Embedding reconstruction methodology for short time series – application to large El Niño events

Abstract

Abstract.We propose an alternative approach for the embedding space reconstruction method for short time series. An m-dimensional embedding space is reconstructed with a set of time delays including the relevant time scales characterizing the dynamical properties of the system. By using a maximal predictability criterion a d-dimensional subspace is selected with its associated set of time delays, in which a local nonlinear blind forecasting prediction performs the best reconstruction of a particular event of a time series. An locally unfolded d-dimensional embedding space is then obtained. The efficiency of the methodology, which is mathematically consistent with the fundamental definitions of the local nonlinear long time-scale predictability, was tested with a chaotic time series of the Lorenz system. When applied to the Southern Oscillation Index (SOI) (observational data associated with the El Niño-Southern Oscillation phenomena (ENSO)) an optimal set of embedding parameters exists, that allows constructing the main characteristics of the El Niño 1982Niño -1983Niño and 1997Niño -1998 events, directly from measurements up to 3 to 4 years in advance.

Temporal–spatial distribution of the predictability limit of monthly sea surface temperature in the global oceans

“…Using the NLLE and its derivatives, the limit of dynamical predictability of chaotic systems can be quantitatively determined. Compared to traditional Lyapunov exponents based on linear error dynamics, the NLLE is more suitable for determining quantitatively the predictability limit of a chaotic system (Astudillo et al , ; Rogberg et al , ). To apply the NLLE method to the study of atmospheric and oceanic predictability, a reasonable and efficient algorithm has been devised to enable estimates of the NLLE and its derivatives based on observational or reanalysis data (Ding et al , ; Li and Ding, ).…”

A simple logistic model to understand the occurrence of flood events into the Biobío River Basin in central Chile

“…Another shortcoming is that our approach assumes seasonality with an unvarying probabilistic relationship. However, the distribution of future flood events is likely to change as ENSO events are non‐linear (Astudillo et al ., ), as are rainfall processes (Ramírez et al ., ). Nevertheless, this is a shortcoming that any statistical relationship found in geophysics would share.…”

“…We are concerned that precipitation (Ramírez et al, 2006) and climatic indices associated with ENSO (El Niño Southern Oscillation) may show a nonlinear behavior (Astudillo et al, 2010). Therefore, we are interested in the probability of occurrence of flood phenomena with different probability distributions.…”

Long-term non-linear predictability of ENSO events over the 20th century

“…The main purpose of this paper is to report that such evidence exists. In a precedent article [17], a methodology for applying Takens's embedding theorem [18] to reconstruct an event is developed, by taking into account just the local information contained in a short time series. The physical basis of the method lies in the definition of the local predictability [19].…”

“…Extending [17] to the problem under consideration, a wavelet decomposition separates the different n state coordinates from the signal equivalently as described in [23] and [24]. Thus liberating us from the tedious searching process for the optimal state spaces as was done previously.…”