Learning Discriminative Bayesian Networks from High-dimensional Continuous Neuroimaging Data

Due to its causal semantics, Bayesian networks (BN) have been widely employed to discover the underlying data relationship in exploratory studies, such as brain research. Despite its success in modeling the probability distribution of variables, BN is naturally a generative model, which is not necessarily discriminative. This may cause the ignorance of subtle but critical network changes that are of investigation values across populations. In this paper, we propose to improve the discriminative power of BN models for continuous variables from two different perspectives. This brings two general discriminative learning frameworks for Gaussian Bayesian networks (GBN). In the first framework, we employ Fisher kernel to bridge the generative models of GBN and the discriminative classifiers of SVMs, and convert the GBN parameter learning to Fisher kernel learning via minimizing a generalization error bound of SVMs. In the second framework, we employ the max-margin criterion and build it directly upon GBN models to explicitly optimize the classification performance of the GBNs. The advantages and disadvantages of the two frameworks are discussed and experimentally compared. Both of them demonstrate strong power in learning discriminative parameters of GBNs for neuroimaging based brain network analysis, as well as maintaining reasonable representation capacity. The contributions of this paper also include a new Directed Acyclic Graph (DAG) constraint with theoretical guarantee to ensure the graph validity of GBN.Comment: 16 pages and 5 figures for the article (excluding appendix

The FKPP wave front as a sensor of perturbed diffusion in concentrated systems

The sensitivity to perturbations of the Fisher and Kolmogorov, Petrovskii, Piskunov front is used to find a quantity revealing perturbations of diffusion in a concentrated solution of two chemical species with different diffusivities. The deterministic dynamics includes cross-diffusion terms due to the deviation from the dilution limit. The behaviors of the front speed, the shift between the concentration profiles of the two species, and the width of the reactive zone are investigated, both analytically and numerically. The shift between the two profiles turns out to be a well-adapted criterion presenting noticeable variations with the deviation from the dilution limit in a wide range of parameter values.Comment: 22 pages, 5 figure

Analysis of Models for Epidemiologic and Survival Data

Mortality statistics are useful tools for public-health statisticians, actuaries and policy makers to study health status of populations in communities and to make plans in health care systems. Several statistical models and methods of parameter estimation have been proposed. In this thesis, we review some benchmark mortality models and propose three alternative statistical models for both epidemiologic data and survival data. For epidemiologic data, we propose two statistical models, a Smoothed Segmented Lee-Carter model and a Smoothed Segmented Poisson Log-bilinear model. The models are modifications of the Lee-Carter (1992) model which combine an age segmented Lee-Carter parameterization with spline smoothed period effects within each age segment. With different period effects across age groups, the two models are fitted by maximizing respectively a penalized least squares criterion and a penalized Poisson likelihood. The new methods are applied to the 1971-2006 public-use mortality data sets released by the National Center for Health Statistics (NCHS). Mortality rates for three leading causes of death, heart diseases, cancer and accidents, are studied. For survival data, we propose a phase type model having features of mixtures, multiple stages or hits and a trapping state. Two parameter estimation techniques studied are a direct numerical method and an EM algorithm. Since phase type model parameters are known to be difficult to estimate, we study in detail the performance of our parameter estimation techniques by reference to the Fisher Information matrix. An alternative way to produce a Fisher Information matrix for an EM parameter estimation is also provided. The proposed model and the best available parameter estimation techniques are applied to a large SEER 1992-2002 breast cancer dataset

Finite sample analysis of profile M-estimators

In dieser Arbeit wird ein neuer Ansatz für die Analyse von Profile Maximierungsschätzern präsentiert. Es werden die Ergebnisse von Spokoiny (2011) verfeinert und angepasst für die Schätzung von Komponenten von endlich dimensionalen Parametern mittels der Maximierung eines Kriteriumfunktionals. Dabei werden Versionen des Wilks Phänomens und der Fisher-Erweiterung für endliche Stichproben hergeleitet und die dafür kritische Relation der Parameterdimension zum Stichprobenumfang gekennzeichnet für den Fall von identisch unabhängig verteilten Beobachtungen und eines hinreichend glatten Funktionals. Die Ergebnisse werden ausgeweitet für die Behandlung von Parametern in unendlich dimensionalen Hilberträumen. Dabei wir die Sieve-Methode von Grenander (1981) verwendet. Der Sieve-Bias wird durch übliche Regularitätsannahmen an den Parameter und das Funktional kontrolliert. Es wird jedoch keine Basis benötigt, die orthogonal in dem vom Model induzierten Skalarprodukt ist. Weitere Hauptresultate sind zwei Konvergenzaussagen für die alternierende Maximisierungsprozedur zur approximation des Profile-Schätzers. Alle Resultate werden anhand der Analyse der Projection Pursuit Prozedur von Friendman (1981) veranschaulicht. Die Verwendung von Daubechies-Wavelets erlaubt es unter natürlichen und üblichen Annahmen alle theoretischen Resultate der Arbeit anzuwenden.This thesis presents a new approach to analyze profile M-Estimators for finite samples. The results of Spokoiny (2011) are refined and adapted to the estimation of components of a finite dimensional parameter using the maximization of a criterion functional. A finite sample versions of the Wilks phenomenon and Fisher expansion are obtained and the critical ratio of parameter dimension to sample size is derived in the setting of i.i.d. samples and a smooth criterion functional. The results are extended to parameters in infinite dimensional Hilbert spaces using the sieve approach of Grenander (1981). The sieve bias is controlled via common regularity assumptions on the parameter and functional. But our results do not rely on an orthogonal basis in the inner product induced by the model. Furthermore the thesis presents two convergence results for the alternating maximization procedure. All results are exemplified in an application to the Projection Pursuit Procedure of Friendman (1981). Under a set of natural and common assumptions all theoretical results can be applied using Daubechies wavelets

Entanglement detection from channel parameter estimation problem

We derive a general criterion to detect entangled states in multi-partite systems based on the symmetric logarithmic derivative quantum Fisher information. This criterion is a direct consequence of the fact that separable states do not improve the accuracy upon estimating one-parameter family of quantum channels. Our result is a generalization of the previously known criterion for one-parameter unitary channel to any one-parameter quantum channel. We discuss several examples to illustrate our criterion. The proposed criterion is extended to the case of open quantum systems and we briefly discuss how to detect entangled states in the presence of decoherence.Comment: 10 page