The present work starts with a study of isotropic turbulence which was introduced by G. I. Taylor in 1935. The different notions of averages are critically examined. The notion of stochastic average is then introduced and the general transport equation is developed. After a detailed study of kinematics of turbulence, the concept of correlation and spectrum, the correspondence between the Karman-Howarth equation and the spectrum equation is made. The turbulence decay is studied. A theory for turbulence decay at large Reynolds number is proposed. In the study of turbulence spectrum, different assumptions on the transfer function are critically discussed and the solution using Heisenberg's assumption is obtained explicitly. The spectrum is further studied by trying to fit the turbulence phenomenon into a general scheme of stochastic processes. In the second part of the work, an entirely different approach to the statistical theory is made. Linearized vorticity transport theory is developed and finally the non-linear effects in turbulence are studied.

Thesis: M.S., Massachusetts Institute of Technology, Department of Electrical Engineering, 1960 / Includes bibliographical references (leaves [60]-62). / by Paul Kinzbruner Buller. / M.S. / M.S. Massachusetts Institute of Technology, Department of Electrical Engineering

In this study, several wave parameters are downloaded or calculated from spectral wave data, and their interdependence is investigated. A literature review is carried out to summarize these dependencies. The literature study is followed by a statistical analysis of the relations between the calculated wave parameters. This statistical analysis is done both for all wave data, resulting in a global expression for the relations, and for the different seas and oceans, resulting in a deeper analysis about local differences and how these differences should be taken into consideration when using wave parameter relations in specific water bodies.

Prior results (from a mathematics competition) show that the expected value of the minimum element in a random sample of size r from {1,2,3,...,n} is (n+1) / (r+1) In my research, I've calculated the more complicated value of the variance of this quantity using standard and novel techniques from mathematical statistics. I will also present results obtained between now and April 24th and indicate directions for further research.

For any company involved in sales, the goal of maximizing profit guides all decision-making. Understanding trends in one's own data, a company can identify potential growth opportunities to help increase profitability. In this talk, we examine general trends in a data set to recommend specific actions a company could implement to improve performance. Additionally, we explain the purpose of predictive modeling that helps companies adapt their decision-making process as the trends in the data change over time.

Differential Privacy is now a gold standard for data privacy in many learning and statistical tasks. It has enjoyed over a decade of intense study, with focus on both upper and lower bounds in different settings for different problems. In the intersection of privacy and statistical estimation (henceforth called, ``private statistical estimation''), our understanding of fundamental problems has grown, but several open questions emerged in the process, which had not received adequate attention. The goal of this dissertation has been to identify and address some of these challenges. We tackle these problems with focus towards reducing the cost of privacy, whilst attaining near optimal accuracy. Specifically, we make progress in answering the following questions.--How to privately estimate the mean of distributions from various families? --How to privately estimate the covariance of high-dimensional Gaussians with both sample and time efficiency? --How to privately estimate the parameters of mixtures of high-dimensional Gaussians? --When the data lies in some low-dimensional subspace, how do we privately learn that subspace with no cost in sample complexity in terms of the ambient dimension? Future directions of private statistical estimation are also discussed.--Author's abstract

Most complex systems in the world are time-dependent and dynamic in nature, many of which are suitable to be modeled as dynamic networks that evolve over time. From the analysis of time-varying social networks to the analysis of functional brain networks in longitudinal study designs, new statistical methods are needed for a better understanding of network dynamics and the underlying complex systems. Our work revolves around statistical modeling, sampling and inference for dynamic networks driven by various applications. Specifically, we develop a class of random graph hidden Markov models (RGHMM) for percolation in noisy dynamic networks to infer the type of phase transitions undergone in epileptic seizures. We also develop a broadly applicable class of coevolving latent space network with attractors (CLSNA) models for characterizing coevolutionary phenomenon in social behaviors, such as flocking and polarization, and use it under the context of American politics to disentangle positive and negative partisanship in affective polarization. Finally, we provide uncertainty quantification in conjunction with estimation of the frequency of motifs in dynamic networks under a certain sampling model, by studying the asymptotics for streaming data applications.

