sign in sign up
<<
Home , Coefficient of variation

Advances in Safety and – Kołowrocki (ed.) © 2005 Taylor & Francis Group, London, ISBN 0 415 38340 4

An analysis of drag forces based on L-moments

P.H.A.J.M. van Gelder Delft University of Technology, Delft, The Netherlands CeSOS, NTNU, Trondheim, Norway M.D. Pandey University of Waterloo, Ontario, Canada

ABSTRACT: Since L- estimators are linear functions of the ordered values, they are virtually unbiased and have relatively small , especially in comparison to the classical coefficients of and . Moreover, estimators of L-moments are relatively insensitive to outliers. Liaw and Zheng (2004) calculated drag forces on cylinders by polynomial approximations in which the coefficients are estimated by least-squares and moment methods. In this paper, the coefficients will be estimated with L-Moment methods. Its advantages will shown to be that: (i) an L-Moments approach leads to a linear solution of the polynomialisation of drag forces and (ii) other distribution types for the turbulent current and wave heights acting on the cylinder can be analysed fairly simple.

1 INTRODUCTION of quantile estimators (Hosking and Wallis, 1987; Rosbjerg et al., 1992). As compared with for exam- Since Hosking (1990), introduced L-moments, they ple the classical method of moments, the robustness have become popular tools for solving various sta- vis-à-vis sample outliers is clearly a characteristic tistical problems related to parameter estimation, of L-moment estimators. However, estimators can be distribution identification, and regionalization. The “too robust” in the sense that large (or small) sam- L-moments are linear functions of probability ple values reflecting important information on the weighted moments (PWM’s) and hence for certain tail of the parent distribution are given too little applications, such as the estimation of distribution weight in the estimation. Hosking (1990) assessed that parameters, serve identical purposes (Hosking, 1986). L-moments weigh each element of a sample according In other situations, however, L-moments have signi- to its relative importance. ficant advantages over PWM’s due to their ability to The present paper briefly describes first the the- summarize a statistical distribution in a more - ory of L-Moments followed by an overview of papers ingful way. Since L-moment estimators are linear with applications of L-moments. The literature review functions of the ordered data values, they are virtually has shown that the theory of L-moments hase mostly unbiased and have relatively small sampling variance. been applied to a regionalized setting combining L-moment ratio estimators also have small bias and data from more than one site. However, in univari- variance, especially in comparison with the classi- ate settings the method of L-moments has not been cal coefficients of skewness and kurtosis. Moreover, investigated in detail. Therefore, this paper presents estimators of L-moments are relatively insensitive to a Monte Carlo in a univariate setting outliers. These often-heard arguments in favor of esti- in order to compare the L-moments method with mation of distribution parameters by L-moments (or the classical parameter estimation methods (MOM, PWM’s)should, nevertheless, not be accepted without MML, and MLS). The performance of these methods any scrutiny. For instance, in the wave height will also be analyzed with respect to inhomogeneous analysis, the interest is in the estimation of a given data. quantile, not in the L-moments themselves. Although L-moments are in fact for prob- the latter may have desirable sampling properties, the ability distributions and data samples. They are anal- same does not necessarily apply to a function of them, ogous to ordinary moments – they provide measures such as a quantile estimator. In fact, several simulation of location, dispersion, skewness, kurtosis, and other studies have demonstrated that for some distribu- aspects of the shape of probability distributions or tions, other estimation methods may be superior to the data samples – but are computed from linear combina- method of L-moments in terms of mean square errors tions of the ordered data values (hence the prefix L).

661 Hosking and Wallis (1997) give an excellent overview of the whole theory of L-moments. Liaw and Zheng (2004) calculated drag forces by polynomial approximations in which the coefficients are estimated by least-squares and moment meth- Hosking (1990) showed that the first few L-moments ods. In this paper, coefficients will be estimated with follow from PWMs via: L-Moment methods, and the its advantages will be shown.

