Modeling and predicting the monthly rainfall in tamilnadu as a seasonal multivariate arima process
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Modeling and predicting the monthly rainfall in tamilnadu as a seasonal multivariate arima process

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Modeling and predicting the monthly rainfall in tamilnadu as a seasonal multivariate arima process Modeling and predicting the monthly rainfall in tamilnadu as a seasonal multivariate arima process Document Transcript

  • International Journal of ComputerComputerand Technology (IJCET), ISSN 0976 – 6367(Print),International Journal of Engineering EngineeringISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEMEand Technology (IJCET), ISSN 0976 – 6367(Print) IJCETISSN 0976 – 6375(Online) Volume 1Number 1, May - June (2010), pp. 103-111 ©IAEME© IAEME, http://www.iaeme.com/ijcet.html MODELING AND PREDICTING THE MONTHLY RAINFALL IN TAMILNADU AS A SEASONAL MULTIVARIATE ARIMA PROCESS M. Nirmala Research Scholar, Sathyabama University Rajiv Gandhi Road, Jeppiaar Nagar, Chennai – 19 Email ID: monishram5002@gmail.com S. M. Sundaram Department of Mathematics, Sathyabama University Rajiv Gandhi Road, Jeppiaar Nagar, Chennai – 19 Email ID: sundarambhu@rediffmail.comABSTRACT: Amongst all weather happenings, rainfall plays the most imperative role inhuman life. The understanding of rainfall variability helps the agricultural management inplanning and decision- making process. The important aspect of this research is to find asuitable time series seasonal model for the prediction of the amount of rainfall inTamilnadu. In this study, Box-Jenkins model is used to build a Multivariate ARIMAmodel for predicting the monthly rainfall in Tamilnadu together with a predictor, SeaSurface Temperature for the period of 59 years (1950 – 2008) with a total of 708readings.Keywords: Seasonality, Sea Surface Temperature, Monthly Rainfall, Prediction,Multivariate ARIMAINTRODUCTION: Time series analysis is an important tool in modeling and forecasting. Among themost effective approaches for analyzing time series data is the model introduced by Boxand Jenkins [1], Autoregressive Integrated Moving Average (ARIMA). Box-JenkinsARIMA modeling has been successfully applied in various water and environmental 103
  • International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print),ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEMEmanagement applications. The association between the southwest and northeast monsoonrainfall over Tamilnadu have been examined for the 100 year period from 1877 – 1976through a correlation analysis by O.N. Dhar and P.R. Rakhecha [2]. The average rainfallseries of Tamilnadu for the northeast monsoon months of October to December and theseason as a whole were analyzed for trends, periodicities and variability using standardstatistical methods by O. N. Dhar, P. R. Rakhecha, and A. K. Kulkarni [3]. BalachandranS., Asokan R. and Sridharan S [4] examined the local and teleconnective associationbetween Northeast Monsoon Rainfall (NEMR) over Tamilnadu and global SurfaceTemperature Anomalies (STA) using the monthly gridded STA data for the period 1901-2004. The trends, periodicities and variability in the seasonal and annual rainfall series ofTamilnadu were analyzed by O. N. Dhar, P. R. Rakhecha, and A. K. Kulkarni [5]. Theannual rainfall in Tamilnadu was predicted using a suitable Box – Jenkins ARIMA modelby M. Nirmala and S.M.Sundaram [6].STUDY AREA AND MATERIALS: Tamilnadu stretches between 8o 5-13o 35 N by latitude and between 78o 18-80o 20 E by longitude. Tamilnadu is in the southeastern portion of the Deccan in India,which extends from the Vindhya mountains in the north to Kanyakumari in the south.Tamilnadu receives rainfall in both the southwest and northeast monsoon. Agriculture ismore dependants on the northeast monsoon. The rainfall during October to Decemberplays an important role in deciding the fate of the agricultural economy of the state.Another important agro – climatic zone is the Cauvery river delta zone, which dependson the southwest monsoon. Tamilnadu should normally receive 979 mm of rainfall everyyear. Approximately 33% is from the southwest monsoon and 48 % is from the northeastmonsoon. 104
