The spectral color of natural and anthropogenic time series and its impact on the statistical significance of cross correlation

The cross-correlation between time series is a common tool to study and quantify the impact of climatic and anthropogenic changes on ecosystems. The traditional method for estimating the statistical significance of correlation relies on the assumption that the data are independent, but time series found in nature are often strongly auto-correlated because of low-frequency environmental variability and ecosystem inertia. Previous authors have used Monte Carlo simulations to study the impact of serial auto-correlation on the significance of cross-correlations. Most studies have used random time series that are often a poor representation of those found in nature, e.g., low-order auto-regressive models with normally distributed noise. Moreover, we are not aware of any tests of the applicability of those methods to anthropogenic time series. Here, we study the effect of serial auto-correlation on the performance of two methods for estimating the significance of cross-correlations determined from Monte Carlo simulations with time series that are generated synthetically based on power-law specification of spectral characteristics. Such time series have an auto-correlation structure defined by a single parameter, their spectral “color”, and are generally more convenient representations of natural time series than the autoregressive models. Our results show that one of the two methods considered here accurately reproduces prescribed error rates for the wide range of spectral colors representative of climatic, ecological and anthropogenic time series. For this, we characterized roughly 1800 observational records in different categories of spectral colors, including climate variability, abundance of vertebrate species, and pollution. We specifically focus on time series with annual sampling over data records of at least 40 years, which are particularly relevant for climate studies. The methodology advocated in this study provides a simple and realistic assessment of the significance of sample estimates of cross correlation for time series with any sample interval and record length.