2 L-MOMENTS FOR DATA SAMPLES

Probability weighted moments, defined by Greenwood et al. (1979), are precursors of L-moments. Sample The coefficients in Eqn. (5) are those of the shifted probability weighted moments, computed from data Legendre polynomials. The first L-moment is the values x1:n,x2:n,…,xn:n, arranged in increasing order, sample mean, a measure of location. The second L- are given by: moment is (a multiple of) Gini’s mean difference (Johnson et al., 1994), 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:

L-moments are certain linear combinations of prob- These are dimensionless quantities, independent of the ability weighted moments that have simple interpre- units of measurement of the data; t is a measure of tations as measures of the location, dispersion and 3 skewness and t4 is a measure of kurtosis – these are shape of the data sample. A sample of size 2 con- respectively the L-skewness and L-kurtosis. They take tains two observations in ascending order x1:2 and values between −1 and +1 (exception: some even- x2:2. The difference between the two observations − order L-moment ratios computed from very small x2:2 x1:2 is a measure of the scale of the distribu- samples can be less than −1). The L-moment ana- tion. A sample of size 3 contains three observations logue of the coefficient of variation ( in ascending order x1:3,x2:3 and x3:3. The differ- divided by the mean), is the L-CV,defined by: ence between the two observations x2:3 − x1:3 and the difference between the two observations x3:3 − x2:3 can be subtracted from each other to have a mea- sure of the skewness of the distribution. This leads It takes values between 0 and 1 (if X ∃ 0). to: (x3:3 − x2:3) − (x2:3 − x1:3) = x3:3 − 2x2:3 + x1:3.A sample of size 4 contains four observations in ascend- ing order x ,x ,x and x . A measure for the 1:4 2:4 3:4 4:4 3 L-MOMENTS FOR PROBABILITY kurtosis of the distribution is given by: x − x −3 4:4 1:4 DISTRIBUTIONS (x3:4 − x2:4). In short: the above linear combinations of the elements of the ordered sample contain informa- For a with cumulative distri- tion about the location, scale, skewness and kurtosis of bution function F(x), probability weighted moments the distribution from which the sample was drawn. A are defined by: natural way to generalize the above approach to sam- ples of size n, is to take all possible sub-samples of size 2 and then take the average of the differences, i.e., (x2:2 − x1:2)/2: L-moments are defined in terms of probability weighted moments, analogously to the sample L- moments:

Furthermore, the skewness and kurtosis are similarly obtained as:

662 L-moment ratios are defined by: in which:

The L-moment analogue of the coefficient of variation, is the L-CV,defined by: So, the expression (15) can be written in terms of an incomplete Beta function as:

Examples (for a complete overview,see theAppendix): Uniform (rectangular) distribution on (0,1):

Indeed; note that with mean 0 and variance 1:

The theory of L-moments has been applied in numer- The probability density function of Xr:n is given by the ous papers. The following work is worth mentioning: first derivative of Eqn. (16): Rao and Hamed (1997), Duan et al. (1998), Ben-Zvi and Azmon (1997), Van Gelder and Neykov (1998), Demuth and Kuells (1997), and Pearson et al. (1991).

Now, the expected value of r-th order statistics can be 4 RELATION OF L-MOMENTS WITH ORDER obtained as STATISTICS

Consider a sample consisting of n observations {x1, x2,…, xn} randomly drawn from a statistical pop- ulation. If the sample values are rearranged in a non-decreasing order of magnitude, x #x #…# Substituting from eqn. (17) into (18) and introducing 1:n 2:n a transformation, u = F(x)orx = F−1(u), 0 ≤ u ≤ 1, xn:n, then the r-th member (xr:n) of this new sequence is called the r-th of the sample (Harter, leads to: 1969). When all the sample values come from a com- mon parent population with cumulative distribution function F(x), the probability distribution (CDF) of the r-th order statistic, i.e., Prob[Xr:n ≤ x], that at least r observations in a sample of n do not exceed Note that x(u) denotes the quantile function of a ran- a fixed value, x. dom variable. The expectation of the maximum and A sample randomly drawn from a distribution is minimum of a sample of size n can be easily obtained analogous to a Bernouilli experiment in which the from eqn. (19) by setting r = n and r = 1, respectively. success is defined by the sampled value being less than the threshold, x. Naturally, the probability of suc- cess in such an experiment is given as p = F(x), and the number of successes, a , follows the binomial distribution. Based on this argument, and the CDF of the r-th order statistic, F(r)(x), can be mathematically expressed as

The probability weighted moment (PWM) of a random variable was formally defined by Greenwood et al. The incomplete Beta function I (a,b) (Kendall and x (1979) as: Stuart 1977) is defined via the Beta function B(a,b) as:

663 The following two forms of PWM are particularly simple and useful: Type 1:

and Type 2:

Comparing eqns. (22) and (23), it can be seen that αk and βk, respectively, are related to the expectations of the minimum and maximum in a sample of size k Figure 1. Comparison of Eqn. (25) and (26).

