Design of Time Series Model for Road Accident Fatal Death in Tamilnadu

Transcription

1 Volume 109 No , ISSN: (printed version); ISSN: (on-line version) url: ijpam.eu Design of Time Series Model for Road Accident Fatal Death in Tamilnadu T. Jai Sankar 1, C. Vijayalakshmi 2 and P. Pushpa 1 1 Department of Statistics, Bharathidasan University, Tiruchirapalli , India tjaisankar@gmail.com, pushbkr@gmail.com 2 SAS, Mathematics Division, VIT University, Chennai , India vijusesha2002@yahoo.co.in June 20, 2017 Abstract This article highlights the design of time series model for road accident fatal death in Tamilnadu. This study considers Autoregressive Integrated Moving Average (ARIMA) processes to select the appropriate times series model for for road accident fatal death in Tamilnadu. Based on ARIMA (p, d, q) and its components ACF, PACF, Normalized BIC and Box-Ljung Q statistics estimated, ARIMA (0, 1, 1) was selected. Based on the chosen model, it could be predicted that the road accident would increase from in 2010 to in 2015 in Tamilnadu. Key Words and Phrases:Road accidents, causes, ARIMA model, forecasting, safety measures. 1 Introduction The World Health Organization (W HO, 2004) has estimated that fatalities per 100, 000 populations in the developing world will grow 225 1

2 from 13.3 in 2000 to 19.0 in 2020, while in the developed world during the same period, they will decline from 11.8 to 7.8. It may be mentioned that India had 10.1 fatalities per 100, 000 populations in Material and methods This study targets at forecasting number fatal deaths due to road accidents in Tamilnadu by using different forecasting techniques. Model parameters were estimated to fit the ARIMA models. An AR(p) autoregressive component of p order is Y t = µ + Φ 1 Y t 1 + Φ 2 Y t Φ p Y t p + ε t MA(q) moving average component of q order is Y t = µ + θ 1 Y t 1 + θ 2 Y t θ p Y t p + ε t Trend Fitting: The reliability statistics viz. RMSE, MAPE, BIC and Q statistics were computed as below: Root Mean Square Error is given by RMSE = [ 1 n i=1 n (Y t Ŷt) 2 ] 1 and Mean Absolute Percentage Error (MAPE) is given by RMSE = 1 n i=1 n Y t Ŷt Y t BIC(p, q) = lnv (p, q) + (p + q)[ln(n)/n] where p and q are the order of AR and MA processes respectively and n is the number of observations in the time series and v* is the estimate of white noise variance σ 2. Q = (n(n + 2) k i=1 rk2 ) where n is the number of residuals and rk (n k) is the residuals autocorrelation at lag k 3 Model Identification: If a data series is stationary then the variance of any major subset of the series will differ from the variance of any other major 226 2

3 subset only by chance (P ankratz, 1983). The stationarity condition ensures that the autoregressive parameters invertible. If this condition is assured then, the estimated model can be forecasted (Hamilton, 1994). The final model can be selected using a penalty function statistics such as the Akaike Information Criterion (AIC) or Bayessian Information Criterion (BIC) (Sakamoto et al, 1986 and Akaike, 1974). Figure 1: ACF and PACF of differenced data Assessing the variable under forecasting was a stationary series, ARIMA model was designed. Hence is done in Figure 2 which reveals that the data used were non-stationary. Again, nonstationarity in mean was corrected through first differencing of the data. The newly constructed variable Y t could now be examined for stationarity. Since, Y t was stationary in mean, the next step was to identify the values of p and q. For this, the autocorrelation and partial autocorrelation coefficients (ACF and PACF) of various orders of Y t were computed and presented in Table 2. Model Estimation: SPSS package is used for estimating the parameters of the ARIMA model and their estimates are represented in Tables 4. Hence the ARIMA (0, 1, 1) has the lowest the lowest normalized BIC value and better R-square value, this model best suits for forecasting number of fatal deaths due to road accidents in Tamilnadu

