Fourier series model

With the data from April 1997 to May 2001, the Fourier series models were generated and from June to December 2001 they were used for the validation of these. The input variable was Julian day, and the output, maximum or minimum temperature.

The equation for the Fourier series presented by ^{Dubbel (1977)} is:

where: 𝑎 0 , 𝑎 𝑖 and 𝑏 𝑖 are the constants of the Fourier series, for i = 1 oscillation or fundamental wave, and for i = 2, 3, ... ..n, higher waves or higher harmonics, w is the fundamental frequency of the signal, n is the number of terms (harmonics) in the series, and in this case: 1≤n≤8; x, is the dependent variable.

Artificial neural networks

In the artificial neural network, the backpropagation algorithm was applied (^{Haykin, 1994}).

For training, evaluation and validation were used the daily data of maximum or minimum temperatures from April 1997 to May 2001 and for the forecast from June to December 2001. The input variables are: Julian day, longitude, latitude and altitude, and output: maximum or minimum temperatures.

The structures of the 96 different scenarios for training the neural network backpropagation are shown in Table 1, combined number of layers and number of neurons in these, we used two different training algorithms.

Statistical evaluation of the results of the models

The average standard error (RMSE) was used, its equation is: RMSE=∑i=1Nai-ti2N1/2, where ai is the data estimated by the model, ti is the observed data and N is the total number of observations, (^{Cai, Liu, Lei, & Pereira, 2007}).

Training, evaluation and validation (1484 data), for the 96 scenarios of the artificial neural network

All the scenarios were determined (Table 1), the ranges of variation of the RMSE of these, for the variable maximum temperature were: 1.2540 {e4} {20} to 2.4858 {e4} {40x20}; 2.1790 {e3} {40x40} to 2.2765 {e4} {8x12x4}; 2.3280 {e1} {40x40x40} to 2.4277 {e4} {16x40x16} and 2.5093 {e3} {40} to 2.6276 {e4} {16x32x16}, for stations Santa Rosa I, Ruiz Cortinez, Batequis and Santa Rosa 2, respectively.

For the minimum temperature they were: 2.2372 {e1} {36x36} to 2.2554 {e4} {32x16}; 2.0152 {e3} {40x40} to 2.17 {e4} {16x40x16}; 2.1523 {e3} {36x36} to 2.2934 {e4} {8x8x4} and 2.01 {e3} {40x20} to 2.0809 {e4} {32x16}, for stations Santa Rosa 1, Ruiz Cortinez, Batequis and Santa Rosa 2 in that disposition.

Forecast (229 data) with RNA

The ranges of the RMSE of all the scenarios (Table 1), first for the maximum temperature, then for the minimum temperature, and for the seasons: Santa Rosa 1, Ruiz Cortinez, Batequis and Santa Rosa 2, in that order were: 2.4448 {e4} {12x4} to 2.5522 {e1} {4x4x4}; 2.4697 {e3} {4x4} to 2.5420 {e2} {8x4}; 2.4915 {e1} {4} to 2.5246 {e1} {4x4x4}; 2.8649 {e1} {12} to 2.9574 {e1} {4x4}, and 2.0223 {e4} {16x32x16} to 2.1713 {e2} {36x20}; 1.9922 {e4} {16x32x16} to 2.09 {e4} {24x16}; 2.0346 {e4} {8} to 2.1776 {e2} {12x4}; 2.3035 {e1} {8} to 2.3865 {e3} {32x16}.

Of all the scenarios presented for the training and the forecast of daily maximum and minimum temperatures, the best estimators are the scenarios {e3} for the first, {e4} and {e1} for the second, for these it is applied in the output layer the function of rigid limit (purelin), except for {e1}, and in the intermediate layers the tansig function. ^{Şahin (2012)} made monthly mean air temperature estimates with remote sensing and artificial neural networks and obtained RMSE’s of 1.254 and 1.263 K in their best settings and used the same transfer functions, in the output and in the hidden layer. Most of the RMSE’s in the present study are not less than 2 °C, since daily temperature values ​​were used.

The differences of the RMSE of all the ranges of the scenarios presented for both the training and the forecast are not greater than 0.16 °C for both temperatures (maximum and minimum) and in the four seasons (Table 2), so that use any scenario The exception is the maximum temperature of the Santa Rosa 1 station (1.2 °C).

The best scenarios of the trainings do not coincide with the best scenarios of the forecasts (Table 3). Also in most of the best workouts, for both temperatures, the largest number of neurons in the hidden layers have an RMSE close to zero, but not necessarily the trainings with the highest number of hidden layers will always be those that get the best adjustments, only one of the scenarios with three hidden layers presented one of the best settings (Batequis station). It is also observed that unlike training, in the forecasts most of the best adjustments occur when the number of neurons in the hidden layers are the lowest, this corroborates what comment ^{Demuth and Beale (2001)} and ^{Tymvios, Michaelides, & Skouteli. (2008)}, that one of the problems that occurs when an RNA is being trained is that the training set is over-adjusted and does not generalize well for a new data set (forecast), as in the present work.

