ACF

The Autocorrelation Function (ACF) is a statistical tool used to identify the presence of a linear relationship between observations at different time lags. It’s particularly used for time series analysis to identify seasonality or the need for ARIMA-based forecasting models.

The formula for the autocorrelation function ( r_k ) at lag ( k ) is given by:

r_k = \frac{\sum_{t=k+1}^{T} (y_t - \bar{y})(y_{t-k} - \bar{y}) }{\sum_{t=1}^{T} (y_t - \bar{y})^2}

Where:

• ( T ) is the total number of observations
• ( y_t ) is the observation at time ( t )
• ( \bar{y} ) is the mean of the observations

For a given time series, you would typically calculate ( r_k ) for ( k = 1, 2, 3, \ldots ) up to some maximum lag value, and then plot these values to observe the pattern of autocorrelation at different lags.

Here’s a step-by-step process to calculate ACF:

1. Calculate the mean of the time series, ( \bar{y} ).
2. For each lag ( k ), calculate ( r_k ) using the formula above.
3. Plot ( r_k ) against ( k ) to get the autocorrelation plot.

In practice, software like R (using the acf function) or Python (using the statsmodels library) can calculate and plot the ACF for you.

If you provide me with a time series dataset, I can guide you through the process or provide more insights on the computed ACF. However, I won’t be able to compute the ACF directly without the actual data.