Numerical Time Series Research Articles

Temporal granulation and its application to signal analysis

In this study, we elaborate on the role of information granulation and the ensuing information granules in description of time series and signal analysis, in general. Information granules are entities of elements (quite commonly, numeric data) that are combined together (aggregated) owing to their vicinity, similarity and alike. Proceeding with a given window of granulation (that is an initial collection of numeric data), we propose an algorithm that produces a complete information granule – fuzzy set. The principle supported by the method leads to the formation of fuzzy sets that are legitimate in terms of experimental data being at the same time maximized with regard to their specificity (compactness). It has been shown that information granules can be are regarded as generic conceptual entities contributing to the description of numeric time series. In this capacity, they are used as building blocks aimed at achieving high level, compact, and comprehensible models of signals. More importantly, the phase of information granulation could be viewed as a prerequisite to more synthetic and abstract processing such as the one witnessed in syntactic pattern recognition

Microscopic chaos from brownian motion?

Gaspard et al.1 have analysed a time series of the positions of a brownian particle in a liquid, and claimed that it provides empirical evidence for microscopic chaos on a molecular scale. An accompanying comment2 emphasized the fundamental nature of the experiment. Here we show that virtually identical results can be obtained by analysing a corresponding numerical time series of a particle in a manifestly microscopically non-chaotic system.

Control of Chaos in the Rössler System

The control method of Ott, Grebogi and Yorke (1990) as applied to the Rössler system, a set of three-dimensional non-linear differential equations, is examined. Using numerical time series data for a single dynamical variable the method was successfully employed to control several of the unstable periodic orbits in a three-dimensional embedding of the data. The method also failed for a number of unstable periodic orbits due to difficulties in linearising about the orbit or the tangential coincidence of the stable manifold and the motion of the orbit with external parameter.

TIME SERIES ANALYSIS OF COMPLEX DYNAMICAL BEHAVIOR CONTAMINATED WITH OBSERVATIONAL NOISE

Diagnostic methods for discovering deterministic chaos based on the instability and the parallelness of nearby trajectories generated from a time series in phase space are applied to numerical time series contaminated with additive random noise. The diagnostic algorithm based on nonlinear forecasting is prone to be fooled when handling chaotic data including observational noise. Such a misdiagnosis can be circumvented by estimating the degrees of parallelness of neighboring trajectories in the phase space. Dynamical properties of global temperature variations and voice signals of Japanese vowel /a/ are examined by the combinational use of the diagnostic algorithms.

Topographic Rossby modes in the Strait of Sicily

In this paper, numerical results of a barotropic circulation model are presented, indicating that topographic Rossby modes can be excited in the area of the Strait of Sicily. The results also suggest that such motions are robust features, since they can be resonantly generated by different forcings. A numerical barotropic shallow‐water model is implemented in the central Mediterranean Sea, and, in order to perform a spectroscopic analysis, the initial value problem is solved, both in the presence of a white wind with spatially constant curl and of a white boundary flow representing an equivalent remote pressure forcing. A spectral analysis in selected points in the region of the strait reveals five peaks with periods ranging from 2 to 5 days. The associated circulation patterns are then studied by forcing the system with monochromatic winds or boundary flows at the periods of the energy peaks. Traveling patterns are recognized, whose spatial/temporal structures and periods are found to compare well with those of analytical Rossby modes, thus obtaining a clear evidence of their dynamical nature. In order to investigate the actual excitation of Rossby modes in realistic conditions, the system is then forced by the 1980 National Meteorological Center momentum flux data. The main mode at nearly 3 days obtained from the preceding spectral analysis is indeed found to be excited in the narrowest section of the strait during winter and autumn months. Rossby modes in the numerical time series are identified by means of the analysis of the energy spectra and of the coherence and phase of current signals between two points along the theoretical propagation direction. Finally, the existence of topographic Rossby modes in the Strait of Sicily is discussed in connection with the local properties of horizontal mixing in that area.

Predictability in time series

We introduce a technique to characterize and measure predictability in time series. The technique allows one to formulate precisely a notion of the predictable component of given time series. We illustrate our method for both numerical and experimental time series data.

