In coatless/ITS: A Tour of Time Series Analysis with R

ARIMA

Differencing Operator

Definition: Differencing Operator

The Differencing Operator is defined as the gradient symbol applied to a time series: [\nabla {x_t} = {x_t} - {x_{t - 1}}]

The differencing operator is helpful when trying to remove trend from the data.

We can take higher moments of differences by: [\begin{aligned} {\nabla ^2}{x_t} &= \nabla \left( {\nabla {x_t}} \right) \ &= \nabla \left( {{x_t} - {x_{t - 1}}} \right) \ &= \left( {{x_t} - {x_{t - 1}}} \right) - \left( {{x_{t - 1}} - {x_{t - 2}}} \right) \ &= {x_t} - 2{x_{t - 1}} + {x_{t - 2}} \ \end{aligned} ]

So, the difference operator has the following properties: [\begin{aligned} {\nabla ^k}{x_t} &= {\nabla ^{k - 1}} \left( {\nabla {x_t}}\right) \hfill \ {\nabla ^1}{x_t} &= \nabla {x_t} \hfill \ \end{aligned} ]

Notice, within the difference operation, we are backshifting the timeseries.

If we rewrite the difference operator to use the backshift operator, we receive: [\nabla {x_t} = {x_t} - {x_{t - 1}} = \left( {1 - B} \right){x_t}]

This holds for later incarnations as well: [\begin{aligned} {\nabla ^2}{x_t} &= {x_t} - 2{x_{t - 1}} + {x_{t - 2}} \hfill \ &= \left( {1 - B} \right)\left( {1 - B} \right){x_t} \hfill \ &= {\left( {1 - B} \right)^2}{x_t} \hfill \ \end{aligned} ]

Thus, we can generalize this to: [{\nabla ^k}{x_t} = {\left( {1 - B} \right)^k}{x_t}]

coatless/ITS documentation built on May 13, 2019, 8:45 p.m.