Abstract
Spatiotemporal modelling is an area of increasing importance in which models and methods have often been developed to deal with specific applications. In this study, a spatiotemporal model was used to estimate daily rainfall data. Rainfall records from several weather stations, obtained from the Agritempo system for two climatic homogeneous zones, were used. Rainfall values obtained for two fixed dates (January 1 and May 1, 2012) using the spatiotemporal model were compared with the geostatisticals techniques of ordinary kriging and ordinary cokriging with altitude as auxiliary variable. The spatiotemporal model was more than 17% better at producing estimates of daily precipitation compared to kriging and cokriging in the first zone and more than 18% in the second zone. The spatiotemporal model proved to be a versatile technique, adapting to different seasons and dates.
Keywords:
spatiotemporal model; rainfall data; ordinary kriging; ordinary cokriging; homogeneous zones
Resumo
A modelagem espaçotemporal é uma área de importância crescente em que modelos e métodos têm sido desenvolvidos para aplicações específicas. Neste estudo, o modelo espaçotemporal foi utilizado na estimativa de dados de precipitação diários. Foram utilizados dados provenientes dos registros pluviométricos de diversas estações meteorológicas, obtidos junto Sistema Agritempo para duas zonas climaticamente homogêneas. Valores de chuva para duas datas fixas (01 de janeiro de 2012 e 01 de maio de 2012) estimados pelo modelo espaçotemporal foram comparados com as técnicas geoestatísticas de krigagem e cokrigagem ordinária, com altitude como variável auxiliar. O modelo espaçotemporal foi superior em mais de 17% em relação às técnicas geoestatísticas de krigagem e cokrigagem para produzir estimativas da precipitação diária para a primeira zona homogênea e mais de 18% para a segunda zona homogênea. O modelo espaçotemporal provou ser uma técnica versátil, adaptandose nas diferentes épocas e datas estudadas.
Palavraschave:
modelos espaçotemporal; dados de precipitação; krigagem ordinária; cokrigagem ordinária; zonas homogêneas
1. Introduction
Research in agrometeorology depends on complete meteorological data collected from weather stations, remote sensors or climate models without spatial or temporal missing data.
The formation of a solid meteorological database often requires reconstruction of time series, which involves quality control methods, including gap filling or data imputation (VicenteSerrano , 2010VICENTESERRANO, S.M.; BEGUERIA, S.; LOPEZMORENO, J.I.; GARCIAVERA, S.M.; BEGUEIRA, S.; LÓPEZMORENO, J.I.; GARCÍAVERA, M.A.; STEPANEK, P. A complete daily precipitation database for northeast Spain: reconstruction, quality control, and homogeneity. Int. J. Climatol., v. 30, p. 11461163, 2010.). In most cases, filling gaps in daily rainfall data is a difficult task. Rainfall data observed at various locations during different times typically show intrinsic spatial and temporal variations.
A typical example would be a network of weather stations in which data are collected at regular, daily, weekly, monthly or yearly intervals. Data analysis must consider the spatial dependence of the seasons, but also that observations at each station are usually not independent from each other, but form a time series. Therefore, temporal and spatial correlations have to be considered in the analysis. Articles of RuizCárdenas (2009)RUIZCÁRDENAS, R.; ASSUNÇÃO, R.M., DEMÉTRIO, C.G.B. SpatioTemporal modelling of coffee berry borer infestation patterns accounting for inflation of zeroes and missing values. Sci. Agric., v. 66, n. 1, p. 100109, 2009. studying pest distributions, Lou and Obradovic, (2011)LOU, Q.; OBRADOVIC, Z. Modeling multivariate spatiotemporal remote sensing data with large gaps. International Joint Conference On Artificial Intelligence 22, 2011, Barcelona, Spain. Proceedings [Barcelona]. developing more accurate spatiotemporal predictors, Haworth and Cheng (2012)HAWORTH, J.; CHENG, T. Nonparametric regression for spacetime forecasting under missing data. Computers Environment band Urban Systems, v. 36, p. 538550, 2012. defining a nonparametric spatiotemporal kernel regression model to forecast the future unit journey time values of road links, Li and Parker (2008)LI, Y.Y.; PARKER, L.E. A spatialtemporal imputation technique for classification with missing data in a wireless sensor network. IEEE International Conference On Intelligent Robots And Systems, 2008, Nice, France. Proceedings [Nice]. developing a novel method to estimate missing observations in wireless sensor networks, Lau et al. (2014) defining a novel approach for diagnosing misspecifications of a general spatiotemporal transmission model, Franciscon (2008)FRANCISCON, L.; RIBEIRO JUNIOR, P.J; KRAINSKI, E.T.; BASSANEZI, R.B.; CZERMAINSKI, A.B. C. Modelo autologístico espaçotemporal com aplicação à análise de padrões espaciais da leprosedoscitros. Pesq. Agropec. Bras., v. 43, n. 12, p. 16771682, 2008. proposing strategies applied to the analysis of citrus leprosis incidence, through the use of spatio temporal autologistic model, have this concern.
