Possibilistic signal processing: How to handle noise?

We propose a novel approach for noise quantifier at each location of a signal. This method is based on replacing the conventional kernel-based approach extensively used in signal processing by an approach involving another kind of kernel: a possibility distribution. Such an approach leads to interval-valued resulting methods instead of point-valued ones. We propose a theoretical justification to this approach and we show, on real and artificial data sets, that the length of the obtained interval and the local noise level are highly correlated. This method is non-parametric and has an advantage over other methods since no assumption about the nature of the noise has to be made, except its local ergodicity. Besides, the propagation of the noise in the involved signal processing method is direct and does not require any additional computation.     

Minimax detection of a signal in ℓp metrics

The problem of the minimax detection of a signal is considered in a Gaussian white noise of intensity 0. The set of the signals represents an ellipsoid in the _p-metric with lengths of semiaxes a_k ^–1 0 as k , from which a ball of radius 0 as 0 in the _p-metric has been removed. Asymptotically minimax tests have been constructed for the cases 1 < p and asymptotically sharp estimates of the minimax efficiency have been obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 184, pp. 152–168, 1990.     

Can CDS indexes signal future turmoils in the stock market? A Markov switching perspective

Single-name Credit Default Swaps (CDS) are considered the main providers of direct information related with a reference entity’s creditworthiness and, for this reason, they have often been the core of news on the current financial crisis. The academic research has focused mainly on the capacity of CDS in anticipating agencies’ official rating changes and—in this respect—on their superior signalling power, compared to bond and stock markets. The aim of this work is, instead, to investigate the ability of fluctuations in CDS indexes in anticipating the occurrence of stock market crises. Our goal is to show that CDS indexes may provide investors and institutions with early warning signals of financial distresses in the stock market. We make use of a Markov switching model with states characterized by increasing levels of volatility and compare the times in which the first switch in a high volatility state occurs, respectively, in CDS and stock market index quotes. The data set consists of daily closing quotes for 5 years CDS and stock market index prices, covering the time period from 2004 to 2010. In order to capture possible geographic differences in CDS index capacity of foreseeing stock market distresses, data referring to two different regions, Europe and United States, are analyzed.     

Chaotic analysis of pressure fluctuation signal in the gas–liquid–solid slurry column

The pressure signal of a slurry column is easily obtained by using a pressure sensor, and a chaotic analysis method is used to analyze these signals in order to indicate the flow pattern of the slurry column. The slopes of the correlation integral curve indicate the flow pattern of the slurry column in various operating conditions. The flow pattern is dispersed bubble regime when the superficial velocity is low and the correlation integral curve has two slopes. The flow pattern changes into transition regime with increase in the superficial velocity, the correlation integral curve has only one slope. In the case of the flow pattern becoming a slugging regime, there are several slopes to the correlation integral curve. So it is convenient to find out the flow pattern in the slurry column by solving the slopes of the correlation integral of the pressure signal. The maximum Lyapunov exponent represents the chaos in a slurry column with various solid holdups. The maximum Lyapunov exponent is nearly similar at different heights when the flow patterns are dispersed bubble regime and slugging regime, but the maximum Lyapunov exponent at the axial height is quite different when the flow pattern is transition regime.     

Determination of the relaxation and retardation spectra – a view from the up-to-date signal processing perspective

The problem of determination of relaxation and retardation spectra (RRS) is considered from the viewpoint of up-to-date signal processing. It is shown that the recovery of RRS represents the Mellin deconvolution problem, which transforms into the Fourier deconvolution problem for data on a logarithmic time or frequency scale, where it can also be treated as the inverse filtering problem. On this basis, discrete deconvolution (inverse) filters operating with geometrically sampled data are proposed to use as RRS estimators. Appropriate frequency responses and algorithms are derived for estimating RRS from eight different material functions. The noise amplification coefficient is suggested to use as a measure for quantifying the degree of ill-posedness and illconditioness of the RRS recovery problem and algorithms. A methodology is developed for designing RRS estimators with a desired noise amplification, producing maximum accurate spectra for available limited input data. Practical algorithms for determining RRS are proposed, and their performance is studied. The algorithms suggested are compared with the so-called moving-average formulae. It is demonstrated that the minimum frequency range for recovering the point estimate of a relaxation spectrum depends on the allowable noise amplification (the degree of ill-conditioness) and is in no way limited by 1.36 decades, as it is stated by the sampling localization theorem.     

