Stability

Reproducibility is imperative for any scientific discovery. More often than not, modern scientific findings rely on statistical analysis of high-dimensional data. At a minimum, reproducibility manifests itself in stability of statistical results relative to "reasonable" perturbations to data and to the model used. Jacknife, bootstrap, and cross-validation are based on perturbations to data, while robust statistics methods deal with perturbations to models. In this article, a case is made for the importance of stability in statistics. Firstly, we motivate the necessity of stability for interpretable and reliable encoding models from brain fMRI signals. Secondly, we find strong evidence in the literature to demonstrate the central role of stability in statistical inference, such as sensitivity analysis and effect detection. Thirdly, a smoothing parameter selector based on estimation stability (ES), ES-CV, is proposed for Lasso, in order to bring stability to bear on cross-validation (CV). ES-CV is then utilized in the encoding models to reduce the number of predictors by 60% with almost no loss (1.3%) of prediction performance across over 2,000 voxels. Last, a novel "stability" argument is seen to drive new results that shed light on the intriguing interactions between sample to sample variability and heavier tail error distribution (e.g., double-exponential) in high-dimensional regression models with $p$ predictors and $n$ independent samples. In particular, when $p/n\rightarrow\kappa\in(0.3,1)$ and the error distribution is double-exponential, the Ordinary Least Squares (OLS) is a better estimator than the Least Absolute Deviation (LAD) estimator.Comment: Published in at http://dx.doi.org/10.3150/13-BEJSP14 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Frame Theory for Signal Processing in Psychoacoustics

This review chapter aims to strengthen the link between frame theory and signal processing tasks in psychoacoustics. On the one side, the basic concepts of frame theory are presented and some proofs are provided to explain those concepts in some detail. The goal is to reveal to hearing scientists how this mathematical theory could be relevant for their research. In particular, we focus on frame theory in a filter bank approach, which is probably the most relevant view-point for audio signal processing. On the other side, basic psychoacoustic concepts are presented to stimulate mathematicians to apply their knowledge in this field

Time-frequency analysis of the restricted three-body problem: transport and resonance transitions

A method of time-frequency analysis based on wavelets is applied to the problem of transport between different regions of the solar system, using the model of the circular restricted three-body problem in both the planar and the spatial versions of the problem.. The method is based on the extraction of instantaneous frequencies from the wavelet transform of numerical solutions. Time-varying frequencies provide a good diagnostic tool to discern chaotic trajectories from regular ones, and we can identify resonance islands that greatly affect the dynamics. Good accuracy in the calculation of time-varying frequencies allows us to determine resonance trappings of chaotic trajectories and resonance transitions. We show the relation between resonance transitions and transport in different regions of the phase space

Frames for the solution of operator equations in Hilbert spaces with fixed dual pairing

For the solution of operator equations, Stevenson introduced a definition of frames, where a Hilbert space and its dual are {\em not} identified. This means that the Riesz isomorphism is not used as an identification, which, for example, does not make sense for the Sobolev spaces $H_0^1(\Omega)$ and $H^{-1}(\Omega)$. In this article, we are going to revisit the concept of Stevenson frames and introduce it for Banach spaces. This is equivalent to $\ell^2$-Banach frames. It is known that, if such a system exists, by defining a new inner product and using the Riesz isomorphism, the Banach space is isomorphic to a Hilbert space. In this article, we deal with the contrasting setting, where $\mathcal H$ and $\mathcal H'$ are not identified, and equivalent norms are distinguished, and show that in this setting the investigation of $\ell^2$-Banach frames make sense.Comment: 23 pages; accepted for publication in 'Numerical Functional Analysis and Optimization

Rapid evaluation of radial basis functions

Over the past decade, the radial basis function method has been shown to produce high quality solutions to the multivariate scattered data interpolation problem. However, this method has been associated with very high computational cost, as compared to alternative methods such as finite element or multivariate spline interpolation. For example. the direct evaluation at M locations of a radial basis function interpolant with N centres requires O(M N) floating-point operations. In this paper we introduce a fast evaluation method based on the Fast Gauss Transform and suitable quadrature rules. This method has been applied to the Hardy multiquadric, the inverse multiquadric and the thin-plate spline to reduce the computational complexity of the interpolant evaluation to O(M + N) floating point operations. By using certain localisation properties of conditionally negative definite functions this method has several performance advantages against traditional hierarchical rapid summation methods which we discuss in detail