Complexity is a result of interactions among individual components of a distributed

system, each with their own dynamical time scale. Statistical techniques

such as fluctuation analysis are used to quantify extent of long range correlations

within time series data by estimating a scaling exponent commonly known as Hurst

exponent. Data from magnetospheric dynamics, physiology and finance are known

to show multi-exponent nature (two exponents in particular) in their fluctuation

analysis with a crossover between the power laws. This correlation crossover can be seen due to the statistical approach taken in the analysis of a range of systems with differing dynamical time scales, particularly due to the nature of interactions with one another. We refer to this property as multi-scale nature in the time series data from complex systems. The main contributions of the thesis are as follows:

Study of crossover in fluctuation analysis of data from magnetosphere, physiology and finance:

We propose an innovative regression scheme, whose mathematical model well describes two exponent nature with an intermediate crossover regime seen in fluctuation analysis - the Hyperbolic regression. Slopes of the asymptotes of the hyperbola are the Hurst exponents, and, center of the resulting hyperbolic fit is an estimate of the correlation crossover time. It is key to note that in this regression, there are no assumptions made about the crossover time, unlike previous approach to crossover fitting. Different data sets corresponding to different

physical processes demonstrate multi-scale nature. However, each data presents

a unique challenge to be addressed, as a result of characterization of its scaling

crossover. Application of hyperbolic regression on the crossover seen in fluctuation

analysis of auroral electrojet index data from magnetosphere resulted in estimation

of Hurst exponents before and after the crossover. Also, the correlation crossover

time scale is now measured by improved modeling of such data using a stochastic

model that demonstrates crossover in fluctuation functions - the OU-Langevin

model. Characterization of nature of crossover seen in fluctuation analysis of generalized volatility within financial index data has shown differing nature of financial markets. Such a study would help characterize individual markets utilizing features which were not used previously. Heart rate variability data from healthy patients and patients with congestive heart failure demonstrate differing extent of crossover within the crossover seen in fluctuation functions. This resulted in proposal of a parameter i.e., the extent of crossover parameter that can be used to distinguish patients with the congestive heart failure ailment from normal cases.

Quantifying predictability of complex systems using Hurst exponents:

Predictability of complex systems suffers due to noise in the data. Long range correlations in noise are seen to cause extreme events or build ups leading to

extreme events in such data. The increased probability of extreme event occurrence makes prediction of resulting time series data difficult. Tail exponent is an exponent resulting out of power laws seen in heavy tailed distributions and is used as a measure of the probability of extreme events in such data. The Hurst exponent which measures the extent of long range correlations using fluctuation analysis has a known relationship with the tail exponent through Taqqu's theorem. Fluctuation analysis of time series data is oftentimes complicated due to existence of trends in the time series data. Dynamical features in data commonly reflect as trends, and, as a result nonlinear dynamical time series prediction techniques can be used to measure these trends. We refer to this method as Fluctuation Analysis after Trend Elimination (FATE) and apply it to data from space weather and finance. Hurst exponent estimated from FATE and its relationship with tail exponent measured by a commonly used Hill estimator technique are shown. Insufficient data sizes limit our ability to robustly estimate the tail exponent from observational data as extreme events are usually rare. This forms the motivation to use Hurst exponent obtained from FATE as a measure of predictability of complex systems demonstrated by application on auroral electrojet index data. Conversely, Hurst exponent can be used to quantify the ability of a technique to predict time series data as demonstrated by application of FATE.