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Tecnología y ciencias del agua
Online version ISSN 20072422
Tecnol. cienc. agua vol.10 n.5 Jiutepec Sep./Oct. 2019 Epub Feb 15, 2020
https://doi.org/10.24850/jtyca20190502
Articles
Best PDF in 19 large annual series of MDP from the San Luis Potosi state, Mexico.
^{1}Profesor jubilado de la Universidad Autónoma de San Luis Potosí, México, campos_aranda@hotmail.com
All hydraulic works are planned and designed based on Floods Design. Without hydrometric information, these predictions are estimated using hydrological methods that yield the sought flows by means of design rainfalls. Design rainfall is estimated based on pluviometer records of annual maximum daily precipitation (MDP) due to the shortage of pluviographs. The probabilistic analysis of the annual MDP series is identical to that of the floods; however, neither adequate probability distribution functions (PDFs) nor those that should be applied by precept have been defined so far, hence the need to try several. First, the best PDF was searched for using the Lratio diagram, which includes six models with three fit parameters. An objective selection is made by using the weighted absolute distance, in the 19 annual MDP records with more than 50 data from the state of San Luis Potosi, Mexico. Then eight descriptive ability (DA) indexes are described and applied to the eight PDFs that were compared, in each of the 19 PMD records. The results are concentrated and analyzed for geographic areas of the state: Potosino Plateau and Middle Zone. Results show that Wakeby PDF is a model having high DA and for that reason, its application is suggested as precept. The two best PDF options are also highlighted in each of the 19 records processed, according to the eight DA indexes. Finally, a comparison of predictions with periods of return of 50, 100, 500 and 1000 years is carried out to explore shallowly the predictive ability of the PDFs found as best options. In each registry four PDFs are applied, the one obtained according to the Lratio diagram; the two best PDFs according to the eight DA indices and the Wakeby distribution. It is concluded that the use of the Lratio diagram and the application of the eight DA indexes are adequate and lead to a good approximation, since it was not difficult to select the adopted predictions, besides the similarity of the predictions calculated in each register promotes confidence in such estimations.
Keywords: Lratio diagram; standard error of fit; relative standard error of fit; mean absolute error; maximum absolute error; Akaike information criterion; QQ correlation coefficient; concordance indexes and predictive ability
Todas las obras hidráulicas se planean y diseñan con base en las crecientes de diseño. Sin información hidrométrica, estas predicciones se estiman con métodos hidrológicos que transforman lluvias de diseño en los gastos buscados. La escasez de pluviógrafos origina que las lluvias de diseño se estimen a partir de los registros de precipitación máxima diaria (PMD) anual de los pluviómetros. El análisis probabilístico de las series de PMD anual es idéntico al de las crecientes; pero aún no se han definido funciones de distribución de probabilidades (FDP) adecuadas o que se deban aplicar bajo precepto, por lo cual es necesario probar varias. Primero se buscó la mejor FDP en el diagrama de cocientes L, que incluye seis modelos de tres parámetros de ajuste. Se realiza una selección objetiva al emplear la distancia absoluta ponderada en los 19 registros de PMD anual con más de 50 datos del estado de San Luis Potosí, México. Después se describen y aplican ocho índices de habilidad descriptiva (HD) a las ocho FDP que fueron contrastadas en cada uno de los 19 registros de PMD. Los resultados se concentran y analizan por áreas geográficas del estado: Altiplano Potosino y Zona Media. Se obtuvo que la FDP Wakeby es un modelo de gran HD y por ello se sugiere que su aplicación se realice bajo precepto. También se definen las dos mejores opciones de FDP en cada uno de los 19 registros procesados de acuerdo con los ocho índices de HD. Por último, se realiza un contraste de predicciones con periodos de retorno de 50, 100, 500 y 1000 años, para explorar de manera somera la habilidad predictiva de las FDP encontradas como mejores opciones. En cada registro se aplican cuatro FDP: la obtenida según el diagrama de cocientes L, las dos mejores FDP según los ocho índices de HD y la distribución Wakeby. Se concluye que el uso del diagrama de cocientes L y la aplicación de los ocho índices de HD son adecuados y conducen a una buena aproximación, pues no se tuvo dificultad para seleccionar las predicciones adoptadas y la similitud que mostraron estas estimaciones en cada registro genera confianza en tales estimaciones.
Palabras clave: diagrama de cocientes L; error estándar de ajuste; error relativo estándar de ajuste; error absoluto medio; error absoluto máximo; criterio de información de Akaike; coeficiente de correlación de QQ; índices de concordancia y habilidad predictiva
Introduction
The Design Floods hydrologically size all the hydraulic works, in the different stages that they cross. In sites of interest and their respective basins, which do not have annual maximum flow information, the Design Floods must be estimated based on hydrological methods that transform a design rainfall into the response hydrograph or the sought peak flow (^{Mujumdar & NageshKumar, 2012}). Design rainfalls come from the IntensityDurationFrequency (IDF) curves that characterize the way it rains in the study area. The shortage of rainfall recorders prevents the construction of the IDF curves and therefore its estimation is used, based on the available records of the pluviometric or raingauge stations of wider coverage and larger records (^{Teegavarapu, 2012}; ^{Johnson & Sharma, 2017}).
The annual maximum daily precipitation records (PMD, for its Spanish initials) are probabilistically processed in an identical way as those of annual maximum flow or floods; going through four stages: (1) search for records, including debugging and verification of their homogeneity; (2) choice of a cumulative probability distribution function or FDP (for its Spanish initials), that is, of the probabilistic model that will allow obtaining the predictions associated with low probabilities of exceedance; (3) application of one or more methods of estimation of the fit parameters of FDP and (4) validation of the adopted FDP and its predictions ^{Rao & Hamed, 2000}; ^{Meylan, Favre, & Musy, 2012}; ^{Stedinger, 2017}).
The objective of the study was to select the best FDPs, which should be applied in the probabilistic analysis of annual PMD series. First, the Lratio diagram is exposed and applied through the weighted absolute distance to objectively adopt the best FDP of the six that it includes. Then, an approach based on eight descriptive ability indexes for selection is followed, among the eight FDPs that are contrasted. The concentrated results of the Potosino Plateau and Middle Zone of the state of San Luis Potosí, Mexico, are exposed and analyzed, in which 9 and 10 annual PMD records were processed, with 50 or more data. Finally, a contrast of predictions with return periods of 50, 100, 500 and 1000 years is made and conclusions are drawn regarding the predictive capacity of the selected FDPs.
