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The L-moments page by JRM Hosking L-moments L-moments are summary statistics for probability distributions and data samples. They are analogous to ordinary moments -- they provide measures of location, dispersion, skewness, kurtosis, and other aspects of the shape of probability distributions or data samples -- but are computed from linear combinations of the ordered data values (hence the prefix L). L-moments have the following theoretical advantages over ordinary moments: For L-moments of a probability distribution to be meaningful, we require only that the distribution have finite mean; no higher-order moments need be finite [J. R. M. Hosking, J. R. Statist. Soc. B, 52 (1990), Theorem 1]. For standard errors of L-moments to be finite, we require only that the distribution have finite variance; no higher-order moments need be finite [Hosking, 1990, Theorem 3]. Although moment ratios can be arbitrarily large, sample moment ratios have algebraic bounds [J. Dalen, Statistics and Probability Letters, 5 (1987)]; sample L-moment ratios can take any values that the corresponding population quantities can [Hosking, 1990, page 115]. In addition, the following properties hold in a wide range of practical situations: Asymptotic approximations to sampling distributions are better for L-moments than for ordinary moments [Hosking, 1990, Figure 4]. L-moments are less sensitive to outlying data values [P. Royston, Statistics in Medicine, 11 (1992), Figure 7; R. M. Vogel and N. M. Fennessey, Water Resources Research, 29 (1993), Figures 3 and 4]. L-moments provide better identification of the parent distribution that generated a particular data sample [Hosking, 1990, Figure 6]. An L-moment ratio diagram. L-moments for data samples Probability weighted moments, defined by J. A. Greenwood et al. [Water Resources Research, 15 (1979), 1049-1054], are precursors of L-moments. Sample probability weighted moments, computed from data values X1, X2, ... Xn, arranged in increasing order, are given by L-moments are certain linear combinations of probability weighted moments that have simple interpretations as measures of the location, dispersion and shape of the data sample. The first few L-moments are defined by (the coefficients are those of the "shifted Legendre polynomials"). The first L-moment is the sample mean, a measure of location. The second L-moment is (a multiple of) Gini's mean difference statistic, a measure of the dispersion of the data values about their mean. By dividing the higher-order L-moments by the dispersion measure, we obtain the L-moment ratios, These are dimensionless quantities, independent of the units of measurement of the data. t3 is a measure of skewness and t4 is a measure of kurtosis -- these are respectively the L-skewness and L-kurtosis. They take values between -1 and +1 (exception: some even-order L-moment ratios computed from very small samples can be less than -1). The L-moment analogue of the coefficient of variation (standard deviation divided by the mean), is the L-CV, defined by It takes values between 0 and 1. L-moments for probability distributions For a probability distribution with cumulative distribution function F(x), probability weighted moments are defined by L-moments are defined in terms of probability weighted moments, analogously to the sample L- moments: L-moment ratios are defined by Examples: Uniform (rectangular) distribution on (0,1): Normal distribution with mean 0 and variance 1: exponential distribution with mean 1: The L-moment ratio diagram can be used to compare the L-skewness--L-kurtosis relations of different distributions and data samples. An example of the use of the L-moment ratio diagram The L-moment ratio diagram can be used to compare the L-skewness--L-kurtosis relations of different distributions and data samples. This gives a visual indication of which distribution may be expected to give a good fit to a data sample or samples. The following example is adapted from J. R. M. Hosking ["L-moments: analysis and estimation of distributions using linear combinations of order statistics", Journal of the Royal Statistical Society, Series B, 52 (1990), 105-124, section 3.5], and appears here