Introduction

The daily global radiation is important in areas such as engineering, agriculture, soil physics, agricultural hydrology modeling crops and estimation of evapotranspiration, as well as: modeling climate and weather, growth monitoring crop and disease control.

The radiation reaching the surface of the earth due to gases, clouds and particles in the atmosphere, they absorb and scatter radiation at different wave levels. The need for records of solar radiation becomes important due to the increase in solar energy applications. Obtain reliable data radiation requires systematic measurements (^{Muribu, 2008}).

The artificial neural networks have proven to be excellent tools for different areas of research; since they are able to handle nonlinear relationships (approximation of nonlinear function), separating data (data classification), find hidden relationships in data groups (clustering) or modeling natural systems (simulation) (^{Demuth et al., 2008})

In estimating global radiation by artificial neural networks is the work of ^{Hasni et al. (2012)} cited by ^{Yadav and Chandel (2014)} estimated the global radiation each time using an artificial neural network using temperature and relative humidity with a feedforward backpropagation algorithm, similarly ^{Jiang (2008)} used an artificial neural network (RNA) of this type to make a prognosis of diffuse solar radiation and compares the results with two empirical models. ^{Rehman and Mohandes (2008)} used an artificial neural network model to estimate recurrent global radiation measured temperature and relative humidity values as inputs. ^{Martínez-Romero et al. (2012)}, used to linear regression models, the model Hargreaves Hargreaves calibrated and RNA to estimate global solar radiation data with average monthly maximum temperature, minimum and extraterrestrial solar radiation. Regarding empirical models ^{Bandyopadhyay et al. (2008)}, used: Hargreaves, calibration of Angstrom-Prescott and Bristow and Campbell (^{Bristow and Campbell, 1984}) to estimate solar radiation.

The aim of this study was to compare the results of estimates of average daily global radiation four stations, located in the Irrigation District 075, Valle del Fuerte, Los Mochis, Sinaloa, made with the model of artificial neural network (RNA) backpropagation against the estimates obtained by empirical models Hargreaves, Hargreaves calibrated, Angstrom-Prescott and Angström-Prescott calibrated. The averages of the data were used by the proximity of a station to another, since there are no significant differences in climate prevailing between stations.

Materials and methods

Results and discussion

The model Hargreaves uncalibrated with k_Rs= 0.19 with 1 484 data obtained an RMSE of 5.6735 and for k_Rs= 0.16 obtained an RMSE of 3.4062 and the same R² (0.59) is preserved for 229 data validation found a RMSE of 6.8224 and 3.3696 for k_Rs 0.19 and 0.16 respectively and R² of 0.73 for both. The values of z and k_Rs model Hargreaves calibrated were 0.2995 and 0.2711, respectively, and the estimated values of global radiation for the 1 484 data showed a RMSE of 3.0068 and an R² of 0.65 and for 229 data validation RMSE of 3.2460 with an R² of 0.75 for this model. As seen with k_Rs= 0.16, better fit than a k_Rs= 0.19, indicating that the irrigation district 075, where the four seasons are located is close to being a locality where land mass dominates according to obtained criteria ^{Allen et al. (1998)} and the calibrated model generated better fit the standard error and the coefficient of determination, however, that the k_Rs the calibrated model is far from 0.16; ^{Martínez-Romero et al. (2012)} found RMSE calibration of 1.43 MJ/m²day for Hargreaves model with k_Rs 0.15928, to estimate monthly average values of global solar radiation, this value k_Rs say the authors hardly differs from the value of 0.16 so they used the latter for the RMSE of spatial and temporal validation, 1.23 and 1.66 MJ/m²day respectively, against a calibration RMSE of 1.17 MJ/m²day for RNA with entries maximum temperature, minimum and extraterrestrial solar radiation.

