Detecting signals from data with noise: Theory and applications

Signal detection from noisy data by rejecting a noise null hypothesis depends critically on a priori assumptions regarding the background noise and the associated statistical methods. Rejecting one kind of noise null hypothesis cannot rule out the possibility that the detected oscillations are generated from the stochastic processes of another kind. This calls for an adaptive null hypothesis based on general characteristics of the noise that is present. In this paper, a new method is developed for identifying signals from data based on the finding that true physical signals in a well-sampled time series cannot be destroyed or eliminated by resampling the time series with fractional sampling rates through linear interpolation. Therefore, the significance of signals could be tested by checking whether the signals persist in the true time-frequency spectral representation during resampling. This hypothesis is based on the general characteristics of noise as revealed by empirical mode decomposition, an adaptive data analysis method without linear or stationary assumptions, and without any predefinition of the background noise. Applications of this method to synthetic time series, solar spot number, and sea surface temperature time series illustrate its power in identifying characteristics of background noise without any a priori knowledge.