ST.R is a code in R language developed for the treatment of time series and the realisation of ARIMA models following the Box-Jenkins methodology [11] . The script is split into two blocks. The first one is a collection of

commands for the numerical and graphic description of the time series, and the development of the ARIMA models. In the second block the commands of different precision measurements are set up, which allow to compare the forecasts made by the models with the actual data with the aim of selecting the model with the most optimal fit to actual observations [13] [14] .

In order to successfully run the ST.R script, the necessary libraries are lmtest and tseries. These libraries are available from the repository Comprehensive R Archive Network (CRAN) at http://CRAN.R-project.org/package=OptGS. In this work, the R “stat” package version 3.3.0 was used, using “ARIMA” argument. The fitting methods are described in the R manual [15] .

1) Graphical representation: graphical representation of the time series to visualise its components (trend, cycle, stationarity and random or irregular component (Figure 2 and Figure 3).

2) Trend analysis: the existence or non-existence of the trend is studied from the graphical results. A linear trend will be removed with first differences. However, for a nonlinear trend two differences are used. The Dickey-Fuller [16] and KPSS [17] tests are used for the analysis (Figure 4).

3) Homocedasticity analysis: This is done from both a visual and a mathematical perspective. From a visual point of view, it is carried out through the study of the thickness of the series. If this thickness remains constant, with no major irregularities observed, the series will be homocedastic; otherwise, the series will be considered heterocedastic. From a mathemathical, it is carried out with the application of the homoscedasticity Breusch- pagan test [18] (Figure 5).

4) Stationarity analysis: As a result of the steps above, when neither seasonal cycle, nor trend, nor a significant thickness alteration of the series are to be perceived, the series is regarded as stationary (Figure 5).

5) Model identification: the most optimal model type is determined from the order of the Autoregressive procedure and moving averages of the constituents, both uniform and seasonal. This choice is made from autocorrelation (FAC) and partial autocorrelation (partial FAC) functions (Figure 5).

6) Estimation of the coefficients of the model: the order of the model having been established, the estimation of its parameters is made. Given it is an iterative calculation process, initial values (pool of models) can be suggested (Figure 5).

7) Detailed error analysis: It is made from the verified differences between values observed empirically and estimated by the model for their final assessment. It is necessary to check an inconsistent regime of them and analyse the existence of significant errors. The Ljung-Box test is applied [19] (Figure 5).

8) Contrast of model validity: the model or models initially selected are quantified and valued using various statistical measures. The measures applied are: R² (coefficient of determination), % SEP (standard error percentage), E₂ (coefficient of efficiency), ARV (average relative variance), AIC (Akaike information criterion), RMSE (root mean square error) and MAE (mean absolute error) (Figure 6).

9) Model selection: based on the results of the previous steps, the model to work on is decided upon (Figure 6).

10) Forecast: the most optimal model will be used as the prediction base tool (Figure 5).

The univariate ARIMA models (p,d,q) [11] try to explain the behaviour of a time series from past observations of the series itself and from past forecast errors. The compact notation of the ARIMA models is as follows:

where p is the number of autoregressive parameters, d is the number of differentiations for the series to be stationary, and q is the number of parameters of moving averages. The Box-Jenkins model (p,q) is represented by the following equation:

The autoregressive part (AR) of the model is, while the part of moving averages of the

from the data, by means of any consistent statistic. The ARIMA models allow fitting the trend plus the stationarity in data. In this case, the model is noted as:

where P is the number of autoregressive parameters in the seasonal part, D is the number of differentiations for the series to be seasonal in the seasonal part, Q is the number of parameters of moving averages in the seasonal part and S is the series frequency.

The Box-Jenkins method provides forecasts without any previous conditions, apart from being parsimonious with regard to coefficients [20] . Once the model has been found, forecasts and comparisons between actual and estimated data for observations from the past can be done immediately [21] .

The identification of the parameters p, q, P, Q and S is done by inspecting the autocorrelation function (ACF) and the partial autocorrelation function (PACF), taking into account differentiation and seasonal differentiation [22] .

To create models, the most suitable values of p, d and q were used, according to the measures of accuracy which are presented in the section of criteria for model selection. The parameters ϕ and θ are set through the use of the function minimisation procedures so that the square sum of residues be minimised.

The time series trend is studied applying the Dickey-Fuller [16] and KPSS [17] tests. The Dickey-Fuller test contrasts the null hypothesis that there is a unit root in the autoregressive polynomial (non-stationary series) against the alternative hypothesis that holds the opposite. The KPSS is another test with the same aim, but not exclusive of autoregressive models, supplementary of the former, which contrasts the null hypothesis that the series is stationary around a deterministic trend against the unit root alternative (non-stationary series). Homoscedasticity is studied through the Breusch-Pagan test [18] , which contrasts the null hypothesis that holds heteroscedasticity exists against its nonexistence.

