Climate change is a reality and efficient use of scarce resources is vital. The challenge of this project is to study the behaviour of humidity in the soil by mathematical/statistical modeling in order to find optimal solutions to improve the efficiency of daily water use in irrigation systems. For that, it is necessary to estimate and forecast weather variables, in this particular case daily maximum and minimum air temperature. These time series present strong trend and high-frequency seasonality. This way, we perform a state space modeling framework using exponential smoothing by incorporating Box-Cox transformations, ARMA residuals, Trend and Seasonality.This research was partially financed by Portuguese funds by the Center for Research and Development in Mathematics and Applications (CIDMA) and the Portuguese Foundation for Science and Technology (”Fundação para a Ciência e a Tecnologia” - FCT), within project UID/MAT/04106 2019. This research was partially financed by Portuguese funds through Portuguese Foundation for Science and Tech nology (”Funda¸c˜ao para a Ciˆencia e a Tecnologia” - FCT), within project UID/MAT/00013/2013. FEDER/ COMPETE/- NORTE2020/ POCI/FCT funds through grants PTDC-EEI-AUT-2933-2014116858-TOCCATA and
To CHAIR - POCI-01-0145-FEDER-028247 Financial support from the Portuguese Foundation for Science and Technology (FCT) within the frame work of Strategic Financing UIDIFIS/04650/2013 is also acknowledge

Costa, Cláudia

Costa, Marco

Gonçalves, A. Manuela

Lopes, Sofia O.

English

Climate change is a reality and efficient use of scarce resources is vital. The challenge of this project is to study the behaviour of humidity in the soil by mathematical/statistical modeling in order to find optimal solutions to improve the efficiency of daily water use in irrigation systems. For that, it is necessary to estimate and forecast weather variables, in this particular case daily maximum and minimum air temperature. These time series present strong trend and high- frequency seasonality. This way, we perform a state space modeling framework using exponential smoothing by incorporating Box-Cox transformations, ARMA residuals, Trend and Seasonality.This research was partially financed by Portuguese funds by the Center for Research and Development in Mathematics and Applications (CIDMA) and the Portuguese Foundation for Science and Technology (”Fundação para a Ciência e a Tecnologia” - FCT), within project UID/MAT/04106 2019. This research was partially financed by Portuguese funds through Portuguese Foundation for Science and Technology (”Fundação para a Ciência e a Tecnologia” - FCT), within project UID/MAT/00013/2013. FEDER/ COMPETE/- NORTE2020/ POCI/FCT funds through grants PTDC-EEI-AUT-2933-2014116858-TOCCATA and To CHAIR - POCI-01-0145-FEDER-028247 Financial support from the Portuguese Foundation for Science and Technology (FCT) within the framework of Strategic Financing UIDIFIS/04650/2013 is also acknowledged.publishe