Data of PMD processed
Debugged series
Based on the Excel file updated until 2015 of the San Luis Potosí weather stations, provided to the author by the Local Office of the National Water Commission (Conagua), all records of annual maximum daily precipitation (PMD) were selected with more than 40 values and scarce missing data, 100 series were obtained (^{CamposAranda, 2018}). Then a ratification of their minimum and maximum extreme values was carried out with the aid of CONAGUA, to obtain the socalled debugged series.
Homogeneity tests applied
The Neumann Von Test was applied to each debugged series as a general test, which detects nonrandomness by deterministic components unspecified and several specific tests: Anderson and Sneyers of persistence, Kendall and Spearman of trend, Bartlett of inconsistency in dispersion and Cramer of changes in the mean. These tests can be found in ^{WMO (1971)}, and ^{Machiwal and Jha (2008)}. It was found that a total of 39 series were random or not, or they presented persistence and/or trend.
Series to be processed
The 39 nonhomogeneous series were eliminated, as well as those with less than 50 data; with these restrictions, 35 records of annual PMD were available in the state of San Luis Potosí. Nine series belong to the Potosino Plateau (AP), ten to the middle zone (ZM) and 16 to the Huasteca Region. In this study the 19 series that are located in the arid and semiarid climates of the geographical areas AP and ZM, whose altitudes are generally higher than one thousand meters, were processed. Table 1 shows their altitude, record width, statistical values and Lmoment ratios (Equations (6) to (8)). The first nive climatological stations belong to the AP and the 10 remaining to the ZM. Figure 1 shows their geographical location in the state of San Luis Potosí; Mexico.
No.  Station name  Altitude (masl^{1}) 
Record  PMD  

Period  n ^{2}  min  Max  
1  Cedral  1702  19462014  66  19.0  315.8 
2  Charcas  2126  19612014  54  12.0  117.0 
3  La Maroma  1900  19652014  50  16.0  140.1 
4  Los Filtros (SLP)  1904  19492014  66  15.9  111.0 
5  Matehuala  1630  19612014  54  25.5  200.0 
6  Mexquitic  1749  19432014  72  12.0  107.0 
7  Peñón Blanco  2099  19502014  57  13.0  235.0 
8  Santo Domingo  1415  19612013  52  19.0  270.0 
9  Vanegas  1746  19642013  50  12.0  90.0 
10  Armadillo de los Infante  1636  19612013  52  22.0  133.0 
11  Cárdenas  1353  19462013  61  21.5  180.5 
12  Lagunillas  908  19542013  53  30.0  210.0 
13  Ojo de Agua  1148  19602013  52  45.0  300.2 
14  Ojo de Agua Seco  1077  19612013  51  26.5  172.5 
15  Paso de San Antonio  1246  19582013  52  26.0  200.0 
16  Rayón  1258  19612013  51  33.5  330.0 
17  Río Verde  987  19612013  52  27.0  126.3 
18  San Francisco  1066  19612013  50  12.0  135.0 
19  San José Alburquerque  1863  19612014  50  21.0  126.5 
Lratio diagram
Lmoments of the sample
Lmoments are linear combinations of the moments of weighted probability (b
_{
r
} ), for that reason they are robust before the dispersed values of sample. Their calculation begins by ordering the available series (x
_{
i
} ) of annual PMD from lowest to highest (
In the previous expression the order number r varies from 0 to 3 and n is the data number of the annual PMD series. It follows that b _{0} is equal to the arithmetic mean. The Lmoments of the sample (l) and their respective quotients (t) of similarity with the coefficients of variation, asymmetry and kurtosis are:
Lratio diagram
It has on the abscissa axis t _{3} and on the ordinate t _{4}. The FDPs of three fit parameters are curved lines with the following polynomialtype equations (^{Hosking & Wallis, 1997}):
Generalized Logistics (LOG):
Generalized Pareto (PAG):
LogNormal (LGN):
Pearson type III (PT3):
and General of Extreme Values (GVE):
being:
Using the logarithms of the data in Equations 1 to 8, the logarithmic L ratios are obtained and then the expression 12 can be used to evaluate the FDP LogPearson type III. Figure 2 shows a portion of the Lmoments ratio diagram from ^{Hosking and Wallis (1997)}.
Absolute weighted distance
One of the recent approaches to choose the best FDP, at the local and regional level, consists of taking to the Lratio diagram the values of the sample (t _{3} and t _{4}) and defining its proximity to any of the curves, in order to obtain the best probabilistic model. This is relatively simple in local analyses, but it is complicated in the regional approach as pointed out by ^{Peel, Wang, Vogel and McMahon (2001)}, since then there is a cloud of points. To avoid subjectivity in the selection of the FDP, it has been proposed to evaluate the Absolute Weighted Distance (DAP) with the following expression (^{Yue & Hashino, 2007}):
where NE is the number of stations that make up the region, n
_{
j
} is the data number of each PMD record,
Distribution functions to be contrasted
Based on the Lratio diagram, six FDP can be tested and as accepted a priori to eliminate the one that was not selected at least once (see Table 3) in the 19 annual PMD records to be processed, the Pearson Type III distribution was not contrasted and then the LOG, PAG, LGN, GVE and LP3 models will be tested.
Sta. No. 
