Four-parameter Kappa Distribution Research Articles

Regression in extremes using the four-parameter Kappa distribution

The four-parameter Kappa distribution (4KD) is known to be flexible and has been used for extreme value analysis, especially in hydrological sciences. This distribution has the great advantage of encompassing the two main distributions of extreme values used in the literature: the Generalized Pareto Distribution (GPD) and the Generalized Extreme Value (GEV) distribution. However, the analyses made in previous works for 4KD in the context of extreme values are based only on models with static parameters, not allowing the capture of covariate effects such as seasonality and location variables in a specific dataset. This study proposes that the inference about the 4KD can be made under the Bayesian paradigm by inserting a regression structure on the parameters to assess possible effects of covariates under the model. This proposal allows us to interpret the parameters and predict extreme events when there are covariates. The estimation of the parameters is performed using Markov Chain Monte Carlo (MCMC) procedures. The model was applied with both approaches in extremes: maximum and exceedance data. In this analysis, the proposed model was compared with models that use a regression structure in the usual extreme values distribution (GEV and GPD), and it presented better results in fit measures.

The Partial L-Moment of the Four Kappa Distribution

Statistical analysis of extreme events such as flood events is often carried out to predict large return period events. The behaviour of extreme events not only involves heavy-tailed distributions but also skewed distributions, similar to the four-parameter Kappa distribution (K4D). In general, this covers many extreme distributions such as the generalized logistic distribution (GLD), the generalized extreme value distribution (GEV), the generalized Pareto distribution (GPD), and so on. To utilize these distributions, we have to estimate parameters accurately. There are many parameter estimation methods, for example, Method of Moments, Maximum Likelihood Estimator, L-Moments, or partial L-Moments. Nowadays, no researchers have applied the partial L-Moments method to estimate the parameters of K4D. Therefore, the objective of this paper is to derive the partial L-Moments (PL-Moments) for K4D, namely the PL-Moments of the K4D in order to estimate hydrological extremes from censored data. The findings of this paper are formulas of parameter estimation for K4D based on the PL-Moments approach. We have derived the Partial Probability-Weighted Moments (PPWMs) of the K4D (β'r) and derive the estimation of parameters when separated by shape parameters (k,h) conditions i.e., case k>-1 and h>0, case k>-1 and h=0 and case -1<k<-1/h and h<0. Finally, we expect that the parameter estimate for K4D from this formula will help to make accurate forecasts. Doi: 10.28991/ESJ-2023-07-04-06 Full Text: PDF

Modeling climate extremes using the four-parameter kappa distribution for [formula omitted]-largest order statistics

Accurate estimation of the T-year return levels of climate extremes using statistical distribution is a critical step in the projection of future climate and in engineering design for disaster response. We show how the estimation of such quantities can be improved by fitting the four-parameter kappa distribution for r-largest order statistics (rK4D), which was developed in this study. The rK4D is an extension of the generalized extreme value distribution for r-largest order statistics (rGEVD), similar to the four-parameter kappa distribution (K4D), which is an extension of the generalized extreme value distribution (GEVD). This new distribution (rK4D) can be useful not only for fitting data when three parameters in the GEVD are not sufficient to capture the variability of the extreme observations, but also in reducing the estimation uncertainty by making use of the r-largest extreme observations instead of only the block maxima. We derive a joint probability density function (PDF) of rK4D and the marginal and conditional cumulative distribution functions and PDFs. To estimate the parameters, the maximum likelihood estimation and the maximum penalized likelihood estimation methods were considered. The usefulness and practical effectiveness of the rK4D are illustrated by the Monte Carlo simulation and by an application to the Bangkok extreme rainfall data. A few new distributions for r-largest order statistics are also derived as special cases of the rK4D, such as the r-largest logistic, the r-largest generalized logistic, and the r-largest generalized Gumbel distributions. These distributions for r-largest order statistics would be useful in modeling extreme values for many research areas, including hydrology and climatology.