The model of Angstrom-Prescott uncalibrated obtained an RMSE data for 1484 5.1948, with an R² of 0.73 for the calibrated model (a_s= -0.5535 and b_s= 1.3824) the RMSE obtained was 2.0889 with an R² of 0.84, this indicates that presented better fit the Angström-Prescott model calibrated the calibrated Hargreaves, corroborating indicated in ^{Cervantes-Osornio et al. (2012)}. ^{Liu et al. (2009)} calibrated the coefficients a and b with daily data, finding a better approximation of the global radiation model AngstromPrescott calibrated the calibrated Hargreaves, unlike what was found by ^{Bandyopadhyay et al. (2008)}, who used the model modified Hargreaves, and Angstrom-Prescott calibrated to estimate monthly solar radiation data and found that the method modified by Hargreaves ^{Annandale et al. (2002)} it was better than the Angstrom-Prescott. ^{Almorox et al. (2008)} and ^{Meza and Varas (2000)}, estimated the monthly global radiation with the equation of Angström-Prescott, resulting in this model with the calibrated coefficients is a useful tool, which corroborates the findings in this paper, calibrate models generates a better fit than not.

In the Table 1 shows the results of the average standard error and average error for the training scenario 2, which was one of those who had values of RMSE and MBE closer to zero, in this observed shown that increasing the number of layers hidden not had an impact on improving the fit of RMSE and MBE, as the closest zero adjustment was training with two hidden layers of 24 x 24 neurons in these, in training with a hidden layer the best fit was the one who had 30 neurons in this and three hidden layers was 9 x 24 x 9 neurons, whose best fit of this last block did not exceed the two layers. Additionally, the increased number of neurons in the hidden layers, does not necessarily improve the fit of these tests, for two and three hidden layers, but e2, with a hidden layer obtained the best fit with the maximum number of neurons (30).

In the Table 2 shows the results of RMSE that are closer to zero, results of different workouts performed with RNA of the three scenarios for one, two and three hidden layers with variations in the number of neurons in these layers are shown. The best fit was with the e6 stage with a hidden layer with 27 neurons in it, with the entries: T_min, T_max, n/N and R_a, with an RMSE of 1.6871. The second best adjustment was made in the e5 stage, with tickets T_min, T_max, n and R_a, RMSE of 1.69 with two hidden layers with 18 neurons in each of them and the third best fit was the e5, with three hidden layers 9, 27 and 9 neurons in each layer and RMSE of 1.7019, and for each of these scenarios has an R² of 0.89. These results confirm the above, with three hidden layers does not improve the fit workouts, with two hidden layers is sufficient to obtain an acceptable RMSE, in this sense, ^{Tymvios et al. (2005)}, say that with an excessive number of hidden layers, often leads to a deterioration in the performance of the RNA, and also causes better adjustment feeding the RNA with the difference maximum and minimum temperature variable, or with the variable n/N.

The coefficients of determination obtained in these scenarios are closer to one than the R² of 0.84, obtained from an empirical model (Angström-Prescott calibrated). ^{Tymvios et al. (2005)}, with an RNA with two hidden and 23 x 46 layers neurons in these they found a RMSE of 5.67%, better fit compared with the RMSE obtained from Angstrom model of 13.36%, overall, this coincides with the findings in the this study, data for 1 484 shows the RNA best fit the model of Angström-Prescott calibrated.

The training with RNA with different inputs (Table 2) showed the closest to zero or better fit RMSE, were the scenes that include the maximum temperature, minimum temperature and hours shine sun, which is also presented in Table 1 (scenario 2), no workouts were performed considering only the maximum and minimum temperature, that is, regardless of the hours varying brightness sun, because as presented in ^{Cervantes-Osornio et al. (2012)}, and as stated by ^{Benghanem et al. (2009)}, the hours sun causes varying brightness improves the accuracy of the estimate of daily global radiation.

It also observed in Table 1 that the N and R_a data is included in the entry itself improves the fit, although not as much as when you include the hours varying brightness sun. In addition, the more input variables have to feed the RNA leads to a better fit, but also causes the computer equipment perform more processing and therefore more resources of all kinds are consumed. The results of RMSE scenario 2 (Table 1), did not exceed the results of RMSE or the coefficient of determination of Angström-Prescott empirical model calibrated (Figure 1), but in a season that has only said three variables entrance, this is enough to train the RNA.

