ISSN Original Article Statistical Models for Forecasting Road Accident Injuries in Ghana.

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1 Available online at International Journal of Research in Environmental Science and Technology Universal Research Publications. All rights reserved ISSN Original Article Statistical Models for Forecasting Road Accident Injuries in Ghana. T. Ofori 1,*, B. Ackah 1 and L. Ephraim 2 1 Department of Mathematics, University of Mines and Technology, Tarkwa, Ghana 2 Faculty of Engineering, University of Mines and Technology, Tarkwa, Ghana Received 09 November 2012; accepted 26 December 2012 Abstract In this paper, we forecast road accident injuries using time-series modeling techniques-: Various Exponential Smoothing and Auto Regressive Integrated Moving Average (ARIMA). On validation of the forecasting Models, ARIMA model performed better than the Damped trend Exponential Smoothing Universal Research Publications. All rights reserved Keywords: Statistical Models, Road Accident, ARIMA model, Akaike Information Criterion (AIC), Bayessian Information Criterion (BIC). 1 INTRODUCTION In almost countries in Africa, Asia, and Latin America, road traffic crashes have become one of the leading causes of deaths in the Older, Children and the economically active adults between ages 30 and 49 years (Murray and Lopaz 1996, Ross et al 1991). Despite this burgeoning problem, little attention has been paid to road traffic injury prevention and treatment in most developing countries of which Ghana is no exception (Afukaar et al 2003). Forecasting of road traffic injuries enables stakeholder and policy makers to take various prevention measures. In this paper, an attempt has been made to forecast road traffic injuries using statistical time-series modeling technique Damped Trend Exponential Smoothing and Auto- Regressive Moving Average (ARIMA). Materials and Methodology Exponential smoothing Exponential smoothing is a procedure for continually revising a forecast in the light of more recent experience. Exponential Smoothing as signs exponentially decreasing weights as the observation get older. In other words, recent observations are given relatively more weight in forecasting than the older observations (Gardner 1985). Single exponential smoothing This is also known as simple exponential smoothing. Simple smoothing is used for short- range fore casting, usually just one month into the future. The model assumes that the data fluctuates around a reasonably stable mean, thus is assumes only level and noise (Brown and Meyers 1961). The specific formula for simple exponential smoothing is: When applied recursively to each successive observation in the series, each new smoothed value (forecast) is computed as the weighted average of the current observation and the previous smoothed observation; the previous smoothed observation was compute d in turn from the previous observed value and the smoothed value before the previous observation, and so on. Thus, ineffect, each smoothed value is the weighted average of the previous observations, where the weights decrease exponentially depending on the value of parameter ( ). If it is equal to 1 (one) then the previous observations are ignored entirely; if it is equal to 0 (zero), the n the current observation is ignored entirely, and the smoothed value consists entirely of the previous smoothed value (which in turn is computed from the smoothed observation before it, and soon; thus all smoothed values will be equal to the initial smoothed value ). In- between values will produce intermediate results. The initial value of plays an important role in computing all the subsequent values. Setting it to is one method of initialization. Another possibility would be to average the first four or five observations. The smaller the value of the more important is the selection of the initial value of. Double Exponential Smoothing (Holt) This method is assumes levels, trend and noise are in the data develops a forecast bases on the following formula of a straight line (Holt 1957). The smoothing equations are: 143

2 smoothed level at time of the series, computed after is observed, smoothing parameter for the level of the series, smoothed trend at the end of the period t and smoothing parameter for trend. The current value of the series is use d to calculate its smoothed value replacement in double exponential smoothing. There are several methods to choose the initial values for and. in general is set to and there are suggestion for. used to smooth seasonality. Thus individual smoothing constant can be set for each time series component based upon particular data pattern. It uses the following formula to forecast: is the forecast for a period that is period in the future from,, is the value for the trend in the period, and is the multiplicative seasonal adjustment for the period. To use this formula, Winters calculates the values, and through the following formulae. The values of and are both constants between 0 and +1 but do not have the same value. Holt s technique smooth both level (with ) and the trend (with ) component in the data. The advantage of the Holt is that it allows levels and trend component of the data to be