Improving the predictability of time series by chaotic parameter estimation

Limited literature regarding parameter estimation of dynamic systems has been identified as the central-most reason for not having parametric bounds in chaotic time series. However, literature suggests that a chaotic system displays a sensitive dependence on initial conditions, and our study reveals that the behavior of chaotic system: is also sensitive to changes in parameter values. Therefore, parameter estimation technique could make it possible to establish parametric bounds on a nonlinear dynamic system underlying a given time series, which in turn can improve predictability. By extracting the relationship between parametric bounds and predictability, we implemented chaos-based models for improving prediction in time series. ^ This study describes work done to establish bounds on a set of unknown parameters. Our research results reveal that by establishing parametric bounds, it is possible to improve the predictability of any time series, although the dynamics or the mathematical model of that series is not known apriori. In our attempt to improve the predictability of various time series, we have established the bounds for a set of unknown parameters. These are: (i) the embedding dimension to unfold a set of observation in the phase space, (ii) the time delay to use for a series, (iii) the number of neighborhood points to use for avoiding detection of false neighborhood and, (iv) the local polynomial to build numerical interpolation functions from one region to another. Using these bounds, we are able to get better predictability in chaotic time series than previously reported. In addition, the developments of this dissertation can establish a theoretical framework to investigate predictability in time series from the system-dynamics point of view. ^ In closing, our procedure significantly reduces the computer resource usage, as the search method is refined and efficient. Finally, the uniqueness of our method lies in its ability to extract chaotic dynamics inherent in non-linear time series by observing its values. ^

Using extreme value theory to value stock market returns

Extreme stock price movements are of great concern to both investors and the entire economy. For investors, a single negative return, or a combination of several smaller returns, can possible wipe out so much capital that the firm or portfolio becomes illiquid or insolvent. If enough investors experience this loss, it could shock the entire economy. An example of such a case is the stock market crash of 1987. Furthermore, there has been a lot of recent interest regarding the increasing volatility of stock prices. ^ This study presents an analysis of extreme stock price movements. The data utilized was the daily returns for the Standard and Poor's 500 index from January 3, 1978 to May 31, 2001. Research questions were analyzed using the statistical models provided by extreme value theory. One of the difficulties in examining stock price data is that there is no consensus regarding the correct shape of the distribution function generating the data. An advantage with extreme value theory is that no detailed knowledge of this distribution function is required to apply the asymptotic theory. We focus on the tail of the distribution. ^ Extreme value theory allows us to estimate a tail index, which we use to derive bounds on the returns for very low probabilities on an excess. Such information is useful in evaluating the volatility of stock prices. There are three possible limit laws for the maximum: Gumbel (thick-tailed), Fréchet (thin-tailed) or Weibull (no tail). Results indicated that extreme returns during the time period studied follow a Fréchet distribution. Thus, this study finds that extreme value analysis is a valuable tool for examining stock price movements and can be more efficient than the usual variance in measuring risk. ^

Three-dimensional brain fiber tracking modeling in diffusion tensor imaging

This dissertation establishes the foundation for a new 3-D visual interface integrating Magnetic Resonance Imaging (MRI) to Diffusion Tensor Imaging (DTI). The need for such an interface is critical for understanding brain dynamics, and for providing more accurate diagnosis of key brain dysfunctions in terms of neuronal connectivity. ^ This work involved two research fronts: (1) the development of new image processing and visualization techniques in order to accurately establish relational positioning of neuronal fiber tracts and key landmarks in 3-D brain atlases, and (2) the obligation to address the computational requirements such that the processing time is within the practical bounds of clinical settings. The system was evaluated using data from thirty patients and volunteers with the Brain Institute at Miami Children's Hospital. ^ Innovative visualization mechanisms allow for the first time white matter fiber tracts to be displayed alongside key anatomical structures within accurately registered 3-D semi-transparent images of the brain. ^ The segmentation algorithm is based on the calculation of mathematically-tuned thresholds and region-detection modules. The uniqueness of the algorithm is in its ability to perform fast and accurate segmentation of the ventricles. In contrast to the manual selection of the ventricles, which averaged over 12 minutes, the segmentation algorithm averaged less than 10 seconds in its execution. ^ The registration algorithm established searches and compares MR with DT images of the same subject, where derived correlation measures quantify the resulting accuracy. Overall, the images were 27% more correlated after registration, while an average of 1.5 seconds is all it took to execute the processes of registration, interpolation, and re-slicing of the images all at the same time and in all the given dimensions. ^ This interface was fully embedded into a fiber-tracking software system in order to establish an optimal research environment. This highly integrated 3-D visualization system reached a practical level that makes it ready for clinical deployment. ^

Algorithms for scheduling parallel batch processing machines with non-identical job ready times