Penalized likelihood approach for the four-parameter kappa distribution

The four-parameter kappa distribution (K4D) is a generalized form of some commonly used distributions such as generalized logistic, generalized Pareto, generalized Gumbel, and generalized extreme value (GEV) distributions. Owing to its flexibility, the K4D is widely applied in modeling in several fields such as hydrology and climatic change. For the estimation of the four parameters, the maximum likelihood approach and the method of L-moments are usually employed. The L-moment estimator (LME) method works well for some parameter spaces, with up to a moderate sample size, but it is sometimes not feasible in terms of computing the appropriate estimates. Meanwhile, using the maximum likelihood estimator (MLE) with small sample sizes shows substantially poor performance in terms of a large variance of the estimator. We therefore propose a maximum penalized likelihood estimation (MPLE) of K4D by adjusting the existing penalty functions that restrict the parameter space. Eighteen combinations of penalties for two shape parameters are considered and compared. The MPLE retains modeling flexibility and large sample optimality while also improving on small sample properties. The properties of the proposed estimator are verified through a Monte Carlo simulation, and an application case is demonstrated taking Thailand’s annual maximum temperature data.

Bayesian Inference in Extremes Using the Four-Parameter Kappa Distribution

Maximum likelihood estimation (MLE) of the four-parameter kappa distribution (K4D) is known to be occasionally unstable for small sample sizes and to be very sensitive to outliers. To overcome this problem, this study proposes Bayesian analysis of the K4D. Bayesian estimators are obtained by virtue of a posterior distribution using the random walk Metropolis–Hastings algorithm. Five different priors are considered. The properties of the Bayesian estimators are verified in a simulation study. The empirical Bayesian method turns out to work well. Our approach is then compared to the MLE and the method of the L-moments estimator by calculating the 20-year return level, the confidence interval, and various goodness-of-fit measures. It is also compared to modeling using the generalized extreme value distribution. We illustrate the usefulness of our approach in an application to the annual maximum wind speeds in Udon Thani, Thailand, and to the annual maximum sea-levels in Fremantle, Australia. In the latter example, non-stationarity is modeled through a trend in time on the location parameter. We conclude that Bayesian inference for K4D may be substantially useful for modeling extreme events.

Stochastic simulation of streamflow and spatial extremes: a continuous, wavelet-based approach

Abstract. Stochastically generated streamflow time series are used for various water management and hazard estimation applications. They provide realizations of plausible but as yet unobserved streamflow time series with the same temporal and distributional characteristics as the observed data. However, the representation of non-stationarities and spatial dependence among sites remains a challenge in stochastic modeling. We investigate whether the use of frequency-domain instead of time-domain models allows for the joint simulation of realistic, continuous streamflow time series at daily resolution and spatial extremes at multiple sites. To do so, we propose the stochastic simulation approach called Phase Randomization Simulation using wavelets (PRSim.wave) which combines an empirical spatio-temporal model based on the wavelet transform and phase randomization with the flexible four-parameter kappa distribution. The approach consists of five steps: (1) derivation of random phases, (2) fitting of the kappa distribution, (3) wavelet transform, (4) inverse wavelet transform, and (5) transformation to kappa distribution. We apply and evaluate PRSim.wave on a large set of 671 catchments in the contiguous United States. We show that this approach allows for the generation of realistic time series at multiple sites exhibiting short- and long-range dependence, non-stationarities, and unobserved extreme events. Our evaluation results strongly suggest that the flexible, continuous simulation approach is potentially valuable for a diverse range of water management applications where the reproduction of spatial dependencies is of interest. Examples include the development of regional water management plans, the estimation of regional flood or drought risk, or the estimation of regional hydropower potential. Highlights. Stochastic simulation of continuous streamflow time series using an empirical, wavelet-based, spatio-temporal model in combination with the parametric kappa distribution. Generation of stochastic time series at multiple sites showing temporal short- and long-range dependence, non-stationarities, and spatial dependence in extreme events. Implementation of PRSim.wave in R package PRSim: Stochastic Simulation of Streamflow Time Series using Phase Randomization.