smoothed at different values of and. A disadvantage to Holts is that different values chosen for and greatly affect accuracy. Also two smoothing constant must now be selected. Damped-Trend Exponential Smoothing This is a modification of Holts Double exponential smoothing. Damped-Trend Exponential Smoothing assumes level and damped trend parameters. This model is appropriate for series with a linear trend that is dying out with no seasonality (Gardner and Mckenzie 1985). Its smoothing parameters are level, trend and damping trend. It describe how a damping parameter, can be used within the Holt s method to give more control over trend extrapolation. The damped trend is given by: smoothed level at time of the series, computed after is observed smoothing parameter for the level of the series, smoothed trend at the end of the period t and smoothing parameter for trend. trend modification or damping parameter If, trend is damped and the forecasts approach an asymptote given the horizontal straight line or plateau: If the, the model is identical to the standard Holt (1957) model and the trend is linear. If the model is identical to standard simple exponential smoothing and if the forecast function has an exponential trend. Triple Exponential Smoothing (Winters) Winters assumes level, trend, seasonality and noise to be present in the data (Winters 1960). It uses exponential smoothing to smooth the values for level, trend and trend individually. The value of is used to smooth level; the value of is used to smooth trend and the value of is smoothed level at time of the series, computed after is observed, smoothing parameter for the level of the series, smoothed trend at the end of the period smoothing parameter for trend. The values of, and are all constants between 0 and 1, but, and do not have the same value. Since Winters considers levels, trend, and seasonality, it has advantage of being more realistic than any of the previous ones discussed. These constants work the same as other smoothing constants. A small value of will dampen out noise, but reacts slowly to changes in level in the data. A small value of dampen out noise in the seasonality component, but reacts slowly to changes in seasonality pattern. A small value of dampen out the noise in the trend component but reacts slowing to changes in the trend component. A large value of will reacts quickly to the changes in level, but will also be adversely affected by noise. The same conclusion can be draw about for seasonality and for trend. The advantage to Winters is its comprehensive nature from Time Series perspective. All the possible time series components are covered. Also it allows the levels, trend and seasonality component present in the data to be smoothed at different rates by setting different values for, and. A disadvantage to Winters is the values of the constants, that greatly affect accuracy and finding the most accurate combination of three different smoothing constants, each of which can have a value anywhere between 0 and 1, can be formidable task. ARIMA MODEL In light of the problem associated with ordinary methods, several researchers have turned to the ARIMA model as a means to better predict accident variables. The ARIMA model is a useful statistical method for analyzing longitudinal data with a correlation among neighboring observations. This method has proven to be very useful in the analysis of multivariate time series. (Sales etal,1980) In ARIMA analysis, there are two simple components for representing the behavior of observed time series processes, namely the autoregressive (AR) and moving average (MA) models (Pankratz, A.1988). The AR model is used to describe a time series in which the current observation depends on its preceding values, whereas the moving average (MA) model is used to describe a time 144

3 series process as a linear function of current and previous random errors (Shumway,1988) It is possible that a time series model will consist of a mixture of AR and MA components. In this case the series is said to be computed by an autoregressive moving average process of order (p, q), where p and q are the orders of the AR and MA components respectively. The selection strategy for such models was developed and selected by the Box and Jenkins method (Box and Jenkins 1976). A general ARIMA model can be written as: is assumed to have white noise, B is the backshift operator, N representing the stochastic part and d is the order of regular differencing needed to achieve time series stationarity. Other parameters in the model are defined as follows: are autoregressive (AR) parameters, are moving average (MA) parameter. ARIMA model development consists of a three-stage iterative process, which comprises of identification of the model, model parameter estimation and diagnostic checking of residuals of the fitted model (Brockwell and Davis 1991). This forecasting model, was proposed by Box and Jenkins in An ARIMA process is a mathematical model used for forecasting. Box-Jenkins modeling involves identifying an appropriate ARIMA process, fitting it to the data, and then using the fitted model for forecasting. One of the attractive features of the Box-Jenkins approach to forecasting is that ARIMA