Statistical parameters^{3}  Lmoment ratios^{4}  


l _{2}  S  Cs  t _{3}  t _{4} 



1  47.1  13.465  38.6  5.501  0.49274  0.41041  0.20245  0.20806 
2  48.8  12.068  22.0  0.974  0.17331  0.18124  0.06888  0.17927 
3  46.6  10.807  21.3  2.006  0.27042  0.21953  0.05484  0.16449 
4  43.0  8.448  15.7  1.315  0.13515  0.16117  0.04764  0.13773 
5  59.3  14.243  29.2  2.471  0.28588  0.23605  0.06420  0.13996 
6  47.8  9.521  17.1  0.540  0.05475  0.15489  0.14445  0.17245 
7  47.8  15.395  40.2  3.669  0.51286  0.44109  0.19048  0.26367 
8  57.6  16.564  37.8  3.685  0.31725  0.27714  0.02817  0.14591 
9  37.6  10.013  18.4  1.028  0.22725  0.16424  0.00150  0.14389 
10  57.5  14.504  27.3  1.287  0.28601  0.18574  0.08089  0.14044 
11  67.7  20.216  38.9  1.457  0.33709  0.18138  0.11049  0.12095 
12  77.8  18.480  35.0  1.525  0.22455  0.15699  0.02475  0.10105 
13  91.4  21.459  46.5  2.586  0.38619  0.31739  0.16556  0.20467 
14  69.3  14.883  28.9  1.694  0.28166  0.21420  0.08873  0.16202 
15  69.3  13.793  27.6  2.219  0.22726  0.23726  0.01936  0.18991 
16  76.4  19.343  45.5  3.765  0.39215  0.34343  0.13435  0.21192 
17  58.4  12.973  23.4  0.934  0.20926  0.09728  0.05010  0.06390 
18  46.7  12.757  24.4  1.461  0.23966  0.24363  0.03897  0.21540 
19  50.1  11.466  21.7  1.434  0.19616  0.16114  0.00396  0.08689 
Symbols:
^{1}meters above sea level.
^{2}number of processed data.
^{3}
l_{1}, l_{2}L moments of order 1 and 2.
Sstandard deviation, in millimeters.
Cscoefficient of asymmetry, dimensionless.
^{4}t _{3}asymmetry Lratio, dimensionless.
t_{4}kurtosis Lratio, dimensionless.
No.  Station  DAP and FDP  

1st option  2nd option  3rd option  
1  Cedral  0.0414  0.0511  0.0717 
LOG  GVE  LP3  
2  Charcas  0.0105  0.0299  0.0350 
LOG  GVE  LGN  
3  La Maroma  0.0081  0.0218  0.0394 
LOG  GVE  LGN  
4  Los Filtros (SLP)  0.0146  0.0207  0.0238 
LP3  LOG  GVE  
5  Matehuala  0.0013  0.0163  0.0296 
LOG  LP3  GVE  
6  Mexquitic  0.0143  0.0297  0.0316 
LOG  LGN  PT3  
7  Peñón Blanco  0.0552  0.0635  0.1078 
LOG  GVE  LGN  
8  Santo Domingo  0.0233  0.0266  0.0517 
LP3  LOG  GVE  
9  Vanegas  0.0011  0.0110  0.0215 
LGN  GVE  LP3  
10  Armadillo de los Infante  0.0012  0.0160  0.0208 
LGN  LP3  GVE  
11  Cárdenas  0.0053  0.0115  0.0132 
LP3  PAG  PT3  
12  Lagunillas  0.0052  0.0169  0.0170 
LGN  GVE  PT3  
13  Ojo de Agua  0.0264  0.0450  0.0733 
LOG  GVE  LP3  
14  Ojo de Agua Seco  0.0102  0.0186  0.0292 
GVE  LOG  LGN  
15  Paso de San Antonio  0.0276  0.0620  0.0674 
LOG  GVE  LP3  
16  Rayón  0.0486  0.0666  0.0838 
LOG  GVE  LP3  
17  Río Verde  0.0148  0.0401  0.0593 
PAG  PT3  LP3  
18  San Francisco  0.0291  0.0622  0.0759 
LOG  GVE  LGN  
19  San José Alburquerque  0.0002  0.0083  0.0258 
GVE  LGN  PT3 
The eight FDPs that were contrasted include the Beta Kappa and Beta Pareto models proposed by ^{Mielke and Johnson (1974)} that are not popular in Mexico, but that were compared in a pioneering work of optimal selection of distributions with three fitting parameters in records of annual PMD, that of ^{Wilks (1993)}, with good results for Betaκ (BEK) in series of maximums and values above a threshold for BetaP (BEP): ^{Nguyen, El Outayek, Lim, and Nguyen (2017)} also include them in their contrast.
Finally, the FDP Wakeby (WAK) was included, which has five fit parameters, it was proposed by ^{Houghton (1978)} and allows the left and right ends of the sample to be modeled separately. ^{Nguyen et al. (2017)} find that the Wakeby distribution is the best in descriptive ability.
To avoid more variables involved in the selection of the best FDPs, in models that have several estimation methods of their fit parameters, the Lmoments method was exclusively applied, which has proven to be consistent and accurate. Such method in the GVE, LOG and PAG models is not exposed, since it has been described in several articles by the author, for example in ^{CamposAranda (2015}; ^{2016}). It can also be consulted in ^{Hosking and Wallis (1997)}, ^{Rao and Hamed (2000)} and ^{Stedinger (2017)}. The FDPs LogNormal (LGN) and Wakeby (WAK) were fitted with the Lmoments method described by ^{Hosking and Wallis (1997)}.
Regarding the LogPearson type III distribution (LP3) it was fitted with the moments method, in the logarithmic and real domains (^{WRC, 1977}; ^{Bobée & Ashkar, 1991}; ^{CamposAranda, 2002}), selecting that of lower standard error of fit (^{Kite, 1977}). The Beta distributions were fitted with the iterative method of maximum likelihood of ^{Mielke and Johnson (1974)}, adopting as initial value of the scale parameter the mean of the PMD record and of the shape parameter a value of five. The maximum of iterations was taken to two thousand.
Descriptive ability indexes
Diagnostic Graphics
The PP and QQ graphs of empirical versus calculated probability and observed versus estimated quantity have become popular (^{Coles, 2001}; ^{Wilks, 2011}) and are a simple and effective way to compare the results of a FDP contrasted. Their disadvantage lies in the subjective appreciation that is made when comparing various FDPs, with such graphs, since a numerical value is not available (^{Nguyen et al., 2017}). ^{CamposAranda (2015)} has exposed such graphs and visualizes more useful the QQ graph, to observe predictions overestimated (for being above the line at 45°) or underestimated (for being below the line at 45°).
Statistical tests
These tests, like the Chisquare, the KolmogorovSmirnov or AndersonDarling, can justify that a sample can be accepted coming from a specific distribution that is analyzed. In the first two tests, their disadvantage lies in having low power and in the second, only being applicable for a specific FDP (^{Meylan et al., 2012}).
Goodnessoffit indexes
They have the advantage of being of easy calculation and commonly involve the difference between the observed values x
_{
i
} and those estimated with the FDP that is contrasted
Empirical probability formula
^{Meylan et al. (2012)} indicate that all the empirical equations that assign probabilities are based on ordering the available sample or series in upward magnitudes, making it possible to associate an order i with each datum and then make use of the following general formula, which ensures symmetry with respect to the median:
in which, c is a constant quantity and n is the size of the record or series of the annual PMD.
