Wavelet Cross-correlation in Bivariate Time-Series Analysis

Abstract

The estimation of the correlation between independent data sets using classical estimators, such as the Pearson coefficient, is well established in the literature. However, such estimators are inadequate for analyzing the correlation among dependent data. There are several types of dependence, the most common being the serial (temporal) and spatial dependence, which are inherent to the data sets from several fields. Using a bivariate time-series analysis, the relation between two series can be assessed. Further, as one time series may be related to an other with a time offset (either to the past or to the future), it is essential to also consider lagged correlations. The cross-correlation function (CCF), which assumes that each series has a normal distribution and is not autocorrelated, is used frequently. However, even when a time series is normally distributed, the autocorrelation is still inherent to one or both time series, compromising the estimates obtained using the CCF and their interpretations. To address this issue, analysis using the wavelet cross-correlation (WCC) has been proposed. WCC is based on the non-decimated wavelet transform (NDWT), which is translation invariant and decomposes dependent data into multiple scales, each representing the behavior of a different frequency band. To demonstrate the applicability of this method, we analyze simulated and real time series from different stochastic processes. The results demonstrated that analyses based on the CCF can be misleading; however, WCC can be used to correctly identify the correlation on each scale. Furthermore, the confidence interval (CI) for the results of the WCC analysis was estimated to determine the statistical significance.