Year: 2018
Type: dissertation
Abstract: Many real world systems consist of multiple parts and processes that nonlinearly interact with each other at different time scales. Data generated by such systems often contain self-similar structures with proportional scaling of statistical properties across time-scales. It is well known that such structural richness is not captured by conventional statistics like the mean and variance. Development of methods of nonlinear dynamics that quantify complexity, self-similarity, and scaling properties of empirical signals have had a profound impact on our understanding of systems in various scientific domains. In particular, our understanding of physiological systems has undergone a paradigm shift from homeostasis to notions of operation far from equilibrium in physiological regulatory networks. A commonly used methodology for quantifying the scaling relationships is fractal analysis, which estimates the power law exponent, H of time-series fluctuations versus time scales. However, many real world time series are determined by a multitude of exponents which has led to development of multifractal analysis methods. The cornerstone of multifractal analysis has traditionally been the multifractal spectrum which is akin to a probability density of all scaling exponents in the signal. Multifractal spectra are useful for comparing the distributions of exponents in signals. However, a multifractal spectrum does not provide any information about the temporal evolution of scaling structures and therefore, it is less useful in studying dynamic scaling structures. I introduce a set of tools for investigation of scaling structures across time. In particular, I propose scaling maps as a tool that augments the multifractal spectrum with temporal information and then develop a methodology for partitioning of a time series into segments of homogeneous scaling structures. In an attempt to quantify the differences in time local scaling properties of groups of time series, I then introduce Holder histograms as a better representation of time-localized scaling properties scaling structures and propose a feature extraction methodology and statistical tests that lend themselves to quantified comparison of multiple groups of time series in terms of determining their scaling regimes and their temporal dynamics. I then empirically demonstrate the application of the developed methodology on a set of simulated and real world datasets. Result show the ability of the method in recovering scaling modes of complex signals and the quantified discriminative capability of local temporal scaling structures.
Source: TSpace (University of Toronto)
Author: Hamidreza Saghir
Language: English
Subfield: Economics and Econometrics
Domain: Social Sciences
Open Access status: green
OpenAlex ID: W3159928333