^{Haktanir (1991)} describes a practical finding of 1971 by J.R. Stipp and G.K. Young who processed 37 annual flood records of exactly 20 data each, all in the USA. They estimated the probability of each maximum and minimum event of each series based on the LogPearson type III distribution and then equated those values with the one obtained from the Equation 15, finding that c had an approximate magnitude of 0.40, whereby:
^{Haktanir (1991)} also points out that years later, Cunnane arrives at the same Equation 16 in a theoretical and independent way, stating that such formula is not specific of a FDP and that their results are unbiased and have a minimum square error. ^{Cunnane (1978)} also finds, with statistical arguments, that the popular Weibull formula (^{Benson, 1962}) is only applicable for a uniform distribution, so it is not convenient to use it in series of extreme hydrological values (floods, droughts, PMD, etc.).
Standard error of fit (EEA)
It is the most common index (^{Chai & Draxler, 2014}), it was established in the midseventies (^{Kite, 1977}) and has been applied in Mexico using the empirical formula of Weibull (^{Benson, 1962}). Now it will be applied using the Cunnane formula (Equation 16), which according to ^{Stedinger (2017)} leads to probabilities of nonexceedance (
in which, x
_{
i
} are the observed data ordered from lowest to highest,
Relative standard error of fit (EREA)
In the EEA all the differences or residuals are squared and this implies giving greater weight to the high values of the sample, to reduce this impact the EREA is applied, which is dimensionless, its equation is (^{Pandey & Nguyen, 1999}; ^{Nguyen et al., 2017}):
Mean Absolute Error (EAM)
Its advantages lie in having the units of the variable, like the EEA, and avoiding that the impact of the scattered values be squared and therefore EEA ≥ EAM (^{Willmott & Matsuura, 2005}). Its expression is (^{Pandey & Nguyen, 1999}; ^{Nguyen et al., 2017}):
Maximum Absolute Error (EAMx)
This index shows the largest of the errors or absolute residuals, for that reason ^{Nguyen et al. (2017)} have indicated that it is related to the statistics of the KolmogorovSmirnov test; its equation is:
Akaike Information Criterion (AIC)
This index uses in its original conception the maximum value reached in the likelihood function during the fit, with such method, of the FDP that is contrasted. In order to apply such index, ^{Nguyen et al. (2017)} use the sum of squared errors (SEC) as indictor of the fitting quality and then its equation is:
In these first five indexes, the lowest value of them indicates the best fit of FDP and its maximum magnitude the worst fit of such contrasted FDP. In the following three indexes occurs the opposite, so that a maximum value indicates a good fit of the FDP and vice versa.
QQ Correlation Coefficient (COC)
This index has been used as the main selection criterion by ^{Zalina et al. (2002)}, it indicates the linear dependence that exists in the QQ graph; whereby, it varies from zero to one. The values of the COC close to the unit indicate a good fit of the FDP that is contrasted; its equation is:
where,
Concordance indexes (d_{1}, d_{2})
According to ^{Legates and McCabe (1999)}, Willmott pointed out since the beginning of the eighties that the COC index is limited and insensitive to the differences and variances between x
_{
i
} and
The numerator is the SEC and the denominator is called the potential error, because it is the maximum value that the difference between x
_{
i
} and
As d _{2} as d _{1} vary from zero to one, with an interpretation equal to COC; usually, d _{2} ≥ d _{1}.
Review of predictive ability
Available approaches
The predictive ability of the FDPs is related to the predictions made to return periods (Tr) higher than the size of the record (n), or to the comparison with the extreme values observed in the record. There are three approaches to test or verify such predictive ability, the first is the simplest and consists of the numerical contrast of the predictions obtained with each FDP for various adopted Tr.
The second approach of verification of predictive ability was exemplified by ^{Haktanir (1992)}, he uses random samples generated with basic models (LP3, GVE and WAK), which use the fit parameters obtained in the records adopted as representative of the geographical regions studied. The best FDPs are adopted based on the lowest EREA.
From the decade of the nineties (^{Wilks, 1993}; ^{Zalina et al., 2002}; ^{Nguyen et al., 2017)} a third approach of contrast has been proposed, based on random samples generated from the historical record, by sampling with replacement (bootstrap technique), which contrasts predictions within the record obtained with the tested FDPs, with the extreme values of such sample. The best FDPs are those that show less dispersion.
Adopted approach
It corresponds to the first and simplest, since it has the best FDP of each annual PMD record, according to the Lrates diagram and according to the eight indexes of descriptive ability applied. It consists of adopting one of the contrasted FDPs and their predictions, following a rule established a priori, for example, adopting the most critical or major values in most of the contrasted Tr; as long as such predictions are similar, which implies accuracy and generates confidence in the magnitudes adopted under such a subjective scheme.
Results according to the Lratio Diagram
The evaluation of the Weighted Absolute Distance (Equation (14)) in each of the six FDPs of the L momentratio diagram (Equations (9) to (13)), making use of the values in Table 2, provided the three minimum values shown in Table 3, thus defining the best FDP and the subsequent two at local level of each series of annual PMD. In Figure 2 the values of t
_{3} and t
_{4} of each record, taken from Table 1, have been indicated. Most of these points define their proximity to an FDP, except for the stations: Los Filtros (No. 4), Santo Domingo (No 8) and Cárdenas (No. 11), with proximity to the LP3 model due to its values of
According to the summary by geographical areas of Table 4, it is concluded that the first or best option of FDP is the Generalized Logistics (LOG) with 10 selections, followed by the LogNormal (LGN) and the LogPearson type III (LP3) with 3 selections and the least suitable one was the Pearson Type III (PT3) with no selection.
Results according to descriptive ability
General observations
Regarding the LogPearson type III (LP3) distribution, a lower standard error of fit was obtained in the real domain in the Charcas, Los Filtros, Mexquitic and Santo Domingo stations of the Potosino Plateau and in the Paso de San Antonio and San Francisco stations from the Middle Zone. In the remaining 13 stations, the best fit was obtained in the logarithmic domain.
