∂r, at a few isolated points on the top surface of the Earth's core from observations of the internal geomagnetic field at the Earth's surface. The theoretical technique rests on combining unaccelerated, gravitationally-driven Boussinesq fluid dynamics of the core with frozen-flux electromagnetism, the mantle being treated as a spherically symmetric insulator.

Insertion into this theory of main field models for epochs 1965, 1975 leads to preliminary values for these quantities of magnitude: (a) j_θ 1 A/m², (b) ∂B_φ/∂r 10⁻⁶ T/m or B_φ/(∂B_φ/∂r) 10 m, (c) ∂j_r/∂r 10⁻⁶ A/m³. Some geophysical implications of these estimates are discussed.     

Unbiased estimation of probability weighted moments and partial probability weighted moments from systematic and historical flood information and their application to estimating the GEV distribution

Unbiased estimators of probability weighted moments (PWM) and partial probability weighted moments (PPWM) from systematic and historical flood information are derived. Applications are made to estimating parameters and quantiles of the generalized extreme value (GEV) distribution. The effect of lower bound censoring, which might be deliberately introduced in practice, is also considered.     

The PWM large quantile estimates of heavy tailed distributions from samples deprived of their largest element / Estimation des grands quantiles de distributions à queue à décroissance lente par la méthode des moments pondérés par les probabilités à partir d'échantillons amputés de leur plus grande valeur

Abstract

Extremes of streamflow are usually modelled using heavy tailed distributions. While scrutinising annual flow maxima or the peaks over threshold, the largest elements in a sample are often suspected to be low quality data, outliers or values corresponding to much longer return periods than the observation period. In the case of floods, since the interest is focused mainly on the estimation of the right-hand tail of a distribution function, sensitivity of large quantiles to extreme elements of a series becomes the problem of special concern. This study investigated the sensitivity problem using the log-Gumbel distribution by generating samples of different sizes and different values of the coefficient of L-variation by means of Monte Carlo experiments. Parameters of the log-Gumbel distribution were estimated by the probability weighted moments (PWM) method, both for complete samples and the samples deprived of their largest element. In the latter case Hosking's concept of the “A” type PWM with Type II censoring was employed. The largest value was censored above the random threshold T corresponding to the non-exceedence probability F _T. The effect of the F _T value on the performance of the quantile estimates was then examined. Experimental results show that omission of the largest sample element need not result in a decrease in the accuracy of large quantile estimates obtained from the log-Gumbel model by the PWM method.     

Magnetic domain observations of nucleation processes in natural pyrrhotite and titanomagnetite

Magnetic domain patterns have been observed on particles of natural pyrrhotite and titanomagnetite undergoing hysteresis. These observations indicate that hysteresis properties are governed by two distinct mechanisms: (1) wall-pinning and (2) nucleation of reverse domains. Particles which are dominated by wall-pinning spontaneously nucleate reverse domains in saturation remanence (J_rs). The coercivity of such grains is determined by the presence of potential wells encountered by the wall in its traverse across the grain. However, many pseudosingle-domain particles (PSD) between 5 and 30 μm in diameter do not nucleate reverse domains in J_rs, but remain as saturated single-domains. These particles require a reverse field H_n to nucleate domain walls. When H_n is sufficiently large, the nucleating field controls magnetization reversal by driving the wall across the particle in a single Barkhausen jump, and the muscopic coercivity is nucleation-dominated.

The proportion P (w=0) of particles of a given size d which fail to nucleate walls in J_rs is found to be given by A exp(−Bd^1/2), where A and B are experimentally determined constants. The nucleation field H_n in pyrrhotite is observed to increase with decreasing grain size, exceeding 500 Oe in 5 μm particles.

The difficulty with which reverse domains are nucleated subsequent to saturation may thus provide a mechanism for achieving the high values of J_rs/J_s and coercive force observed in fine, pseudosingle-domain particles.     

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.     

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.     

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.     

Further research on application of probability weighted moments in estimating parameters of the Pearson type three distribution

This paper contains further research results on the recently developed method for estimating parameters of P-III distribution by using probability weighted moments (PWM). The computing procedure can be largely simplified and the PWM method is appropriate and functional for various conditions of the skewness coefficient. Statistical experiments made based on the new procedure have shown that PWM estimators are almost unbiased and compare favorably with those by the conventional moment method.