  • International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print),ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEME Figure 1 Geographical location of area of study, Tamilnadu A dataset containing a total of 59 years (1950 - 2008) monthly rainfall totals ofTamilnadu was obtained from Indian Institute of Tropical Meteorology (IITM), Pune,India. The monthly Sea Surface Temperature of Nino 3.4 indices were obtained fromNational Oceanic and Atmospheric Administration, United States, for a period of 59years (1950 – 2008) with 708 observations.METHODOLOGY:BOX-JENKINS SEASONAL MULTIVARIATE ARIMA MODEL: Univariate time series analysis using Box-Jenkins ARIMA model is a major toolin hydrology and has been used extensively, mainly for the prediction of such surfacewater processes as precipitation and stream flow events. It is basically a linear statisticaltechnique and most powerful for modeling the time series and rainfall forecasting due toease in its development and implementation. The ARIMA models are a combination ofautoregressive models and moving average models [1]. The autoregressive models AR(p)base their predictions of the values of a variable xt, on a number p of past values of the 105
  • International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print),ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEMEsame variable number of autoregressive delays xt−1,xt−2,. . . xt−p and include a randomdisturbance et. The moving average models MA(q) generate predictions of a variable xtbased on a number q of past disturbances of the same variable prediction errors of pastvalues et−1,et−2,. . ., et−q. The combination of the auto regressive and moving averagemodels AR(p) and MA(q) generates more flexible models called ARMA(p,q) models.The stationarity of the time series is required for the implementation of all these models.In 1976, Box and Jenkins proposed the mathematical transformation of the non-stationarytime series into stationary time series by a difference process defined by an order ofintegration parameter d. This transforms ARMA (p,q) models for non stationarytransformed time series as the ARIMA (p,d,q) models. The ARIMA model building strategy includes iterative identification, estimation,diagnosis and forecasting stages [7]. Identification of a model may be accomplished onthe basis of the data pattern, time series plot and using their autocorrelation function andpartial autocorrelation function. The parameters are estimated and tested for statisticalsignificance after identifying the tentative model. If the parameter estimates does notmeet the stationarity condition then a new model should be identified and its parametersare estimated and tested. After finding the correct model it should be diagnosed. In thediagnosis process, the autocorrelation of the residuals from the estimated model shouldbe sufficiently small and should resemble white noise. If the residuals remainsignificantly correlated among themselves, a new model should be identified estimatedand diagnosed. Once the model is selected it is used to forecast the monthly rainfallseries. Time series analysis provides great opportunities for detecting, describing andmodeling climatic variability and impacts. Ultimately, to understand the meteorologicalinformation and integrate it into planning and decision making process, it is important tostudy the temporal characteristic and predict lead times of the rainfall of a region. Thiscan be done by identifying the best time series model using Box – Jenkins SeasonalARIMA modeling techniques. The Seasonal ARIMA (p,d,q)(P,D,Q)s model is defined asφ p ( B)Φ P ( B s )∇ d ∇ s D yt = Θ Q ( B s )θ q ( B)ε t …………… [1] 106
  • International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print),ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEMEWhereφ p ( B ) = 1 − φ1 B − ........... − φ p B p , θ q ( B ) = 1 − θ1 B − ............ − θ q B q …..[2]Φ P ( B s ) = 1 − Φ 1 B s − ........ − Φ P B sP , Θ Q ( B s ) = 1 − Θ1 B s − ....... − Θ Q B sQεt denotes the error term, φ’s and Φ’s are the non seasonal and seasonal autoregressiveparameters and θ’s and Θ’s are the non seasonal and seasonal moving averageparameters.RESULTS AND DISCUSSIONS: A time series is said to be stationary if its underlying generating process is basedon constant mean and constant variance with its autocorrelation function (ACF)essentially constant through time. The ACF is a measure of the correlation between twovariables composing the stochastic process, which are k temporal lags far away and thePartial Autocorrelation Function (PACF) measures the net correlation between twovariables, which are k temporal lags far away. TIME SERIES POLT OF MONTHLY RAINFALL IN TAMILNADU 4500 MONTHLY RAINFALL 4000 3500 3000 2500 2000 1500 1000 500 0 1 34 67 100 133 166 199 232 265 298 331 364 397 430 463 496 529 562 595 628 661 694 MONTH Figure 2 Time Series Plot of Monthly Rainfall in Tamilnadu (1950 - 2008) The visual plot of the time series plot (Figure 2) is often enough to convince astatistician that the series is stationary or non – stationary [6]. The visual inspection ofthe sample autocorrelation function shows that the rainfall series is stationary. If theAutocorrelation function dies out rapidly, that is, it reaches zero within one or two lagperiods then it indicates that the time series is stationary. From the pattern of ACF andPACF plots, the monthly rainfall series is stationary but with seasonality. 107