The terms li and ki in Eqn. (25) and (26) are com- pared in Figure 1 and we notice indeed a very close similarity. Reiss (1989) derived more approximate distributions of order statistics and Durrans (1992) derived distributions of fractional order statistics. In essence, PWM’s are the normalized expectations Summarizing: L-moments are certain linear com- of maximum/minimum of k random observations; the binations of probability weighted moments that are normalization is done by the sample size (k) itself. analogous to ordinary moments in a sense that they = ∃ − From Eqn. (24), we notice that E(Xn:n) n n 1 and also provide measures of location, dispersion, skew- from Eqn. (8) we have: ness, kurtosis, and other aspects of the shape of probability distributions or data samples. An rth order L-moment is mathematically defined as:

So

∗ where pr,k represents the coefficients of shifted Leg- On the other hand, using Eqn. (1), we have endre polynomials (Hosking 1990).The following nor- malized form of higher order L-moments is convenient to work with:

From this it indeed follows that bn−1 is an unbiased estimator of ∃n−1. Landwehr et al. (1979) gave a proof The normalized fourth order L-moment, τ4, is referred that br is an unbiased estimator of ∃r for other values of r. to as the L-kurtosis of a distribution. Hosking and Wallis (1997) showed that L-moments are very effi- The expression for ∃r in Eqn. (8) can numerically calculated by using a plotting-position formula as cient in estimating parameters of a wide of follows: distributions from small samples. The required com- putation is fairly limited as compared with other traditional techniques, such as maximum likelihood and . In the Appendix the L-moment for- mulae are given for a selection of PDFs. Apart from the well-known moments diagrams, also L-Moment Notice that the expression looks almost the same as diagrams exist in which L-skewness and L-kurtosis Eqn. (1) by writing: are plotted against eachother. However, the L-Moment diagrams do not form a complete class; that is to say points in the diagram may correspond to more than one probability distribution. This is in contrast to

664 the ordinary moment diagram and also to the ∗1 −∗2 Table 1. Matrix B with numerical evaluations of diagram of Halphen distributions (Bobee et al. 1993).

5 L-MOMENTS OF A POLYNOMIAL 0 1 2 3 4 5 FUNCTION OF RANDOM VARIABLES X X X X X X

β0 101030 The i-th PWM of a random variable X with quantile β 1 0.282 0.5 0.705 1.5 3.032 function x(u) is given by: 1 β2 1 0.282 0.425 0.705 1.400 3.032 β3 1 0.257 0.388 0.675 1.350 2.969 β4 1 0.233 0.360 0.650 1.305 2.907 β5 1 0.211 0.337 0.618 1.266 2.848

The quantile function of the random variable Y = Xm follows from a transformation y(u) = xm(u): in which y follows a standard normal distribution and c is a deterministic current. It is proposed to use Monte Carlo simulations to generate the samples of the drag force, (y + c)|y + c|, and then compute sample L-moments using Eqn. (5) for each sample. Finally, the average of L-moments and over all the simulation samples is taken. The polynomial coefficients ai will be solved from a set of linear equations:

Therefore, the i-th PWM of Xm is given by:

Finally, the coefficient vector ai (i = 0, … , 4) is solved by: In particular, if X is a standard normally distributed variable, then the following PWM’s can be calculated numerically as shown in Table 1. in which matrix B is given in Table 1. In particular, if X is exponentially distributed, then The above expression (34) is a normalised version the following PWM’s can be calculated analytically as of the well known drag force equations: shown in Table 2. L-Moments are linear combinations of the PWM’s, given by the matrix multiplication λ = Aβ in which and

Normalisation of the inundation drag force in Eqn. Furthermore, L-Moments are linear combinations (38) is: of observations and therefore the L-Moment of the summation of two random variables is given by the summation of the L-Moments of the random variables separately. Estimation of the polynomial coefficients by Least Squares is based upon the minimisation of the error: 6 APPROXIMATION OF THE DISTRIBUTED DRAG TERM R BY A POLYNOM

The aim is to derive a polynomial expression for the drag force on a cylinder. A Moment-based approximation of the polynomial coefficients follows by equating the central moments

665 1 m m k Table 2. Matrix B with numerical evaluations of βk (X ) = (ξ − α log(1 − u)) u du 0 X0 X1 X2 X3 X4 X5

β0 1 ξ + αξˆ2+2*αˆ2+2*ξ*αξˆ3+6*αˆ3+6*ξ*αˆ2+3*ξˆ2*α 4*ξˆ3*α+12*ξˆ2*αˆ2+ 120*ξ*αˆ4+5*ξˆ4*α+20*ξˆ3*αˆ2+ 24*ξ*αˆ3+ξˆ4+24*αˆ4 60*ξˆ2*αˆ3+120*αˆ5+ξˆ5