Repositório Institucional da Universidade de Aveiro

Forecasting Temperature Time Series forIrrigation Planning ProblemsCla´udia Costa1, A. Manuela Gonc¸alves2, Marco Costa3, SofiaO. Lopes41 Department of Mathematics, University of Minho, Guimara˜es, Portugal2 Department of Mathematics and Center of Mathematics, University of Minho,Guimara˜es, Portugal3 A´gueda School of Technology and Management and Center for Research andDevelopment in Mathematics and Applications, University of Aveiro, Aveiro,Portugal4 Department of Mathematics and Center of Physics, University of Minho, Guim-ara˜es, Portugal and SYSTEC - Research Center for Systems and Technologies,University of Porto, Porto, PortugalE-mail for correspondence: mneves@math.uminho.ptAbstract: Climate change is a reality and efficient use of scarce resources is vital.The challenge of this project is to study the behaviour of humidity in the soil bymathematical/statistical modeling in order to find optimal solutions to improvethe efficiency of daily water use in irrigation systems. For that, it is necessary toestimate and forecast weather variables, in this particular case daily maximumand minimum air temperature. These time series present strong trend and high-frequency seasonality. This way, we perform a state space modeling frameworkusing exponential smoothing by incorporating Box-Cox transformations, ARMAresiduals, Trend and Seasonality.Keywords: Forecasting; Irrigation; TBATS; Temperature; Time Series Model-ing.1 IntroductionIn this work, we address the daily irrigation problem of minimizing waterconsumption. Our data source are the records of temperatures observedin a farm located in Vila Real County in northern Portugal (Figure 1),registered in the period from January 23rd, 2015 to August 11th, 2018, ona daily basis. The main goal is to forecast the temperature at a location inThis paper was published as a part of the proceedings of the 34th Interna-tional Workshop on Statistical Modelling (IWSM), Guimara˜es, Portugal, 7–12July 2019. The copyright remains with the author(s). Permission to reproduce orextract any parts of this abstract should be requested from the author(s).2 Forecasting Temperature Time Series for Irrigation Planning Problemsorder to determine, in particular, the evapotranspiration which is directlyrelated to the irrigation planning problem. The location, in this case, is thefarm where there are historical observations, but current measurements arenot available, including various steps for forecasting.FIGURE 1. Farm in Vila Real County in northern Portugal (left); Maximum andMinimum Temperature daily time series distributions (right).2 MethodologyIn this study were used Trigonometric Seasonal, Box-Cox Transformation,ARMA errors, Trend and Seasonal Components method given by desig-nation TBATS(ω, φ, p, q, {m1, k1} , {m2, k2} , ..., {mT , kT }), where ω is theparameter of the Box-Cox transformation, φ is the damping parameter,the errors are modeled as ARMA(p, q) process, m1, ...,mT are the seasonalperiods and k2, ..., kT are the numbers corresponding to the Fourier termsused for each seasonal period. The smoothing parameters are given by α, βand γi for i = 1, ..., T . This method has the ability to deal with time serieswith high-frequency seasonality. The collected data were divided into twosets: training data (in-sample data) and testing data (out-of-sample data)in order to testify the accuracy of the suggested forecasting models.FIGURE 2. Daily time series of Maximum and Minimum Temperature distribu-tions in the farm during the observed period.Costa et al. 3FIGURE 3. Box-plots of the daily Maximum and Minimum Temperature distri-butions in the farm, by month.The selected training period was from January 23rd, 2015 to August 11th,2018 (first 1317 observations), and the test period with the last 10 obser-vations. Figure 2 and Figure 3 show plots of the Temperature series overtime and the distributions by month, respectively.3 ResultsThe model obtained for Maximum Temperature was TBATS (1, {3, 0} , 0.92,{365, 1}) and for Minimum Temperature TBATS(1, {2, 1} ,−, {365, 8}).Figure 4 (left) shows the result for the fit of the TBATS model for Maxi-mum and Minimum Temperature time series, respectively. Figure 4 (right)presents the estimates, and the forecasts (from August 2nd, 2018 to August11th, 2018). The values of Sigma are 2.882 and 2.253 for Maximum andMinimum Temperature time series modeling, respectively.FIGURE 4. Estimates (left) and estimates and forecasts for Maximum and Min-imum Air Temperature time series (right-last 250 observations).The results are satisfactory and allow assuring the capability and optimalperformance of the TBATS model, and suggest that the proposed frame-work is promising for future studies (see Table 1 and Table 2).4 Forecasting Temperature Time Series for Irrigation Planning ProblemsTABLE 1. Estimated Parameters using TBATS method.ω φ α β γ1 γ2 ϕ1 ϕ2 ϕ3 θ1Max. Temp. 1 0.91999 0.23332 -0.02287 3.97795e-05 0.00020 0.62759 -0.15828 -0.06488 -Min. Temp. 1 - 0.00552 - -0.00267 0.00046 0.83458 -0.18014 - -0.17154TABLE 2. Performance metric (different criteria: Mean Error (ME), Root MeanSquared Error (RMSE), Mean Absolute Error (MAE), Mean Absolute ScaledError (MASE), and Autocorrelation Function for first lag (ACF1)).ME RMSE MAE MASE ACF1Max. Temp. 0.03972 2.88222 2.30470 0.94509 -0.02451Min. Temp. 0.06769 2.25386 1.78015 0.88824 -0.00053Acknowledgments: This research was partially financed by Portuguesefunds by the Center for Research and Development in Mathematics andApplications (CIDMA) and the Portuguese Foundation for Science andTechnology (”Fundac¸a˜o para a Cieˆncia e a Tecnologia” - FCT), withinproject UID/MAT/04106 2019. This research was partially financed byPortuguese funds through Portuguese Foundation for Science and Tech-nology (”Fundac¸a˜o para a Cieˆncia e a Tecnologia” - FCT), within projectUID/MAT/00013/2013. FEDER/ COMPETE/- NORTE2020/ POCI/FCTfunds through grants PTDC-EEI-AUT-2933-2014116858-TOCCATA andTo CHAIR - POCI-01-0145-FEDER-028247 Financial support from thePortuguese Foundation for Science and Technology (FCT) within the frame-work of Strategic Financing UIDIFIS/04650/2013 is also acknowledged.ReferencesAlpuim, T. and El-Shaarawi, A. (2009). Modeling monthly temperature da-ta in Lisbon and Prague. Environmetrics, 20, 835 – 852.Costa, M. and Monteiro, M. (2017). Statistical Modeling of an Air Tem-perature Time Series of European Cities. In Advances in Environ-mental Research, vol. 59, ed. Justin A. Daniels, 213 - 236. ISBN:978-1-53612-701-0. New York: Nova Science Publishers.Hyndman, R.J., Koehler, A.B., Ord, J.K. and Snyder, R.D. (2008). Fore-casting with Exponential Smoothing: the State Space Approach. Berlin:Springer.Lopes, S.O., Fontes, F.A.C.C., Pereira, R.M.S., Pinho, MdR and Gonc¸alves, A.M.(2016). Optimal Control Applied to an Irrigation Planning Problem.Mathematical Problems in Engineering, Volume 2016, Article ID5076879, 10 pages.