Renormalized radial large-data solutions to the higher-dimensional Keller–Segel system with singular sensitivity and signal absorption

The chemotaxis system

$\{\begin{array}{c}{u}_t=\mathrm{\Delta }u?\mathrm{?}?\left(\frac{u}{v}\mathrm{?}v\right),\hfill \\ v_t=\mathrm{\Delta }v?uv,\hfill \end{array}(?)$

is considered under homogeneous Neumann boundary conditions in the ball $\mathrm{\Omega }=B_R(0)?{\mathbb{R}}^n$, where $R>0$ and $n\ge 2$.Despite its great relevance as a model for the spontaneous emergence of spatial structures in populations of primitive bacteria, since its introduction by Keller and Segel in 1971 this system has been lacking a satisfactory theory even at the level of the basic questions from the context of well-posedness; global existence results in the literature are restricted to spatially one- or two-dimensional cases so far, or alternatively require certain smallness hypotheses on the initial data.For all suitably regular and radially symmetric initial data $(u_0,v_0)$ satisfying $u_0\ge 0$ and $v_0>0$, the present paper establishes the existence of a globally defined pair $(u,v)$ of radially symmetric functions which are continuous in $(\overline{\mathrm{\Omega }}?\{0\})\times [0,\infty )$ and smooth in $(\overline{\mathrm{\Omega }}?\{0\})\times (0,\infty )$, and which solve the corresponding initial-boundary value problem for (?) with $(u(?,0),v(?,0))=(u_0,v_0)$ in an appropriate generalized sense. To the best of our knowledge, this in particular provides the first result on global existence for the three-dimensional version of (?) involving arbitrarily large initial data.     

Combined effects of correlated bounded noises and weak periodic signal input in the modified FitzHugh–Nagumo neural model

We study the dynamics of neurons via a bistable modified stochastic FitzHugh–Nagumo model having two stable fixed points separated by one unstable fixed point. Due to the ability of a neuron to detect and enhance weak information transmission, we show numerically that starting from the resting potential, we get firing activities (spiking) when operating slightly beyond the supercritical Hopf bifurcation. For real biological systems which are sometimes embedded in the complex environment, we observe that a gradual increase or decrease noise intensities did not result in a gradual change of the membrane potential distribution thanks to noise induced transition phenomena. We shown analytically that for zero correlation between two sine Wiener noises, additive noise has no effect on the transition between monostable and bistable phase on the neural model. We adapted a general expression of the signal-to-noise ratio for a general two-state theory extended in the asymmetric case and non-Gaussian noises in our model to study the influence of noise strength in stochastic resonance. Our investigation revealed that in the evolution of excitable system, neurons may use noises to their advantage by enhancing their sensitivity near a preferred phase to detect external stimuli or affect the efficiency and rate of information processing.     

On verifiable sufficient conditions for sparse signal recovery via <Emphasis Type="Italic">ℓ</Emphasis><Subscript>1</Subscript> minimization

We discuss necessary and sufficient conditions for a sensing matrix to be “s-good”—to allow for exact ℓ ₁-recovery of sparse signals with s nonzero entries when no measurement noise is present. Then we express the error bounds for imperfect ℓ ₁-recovery (nonzero measurement noise, nearly s-sparse signal, near-optimal solution of the optimization problem yielding the ℓ ₁-recovery) in terms of the characteristics underlying these conditions. Further, we demonstrate (and this is the principal result of the paper) that these characteristics, although difficult to evaluate, lead to verifiable sufficient conditions for exact sparse ℓ ₁-recovery and to efficiently computable upper bounds on those s for which a given sensing matrix is s-good. We establish also instructive links between our approach and the basic concepts of the Compressed Sensing theory, like Restricted Isometry or Restricted Eigenvalue properties.