Some Computational Aspects of Gaussian CARMA Modelling

Representation of continuous-time ARMA, CARMA, models is reviewed. Computational aspects of simulating and calculating the likelihood-function of CARMA are summarized. Some numerical properties are illustrated by simulations. Some real data applications are shown.CARMA, maximum-likelihood, spectrum, Kalman filter, computation

Blind deconvolution of sparse pulse sequences under a minimum distance constraint: a partially collapsed Gibbs sampler method

For blind deconvolution of an unknown sparse sequence convolved with an unknown pulse, a powerful Bayesian method employs the Gibbs sampler in combination with a Bernoulli–Gaussian prior modeling sparsity. In this paper, we extend this method by introducing a minimum distance constraint for the pulses in the sequence. This is physically relevant in applications including layer detection, medical imaging, seismology, and multipath parameter estimation. We propose a Bayesian method for blind deconvolution that is based on a modified Bernoulli–Gaussian prior including a minimum distance constraint factor. The core of our method is a partially collapsed Gibbs sampler (PCGS) that tolerates and even exploits the strong local dependencies introduced by the minimum distance constraint. Simulation results demonstrate significant performance gains compared to a recently proposed PCGS. The main advantages of the minimum distance constraint are a substantial reduction of computational complexity and of the number of spurious components in the deconvolution result

Variational Bayesian Inference of Line Spectra

In this paper, we address the fundamental problem of line spectral estimation in a Bayesian framework. We target model order and parameter estimation via variational inference in a probabilistic model in which the frequencies are continuous-valued, i.e., not restricted to a grid; and the coefficients are governed by a Bernoulli-Gaussian prior model turning model order selection into binary sequence detection. Unlike earlier works which retain only point estimates of the frequencies, we undertake a more complete Bayesian treatment by estimating the posterior probability density functions (pdfs) of the frequencies and computing expectations over them. Thus, we additionally capture and operate with the uncertainty of the frequency estimates. Aiming to maximize the model evidence, variational optimization provides analytic approximations of the posterior pdfs and also gives estimates of the additional parameters. We propose an accurate representation of the pdfs of the frequencies by mixtures of von Mises pdfs, which yields closed-form expectations. We define the algorithm VALSE in which the estimates of the pdfs and parameters are iteratively updated. VALSE is a gridless, convergent method, does not require parameter tuning, can easily include prior knowledge about the frequencies and provides approximate posterior pdfs based on which the uncertainty in line spectral estimation can be quantified. Simulation results show that accounting for the uncertainty of frequency estimates, rather than computing just point estimates, significantly improves the performance. The performance of VALSE is superior to that of state-of-the-art methods and closely approaches the Cram\'er-Rao bound computed for the true model order.Comment: 15 pages, 8 figures, accepted for publication in IEEE Transactions on Signal Processin

Atomic norm denoising with applications to line spectral estimation

Motivated by recent work on atomic norms in inverse problems, we propose a new approach to line spectral estimation that provides theoretical guarantees for the mean-squared-error (MSE) performance in the presence of noise and without knowledge of the model order. We propose an abstract theory of denoising with atomic norms and specialize this theory to provide a convex optimization problem for estimating the frequencies and phases of a mixture of complex exponentials. We show that the associated convex optimization problem can be solved in polynomial time via semidefinite programming (SDP). We also show that the SDP can be approximated by an l1-regularized least-squares problem that achieves nearly the same error rate as the SDP but can scale to much larger problems. We compare both SDP and l1-based approaches with classical line spectral analysis methods and demonstrate that the SDP outperforms the l1 optimization which outperforms MUSIC, Cadzow's, and Matrix Pencil approaches in terms of MSE over a wide range of signal-to-noise ratios.Comment: 27 pages, 10 figures. A preliminary version of this work appeared in the Proceedings of the 49th Annual Allerton Conference in September 2011. Numerous numerical experiments added to this version in accordance with suggestions by anonymous reviewer

Identification of periodic bursts in surface EMG: Applications to the erector spinae muscles of sitting violin players

Objective: This work compares two known and one novel techniques for the detection of surface EMG (sEMG) quasi-periodic burst-like signals and the estimation of their frequency. The novel method (ES) is based on the spectral analysis of the envelope signal, the other two methods use a fixed (FT) or automatically selected optimal threshold (OT). Methods: The methods are compared using both simulated signals and samples of High Density sEMG experimental signals collected using electrode arrays applied to the erector spinae muscles of violinists. Results: The ES method does not require thresholds. It detects presence/absence of bursts and their frequency, even in cases of a few missing bursts. It does not provide their duration. The FT method requires the selection of a fixed threshold value, estimates burst duration but is applicable only if bursts are present. The OT method identifies an optimal threshold, estimates burst duration but behaves irregularly when bursts are small or absent. Conclusions: The ES method provides the estimates closest to those of an expert human counter and is not sensitive to amplitude fluctuations. It is suitable when the general bursts periodicity is of interest even if some bursts may be missing. The FT and OT methods are sensitive to amplitude fluctuations and identify random threshold crossings as bursts even when burst activity is absent. Significance: Postural muscles are often activated in a burst-like fashion. The proposed ES method identifies presence/absence of bursts and their frequency, which is important for studying the neurophysiological mechanism generating them

Dynamic analysis of unevenly sampled data with applications to statistical process control

Dynamic analysis involves describing how a process changes over time. Applications of this type of analysis can be implemented in industrial settings in order to control manufacturing processes and recognize when they have changed significantly. The primary focus of this work is to construct methods to detect the onset of periodic behavior in a process which is being monitored using a scheme where data is sampled unevenly. Techniques that can be used to identify statistically significant periodic structure using the periodogram will be reviewed and developed. The statistical properties of the periodogram for unevenly sampled data will be calculated. These statistics reveal that standard methods applied to randomly sampled data give incorrect results, especially for small sample sizes. These standard tests are not designed specifically for data collected at random times. Monte Carlo methods are used to adjust the critical values used for testing the significance of spectral peaks. The effectiveness of the tests for determining periodic behavior are compared using the standard critical values and the adjusted values. The adapted test is then extended into a control chart which will signal when periodic behavior enters into an irregularly sampled process. The new methods are applied to an industrial example from a silicon wafer coating process. The data was collected irregularly and the underlying dynamics of the process were investigated. Interesting periodic behavior was uncovered in the analysis. When data has complicated oscillatory behavior, methods of nonlinear dynamic analysis can be used to make predictions. A new toroidal reconstruction technique is developed for data that appears to be driven predominantly by two or three frequencies. Comparisons between the new method and a standard time delay reconstruction utilizing nonlinear dynamic forecasting methods are made using simulated and real-world data collected from a vibrating warehouse air duct