Forecasting temperature time series for irrigation planning problems

Universidade do Minho: RepositoriUM

Forecasting Temperature Time Series forIrrigation Planning ProblemsCláudia Costa1, A. Manuela Gonçalves2, Marco Costa3, SofiaO. Lopes41 Department of Mathematics, University of Minho, Guimarães, Portugal2 Department of Mathematics and Center of Mathematics, University of Minho,Guimarães, Portugal3 Águeda School of Technology and Management and Center for Research andDevelopment in Mathematics and Applications, University of Aveiro, Aveiro,Portugal4 Department of Mathematics and Center of Physics, University of Minho, Guim-arães, Portugal and SYSTEC - Research Center for Systems and Technologies,University of Porto, Porto, PortugalE-mail for correspondence: mneves@math.uminho.ptAbstract: Climate change is a reality and efficient use of scarce resources is vital.The challenge of this project is to study the behaviour of humidity in the soil bymathematical/statistical modeling in order to find optimal solutions to improvethe efficiency of daily water use in irrigation systems. For that, it is necessary toestimate and forecast weather variables, in this particular case daily maximumand minimum air temperature. These time series present strong trend and high-frequency seasonality. This way, we perform a state space modeling frameworkusing exponential smoothing by incorporating Box-Cox transformations, ARMAresiduals, Trend and Seasonality.Keywords: Forecasting; Irrigation; TBATS; Temperature; Time Series Model-ing.1 IntroductionIn this work, we address the daily irrigation problem of minimizing waterconsumption. Our data source are the records of temperatures observedin a farm located in Vila Real County in northern Portugal (Figure 1),registered in the period from January 23rd, 2015 to August 11th, 2018, ona daily basis. The main goal is to forecast the temperature at a location inThis paper was published as a part of the proceedings of the 34th Interna-tional Workshop on Statistical Modelling (IWSM), Guimarães, Portugal, 7–12July 2019. The copyright remains with the author(s). Permission to reproduce orextract any parts of this abstract should be requested from the author(s).2 Forecasting Temperature Time Series for Irrigation Planning Problemsorder to determine, in particular, the evapotranspiration which is directlyrelated to the irrigation planning problem. The location, in this case, is thefarm where there are historical observations, but current measurements arenot available, including various steps for forecasting.FIGURE 1. Farm in Vila Real County in northern Portugal (left); Maximum andMinimum Temperature daily time series distributions (right).2 MethodologyIn this study were used Trigonometric Seasonal, Box-Cox Transformation,ARMA errors, Trend and Seasonal Components method given by designa-tion TBATS(ω, φ, p, q, {m1, k1}, {m2, k2}, ..., {mT , kT }), where ω is theparameter of the Box-Cox transformation, φ is the damping parameter,the errors are modeled as ARMA(p, q) process, m1, ...,mT are the seasonalperiods and k2, ..., kT are the numbers corresponding to the Fourier termsused for each seasonal period. The smoothing parameters are given by α, βand γi for i = 1, ..., T . This method has the ability to deal with time serieswith high-frequency seasonality. The collected data were divided into twosets: training data (in-sample data) and testing data (out-of-sample data)in order to testify the accuracy of the suggested forecasting models.FIGURE 2. Daily time series of Maximum and Minimum Temperature distribu-tions in the farm during