This work presents a periodic state space model to model monthly temperature data. Additionally, some issues are discussed, as the parameter estimation or the Kalman filter recursions adapted to a periodic model. This framework is applied to monthly long-term temperature time series of Lisbon

Magda Monteiro

Marco Costa

Costa, Marco

Monteiro, Magda

Repositório Institucional da Universidade de Aveiro

II Encontro Galaico-Portugueˆs de BiometriaSantiago de Compostela, 30 de Junho, 1 e 2 de Julho de 2016A Periodic State Space Model to Monthly Long-term Temperature DataMarco Costa1,2 and Magda Monteiro1,21ESTGA - School of Technology and Management of A´gueda, University of Aveiro, Portugal2CIDMA - Center for Research & Development in Mathematics and Applications, University ofAveiro, PortugalABSTRACTThis work presents a periodic state space model to model monthly temperature data.Additionally, some issues are discussed, as the parameter estimation or the Kalmanfilter recursions adapted to a periodic model. This framework is applied to monthlylong-term temperature time series of Lisbon.Key-words: State space model, Kalman filter, periodic data, temperature.1. INTRODUCTIONThe rise in global temperature has been an increasing concern of several authorities. Accordingto the Intergovernmental Panel on Climate Change, the world’s greenhouse gas emissions are stillincreasing and on the present path, the global temperature rise will substantial exceed the limitgoal of two degrees Celsius that countries have agreed upon in order to avoid the most dangerousimpacts of climate change. In the United Nations (UN) Framework on Climate Change, it wasadopted the Paris Agreement, on 12 December 2015, at the UN Climate Change Conference inFrance where parties committed to take ambitious actions to keep global temperature rise below2 degrees Celsius by the end of the century [9].Climate change monitoring is performed with multidisciplinar approaches. However, the statisticalanalysis has been a key in this theme. Usually, monitoring climate change is performed based onregional or global temperature long-term data. Indeed, temperature data is considered in severalstudies of climate change or as the main variable under analysis or as a covariate on modeling otherenvironmental variables, for instance in rainfall modeling. Thus, long-term temperature time seriesare, par excellence, the most appropriate data to study climate change, mainly in the global orlocal warming.Indeed, modeling and foresight scenarios of environmental and climate variables are based on theselection of climate models using the analysis of other variables as rainfall and temperature [5].On the other hand, the impact of changes in environmental variables changes, as the temperatureor the precipitation, on environmental issues have been assessed and analyzed in order to explorethe implications under a future climate [3].The most applied statistical models in order to analyze climate time series, in particular of temper-atures data sets, are the linear models and the usual time series models as ARIMA models [1, 2, 4].These models are largely known as simple and they have well-known optimal properties under theusual assumptions (normality of errors, etc.). Some extensions and combinations of these models63have been proposed in environmental data analysis in order to improve models performance andto increase the predictions accuracy.State space models have been largely considered an extension of the usual linear models whichincorporate a versatile stochastic structure at the same time that integrate temporal dependence.The goal of this work is to analyze monthly long-term temperature time series of the Portuguesecity of Lisbon. In [7] propose and implement a methodology for detection and correction of seriesof monthly temperatures for Lisbon, Oporto and Coimbra, providing the homogenization of thesePortuguese long-term temperature data series (data in [8]). For instance, see Figure 1 related toLisbon homogenized monthly temperature data.Figure 1: Lisbon homogenized monthly temperature data.State space models have been largely applied in several areas of applied statistics. In particular,the linear state space models have desirable properties and they have a huge potential in timeseries modeling that incorporates latent processes.Once a model is placed in the linear state space form, the most usual algorithm to predict the latentprocess, the state, is the Kalman filter algorithm. This algorithm is a procedure for computing,at each time t (t = 1, 2, . . .), the optimal estimator of the state vector based on the availableinformation until t and its success lies on the fact that is an online estimation procedure.The main goal of the Kalman filter algorithm is to find predictions for the unobservable variablesbased on observable variables related to each other through a set of equations forming the statespace model. Indeed, in the context