Agriculture is the human activity most dependent on weather and climate conditions. Weather conditions affect all stages of agricultural activities and climate adversities often lead to serious social impacts and huge economic losses that are often difficult to quantify. As adverse weather conditions are common and difficult to control, agriculture constitutes a highrisk activity (Pereira , 2002PEREIRA, A.R.; ANGELOCCI, L.R.; SENTELHAS, P.C. Agrometeorologia: fundamentos e aplicações práticas. Guaíba: Livraria e Editora Agropecuária, 2002. 478 p.). Therefore, climate monitoring and weather forecasting are becoming increasingly important for agricultural decisions.
Due to the usuallysequential nature of the calculations in crop simulation models, interruptions or lack of data at any point in a time series of a crop cycle prevents completion of the simulation of that cycle or harvest. Therefore, such errors, will often result in the loss of results for an entire crop cycle or year of simulation, at a given site.
Meteorological databases maintained by several Brazilian institutions such as the Institute of Meteorology INMET and the Weather Center and Climate Studies CPTEC, for example, often have missing data. This requires that series with gaps be rebuilt for enduser applications and further analysis.
A number of approaches have been used to produce complete time series including multiple discriminant analysis (Young, 1992YOUNG, K.C. A threeway model for interpolating for monthly precipitation values. Mon. Weather Rev., v. 120, p. 25612569, 1992.) nearest neighbor analysis (VicenteSerrano 2010VICENTESERRANO, S.M.; BEGUERIA, S.; LOPEZMORENO, J.I.; GARCIAVERA, S.M.; BEGUEIRA, S.; LÓPEZMORENO, J.I.; GARCÍAVERA, M.A.; STEPANEK, P. A complete daily precipitation database for northeast Spain: reconstruction, quality control, and homogeneity. Int. J. Climatol., v. 30, p. 11461163, 2010.), regression methods (Lo Presti , 2010LO PRESTI, R.;BARCA, E.; PASSARELLA, G.A. methodology for treating missing data applied to daily rainfall data in the Candelaro River Basin (Italy). Environmental Monitoring and Assessment, v. 160, n. 14, p. 122, 2010.), geostatistical methods (Bajat , 2013BAJAT, B.; PEJOVIC, M.; LUKOVIC, J.; MANOJLOVIC, P.; DUCIC, V.; MUSTAFIC, S. Mapping average annual precipitation in Serbia (19611990) by using regression kriging. Theoretical and Applied Climatology, v. 112, n. 12, p. 113, 2013.), multiple linear regressions or neural networks (Fowler , 2007FOWLER, H.J.; BLENKINSOP, S.; TEBALDI, C. Linking climate change modelling to impacts studies: recent advances in downscaling techniques for hydrological modelling. Int. J. Climatol., v. 27, n. 12, p. 15471578, 2007.).
Often, the primary interests in the analysis of spatiotemporal data is a prediction of development time of response variable in a given spatial domain (Lasinio , 2007LASINIO, G.J.; SAHU, S.K.; MARDIA, K.V. Modeling rainfall data using a Bayesian KrigedKalman model. In: UPADHYAY, S. K.; SINGH, UMESH; Dey, Dipak K. (Ed.). Bayesian statistics and its applications. Tunbridge Wells, UK: Anshan, 2007. p. 6186.). In recent years, there has been an increase in research on statistical models and techniques to solve this problem.