processes are very rich class of possible models and it is usually possible to find a process which provides an adequate description to the data. The original Box-Jenkins modeling procedure involved an iterative three-stage process of model selection, parameter estimation and model checking. Recent explanations of the process (e.g., Makridakis, Wheelwright and Hyndman, 1998) often add a preliminary stage of data preparation and a final stage of model application (or forecasting). Data preparation This involves transformations and differencing. Transformations of the data (such as square roots or logarithms) can help stabilize the variance in a series where the variation changes with the level. This often happens with business and economic data. Then the data are differenced until there are no obvious patterns such as trend or seasonality left in the data. Differencing means taking the difference between consecutive observations, or between observations a year apart. The differenced data are often easier to model than the original data. Model selection (Identification) Here the Box-Jenkins framework uses various graphs based on the transformed and differenced data to try to identify potential ARIMA processes which might provide a goodfit to the data. In the identification stage of model building, we determine the possible models based on the data pattern. But before we can begin to search for the best model for the data, the first condition is to check whether the series is stationary or not. The ARIMA model is appropriate for stationary time series data (i.e. the mean, variance, and autocorrelation are constant through time). If a time series is stationary then the mean of any major subset of the series does not differ significantly from the mean of any other major subset of the series. Also 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 subset only by chance (see Pankratz, 1983). The stationarity condition ensures that the autoregressive parameters invertible. If this condition is assured then, the estimated model can be forecasted (see Hamilton, 1994). To check for stationarity, we usually test for the existence or nonexistence of what we called unit root or observed the trend of the graph of the data. Unit root test is performed to determine whether a stochastic or a deterministic trend is present in the series. The ACF and the PACF are also so used to determined the stationarity of the data and the order of the model. Though the ACF and PACF assist in determine the order of the model but this is just a suggestion on where the model can be build from. It becomes necessary to build the model around the suggested order. In this case several models with different order can be considered. The final model can be selected using a penalty function statistics such as the Akaike Information Criterion (AIC) or Bayessian Information Criterion (BIC). See Sakamoto et al (1986), Akaike (1974). The AIC and the BIC which are measure of goodness of fit with others like the MAPE, MAE are used to select the best model. Given a data set, several competing models may be ranked according to their AIC, AICc or BIC with the one having the lowest information criterion value being the best. In the general case, the AIC and BIC take the form as shown below: Model estimation Model estimation means finding the values of the model coefficients which provide the best fit to the data. At the identification state one or more models are tentatively chosen that seen to provide statistically adequate representations of the available data. At this stage we get precise estimates of the coefficients of the model chosen at the identification stage. That is we fit the chosen model to our time series data to get estimates of the coefficients. This stage provides some warning signals about the adequacy of our model. In particular, if the estimated coefficients do not satisfy certain mathematical inequality conditions, that model is rejected.. There are sophisticated computational algorithms designed to do this. Model checking (Diagnosis) This involves testing the assumptions of the model to identify any areas where the model is inadequate. If the model is found to be inadequate, it is necessary to go back to Step 2 and try to identify a better model. Ideally, a model should extract all systematic information from the data. The 145

4 part of the data unexplained by the model (i.e., the residuals) should be small. These checks are usually based on the residuals of the model. One assumption of the ARIMA model is that, the residuals of the model should be white noise.. For a white noise series the, all the ACF are zero. In practice if the residuals of the model is white noise, then the ACF of the residuals are approximately zero. If the assumption of are not fulfilled then different model for the series must be search for. A statistical tool such as LjungBox Q statistic can be used to determine whether the series is independent or not. The test stastistics Q is (e) = the residual correlation at lag k. m = the number of times lags includes in the test n = the number of residuals The p-value associated with the Q- statistics is small ( p- value < ) the model is considered inadequate. Moreover we can