Technical note: Stochastic simulation of streamflow time series using phase randomization

Abstract. Stochastically generated streamflow time series are widely used in water resource planning and management. Such series represent sets of plausible yet unobserved streamflow realizations which should reproduce the main characteristics of observed data. These characteristics include the distribution of daily streamflow values and their temporal correlation as expressed by short- and long-range dependence. Existing streamflow generation approaches have mainly focused on the time domain, even though simulation in the frequency domain provides good properties. These properties comprise the simulation of both short- and long-range dependence as well as extension to multiple sites. Simulation in the frequency domain is based on the randomization of the phases of the Fourier transformation. We here combine phase randomization simulation with a flexible, four-parameter kappa distribution, which allows for the extrapolation to as yet unobserved low and high flows. The simulation approach consists of seven steps: (1) fitting the theoretical kappa distribution, (2) normalization and deseasonalization of the marginal distribution, (3) Fourier transformation, (4) random phase generation, (5) inverse Fourier transformation, (6) back transformation, and (7) simulation. The simulation approach is applicable to both individual and multiple sites. It was applied to and validated on a set of four catchments in Switzerland. Our results show that the stochastic streamflow generator based on phase randomization produces realistic streamflow time series with respect to distributional properties and temporal correlation. However, cross-correlation among sites was in some cases found to be underestimated. The approach can be recommended as a flexible tool for various applications such as the dimensioning of reservoirs or the assessment of drought persistence. Highlights. Stochastic simulation of streamflow time series for individual and multiple sites by combining phase randomization and the kappa distribution. Simulated time series reproduce temporal correlation, seasonal distributions, and extremes of observed time series. Simulation procedure suitable for use in water resource planning and management.

On the probability distribution of daily streamflow in the United States

Abstract. Daily streamflows are often represented by flow duration curves (FDCs), which illustrate the frequency with which flows are equaled or exceeded. FDCs have had broad applications across both operational and research hydrology for decades; however, modeling FDCs has proven elusive. Daily streamflow is a complex time series with flow values ranging over many orders of magnitude. The identification of a probability distribution that can approximate daily streamflow would improve understanding of the behavior of daily flows and the ability to estimate FDCs at ungaged river locations. Comparisons of modeled and empirical FDCs at nearly 400 unregulated, perennial streams illustrate that the four-parameter kappa distribution provides a very good representation of daily streamflow across the majority of physiographic regions in the conterminous United States (US). Further, for some regions of the US, the three-parameter generalized Pareto and lognormal distributions also provide a good approximation to FDCs. Similar results are found for the period of record FDCs, representing the long-term hydrologic regime at a site, and median annual FDCs, representing the behavior of flows in a typical year.

On the use of a four-parameter kappa distribution in regional frequency analysis

ABSTRACTNew developments are presented enabling the using a four-parameter kappa distribution with the widely used regional goodness-of-fit methods as part of an index flood regional frequency analysis based on the method of L-moments. The framework was successfully applied to 564 pooling groups and was found to significantly improve the probabilistic description of British flood flow compared to existing procedures. Based on results from an extensive data analysis it is argued that the successful application of the kappa distribution renders the use of the traditional three-parameter distributions such as the generalized extreme value (GEV) and generalized logistic (GLO) distributions obsolete, except for large and relatively dry catchments. The importance of these findings is discussed in terms of the sensitivity of design floods to distribution choice.