β1 1/2 1/2*ξ+3/4*α 3/2*ξ*α+7/ 21/4*ξ*αˆ2+1/2*ξˆ3+45/ 1/2*ξˆ4+93/4*αˆ4+3*ξˆ3*α+21/ 465/4*ξ*αˆ4+15/4*ξˆ4*α+35/ 4*αˆ2+1/2*ξˆ2 8*αˆ3+9/4*ξˆ2*α 2*ξˆ2*αˆ2+45/2*ξ*αˆ3 2*ξˆ3*αˆ2+225/4*ξˆ2*αˆ3+ 945/8*αˆ5+1/2*ξˆ5

β2 1/3 1/3*ξ+11/18*α 11/9*ξ*α+85/ 11/6*ξˆ2*α+575/108*αˆ3+1/ 575/27*ξ*αˆ3+3661/162*αˆ4+22/ 18305/162*ξ*αˆ4+55/18*ξˆ4*α+ + + + + + 666 54*αˆ2 1/3*ξˆ2 3*ξˆ3 85/18*ξ*αˆ2 9*ξˆ3*α 85/9*ξˆ2*αˆ2 1/3*ξˆ4 425/27*ξˆ3*αˆ2+2875/54*ξˆ2*αˆ3 113155/972*αˆ5+1/3*ξˆ5

β3 1/4 1/4*ξ+25/48*α 25/24*ξ*α+415/ 1/4*ξˆ3+25/16*ξˆ2*α+5845/ 415/48*ξˆ2*αˆ2+5845/288*ξ*αˆ3+ 380555/3456*ξ*αˆ4+125/48*ξˆ4*α+ 288*αˆ2+1/4*ξˆ2 1152*αˆ3+415/96*ξ*αˆ2 76111/3456*αˆ4+25/12*ξˆ3*α+ 2075/144*ξˆ3*αˆ2+ 29225/576*ξˆ2*αˆ3+ 1/4*ξˆ4 4762625/41472*αˆ5+1/4*ξˆ5

β4 1/5 1/5*ξ+137/300*α 1/5*ξˆ2+137/ 1/5*ξˆ3+137/100*ξˆ2*α+12019/ 137/75*ξˆ3*α+12019/1500*ξˆ2*αˆ2+ 58067611/540000*ξ*αˆ4+137/ 150*ξ*α+12019/ 3000*ξ*αˆ2+874853/180000*αˆ3 874853/45000*ξ*αˆ3+ 1/5*ξˆ4+ 60*ξˆ4*α+12019/900*ξˆ3*αˆ2+ 874853/ 9000*αˆ2 58067611/2700000*αˆ4 18000*ξˆ2*αˆ3+3673451957/ 32400000*αˆ5+1/5*ξˆ5

β5 1/6 1/6*ξ+49/120*α 49/60*ξ*α+13489/ 49/40*ξˆ2*α+1/6*ξˆ3+13489/ 1/6*ξˆ4+68165041/3240000*αˆ4+49/ 68165041/648000*ξ*αˆ4+49/ 10800*αˆ2+1/6*ξˆ2 3600*ξ*αˆ2+336581/72000*αˆ3 30*ξˆ3*α+13489/1800*ξˆ2*αˆ2+ 24*ξˆ4*α+13489/1080*ξˆ3*αˆ2+ 336581/ 336581/18000*ξ*αˆ3 7200*ξˆ2*αˆ3+483900263/ 4320000*αˆ5+1/6*ξˆ5 8 CONCLUSIONS

Liaw and Zheng (2004) calculated drag forces on cylinders by polynomial approximations in which the coefficients are estimated by least-squares and moment methods. In this paper, the coefficients will be estimated with L-Moment methods. Its advantages were shown to be: • An L-Moments approach leads to a linear solution of the polynomialisation of drag forces; • Other distribution types for the turbulent current and wave heights acting on the cylinder can be analysed fairly simple.

