3.1 Introduction
ARIMA (p, d, q) model was first introduced by Box and Jenkin in 1976, it can be used for forecasting the non-seasonal stationary time-series data [16]. An ARIMA model is characterized by 3 terms: p, d, q where p is the order of the Auto-Regression (AR) term, q is the order of the Moving Average (MA) term, d is the order of differencing required to make the time-series stationery. Auto-Regression is nothing but the regression of the variable against itself to forecast the variable of interest. It correlates the pattern of the one-time period to its previous periods. MA is a regression-like model that uses the errors associated with the forecast at a previous time-step to forecast a variable later step. The following are the generalized equations of pth order AR model and qth order MA model.
y t = C + φ1yt − 1 + φ2yt − 2 +……. + φpyt − p + Et ……...(1)
yt = C + Et + θ1Et − 1 + θ2Et − 2 + +θqEt – q………... (2)
ARIMA models are built upon incorporating the AR model, integration (I), and the MA model. The integration (I) is the reverse process of differencing to generate the fore- cast. The generalized ARIMA model is mathematically represented as
y t = C + φ1yt + φ2yt − p +……. + φnyt − n + θ1Et − 1 + θq Et − q + Et (3)
where C is an intercept, φ (i = 1, 2... p) is auto-regressive model parameters, θi (i = 1, 2... p) is moving average model parameters, yt is current time-series value, yt−1, yt−2, …, yt−p is past values and Et is a random error or residual term for the tth day, and it is given by the following equation
E t = yt − yt−1
3.1.1 Variables of the study
The major variables utilized in the current study were the number of injuries, the number of fatal RTAs, and the total number of RTAs observed during the study period, i.e., January 2017 to December 2019 for Hilly state of India. The current study aimed at modeling all the above variables using the appropriate time series model.
3.1.2 Statistical analysis
Along with descriptive statistics, time series analysis was conducted for data analysis purposes. The data analysis was carried out using the R-Statistical software package. One of the main objectives of statistics is to forecast the future levels of different processes by studying the behavior of the data in the past. The most important techniques of making inferences about the future based on what has happened in the past are the analysis of time series, which may be defined as a set of observations taking at specified times, usually at equal intervals.
3.1.3 Components of time series
In analyzing time series, we may take the observed composite series for study or study one by one respectively the components. The components are seasonal trends, random or irregular variations.
3.2 Steps involved in Time Forecasting
3.2.1 Identification
The first step in developing an ARIMA model is to determine if the series is stationary. If the model is found to be non-stationary, stationarity could be achieved mostly by differencing the series or going for the dickey fuller test.
3.2.2 Estimation
Once the preliminary model is chosen, the estimation stage begins. The purpose of estimation is to find the parameter estimates that minimize the mean square error. In this method, the R statistical package was used in the estimation.
3.2.3 Diagnostics checking
Residuals from the model are examined to ensure that the model is adequate (random). The following diagnostics are made Time plot of the residuals Plot of the residual ACF Normal Quantile (QQ) Plot.
3.2.4 Forecasting
When a satisfactory ARIMA model is adequate, then we proceed to forecast or predict for a period or several periods ahead. However, chances of forecast errors are inevitable as the period advances.
3.3 Time Forecasting series for Nagaland
Using data collected from last 3 years, monthly time plot curve is obtained in R studio. Time plot curve shows scattered and non-seasonal variation. To fix it change in accident is obtained using appropriate code in R studio. Seasonal plot shows accident is higher in March. It is found that road accidents in Nagaland can be fitted ARIMA (0,1,1) (1,1,0) [12] which is shown in Fig. 6.
3.4 Time Forecasting series for Meghalaya
Using data collected from last 3 years, monthly time plot curve is obtained in R studio. Time plot curve shows scattered and non-seasonal variation. To fix it change in accident is obtained using appropriate code in R studio. Seasonal plot shows accident is higher in November. It is found that road accidents in Meghalaya can be fitted ARIMA (0,1,1) (0,1,0) [12] which is shown in Fig. 10.
3.5 Time Forecasting series for Sikkim
Using data collected from last 3 years, monthly time plot curve is obtained in R studio. Time plot curve shows scattered and non-seasonal variation. To fix it change in accident is obtained using appropriate code in R studio. Seasonal plot shows accident is higher in March. It is found that road accidents in Sikkim can be fitted ARIMA (2,1,0) (1,1,0) [12] which is shown in Fig. 14.
3.6 Time Forecasting series for Sikkim
Using data collected from last 3 years, monthly time plot curve is obtained in R studio. Time plot curve shows scattered and non-seasonal variation. To fix it change in accident is obtained using appropriate code in R studio. Seasonal plot shows accident is higher in March. It is found that road accidents in Sikkim can be fitted ARIMA (2,1,0) (1,1,0) [12] which is shown in Fig. 18.
3.7 Time Forecasting series for Manipur
Using data collected from last 3 years, monthly time plot curve is obtained in R studio. Time plot curve shows scattered and non-seasonal variation. To fix it change in accident is obtained using appropriate code in R studio. Seasonal plot shows accident is higher in November. It is found that road accidents in Manipur can be fitted ARIMA (0,1,1) (0,1,0) [12] which is shown in Fig. 22.
3.8 Time Forecasting series for Jammu & Kashmir
Using data collected from last 3 years, monthly time plot curve is obtained in R studio. Time plot curve shows scattered and non-seasonal variation. To fix it change in accident is obtained using appropriate code in R studio. Seasonal plot shows accident is higher in March. It is found that road accidents in Jammu & Kashmir can be fitted ARIMA (0,1,1) (0,1,0) [12] which is shown in Fig. 26.