A three-parameter kappa distribution with hydrologic application: a generalized gumbel distribution

A three-parameter kappa distribution (K3D) as a special case of the four-parameter kappa distribution is studied. This distribution (K3D) is viewed as a generalized Gumbel distribution. A description of the mathematical properties of K3D is provided. We consider the moment generating function, the moments, the L-moments, LH-moments, the asymptotic distribution of the extreme order statistics, and its graphical illustrations. Parameters are estimated by using the method of L-moments and maximum likelihood method. A Monte-Carlo simulation study is given to compare the estimation methods. To illustrate its applicability, heavy rainfall data from Korea is analyzed, and the fitness is compared with Gumbel, generalized extreme value and four-parameter kappa distributions.

Cramér-von Mises and Anderson-Darling Goodness-of-Fit Tests for the Two-Parameter Kappa Distribution

AbstractIn hydrology, one often needs to use goodness-of-fit tests to describe how well a statistical frequency model fits a set of observations. The two-parameter kappa (KAP2) distribution is a frequency model that has already shown useful hydrological applications. The Cramer-von Mises and the Anderson-Darling tests are among the most frequently used goodness-of-fit tests in statistical practice. To apply these tests to the KAP2 distribution, statistical tables of critical values are needed. Such tables already exist for various other statistical distributions, but have not yet been developed for KAP2. This paper fills the gap by developing such tables for the KAP2 distribution. These tables are based on large sample theory, but correction formulas are also provided that make them applicable to small samples. KAP2 and the generalized Pareto distribution (GPD) are two subfamilies of the four-parameter kappa distribution that have been used in extreme value hydrological applications in the past. A detaile...

A Rainfall Distribution for the Lampao Site in the Chi River Basin, Thailand

In this study, the four-parameter kappa distribution with L-Moments estimation has been used to fit the distribution of weekly rainfall data at Lampao in the Chi River Basin, Thailand. The weekly precipitations with probabilities 0.75 were estimated, and the extreme rainfall estimates obtained can be used for water and agriculture management.

Regional Flood Frequency Analysis Using L-Moments for the West Mediterranean Region of Turkey

The aim of this study is to investigate and determine hydrologically homogeneous regions and to derive regional flood frequency estimates for 47 gauged sites in the West Mediterranean River Basins in Turkey, using an index flood method with L-moments parameter estimation. Screening of the data of the gauged site is carried out based on a discordancy measure in terms of the L-moments. Initial candidate regions are established by the cluster analysis of first five L-moment statistics, using k-means method. Homogeneity of the basins is tested using simulation with a four-parameter Kappa distribution and an L-moments based heterogeneity measure. Three subregions are defined, namely the Antalya subregion, the Lower West Mediterranean subregion, and the Upper West Mediterranean subregion. Comparative regional flood frequency estimates are made for each subregion using various distributions, namely the generalized logistic, general extreme value, generalized normal, Pearson type III, generalized Pareto, kappa, and Wakeby distributions. Based on an L-moments goodness-of-fit statistic, the Pearson type III distribution is identified as the best-fit distribution for the Antalya and Lower-West Mediterranean subregions, and the Generalized Logistic for the Upper-West Mediterranean subregion. Monte Carlo simulation is used to evaluate the accuracy of the quantile estimates on the basis of the relative root-mean-square error and relative bias.

Fisher information matrix for a four-parameter kappa distribution

In this paper, the exact form of Fisher information matrix for a four-parameter kappa distribution (K4D) is determined. The K4D is so general that includes a variety of distributions as special cases. The necessary condition for the existence of Fisher information matrix is { 0 < h < 1 2 , k < 1 2 } ∪ { h < 0 , 1 2 h < k < 1 2 } for k ≠ 0 , h ≠ 0 .

