Fig. 2: Identifying revolutions using Foote novelty in simulated series.

a Evolution of 20 simulated stationary time series with a revolution in the middle. b Evolution of 20 simulated undirected random walk time series with a revolution in the middle. In both sets of simulations, the standard deviation of series perturbations in non-revolutionary periods is set at \(\sigma\,=\,1\). During the revolutions, which start at time point 40, the size of the change in each time point is increased until time point 50, when the revolution ends. The amount by which each variable, i, changes during during a revolution is drawn from a normal distribution. First row from top: Evolution of the time series. Second row: Distance matrices among time points: dark reds are increasingly dissimilar. Third row: The rate of change index, \({R}_{i}\), which is the sum of the \({F}_{i}^{k}\) values for any time point i over all k, relative to the sum of the mean \({F}_{i}^{k}\) values over all time points. Fourth row: Identifying revolutions by Foote novelty. Each cell represents the \({F}_{i}^{k}\) estimate for a given half-width, k and time point; the color of the cell gives the relative \({F}_{i}^{k}\) value, light gray being low and dark gray being high. Note that this color scale is only comparable within any given plot. Statistically significant (\(\alpha\,=\,0.05/2\)) revolutionary periods are overlain in red; conservative periods are blue. In both cases, we identify a revolution in the correct region, but at larger half-widths, the resolution becomes coarser. Statistically significant time points which are not contiguous with the simulated revolution are false positives. Note that, for the random walk series, the undifferenced data are shown but the distance matrices, \({R}_{i}\,{\mathrm{and}}\,{F}_{i}^{k}\) values are all based on first differences. This means that only revolution boundaries are expected to have high \({F}_{i}^{k}\) values.