by permission of the Royal Statistical Society. Annual maximum hourly rainfall data at 689 raingages in California was analysed by the State of California ["Rainfall depth-duration-frequency for California", Technical Report, State of California Department of Water Resources, 1981]. It was suggested that a gamma distribution was appropriate for the data because the average values of sample skewness and kurtosis were consistent with the relationship between the population skewness and kurtosis of gamma distributions. This inference is valid only if sample moments are accurate estimators of population moments. For the California data, this is dubious, because the sample sizes are small - - only 12 gauges have records as long as 50 years. L-moments tell a different story. The sample L-skewness and L-kurtosis values for the 68 sites with at least 20 years of record in the Central Valley of California are shown in the first diagram. The data are on average closer to the population L-moments of a generalized extreme-value (GEV) distribution rather than a gamma distribution. The second diagram shows sample L-moments of data simulated from independent GEV distributions each with population L-skewness 0.24 (the average of the 68 sample L-skewness values) and the same record lengths as the actual rainfall records. The scatter of the points is similar to the first diagram. This is what one would expect if the actual data did in fact follow a GEV distribution. Sample L-moments of data simulated from a gamma distribution are shown in the third diagram. This has an appearance rather different from the first two diagrams: there are many fewer points above the "GEV" line and many more below the "gamma" line. We therefore conclude that the Central Valley hourly rainfall data may be well described by a GEV distribution but seem most unlikely to follow a gamma distribution. Approximations for use in constructing L-moment ratio diagrams To construct an L-moment ratio diagram it is convenient to have simple explicit expressions for tau_4 (L-kurtosis) in terms of tau_3 (L-skewness) for some commonly used probability distribtions. Polynomial approximations of the form 2 8 tau_4 = A0 + A1 * (tau_3) + A2 * (tau_3) + ... + A8 * (tau_3) have been obtained, and the coefficients are given in the table below. "Overall lower bound" is the lower bound on tau_4 for all distributions [Hosking, J. R. Statist. Soc. B, 1990, eq.(2.7)]. For given tau_3, the approximations yield values of tau_4 that are accurate to within 0.0005 provided that tau_3 is in the range -0.9 to +0.9, except that for the generalized extreme-value distribution 0.0005 accuracy is attained only when tau_3 is between -0.6 and +0.9, The approximations are not intended for detailed analytical calculations -- for that purpose, use the routines in the LMOMENTS software package -- but they are sufficiently accurate for use in plotting theoretical L-moment relationships on an L-moment ratio diagram. ----------------------------------------------------------------------------- -- Generalized Generalized Generalized Pearson Overall logistic extreme-value Pareto Lognormal type III lower bound ----------------------------------------------------------------------------- -- A0 0.16667 0.10701 0. 0.12282 0.12240 -0.25 A1 . 0.11090 0.20196 . . . A2 0.83333 0.84838 0.95924 0.77518 0.30115 1.25 A3 . -0.06669 -0.20096 . . . A4 . 0.00567 0.04061 0.12279 0.95812 . A5 . -0.04208 . . . . A6 . 0.03673 . -0.13638 -0.57488 . A7 . . . . . . A8 . . . 0.11368 0.19383 . ----------------------------------------------------------------------------- -- This material is based on an IBM Research Report by J. R. M. Hosking ["Approximations for use in constructing L-moment ratio diagrams", Research Report RC 16635, IBM Research Division, Yorktown Heights, N.Y., 1991]. The approximations are also given in Appendix A.12 of the book Regional frequency analysis: an approach based on L-moments, by J. R. M. Hosking and J. R. Wallis. L-moments can be used as the basis of a unified approach to the statistical analysis of univariate probability distributions. They can be defined for any random variable whose mean exists and form the basis of a general theory that covers: the summarization