ARIMA methodology proposed by Box Jenkin (1976) is one of the popular methods to uncover the hidden characteristics in the time series data, and to generate forecasts. The model is built based on the plot of the autocorrelation and partial autocorrelation functions of the dependent time series. The plot provides the information to determine which autoregressive or moving average component should be used in the model. Basically, ARIMA(*p*,*q*) model includes two independent processes. They are autoregressive process (AR(*p*)) and Moving average Process (MA(*q*)).

#### Autoregressive Process

Autoregressive Process can be represented as AR(*p*) that can be interpreted as a linear combination of prior observations, AR(*p*) can be summarized as:*x*_{t} = *ζ* + *φ*_{1}x_{t}_{−1} + *φ*_{2}x_{t}_{−2} + … *φ*_{k}x_{t}_{−}_{p} + *ε**(1)* where *p* is order of autoregressive model, *ζ* is the constant of the model, and *φ*_{1}, *φ*_{2}, …, *φ*_{p} are the autoregressive model parameters.

#### 3Moving Average Process

It can be represented as MA(*q*). In the Moving average model, the observation can be affected by its previous error, MA(*q*) can be written as:*x*_{t} = *μ* + *ε*_{t} − *θ*_{1}ε_{t}_{−1} − *θ*_{2}ε_{t}_{−2} − … − *θ*_{3}ε_{t}_{−}_{q}*(2)* where q is the order of the moving average model, *μ* is a constant and *θ*_{1},*θ*_{2},…,*θ*_{q} are the moving average model parameters

We suppose the stochastic process {*x*_{t} : *t* = 0, ±1, ±2,…}, and its mean function is defined as *μ*_{t} = *E*(*x*_{t}) for *t* = 0, ±1, ±2,…. In general, *μ*_{t} differs at different *t*. Autocorrelation is defined as:

* (3)* where

*Cov*(

*x*_{t},

*x*_{s} ) is autocovariance function,

*Cov*(

*x*_{t},

*x*_{s} ) =

*E*[(

*x*_{t} − μ_{t})(

*x*_{s} − μ_{s})] =

*E*(

*x*_{t}x_{s})

*− μ*_{t}μ_{s}The partial autocorrelation at lag k (*ϕ*_{kk}) is defined as:*ϕ*_{kk} = Corr(*x*_{t}, *x*_{t− k} | *x*_{t−}_{1}, *x*_{t−}_{2}, …, *x*_{t− k+1}) that can be interpreted as the correlation between *x*_{t} and *x*_{t− k} after removing the effect of the intervening variables *x*_{t− 1}, *x*_{t− 2}, …, *x*_{t− k=}_{1}