  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.
