Simulated white-noise and pink-noise time series and associated time-series analyses: The autocorrelation function and the power spectrum.

Both the simulated white-noise (top left panel) and pink-noise (bottom left panel) time series contain 1000 events of varying amplitude levels. The autocorrelation reflects the correlation of a time series with itself at different lags. For example, a lag-0 autocorrelation reflects the correlation of each value in the time series with itself. In that case, the correlation will always be r = 1. The lag-1 autocorrelation reflects the correlation of each time-series value with the very next time-series value; the lag-100 autocorrelation is the correlation of each time-series value with the value 100 events into the future. In the current figure, the autocorrelation spans lag 1 to lag 100. Another method of time-series analysis is spectral analysis: The power spectrum (right panels) decomposes a signal into component sinusoidal frequencies and assigns a power value scaled to the magnitude of each frequency’s contribution to the time series. For white noise, by definition, each observation in the time series is independent of every other event in the time series, and across all autocorrelation lags the correlations hover around a value of zero (top middle panel). For the white-noise power spectrum, each frequency in the spectrum contributes similar levels of power to the signal; as reflected by the slope of the frequency-power regression equation, there is essentially no change in power as a function of frequency (top right panel). In contrast, for the pink-noise time series one may observe low-frequency wavelike oscillations that reduce in magnitude as frequency increases (bottom left panel). That gives rise to elevated autocorrelations (bottom middle panel) and an inversely proportional reduction in power as a function of frequency: The slope of the frequency-power regression equation is essentially −1 (bottom left panel).