329 research outputs found
Variable order smoothness priors for ill-posed inverse problems
No full textIn this article we discuss ill-posed inverse problems, with an emphasis on hierarchical variable order regularization. Traditionally, smoothness penalties in Tikhonov regularization assume a fixed degree of regularity of the unknown over the whole domain. Using a Bayesian framework with hierarchical priors, we derive a prior model, formally represented as a convex combination of autoregressive (AR) models, in which the parameter controlling the mixture of the AR models can dynamically change over the domain of the signal. Moreover, the mixture parameter itself is an unknown and is to be estimated using the data. Also, the variance of the innovation processes in the AR model is a free parameter, which leads to conditionally Gaussian priors that have been previously shown to be much more flexible than the traditional Gaussian priors, capable, e.g., to deal with sparsity type prior information. The suggested method, the Weighted Variable Order Autoregressive model (WVO-AR) is tested with a computed example.Fil: Calvetti, Daniela. Case Western Reserve University; Estados UnidosFil: Somersalo, Erkki. Case Western Reserve University; Estados UnidosFil: Spies, Ruben Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Santa Fe. Instituto de Matemática Aplicada "Litoral"; Argentin
Brain Activity Mapping from MEG Data via a Hierarchical Bayesian Algorithm with Automatic Depth Weighting
No full textA recently proposed iterated alternating sequential (IAS) MEG inverse solver algorithm, based on the coupling of a hierarchical Bayesian model with computationally efficient Krylov subspace linear solver, has been shown to perform well for both superficial and deep brain sources. However, a systematic study of its ability to correctly identify active brain regions is still missing. We propose novel statistical protocols to quantify the performance of MEG inverse solvers, focusing in particular on how their accuracy and precision at identifying active brain regions. We use these protocols for a systematic study of the performance of the IAS MEG inverse solver, comparing it with three standard inversion methods, wMNE, dSPM, and sLORETA. To avoid the bias of anecdotal tests towards a particular algorithm, the proposed protocols are Monte Carlo sampling based, generating an ensemble of activity patches in each brain region identified in a given atlas. The performance in correctly identifying the active areas is measured by how much, on average, the reconstructed activity is concentrated in the brain region of the simulated active patch. The analysis is based on Bayes factors, interpreting the estimated current activity as data for testing the hypothesis that the active brain region is correctly identified, versus the hypothesis of any erroneous attribution. The methodology allows the presence of a single or several simultaneous activity regions, without assuming that the number of active regions is known. The testing protocols suggest that the IAS solver performs well with both with cortical and subcortical activity estimation
Sparse reconstructions from few noisy data: analysis of hierarchical Bayesian models with generalized gamma hyperpriors
No full textSolving inverse problems with sparsity promoting regularizing penalties can be recast in the Bayesian framework as finding a maximum a posteriori (MAP) estimate with sparsity promoting priors. In the latter context, a computationally convenient choice of prior is the family of conditionally Gaussian hierarchical models for which the prior variances of the components of the unknown are independent and follow a hyperprior from a generalized gamma family. In this paper, we analyze the optimization problem behind the MAP estimation and identify hyperparameter combinations that lead to a globally or locally convex optimization problem. The MAP estimation problem is solved using a computationally efficient alternating iterative algorithm. Its properties in the context of the generalized gamma hypermodel and its connections with some known sparsity promoting penalty methods are analyzed. Computed examples elucidate the convergence and sparsity promoting properties of the algorithm
Computational issues in linear multistep method particle filtering
No full textThe LMM PF is a methodology for solving the state and parameter estimation problem for ODEs system, rooted into a Bayesian framework and for it we consider a test problem with known solution to investigate in depth the error sources and how they depend on the choice of LMM used for the numerical integration. We conclude by looking at the effect on the error of replacing LMM with Runge-Kutta (RK) class integration methods
Exploring muscle recruitment by Bayesian methods during motion
No full textThe human musculoskeletal system is characterized by redundancy in the sense that the number of muscles
exceeds the number of degrees of freedom of the musculoskeletal system. In practice, this means that a given
motor task can be performed by activating the muscles in infinitely many different ways. This redundancy is
important for the functionality of the system under changing external or internal conditions, including different
diseased states. A central problem in biomechanics is how, and based on which principles, the complex of
central nervous system and musculoskeletal system selects the normal activation patterns, and how the patterns
change under various abnormal conditions including neurodegenerative diseases and aging. This work lays the
mathematical foundation for a formalism to address the question, based on Bayesian probabilistic modeling of
the musculoskeletal system. Lagrangian dynamics is used to translate observations of the movement of a subject
performing a task into a time series of equilibria which constitute the likelihood model. Different prior models
corresponding to biologically motivated assumptions about the muscle dynamics and control are introduced.