Spatiotemporal models have been successfully applied in several areas such as hydrology (Rouhani and Myers, 1990ROUHANI, S.; MYERS, D.E. Problems in spacetime kriging of geohydrological data. Mathematical Geology, v. 22, n. 5, p. 611623, 1990.; Amisigo and Giesen, 2005AMISIGO, B.A.; GIESEN, N.C. van de. Using a spatiotemporal dynamic statespace model with the EM algorithm to patch gaps in daily riverflow series. Hydrology and Earth System Sciences, v. 9, p. 209224, 2005.), meteorology (Soares , 2014SOARES, F.S.; FRANCISCO, C.N.; SENNA, M.C.A. Distribuição espaçotemporal da precipitação na Região Hidrográfica da Baía da Ilha Grande  RJ. Revista Brasileira de Meteorologia, v. 29, n. 1, p. 125138, 2014.; Haslett, 1989HASLETT, J. Spacetime modelling in meteorology  a review. Bulletin Statistical Institute, v. 51, p. 229246, 1989.) and environmental systems (Goodall and Mardia, 1994GOODALL, C.; MARDIA, K.V. Challenges in multivariate spatiotemporal modeling. In: International Biometric Conference, 17, 1994, Ontario, Canada. Proceedings. [Ontario]: McMaster University Press, 1994. p. 117.; Mardia , 1998MARDIA, K.V.; GOODALL, C.; REDFERN, E.J.; ALONSO, F.J. The Kriged Kalman filter (with discussion). Test, v. 7, p. 217252, 1998.; Fassò and Cameletti, 2009FASSÒ, A.; CAMELETTI, M.A. The EM algorithm in a distributed computing environment for modelling environmental spacetime data. Environmental Modelling&Software, v. 24, n. 9, p. 10271035, 2009., ^{2010}FASSÒ, A.; CAMELETTI, M.A. A unified statistical approach for simulation, modeling, analysis and mapping of environmental data. Simulation: Transactions of the Society for Modeling and Simulation International, v.86, n. 3, p. 139154, 2010.).
These models can be represented in statespace form and their parameters can be estimated using a Kalman filter (Cressie and Wikle, 2002CRESSIE, N.; WIKLE, C. Spacetime kalman filter. In: EL SHAARAWI, A.; PIEGORSCH, W. (Ed.). Encyclopedia of environmetrics. New York: Wiley, 2002. v. 4, p. 20452049.). However, for the most common configuration, where the model parameters are unknown, the standard approach uses the EM algorithm (ExpectationMaximization) to estimate the model parameters (Shumway and Stoffer, 1982SHUMWAY, R.H.; STOFFER, D.S. An approach to time series smoothing and forecasting using the EM algorithm. Journal of Time Series Analysis, v. 3, n. 4, p. 253264, 1982.).
In this work, a spatiotemporal model was used to estimate daily rainfall data. We used data from weather stations located in two rainfall homogeneous zones in Brazil. The results were compared in two specific days with geostatistical kriging and cokriging techniques.
2. Material and Methods
We used rainfall data from several weather stations, in two homogeneous areas in Brazil, identified according to Keller Filho (2005)KELLER FILHO, T.; ASSAD, E.D.; LIMA, P.R.S.R. Regiões pluviometricamente homogêneas no Brasil. Pesq. Agropec. Bras., v. 40, n. 4, p. 311322, 2005., obtained through the Agritempo System for all Brazilian regions (Embrapa Informática Agropecuária, 2014EMBRAPA INFORMÁTICA AGROPECUÁRIA. Agritempo Sistema de Monitoramento Agrometeorológico. Campinas, 2014. Available in: www.agritempo.gov.br/. Access in: Oct. 2014.
www.agritempo.gov.br/...
). Agritempo is an agrometeorological monitoring system that allows users to access weather and agrometeorological information from various Brazilian municipalities and states via the Internet.
