This work sits at the fringe of the knowledge map: its nearest neighbors deal with cosmological constants, cosmic microwave background, and dark energy, but in essence it is closer to regime-switching time series models used in geophysics, economics, and signal processing. Unlike neighbors that work with physical fields and equations of state, here the object is a discrete sequence of observations, and the task is statistical inference. This is the first work to systematically introduce fractional delays into Markov-switching models and provide an estimation scheme for them.
The authors construct the cDN-ARMS(L) model: at each step, the state depends on the previous value and on a value taken with a delay characteristic of the current hidden regime. Delays can be fractional, in which case the missing value is obtained by linear interpolation of neighboring samples. For parameter estimation, a space-alternating EM algorithm is used: in the E-step, posterior probabilities of hidden states are computed via the forward-backward algorithm; in the M-step, some parameters are updated exactly, while others are updated one by one using accelerated random search. The number of regimes is selected by penalized likelihood.
On synthetic data with two and three regimes, the penalized likelihood correctly identified the number of layers. For series of length 1000 points, integer delays were correctly identified in 95% of simulations. For fractional delays (3.5 and 9.5 months), the median normalized error was about 0.01, but it remained constant as series length increased, indicating a systematic bias due to interpolation. On real El Niño data (860 months, 1950–2021), the model with two nonlinear regimes yielded delays of 2.386 and 7.301 months, while the three-regime model gave 2.000, 2.153, and 13.844 months. Nonlinear models reproduced the autocorrelation up to 15 months and the central quantiles of the distribution better than linear AR counterparts.
This work differs from its neighbors on the knowledge map by transferring the idea of hidden states with continuous delays into the realm of time series statistics, where previously only integer lags were considered. This opens the way to more accurate descriptions of systems with delays observed on irregular grids. For geophysics, it means that El Niño models can be tuned to real data without assuming that delays are multiples of the observation step.
The topic could evolve toward Bayesian methods with prior distributions on delays, allowing uncertainty quantification. Another direction is adapting the algorithm to multivariate series and to more general nonlinear functions. Application to other climate oscillations, such as the North Atlantic Oscillation, is also possible.
The method will be useful in climatology for analyzing El Niño and similar phenomena, as well as in economics and engineering where regime switching with delays occurs.
The next step is to reduce the bias in fractional delay estimates, possibly through more accurate interpolation or joint estimation of the observation step. It is also worth testing the method on longer series and with more regimes.
In time series statistics, the question of consistency of regime number estimates for small samples and the effect of interpolation on asymptotics remains open. The work does not solve the problem of local likelihood maxima, which requires multiple runs of the algorithm.
El Niño literally means 'the boy' in Spanish, 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.