The ocean doesn't always stick to the same schedule. Sometimes the warm El Niño current strengthens, sometimes it weakens, and each regime dictates its own reaction speed to the past. These hidden regimes are like shifts in currents that remember their path with different depths. Scientists want to catch these shifts, but data arrive unevenly from the shore: measurements are taken once a month, while the physical delay might be, say, two and a half months. Then conventional models that only work with integer steps start to lie.
A new model cDN-ARMS(L) allows delays to be fractional. If you need to know the temperature two and a half months ago, but you only have records for two and three months, the model takes the intermediate value as a point on the line between neighboring samples. It's like wanting to recall what happened midweek when your diary only has Monday and Friday entries: you mentally fill in the gap. The hidden regime switches according to a probabilistic rule, like a current that changes direction not on a schedule but under random gusts of wind.
To estimate the parameters of such a model, the authors used a space-alternating EM algorithm. In the first step, the algorithm guesses which regime the ocean was in at each time point; in the second, it adjusts parameters so the data better match those guesses. Some parameters are updated with exact formulas, while fractional delays are sought by random search: the algorithm tries different values and keeps those that increase the likelihood, that is, the agreement between model and data. This resembles tuning a radio: you turn the knob, listening for when the signal becomes clearer.
On synthetic data with two and three regimes, the method correctly identified their number. When the series consisted of a thousand points, integer delays were recognized in 95% of cases. For fractional delays of 3.5 and 9.5 months, the median error was about 0.01, but it did not decrease with increasing series length. This indicates a systematic bias due to interpolation: filling in intermediate values introduces a small but constant inaccuracy, like always falling slightly short of the desired mark on a ruler.
Previously, Markov-switching models only allowed integer delays, which did not match real observations with arbitrary sampling intervals. Now a class of models with fractional delays and an algorithm that estimates them along with other parameters from data has emerged. This opens the way to more accurate descriptions of systems with delay observed on irregular grids. For geophysics, it means El Niño models can be tuned to real data without assuming delays are multiples of the observation step. And then the ocean ceases to be a black box and becomes a conversational partner whose pauses we have finally learned to measure.
El Niño literally means "the boy" in Spanish, and the phenomenon was named by Peruvian fishermen who noticed warm water around Christmas. The delay in El Niño models reflects the time it takes for a warm water wave to cross the Pacific Ocean.