of linear state space models, the Kalman filter produces thebest linear unbiased estimators. When the errors and the initial state are Gaussian, the Kalmanfilter predictors are the best unbiased estimators in the sense of the minimum mean square error.However, the optimal properties only can be guaranteed when all model’s parameters are known[6].2. THE PERIODIC STATE SPACE MODEL - PSSMIn order to accommodate data properties as temporal correlation and the periodic behavior, aperiodic state space model is proposed. This model is defined asYs,n = [S(n− 1) + s]Xs,n +Ds,n + es,n (1)Xs,n = µs + s(Xs−1,n − µs−1) +  s,n (2)where• s is the season of the year with s = 1, 2, ..., S;• n is the year with n = 1, 2, ..., N ;• Ys,n represents the time series observation in the sth season of the nth year, that is, the[S(n− 1) + s]th observation of the time series;64•  = [ 1 2... S ]  are unknown parameters representing the fixed effects in the model;• Ds,n is an 1× S matrix of known values, a design matrix;• the error process (es,n) is a white noise disturbance, which is assumed to have var(es,n) = 2e ;• µs is the mean of the process (Xs,n) for the sth season, s is the autoregressive parameterfor season s and  s,n is the white noise disturbance;• the error process ( s,n) are distributed IID(0, 2s) and it is assumed to have an covariancedefined bycov( s,n,  s−i,n) =2 ,s i = 00 i  = 0 for i = 1, 2, ..., S.The first equation is referred as the observation equation and the second is called the stateequation.4. GAUSSIAN LIKELIHOOD ESTIMATIONUnder the assumptions that the initial state, the state noise  s,n and the observation noise es,n aremutually independent and normally distributed, for a given realization Y = (Y1,1, Y2,1, ..., YS,N ) the logarithm of the Gaussian likelihood function is computed as followslogL( ;Y) = −NS2log(2 )− 12NXn=1SXs=1log( s,n)− 12NXn=1SXs=12s,n s,nwhere = ( , 2e , µ1, ..., µS , 1, ..., S ,21, , ...,2S, ) , s,n = Ys,n −  Ys−1,n is the innovation and s,n is the mean square error of the innovation. The forecast one-step-ahead of Ys,n,  Ys−1,n,is obtained by the Kalman filter equations reformulated to the model (1)-(2). The maximumlikelihood estimates are obtained through numerical procedures.5. SOME RESULTSThe modeling procedure and the Kalman filter algorithm allow obtaining one-step-ahead forecastsand some overall results. In the first case, Figure 2 shows the one-step-ahead forecasts in the lastdecade of the period analyzed in Lisbon. It seems that the model has a good adjustment to thereal data.Figure 2: One-step-ahead forecasts in the last decade of the period analyzed in Lisbon.On the other hand, the model’s parameters allow to concluded that there is a global temperaturerise. However, the months with the highest average rates of temperature increase by century areJanuary (+0,9oC), March (+1,1oC), October (+0,8oC) and December (+0,8oC).ACKNOWLEDGEMENTSAuthors were partially supported by Portuguese funds through the CIDMA - Center for Researchand Development in Mathematics and Applications, and the Portuguese Foundation for Science andTechnology (”FCT– Fundac¸a˜o para a Cieˆncia e a Tecnologia”), within project UID/MAT/04106/2013.65References[1] Alpuim, T., El-Shaarawi, A. (2009). Modeling monthly temperature data in Lisbon and Prague.Environmetrics 20 835–852.[2] Blizˇnˇa´k, V., Valente, M.A., Bethke, J.E. (2015). Homogenization of time series from Portugaland its former colonies for the period from the late 19th to the early 21st century. InternationalJournal of Climatology 35 2400–2418.[3] Flannigan, M.D., Wotton, B. M., Marshall, G.A., Groot, W.J., Johnston, J., Jurko, N., Cantin,A.S. (2016). Fuel moisture sensitivity to temperature and precipitation: climate change impli-cations Climatic Change 134 59–71.[4] Freitas, C.R., Dedekind, M.O., Brill, B.E. (2015). A Reanalysis of Long-Term Surface AirTemperature Trends in New Zealand Environmental Modeling & Assessment 20 399-410.[5] Guerreiro, S.B., Kilsby, C. G., Fowler, H.J. (2016). Rainfall in Iberian transnational basins: adrier future for the Douro, Tagus and Guadiana? Climatic Change DOI:10.1007/s10584-015-1575-z.[6] Harvey, A.C. (1996). Forecasting Structural Time Series Models and the Kalman Filter,Cambridge University Press, Cambridge.[7] A.L. Morozova, M.A. Valente, Homogenization of Portuguese long-term temperature data se-ries: Lisbon, Coimbra and Porto, Earth Syst. Sci. Data 4: 187–213 (2012).[8] A.L. Morozova, M.A. Valente, Homogenization of Portuguese long-term temperature data se-ries: Lisbon, Coimbra and Porto, doi:10.1594/PANGAEA.785377 (2012).[9] United Nations (2016). http://www.un.org/sustainabledevelopment/climate-change/ (ac-cessed in January 22, 2016).66

A Periodic State Space Model to Monthly Long-term Temperature Data

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A Periodic State Space Model to Monthly Long-term Temperature Data

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