The homogeneous zones are identified according to the similarity of their rainfall probability distributions and delimited using the hierarchical cluster analysis obtaining 25 rainfall homogeneous zones in Brazil. Two homogeneous zones were chosen. The first zone covers the states of São Paulo within a rectangle with latitude ranging from −23.0 to −20.5 and longitude of −49.5 and −47.5 with 103 weather stations. The second zone included 51 weather stations located in the northeast of Brazil, within the rectangle defined with latitude ranging from −14.0 to −8.0 and longitude between −46.5 to −43.0. The first homogeneous zone is a climatic transition zone with average rainfall. In the second homogeneous area dominates the semiarid tropical climate of low rainfall.
The spatiotemporal model for each set of data from each zone was adjusted using programs developed in R language (R Development Core Team, 2011R DEVELOPMENT CORE TEAM. R: A language and environment for statistical computing. Vienna: The R Foundation for Statistical Computing, 2011. Available in: http://www.gbif.org/resource/81287. Access in: Jan. 2015.
http://www.gbif.org/resource/81287...
) with the support of the Stem library (Spatiotemporal models in R) (Cameletti, 2009CAMELETTI, M. Spatiotemporal models in R. 2009. Availabe in: http://cran.rproject.org/web/packages/Stem/index.html. Access in: Nov. 2014.
http://cran.rproject.org/web/packages/S...
). In order to improve the estimates of the model parameters, the altitude of the stations was used as a covariate.
The spatiotemporal model used is discussed following Fassò and Cameletti (2009)FASSÒ, A.; CAMELETTI, M.A. The EM algorithm in a distributed computing environment for modelling environmental spacetime data. Environmental Modelling&Software, v. 24, n. 9, p. 10271035, 2009.. Let Z (s, t) be a scalar spatiotemporal process observed at time t and geographical location s. Let Z_{t} = {(s_{1}, t), …, Z(sn, t)} be a dataset at time t for n geographic locations s_{1}, …, s_{n}. Moreover, let Y_{t} = {Y_{1}(t), …, Y_{p} (t)} be a vector of p dimension of a notobserved temporal process at time t with p ≤ n. The hierarchical three stage model for t = 1,…, T is defined as follows:
In Eq. (1), the error ε_{t} is introduced so that U_{t} is regarded as a smoothed version of the spactiotemporal process Z_{t}. In Eq. (2), the notobserved spatiotemporal process U_{t} is defined as the sum of three components: the X_{t} matrix of observed covariates for the time t for n locations, the latent spatiotemporal process Y_{t} and the ω_{t} error model. Finally, the Eq. (3) is modeled as an autoregressive process where G is the matrix of transition and η_{t} is the innovation error. The errors ε_{t}, ω_{t} and η_{t} have zero mean and are timeindependent and independent from each other. Substituting Eq. (2) into Eq. (1) results in the following hierarchical two stages model:
Equations (4) and (5) are the statespace model equations (Durbin and Koopman, 2001DURBIN, J.; KOOPMAN, S. Time series analysis by state space methods. New York: Oxford University Press, 2001. 346 p.; Carvalho 2011CARVALHO, J.R.P de; ASSAD, E.D.; PINTO, H.S. Kalman filter and correction of the temperatures estimated by PRECIS model. Atmospheric Research, v. 102, n. 12, p. 218226, 2011.; Chui and Chen, 2009CHUI, C.K.; CHEN, G. Kalman filtering with realtime applications. Berlin: SpringerVerlag Berlin Heidelberg, 2009. 229 p.). Eq. (4) is the equation of measurements and Eq. (5) is the equation of state. The Y_{t} temporal process can be estimated using the Kalman filter or Kalman smoother.
