📂Statistical Analysis

Seasonal ARIMA Model

Model ¹

1. An operator defined as $\nabla_{s} Y_{t} := Y_{t} - Y_{t-s}$ is called the Seasonal Difference.
2. If $W_{t} := \nabla^{d} \nabla_{s}^{D} Y_{t}$ is defined as $\left\{ W_{t} \right\}_{t \in \mathbb{N}}$, and if $ARMA(P,Q)$ and $\left\{ Y_{t} \right\}_{t \in \mathbb{N}}$ is $ARMA(p,q)$, then $\left\{ Y_{t} \right\}_{t \in \mathbb{N}}$ is called a Seasonal ARIMA process $ARIMA(p,d,q)\times(P,D,Q)_{s}$. This form is called the Seasonal ARIMA model.

Explanation

Today’s temperature is, of course, mostly influenced by yesterday’s temperature, but fundamentally, it is hot in the summer and cold in the winter. There may be unusually cold days or exceptionally warm winters, but globally, it must follow the seasons. Such a macroscopic cyclical characteristic is called Seasonality.

For example, AirPassenger is data on the number of airline passengers monthly from 1949 to 1960, showing a pattern of peaking during the vacation season in summer and gradually decreasing. There are years with more or less traffic, and there is a tendency for an increase over the years, but still, there is a clear seasonality on an annual cycle.

Of course, seasonality needn’t necessarily be related to the ‘seasons’; it can refer to weekly trends depending on the day of the week, daily cycles, or even turns in a board game. If there is a sequence, it can eventually be seen as time series data, but abstractly approached, it does not even need to relate to time properties.

The definition of the Seasonal ARIMA process might look quite complex at first glance, but its concept doesn’t go beyond that of an ARIMA process. Mathematically, it only looks messy due to the extended differences; obtaining some elegant insights is difficult. It’s reasonable to just consider it as ’there’s a larger flow’.