In relation to the Wakeby distribution (WAK), in a total of nine records it was obtained that the location parameter (ξ) was slightly higher than the minimum value of the series, which is strictly incorrect. In these nine records, the Wakeby distribution was fitted with ξ = 0, according to the procedure of ^{Hosking and Wallis (1997)} and its results (descriptive ability and predictions indexes) were compared against the previous fits. Only in the San José Alburquerque station it was found more adequate according to the EAMx and COC indexes, as well as less dispersed predictions, that is, it improved its predictive ability.
Results in the Potosino Plateau
Concentrate of numerical indexes
The three characteristic values (minimum, medium and maximum) of each index of descriptive ability (HD) obtained with each of the eight contrasting FDPs in the 9 raingauge stations of the Potosino Plateau of the state of San Luis Potosí, Mexico, have been concentrated in Table 4. Table 5 summarizes the eight unexposed tabulations of the results of each index with the eight FDPs contrasted in the 9 annual PMD records of the Potosino Plateau. Exclusively for the mean values, the best respective index value is indicated with circular parenthesis, pointing out such magnitude the best PDF at regional level. The worst indexes at regional level are also marked with rectangular parenthesis.
DA Index  FDP contrasted  

BEK  BEP  LGN  GVE  LOG  PAG  LP3  WAK  
EEA mín  2.46  1.85  2.10  2.32  1.67  2.74  2.25  1.45 
EEA med  8.13  7.19  7.10  6.98  (6.37)  [8.35]  6.54  7.06 
EEA máx  20.40  16.42  15.68  14.85  14.80  15.96  13.40  15.73 
EREA mín  0.041  0.038  0.033  0.035  0.041  0.073  0.048  0.040 
EREA med  0.066  (0.060)  0.069  0.065  0.065  [0.108]  0.078  0.065 
EREA máx  0.125  0.110  0.149  0.124  0.122  0.156  0.138  0.131 
EAM mín  1.473  1.227  1.268  1.364  1.182  2.334  1.650  1.397 
EAM med  3.107  2.669  3.013  2.802  2.618  [3.886]  3.127  (2.555) 
EAM máx  6.705  5.409  5.498  4.978  4.920  6.737  6.989  5.106 
EAMx mín  9.6  7.8  7.4  5.3  6.6  2.9  7.7  5.4 
EAMx med  50.5  46.9  46.1  45.3  43.4  [52.5]  (39.7)  45.3 
EAMx máx  161.2  129.5  120.2  116.0  116.1  122.8  100.5  115.4 
AIC mín  311.5  302.2  302.2  313.8  309.7  300.0  306.4  293.6 
AIC med  441.9  426.1  430.5  432.0  (421.8)  [463.5]  424.9  423.1 
AIC máx  678.2  649.5  643.4  636.2  635.8  645.7  620.7  639.2 
COC mín  0.916  0.944  0.935  0.945  0.953  0.914  0.938  0.951 
COC med  0.971  0.976  0.972  0.975  0.978  [0.959]  0.976  (0.980) 
COC máx  0.997  0.995  0.996  0.996  0.995  0.989  0.995  0.997 
d_{2} mín  0.889  0.937  0.946  0.951  0.951  0.943  0.963  0.944 
d_{2} med  [0.967]  0.979  0.981  0.982  0.983  0.974  (0.985)  0.980 
d _{2} máx  0.997  0.997  0.998  0.997  0.997  0.995  0.997  0.998 
d_{1} mín  0.839  0.874  0.879  0.890  0.888  0.864  0.864  0.886 
d_{1} med  0.913  0.927  0.921  0.925  0.929  [0.897]  0.918  (0.933) 
d_{1} máx  0.956  0.957  0.953  0.951  0.957  0.923  0.952  0.965 
Concentrate by raingauge stations
In Table 6, the results of the last columns of each tabulation not exposed of the analyzed index are integrated, that is, the best FDPs are obtained in each station according to each index. When two or more FDP showed equal value for the index analyzed at a certain station, the best FDP was chosen based on the best average value (last line of each tabulation not exposed).
Station  Descriptive ability indexes  Best two FDP* 


EEA  EREA  EAM  EAMx  AIC  COC  d_{2}  d_{1}  
Cedral  LP3  WAK  WAK  LP3  LP3  WAK  LP3  WAK  WAK(4),LP3(4) 
Charcas  WAK  WAK  WAK  GVE  WAK  GVE  WAK  WAK  WAK(6),GVE(2) 
La Maroma  BEP  BEP  BEP  BEP  BEP  BEP  BEP  BEP  BEP(8) 
Los Filtros  BEK  LGN  WAK  BEK  BEK  BEK  BEK  WAK  BEK(5),WAK(2) 
Matehuala  BEK  WAK  WAK  BEK  BEK  BEK  BEK  WAK  BEK(5),WAK(3) 
Mexquitic  WAK  LGN  WAK  WAK  WAK  BEK  LOG  WAK  WAK(5),BEK(1) 
Peñón Bco.  LP3  WAK  GVE  LP3  LP3  LP3  LP3  GVE  LP3(5),GVE(2) 
S. Domingo  BEK  WAK  WAK  LP3  BEK  BEK  BEK  LOG  BEK(4),WAK(2) 
Vanegas  PAG  LP3  LP3  PAG  PAG  PAG  PAG  LP3  PAG(5),LP3(3) 
regional  LOG  BEP  WAK  LP3  LOG  WAK  LP3  WAK  WAK(3),LP3(2) 
*Between parenthesis the number of times that occur.
Table 6 shows that the Wakeby distribution (WAK) appears in all the columns, with one occurrence in the EAMx and COC indexes and up to six in the EAM index and five in the EREA and d _{1} indices. The Wakeby probabilistic model is the best in 31.9% of cases. These results guide the definition of the FDP Wakeby as a model that should always be applied when processing annual PMD records in arid and semiarid climates. In the last row of Table 6, the second option of the regional FDP can be the LOG and LP3 distributions with two occurrences each, the second is chosen because it is better in relation to two noncorrelated indexes (EAMx and d _{2}).
By suppressing the Wakeby distribution of Table 6, the next best is sought and then Table 6 is made, whose final column indicates the two best FDPs and their number of occurrences in each processed record. As a summary of the results of Table 7 for the Potosino Plateau, it can be indicated that the Beta FDPs are the best option in four stations, then the LP3 and LOG models follow in two stations each and finally, the Pareto Generalized distribution is the best option of one station.
