We have recently introduced the ``thermal optimal path (TOP) method to investigate the real-time lead-lag structure between two time series. The TOP method consists in searching for a robust noise-averaged optimal path of the distance matrix along which the two time series have the greatest similarity. Here, we generalize the TOP method by introducing a more general definition of distance which takes into account possible regime shifts between positive and negative correlations. This generalization to track possible changes of correlation signs is able to identify possible transitions from one convention (or consensus) to another. Numerical simulations on synthetic time series verify that the new TOP method performs as expected even in the presence of substantial noise. We then apply it to investigate changes of convention in the dependence structure between the historical volatilities of the USA inflation rate and economic growth rate. Several measures show that the new TOP method significantly outperforms standard cross-correlation methods.
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