Iterative Thresholding for Sparse Approximations

Sparse signal expansions represent or approximate a signal using a small number of elements from a large collection of elementary waveforms. Finding the optimal sparse expansion is known to be NP hard in general and non-optimal strategies such as Matching Pursuit, Orthogonal Matching Pursuit, Basis Pursuit and Basis Pursuit De-noising are often called upon. These methods show good performance in practical situations, however, they do not operate on the ? ₀ penalised cost functions that are often at the heart of the problem. In this paper we study two iterative algorithms that are minimising the cost functions of interest. Furthermore, each iteration of these strategies has computational complexity similar to a Matching Pursuit iteration, making the methods applicable to many real world problems. However, the optimisation problem is non-convex and the strategies are only guaranteed to find local solutions, so good initialisation becomes paramount. We here study two approaches. The first approach uses the proposed algorithms to refine the solutions found with other methods, replacing the typically used conjugate gradient solver. The second strategy adapts the algorithms and we show on one example that this adaptation can be used to achieve results that lie between those obtained with Matching Pursuit and those found with Orthogonal Matching Pursuit, while retaining the computational complexity of the Matching Pursuit algorithm.     

Asymmetric response and interaction of U.S. and local news in financial markets

This paper examines the extent to which financial returns on market indices exhibit mean and volatility asymmetries, as a response to past information from both the U.S. market and the local market itself. In particular, we wish to assess the asymmetric effect of a combination of local and U.S. market news on volatility. To the best of the authors knowledge, this joint effect has not been considered previously. We propose a double threshold non‐linear heteroscedastic model, combined with a GJR‐GARCH effect in the conditional volatility equation, to capture jointly both mean and volatility asymmetric behaviours and the interactive effect of U.S. and local market news. In an application to five major international market indices, clear evidence of threshold non‐linearity is discovered, supporting the hypothesis of an uneven mean‐reverting pattern and volatility asymmetry, both in reaction to U.S. market news and news from the local market itself. Significant, but somewhat different, interactive effects between local and U.S. news are observed in all markets. An asymmetric pattern in the exogenous relationship between the local market and the U.S. market is also found. Copyright © 2005 John Wiley & Sons, Ltd.     

Pragmatic inquiry

Inquiry into how people might act to solve a problem, which involves powerful communities of stakeholders, cannot be undertaken using only the principles of scientific inquiry. Science is useful for inquiry into physical activity but less so for inquiry into designing appropriate social systems such as a strategic plan or the appropriate solution to a social conflict. This paper argues that complex problem solving needs to draw on a set of pragmatic principles, which it explains. Pragmatism is an epistemology intended to deal with alternative interpretations of the same physical reality. It is increasingly coming back into the attention of soft operational researchers. After briefly explaining which thread of pragmatism is being drawn upon, this paper identifies five interrelated principles for undertaking a pragmatic inquiry. To explain these interrelations, the principles are designed into a linked system of pragmatic inquiry which is applied in a small case study.     

Ideals containing the squares of the variables

We study the Betti numbers of graded ideals containing the squares of the variables, in a polynomial ring. We prove the lex-plus-powers conjecture for such ideals.     

Announcing the Burtin graphic contest winners

The graphics contest announced in Volume 23, Issue 4, of CHANCE asked readers to create a display of a 1951 data set on antibiotic effectiveness. The three winners of the contest are Mark Nicolich, Brian Schmotzer, and Dibyojyoti Haldar. The winners’ graphs are different, yet communicate the same information—each compares the antibiotics and bacteria and succinctly summarizes the influence of the gram staining factor.