Station  Descriptive ability indexes  Best two FDP* 


EEA  EREA  EAM  EAMx  AIC  COC  d_{2}  d_{1}  
Cedral  LP3  LOG  LOG  LP3  LP3  LP3  LP3  LOG  LP3(5),LOG(3) 
Charcas  LGN  BEK  LOG  GVE  LGN  GVE  LP3  LOG  LOG(2),GVE(2) 
La Maroma  BEP  BEP  BEP  BEP  BEP  BEP  BEP  BEP  BEP(8) 
Los Filtros  BEK  LGN  BEP  BEK  BEK  BEK  BEK  BEP  BEK(5),BEP(2) 
Matehuala  BEK  GVE  LOG  BEK  BEK  BEK  BEK  BEP  BEK(5),LOG(1) 
Mexquitic  LOG  LGN  LOG  LOG  LOG  BEK  LOG  LOG  LOG(6),BEK(1) 
Peñón Bco.  LP3  BEP  GVE  LP3  LP3  LP3  LP3  GVE  LP3(5),GVE(2) 
S. Domingo  BEK  LOG  BEP  LP3  BEK  BEK  BEK  LOG  BEK(4),LOG(2) 
Vanegas  PAG  LP3  LP3  PAG  PAG  PAG  PAG  LP3  PAG(5),LP3(3) 
*Between parenthesis the number of times that occur.
Figure 3 shows the QQ graph of the Mexquitic station, whose estimated annual PMD values were obtained with the Wakeby FDP. This fit corresponds to an EEA of 1.45 millimeters, which was the minimum found in the stations of the Potosino Plateau (Table 5).
Results in the middle zone
Concentrate of numerical indexes
The three characteristic values (minimum, medium and maximum) of each index of descriptive ability (HD) obtained with each of the eight contrasting FDPs in the 10 raingauge stations of the Middle Zone of the state of San Luis Potosí, Mexico have been integrated in Table 8. Table 8 is similar to Table 5.
DA Index  contrasted FDP  

BEK  BEP  LGN  GVE  LOG  PAG  LP3  WAK  
EEA mín  3.52  3.48  3.03  3.26  3.45  2.30  3.38  2.27 
EEA med  [8.31]  8.19  6.46  6.41  6.57  7.24  (6.15)  7.47 
EEA máx  15.52  20.78  14.40  13.34  13.00  15.64  12.37  22.68 
EREA mín  0.043  0.043  0.045  0.048  0.044  0.034  0.045  0.0302 
EREA med  0.068  0.070  (0.065)  0.063  0.065  0.086  0.063  [0.108] 
EREA máx  0.093  0.125  0.098  0.091  0.091  0.161  0.092  0.571 
EAM mín  2.361  2.382  2.112  2.312  2.424  1.768  2.225  1.653 
EAM med  3.784  3.676  3.615  (3.483)  3.575  4.289  3.579  [5.064] 
EAM máx  5.247  6.292  5.437  4.864  5.638  6.611  4.870  28.199 
EAMx mín  8.8  8.4  8.7  7.7  8.3  6.1  10.6  5.8 
EAMx med  [47.6]  46.5  32.1  31.8  32.6  35.4  29.3  (26.5) 
EAMx máx  105.5  150.5  94.0  88.5  87.0  100.8  78.9  79.4 
AIC mín  325.1  324.0  324.4  326.0  323.2  295.8  335.2  297.1 
AIC med  [423.6]  417.4  396.9  396.9  400.3  406.8  (394.3)  401.9 
AIC máx  579.3  624.6  511.6  529.3  543.8  484.8  518.0  514.4 
COC mín  0.950  0.921  0.959  0.968  0.964  0.943  0.970  0.971 
COC med  0.975  [0.974]  0.982  0.982  0.982  0.974  0.983  (0.987) 
COC máx  0.991  0.991  0.992  0.991  0.991  0.995  0.991  0.992 
d _{2} mín  0.964  0.944  0.971  0.975  0.976  0.965  0.979  0.800 
d _{2} med  0.982  0.982  0.989  0.989  0.989  0.985  (0.990)  [0.973] 
d _{2} máx  0.995  0.995  0.996  0.995  0.995  0.998  0.995  0.998 
d _{1} mín  0.886  0.894  0.896  0.895  0.893  0.881  0.901  0.453 
d _{1} med  0.920  0.922  0.924  (0.926)  0.923  0.911  0.924  [0.891] 
d _{1} máx  0.946  0.941  0.946  0.948  0.944  0.956  0.944  0.954 
Concentrate by raingauge stations
Table 9 shows for each record of annual PMD processed, which is the best FDP according to each index of descriptive ability. It is observed that the Wakeby distribution is the best option in six stations (Ojo de Agua to San Francisco); in total, it is the best FDP in 42.5% of cases. Based on the results of Table 9, it is concluded that the Wakeby model should always be tested when analyzing annual PMD records of warmsubhumid climates. In the last line of Table 9, the WAK and GVE models can be selected as the second best regional FDP option; the first model was chosen due to its greater number of local occurrences.
Station  Descriptive ability indexes  Best two FDP* 


EEA  EREA  EAM  EAMx  AIC  COC  d _{2}  d _{1}  
Armadillo de los Infante  PAG  LGN  LGN  PAG  PAG  PAG  PAG  WAK  PAG(5), LGN(2) 
Cárdenas  PAG  LGN  PAG  PAG  PAG  PAG  PAG  PAG  PAG(7), LGN(1) 
Lagunillas  BEP  LGN  WAK  BEP  BEP  BEP  BEP  WAK  BEP(5), WAK(2) 
Ojo de Agua  LP3  WAK  WAK  LP3  BEP  BEP  LP3  WAK  WAK(3), LP3(3) 
Ojo de Agua Seco  WAK  WAK  WAK  BEP  GVE  GVE  BEK  WAK  WAK(4), GVE(2) 
Paso de San Antonio  WAK  BEP  BEP  WAK  WAK  WAK  WAK  LOG  WAK(5), BEP(2) 
Rayón  WAK  WAK  WAK  LP3  LP3  WAK  WAK  WAK  WAK(6), LP3(2) 
Río Verde  WAK  WAK  WAK  PAG  PAG  WAK  PAG  WAK  WAK(5), PAG(3) 
San Francisco  WAK  BEK  WAK  WAK  WAK  WAK  WAK  WAK  WAK(7), BEK(1) 
San José Alburquerque  LP3  LP3  LP3  WAK  LP3  LP3  LP3  LP3  LP3(7), WAK(1) 
Regional  LP3  LGN  GVE  WAK  LP3  WAK  LP3  GVE  LP3(3), WAK(2) 
*Between parenthesis the number of times that occur.