Acoustic detection and location of blue whales (Balaenoptera musculus) from SOSUS data by matched filtering

Blue whale calls were recorded off central California in the Fall of 1993. These calls were characterized as to duration, frequency downsweep, intercall interval, and sound-pressure level. Average values were determined from 303 calls, including up to three harmonics (fundamental downsweep from 18.9 to 17.2 Hz over 16 s). These frequency-domain characterizations were then used to develop numerical time series (kernels) that, when convolved with the original time series, produce correlation peaks indicating the presence of blue whale calls (matched filter). When harmonics were present in the data, a combined kernel, including the fundamental frequency and first harmonic, improved the signal to noise ratio over use of the fundamental kernel alone. These matched filters were able to detect blue whale calls even in very ‘‘noisy’’ time series. When applied to hydrophone recordings from three U.S. Navy SOSUS (SOund SUrveillance System) arrays, it is possible to produce locations for blue whale calls by timing the arrival of individual calls and applying least-squares techniques. This information can be used to increase our knowledge of blue whale distribution in the northeast Pacific. The methods described here may also be extended to other species that employ low-frequency vocalizations or to other ocean areas.

Application of the Extended Kalman Filter for Reconstructing Systems from Chaotic Numerical Time Series

Application of the Extended Kalman Filter for Reconstructing Systems from Chaotic Numerical Time Series

Construction of phenomenological models from numerical scalar time series

The problem of reconstruction of ordinary differential equations from numerical scalar time series is discussed. Techniques are exemplified for Rössler and Lorenz chaotic attractors, with emphasis on improved algorithms with respect to those previously published. The steps which are still required in order to investigate experimental noisy data are discussed.

Reconstruction of vector fields: The case of the Lorenz system.

Ordinary differential equations of continuous dynamical systems, or at least of equivalent systems, can be reconstructed from numerical scalar time series. Corresponding methods have been previously discussed for a R\"ossler band. This paper is devoted to the case of the Lorenz system. Several kinds of equivalent systems are being investigated, and a classification of them is being introduced. Validations may rely (i) qualitatively with comparisons between graphical displays of exact and corresponding reconstructed attractors and (ii) quantitatively with comparisons between invariants such as generalized dimension spectra.

Reconstruction of standard and inverse vector fields equivalent to a Rössler system.

Ordinary diffential equations of continuous dynamical systems, or at least of equivalent systems, can be reconstructed from numerical scalar time series. Methods are exemplified for a R\"ossler band. Equivalent systems are standard and inverse systems, which are systematically investigated. Validations rely (i) qualitatively on comparisons between phase portraits and (ii) quantitatively on comparisons between generalized dimension spectra. By-products of the work are an information-compression scheme for time-series encoding and the introduction of squeezed systems that facilitate evaluations of generalized dimensions of small order q1.

Reconstruction of the vector fields of continuous dynamical systems from numerical scalar time series.

We emphasize that the ordinary differential equations of a continuous dynamical system, or at least of equivalent systems, can be reconstructed from numerical scalar time series. Methods are exemplified in the case of a strange, chaotic attractor generated by a mathematical model, namely, a R\"ossler band. Resultant validations rely (i) qualitatively on comparisons between original and reconstructed phase portraits, and (ii) quantitatively on comparisons between generalized dimensions ${\mathit{D}}_{\mathit{q}}$ of original and reconstructed attractors. Some of the many lines of research offered by the presented results are discussed to stress potentialities of this kind of reconstruction.

The harmonic analysis of tidal model time series

The harmonic analysis of tides is reviewed and guidelines for the analysis of tidal model time series are presented. The analysis technique introduced by Godin and circulated by Foreman is summarized. The related issues of constituent selection, time series length, confidence regions, and inference are discussed. It is shown that since numerical model time series have a much lower non-tidal content than observational time series, constituents can be separated with shorter record lengths.

Use of correlation analysis to interpret continental margin ECOOE refraction data

Numerical time series analysis has been applied in the study of a set of refraction records obtained from a series of explosions detonated on the continental margin east of the Maryland coast. Correlation filters made it possible not only to find arrivals that were buried in the noise but also to distinguish between arrivals that were nearly coincident in time. Since this technique enabled us to obtain more arrivals on the seismic record than could be found visually, the travel-time curve is more complete for the machine picks than for the visual picks. Velocity-depth calculations were made and compared, using the intercept times and the velocities obtained from both travel-time curves (visually picked and machine-picked data). It was calculated that the Moho dips toward the continent at about 4° and has a depth in the Maryland area of nearly 25 km. It was also found that the definition of the Mono has considerable north-south dependence, both in energy and in frequency content of its refraction arrival. The crustal layer overlying the Moho was found to dip west both on the continental shelf area and under Maryland, whereas the shallowest observed refractor was found to change direction of dip from west on the shelf to east under the coast. Both these layers strike approximately north-south in each region.