A statistical test of phase difference via wavelet method and its application to the spread of air pollution

Accurate measurement and inference of phase difference between two time series are critical across several fields, including signal processing, economic dynamics, and air pollution research. Wavelet methods offer advantages over traditional approaches by allowing time–frequency localization and adaptability to non-stationary signals, which makes them widely used for phase difference estimation. However, existing methods do not provide a statistical test to determine whether a measured phase difference reflects a true underlying relationship between the signals or is merely an artifact of measurement errors or randomness. In this paper, we propose a bootstrap method to fill this gap. Our method is particularly suited to the analysis of non-standard data distributions and complex temporal dependencies. Extensive simulations demonstrate its desirable power and control of type-I error. Furthermore, we apply the method to study air pollution dispersion in China and elucidate the factors influencing phase differences.