the observed period.Costa et al. 3FIGURE 3. Box-plots of the daily Maximum and Minimum Temperature distri-butions in the farm, by month.The selected training period was from January 23rd, 2015 to August 11th,2018 (first 1317 observations), and the test period with the last 10 obser-vations. Figure 2 and Figure 3 show plots of the Temperature series overtime and the distributions by month, respectively.3 ResultsThe model obtained for Maximum Temperature was TBATS (1, {3, 0} , 0, 92,{(365, 1)}) and for Minimum Temperature TBATS(1, {2, 1} ,−, {(365, 8)}).Figure 4 (left) shows the result for the fit of the TBATS model for Maxi-mum and Minimum Temperature time series, respectively. Figure 4 (right)presents the estimates, and the forecasts (from August 2nd, 2018 to August11th, 2018). The values of Sigma are 2.882 and 2.253 for Maximum andMinimum Temperature time series modeling, respectively.FIGURE 4. Estimates (left) and estimates and forecasts for Maximum and Min-imum Air Temperature time series (right-last 250 observations).The results are satisfactory and allow assuring the capability and optimalperformance of the TBATS model, and suggest that the proposed frame-work is promising for future studies (see Table 1 and Table 2).4 Forecasting Temperature Time Series for Irrigation Planning ProblemsTABLE 1. Estimated Parameters using TBATS method.ω φ α β γ1 γ2 ϕ1 ϕ2 ϕ3 θ1Max. Temp. 1 0,91999 0,23332 -0,02287 3,97795e-05 0,00020 0,62759 -0,15828 -0,06488 -Min. Temp. 1 - 0,00552 - -0,00267 0,00046 0,83458 -0,18014 - -0,17154TABLE 2. Performance metric (different criteria: Mean Error (ME), Root MeanSquared Error (RMSE), Mean Absolute Error (MAE), Mean Absolute ScaledError (MASE), and Autocorrelation Function for first lag (ACF1)).ME RMSE MAE MASE ACF1Max. Temp. 0.03972 2.88222 2.30470 0.94509 -0.02451Min. Temp. 0.06769 2.25386 1.78015 0.88824 -0.00053Acknowledgments: This research was partially financed by Portuguesefunds by the Center for Research and Development in Mathematics andApplications (CIDMA) and the Portuguese Foundation for Science andTechnology (”Fundação para a Ciência e a Tecnologia” - FCT), withinproject UID/MAT/04106 2019. This research was partially financed byPortuguese funds through Portuguese Foundation for Science and Tech-nology (”Fundação para a Ciência e a Tecnologia” - FCT), within projectUID/MAT/00013/2013. FEDER/ COMPETE/- NORTE2020/ POCI/FCTfunds through grants PTDC-EEI-AUT-2933-2014116858-TOCCATA andTo CHAIR - POCI-01-0145-FEDER-028247 Financial support from thePortuguese Foundation for Science and Technology (FCT) within the frame-work of Strategic Financing UIDIFIS/04650/2013 is also acknowledged.ReferencesAlpuim, T. and El-Shaarawi, A. (2009). Modeling monthly temperature da-ta in Lisbon and Prague. Environmetrics, 20, 835 – 852.Costa, M. and Monteiro, M. (2017). Statistical Modeling of an Air Tem-perature Time Series of European Cities. In Advances in Environ-mental Research, vol. 59, ed. Justin A. Daniels, 213 - 236. ISBN:978-1-53612-701-0. New York: Nova Science Publishers.Hyndman, R.J., Koehler, A.B., Ord, J.K. and Snyder, R.D. (2008). Fore-casting with Exponential Smoothing: the State Space Approach. Berlin:Springer.Lopes, S.O., Fontes, F.A.C.C., Pereira, R.M.S., Pinho, MdR and Gonçalves, A.M.(2016). Optimal Control Applied to an Irrigation Planning Problem.Mathematical Problems in Engineering, Volume 2016, Article ID5076879, 10 pages.

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Forecasting temperature time series for irrigation planning problems

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