check the properties of of the residual with the following graph. The normality of the residual can be check by considering the normal probability plot or the p-value from the One- Sample Kolmogorov-Smirrno test. Also we can check the randomness of the residual by considering the graph of the ACF and PACF of the residual. The individual residual autocorrelation should be small and generally within of zero. Forecasting This is what the whole procedure is designed to accomplish. Once the model has been selected, estimated and checked, it is usually a straight forward task to compute forecasts. Forecasting is the process of making statements about events whose actual outcomes have not yet been observed. It is an important application of time series. If a suitable model for the data generation process of a given time series has been found, it can be used for forecasting the future development of the variable under consideration. To choose a final model for forecasting the accuracy of the model must be higher than that of all the competing models. The accuracy for each model can be checked to determine how the model performed in terms of in-sample forecast. Usually in time series forecasting, some of the observations are left out during model building in other to access models in terms of out of sample forecasting also. The accuracy of the models can be compared using some statistic such as mean error (ME), root mean square error. (RMSE), mean absolute error (MAE), mean percentage error (MPE), mean absolute percentage error (MAPE) etc. A model with a minimum of these statistics is considered to be the best for forecasting. Error Magnitude Measurement The error magnitude measures allow forecasters to evaluate and compare the performance of various models across different time periods. It relates to forecast error and is defined by = the forecast error, = actual number of period t, = Forecast value. In a summary of visitation trend forecasts, Witt and Witt (1995) found that accuracy is the most important evaluation criterion. Because of their clear definitions, the Mean Absolute Percentage Error (MAPE), Root Mean Square Error (RMSE) was selected for this project. The MAPE is written as = number of time periods, = forecast errors in time period t, = actual number of inflation at time period t, Lower MAPE values are better because they indicate that smaller percentages errors are produced by the forecasting model. The following interpretation of MAPE values was suggested by Lewis (1982) as follows: Less than 10% is highly accurate forecasting, 10% to 20% is good forecasting, 21% to 50% is reasonable forecasting and 51% and above is inaccurate forecasting. The Root Mean Square Error (RMSE) is also an accuracy measure which was considered. This approach is capable of comparing actual rates of changes in time series data and computes the averages forecast errors. It is given by: ANALYSIS OF THE COLLECTED DATA Table (1) shows the yearly injuries resulting from road motor accidents from 1991 to Time plot (fig. 1) of total number of injuries data reveal that there is an increasing trend in the number of accident injuries. Table 1 Data on yearly accident injuries in Ghana for the period 1991 to 2010 SN. YEAR INJURIES Source: MTTU Annual report

5 Table 2 above shows the exponential smoothing model parameters. It has an value of 0.321, value of and value of It can be seen that they are all significant. Thus their p-values are all less 5%. The fitted model is given by: Fig. (1) Graph of Road Traffic Injuries For the smoothing of the data the damped trend exponential smoothing technique was found to be the most appropriate. Various of combination of, and values were tried. The various smoothing techniques were applied to smooth the data it was found that the Damped Trend gives accurate forecast based on the set criterion of least values of their RMSE and MAPE. m = 1, 2,...,5 and the initial values of for level and trend are 2667 and 1093 respectively. Model Identification At the identification stage, the appropriate AR and MA parameters are found by examining Autocorrelation function (ACF) and partial autocorrelation function (PACF) of the time series. Autocorrelation and partial autocorrelation analyses were conducted on the stationary time series. Autocorrelation measures the unconditional relationship of values between time lags, while partial autocorrelation measures the conditional relationship. EXPONENTIAL SMOOTHING MODEL Table 2 Exponential Smoothing Model Parameters Model Estimate SE t Sig. Injuries-Model_1 Table 3 Model Statistics Model Square Root Model Fit statistics Alpha (Level) Gamma (Trend) Phi (Trend damping factor) Ljung-Box Q(18) R-squared RMSE MAPE Statistics DF Sig. Injuries-Model_ E The stationary check of the series showed that it was non stationary. Merely by using the first differencing technique it was made stationary as in figure 2. By implication the mean value is zero and all the variances are constant. Figures 3 and 4 shows the correlogram of the both the ACF and the PACF of the first difference of the data. It can be seen that the ACF has one significant cut off at the first lag showing the presence of white noise and the PACF also shows two major spike. The graphs of the sample ACF and PACF were plotted (Fig. 3 and 4). Fig. 3 Graph of ACF of first difference Fig. 2 Graph of first difference Fig. 4 Graph of PACF of first difference 147

