Population L Moments

Statistical moments are used to characterize a series of observed data or a PDF. The L moments and trimmed L moments were established as alternative procedures to conventional moments, since they have lower sampling variance and are more robust when there are disperse values. Like conventional moments, those of order one to four characterize the location, scale, asymmetry, and kurtosis (^{Hosking, 1990}; ^{Elamir and Seheult, 2003}).

According to ^{Karvanen (2006)}, the concept of L moments was created at the end of the 1960’s, with various isolated results on linear combinations of the order statistics, which culminated with the work by ^{Greenwood et al. (1979)}. ^{Hosking (1990)} unified the theory of L moments and provided guides for their practical use, which have extended to the areas of hydrology, meteorology, and other disciplines of engineering.

moments are related to the expected values of the order statistics. Let X a random variable and Xj:n its order statistic, another random variable distributed as its j-th smallest element from a random sample with size n, extracted from the PDF of X. Also, let Q(u) the so-called quantile function, i.e., the reverse solution of the PDF equation, is then a function that is increased in the interval of uϵ[0,1]. The first four population L moments are (^{Hosking, 1990}; ^{Hosking y Wallis, 1997}; ^{Karvanen, 2006}):

its general expression is:

where the second parenthesis is a quotient of factorials that defines the number of possible combinations of the m terms, taking q in each arrangement; its general expression is (^{Asquith, 2011}):

By conventions 0! = 1 and the Gamma factorial function is Г(m+1)=m! The values expected of the order statistics of Equation 5 are estimated using the following expression (^{Hosking and Wallis, 1997}; ^{Kottegoda and Rosso, 2008}; ^{Asquith, 2011}):

The above equation establishes the relation between the probability weighted moments by ^{Greenwood et al. (1979)} and the L moments. The quotients of L moments of asymmetry for r=3 and kurtosis for r=4 are:

Trimmed population L moments (1,1)

These moments, designated as “TL” for “Trimmed,” which means depurated, cut, or truncated, give no weight to extreme observations and are therefore robust generalizations of the L moments. Its population expressions are (^{Elamir and Seheult, 2003}; ^{Hosking, 2007}):

its general expression is:

Trimmed L moments (1,1) of the sample

In a sample or series of data sized n, ordered progressively in such a way that X1:n≤X2:n≤...≤Xn:n, the estimation is not biased for l _r ^(t) , because s=t, it will be (^{Elamir and Seheult, 2003}; ^{Hosking, 2007}):

where l _r ^(t) is a symmetrically pondered linear combination of a trimmed or truncated sample sized (n-2t) of values ordered X _t+1:n , ..., X _n-t:n . The quotients of trimmed L moments (1,1) of the sample shall be: t ₃ ^(1,1) and t ₄ ^(1,1), evaluated with Equation 8.

Fittings with the L moments of the distributions GEV, GLO, and GPA

^{Hosking and Wallis (1997)} point out in their Table 5.1 that these three PDFs, when their shape parameters are negative (k<0), have ticker or more dense right tails than all other PDFs commonly used in the FFA. Due to this, they have gained acceptance in the frequency analysis of hydrological extreme data (^{El Adlouni et al. 2008}). Such PDFs, when k=0, define functions with two fitting parameters known as Gumbel, logistic, and exponential. These PDFs also coincide in having a top limit when k>0.

Below are the formula F(x), and quantile function x(F) used for distributions GEV, GLO, and GPA, to estimate the predictions that relate to a certain probability of non-exceedance (F) and the equations that help estimate its three fitting parameters (k, a, u) for form, scale, and location, using the method of L moments.

GEV distribution (^{Stedinger et al., 1993}): interval of x: u+a/k≤x<¥ si k<0; -¥<x<¥ si k=0; -¥<x≤u+a/k si k>0.

where y is the reduced variable:

where:

To evaluate the Gamma function, the Stirling formula (^{Davis, 1965}) was used:

GLO distribution (^{Rao and Hamed, 2000}): interval of x, identical to that of GVE.

where y equals Equations 16 and 17.

GPA distribution (^{Hosking and Wallis, 1997}): interval of x:u≤x<∞ si k≤0;u≤x≤u+a/k si k>0

where y equals Equations 16 and 17.

Using the L moments and quotients displayed by ^{Campos- Aranda (2014)}, the three fitting parameters of distributions GEV, GLO, and GPA were estimated using Equations 20 to 24, 28 to 30, and 34 to 36, respectively.

Fitting with trimmed L moments (1,1) of the distributions GEV, GLO, and GPA

For the GEV distribution, ^{Deka and Borah (2011)} developed an equation of t₃ ^(1,1) based on the parameter of shape (k), whose reverse empirical expression is:

which is valid when -0.70≤k≤0.50 and has a determination coefficient close to one and a standard error of estimation of 0.00261. Equations 38 to 45 come from ^{Deka and Borah (2011)}, ^{Shabri et al. (2011)}, and ^{Ahmad et al. (2013)}.

For the GLO distribution, we have:

and for GPA distribution:

Diagram of quotients for trimmed L moments (1,1)

^{Hosking and Wallis (1997)} displayed the graphic and numeric relation (Equation 46) between the quotient of asymmetry L moments (t₃) and that for kurtosis (t₄) in five PDFs, which constitutes the diagram for L moment quotients. This graph can be used to select the best PDF, according to the values of the quotients for asymmetry and kurtosis L moments (t ₃ and t ₄) of the local sample, or to its regional pondered estimation. To obtain the diagram of quotients of trimmed L moments (1,1), a relation must be found between t₃ ^(1,1) and t₄ ^(1,1) for each PDF. ^{Shabri et al. (2011)} and ^{Deka and Borah (2011)} obtained such theoretical relations and represented them with fourth-degree polynomials for the distributions GVE and GPA, and second-degree polynomials for the GLO; the coefficients of Equation 46 of each polynomial are in Table 1.