Regional precipitation-frequency analysis and spatial mapping for 24-hour and 2-hour durations for Washington State

Abstract. This study is an update of the information contained in the precipitation-frequency atlas published by the US National Weather Service in 1973. Data collection for the NWS study ended in 1966 while this study uses the current data base which more than doubles the record length used in the NWS study. Washington State has highly variable topography and climate; in particular Mean Annual Precipitation (MAP) varies from over 260 inches a year to less than 7 inches. Steep high mountain ranges provide very wet slopes as well as pronounced rain shadows with large climate changes occurring in relatively short distances. In addition there are four distinct sources for the atmospheric moisture needed for precipitation which gives rise to complex seasonal and spatial interactions. The PRISM mapping system used in this study has greatly improved the spatial mapping of precipitation and increased the reliability of estimates of precipitation in the broad areas between precipitation measurement stations. Further, the development and use of regional L-Moments has greatly improved the reliability of precipitation magnitude-frequency estimates, particularly for the rarer and more extreme storms. Washington State could be specified adequately by 12 regions for the purposes of estimating the 2-hour and 24 hour precipitation frequencies. Within each region algorithms were developed for L-CV and L-Skewness expressed as functions of the MAP. The GEV distribution was acceptable statistically for all regions up to the 1 in 500 recurrence interval, beyond which the four-parameter Kappa distribution is recommended. The estimated changes in precipitation magnitudes for a given frequency as regional boundaries were crossed were found to be small, and well within the expected differences likely from sampling errors. An interesting transition zone was observed at the eastern edge of the Cascade foothills, associated with the maxima having a seasonal change from autumn–winter synoptic scale general storms in the west to spring–summer maxima in the east that were produced by a mix of storm types. (Comment: storms were a mix of general storms and more-isolated convective storms).

Entropy-Based Parameter Estimation for Kappa Distribution

An entropy-based method is developed for estimating parameters of the four-parameter kappa distribution. Using four sets of data on annual maximum rainfall and on annual peak flow discharge, the entropy-based method is evaluated and compared with the methods of moments, L-moments, probability-weighted moments, and the maximum likelihood estimation. The results of estimation show that both the entropy method and the L-moment method enable the four-parameter kappa distribution to fit the data well and a combination of the two methods can further improve estimation.

Maximum likelihood estimation of the four-parameter Kappa distribution using the penalty method

Maximum likelihood estimation of the four-parameter Kappa distribution using the penalty method

More on the four-parameter kappa distribution

The generalized extreme-value has been the distribution of choice for modeling available maxima (or minima) data since theory has shown it to be the limiting form of the distribution of extremes. However, fits to finite samples are not always adequate. Hosking (1994) and Parida (1999) suggest the four-parameter Kappa distribution as an alternative. Hosking (1994) developed an L-moment procedure for estimation. Some compromises must be made in practice however, as seen in Parida (1999). L-moment estimators of the four-parameter Kappa distribution are not always computable nor feasible. A simulation study in this paper quantifies the extent of each problem. Maximum likelihood is investigated as an alternative method of estimation and a simulation study compares the performance of both methods of estimation. Finally, further benefits of maximum likelihood are shown when wind speeds From the Tropical Pacific are examined and the weekly maxima for 10 buoys in the area are analyzed.

Modelling of Indian summer monsoon rainfall using a four-parameter Kappa distribution

In attempting to model the random behaviour of the Indian summer monsoon, the possibility of using a generalized four-parameter Kappa distribution representing a family of distributions has been explored. An L-Moment procedure developed by Hosking (1994) IBM J. Res. Dev., 38(3), 251–258, has been used for estimation of reliable rainfall quantiles. Using these estimates, isopluvial maps have been developed for some commonly used recurrence intervals (viz. 20, 50, 100, 200, 500 and 1000 years) which can be used by operational hydro-meteorologists. Copyright © 1999 Royal Meteorological Society

The four-parameter kappa distribution

Many common probability distributions, including some that have attracted recent interest for flood-frequency analysis, may be regarded as special cases of a four-parameter distribution that generalizes the three-parameter kappa distribution of P.W. Mielke. This four-parameter kappa distribution can be fitted to experimental data or used as a source of artificial data in simulation studies. This paper describes some of the properties of the four-parameter kappa distribution, and gives an example in which it is applied to modeling the distribution of annual maximum precipitation data.