The ice cover on a river looks solid, but beneath it lie cracks that alter the flow. So too with a time series: behind the smooth surface of numbers hide moments when the rules of the game change. These moments are called change points, and finding them is like tapping the ice to locate weak spots.

Classical methods look for cracks by assuming the ice is homogeneous, meaning the data are independent. But in reality, neighboring values in a series are connected, like adjacent ice floes: today's temperature depends on yesterday's. The authors proposed splitting the series into windows, estimating autoregressive coefficients (the dependence on the past) within each window, and then obtaining a sequence of nearly independent points. Fast change-point algorithms can then be applied to these points.

To determine how many cracks truly exist rather than how many are imagined, a penalized criterion is used. It acts like a judge who fines you for every extra crack: the sum of squared losses within segments plus a penalty for each change point. The authors proved that if the penalty grows as the logarithm of the logarithm of the sample size, the method almost surely selects the true number of change points. On real data, the method worked precisely: in eastern U.S. temperatures, it found change points in 1901, 1929, 1944, 2009, and 2012, and in the El Niño index, a shift around 1979. The ice cracked exactly where climatologists expected.