ARIMA
References:
Sections:
- Conceptual Approach
- Assumptions
- Limitations / Disadvantages
- Example Code
1. Conceptual Approach
ARIMA stands for Auto Regressive Integrated Moving Average. There are 2 parameters for a basic ARIMA model: There are multiple parts to ARIMA:
- Auto Regressive
- Residuals
There are two widely used linear time series models in literature, viz. Autoregressive (AR) and Moving Average (MA) models. ARIMA combines the two into a single model.
An AR model, as the name implies, is a regression on the previous values of the series, to predict the next value.
A $AP(p)$ model is given by
$$ X_t = c + \Sigma{_{i=1}^{p}\phi_i X_{t-i} }$$The notation $MA(q)$ refers to the moving average model of order q: $$ X_t = \mu + \Sigma{_{i=1}^{q}\theta_i \varepsilon_{t-i} }$$
where $\mu$ is the mean of the series, the $\theta_1, ..., \theta_q$ are the parameters of the model and the $\varepsilon_t, \varepsilon_{t−1},..., \varepsilon_{t−q}$ are white noise error terms
AR and MA models can be effectively combined together to form a general and useful class of time series models, known as the ARMA models.
$$ X_t = c + \mu + \Sigma{_{i=1}^{p}\phi_i X_{t-i} } + \Sigma{_{i=1}^{q}\theta_i \varepsilon_{t-i} } $$The lag or backshift operator is defined as $L_{y_t} = y_{t-1} $
The ARMA models, described above can only be used for stationary time series data. However in practice many time series such as those related to socio-economic and business show non-stationary behavior. Time series, which contain trend and seasonal patterns, are also non-stationary in nature. Thus from application view point ARMA models are inadequate to properly describe non-stationary time series, which are frequently encountered in practice. For this reason the ARIMA model is proposed, which is a generalization of an ARMA model to include the case of non-stationarity as well. In ARIMA models a non-stationary time series is made stationary by applying finite differencing of the data points. The mathematical formulation of the ARIMA(p,d,q) model using lag polynomials is given below.
In [6]:
library(forecast)
library(fpp)
In [7]:
fit <- Arima(usconsumption[,1], order=c(0,0,3))
In [8]:
plot(forecast(fit))