Eisner (2007) uses the method of Granger causality to determine phase relationships between time series, and finds convincing evidence for mechanistic relationships between Atlantic SSTs and global temperatures. In contrast with Granger causality methods that work in the time-domain, here we use wavelet methods. The method we use (Moore, Grinsted and Jevrejeva, 2007; Moore, Grinsted and Jevrejeva, 2008) determines the non-linear interactions between the two time series that may be chaotic. Briefly we extract the phase expression of the time series derived from the Continuous Wavelet Transform (CWT) of a time series (e.g. Grinsted, Moore and Jevrejeva, 2004; Torrence and Compo, 1998). Here we apply broad band pass wavelet (the Paul wavelet of order 4) to filter the time series. The centre frequency of the Paul wavelet, l, is an important parameter in the analysis.

The wavelet is stretched in time by varying its scale (s), so that z = s-t, and normalizing it to have unit energy. The CWT of a time series X, {xn, n = 1,...,N} with uniform time steps 8t, is defined as the convolution of xn with the scaled and normalized wavelet.

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