In this work, we present time series of monthly average temperatures in several Euro-
pean locations which were statistically analyzed using a state space approach, where it
is considered a model with a deterministic seasonal component and a stochastic trend.
The analysis of smoother prediction of the stochastic trend and its comparison in a tem-
poral viewpoint can reveal patterns about warming in Europe. The temperature rise
rates in Europe seem to have increased in the last decades when compared with longer
periods, hence a change point detection method is applied to the trend component in
order to identify these possible changes in the monthly temperature rise rates. The
adopted methodology pointed out, for most series a change point in the late eighties.publishe

Costa, Marco

Monteiro, Magda

English

Repositório Institucional da Universidade de Aveiro

XIV Congreso Galego de Estat́ıstica e Investigación de OperaciónsVigo, 24, 25 e 26 de outubro de 2019CHANGE POINT ANALYSIS IN A STATE SPACE FRAMEWORK TOMONTHLY TEMPERATURE DATA IN EUROPEAN CITIESMagda Monteiro1,2, Marco costa1,21 ESTGA - Águeda School of Technology and Management, University of Aveiro2 CIDMA - Center for Research & Development in Mathematics and Applications, University ofAveiroAbstractIn this work, we present time series of monthly average temperatures in several Euro-pean locations which were statistically analyzed using a state space approach, where itis considered a model with a deterministic seasonal component and a stochastic trend.The analysis of smoother prediction of the stochastic trend and its comparison in a tem-poral viewpoint can reveal patterns about warming in Europe. The temperature riserates in Europe seem to have increased in the last decades when compared with longerperiods, hence a change point detection method is applied to the trend component inorder to identify these possible changes in the monthly temperature rise rates. Theadopted methodology pointed out, for most series a change point in the late eighties.Keywords : climate change, monthly data, temperature data, time series analysis, state spacemodeling, Change point detection1. INTRODUCTIONGlobal warming and the associated climate changes are topics in the world’s agenda, which arereceiving much attention from the scientific community, due to their major social, economic andhealth impacts on human life.These changes are already being felt all over the world, with the increase of natural catastrophessuch as droughts, floods, tropical storms, among others. Nevertheless, global warming effects varyaround the world, therefore this phenomenon must be monitored at a smaller scale. In particular,given our location, we are interested in understanding these phenomena at Europe scale. In thissense, the analysis of local temperature time series is an important topic that can contribute to abetter understanding of climate dynamics at smaller scales, allowing for the identification of theirchange patterns. Thus, this analysis can be very useful for the application of efficient environmentmonitoring processes.This work is a preliminary analysis of long monthly average temperature time series in fifteenEuropean cities in order to find change point associated with rise rates changes of temperature.The Data is available at the Climate Data Online and comprises the period between January 1900and December 2017, making a total of 1416 observations. The location of the cities and datadescriptive statistics can be found in Table 1. The major data of this work has already beenanalyzed in Costa and Monteiro(2017), where state space models associated to the Kalman filteralgorithm and also a cluster procedure to smoother trend levels were applied in order to investigatepatterns on the temperature rise in Europe. A conclusion therein is that temperature rise rates inEurope seem to have increased in the last decades being important to find the change points anduse the information from there to get accurate temperature rise rates. Therefore, in the presentwork it is applied a change point detection procedure of maximum type in order to assess if thereis a change point in the temperature trend pattern for each city under analysis.There are several works on change point detection in climate or environmental settings. Forinstance, Reeves et al.