The correlation between the actual and forecast data for a variable (x) is expressed by using the correlation coefficient. The coefficient of determination (R²) describes the proportion of total variation in the actual data, which can be explained by the model. The coefficient of determination shows a range of variation [0-1]. If R² = 1, it means a perfect linear fit, that is to say the proportion of total variation in the actual data is explained by the model. Instead if R² = 0, the model does not explain anything of the proportion of total variation in the actual data [23] .

Other selection measures applied in R are the standard error of prediction percentage (% SEP) [24] , the efficiency coefficient (E₂) [25] [26] , the average relative variance (ARV) [27] and the Akaike information criterion (AIC) [28] . The first four estimators are unbiased estimators which are used in order to check to what extent the model is able to explain the total variance of the data, while the AIC uses the maximum likelihood function to select the model which best fits data. Moreover, it is advisable to quantify the error in the same units as the studied variable.

These measures, or absolute error measures, include the root mean squared error (RMSE) and the mean absolute error (MAE), both expressed as follows:

where is the variable observed at moment t, is the estimated variable at the same moment t and N is the

total number of observations of the validation set.

The standard error of forecast percentage, % SEP, is defined as:

where is the average of the variable observed of the validation set. The main advantage of %SEP is its non-dimensionality, which allows to compare the forecasts of the different models on the same base.

The efficiency coefficient (E₂) and the average relative variance (ARV) are used to verify how the model explains the total variance of data and to represent the proportion of the variation of the data observed considered for the model. E₂ and ARV are defined as:

, (7)

The sensitivity to the atypical values due to squaring the terms of the difference is associated to E₂ or to ARV. The Akaike information criterion (AIC) combines the maximum likelihood theory, theoretical information and information entropy [29] , and is defined by the following equation [30] [31] :

where N is the total number of observations of the validation set, k is the number of the parameters of the estimated model, MSE is the mean square error estimated, which is defined by the following equation [30] [31] :

where N is the total number of observations of the validation set, k is the number of parameters of the estimated

Depending on the fit, a model which explains a high variance level (R², ARV, E₂) in the validation period is associated to low absolute error (RMSE, MAE), relative (% SEP) and Akaike (AIC) values. Hence, the hypothesis is validated that when using AIC the best model will be that which presents the lowest value, since its likelihood function will fit the data more accurately [28] .

The nature of information differs now from that of information in the past. Due to the vast amount of measuring devices (sensors, microphones, cameras, medical scanners, images, etc.), the data generated by these elements are the largest of the entire available information spectrum. For this reason, the analysis of the wealth of time series has been carried out in a continuous and frequent way [33] in order to obtain the prediction variables and thus to be able to warn behaviour in the environment these occur.

The analyses of time series take into account the degree of dependence between observations and allow to obtain valid inferences without violating basic assumptions of the statistical model or introducing variations in order to overlook this problem; this way, the model further fits the real behaviour of the series.

Since time series are currently employed in different and various fields of knowledge―telecommunications [34] , fisheries [35] , medicine [36] , etc.―it is important to perform a script that allows to give a global and integrated vision on the treatment of time series grouping all the relevant information with the characteristics of the series and prediction models.

Treatment and analysis of time series using free software such as R presents advantages and disadvantages in comparison with private software. On the one hand R has been used in this work as a free and cross-platformer software, making it easy to work with different operating systems. As it has an open source, it is continuously updated by users, not to mention its great graphical power. On the other hand we are aware that the development of this script in the R programming environment presents a number of drawbacks, such as abundant but unstructured help information or packages and functions that make it difficult to locate specific information in a given search. Error messages do not show clearly where in the development of the script the bug is committed, which creates problems for users with little experience in this programming environment making the initiation tedious. R is a programming language in lines of commands, which does not use menus as other statistical programs (e.g., Statgraphics) interfaces. However this can also be an advantage since R advanced users are able to schedule the treatment and analysis of data, in order to understand the basis of the statistical development and data analysis.

To this aim the ST.R script has been created, whose main objective is the analysis and development of forecasting models for time series. It can be established that time series models allow to estimate the degree of significance of a level change which is operated as a result of the application of a treatment [37] . These models not only allow to obtain statistical inferences on treatment action, but also solve the problem of dependence inherent to this type of designs which use a single subject.

In this work, Excel has been used for the database structure management. We know that this system is not sufficiently solvent to support the current data productions [38] . Although Excel is satisfactory for time series management since this working field is univariate based, Excel has also the advantage of being user friendly and accessible for most users. Then this system is considered an efficient tool when it comes to structuring univariate time series.