In Eq. (4), the error
In geostatistics, the error
Among the known approaches to perform parameter estimations, one can cite the maximum likelihood techniques involving the use of scoring techniques or the NewtonRaphson method for solving nonlinear equations arising from the differentiation of the log likelihood function. The likelihood methods usually have several undesirable characteristics such as inversion of large Hessian matrices, the instability of the numerical maximization process and the resulting nonpositive definite matrices (Fassò and Cameletti, 2009FASSÒ, A.; CAMELETTI, M.A. The EM algorithm in a distributed computing environment for modelling environmental spacetime data. Environmental Modelling&Software, v. 24, n. 9, p. 10271035, 2009., ^{2010}FASSÒ, A.; CAMELETTI, M.A. A unified statistical approach for simulation, modeling, analysis and mapping of environmental data. Simulation: Transactions of the Society for Modeling and Simulation International, v.86, n. 3, p. 139154, 2010.). To avoid these problems, the Stem library uses the EM algorithm (Shumway and Stoffer, 1982SHUMWAY, R.H.; STOFFER, D.S. An approach to time series smoothing and forecasting using the EM algorithm. Journal of Time Series Analysis, v. 3, n. 4, p. 253264, 1982.; Mclachlan and Krishnan, 1997MCLACHLAN, G.J.; KRISHNAN, T. The Em algorithm and extensions. New York: Wiley, 1997. 400 p.), which is widely used for problems with missing values, as it is the case for Eqs. (4) and (5) where the component of missing values is given by the latent variable Y_{t}.
Since the EM algorithm does not use a Hessian matrix in the loglikelihood function, it does not provide standard errors for use with the estimated parameters, as the NewtonRaphson algorithm does. Hence, the bootstrap method is used mainly for the estimation of EM to obtain an estimate of the standard errors.
Bootstrap methods are computationally intensive methods of statistical analysis that uses simulation to calculate standard errors and confidence intervals. These methods are applicable to any level of modeling, and thus can be used in both parametric and non parametric analysis (Efron and Tibshirani, 2003EFRON, B.; TIBSHIRANI, R.J. An introduction to the bootstrap. London: Chapman and Hall, 2003. 436 p.).
To validate the values obtained using the spatiotemporal model, for the study zones, two dates were chosen, January 1, 2012 and May 1, 2012. For these fixed times, ordinary kriging and cokriging geostatistical techniques (Yamamoto and Landim, 2013YAMAMOTO, J.K.; LANDIM, P.M.B. Geoestatística: conceito e aplicações. São Paulo: Oficina de Textos, 2013. 215 p.) were used with altitude as a covariate, to estimate missing values by crossvalidation assuming that one of the sample elements, was not observed.
The first date corresponds to the rainy season in the first homogeneous zone of Brazil. In the second homogeneous zone, this date is during the dry season. The second date, represents the dry season in the first homogeneous zone of the study area and the rainy season for the second homogeneous zone. According to these criteria, the best interpolation for each variable, is one that has the lowest value for the Mean Squared Error (MSE), that is, the ratio of the squared difference between the observed value and the estimated value divided by the number of observations. It is expected that by setting two distinct dates, the best performance of the estimates are obtained by the spatiotemporal model.
It is quite common in verification studies to use the statistic SS (Skill Score) to summarize the quality of the forecasting system. This score quantifies the relative variation of the mean square error of the spatiotemporal model (MSE_{mod}) with regard to the kriging and cokriging (MSE_{krig} and MSE_{cockri}). Positive values of SS indicate that the model improved the predictions (Carvalho , 2011CARVALHO, J.R.P de; ASSAD, E.D.; PINTO, H.S. Kalman filter and correction of the temperatures estimated by PRECIS model. Atmospheric Research, v. 102, n. 12, p. 218226, 2011.; Libonati , 2008LIBONATI, R.; TRIGO, I., DACAMARA, C.C. Corrections of 2 mtemperature forecasts using Kalman filtering technique. Atmospheric Research, v. 87, n. 2, p. 183197, 2008.).
3. Results and Discussion
For 2012, 103 weather stations with complete data series were selected in the first homogeneous zone and 51 weather stations for the second homogeneous zone (365 days of precipitation values). Figure 1 shows the spatial distribution of rainfall observations used in this study by zone.
For the first and second homogeneous zones, random data samples of six weather stations were taken out and estimates of the same values for the spatiotemporal pattern and geostatistical interpolation using values of neighboring points were obtained. Estimates of daily rainfall for January 1, 2012 and May 1, 2012 using the spatiotemporal model, kriging and cokriging methods are shown in Tables 1 and 2 respectively.
