Given the ubiquity of time series data, and the exponential growth of databases, there has recently been an explosion of interest in time series data mining. Finding similar trends and patterns among time series data is critical for many applications ranging from financial planning, weather forecasting, stock analysis to policy making. With time series being high-dimensional objects, detection of similar trends especially at the granularity of subsequences or among time series of different lengths and temporal misalignments incurs prohibitively high computation costs. Finding trends using non-metric correlation measures further compounds the complexity, as traditional pruning techniques cannot be directly applied. My dissertation addresses these challenges while meeting the need to achieve near real-time responsiveness. First, for retrieving exact similarity results using Lp-norm distances, we design a two-layered time series index for subsequence matching. Time series relationships are compactly organized in a directed acyclic graph embedded with similarity vectors capturing subsequence similarities. Powerful pruning strategies leveraging the graph structure greatly reduce the number of time series as well as subsequence comparisons, resulting in a several order of magnitude speed-up. Second, to support a rich diversity of correlation analytics operations, we compress time series into Euclidean-based clusters augmented by a compact overlay graph encoding correlation relationships. Such a framework supports a rich variety of operations including retrieving positive or negative correlations, self correlations and finding groups of correlated sequences. Third, to support flexible similarity specification using computationally expensive warped distance like Dynamic Time Warping we design data reduction strategies leveraging the inexpensive Euclidean distance with subsequent time warped matching on the reduced data. This facilitates the comparison of sequences of different lengths and with flexible alignment still within a few seconds of response time. Comprehensive experimental studies using real-world and synthetic datasets demonstrate the efficiency, effectiveness and quality of the results achieved by our proposed techniques as compared to the state-of-the-art methods.

Dynamic compressors are susceptible to aerodynamic instabilities while operating at low mass flow rates. These instabilities, rotating stall and surge, are detrimental to engine life and operational safety, and are thus undesirable. In order to prevent stability problems, a passive technique, involving fuel flow scheduling, is currently employed on gas turbines. The passive nature of this technique necessitates conservative stability margins, compromising performance and/or efficiency. In the past, model based active control has been proposed to enable reduction of margin requirements. However, available compressor stability models do not predict the different stall inception patterns, making model based control techniques practically infeasible. This research presents active stability management as a viable alternative. In particular, a limit detection and avoidance approach has been used to maintain the system free of instabilities. Simulations show significant improvements in the dynamic response of a gas turbine engine with this approach. A novel technique has been developed to enable real-time detection of stability limits in axial compressors. It employs a correlation measure to quantify the chaos in the rotor tip region. Analysis of data from four axial compressors shows that the value of the correlation measure decreases as compressor loading is increased. Moreover, sharp drops in this measure have been found to be relevant for stability limit detection. The significance of these drops can be captured by tracking events generated by the downward crossing of a selected threshold level. It has been observed that the average number of events increases as the stability limit is approached in all the compressors studied. These events appear to be randomly distributed in time. A stochastic model for the time between consecutive events has been developed and incorporated in an engine simulation. The simulation has been used to highlight the importance of the threshold level tosuccessful stability management. The compressor stability management concepts have also been experimentally demonstrated on a laboratory axial compressor rig. The fundamental nature of correlation measure has opened avenues for its application besides limit detection. The applications presented include stage load matching in a multi-stage compressor and monitoring the aerodynamic health of rotor blades.

This thesis examines a multivariate point process in time with focus on a mu- tual relations of its marginal point processes. The first chapter acquaints the re- ader with the theoretical background of multivariate point processes and their properties, especially the higher-order cumulant-correlation measures. Later on, several models of multivariate point processes with different dependence structu- res are characterized, such as the random superposition model, a Poisson depen- dent superposition point process, a jitter Poisson dependent superposition point process orrenewal processes models. Simulations of each of them are provided. Furthermore, two statistical methods for higher-order correlations are presented; the cumulant based inference of higher-order correlations, and the extended til- ling coefficient. Finally, the introduced methods are applied not only on the data from simulations, but also on the real, simultaneously recorded nerve cells spike train data. The results are discussed. 1