Variability analysis of observational time series and its application to the characterization of complex dynamics in geoscience

Context and Program

This one-day workshop aims to gain an understanding into the behaviour of a physical system from observational data sequences. The methodology presented is structured around two points: the first point consists in studying the variability of observational time series by using decomposition methods with the objectives of removing noise and determining the dimension of the underlying dynamics; the second point consists of representing the evolution of the observed physical system in the form of a trajectory in a space (called the state space) whose dimension is equal to the dimension of the underlying dynamics with the objective to view geometric structure of the evolution observed system.

Workshop organization

The organization of this workshop will include lectures providing a detailed description of the techniques used with their algorithms in the form of scripts written in R language as well as sessions during which the R scripts will be applied to observational time series. Therefore, this workshop will enable geoscience students and practicing professionals to become familiar with decomposition time series technics as well as technics for reconstructing the underlying dynamics in the state space.

The one-day course will teach you: 1) to decompose observational time series into several independent components, then identify the noise related components, select the relevant free-noise components, extracting useful information and determine the dimensionality of the observed dynamic; 2) to visualize the underlying dynamics to observational time series in the state space.

By the end of the workshop, you will have a firm understanding of:

• Time series decomposition and its representation in the time frequency domain
• Noise removing and dimension reduction
• Visualisation of underlying dynamics in the state space

Working environment

RSTUDIO environment and R language

The main objective in the analysis of a physical system involving one or several observational time series is to characterize and even model the underlying dynamics. The first two major steps are first to improve the quality of the data as noise removing or process missing data and outliers and second to determine the dimension of the observed dynamics.

The irregular behavior of a physical system seen through a set of observational data results from the interaction of a certain number of degrees of freedom (d-o-f) and the question that must be asked is how many d-o-f are needed to reproduce the observed dynamics. On the other hand, reconstructing the observed dynamics of a data set in the state space enables to acquire a deep understanding of the observed physical system irregularities.

The method commonly used in the literature is to decompose a time series into several independent components, then identify the noise-related components and select the relevant free noise components that contribute significatively to the signal.

The number of relevant free noise components represents the statistical dimension of the observed dynamics, which is an upper bound on the dimension of the underlying dynamics. Most of observational time series have characteristics of non-stationarity and present fluctuations at all time scales. In this context, the most widely used decomposition methods are adaptive and data driven and the most cited in the literature are the empirical mode decomposition (EMD), the empirical wavelet transform (EWT) and the singular spectrum analysis (SSA).

The non-stationary nature of observational data sequences can be interpreted physically as arising from the interaction of several physical processes at several time scales. To analyze the resulting dynamics and, more specifically, its irregularities, we need to reconstruct the observed dynamics in the state space. Two popular methods of reconstruction are the method of delays (MOD) and the singular spectrum analysis (SSA). This workshop aims to provide a description of the decomposition methods mentioned above and to evaluate their effectiveness and their ability to remove noise and to identify components corresponding to the physical processes involved in the evolution of the observed system and deduce the dimensionality of the associated dynamics. The MOD method will be described and it will be explained why the SSA method can be used in the same way to reconstruct the observed dynamics in the state-space. All of the methods mentioned will be applied to experimental total ozone columns and rainfall time series and results obtained will be discussed and compared.