Algorithms for nonnegative matrix factorization with the beta-divergence

This paper describes algorithms for nonnegative matrix factorization (NMF) with the beta-divergence (beta-NMF). The beta-divergence is a family of cost functions parametrized by a single shape parameter beta that takes the Euclidean distance, the Kullback-Leibler divergence and the Itakura-Saito divergence as special cases (beta = 2,1,0, respectively). The proposed algorithms are based on a surrogate auxiliary function (a local majorization of the criterion function). We first describe a majorization-minimization (MM) algorithm that leads to multiplicative updates, which differ from standard heuristic multiplicative updates by a beta-dependent power exponent. The monotonicity of the heuristic algorithm can however be proven for beta in (0,1) using the proposed auxiliary function. Then we introduce the concept of majorization-equalization (ME) algorithm which produces updates that move along constant level sets of the auxiliary function and lead to larger steps than MM. Simulations on synthetic and real data illustrate the faster convergence of the ME approach. The paper also describes how the proposed algorithms can be adapted to two common variants of NMF : penalized NMF (i.e., when a penalty function of the factors is added to the criterion function) and convex-NMF (when the dictionary is assumed to belong to a known subspace).Comment: \`a para\^itre dans Neural Computatio

Bayesian interpretation of periodograms

The usual nonparametric approach to spectral analysis is revisited within the regularization framework. Both usual and windowed periodograms are obtained as the squared modulus of the minimizer of regularized least squares criteria. Then, particular attention is paid to their interpretation within the Bayesian statistical framework. Finally, the question of unsupervised hyperparameter and window selection is addressed. It is shown that maximum likelihood solution is both formally achievable and practically useful

Risk aversion and Uncertainty in European Sovereign Bond Markets

Risk aversion and uncertainty are often both at play in market price determination, but it is empirically challenging to disentangle one from the other. In this paper we set up a theoretical model particularly suited for opaque over-the-counter markets that is shown to be empirically tractable. Based on high frequency data, we thus propose an evaluation of risk aversion and uncertainty inherent to the government bond markets in the euro area between 2007 and 2011. We particularly examine the impact of the European Central Bank Securities Markets Programme [SMP] implemented in May 2010 and re- activated in August 2011 to ease the pressure on the European sovereign bond markets. We show how this programme has killed market uncertainty but raised risk aversion for all countries except Greece in a risk-pooling mechanism: this can therefore weaken the impact of market interventions over the long-term.MES, systemic risk, tail correlation, balance sheet ratios, panel.

Stock exchanges industry consolidation and shock transmission.

Stock exchange industry consolidation is at work since many years and has recently accelerated through competition for order flows, agreements and mergers. However, consolidation may not mean that all shocks are transmitted to every place. Therefore, following Forbes and Rigobon (2002) we distinguish convergence (as interdependence) from contagion. Long run interdependence is analyzed through overlapping rolling cointegration and shocks on correlations through multivariate GARCH models. The models are estimated on daily data from January 1 1994 and April 30 2006. We consider the DAX30, the CAC40, the FTSE100 and the NYSE indexes. We identify stock exchanges convergence between European places. However we mainly witness a leading role of the US market even after the euro area creation. Finally, dynamic correlations still exert local shocks while others are effectively transmitted.Equity market integration ; Cointegration ; Multivariate GARCH models.

Probability of informed trading on the euro overnight market rate: an update

In this paper the probability of informed trading (PIN) model developed by Easley and O’Hara (1992) is applied to analyze the role and impact of heterogeneities in euro overnight unsecured market. The empirical assessment of the functioning of this market is based on the PIN which measures the ability of traders to interpret signals on the expected evolution of the overnight rate. Results show that between 2000 and 2004 a heterogeneous learning process of market mechanisms within participants could be observed, whereas such asymmetries have been sharply decreasing since 2005. This is reviewed against some significant events that occurred in the euro money market, such as the reform of the Eurosystem’s operational framework in March 2004 and the recent financial market turmoil, which has represented a break in the steady decline of asymmetries as evidence suggest. JEL Classification: G14, E52Microstructure, money markets, PIN model

Probability of informed trading: an empirical application to the euro overnight market rate.

