2.4.2. Seasonal Model

General multiplicative seasonal ARIMA model, which is known as SARIMA model defined as ARIMA (*p*, *d*, *q*)(*P*, *D*, *Q*)*<sup>s</sup>* where (*p*, *d*, *q*) the non-seasonal part of model is and (*P*, *D*, *Q*)*<sup>s</sup>* is the seasonal part of the model is given as:

$$
\beta\_p(\mathbb{C})\Phi\_p(\mathbb{C}^\delta)\nabla^d\nabla\_s^D z\_t = a\_q(\mathbb{C})\Theta\_Q(\mathbb{C}^\delta)a\_t \tag{4}
$$

where *p* is the order of non-seasonal auto regression, *d* is the number of regular differencing, *q* is the order of non-seasonal MA, *P* is the order of seasonal auto regression, *D* is the number of seasonal differencing, *Q* is the order of seasonal MA, *s* is the length of season, Φ*<sup>p</sup>* is the seasonal AR parameter of order *P*, and Θ*<sup>Q</sup>* is the seasonal MA parameter of order *Q* [45].

#### *2.5. Model Identification*

Model identification comprises recognizing the possible ARIMA model that depicts the nature of time series. In order to ascertain the order of model, the Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) were utilized for assistance. The

information obtained through the utility of ACF and PACF was also helpful in advocating various types of new models that could be established. The selection of the ultimate model is performed by employing the penalty function statistics through the Akaike Information Criterion (AIC) and Schwarz Bayesian Criterion (SBC). These criteria assist in ranking the models, where the models which have the least value of criterion are considered the best. AIC is an estimator of prediction error and thereby relative quality of statistical models for a given set of data. Given a collection of models for the data, AIC estimates the quality of each model, relative to each of the other models. Thus, AIC provides a means for model selection.

$$\text{AIC} = 2k - 2\ln(\mathcal{L})\,\,\mathcal{L} \tag{5}$$

Let *k* be the number of estimated parameters in the model. Let *L*ˆ be the maximum value of the likelihood function for the model.

In statistics, SBC is a criterion for model selection among a finite set of models; the model with the lowest BIC is preferred. It is based, in part, on the likelihood function and it is closely related to the AIC.

$$\text{SBC} = -2\ln(\hat{L}) + k\ln(n) \tag{6}$$

Here *k* represents number of parameters in the model, (*p* + *q* + *P* + *Q*); whereas, *L* depicts likelihood function of ARIMA model. Additionally, *n* shows number of observations [46,47].