(2007) makes a review of this topic for models with independent and iden-tically distributed (i.i.d.) errors and Jarušková and Antoch(2018) applied several change pointsprocedures to Klementinum temperature. In line with this last work, it will be considered basicstatistical tests with the applications of maximum type statistics to the stochastic trend of thetemperature time series in Europe.city (country) coord. min max average average missing1900 1900 2000 valuesto 2017 to 1917 to 2017 (%)Berlin (Germany) 52.47N, 13.30E -10.7 23.0 8.7 9.9 2.3Bucharest (Romania) 44.40N, 26.10E -11.5 27.5 10.1 11.1 2.6Copenhagen (Denmark) 55.68N, 12.55E -6.7 21.6 8.6 9.6 0.4Dublin (Ireland) 53.43N, 6.25W 0.0 17.1 9.2 9.6 2.2Kiev (Ukraine) 50.40N, 30.57E -15.3 26.0 8.1 9.4 0.9Lisbon (Portugal) 38.72N, 9.15W 8.0 25.1 16.9 17.4 3.4Minsk (Belarus) 53.93N, 27.63E -16.8 22.6 6.1 7.4 1.8Nantes (France) 47.17N, 1.60W -1.2 23.3 12.0 12.5 0.4Prague (Czech Rep.) 50.10N, 14.25E -13.1 22.0 7.9 9.1 8.2S. Petersburg (Russia) 59.97N, 30.30E -18.3 24.4 5.3 6.5 0.0Talin (Estonia) 59.42N, 24.80E -15.5 21.5 5.3 6.5 5.4Vienna (Austria) 48.25N, 16.37E -9.5 24.4 10.1 11.3 0.2Vilnius (Lithuania) 54.63N, 25.10E -17.1 21.8 5.8 7.3 2.6Zagreb (Croatia) 45.82N, 15.98E -7.0 25.9 11.8 13.0 1.4Zurich (Switzerland) 47.38N, 8.57E -8.7 22.4 8.7 9.8 0.0Table 1: Characterization of time series data set.2. MODEL DESCRIPTIONIn order to model, in each city, the monthly temperature time series, it is considered a modelwith a deterministic seasonal component and a stochastic trend, defined byYt = τt +12∑i=1βiSt,i + e1,t (1)τt = µ+ φ(τt−1 − µ) + e2,t (2)where {τt} is the trend component that follows a stationary autoregressive process with mean µand the autoregressive parameter |φ| < 1; the error ei,t, with i = 1, 2, follows a white noise process(E(ei,t) = 0, var(e2i,t) = σ2i and E(ei,tei,r) = 0, for t 6= r), and are uncorrelated errors, that isE(e1,te2,r) = 0, ∀t, r. The error e1,t is called the observation error and it can be seen as a measureerror, whereas the error e1,t is called the state error and translates the randomness of the trendcomponent.Note that state τt is an unobservable variable and its predictions must be obtained. For that,the state process {τt} in the model (1)-(2) can be predicted by Kalman filter and Kalman smootherestimators.This model was also used in Costa and Monteiro(2017) to which a clustering procedure wasapplied to understand the patterns associated to the climate in Europe. In terms of parameterestimation, similar to what was done in this previous work, a classical decomposition approach wasused combining the least squares estimation of the seasonal parameters βi, with i = 1, ..., 12, witha distribution-free estimators developed to state space models in order to estimate the remainingparameters.2. CHANGE POINT DETECTIONIn order to discover an abrupt change in the behaviour of the temperature rise rates, a basicstatistical test applying maximum type statistics to detect a change in location is performed. Ifthere are no structural changes in the temperature trend, it is expected that τt has no changesover time in its mean. Therefore, it is applied a statistical test to the smoother predictions, τt|n, ofτt having into into account that τ1, τ2, ..., τn are an AR(1) process and that smoother predictionsare the predictors of the non-observable component that present lower mean square error whencompared with the other Kalman filter predictions.Briefly, if we are in the presence of an AR(1) process, {Xt}, the hypotheses of the test ofmaximum type areH0: Xt are an AR(1) sequence normally distributed variables according to the sameN(µ, σ2)vsH1: there is a time point k ∈ {1, ..., n− 1} such that X1, ..., Xk are distributed accordingto N(µ1, σ2) and Xk+1, ..., Xn are distributed according to N(µ2, σ2).Assuming σ2 is unknown, then the test statistic T (n) is the maximum of the absolute valuesof two sample t-test statisticsT (n) = max1≤k<n|Tk| = max1≤k<n√(n− k)kn∣∣∣Xk −X∗k∣∣∣skwhereXk =1kk∑i=1Xi, X∗k =1n− kn∑i=k+1Xi and sk =√√√√ 1n− 2[k∑i=1(Xi −Xk)2+n∑i=k+1(Xi −X∗k)2].The null hypothesis can be rejected if the statistic T (n) is greater than the critical value.The exact distribution of T (n), for independent variables, is very complex and Worsley(1979)was able to calculate the true critical values only for the number of observations n less than 10.Alternatively, approximate critical values can be computed by other methods as the Bonferroniinequality, simulation or the asymptotic distribution.Antoch et al.