Figure 1 RMSE and R² for estimates of Angström-Prescott models Hargreaves, Hargreaves calibrated with kRS= 0.16 and RNA e6{27} of the 1 484 average global radiation data. 

In the Tables 2 and 3, it is observed that the best settings, mostly correspond to e5 scenarios and e6, which are those with more input variables, but also noted that a small number of input variables, such as maximum temperature and minimum temperature can make an acceptable prognosis of daily global radiation, with a variation of thousandths in the RMSE, but since there are weather stations that do not have an extensive range of instruments to measure all of climatological variables requiring e6 or e5 scenario, then a stage e2 is sufficient for prognosis. The Table 3 was formed with the same scenarios in Table 2, the first, second and third best fit observed in Table 3 were presented as follows: with three hidden layers, setting e15 with T_max-T_min, n/N, R_a, with 9x18x9 neurons in the hidden layers and a RMSE of 1.9847; the e2 entries with T_min, T_max, n, , with 9x24x9 in the hidden layers with a RMSE of 1.9918, and e15 with T_max-T_min, n/N, R_a with a hidden layer, and three neurons in it with a RMSE of 1.9939, although these do not match the best fits obtained in practice with 1 484 data. ^{Jiang (2008)} to estimate the diffuse solar radiation monthly average determined empirical models RMSE's of 0.783, 0.871 where the neural network model used, obtained a RMSE of 0.746, with the variables: index of daylight, percentage of hours brightness sun and output the average daily diffuse fraction.

In Figures 1 and 2 shows that the maximum and minimum temperature, with the figure for hours shine sun divided by photoperiod and extraterrestrial radiation (e6{27}) the best fit for training with 1484 vectors is reached data (R² of 0.89 and RMSE of 1.6871), and the RNA and trained a new estimate of 229 data R_s (R² of 0.88 and RMSE of 2.5475), which as shown in Table 3 is not for the best fit is performed results for global data 229, in this connection ^{Senkal and Kuleli (2009)} found correlation coefficient values for the training set and evaluation 0978 and 0971 respectively; and ^{Fadare (2009)} found values of the correlation coefficient and RMS (root mean square) for training 99.36% and 2.32 MJ m2 and for evaluation of 88.39% and 3.94 MJ/m2 respectively, indicating that all evaluation error values tend to grow.

Figure 2 RMSE and R² for estimates of Angström-Prescott models Hargreaves, Hargreaves calibrated with kRS= 0.16 and RNA e6{27} of the 229 average global radiation data. 

Additionally in Figure 2 shows that both models calibrated Angström-Prescott models Hargreaves, Hargreaves calibrated and RNA including backpropagation overestimate the data observed daily global radiation. ^{Tymvios et al. (2005)} used three model variations Angström with different calibrated values for b and RNA performing training with different input variables including: month duration of hours brightness sun, maximum temperature and duration of hours brightness theoretical sun (photoperiod), their results they are similar to those found in this work; the artificial neural network model that used ^{Tymvios et al. (2005)} with the maximum temperature inputs, sun shine hours photoperiod measures and obtained an RMSE of 5.67%, followed closely by one of the models of Angström with RMSE of 5.81%. However ^{Wan et al. (2008)} found that the model of Angstrom calibrated in nine warm-seven areas with sunny weather in China, obtained a similar performance in the statistics RMSE and MBE fit with those found in this work, for 229 data, compared with the setting a neural network.

Conclusions

Invariably all workouts containing the variable sun shine hours (n) have a good fit.

The increase in hidden layers does not improve the setting value of RMSE, with one and two layers is sufficient to train an RNA for estimating daily global radiation.

The variables minimum temperature, maximum temperature and sun shine hours (n) are essential for a good prognosis of daily global radiation. Variable theoretical extraterrestrial radiation (R_a) is not relevant in the prognosis of daily global radiation.

For 1 484 average data, artificial neural network backpropagation in your scenario e6{27} presents the best estimate of daily global radiation (R_s), and is better than the empirical models, however for 229 forecast data, the Angström-Prescott calibrated model provides an estimate of R_s slightly better e6{27}.