By eliminating the Wakeby distribution from Table 9 and looking for the next best FDP option, the Table 10 is integrated, whose results for the annual PMD records of the Middle Zone place in first and downward order the Generalized Pareto distributions in three stations; LogPearson type III, Pareto Beta and Generalized Logistics in two stations. The FDP General of Extreme Values is the best option in one station.
Station  Descriptiva Ability Indexes  Best two FDP* 


EEA  EREA  EAM  EAMx  AIC  COC  d _{2}  d _{1}  
Armadillo de los Infante  PAG  LGN  LGN  PAG  PAG  PAG  PAG  LGN  PAG(5), LGN(3) 
Cárdenas  PAG  LGN  PAG  PAG  PAG  PAG  PAG  PAG  PAG(7), LGN(1) 
Lagunillas  BEP  LGN  LP3  BEP  BEP  BEP  BEP  LP3  BEP(5), LP3(2) 
Ojo de Agua  LP3  LOG  BEP  LP3  BEP  BEP  LP3  BEP  BEP(4), LP3(3) 
Ojo de Agua Seco  GVE  BEK  GVE  BEP  GVE  GVE  BEK  GVE  GVE(5), BEK(2) 
Paso de San Antonio  LOG  BEP  BEP  LOG  BEP  LOG  LOG  LOG  LOG(5), BEP(3) 
Rayón  LP3  LOG  BEP  LP3  LP3  LOG  LP3  LOG  LP3(4), LOG(3) 
Río Verde  PAG  PAG  PAG  PAG  PAG  PAG  PAG  PAG  PAG(8) 
San Francisco  LOG  BEK  LOG  GVE  LOG  LOG  GVE  LOG  LOG(5), GVE(2) 
San José Alburquerque  LP3  LP3  LP3  LP3  LP3  LP3  LP3  LP3  LP3(8) 
*Between parenthesis the number of times that occur.
Results according to predictive ability
FDPs applied
For each climatological station or annual PMD record, four FDPs were chosen to be contrasted. The first corresponds to the best option in Table 3, that is, it is the most appropriate FDP according to the results of the Lratio diagram. The following two FDPs to be applied were those obtained as best options according to the eight indexes of descriptive ability, which were concentrated in Table 7 and Table 10. Finally, the Wakeby FDP was applied, due to its great descriptive capacity, which was shown in Table 6 and Table 9; therefore, it is suggested to be applied under precept. In Table 11 and Table 12 of calculated and adopted predictions, the following three stations have been highlighted with bold letters: La Maroma, Río Verde and San José Alburquerque, because in them only three FDPs are contrasted, since in Table 7 and Table 10 report only a better FDP in the eight descriptive ability indexes applied.
Station Best FDP 
P_{M} (P_{M } /P _{50}) 
Return periods in years  

50  100  500  1 000  
Cedral  315.8  
GVE [2]  2.21  143  192  384  520 
LP3 (5)  2.10  (151)  (206)  (431)  (596) 
LOG (3)  2.24  141  191  400  555 
WAK (4)  2.32  136  191  443  2326 
Charcas  117.0  
LGN [3]  1.10  106  119  147  202 
LOG (2)  1.07  (109)  (126)  (174)  (199) 
GVE (2)  1.07  109  121  150  162 
WAK (6)  1.07  109  122  150  198 
La Maroma  140.1  
LOG [1]  1.30  108  129  197  236 
BEP (8)  1.25  (112)  (136)  (213)  (258) 
WAK (0)  1.29  109  126  172  279 
Los Filtros  111.0  
LP3 [1]  1.32  84  93  114  123 
BEK (5)  1.32  (84)  (95)  (127)  (143) 
BEP (2)  1.35  82  91  117  129 
WAK (2)  1.34  83  93  119  176 
Matehuala  200.0  
LP3 [2]  1.39  144  170  245  285 
BEK (5)  1.29  (155)  (191)  (309)  (380) 
LOG (1)  1.41  142  171  266  323 
WAK (3)  1.48  135  170  316  1211 
Mexquitic  107.0  
LGN [2]  1.24  86  91  104  124 
LOG (6)  1.22  88  97  117  127 
BEK (1)  1.32  81  89  111  122 
WAK (5)  1.20  (89)  (98)  (120)  (163) 
Peñón Blanco  235.0  
LOG [1]  1.51  156  216  473  667 
LP3 (5)  1.39  (169)  (233)  (490)  (676) 
GVE (2)  1.49  158  217  457  631 
WAK (1)  1.51  156  221  510  2521 
Santo Domingo  270.0  
LP3 [1]  1.61  168  201  293  340 
BEK (4)  1.70  (159)  (198)  (327)  (406) 
LOG (2)  1.72  157  195  322  400 
WAK (2)  2.03  133  175  400  2770 
Vanegas  90.0  
LGN [1]  1.00  (90)  (102)  (132)  (197) 
PAG (5)  1.06  85  92  103  107 
LP3 (3)  1.01  89  101  130  143 
WAK (0)  1.01  89  99  118  146 
Station Best FDP 
P_{M} (P _{ M } /P _{50}) 
Return periods in years  

50  100  500  1 000  
Armadillo de los I.  133.0  
LP3 [2]  0.95  140  163  226  258 
PAG (5)  0.99  135  150  180  191 
LGN (3)  0.96  (139)  (162)  (220)  (354) 
WAK (1)  0.96  138  156  196  267 
Cárdenas  180.5  
LP3 [1]  0.95  (191)  (231)  (345)  (406) 
PAG (7)  0.97  186  215  282  311 
LGN (1)  0.95  191  228  330  586 
WAK (0)  0.97  187  216  285  418 
Lagunillas  210.0  
LGN [1]  1.21  173  196  251  368 
BEP (5)  1.16  (181)  (214)  (317)  (375) 
LP3 (2)  1.19  176  201  265  295 
WAK (2)  1.22  172  200  289  579 
Ojo de Agua  300.2  
LOG [1]  1.31  (229)  (290)  (510)  (656) 
BEP (4)  1.29  233  293  498  625 
LP3 (3)  1.27  236  292  474  583 
WAK (3)  1.29  233  298  533  1623 
Ojo de Agua Seco  172.5  
LOG [2]  1.11  (155)  (185)  (283)  (341) 
GVE (5)  1.12  154  180  252  289 
BEK (2)  1.11  155  185  274  325 
WAK (4)  1.11  155  178  234  356 
Paso de S. Antonio  200.0  
GVE [2]  1.41  142  161  208  230 
LOG (5)  1.39  (144)  (167)  (238)  (277) 
BEP (3)  1.42  141  163  228  263 
WAK (5)  1.36  147  172  246  463 
Rayón  330.0  
GVE [2]  1.63  203  254  427  533 
LP3 (4)  1.57  210  265  447  558 
LOG (3)  1.63  (202)  (257)  (461)  (596) 
WAK (6)  1.63  203  267  515  1834 
Río Verde  126.3  
LP3 [3]  1.00  (126)  (142)  (184)  (204) 
PAG (8)  1.07  118  125  137  141 
WAK (5)  1.07  118  126  140  153 
San Francisco  135.0  
LGN [3]  1.18  114  131  172  260 
LOG (5)  1.15  117  139  208  247 
GVE (2)  1.17  115  134  181  204 
WAK (7)  1.13  (119)  (141)  (198)  (342) 
S. J. Alburquerque  126.5  
GVE [1]  1.17  108  121  153  167 
LP3 (8)  1.13  (112)  (127)  (167)  (186) 
WAK (1)  1.00  127  137  156  182 
