Computing the multifractal spectrum from time series: an algorithmic approach

Exploring the multi-fractal nature of the air flow and pollutant dispersion in a turbulent urban atmosphere and its implications for long range pollutant transport

This work investigates the multi-fractal nature of a turbulent urban atmosphere using high-resolution atmospheric data. Meteorological and concentration measurements of passive and reactive pollutants collected over a 3-year period in a sub-urban high-Reynolds number atmospheric field were analyzed. Scaling laws characterizing the self-similarity and thereby depicting the multi-fractal nature are determined by calculating the singularity spectra, where a range of Hölder exponents, h, are estimated. In doing so, the complexity of the urban atmosphere entailing different stability regimes was addressed. Using the Monin-Obukhov length (L_MO) as a marker of atmospheric stability and thereby an indication of the magnitude of anisotropy, we find where and how self-similarity is manifested relative to the different regimes and we estimate corresponding appropriate scaling laws. We find that the wind speed obeys the -5/3 law suggested by Kolmogorov only when the atmosphere lies within the stable regime as defined by Monin-Obukhov theory. Specifically, when the ratio of the atmospheric boundary layer height (H_b.l) over L_MO is greater than 15, and at the same time, the ratio of the height above ground of the wind measurements (z₀) over L_MO is higher than 3 (i.e., in stable regime), then the singularity spectra of wind speed time series indicate that the dominant Hölder exponent, h_max, coincides with Kolmogorov's second hypothesis. On the contrary under unstable regimes in the atmosphere where the anisotropy is approached, different scaling laws are estimated. In detail, when z₀/L_MO<0, the dominant Hölder exponent, h_max, of the singularity spectra of the wind speed time series is either negative or close to zero, which is an indication of an impulse-like singularity, that is associated with rapid changes. For the ambient temperature and air quality measurements such as of carbon monoxide and particulate matter concentrations, it was found that they obey different laws, which are related with the long-term correlation of their data fluctuation.

Multifractal analysis of financial markets: a review

Multifractality is ubiquitously observed in complex natural and socioeconomic systems. Multifractal analysis provides powerful tools to understand the complex nonlinear nature of time series in diverse fields. Inspired by its striking analogy with hydrodynamic turbulence, from which the idea of multifractality originated, multifractal analysis of financial markets has bloomed, forming one of the main directions of econophysics. We review the multifractal analysis methods and multifractal models adopted in or invented for financial time series and their subtle properties, which are applicable to time series in other disciplines. We survey the cumulating evidence for the presence of multifractality in financial time series in different markets and at different time periods and discuss the sources of multifractality. The usefulness of multifractal analysis in quantifying market inefficiency, in supporting risk management and in developing other applications is presented. We finally discuss open problems and further directions of multifractal analysis.

Detecting abnormality in heart dynamics from multifractal analysis of ECG signals

The characterization of heart dynamics with a view to distinguish abnormal from normal behavior is an interesting topic in clinical sciences. Here we present an analysis of the Electro-cardiogram (ECG) signals from several healthy and unhealthy subjects using the framework of dynamical systems approach to multifractal analysis. Our analysis differs from the conventional nonlinear analysis in that the information contained in the amplitude variations of the signal is being extracted and quantified. The results thus obtained reveal that the attractor underlying the dynamics of the heart has multifractal structure and the variations in the resultant multifractal spectra can clearly separate healthy subjects from unhealthy ones. We use supervised machine learning approach to build a model that predicts the group label of a new subject with very high accuracy on the basis of the multifractal parameters. By comparing the computed indices in the multifractal spectra with that of beat replicated data from the same ECG, we show how each ECG can be checked for variations within itself. The increased variability observed in the measures for the unhealthy cases can be a clinically meaningful index for detecting the abnormal dynamics of the heart.

Series-NonUniform Rational B-Spline (S-NURBS) model: a geometrical interpolation framework for chaotic data

Time series is widely exploited to study the innate character of the complex chaotic system. Existing chaotic models are weak in modeling accuracy because of adopting either error minimization strategy or an acceptable error to end the modeling process. Instead, interpolation can be very useful for solving differential equations with a small modeling error, but it is also very difficult to deal with arbitrary-dimensional series. In this paper, geometric theory is considered to reduce the modeling error, and a high-precision framework called Series-NonUniform Rational B-Spline (S-NURBS) model is developed to deal with arbitrary-dimensional series. The capability of the interpolation framework is proved in the validation part. Besides, we verify its reliability by interpolating Musa dataset. The main improvement of the proposed framework is that we are able to reduce the interpolation error by properly adjusting weights series step by step if more information is given. Meanwhile, these experiments also demonstrate that studying the physical system from a geometric perspective is feasible.

Characterization of complexities in combustion instability in a lean premixed gas-turbine model combustor

We characterize complexities in combustion instability in a lean premixed gas-turbine model combustor by nonlinear time series analysis to evaluate permutation entropy, fractal dimensions, and short-term predictability. The dynamic behavior in combustion instability near lean blowout exhibits a self-affine structure and is ascribed to fractional Brownian motion. It undergoes chaos by the onset of combustion oscillations with slow amplitude modulation. Our results indicate that nonlinear time series analysis is capable of characterizing complexities in combustion instability close to lean blowout.