Search results for: A. Bourdier

Authors: Olivier Delage, Hassan Bencherif, Alain Bourdier

Abstract:

Signal decomposition approaches represent an important step in time series analysis, providing useful knowledge and insight into the data and underlying dynamics characteristics while also facilitating tasks such as noise removal and feature extraction. As most of observational time series are nonlinear and nonstationary, resulting of several physical processes interaction at different time scales, experimental time series have fluctuations at all time scales and requires the development of specific signal decomposition techniques. Most commonly used techniques are data driven, enabling to obtain well-behaved signal components without making any prior-assumptions on input data. Among the most popular time series decomposition techniques, most cited in the literature, are the empirical mode decomposition and its variants, the empirical wavelet transform and singular spectrum analysis. With increasing popularity and utility of these methods in wide ranging applications, it is imperative to gain a good understanding and insight into the operation of these algorithms. In this work, we describe all of the techniques mentioned above as well as their ability to denoise signals, to capture trends, to identify components corresponding to the physical processes involved in the evolution of the observed system and deduce the dimensionality of the underlying dynamics. Results obtained with all of these methods on experimental total ozone columns and rainfall time series will be discussed and compared

Keywords: denoising, empirical mode decomposition, singular spectrum analysis, time series, underlying dynamics, wavelet analysis

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Authors: O. Delage, H. Bencherif, T. Portafaix, A. Bourdier

Abstract:

The main purpose of time series analysis is to learn about the dynamics behind some time ordered measurement data. Two approaches are used in the literature to get a better knowledge of the dynamics contained in observation data sequences. The first of these approaches concerns time series decomposition, which is an important analysis step allowing patterns and behaviors to be extracted as components providing insight into the mechanisms producing the time series. As in many cases, time series are short, noisy, and non-stationary. To provide components which are physically meaningful, methods such as Empirical Mode Decomposition (EMD), Empirical Wavelet Transform (EWT) or, more recently, Empirical Adaptive Wavelet Decomposition (EAWD) have been proposed. The second approach is to reconstruct the dynamics underlying the time series as a trajectory in state space by mapping a time series into a set of Rᵐ lag vectors by using the method of delays (MOD). Takens has proved that the trajectory obtained with the MOD technic is equivalent to the trajectory representing the dynamics behind the original time series. This work introduces the singular spectrum decomposition (SSD), which is a new adaptive method for decomposing non-linear and non-stationary time series in narrow-banded components. This method takes its origin from singular spectrum analysis (SSA), a nonparametric spectral estimation method used for the analysis and prediction of time series. As the first step of SSD is to constitute a trajectory matrix by embedding a one-dimensional time series into a set of lagged vectors, SSD can also be seen as a reconstruction method like MOD. We will first give a brief overview of the existing decomposition methods (EMD-EWT-EAWD). The SSD method will then be described in detail and applied to experimental time series of observations resulting from total columns of ozone measurements. The results obtained will be compared with those provided by the previously mentioned decomposition methods. We will also compare the reconstruction qualities of the observed dynamics obtained from the SSD and MOD methods.

Keywords: time series analysis, adaptive time series decomposition, wavelet, phase space reconstruction, singular spectrum analysis

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