Markov processes

One of the limiting factors restricting aircraft landings at maJor airports is the
minimum spacing requirements due to vortex wake avoidance. If it can be shown that the
separation requirements are too conservative, then it may be possible to increase the rate
of landings on a given runway. During August/September 2003, NASA and the (United
States Department of Transportation) USDOT sponsored a wake acoustics test at the
Denver International Airport. The central instrument of the test was a large microphone
phased array. Different types of aircrafts were recorded during landing and the acoustic
data obtained was stored. From acoustic data the spectrograms were generated using the
technique of AutoRegressive (AR) spectral estimation from multitaper autocorrelation
estimates.
Several sources of sound that are recorded in the audio files can be observed in the
spectrograms. Some these signals, such as the noise generated from the aircraft engine can be identified easily because of their strength and the Doppler shift they undergo. In
contrast to this, the wake vortex signal is weaker and does not exhibit a Doppler shift
because it's stationary in space. Therefore it may not be identified easily because of the
existence of stronger signals. The motive in our research is to develop methods to
determine these strong signals that appear as spectral lines in the spectrogram. In the
future, the results obtained in this work can be used to eliminate these strong signals from
the spectrogram thus allowing us to see and identify wake vortex signal which is more
important to us.

With rapid growth of the World Wide Web, web performance becomes increasingly important for modern businesses, especially for e-commerce. As we all know, web server logs contain potentially useful empirical data to improve web server performance. In this thesis, we discuss some topics related to the analysis of a website's server logs for enhancing server performance, which will benefit some applications in business. Markov chain models are used and allow us to dynamically model page sequences extracted from server logs. My experimental studies contain three major parts. First, I present a workload characterization study of the website used for my research. Second, Markov chain models are constructed for both page request and page-visiting sequence prediction. Finally, I carefully evaluate the constructed models using an independent test data set, which is from server logs on a different day. The research results demonstrate the effectiveness of Markov chain models for characterizing page-visiting sequences.

The purpose of sequence alignment is to detect mutual similarity, characterized by the so-called "alignment score", between sequences compared. To quantitatively assess the confidence level of an alignment result requires the knowledge of alignment score statistics under a certain null model and is the central issue in sequence alignment. In this thesis, the score statistics of Markov null model were revisited and the score statistics of non-Markov null model were investigated for two state-of-the-art algorithms, namely, the gapless Smith-Waterman and Hybrid algorithms. These two algorithms were further used to find highly related signals in unrelated sequences and in weakly related sequences corresponding, respectively, to Markov null model and non-Markov null model. The confidence levels of these models were also studied. Since the sequence similarity we are interested in comes from evolutionary history, we also investigated the relationship between sequence alignment, the tool to find similarity, and evolution. The average evolution distance between the daughter sequences was found and compared with their expected values, for individual trees and as an average over many trees.

The present dissertation is focused on the numerical method of path integration for stochastic systems. The existing procedures of numerical path integration are re-examined. A comparison study is made of the results obtained using various interpolation schemes. The amounts of computation time and relative accuracies of the existing procedures are tested with different mesh sizes and different time step sizes. A new numerical procedure based on Gauss-Legendre integration formula is proposed, which requires no explicit numerical interpolation. The probability evolution is represented in terms of the transition probabilities among Gauss points in various sub-intervals. Each transition probability is assumed to be Gaussian, and it can be obtained from the moment equations. Gaussian closure is used to truncate the moment equations in the case of a nonlinear system. The computation parameters of the new procedure, such as size of time-step and number of sub-intervals, can be determined in a systematic manner. The approximate Gaussianity of the transition probability obtained from the moment equations is first tested by comparing it with the simulation results, from which a proper time-step size is selected. The standard deviation of the transition probability in each direction of the state space can then be obtained from the moment equations, and is used to determine the size of the sub-intervals in that direction. The new numerical path integration procedure is applied to several one-dimensional and two-dimensional stochastic systems, for which the responses are homogeneous Markov processes. It is shown that the new procedure is not only accurate and efficient, but also numerically stable and highly adaptable. The new procedure is also applied to a nonlinear stochastic system subjected to both sinusoidal and random excitations. The system response in this case is a non-homogeneous Markov process. The algorithm is adapted for this case, so that re-computation of the transition probability density at every time step can be avoided.