Based on Equation 46 and its coefficients (Table 1), a diagram was created for trimmed L moments (1,1) for the distributions GEV, GLO, and GPA (Figure 1).

Standard Error of Fit

In the mid-1960’s, the standard error of fit (SEF) was established as a quantitative statistical indicator, since it evaluates the standard deviation of the differences between the values observed and estimates with the PDF tested. Its expression is as follows (^{Kite, 1977}):

where n and np are the number of data in the samples and fitting parameter, in this case, 3; x _i are the data arranged from lowest to greatest, and x^1 are the values estimated with the reverse solution x(F) for a probability of non-exceedance estimated using Weibull’s formula (^{Benson, 1962}):

In which m is the number of the order of data, with 1 for lowest, and n for the greatest.

Qualitative indicators of the fitting

The diagnostic diagrams for probability and for quantities have become popular (^{Coles, 2001}; ^{Wilks, 2011}); the first uses in the horizontal axis the empirical probability estimated using Equation 48, and in the vertical axis, the probability that defines the PDF adjusted for all data available, ordered by magnitude (x ₁≤x ₂≤...≤x _n ), which is estimated with Equations 15, 25, and 31 for distributions GEV, GLO, and GPA. The graph for quantities indicates, on the horizontal axis, the value of the data observed x ₁≤x ₂≤...≤x _n , and on the vertical, the value estimated using the reverse solution of the PDF using Equations 18, 26, and 32 for functions GEV, GLO, and GPA, and the empirical probability, estimated using Equation 48. In sum, the coordinates for x and y of each diagram are (^{Coles, 2001}):

Records processed of annual floods

Among the records for maximum annual discharge (m³·s^-1) in the hydrometric stations of the Hydrological Region No. 10 (Sinaloa; Mexico), 21 have no runoff regime affected by reservoirs, drastic physical changes or both in their basins, and they display data of over twenty years in the BANDAS system (^{IMTA, 2002}). The records in the hydrometric stations Huites and Guamuchil reach up to the year in which the construction of the respective reservoir affected its runoff. The data from the San Francisco station come from the Hydrological Bulletin No. 36 (^{SRH, 1975}). ^{Campos-Aranda (2014)} verified the statistical quality of the records cited and presents the L quotients and moments, and displays a map with the location of the 21 hydrometric stations and their respective basins.

The shortest record have 19 years old and the longest, 56, with an average value of 33 years (Table 2). The fitted L moments (1,1) estimated using Equation 14 and the quotients of the moments calculated with Equation 8 are shown in Table 2.

Selection of the most convenient PDF

Each record of floods was taken into the diagram of quotients of trimmed L moments (1,1) to obtain the most convenient PDF according to its L quotients for asymmetry and kurtosis (Table 2). Eleven records came close to the GPA function, seven to GLO, and three could be represented with the GEV distribution (Figure 1) (Table 3).

Standard error of fit

Using the reverse solutions of PDF (Equations 18, 26, and 32) the x^1 estimations were obtained to apply Equation 47 of the SEF (Table 3, columns moL).

With the trimmed L moments and quotients (1,1) (Table 2) and Equations 37 to 45, the three fitting parameters were obtained for functions GEV, GLO, and GPA. Afterwards, using their reverse solutions (Equations 18, 26, and 32) the values of x^1 were estimated to calculate the respective SEFs (Table 3, columns moLD).

The lowest SEF of the six calculated was identified (Table 3, in circular parentheses) and equal values occurred in the records for Chinipas and Zopilote, selecting the adjustment that provided the largest predictions (Table 3, in rectangular parentheses). The minimum SEFs defined by distributions GEV, GLO, and GPA coincided with PDFs in eleven records (Figure 1): San Francisco, Ixpalino, La Huerta, Tamazula, Naranjo, Acatitán, Zopilote, Chico Ruiz, El Bledal, La Tina, and Bamícori.

The fitting of distributions GEV, GLO, and GPA, based on the trimmed L moments (1,1) (Table 3, last line) is an option that can systematically take down the SEF of the method of L moments, since this took place in ten records. The fitting using this method of the GLO function stands out, showing the lowest SEFs in seven records. The GPA distribution also stood out, which led to the highest fittings in eleven of the 21 records processed, nine of them with the method of L moments.

The final result must not guide to the exclusive application of such PDF, since the records for Santa Cruz, Palo Dulce, Tamazula, Choix, Badiraguato, and La Tina, with the other functions, gave minimum SEFs, nearly equal, and in several records, such as Ixpalino, Chinipas, Guamuchil, El Quelite, and Pericos, similar SEFs were obtained. Due to this, we recommend the application of under precepts to the distributions GEV and GLO.

Predictions obtained with the best fit

The predictions with each of the best adjustments in the records processed (third column in Table 4) were obtained according to the moLD results (Table 3). For the record of the Jaina hydrometric station, the method of L moments produced SEF of around 360 m³·s^-1 with distributions GEV, GLO, and GPA; on the other hand, using the procedure of the trimmed L moments (1,1) this was reduced to 270 m³·s^-1 with the function GLO. For this fitting in the respective diagnostic diagrams, the fitting reached was verified (Figures 2 and 3).