Propagation of coherent waves through disordered media, whether optical, acoustic, or radio waves, results in a spatially redistributed random intensity pattern known as speckle -- a statistical phenomenon. The subject of this dissertation is the statistics of monochromatic coherent light traversing disordered photonic lattices and its dependence on the disorder class, the level of disorder and the excitation configuration at the input. Throughout the dissertation, two disorder classes are considered, namely, diagonal and off-diagonal disorders. The latter exhibits disorder-immune chiral symmetry -- the appearance of the eigenmodes in skew-symmetric pairs and the corresponding eigenvalues in opposite signs. When a disordered photonic lattice, an array of evanescently coupled waveguides, is illuminated with an extended coherent optical field, discrete speckle develops. Numerical simulations and analytical modeling reveal that discrete speckle shows a set of surprising features, that are qualitatively indistinguishable in both disorder classes. First, the fingerprint of transverse Anderson localization -- associated with disordered lattices, is exhibited in the narrowing of the spatial coherence function. Second, the transverse coherence length (or speckle grain size) freezes upon propagation. Third, the axial coherence depth is independent of the axial position, thereby resulting in a coherence voxel of fixed volume independently of position. When a single lattice site is coherently excited, I discovered that a thermalization gap emerges for light propagating in disordered lattices endowed with disorder-immune chiral symmetry. In these systems, the span of sub-thermal photon statistics is inaccessible to the input coherent light, which -- once the steady state is reached -- always emerges with super-thermal statistics no matter how small the disorder level. An independent constraint of the input field for the chiral symmetry to be activated and the gap to be observed is formulated. This unique feature enables a new form of photon-statistics interferometry: by exciting two lattice sites with a variable relative phase, as in a traditional two-path interferometer, the excitation-symmetry of the chiral mode pairs is judiciously broken and interferometric control over the photon statistics is exercised, spanning sub-thermal and super-thermal regimes. By considering an ensemble of disorder realizations, this phenomenon is demonstrated experimentally: a deterministic tuning of the intensity fluctuations while the mean intensity remains constant. Finally, I examined the statistics of the emerging light in two different lattice topologies: linear and ring lattices. I showed that the topology dictates the light statistics in the off-diagonal case: for even-sited ring and linear lattices, the electromagnetic field evolves into a single quadrature component, so that the field takes discrete phase values and is non-circular in the complex plane. As a consequence, the statistics become super-thermal. For odd-sited ring lattices, the field becomes random in both quadratures resulting in sub-thermal statistics. However, this effect is suppressed due to the transverse localization of light in lattices with high disorder. In the diagonal case, the lattice topology does not play a role and the transmitted field always acquires random components in both quadratures, hence the phase distribution is uniform in the steady state.

View Synthesis is the challenging problem of predicting a new view or pose of an object given an exemplar view or set of views. This thesis presents a novel approach for the problem of view synthesis. The proposed method uses global features rather than local geometry to achieve an effect similar to that of the well known view morphing method . While previous approaches to the view synthesis problem have shown impressive results, they are highly dependent on being able to solve for epipolar geometry and therefore have a very precise correspondence between reference images. In cases where this is not possible such as noisy data, low contrast data, or long wave infrared data an alternative approach is desirable. Here two problems will be considered. The proposed view synthesis method will be used to synthesis new views given a set of reference views. Additionally the algorithm will be extended to synthesis new lighting conditions and thermal signatures. Finally the algorithm will be applied toward enhancing the ATR problem by creating additional training data to increase the likelihood of detection and classification.

This thesis has dual purpose of fulfilment of the requirements for the degree of Master of Science as well as a statistical report to Blair Research Laboratory, Salisbury, to whom I am indebted for allowing me to use the data they had collected. I am grateful to the director, Dr. V. de V. Clarke, for his help and co-operation. I thank all the members of the Department of Mathematical Statistics, University of Cape Town, for being most willing to help me, and in icu for the guidance given by my supervisor Professor C.G. Troskie as well as the help given by Associate Professor .H. Money. I a grateful to Dr. W. Hatchuel for arranging, and to Pfizer Laboratories (Pty) Ltd, for supplying I financial assistance which together with my bursary from the C. S. I. R. made it possible to complete this thesis. Lastly, but by no means least, I am indebted to Mrs. M.I. Cousins for so skilfully typing this thesis despite all the tables it contains.