and description of theoretical probability distributions; the summarization and description of observed data samples; estimation of parameters and quantiles of probability distributions; hypothesis tests for probability distributions. A short summary of the theory and applications of L-moments can be found in the article "L- moments" in the Encyclopedia of statistical sciences, Update Volume 2, ed. S. Kotz, C. Read and D. L. Banks, Wiley, New York, 1998, pp. 357-362. More details are given in the principal references, the first two items in the following list of publications. Book publication related to L-moments Hosking, J. R. M., and Wallis, J. R. (1997). Regional frequency analysis: an approach based on L-moments. Cambridge University Press, Cambridge, U.K. Extreme environmental events, such as floods, droughts, rainstorms, and high winds, have severe consequences for human society. How frequently an event of a given magnitude may be expected to occur is of great importance. Planning for weather-related emergencies, design of civil engineering structures, reservoir management, pollution control, and insurance risk calculations, all rely on knowledge of the frequency of these extreme events. Estimation of these frequencies is difficult because extreme events are by definition rare and data records are often short. Regional frequency analysis resolves this problem by 'trading space for time': data from several sites are used in estimating event frequencies at any one site. L-moments are a recent development within statistics. They form the basis of an elegant mathematical theory in their own right, and can be used to facilitate the estimation process in regional frequency analysis. L- moment methods are demonstrably superior to those that have been used previously, and are now being adopted by major organizations worldwide. This book is the first complete account of the L-moment approach to regional frequency analysis. It brings together results that previously were scattered among academic journals, and also includes much new material. Regional frequency analysis comprehensively describes the theoretical background to the subject, is rich in practical advice for users, and contains detailed examples that illustrate the approach. This book will be of great value to hydrologists, atmospheric scientists and civil engineers, concerned with environmental extremes. ISBN 0-521- 43045-3. Published May 1997. Papers describing L-moments and their properties Chen, G., and Balakrishnan, N. (1995). The infeasibility of probability weighted moments estimation of some generalized distributions. In Recent advances in life-testing and reliability, ed. N. Balakrishnan, 565-573. Boca Raton, Fla.: CRC Press. Donaldson, R. W. (1996). Calculating inverse Cv, skew and PWM functions for Pearson-3, log- normal, extreme-value, and log-logistic distributions. Communications in Statistics - Simulation and Computation, 25, 741-747. Elamir, E. A. H., and Seheult, A. H. (2003). Trimmed L-moments. Computational Statistics and Data Analysis, 43, 299-314. Gingras, D., and Adamowski, K. (1994). Performance of L-moments and nonparametric flood frequency analysis. Canadian Journal of Civil Engineering, 21, 856-862. Guttman, N. B. (1994). On the sensitivity of sample L-moments to sample size. Journal of Climate, 7, 1026-1029. Haktanir, T., and Bozduman, A. (1995). A study on sensitivity of the probability-weighted moments method on the choice of the plotting position formula. Journal of Hydrology, 168, 265- 281. Lu, L.-H., and Stedinger, J. R. (1992). Sampling variance of normalized GEV/PWM quantile estimators and a regional homogeneity test. Journal of Hydrology, 138, 223-245. Lu, L.-H., and Stedinger, J. R. (1992). Variance of two- and three-parameter GEV/PWM quantile estimators: formulae, confidence intervals, and a comparison. Journal of Hydrology, 138, 247-267. Pandey, M. D., van Gelder, P. H. A. J. M., and Vrijling, J. K. (2001). Assessment of an L- kurtosis-based criterion for quantile estimation. Journal of Hydrologic Engineering, 6, 284-292. Peel, M. C., Wang, Q. J., Vogel, R. M., and McMahon, T. A. (2001). The utility of L-moment ratio diagrams for selecting a regional probability distibution. Hydrological Sciences Journal, 46, 147-155. Royston, P. (1992). Which measures of skewness and kurtosis are best? Statistics in Medicine, 11, 333-343. Sankarasubramanian, A., and Srinivasan, K. (1999). Investigation and comparison of sampling properties of L-moments and conventional moments. Journal of Hydrology, 218, 13-34. Stedinger, J. R., Vogel, R. M., and Foufoula-Georgiou, E. (1992). Frequency analysis of extreme events. In Handbook of Hydrology, ed. D. 