4 Table 1: BIC values of ARIMA (p, d, q) ARIMA (p,d,q) BIC Values 0, 1, , 1, , 1, , 1, , 1, , 1, , 1, , 1, , 1, Table 2: Estimated ARIMA model and fit statistics Estimate SE T Sig. Constant MA Fit Stastistic Mean Mean Stationary R-squared R-squared RMSE MAPE Normalized BIC

5 Diagnostic Checking: The model verification is concerned with checking the residuals of the model to see if they contained any systematic pattern which still could be removed to improve the chosen ARIMA, which has been done through examining the autocorrelations and partial autocorrelations of the residuals of various orders. For this purpose, various autocorrelations up to 16 lags were computed and the same along with their significance tested by Box-Ljung statistic are provided in Table 5. As the results indicate, none of these autocorrelations was significantly different from zero at any reasonable level. This proved that the selected ARIMA model was an appropriate model for forecasting number of road accidents in Tamilnadu. Table 3: ACF and PACF of No. of road accidents in Tamilnadu Lag ACF - Mean ACF - SE PACF - Mean PACF -SE Lag Lag Lag Lag Lag Lag Lag Lag Lag Lag Lag Lag Lag Lag Lag Lag The ACF and PACF of the residuals are given in Figure 3, which also indicated the good fit of the model. Hence, the fitted ARIMA model for the number of fatal deaths due to road accidents in Tamilnadu was Y t = ε t 1 + ε t 229 5

6 Figure 2: Residuals of ACF and PACF Figure 3: Actual and estimated number of deaths 230 6

7 4 Conclusion Conclusion: The fitted ARIMA model gives the forecasted number of fatal deaths due to road accidents in Tamilnadu in the years 2011 to 2015 and the counts where 16523, 17562, 18647, and respectively and the UCL and LCL values of the forecasted number of accidents is provided in T able6. The graph of estimated number of fatal deaths due to road accidents is shown in F igure4. Transportation is one of the basic things that people use and need in their everyday lives. The only solution to avoid road accidents is to develop a road safety culture in general people so that they follow the safety rules and obey the laws of Traffics. The government should focus on the usage of helmets must be compulsory, separate lanes for heavy vehicles and warning signs should be installed and correctly placed wherever needed. References [1] Akaike, H. (1970), Statistical Predictor Identification, Annals of Institute of Statistical Mathematics, 22: [2] Akaike, H. (1974), A New Look at the Statistical Model Identification, IEEE Transactions on Automatic Control. 19(6): [3] Box, G.E.P. and Jenkins, G.M. (1976), Time Series Analysis: Forecasting and Control, Holden-Day, San Francisco. [4] Brockwell, P.J. and Davis, R.A. (1991), Time series: Theory and methods, Springer-Verlag, (2nd Eds.) New York. [5] Elvik, R. (1993), Quantified Road Safety Targets: A Useful Tool for Policy Making?, Accident Analysis and Prevention, 25(5): [6] Hamilton, J.D. (1994), Time Series Analysis, Princeton Univ. Press, Princeton New Jersey. [7] Jai Sankar, T. (2011), Forecasting Fish Products Exports in Tamilnadu - A Stochastic Model Approach, Journal of Recent Research in Science and Technology, 3(7):

8 [8] Makridakis, S., Wheelwright, S. and Hyndman, R. (1988), Forecasting: Methods and Applications, Third edition, John Wiley and Sons. [9] Mandal, B.N. (2004), Forecasting sugarcane production in India with ARIMA model, IASRI, New Delhi. [10] Pankratz, A. (1983), Forecasting with Univariate Box-Jenkins Models: Concepts and Cases, Wiley, New York. [11] Sakamoto, Y., Ishiguro, M. and Kitagawa, G. (1986), Akaike Information Criterion Statistics, D. Reidel Publishing Company. [12] Slutzky, E. (1973), The summation of random causes as the source of cyclic processes, Econometrica, 5: [13] WHO (2004), Annual Report. Geneva, World Health Organization 232 8