Autocorrelation Calculator

Autocorrelation, also known as serial correlation, measures the correlation of a time series with a delayed (lagged) copy of itself. It quantifies how similar observations are to each other as a function of the time gap between them. A high positive autocorrelation at lag 1 means consecutive values tend to move in the same direction.

The autocorrelation coefficient at lag k is computed using the Pearson correlation formula between the original series and the same series shifted by k time periods. Values range from -1 to +1, where +1 is perfect positive correlation, -1 is perfect negative correlation, and 0 indicates no linear relationship.

A positive autocorrelation means that high values tend to be followed by high values and low values by low values — the series has momentum or persistence. A negative autocorrelation means high values tend to be followed by low values, indicating mean-reverting or alternating behavior.

A lag is the number of time periods by which the series is shifted before computing the correlation with itself. Lag 1 compares each observation with the immediately preceding one; lag 2 compares each observation with the one two periods earlier, and so on.

As a general rule, you should have at least 30–50 observations for meaningful autocorrelation estimates. For a lag of k, only n−k pairs are available, so analysis with large lags on short series becomes unreliable. Keep your maximum lag below roughly n/4 for stable results.

The ACF is the collection of autocorrelation coefficients computed at multiple lag values (lag 1, 2, 3, …). Plotting the ACF as a bar chart helps visually identify patterns — a slowly decaying ACF suggests a trend or non-stationarity, while sharp cut-offs or periodic spikes can indicate seasonality or an MA process.

Autocorrelation at lag k captures the total linear relationship between a series and its k-period lag, including indirect effects through intermediate lags. Partial autocorrelation (PACF) measures only the direct relationship at lag k after removing the contributions of shorter lags. Both are used together when fitting ARIMA models.

Yes. If a time series has seasonality with period s (e.g. monthly data with annual seasonality at s=12), the ACF will show pronounced spikes at lags that are multiples of s. Identifying these spikes is one of the primary uses of autocorrelation analysis in time series decomposition.