  • International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print),ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEME Figure 3 ACF and PACF correlograms Seasonality is defined as a pattern that repeats itself over fixed intervals of times.For a stationary data, seasonality can be found by identifying those autocorrelationcoefficients of more than two or three time lags that are significantly different from zero[8]. Since the monthly rainfall series consists seasonality of order s = 12, the modelconsidered here is a seasonal multivariate ARIMA model. Table 1: ACF and PACF coefficients for the first five lags Lag ACF coefficient PACF coefficient 1 0.457 0.457 2 0.123 -0.108 3 -0.155 -0.215 4 -0.268 -0.127 5 -0.250 -0.069 The ACF and PACF correlograms (figure 3) and the coefficients are analyzedcarefully and the tentative multivariate ARIMA model chosen is ARIMA (1,0,1)(1,1,1)12.The parameter’s estimates are tabulated in table 2. Table 2: Parameter’s Estimates of ARIMA (1,0,1)(1,1,1)12 model Model Parameters Parameter’s Estimates ARIMA AR 0.475 (1,0,1)(1,1,1)12 MA 0.392 AR, Seasonal 0.072 MA, Seasonal 0.965 108
  • International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print),ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEME After fitting the appropriate ARIMA model, the goodness of fit can be examinedby plotting the ACF of residuals of the fitted model. If most of the autocorrelationcoefficients of the residuals are within the confidence intervals then the model is a goodfit. Figure 4 Residual plots of ACF and PACF of ARIMA (1,0,1)(1,1,1)12 model Since the coefficients of the residual plots of ACF and PACF are lying within theconfidence limits, the fit is a good fit and the error measures obtained through this modelis tabulated in the Table 3. The graph showing the observed and fitted values is shown inFigure 5. Table 3: Error Measures obtained for the model ARIMA (1,0,1)(1,1,1)12 Error Measures Value RMSE 1.082 MAPE 16.081 R squared 0.547 109
  • International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print),ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEME Figure 5: Graph showing the observed and the fitted values obtained in ARIMA (1,0,1)(1,1,1)12 modelCONCLUSION: In the modern world, there is an ever-increasing demand for more accurateweather forecasts. Accurate weather forecasting happens to be the ultimate target ofatmospheric research. In this article, an attempt was made to predict the monthly rainfallin Tamilnadu with a predictor, sea surface temperature through Box – JenkinsMultivariate ARIMA model. The error measures show the model fitted is a good model.REFERENCE: 1. George E. P. Box, Gwilym M. Jenkins and Gregory C. Reinsel (1994). Time Series Analysis – Forecasting and Control, 3rd Edition, Pearson Education, Inc. 2. O. N. Dhar and P. R. Rakhecha (1983). Foreshadowing Northeast Monsoon Rainfall over Tamilnadu, Monthly Weather Review, Vol.3, Issue 1, pp. 109 – 112. 3. O. N. Dhar, P. R. Rakhecha, A. K. Kulkarni (1982). Fluctuations in northeast monsoon rainfall of Tamilnadu, International Journal of Climatology Volume 2, Issue 4, pp. 339 – 345. 4. Balachandran S., Asokan R. and Sridharan S (2006). Global surface temperature in relation to northeast monsoon rainfall over Tamilnadu, Journal of Earth System Science, vol. 115, no.3, pp. 349-362. 5. O. N. Dhar, P. R. Rakhecha, A. K. Kulkarni (1982). Trends and fluctuations of seasonal and annual rainfall of Tamilnadu, Proceedings of Indian Academy of Sciences (Earth planetary Sciences), Vol. 91, No.2, pp. 97 –104. 110
  • International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print),ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEME 6. Nirmala M and Sundaram S.M. (2009), Rainfall Predicting Model: An Application to Tamilnadu Rainfall Series, Proceedings of National Conference on Effect of Climatic Change and Sustainable Resource Management, SRM University, and Meteorological Department, Chennai India. 7. Damodar N. Gujarati and Sangeetha (2007). Basic Econometrics, 4th edition, the Tata Mcgraw – Hill Publishing Company Limited. 8. Meaza Demissie (2003). Characterizing trends and identification of rainfall predicting model: An application to Melksa observatory Nazreth rainfall series, Proceedings of Biometric society, pp.132 – 134. 9. Nirmala M and Sundaram S.M. (2010), Comparison of Learning Algorithms with the ARIMA Model for the Forecasting of Annual Rainfall in Tamilnadu, International Journal on Information Sciences and Computing, Vol.4, No.1, pp.64 – 71. 111