This research is motivated by a practical application observed at a printed circuit board (PCB) manufacturing facility. After assembly, the PCBs (or jobs) are tested in environmental stress screening (ESS) chambers (or batch processing machines) to detect early failures. Several PCBs can be simultaneously tested as long as the total size of all the PCBs in the batch does not violate the chamber capacity. PCBs from different production lines arrive dynamically to a queue in front of a set of identical ESS chambers, where they are grouped into batches for testing. Each line delivers PCBs that vary in size and require different testing (or processing) times. Once a batch is formed, its processing time is the longest processing time among the PCBs in the batch, and its ready time is given by the PCB arriving last to the batch. ESS chambers are expensive and a bottleneck. Consequently, its makespan has to be minimized. ^ A mixed-integer formulation is proposed for the problem under study and compared to a formulation recently published. The proposed formulation is better in terms of the number of decision variables, linear constraints and run time. A procedure to compute the lower bound is proposed. For sparse problems (i.e. when job ready times are dispersed widely), the lower bounds are close to optimum. ^ The problem under study is NP-hard. Consequently, five heuristics, two metaheuristics (i.e. simulated annealing (SA) and greedy randomized adaptive search procedure (GRASP)), and a decomposition approach (i.e. column generation) are proposed—especially to solve problem instances which require prohibitively long run times when a commercial solver is used. Extensive experimental study was conducted to evaluate the different solution approaches based on the solution quality and run time. ^ The decomposition approach improved the lower bounds (or linear relaxation solution) of the mixed-integer formulation. At least one of the proposed heuristic outperforms the Modified Delay heuristic from the literature. For sparse problems, almost all the heuristics report a solution close to optimum. GRASP outperforms SA at a higher computational cost. The proposed approaches are viable to implement as the run time is very short. ^

Minimizing makespan for hybrid flowshops with batch, discrete processing machines and arbitrary job sizes

This research aims at a study of the hybrid flow shop problem which has parallel batch-processing machines in one stage and discrete-processing machines in other stages to process jobs of arbitrary sizes. The objective is to minimize the makespan for a set of jobs. The problem is denoted as: FF: batch1,sj:Cmax. The problem is formulated as a mixed-integer linear program. The commercial solver, AMPL/CPLEX, is used to solve problem instances to their optimality. Experimental results show that AMPL/CPLEX requires considerable time to find the optimal solution for even a small size problem, i.e., a 6-job instance requires 2 hours in average. A bottleneck-first-decomposition heuristic (BFD) is proposed in this study to overcome the computational (time) problem encountered while using the commercial solver. The proposed BFD heuristic is inspired by the shifting bottleneck heuristic. It decomposes the entire problem into three sub-problems, and schedules the sub-problems one by one. The proposed BFD heuristic consists of four major steps: formulating sub-problems, prioritizing sub-problems, solving sub-problems and re-scheduling. For solving the sub-problems, two heuristic algorithms are proposed; one for scheduling a hybrid flow shop with discrete processing machines, and the other for scheduling parallel batching machines (single stage). Both consider job arrival and delivery times. An experiment design is conducted to evaluate the effectiveness of the proposed BFD, which is further evaluated against a set of common heuristics including a randomized greedy heuristic and five dispatching rules. The results show that the proposed BFD heuristic outperforms all these algorithms. To evaluate the quality of the heuristic solution, a procedure is developed to calculate a lower bound of makespan for the problem under study. The lower bound obtained is tighter than other bounds developed for related problems in literature. A meta-search approach based on the Genetic Algorithm concept is developed to evaluate the significance of further improving the solution obtained from the proposed BFD heuristic. The experiment indicates that it reduces the makespan by 1.93 % in average within a negligible time when problem size is less than 50 jobs.

Foundations of quantitative information flow: Channels, cascades, and the information order

Secrecy is fundamental to computer security, but real systems often cannot avoid leaking some secret information. For this reason, the past decade has seen growing interest in quantitative theories of information flow that allow us to quantify the information being leaked. Within these theories, the system is modeled as an information-theoretic channel that specifies the probability of each output, given each input. Given a prior distribution on those inputs, entropy-like measures quantify the amount of information leakage caused by the channel. ^ This thesis presents new results in the theory of min-entropy leakage. First, we study the perspective of secrecy as a resource that is gradually consumed by a system. We explore this intuition through various models of min-entropy consumption. Next, we consider several composition operators that allow smaller systems to be combined into larger systems, and explore the extent to which the leakage of a combined system is constrained by the leakage of its constituents. Most significantly, we prove upper bounds on the leakage of a cascade of two channels, where the output of the first channel is used as input to the second. In addition, we show how to decompose a channel into a cascade of channels. ^ We also establish fundamental new results about the recently-proposed g-leakage family of measures. These results further highlight the significance of channel cascading. We prove that whenever channel A is composition refined by channel B, that is, whenever A is the cascade of B and R for some channel R, the leakage of A never exceeds that of B, regardless of the prior distribution or leakage measure (Shannon leakage, guessing entropy leakage, min-entropy leakage, or g-leakage). Moreover, we show that composition refinement is a partial order if we quotient away channel structure that is redundant with respect to leakage alone. These results are strengthened by the proof that composition refinement is the only way for one channel to never leak more than another with respect to g-leakage. Therefore, composition refinement robustly answers the question of when a channel is always at least as secure as another from a leakage point of view.^