A Direct Estimation Approach to Sparse Linear Discriminant Analysis

This paper considers sparse linear discriminant analysis of high-dimensional data. In contrast to the existing methods which are based on separate estimation of the precision matrix \O and the difference \de of the mean vectors, we introduce a simple and effective classifier by estimating the product \O\de directly through constrained $\ell_1$ minimization. The estimator can be implemented efficiently using linear programming and the resulting classifier is called the linear programming discriminant (LPD) rule. The LPD rule is shown to have desirable theoretical and numerical properties. It exploits the approximate sparsity of \O\de and as a consequence allows cases where it can still perform well even when \O and/or \de cannot be estimated consistently. Asymptotic properties of the LPD rule are investigated and consistency and rate of convergence results are given. The LPD classifier has superior finite sample performance and significant computational advantages over the existing methods that require separate estimation of \O and \de. The LPD rule is also applied to analyze real datasets from lung cancer and leukemia studies. The classifier performs favorably in comparison to existing methods.Comment: 39 pages.To appear in Journal of the American Statistical Associatio

Adaptive Thresholding for Sparse Covariance Matrix Estimation

In this paper we consider estimation of sparse covariance matrices and propose a thresholding procedure which is adaptive to the variability of individual entries. The estimators are fully data driven and enjoy excellent performance both theoretically and numerically. It is shown that the estimators adaptively achieve the optimal rate of convergence over a large class of sparse covariance matrices under the spectral norm. In contrast, the commonly used universal thresholding estimators are shown to be sub-optimal over the same parameter spaces. Support recovery is also discussed. The adaptive thresholding estimators are easy to implement. Numerical performance of the estimators is studied using both simulated and real data. Simulation results show that the adaptive thresholding estimators uniformly outperform the universal thresholding estimators. The method is also illustrated in an analysis on a dataset from a small round blue-cell tumors microarray experiment. A supplement to this paper which contains additional technical proofs is available online.Comment: To appear in Journal of the American Statistical Associatio

A Constrained L1 Minimization Approach to Sparse Precision Matrix Estimation

A constrained L1 minimization method is proposed for estimating a sparse inverse covariance matrix based on a sample of $n$ iid $p$-variate random variables. The resulting estimator is shown to enjoy a number of desirable properties. In particular, it is shown that the rate of convergence between the estimator and the true $s$-sparse precision matrix under the spectral norm is $s\sqrt{\log p/n}$ when the population distribution has either exponential-type tails or polynomial-type tails. Convergence rates under the elementwise $L_{\infty}$ norm and Frobenius norm are also presented. In addition, graphical model selection is considered. The procedure is easily implementable by linear programming. Numerical performance of the estimator is investigated using both simulated and real data. In particular, the procedure is applied to analyze a breast cancer dataset. The procedure performs favorably in comparison to existing methods.Comment: To appear in Journal of the American Statistical Associatio

Modeling Continuous IED Supply Chains

Improvised Explosive Devices (IEDs) continue to be a main weapon used by terrorists against coalition forces overseas. This thesis intends to provide methods that can give coalition forces a new perspective on fighting IEDs. We begin by first developing a model of the supply chain terrorists use to develop, emplace and detonate IEDs. Our model contains four states in which IEDs can exist in: construction (C), emplaced (E), detonated (D) and found by coalition forces (F). We also have rate parameters representing the flow rates of IEDs. Over a given period of time, coalition forces can collect data on the number of IEDs that they find as well as the number of IEDs that detonate. From here, we apply a least squares method to obtain the parameter set for our supply chain model that best fits the collected IED data. Minimizing our least squares equation allows us to estimate where the IEDs are located as well as how fast they are being moved throughout the entire supply chain. Using this, we can judge the impact of our past efforts in stopping IEDs and determine how to best move forward

A real-time computer vision library for heterogeneous processing environments

Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 69-70).With a variety of processing technologies available today, using a combination of different technologies often provides the best performance for a particular task. However, unifying multiple processors with different instruction sets can be a very ad hoc and difficult process. The Open Component Portability Infrastructure (OpenCPI) provides a platform that simplifies programming heterogeneous processing applications requiring a mix of processing technologies. These include central processing units (CPU), graphics processing units (GPU), field-programmable gate arrays (FPGA), general-purpose processors (GPP), digital signal processors (DSP), and high-speed switch fabrics. This thesis presents the design and implementation of a computer vision library in the OpenCPI framework, largely based on Open Source Computer Vision (OpenCV), a widely used library of optimized software components for real-time computer vision. The OpenCPI-OpenCV library consists of a collection of resource-constrained C language (RCC) workers, along with applications demonstrating how these workers can be combined to achieve the same functionality as various OpenCV library functions. Compared with applications relying solely on OpenCV, analogous OpenCPI applications can be constructed from many workers, often resulting in greater parallelization if run on multi-core platforms. Future OpenCPI computer vision applications will be able to utilize these existing RCC workers, and a subset of these workers can potentially be replaced with alternative implementations, e.g. on GPUs or FPGAs.by Tony J. Liu.M.Eng

Develop Guidelines for Pavement Preservation Treatments and for Building a Pavement Preservation Program Platform for Alaska

INE/AUTC 12.0