Bayesian Spectral Analysis with Student-t Noise

PublishedArticleWe introduce a Bayesian spectral analysis model for one-dimensional signals where the observation noise is assumed to be Student-t distributed, for robustness to outliers, and we estimate the posterior distributions of the Student-t hyperparameters, as well as the amplitudes and phases of the component sinusoids. The integrals required for exact Bayesian inference are intractable, so we use variational approximation. We show that the approximate phase posteriors are Generalised von Mises distributions of order 2 and that their spread increases as the signal to noise ratio decreases. The model is demonstrated against synthetic data, and real GPS and Wolf’s sunspot data

The degeneracy between dust colour temperature and spectral index. Comparison of methods for estimating the beta(T) relation

Sub-millimetre dust emission provides information on the physics of interstellar clouds and dust. Noise can produce anticorrelation between the colour temperature T_C and the spectral index beta. This must be separated from the intrinsic beta(T) relation of dust. We compare methods for the analysis of the beta(T) relation. We examine sub-millimetre observations simulated as simple modified black body emission or using 3D radiative transfer modelling. In addition to chi^2 fitting, we examine the results of the SIMEX method, basic Bayesian model, hierarchical models, and one method that explicitly assumes a functional form for beta(T). All methods exhibit some bias. Bayesian method shows significantly lower bias than direct chi^2 fits. The same is true for hierarchical models that also result in a smaller scatter in the temperature and spectral index values. However, significant bias was observed in cases with high noise levels. Beta and T estimates of the hierarchical model are biased towards the relation determined by the data with the highest S/N ratio. This can alter the recovered beta(T) function. With the method where we explicitly assume a functional form for the beta(T) relation, the bias is similar to the Bayesian method. In the case of an actual Herschel field, all methods agree, showing some degree of anticorrelation between T and beta. The Bayesian method and the hierarchical models can both reduce the noise-induced parameter correlations. However, all methods can exhibit non-negligible bias. This is particularly true for hierarchical models and observations of varying signal-to-noise ratios and must be taken into account when interpreting the results.Comment: Submitted to A&A, 18 page

Bayesian separation of spectral sources under non-negativity and full additivity constraints

This paper addresses the problem of separating spectral sources which are linearly mixed with unknown proportions. The main difficulty of the problem is to ensure the full additivity (sum-to-one) of the mixing coefficients and non-negativity of sources and mixing coefficients. A Bayesian estimation approach based on Gamma priors was recently proposed to handle the non-negativity constraints in a linear mixture model. However, incorporating the full additivity constraint requires further developments. This paper studies a new hierarchical Bayesian model appropriate to the non-negativity and sum-to-one constraints associated to the regressors and regression coefficients of linear mixtures. The estimation of the unknown parameters of this model is performed using samples generated using an appropriate Gibbs sampler. The performance of the proposed algorithm is evaluated through simulation results conducted on synthetic mixture models. The proposed approach is also applied to the processing of multicomponent chemical mixtures resulting from Raman spectroscopy.Comment: v4: minor grammatical changes; Signal Processing, 200

Semi-supervised linear spectral unmixing using a hierarchical Bayesian model for hyperspectral imagery

This paper proposes a hierarchical Bayesian model that can be used for semi-supervised hyperspectral image unmixing. The model assumes that the pixel reflectances result from linear combinations of pure component spectra contaminated by an additive Gaussian noise. The abundance parameters appearing in this model satisfy positivity and additivity constraints. These constraints are naturally expressed in a Bayesian context by using appropriate abundance prior distributions. The posterior distributions of the unknown model parameters are then derived. A Gibbs sampler allows one to draw samples distributed according to the posteriors of interest and to estimate the unknown abundances. An extension of the algorithm is finally studied for mixtures with unknown numbers of spectral components belonging to a know library. The performance of the different unmixing strategies is evaluated via simulations conducted on synthetic and real data

Bayesian Lower Bounds for Dense or Sparse (Outlier) Noise in the RMT Framework

Robust estimation is an important and timely research subject. In this paper, we investigate performance lower bounds on the mean-square-error (MSE) of any estimator for the Bayesian linear model, corrupted by a noise distributed according to an i.i.d. Student's t-distribution. This class of prior parametrized by its degree of freedom is relevant to modelize either dense or sparse (accounting for outliers) noise. Using the hierarchical Normal-Gamma representation of the Student's t-distribution, the Van Trees' Bayesian Cram\'er-Rao bound (BCRB) on the amplitude parameters is derived. Furthermore, the random matrix theory (RMT) framework is assumed, i.e., the number of measurements and the number of unknown parameters grow jointly to infinity with an asymptotic finite ratio. Using some powerful results from the RMT, closed-form expressions of the BCRB are derived and studied. Finally, we propose a framework to fairly compare two models corrupted by noises with different degrees of freedom for a fixed common target signal-to-noise ratio (SNR). In particular, we focus our effort on the comparison of the BCRBs associated with two models corrupted by a sparse noise promoting outliers and a dense (Gaussian) noise, respectively