In addition, (Table 3) there is no specific scenario that has been the best, both in training and in the forecast.

What matters are the best forecast scenarios, since these guarantee that the estimated data are closer to the observed ones (RMSE's close to zero). As the best forecast scenarios do not correspond with the best ones of the training, the decision was made to work with the forecast ones presented in the last two columns (Table 3).

Comparison of the best scenarios of the RNA forecast with the best fit Fourier Series models

The best forecast scenarios of the RNA were compared with the best Fourier Series models obtained for each of the stations, in Figure 1 (Santa Rosa I) shows the behavior of the differences of the observed values of maximum temperature with the predicted values with each of these two models. These differences present the same trends and are far from the line of zero value in the same proportion. In addition, it is observed that the range of the differences is between 10 °C and -6 °C. This comment is also presented in the other three stations and for the minimum temperature (Figure 2).

Figure 1 Differences between the RNA and Fourier series models with the observed maximum temperature data for Santa Rosa I AC; Fourier model with n = 5: f(x) = 31.86 - 0.6201 cos(0.01727x) + 5.871 sin (0.01727x) + 0.8935 cos(2*0.01727x) - 0.0709 sin(2*0.01727x) - 0.745 cos(3*0.01727x) + 0.6926 sin(3*0.01727x) + 0.3822 cos(4*0.01727x) + 0.1923 sin(4*0.01727x) - 0.227 cos(5*0.01727x) + 0.1547 sin (5*0.01727x). 

Figure 2 Differences of the observed data of minimum temperatures with the RNA and Fourier series models, for Ruiz Cortínez; Fourier model with n = 8: f(x) = 15.98 - 2.83 cos(0.0172x) + 8.476 sin (0.0172x) - 0.5298 cos(2*0.0172x) - 1.121 sin(2*0.0172x) - 1.142 cos(3*0.0172x) + 0.3966 sin(3*0.0172x) + 0.1746 cos(4*0.0172x) - 0.6291 sin(4*0.0172x) + 0.1938 cos(5*0.0172x) - 0.1892 sin (5*0.0172x) - 0.04237 cos(6*0.0172x) + 0.09564 sin(6*0.0172x) + 0.2258 cos(7*0.0172x) + 0.05545 sin(7*0.0172x) - 0.4136 cos(8*0.0172x) - 0.02458 sin(8*0.0172x). 

In the estimation of the maximum and minimum temperature variables, the best Fourier series models for only two stations are described under Figures 1 and 2. The RMSE's obtained, for the adjustment (1484) and the validation (forecast) of the model (229), for the maximum temperature were: Santa Rosa 1, with n =5, 2.553, 2.6213; Ruiz Cortinez, with n =7, 2.266, 2.5321; Batequis, with n =7, 2.417, 2.6847 and Santa Rosa 2, with n =7, 2.555, 3.0855. And for the minimum temperature: Santa Rosa 1, n =7, 2.379, 2.1415; Ruiz Cortinez, n =8, 2.163, 2.1759; Batequis, n =8, 2.289, 2.1822 and Santa Rosa 2, n =7, 2.135, 2.5005.

When comparing the RMSE's (Table 3) of the RNA with those of the Fourier series model for maximum temperature in the four seasons for the forecast, the first ones are better, by hundredths, the same happens when comparing the RMSE's for the minimum temperature. By considering the temperatures are recorded to tenths of a degree, the RNA present better performance (lower RMSE's) in maximum three tenths, than the Fourier model for all seasons. However, to train this type of neural network (backpropagation) you need to know four input variables (Julian day, latitude, longitude and altitude) and for the Fourier series model only the series of time in days.

The obtained in this work corroborates the findings of ^{Şahin (2012)}, which proposes the artificial neural network as an alternative method to estimate the air temperature with sufficient accuracy. The importance of the models of Fourier series obtained is as commented by ^{Coronas and Baldasano (1984)}, that the equations are representative of annual form for where they were generated and their advantage is that they represent a large amount of data in a single equation and new data is easily incorporated. Finally, ^{Almonacid, Perez-Higueras, Rodrigo & Hontoria (2013)} also made a comparison of three methods to estimate hourly temperatures for Spanish localities: that of artificial neural networks provided the best results, the transfer function they used was purelin in the layer of output and sigmoid tangent in the hidden layer, obtaining RMSE's in a range of 0.53 - 1.20 °C, these functions were those that performed better in the present work both in the adjustments, as in the forecasts.