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Imagine you have two time series of audio signals. You run a time lagged cross correlation analysis and find there is a significant correlation between them at lag = 0 and lag = -1. The correlation at lag = 0 is 90%, and -99% at lag = -1.

My inclination is to settle with the strongest lag at -1, and conclude the two series are most in phase at that lag. However, there is a seemingly contradictory correlation at lag = 0, suggesting the two series are in phase but negatively so.

How does one interpret this? I’m struggling with whether or not to stick with the largest correlation, or speaking to both, but I can’t figure out how to explain the significance of having two significant correlations at different lags with different signs (positive correlation and negative correlation.

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If you have $\hat{\rho}_0 \approx 1$ and $\hat{\rho}_1 \approx -1$ then (without more information) that is consistent with two signals that are in-phase with period roughly equal to two. Indeed, we get exactly $\hat{\rho}_0 = 1$ and $\hat{\rho}_1 = -1$ for the signals:

$$x_t = y_t = \sin(a + \pi t) \quad \quad \quad \quad \quad a \neq 0.$$

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