Mean Square Error for estimates of gaps obtained from the spatiotemporal model (MSE_{mod}), kriging (MSE_{krig}) and cokriging (MSE_{cokrig}) on January 1, 2012 – SS_{1} and SS_{2} are the SkillScore statistics when comparing the model with kriging and cokriging, respectively.
Mean Square Error for estimates of gaps obtained by the spatiotemporal model (MSE_{mod}), kriging (MSE_{krig}) and cokriging (MSE_{cokrig}) on May 1, 2012  SS_{1} and SS_{2} are the SkillScore statistics when comparing the model with kriging and cokriging, respectively.
The statistics Skill Score (SS_{1} and SS_{2}) (Eqs. (9)  (10), respectively) are used to quantify, as a percentage, the improvement that occurred in the estimation of daily rainfall data for the two dates, using the spatiotemporal model in relation to the estimates obtained by kriging and cokriging. For the first date in the first zone, the estimate obtained by the spatiotemporal model was 24.41% better than the estimate obtained by kriging (SS_{1}) and 17.77% better than the estimate obtained by cokriging (SS_{2}). For the second date in the same zone, the spatiotemporal model was 32.12% and 26.16% better. The statistical skill score has been used in several studies with good results, as in Carvalho (2011)CARVALHO, J.R.P de; ASSAD, E.D.; PINTO, H.S. Kalman filter and correction of the temperatures estimated by PRECIS model. Atmospheric Research, v. 102, n. 12, p. 218226, 2011. in Kalman filter and correction of the temperatures estimated by PRECIS model.
For the second zone, the estimates obtained by the spatiotemporal model were always better than the estimates obtained by kriging and cokriging. In all cases, the mean squared errors obtained by the model (MSE_{mod}) were considerably less than the mean squared errors obtained by kriging (MSE_{krig}) and cokriging (MSE_{cokrig}).
In both zones the result obtained by the spatiotemporal model is always better indicating that neither the year nor the station influences the results. However, the results of the three methods for second zone are considerably lower compared to the first zone. Amisigo and Giesen (2005)AMISIGO, B.A.; GIESEN, N.C. van de. Using a spatiotemporal dynamic statespace model with the EM algorithm to patch gaps in daily riverflow series. Hydrology and Earth System Sciences, v. 9, p. 209224, 2005. also obtained better results by estimating gaps in daily riverflow series using spatiotemporal modelling. As well as, Franciscon (2008)FRANCISCON, L.; RIBEIRO JUNIOR, P.J; KRAINSKI, E.T.; BASSANEZI, R.B.; CZERMAINSKI, A.B. C. Modelo autologístico espaçotemporal com aplicação à análise de padrões espaciais da leprosedoscitros. Pesq. Agropec. Bras., v. 43, n. 12, p. 16771682, 2008. when she modeled the incidence of citrus leprosis using spatiotemporal autologistic model and RuizCárdenas (2009)RUIZCÁRDENAS, R.; ASSUNÇÃO, R.M., DEMÉTRIO, C.G.B. SpatioTemporal modelling of coffee berry borer infestation patterns accounting for inflation of zeroes and missing values. Sci. Agric., v. 66, n. 1, p. 100109, 2009. using the same model to estimate better values when the temporal series is inflated with zeroes and missing values.
4. Conclusions

The application of a spatiotemporal model produced better estimates of daily precipitation values compared with results obtained by geostatistical kriging and cokriging models for the study period.

The spatiotemporal model proved to be a versatile technique, adapting to different seasons, and should be considered as an alternative to fill gaps in time series.

The mean squared errors obtained by the model were considerably less than the mean squared errors obtained by kriging and cokriging.

For the two time periods studied, predictions using the spatiotemporal model were more than 17% better for first zone and more than 26% better for second zone, compared to the forecasts obtained by geostatistical techniques.
Acknowledgments
The authors would like to thank the anonymous reviewers for the considerable contribution.
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Publication Dates

Publication in this collection
22 AprJun 2016
History

Received
09 Mar 2015 
Accepted
17 Aug 2015