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Differencing In Time Series

Differencing In Time Series


Time Series Modeling and Forecasting are done with stationary series and thus, for a non stationary series, it will have to be made stationary using mathematical transformations and computations like Differencing.

Differencing is used to eliminate trend, to control the Auto-correlation and partial auto-correlations making the series stationary. Differencing is the only mathematical transformation outlined by Box-Jenkins, when modeling Time Series using the ARIMA technique.

The difference between consecutive observations of the series is called differencing, it helps to stabilize the mean and variance. Hence, eliminating trend and seasonality
Given a time series with (n) observations, differencing is the change in the observations in the original series.

Let
y¹(t) - denote the differenced series 

y(t) - the original series

y(t-1) - the original series with one less value from the original one (n-1)

Below is the mathematical representation of differencing (first order)

y¹(t) = y(t) - y(t-1)

y¹(t), the differenced series will have (n-1) number of observations.

When the differenced series is white noised, the Model for the original series will be given as

y(t) = y(t-1) + e(t)

Where e(t) is White Noise

In some cases, one difference might not be sufficient, hence, Second Order Differencing is employed to obtain a stationary series
Let y²(t) - denote the series differenced twice
Them the mathematical representation of Second Order Differenced data

y²(t) = y¹(t) - (y(t-1) - y(t-2))

Where y²(t) has (n-2) number of observations, but in practice, it is never necessary to go beyond Second Order Differencing.

Seasonal differencing is also employed if the data shows characteristics of seasonality, with the goal of detecting and eliminating all its traces. It is also employed to determine the order for the seasonal Auto regressive and seasonal moving average terms when modeling using ARIMA..

For a series whose period is know, the mathematical representation of Seasonal differencing is given by

y¹(t) = y(t) - y(t-m)
Where (m) denotes the number of seasons

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