The posterior distributions of muscle activations are derived and explored by using Markov chain Monte Carlo
(MCMC) sampling techniques. The different priors can be analyzed by comparing the model predictions with
actual observations
Myobolica: a stochastic approach to estimate physiological muscle control variability
No full text: The inherent redundancy of the musculoskeletal systems is traditionally solved by optimizing a cost function. This approach may not be correct to model non-adult or pathological populations likely to adopt a "non-optimal" motor control strategy. Over the years, various methods have been developed to address this limitation, such as the stochastic approach. A well-known implementation of this approach, Metabolica, samples a wide number of plausible solutions instead of searching for a single one, leveraging Bayesian statistics and Markov Chain Monte Carlo algorithm, yet allowing muscles to abruptly change their activation levels. To overcome this and other limitations, we developed a new implementation of the stochastic approach (Myobolica), adding constraints and parameters to ensure the identification of physiological solutions. The aim of this study was to evaluate Myobolica, and quantify the differences in terms of width of the solution band (muscle control variability) compared to Metabolica. To this end, both muscle forces and knee joint force solutions bands estimated by the two approaches were compared to one another, and against (i) the solution identified by static optimization and (ii) experimentally measured knee joint forces. The use of Myobolica led to a marked narrowing of the solution band compared to Metabolica. Furthermore, the Myobolica solutions well correlated with the experimental data (R2 = 0.92, RMSE = 0.3 BW), but not as much with the optimal solution (R2 = 0.82, RMSE = 0.63 BW). Additional analyses are required to confirm the findings and further improve this implementation
Mining the mind: linear discriminant analysis of MEG source reconstruction time series supports dynamic changes in deep brain regions during meditation sessions
No full textMeditation practices have been claimed to have a positive effect on the regulation of mood and emotions for quite some time by practitioners, and in recent times there has been a sustained effort to provide a more precise description of the influence of meditation on the human brain. Longitudinal studies have reported morphological changes in cortical thickness and volume in selected brain regions due to meditation practice, which is interpreted as an evidence its effectiveness beyond the subjective self reporting. Using magnetoencephalography (MEG) or electroencephalography to quantify the changes in brain activity during meditation practice represents a challenge, as no clear hypothesis about the spatial or temporal pattern of such changes is available to date. In this article we consider MEG data collected during meditation sessions of experienced Buddhist monks practicing focused attention (Samatha) and open monitoring (Vipassana) meditation, contrasted by resting state with eyes closed. The MEG data are first mapped to time series of brain activity averaged over brain regions corresponding to a standard Destrieux brain atlas. Next, by bootstrapping and spectral analysis, the data are mapped to matrices representing random samples of power spectral densities in α, β, γ, and θ frequency bands. We use linear discriminant analysis to demonstrate that the samples corresponding to different meditative or resting states contain enough fingerprints of the brain state to allow a separation between different states, and we identify the brain regions that appear to contribute to the separation. Our findings suggest that the cingulate cortex, insular cortex and some of the internal structures, most notably the accumbens, the caudate and the putamen nuclei, the thalamus and the amygdalae stand out as separating regions, which seems to correlate well with earlier findings based on longitudinal studies
Computational issues and numerical experiments for Linear Multistep Method Particle Filtering
No full textThe Linear Multistep Method Particle Filter (LMM PF) is a method for
predicting the evolution in time of a evolutionary system governed by a system
of differential equations. If some of the parameters of the governing equations
are unknowns, it is possible to organize the calculations so as to estimate
them while following the evolution of the system in time. The underlying
assumption in the approach that we present is that all unknowns are modelled as
random variables, where the randomness is an indication of the uncertainty of
their values rather than an intrinsic property of the quantities. Consequently,
the states of the system and the parameters are described in probabilistic
terms by their density, often in the form of representative samples. This
approach is particularly attractive in the context of parameter estimation
inverse problems, because the statistical formulation naturally provides a
means of assessing the uncertainty in the solution via the spread of the
distribution. The computational efficiency of the underlying sampling technique
is crucial for the success of the method, because the accuracy of the solution
depends on the ability to produce representative samples from the distribution
of the unknown parameters. In this paper LMM PF is tested on a skeletal muscle
metabolism problem, which was previously treated within the Ensemble Kalman
filtering framework. Here numerical evidences are used to highlight the
correlation between the main sources of errors and the influence of the linera
multistep method adopted. Finally, we analyzed the effect of replacing LMM with
Runge-Kutta class integration methods for supporting the PF technique
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