REFERENCES

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. v 198 n 1–4 Nov 1997. p 245–259. Bobée, B., Ashkar, F., and Perreault, L., 1993. Two kinds of moment ratio diagrams and their applications in hydrol- ogy, Stochastic Hydrol. Hydraul., 7, 41–65. Chih Young Liaw and Xiang Yuan Zheng, Least-Squares, Moment-Based, and Hybrid Polynomializations of Drag Forces, 294/Journal of Engineering Mechanics © ASCE/March 2004. Demuth, S., and Kuells, C., 1997. Probability analysis and regional aspects of droughts in southern Germany. Symp 1: Sustainability of Water Resources under Increasing Uncertainty IAHS Publication (International Association of Hydrological Sciences). n 240. Duan, J., Selker, J., and Grant, G.E., 1998. Evaluation = Figure 2. a and b: Comparison of r (c 0.2) and polyno- of probability density functions in precipitation mod- = = = mial approximation of r (a0 0.114; a1 1.13; a2 0.74; els for the Pacific Northwest, Journal of the Ameri- =− =− a3 0.14; a4 0.17). can Water Resources Association. v 34 n 3 Jun 1998. p 617–627. of r to those of r: Durrans, S.R., 1992. Distributions of fractional order statis- tics in hydrology, Water Resour. Res., 28(6), 1649–1655. Greenwood, J.A., Landwehr, J.M., Matalas, N.C., and Wallis, J.R., 1979. Probability weighted moments: Def- inition and relation to parameters of several distributions 7 EXAMPLE expressable in inverse form. Water Resources Research, 15(5), 1049–1054. = Harter, H.L., 1969. Order Statistics and Their Use in Testing If we take c 0.2, then the polynomial coefficients. ao and Estimation. Volume 2, Aerospace Research Labora- –a4 as given in eqn. (34) are obtained as: tories, U.S. Air Force, Washington DC, USA. 0.1140 Hosking, J.R.M., Wallis, J.R., and Wood, E.F., 1985. Estima- 1.1349 tion of the Generalized Extreme Value Distribution by the Method of Probability Weighted Moments, Technomet- 0.7380 − rics, August 1985, Vol. 27, No.3, pp. 251–261. 0.1408 Hosking, J.R.M., 1986. The theory of probability weighted −0.1699 moments, Research Rep. RC 12210, 160 pp., IBM 0.0481 Research Division, Yorktown Heights, NY. Hosking, J.R.M., and Wallis, J.R., 1987. Parameter and quan- The exact and approximate polynomial relation for the tile estimation for the generalized Pareto distribution, drag force are graphically compared in Fig. 2a and 2b. Technometrics, 29(3), 339–349. The accuracy of approximation reduces for the higher Hosking, J.R.M., 1990. L-moments: Analysis and estima- values of y. The coefficients ai are estimated with the tion of distributions using linear combinations of order L-moment-based method. statistics. J. R. Stat. Soc. Ser. B., vol. 52, 105–124.

667 Hosking, J.R.M., 1992. Moments or L-moments? An exam- data using L-moments. International Hydrology andWater ple comparing two measures of distribution shape. The Resources Symposium 1991 Part 2. v2n91pt22.p American , 46(3), 186–189. 631–632. Hosking, J.R.M., 1997. Fortran Routines for Use with the Rao, A.R., and Hamed, K.H., 1997. Regional frequency anal- Method of L-Moments. IBM Research Report, RC20525, ysis of Wabash River flood data by L-moments, Journal Yorktown Heights, NY. of Hydrologic Engineering. v 2 n 4 Oct 1997. p 169–179. Hosking, J.R.M., and Wallis, J.R., 1997. Regional Frequency Reiss, R.-D., 1989. Approximate Distributions of Order Analysis: An Approach based on L-Moments. Cambridge Statistics, With Applications to , University Press, Cambridge, UK. Spinger Verlag Series in Statistics. Johnson, N.L., Kotz, S., and Balakrishnan, N., 1994. Contin- Rosbjerg, D., Madsen, H., and Rasmussen, P.F., 1992. Pre- uous univariate distributions. Vol. 1 Publisher New York: diction in partial duration series with generalized Pareto- Wiley, 1994 ISBN 0-471-58495-9, 756 pages. distributed exceedances, Water Resour. Res., 28(11), p. Landwehr, J.M., Matalas, N.C., and Wallis, J.R., 1979. Proba- 3001–3010. bility weighted moments compared with some traditional Van Gelder, P.H.A.J.M., and Neykov, N.M., 1998. Regional techniques in estimating Gumbel parameters and Quan- frequency analysis of extreme water levels along the tiles, Water Resources Research, Vol. 15., No.5, October Dutch coast using L-moments: A preliminary study, In: 1979, pp. 1055–1064. Stochastic models of hydrological processes and their Pearson, C.P., McKerchar, A.I., and Woods, R.A., 1991. applications to problems of environmental preservation, Regional flood frequency analysis of Western Australian pp. 14–20.

668