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 (^{Campos-Aranda, 2018}). Then a ratification of their minimum and maximum extreme values was carried out with the aid of CONAGUA, to obtain the so-called debugged series.

Homogeneity tests applied

The Neumann Von Test was applied to each debugged series as a general test, which detects non-randomness 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 non-homogeneous 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 semi-arid 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 L-moment 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.

Figure 1 Geographical location of the 19 series of annual PMD processed from the state of San Luis Potosí, Mexico. 

L-moments of the sample

L-moments 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 (x1≤x2≤⋯≤xn) and then (^{Hosking & Wallis, 1997}; ^{Rao & Hamed, 2000}; ^{Asquith, 2011}; ^{Stedinger, 2017}):

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 L-moments of the sample (l) and their respective quotients (t) of similarity with the coefficients of variation, asymmetry and kurtosis are:

L-ratio 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 polynomial-type equations (^{Hosking & Wallis, 1997}):

Generalized Logistics (LOG):

Generalized Pareto (PAG):

Log-Normal (LGN):

Pearson type III (PT3):

and General of Extreme Values (GVE):

being:SF=0.00567·t34-0.04208·t35+0.03763·t36

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 Log-Pearson type III. Figure 2 shows a portion of the L-moments ratio diagram from ^{Hosking and Wallis (1997)}.

Figure 2 L-ratio diagram and punctual values corresponding to the 19 series of annual PMD processed of the state of San Luis Potosí, Mexico.  

Absolute weighted distance

One of the recent approaches to choose the best FDP, at the local and regional level, consists of taking to the L-ratio 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, t3obsj and t4obsj are the asymmetry and kurtosis L-ratios of each series and t4t3obsj is the theoretical value of the kurtosis L-ratio calculated with each FDP (Equations 9 to 13), for the observed value of the asymmetry L-ratio. A FDP with the least value of the DAP is the best for the local (NE=1) or regional data.

Diagnostic Graphics

The P-P and Q-Q 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}). ^{Campos-Aranda (2015)} has exposed such graphs and visualizes more useful the Q-Q 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 Chi-square, the Kolmogorov-Smirnov or Anderson-Darling, 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}).

Goodness-of-fit 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 x^i (^{Pandey & Nguyen, 1999}; ^{Zalina, Desa, Nguyen, & Kassim, 2002}). ^{Meylan et al. (2012)} point out two disadvantages of these indexes: (1) they require the application of an empirical formula to estimate the probability of each data and (2) do not provide an estimate of the probability involved in the rejection or acceptance of the FDP and perhaps this is its advantage when comparing several probabilistic models. In this study, eight indexes of goodness-of-fit were applied. ^{Nguyen et al. (2017)} apply five defined in the Equation 17 to 22, except for 19 and include another similar to the AIC (Equation 21).

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 Log-Pearson 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 mid-seventies (^{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 non-exceedance (pi:n) approximately unbiased for many FDPs, that is why it was adopted by ^{Nguyen et al. (2017)}. The expression of the EEA is:

in which, x _i are the observed data ordered from lowest to highest,x^i the estimated ones, for the estimated probability with Equation 16 and the FDP that is contrasted. np is the number of fit parameters of the FDP, with five for the Wakeby and three for the rest of the contrasted ones. The numerator of this equation is the sum of squared errors (SEC). EEA has the units of the variable x _i .

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 Kolmogorov-Smirnov 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.

Q-Q 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 Q-Q 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, x- and x^im are the average values of the data and estimated values.

Concordance indexes (d₁, d₂)

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 x^i, by using standardized values with their means and therefore proposes the adimensional concordance index (d ₂) , with the expression:

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 x^i can reach in each datum. ^{Legates and McCabe (1999)} point out that the concordance index (d ₂) is also sensitive to scattered values by using the differences between x _i and x^i squared and therefore propose the modified concordance index (d ₁), in which the numerator and the denominator are not squared, but with absolute value, this is:

As d ₂ as d ₁ vary from zero to one, with an interpretation equal to COC; usually, d ₂ ≥ d ₁.

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 L-rates 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.

General observations

Regarding the Log-Pearson 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 by rain-gauge 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).

Table 6 Best FDP according descriptive ability indexes in the 9 series of annual PMD processed of the Potosino Plateau of the state of San Luis Potosí, Mexico. 

Station	Descriptive ability indexes								Best two FDP*
	EEA	EREA	EAM	EAMx	AIC	COC	d2	d1
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 semi-arid 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 non-correlated 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.

Table 7 Best FDP (excluding the Wakeby) according to the descriptive ability indexes in the 9 series of annual PMD processed of the Potosino Plateau of the state of San Luis Potosí, Mexico. 

Station	Descriptive ability indexes								Best two FDP*
	EEA	EREA	EAM	EAMx	AIC	COC	d2	d1
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 Q-Q 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).

Figure 3 Q-Q graph of the Mexquitic station obtained with the FDP Wakeby. 

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 rain-gauge 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.

Concentrate by rain-gauge 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 warm-subhumid 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.

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; Log-Pearson type III, Pareto Beta and Generalized Logistics in two stations. The FDP General of Extreme Values is the best option in one station.

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 L-ratio 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.

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 L-ratio 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 Beta-P, Generalized Pareto and Log-Pearson 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 Beta-κ was the model adopted in three stations and the Log-Pearson 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.