Cross-Correlation Explorer: Finding Lead-Lag Relationships Between Two Time Series

Generate Two Related Series

Hidden lag L (steps) 5

Coupling strength α 0.80

Series length N 220

CCF search window ±K 20

Live Statistics

Detected lag (argmax |CCF|)

True lag L

Peak |CCF(k)|

0.00

Series A (leader) Series B (follower) CCF (+) CCF peak

How it works

Series A is a smooth AR(1)-driven signal. Series B is built from a lagged, noisy copy of A:

B_t = α · A_{t-L} + √(1-α²) · ε_t

where L is the hidden lag (A leads B when L > 0) and α controls how strongly B tracks the lagged A versus independent noise ε_t. The cross-correlation function (CCF) tests every candidate lag k in the search window and measures how well A_t predicts B_{t+k}:

CCF(k) = Σ (A_t - Ā)(B_{t+k} - B̄) / √[Σ(A_t - Ā)² · Σ(B_t - B̄)²]

Unlike the autocorrelation function (which compares a series to itself), the CCF compares two different series shifted relative to each other — this is the tool used to detect which of two related signals leads and by how much, e.g. rainfall leading river flow, or a marketing spend series leading sales. The bar whose |CCF(k)| is largest marks the detected lag; if the model is right it should land close to the true lag L used to generate B.

L slider — the true, hidden lag between the series (negative means B actually leads A instead).
α slider — coupling strength; lower α buries the lag relationship in noise, making the CCF peak flatter and harder to pinpoint.
N slider — sample length; longer series sharpen the CCF peak around the true lag.
±K slider — how many lags on each side of zero the CCF bar chart searches.

L = 5, α = 0.80, N = 220 detected lag = 0 · peak |CCF| = 0.00

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