6 From figures 3 and 4 two models were selected based on their ability of reliable prediction. Four criterions namely the Normilsed Bayessian Information Criteria (BIC), the R- square, Root Mean Square Error (RMSE) and the Mean Table 4 Suggested Models Absolute Percentage Error (MAPE) were used. Lower values of the BIC, MAPE, RMSE and high value of R- square were preferable MODEL MAPE BIC RMSE R 2 ARIMA (2,1,1) ARIMA (1,1,1) From the above table and the set criterion it was found that ARIMA (1, 1, 1) was the best fitted model. Model Estimation Table 5 ARIMA Model Parameters Injuries-Model_1 Table 6 Model Statistics Model Injuries No Transformation Estimate SE T Sig. Constant 2.424E E AR Lag Difference 1 MA Lag Years No Transformation Numerator Lag E E Model Fit statistics Ljung-Box Q(18) R-squared RMSE MAPE Normalized BIC Statistics DF Sig. Injuries-Model_ The Ljung-Box statistic of the model is not significantly different from 0, with a value of for 16 DF and associated p- value of 0.003, thus failing to reject the null hypothesis of no remaining significant autocorrelation in the residuals of the model. This indicates that the model adequately captured the correlation in the time series. Moreover the low RMSE indicates a good fit for the model and the high value of the R-squared shows a perfect prediction over the mean. The coefficient of both the AR and MA are all less than 1. Model Diagnostics From the graphs of the residual ACF and PACF below it can be seen that all points are randomly distributed and it can be conclude that there is an irregular pattern which means the model is adequate. Also the individual residual autocorrelation are very small and are generally within ± 2/ n of zero. The model is adequate in the sense that the graphs of the residual ACF and PACF shows a random variation, thus from the origin zero (0) the points below and above the mean are all uneven, hence the model fitted. Fig. 6 Graph of residual PACF Forecast Fig. 5 Graph of residual ACF Fig. 7 Graph of ARIMA model 148

7 Once our model has been found and its parameters has been estimated and checked, we can therefore do our forecasting with 95% confidence interval. Now the forecast of these models were compared by validating. On comparison of the results revealed that among the models fitted ARIMA (1, 1, 1 ) model came out to be performing better when the forecast was validated. Table 7: Forecast Table Year Forecast of injuries Observed Damped ARIMA Injuries Trend (1,1,1) MAPE RMSE Acknowledgement: We thank Prof. Adetunde, I.A, Dean of Faculty of Engineering, University of Mines and Technology, Tarkwa, Ghana for fruitful discussions, his criticism, suggestions and making this article publishable. Reference Afukaar et al (2003). Pattern of road traffic injuries in Ghana: Implication for Control. Injury Control and Safety Promotion. Vol 10 Akaike, H.(1974). A New Look at the Statistical Model Identification. IEEE Transactions on Automatic Controll 19 (6): Box, G.E.P. and Jenkins, G.M. (1976). Time Series Analysis: Forecasting and Control. Holden-Day, San Francisco, C.A Brockwell P.J. and Davis R.A. (1991). Time series: Theory and methods. Springer-Verlag, (2nd Eds.) New York. Brown, R. G and Meyers R.S (1961). The Fundamentals Theorem of Exponential Smoothing. Operational Research 9 (No. 5): pp Gardner, E.S (1985). Exponential Smoothing The State of the art. Journal Forecasting, Gardner, E.S and Mckenzie, E (1985). Damped trend exponential smoothing: A modeling Viewpoint, International Journal of Forecasting, Vol. 26. pp Hamilton, J. D. (1994). Time Series Analysis. Princeton Univ. Press, Princeton New Jersey. Holt, C.C (1957), Forecasting Seasonal and Trends by exponential weighted moving averages, Office of Naval Research, Research Memorandum No. 52. Makridakis,S.,S.C. Wheelwright, and R.J. Hyndman (1998) Forecasting: methods and applications, New York: John Wiley & Sons Murray C.J and Lopez A. (1996) The Global Burden of Diseases, vol. 1: A Comprehensive Assessment of Mortality and Disability from Diseases, Injuries and Risk Factors in 1990 and projected to Cambridge: Harvard University Press Pankratz, A.(1988) Forecasting with Univariate Box- Jenkins Models: Concepts and Cases. Wiley, New York. Sakamoto, Y., Ishiguro, M., and Kitagawa G. (1986). Akaike Information Criterion Statistics. D. Reidel Publishing Company. Sales, J.D., Delleur, J.W., Yevjevich, V. and Lane, W.L.(1980) Applied Modeling of Hydrologic Time Series. Water Resources Publication, Littleton, CO. Shumway, R.H. (1988) Applied Statistical Time Series Analysis. PrenticeHall, Englewood Cliffs, NJ. Ross A. et al (1991) Towards safer Road in Developing Countries;A Guild for Planners and Engineers. Transport Research Laboratories. Crowthrone UK. Witt, S.F, and Witt C.A (1995). Forecasting tourism demand: A review of empirical research, International Journal of forecasting Vol. 2 (No.3): pp Winters, P.R (1960), Weighted Moving Averages. Management Science 6: pp zzzzzzzzzzzzzzzzzzz Source of support: Nil; Conflict of interest: None declared 149