(1997) shows that when random variables X1, X2, ..., Xn are not independentbut form an ARMA sequence then the asymptotic critical values of the test statistics consideringindependence have to be multiplied by√2πf(0)/γ, where γ = varXt and f(·) denotes the spectraldensity function of the corresponding ARMA process. Especially for an AR(1) sequence, the criticalvalues should be multiplied by[(1 + φ)(1− φ)−1]1/2where φ is the first autoregressive coefficient,(Jarušková,1997).The present work uses critical values obtained by the asymptotic distribution which expressioncan be found in Jarušková (1997) and multiplyed by the AR(1) factor with the estimates of φ.For each city under study, these basic change point procedure detected a change and Table2 summarizes these changes and also present the mean of smoother predictors before and afterchanges. The majority of temperature time series presents a change point in the late eighties.Lisbon presents a change point much earlier, in the previous decade, and Bucharest presents achange point about twenty years later.Note that western cities present the lowest mean values of τt|n and because of that the shorterdistances between averages before and after the detected change point with differences below 0.5oC. In the opposite site are the major eastern cities with differences above 1oC.This work is at an early stage and there are some issues that need further research. Forinstance we are applying the change point test not directly to the stochastic trend component,that is unobservable, but to the smoother predictions. The way the Kalman smoother predictionsare built induces a strong dependence between consecutive values and their use to evaluate thetest assumptions associated with the trend component might be misleading, so it is necessary tostudy this subject in depth. Another issue to be studied is the possibility of other change points,since it was only analyzed the at most one change point case. Finally, the confrontation betweenthe obtained results and other different change points methodologies will be the target of futurework.change point mean bef. meancity (country) month/year c.–p. after c.–p.Berlin (Germany) 9/1987 0.491 1.426Bucharest (Romania) 9/2006 0.517 1.732Copenhagen (Denmark) 9/1987 0.416 1.376Dublin (Ireland) 10/1988 0.281 0.721Kiev (Ukraine) 12/1989 0.506 1.479Lisbon (Portugal) 7/1978 0.357 0.843Minsk (Belarus) 11/1987 0.659 1.814Nantes (France) 7/1987 0.359 0.827Prague (Czech Rep.) 8/1987 0.563 1.457S. Petersburg (Russia) 12/1988 0.572 1.643Talin (Estonia) 12/1988 0.507 1.596Vienna (Austria) 8/1987 0.520 1.297Vilnius (Lithuania) 11/1987 0.715 2.325Zagreb (Croatia) 12/1992 0.389 1.274Zurich (Switzerland) 8/1987 0.578 1.329Table 2: Summary of change point detection.ACKNOWLEDGEMENTSAuthors were partially supported through Portuguese funds by the Center for Research and Develop-ment in Mathematics and Applications (CIDMA) and the Portuguese Foundation for Science and Tech-nology (“Fundação para a Ciência e a Tecnologia” [FCT]), within project UID/MAT/04106/2019.REFERENCESAlpuim, T. and El-Shaarawi, A. (2009). Modeling monthly temperature data in Lisbon and Prague.Environmetrics, 20: 835–852.Antoch, J., Hušková, M. and Z. Práškova (1997). Effect of dependence on statistics for determi-nation of change. Journal of Statistical Planning and Inference. 60, 291–310.Costa, M. and Monteiro, M. (2016). Bias-correction of Kalman filter estimators associated to alinear state space model with estimated parameters. Journal of Statistical Planning and Inference176: 22 – 32.Costa, M. and Monteiro, M. (2017). Statistical Modeling of an Air Temperature Time Series ofEuropean Cities, In Advances in Environmental Research. vol. 59, ed. Justin A. Daniels, 213 –236. ISBN: 978-1-53612-701-0. New York: Nova Science Publishers.Costa M. and Monteiro M. (2018). A periodic mixed linear state-space model to monthly long-termtemperature data. Environmetrics e2550 https://doi.org/10.1002/env.2550Jarušková, D. (1997). Some problems with application of change point detection methods toenvironmental data. Environmetrics, 8:469–483.Jarušková, D. and Antoch, J. (2018). Changepoint analysis of Klementinum temperature series.Environmetrics. 2019;e2570. https://doi.org/10.1002/env.2570.Reeves, J. Chen, J., Wang, X.L., Lund, R.B. and Lu, Q. (2007). A review and comparison ofchangepoint detection techniques for climate data. Journal of Applied Meteorology and Climato-logy. 46, 900–915.Worsley, K. J. (1979). On the likelihood ratio test for a shift in location of normal populations.Journal of the American Statistical Association. 74: 365–367.

Change point analysis in a state space framework to monthly temperature data in European cities

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Change point analysis in a state space framework to monthly temperature data in European cities

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