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Regional analysis of annual maximum and partial duration flood data by nonparametric and L-moment methods. Journal of Hydrology, 229, 219-231. Anctil, F., Larouche, W., and Hoang, V. D. (2000). Analyse regionale des etiages 7-jours de la province de Quebec. Water Quality Research Journal of Canada, 35, 125-146. Awadallah, A. G., Rousselle, J., and Leconte, R. (1999). Evolution du risque hydrologique sur la riviere Chateauguay. Canadian Journal of Civil Engineering, 26, 510-523. Benson, C. (1993). Probability distributions for hydraulic conductivity of compacted soil liners. Journal of Geotechnical Engineering, 119, 471-486. Ben-Zvi, A., and Azmon, B. (1997). Joint use of L-moment diagram and goodness-of-fit test: a case study of diverse series. Journal of Hydrology, 198, 245-259. Bradley, A. A. (1998). Regional frequency analysis methods for evaluating changes in hydrologic extremes. Water Resources Research, 34, 741-750. Burn, D. H. (2002). The use of resampling for estimating confidence intervals for single site and pooled frequency analysis. Hydrological Sciences Journal, 48, 25-38. Castellarin, A., Burn, D. H., and Brath, A. (2001). Assessing the effectiveness of hydrological similarity measures for flood frequency analysis. Journal of Hydrology, 241, 270-285. Clausen, B., and Pearson, C. P. (1995). Regional frequency analysis of annual maximum streamflow drought. Journal of Hydrology, 173, 111-130. Cong, S., Li, Y., Vogel, J. L., and Schaake, J. C. (1993). Identification of the underlying distribution form of precipitation by using regional data. Water Resources Research, 29, 1103- 1111. Cunderlik, J. M., and Burn, D. H. (2003). Non-stationary pooled flood frequency analysis. Journal of Hydrology, 276, 210-223. Daviau, J.-L., Adamowski, K., and Patry, G. G. (2000). Regional flood frequency using GIS, L- moment and geostatistical methods. Hydrological Processes, 14, 2731-2753. Dewar, R. E., and Wallis, J. R. (1999). Geographical patterning of interannual rainfall variability in the tropics and near tropics: an L-moments approach. Journal of Climate, 12, 3457-3466. Duan, J., Selker, J., and Grant, G. E. (1998). Evaluation of probability density functions in precipitation models for the Pacific northwest. Journal of the American Water Resources Association, 34, 617-627. Fill, H. D., and Stedinger, J. R. (1995). L moment and PPCC goodness-of-fit tests for the Gumbel distribution and impact of autocorrelation. Water Resources Research, 31, 225-229. Fill, H. D., and Stedinger, J. R. (1999). Using regional regression procedures within index flood procedures and an empirical Bayesian estimator. Journal of Hydrology, 210, 128-145. Gellens, D. (2003). Combining regional approach and data extension procedure for assessing GEV distribution of extreme precipitation in Belgium. Journal of Hydrology, 268, 113-126. Gingras, D., Adamowski, K., and Pilon, P. J. (1994). Regional flood equations for the provinces of Ontario and Quebec. Water Resources Bulletin, 30, 55-67. Glaves, R., and Waylen, P. R. (1997). Regional flood frequency analysis in southern Ontario using L-moments. The Canadian Geographer, 41, 178-193. Gottschalk, L., Tallaksen, L. M., and Perzyna, G. (1997). Derivation of low flow distribution functions using recession curves. Journal of Hydrology, 194, 239-262. Guttman, N. B. (1993). The use of L-moments in the determination of regional precipitation climates. Journal of Climate, 6, 2309-2325. Guttman, N. B. (1996). Statistical characteristics of US historical climatology network temperature distributions. Climate Research, 6, 33-43. Heth, C. D., and Cornell, E. H. (1998). Characteristics of travel by persons lost in Albertan wilderness areas. Journal of Environmental Psychology, 18, 223-235. Kachroo, R. K., Mkhandi, S. H., and Parida, B. P. (2000). Flood frequency analysis of southern Africa: I. Delineation of homogeneous regions. 