Hierarchical adaptive polynomial chaos expansions

Polynomial chaos expansions (PCE) are widely used in the framework of uncertainty quantification. However, when dealing with high dimensional complex problems, challenging issues need to be faced. For instance, high-order polynomials may be required, which leads to a large polynomial basis whereas usually only a few of the basis functions are in fact significant. Taking into account the sparse structure of the model, advanced techniques such as sparse PCE (SPCE), have been recently proposed to alleviate the computational issue. In this paper, we propose a novel approach to SPCE, which allows one to exploit the model's hierarchical structure. The proposed approach is based on the adaptive enrichment of the polynomial basis using the so-called principle of heredity. As a result, one can reduce the computational burden related to a large pre-defined candidate set while obtaining higher accuracy with the same computational budget

Structured variable selection and estimation

In linear regression problems with related predictors, it is desirable to do variable selection and estimation by maintaining the hierarchical or structural relationships among predictors. In this paper we propose non-negative garrote methods that can naturally incorporate such relationships defined through effect heredity principles or marginality principles. We show that the methods are very easy to compute and enjoy nice theoretical properties. We also show that the methods can be easily extended to deal with more general regression problems such as generalized linear models. Simulations and real examples are used to illustrate the merits of the proposed methods.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS254 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

The moderating effects of governance on the relationship between investment opportunites, leverage and ownership indentity with firm performance in the UAE

This study evaluates corporate governance practices of listed firms in the UAE and examines the hypothesized influence of investment opportunities, leverage, foreign and institutional ownership on firm performance. Corporate governance strength is also investigated as a moderator between investment opportunities, leverage, foreign, institutional ownership and firm performance. The moderating impact of corporate governance strength is also examined during the global financial crisis. After constructing an index to measure corporate governance strength, the fixed effects regression in panel data was used to analyze the data. The data included 101 firms with a total of 501 firm-year observations that spanned the period 2008 to 2012, of all the firms listed on the Abu Dhabi Stock Exchange and the Dubai Financial Market. The results show a significant influence of investment opportunities, leverage and institutional ownership on firm performance represented by Return on Assets (ROA) and Refined Economic Value Added (REVA). However, the results find no influence of foreign ownership on ROA, and a negative influence on REVA. The governance index shows a dramatic improvement in the corporate governance practices over time. In addition, corporate governance strength is found to significantly moderate the relationship between investment opportunities, leverage, foreign and institutional ownership with ROA, but only moderates the relationship between leverage and REVA. During the crisis, corporate governance strength appears to play a more efficient moderating role. The findings of this study provide some insights to the regulators and other related parties about the status of corporate governance practices in the UAE and show that good corporate governance is indirectly able to improve the performance of firms during different time periods

Recent Developments in Nonregular Fractional Factorial Designs

Nonregular fractional factorial designs such as Plackett-Burman designs and other orthogonal arrays are widely used in various screening experiments for their run size economy and flexibility. The traditional analysis focuses on main effects only. Hamada and Wu (1992) went beyond the traditional approach and proposed an analysis strategy to demonstrate that some interactions could be entertained and estimated beyond a few significant main effects. Their groundbreaking work stimulated much of the recent developments in design criterion creation, construction and analysis of nonregular designs. This paper reviews important developments in optimality criteria and comparison, including projection properties, generalized resolution, various generalized minimum aberration criteria, optimality results, construction methods and analysis strategies for nonregular designs.Comment: Submitted to the Statistics Surveys (http://www.i-journals.org/ss/) by the Institute of Mathematical Statistics (http://www.imstat.org

A lasso for hierarchical interactions

We add a set of convex constraints to the lasso to produce sparse interaction models that honor the hierarchy restriction that an interaction only be included in a model if one or both variables are marginally important. We give a precise characterization of the effect of this hierarchy constraint, prove that hierarchy holds with probability one and derive an unbiased estimate for the degrees of freedom of our estimator. A bound on this estimate reveals the amount of fitting "saved" by the hierarchy constraint. We distinguish between parameter sparsity - the number of nonzero coefficients - and practical sparsity - the number of raw variables one must measure to make a new prediction. Hierarchy focuses on the latter, which is more closely tied to important data collection concerns such as cost, time and effort. We develop an algorithm, available in the R package hierNet, and perform an empirical study of our method.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1096 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Least angle and ℓ1\ell_1 penalized regression: A review

Least Angle Regression is a promising technique for variable selection applications, offering a nice alternative to stepwise regression. It provides an explanation for the similar behavior of LASSO ($\ell_1$-penalized regression) and forward stagewise regression, and provides a fast implementation of both. The idea has caught on rapidly, and sparked a great deal of research interest. In this paper, we give an overview of Least Angle Regression and the current state of related research.Comment: Published in at http://dx.doi.org/10.1214/08-SS035 the Statistics Surveys (http://www.i-journals.org/ss/) by the Institute of Mathematical Statistics (http://www.imstat.org