This paper presents a microstructure model for the unsecured overnight euro money market, similar to that developed for stock markets by Easley and O'Hara (1992). More specifically, this paper studies the role of heterogeneity in the population of banks participating on this market, and the influence of the institutional framework and market organizational aspects of the overnight deposit market. A first empirical assessment of the functioning of this market is based on the probability of informed trade which measures the ability of traders (banks) to interpret signals on the expected evolution of the overnight rate. This indicator is estimated on real-time data publicly available to market participants. Results show that between 2000 and 2004 a heterogeneous learning process of market mechanisms within participants could be observed. From 2005 onwards, however, heterogeneity in the learning process sharply decreased. Moreover, the empirical evidence show that the March 2004 changes in Eurosystem's operational framework have modified the informational patterns of order flow in the euro area money market: informed trades became even more predominant between the last main refinancing operation and the end of the reserves maintenance period than they were before March 2004.Euro overnight market ; PIN models ; Microstructure, Monetary policy.

Efficient Gaussian Sampling for Solving Large-Scale Inverse Problems using MCMC Methods

The resolution of many large-scale inverse problems using MCMC methods requires a step of drawing samples from a high dimensional Gaussian distribution. While direct Gaussian sampling techniques, such as those based on Cholesky factorization, induce an excessive numerical complexity and memory requirement, sequential coordinate sampling methods present a low rate of convergence. Based on the reversible jump Markov chain framework, this paper proposes an efficient Gaussian sampling algorithm having a reduced computation cost and memory usage. The main feature of the algorithm is to perform an approximate resolution of a linear system with a truncation level adjusted using a self-tuning adaptive scheme allowing to achieve the minimal computation cost. The connection between this algorithm and some existing strategies is discussed and its efficiency is illustrated on a linear inverse problem of image resolution enhancement.Comment: 20 pages, 10 figures, under review for journal publicatio

Long term vs. short term comovements in stock markets: the use of Markov-switching multifractal models.

Empirical techniques to assess market comovements are numerous from cointegration to dynamic conditional correlations. This paper uses the fractal properties of asset returns and presents estimations of Markov switching multifractal models [as MSM] to give new insights about short and long run dependencies in stock returns. The main advantage of the model is to allow for the derivation of several indicators of comovements on heterogenous lasting horizons. Empirical applications are performed for four stock indices (CAC DAX FTSE NYSE) at daily frequency between 1996 and 2008.Multivariate volatility models ; Markov switching multifractal model transmission, comovements.

Exact Recovery Conditions for Sparse Representations with Partial Support Information

We address the exact recovery of a k-sparse vector in the noiseless setting when some partial information on the support is available. This partial information takes the form of either a subset of the true support or an approximate subset including wrong atoms as well. We derive a new sufficient and worst-case necessary (in some sense) condition for the success of some procedures based on lp-relaxation, Orthogonal Matching Pursuit (OMP) and Orthogonal Least Squares (OLS). Our result is based on the coherence "mu" of the dictionary and relaxes the well-known condition mu<1/(2k-1) ensuring the recovery of any k-sparse vector in the non-informed setup. It reads mu<1/(2k-g+b-1) when the informed support is composed of g good atoms and b wrong atoms. We emphasize that our condition is complementary to some restricted-isometry based conditions by showing that none of them implies the other. Because this mutual coherence condition is common to all procedures, we carry out a finer analysis based on the Null Space Property (NSP) and the Exact Recovery Condition (ERC). Connections are established regarding the characterization of lp-relaxation procedures and OMP in the informed setup. First, we emphasize that the truncated NSP enjoys an ordering property when p is decreased. Second, the partial ERC for OMP (ERC-OMP) implies in turn the truncated NSP for the informed l1 problem, and the truncated NSP for p<1.Comment: arXiv admin note: substantial text overlap with arXiv:1211.728