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Application of L moments to maximum river flows. The New Zealand Statistician, 28, 2-10. Pearson, C. P. (1995). Regional frequency analysis of low flows in New Zealand rivers. Journal of Hydrology (NZ), 33, 94-122. Pilon, P. J., and Adamowski, K. (1992). The value of regional information to flood frequency analysis using the method of L-moments. Canadian Journal of Civil Engineering, 19, 137-147. Potter, K. W., and Lettenmaier, D. P. (1990). A comparison of regional flood frequency estimation methods using a resampling method. Water Resources Research, 26, 415-424. Reed, D. W., and Stewart, E. J. (1994). Inter-site and inter-duration dependence of rainfall extremes. In Statistics for the environment 2: water related issues, eds. V. Barnett and K. F. Turkman, 125-143. New York: Wiley. Rulli, M. C., and Rosso, R. (2002). An integrated simulation method for flash-flood risk assessment: 1. Frequency predictions in the Bisagno River by combining stochastic and deterministic methods. Hydrology and Earth System Sciences, 6, 267-283. Schaefer, M. G. (1990). Regional analyses of precipitation annual maxima in Washington State. Water Resources Research, 26, 119-131. Smithers, J. C. (1996). Short-duration rainfall frequency model selection in Southern Africa. Water S.A., 22, 211-217. Smithers, J. C., and Schulze, R. E. (2001). A methodology for the estimation of short duration design storms in South Africa using a regional approach based on L-moments. Journal of Hydrology, 241, 42-52. Vicente-Serrano, S. M., and Begueria-Portugues, S. (2003). Estimating extreme dry-spell risk in the middle Ebro valley (northeastern Spain): a comparative analysis of partial duration series with a general Pareto distribution and annual maxima series with a Gumbel distribution. International Journal of Climatology, 23, 1103-1118. Vogel, R. M., and Wilson, I. (1996). Probability distribution of annual maximum, mean, and minimum streamflows in the United States. Journal of Hydraulic Engineering, 1, 69-76. Wallis, J. R. (1993). Regional frequency studies using L-moments. In Concise encyclopedia of environmental systems, ed. P. C. Young, 468-476. Oxford: Pergamon. Waylen, P., and Laporte, M. S. (1999). Flooding and the El Nino-Southern Oscillation phenomenon along the Pacific coast of Costa Rica. Hydrological Processes, 13, 2623-2638. Waylen, P. R., and Zorn, M. R. (1998). Prediction of mean annual flows in north and central Florida. Journal of the American Water Resources Association, 34, 149-157. Zaidman, M. D., Keller, V., Young, A. R., and Cadman, D. (2003). Flow-duration-frequency behaviour of British rivers based on annual minima data. Journal of Hydrology, 277, 195-213. Zrinji, Z., and Burn, D. H. (1994). Flood frequency analysis for ungauged sites using a region of influence approach. Journal of Hydrology, 153, 1-21. Software The LMOMENTS package contains Fortran routines for L-moment computations and regional frequency analysis. Version 3.03, containing 63 routines, is available from the StatLib software repository at Carnegie Mellon University. IBM software disclaimer LMOMENTS: Fortran routines for use with the method of L-moments Permission to use, copy, modify and distribute this software for any purpose and without fee is hereby granted, provided that this copyright and permission notice appear on all copies of the software. The name of the IBM Corporation may not be used in any advertising or publicity pertaining to the use of the software. IBM makes no warranty or representations about the suitability of the software for any purpose. It is provided "AS IS" without any express or implied warranty, including the implied warranties of merchantability, fitness for a particular purpose and non-infringement. IBM shall not be liable for any direct, indirect, special or consequential damages resulting from the loss of use, data or projects, whether in an action of contract or tort, arising out of or in connection with the use or performance of this software. Source code (284 KB) Documentation (PostScript file, 227 KB)