DRUM Community: Mathematics
http://hdl.handle.net/1903/2261
2015-03-02T13:46:59ZFUNCTIONAL PRINCIPAL COMPONENT ANALYSIS WITH APPLICATION TO VIEWERSHIP OF MOTION PICTURES
http://hdl.handle.net/1903/16273
Title: FUNCTIONAL PRINCIPAL COMPONENT ANALYSIS WITH APPLICATION TO VIEWERSHIP OF MOTION PICTURES
Authors: Tian, Yue
Abstract: Principal Component Analysis (PCA) is one widely used data processing technique in application, especially for dimensionality reduction. Functional Principal Component Analysis (fPCA) is a generalization of ordinary PCA, which focuses on a sample of functional observations and projects the original functional curves to a new space of orthogonal dimensions to capture the primary features of original functional curves. While, fPCA suffers from two potential error sources. One error source is originated from truncation when we approximate the functional subject's expansion; The other stems from estimation when we estimate the principal components from the sample. We first introduce a generalized functional linear regression model and propose it in the Quasi-likelihood setting. Asymptotic inference of the proposed functional regression model is developed.
We also utilize the proposed model to help marketing operational decision process by analyzing viewership of motion pictures. We start with discussing customer reviews effect on movie box office sales. We use the functional regression model with function interactions to measure the effect of Word-of-Mouth on movie box office sales. One main challenge of modeling with functional interactions is the interpretation of model estimate results. We demonstrate one method to help us get important insights from model results by plotting and controlling a re-labbeld 3-D plot.
Apart from movie performance in theater, we also employ functional regression model to predict movie pre-release demand in Video-on-Demand (VOD) channel. As its growing popularity, VOD market attracts much attention in marketing research. We analyze the prediction accuracy of our proposed functional regression model with spatial components and find that our proposed model gives us the best predictive accuracy.
In summary, the dissertation develops asymptotic properties of a generalized functional linear regression model, and applies the proposed model in analyzing viewership of motion picture both in theater and Video-on-Demand channels. The proposed model not only advances our understanding of motion picture demand, but also helps optimize business decision making process.2014-01-01T00:00:00ZA Fokker-Planck Study Motivated by a Problem in Fluid-Particle Interactions
http://hdl.handle.net/1903/16260
Title: A Fokker-Planck Study Motivated by a Problem in Fluid-Particle Interactions
Authors: Markou, Ioannis
Abstract: This dissertation is a study of problems that relate to a Fokker-Planck (Klein-Kramers)
equation with hypoelliptic structure. The equation describes the statistics of motion
of an ensemble of particles in a viscous fluid that follows the Stokes ’ equations of
fluid motion. The significance in this problem is that it relates to a variety of phenomena
besides its obvious connection to the study of macromolecular chains that are composed by
particle “ units ” in creeping flows. Such phenomena range from Kramers escape probability
(for a particle trapped in a potential well), to stellar dynamics. The problem can also be
seen as a simplified version of the Vlasov-Poisson-Fokker-Planck system that mainly describes
electrostatic models in plasma physics and gravitational forces between galaxies.
Well-posedeness of the equation has been studied by many authors, including the
case of irregular coefficients (Lions-Le Bris). The study of Sobolev regularity
is interesting in its own right and can be performed with fairly elementary tools (He\'rau,Villani,…).
We are interested here with short time estimates and with how smoothing proceeds in time.
Different types of Lyapunov functionals can be constructed depending on the
type of initial data to show regularization. Of particular interest is a recent
technique developed by C.Villani that builds upon a system of differential inequalities
and is being implemented here for the slightly more involved case of non constant friction.
The question of asymptotic convergence to a stationary state is also discussed, with
techniques that are similar to certain extend to the ones used in regularization but which in general
involve more computations.
Finally, we examine the hydrodynamic (zero mass) limit of the parametrized version of the
Fokker-Planck equation. We discuss two different approaches of hydrodynamic convergence.
The first uses weak compactness principles
of extracting subsequences that are shown to converge to a solution of the limit problem, and
works with initial data in weighted L^{2} setting. The second is based
on the study of relative entropy, gives L^{1} convergence to a solution of the limit problem, and
uses entropic initial data.2014-01-01T00:00:00ZAsymptotic problems for stochastic processes and corresponding partial differential equations
http://hdl.handle.net/1903/16158
Title: Asymptotic problems for stochastic processes and corresponding partial differential equations
Authors: Tcheuko, Lucas Simplice
Abstract: We consider asymptotic problems for diffusion processes that rely on large deviations.
In Chapter 2, we study the long time behavior (at times of order exp(λ/&Alt 238; &Alt253;) of solutions to quasi-linear
parabolic equations with a small parameter &Alt238;&Alt253 at the diffusion term. The solution to a partial differential equation (PDE) can be expressed in terms of diffusion processes, whose coefficients, in turn, depend on the unknown solution. The notion of a hierarchy of cycles for diffusion processes was introduced by Freidlin and Wentzell and applied to the study of the corresponding linear equations.
In the quasi-linear case, it is not a single hierarchy that corresponds to an equation, but rather a family of hierarchies that depend on the time scale λ. We describe the evolution of the hierarchies with respect to λ in order to gain information on the limiting behavior of the solution of the PDE.
In chapter 3, we study the asymptotic behavior of diffusion processes, with a small diffusion term. This process is constrained to move within some bounded domain D with instantaneous reflection on hitting the boundary ∂D of D. Such processes have applications in asymptotic questions related to linear parabolic PDEs with the Neumann boundary condition. Similar problems were previously studied by Anderson and Orey. We expand on Anderson's and Orey's work by considering different equilbra in the interior of D, similarly to the problem studied in chapter 2. However, some equilibra also appear on the boundary ∂D. We use the results of Anderson and Orey together with the work of Freidlin and Wentzell to investigate the invariant measure of the process and describe the transitions of the process between the attractors. The knowledge of the invariant measure of the process and the transition rates (in the logarithmic scale) allow us to study the long term behavior of the solution to the corresponding linear parabolic PDE as the diffusion parameter goes to zero.2014-01-01T00:00:00ZOptimal Space-Time-Frequency Design of Microphone Networks
http://hdl.handle.net/1903/16152
Title: Optimal Space-Time-Frequency Design of Microphone Networks
Authors: Lai, Yenming
Abstract: Consider a sensing system using a large number of N microphones placed in multiple dimensions to monitor a acoustic field. Using all the microphones at once is impractical because of the amount data generated. Instead, we choose a subset of D microphones to be active. Specifically, we wish to find the D set of microphones that minimizes the largest interference gain at multiple frequencies while monitoring a target of interest. A direct, combinatorial approach - testing all N choose D subsets of microphones - is impractical because of problem size. Instead, we use a convex optimization technique that induces sparsity through a l1-penalty to determine which subset of microphones to use. Our work investigates not only the optimal placement (space) of microphones but also how to process the output of each microphone (time/frequency). We explore this problem for both single and multi-frequency sources, optimizing both microphone weights and positions simultaneously. In addition, we explore this problem for random sources where the output of each of the N microphones is processed by an individual multirate filterbank. The N processed filterbank outputs are then combined to form one final signal. In this case, we fix all the analysis filters and optimize over all the synthesis filters. We show how to convert the continuous frequency problem to a discrete frequency approximation that is computationally tractable. In this random source/multirate filterbank case, we once again optimize over space-time-frequency simultaneously.2014-01-01T00:00:00Z