When one of the two best FDPs in Table 7 or Table 10 coincided with the first applied distribution, the latter was changed, by its second and/or third option in Table 3. The option that has the first applied FDP was indicated in rectangular parentheses in Table 11 and Table 12 of calculated and selected predictions. For the two best FDPs in Table 7 and Table 10, the number of descriptive ability indexes in which they are the best are indicated in round brackets. The same is indicated for the Wakeby distribution, but such datum comes from Table 6 and Table 9.
Obtained statistics
Table 1 shows that most of the records processed from annual PMD have amplitude of 50 years or more, due to which the quotient between the maximum value of the record (P_{ M } ) and the prediction of the 50 year return period (P _{50}) was calculated. This quotient is indicated in columns 2 of Table 11 and Table 12, and when it is close to the unit it indicates that the record does not have extreme scattered values (outliers) that deviate from the natural trend of the data. On the other hand, when it exceeds 1.50, there is one or more scattered values that is the case of the following four stations: Cedral, Peñón Blanco, Santo Domingo and Rayón.
In the four stations mentioned, the Wakeby distribution, due to its extraordinary flexibility given by its five fitting parameters, leads to very high predictions in the return period of 1000 years; as observed when comparing them with those obtained with the other contrasted FDP. In none of the cases mentioned, the FDP Wakeby was adopted, because its predictions were considered exaggerated, as it did not coincide with those of the other three probabilistic models contrasted in that station. ^{Nguyen et al. (2017)} also find that the Wakeby distribution has low predictive ability, by showing great variability in their predictions.
In Table 11 and Table 12 of predictions calculated and adopted in the stations of the Potosino Plateau and the Middle Zone, the descriptive and predictive abilities of each of the contrasted FDP are taken into account implicitly; therefore, the following conclusions are considered globally in the study.
It was obtained that in 12 stations the adopted values come from the two FDPs that were the best option according to the eight indexes of descriptive ability. In five stations the adopted predictions were calculated with the FDP best options according to the Lratio diagram and only in two stations the predictions calculated with the Wakeby distribution were adopted.
As already indicated, exclusively in three stations; La Maroma, Río Verde and San José Alburquerque, a total concordance was obtained in the eight indexes of descriptive ability, for the FDP BetaP, Generalized Pareto and LogPearson type III, respectively. These stations have been highlighted in bold in Table 11 and Table 12.
By geographic areas, in the Potosino Plateau of nine processed records, the FDP Betaκ was the model adopted in three stations and the LogPearson type III distribution in two stations. In the Middle Zone of 10 processed records, the FDP Generalized Logistics with four stations had a preponderance of adoption and was followed by the LP3 distribution with three stations.
Conclusions
The Wakeby distribution, fitted with the Lmoment method is a model of excellent descriptive ability and therefore, it is suggested to be applied under precept in the probabilistic analyzes of annual PMD records of the arid and semiarid climates of the Potosino Plateau (AP) and of the warmsubhumid climate of the Middle Zone (ZM) of the state of San Luis Potosí, Mexico.
The Betaκ and BetaP distributions, fitted with the maximum likelihood method, are models not applied in Mexico that are suggested to be tested, since for four annual PMD records of the AP (Table 7) and two of the ZM (Table 10), lead to the best descriptive ability indexes.
Regarding the distributions that are applied under precept in the USA and England, it was obtained (Table 7 and Table 10): (1) the FDP LogPearson type III that proved to be the best option in two stations of the AP and ZM; (2) the FDP General of Extreme Values only in one station of the ZM was a better option; (3) the FDP Generalized Logistics was the best option in two stations of the AP and ZM. (4) In the AP the Generalized Logistics stands out as the second best option and in the ZM the LogNormal and LogPearson models type III.
Regarding the FDP Generalized Pareto, which is commonly applied together with the LOG and GVE models; it was a better option in one station of the AP and three in the ZM. These results confirm the systematic application or under precept of LP3, GVE, LOG and PAG distributions in annual PMD series of arid, semiarid and warmsubhumid climates.
Regarding the calculated predictions (Table 11 and Table 12) in the return periods of 50, 100, 500 and 1000 years, they generally show similar values and this generates confidence in the adopted values. Dispersion was exclusively found in the predictions of the FDP Wakeby, in the stations or records of annual PMD with extreme scattered value (outlier), in case of the stations: Cedral, Peñón Blanco, Santo Domingo and Rayón.
Regarding the adopted predictions (Table 11 and Table 12) it is concluded that the search procedures for the best FDP to be applied to the annual PMD records, based on the Lratio diagrams and on the eight descriptive ability indexes, are adequate and lead to a good approximation, since there was not difficulty to select the adopted predictions.
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Received: November 16, 2017; Accepted: November 22, 2018