A Semi-parametric Technique for the Quantitative Analysis of Dynamic Contrast-enhanced MR Images Based on Bayesian P-splines

Dynamic Contrast-enhanced Magnetic Resonance Imaging (DCE-MRI) is an important tool for detecting subtle kinetic changes in cancerous tissue. Quantitative analysis of DCE-MRI typically involves the convolution of an arterial input function (AIF) with a nonlinear pharmacokinetic model of the contrast agent concentration. Parameters of the kinetic model are biologically meaningful, but the optimization of the non-linear model has significant computational issues. In practice, convergence of the optimization algorithm is not guaranteed and the accuracy of the model fitting may be compromised. To overcome this problems, this paper proposes a semi-parametric penalized spline smoothing approach, with which the AIF is convolved with a set of B-splines to produce a design matrix using locally adaptive smoothing parameters based on Bayesian penalized spline models (P-splines). It has been shown that kinetic parameter estimation can be obtained from the resulting deconvolved response function, which also includes the onset of contrast enhancement. Detailed validation of the method, both with simulated and in vivo data, is provided

Function estimation with locally adaptive dynamic models

We present a nonparametric Bayesian method for fitting unsmooth and highly oscillating functions, which is based on a locally adaptive hierarchical extension of standard dynamic or state space models. The main idea is to introduce locally varying variances in the state equations and to add a further smoothness prior for this variance function. Estimation is fully Bayesian and carried out by recent MCMC techniques. The whole approach can be understood as an alternative to other nonparametric function estimators, such as local or penalized regression with variable bandwidth or smoothing parameter selection. Performance is illustrated with simulated data, including unsmooth examples constructed for wavelet shrinkage, and by an application to sales data. Although the approach is developed for classical Gaussian nonparametric regression, it can be extended to more complex regression problems

Locally Adaptive Nonparametric Binary Regression

A nonparametric and locally adaptive Bayesian estimator is proposed for estimating a binary regression. Flexibility is obtained by modeling the binary regression as a mixture of probit regressions with the argument of each probit regression having a thin plate spline prior with its own smoothing parameter and with the mixture weights depending on the covariates. The estimator is compared to a single spline estimator and to a recently proposed locally adaptive estimator. The methodology is illustrated by applying it to both simulated and real examples.Comment: 31 pages, 10 figure

Locally Adaptive Bayesian P-Splines with a Normal-Exponential-Gamma Prior

The necessity to replace smoothing approaches with a global amount of smoothing arises in a variety of situations such as effects with highly varying curvature or effects with discontinuities. We present an implementation of locally adaptive spline smoothing using a class of heavy-tailed shrinkage priors. These priors utilize scale mixtures of normals with locally varying exponential-gamma distributed variances for the differences of the P-spline coefficients. A fully Bayesian hierarchical structure is derived with inference about the posterior being based on Markov Chain Monte Carlo techniques. Three increasingly flexible and automatic approaches are introduced to estimate the spatially varying structure of the variances. In an extensive simulation study, the performance of our approach on a number of benchmark functions is shown to be at least equivalent, but mostly better than previous approaches and fits both functions of smoothly varying complexity and discontinuous functions well. Results from two applications also reflecting these two situations support the simulation results

Meta-analysis of functional neuroimaging data using Bayesian nonparametric binary regression

In this work we perform a meta-analysis of neuroimaging data, consisting of locations of peak activations identified in 162 separate studies on emotion. Neuroimaging meta-analyses are typically performed using kernel-based methods. However, these methods require the width of the kernel to be set a priori and to be constant across the brain. To address these issues, we propose a fully Bayesian nonparametric binary regression method to perform neuroimaging meta-analyses. In our method, each location (or voxel) has a probability of being a peak activation, and the corresponding probability function is based on a spatially adaptive Gaussian Markov random field (GMRF). We also include parameters in the model to robustify the procedure against miscoding of the voxel response. Posterior inference is implemented using efficient MCMC algorithms extended from those introduced in Holmes and Held [Bayesian Anal. 1 (2006) 145--168]. Our method allows the probability function to be locally adaptive with respect to the covariates, that is, to be smooth in one region of the covariate space and wiggly or even discontinuous in another. Posterior miscoding probabilities for each of the identified voxels can also be obtained, identifying voxels that may have been falsely classified as being activated. Simulation studies and application to the emotion neuroimaging data indicate that our method is superior to standard kernel-based methods.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS523 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Locally adaptive smoothing with Markov random fields and shrinkage priors

We present a locally adaptive nonparametric curve fitting method that operates within a fully Bayesian framework. This method uses shrinkage priors to induce sparsity in order-k differences in the latent trend function, providing a combination of local adaptation and global control. Using a scale mixture of normals representation of shrinkage priors, we make explicit connections between our method and kth order Gaussian Markov random field smoothing. We call the resulting processes shrinkage prior Markov random fields (SPMRFs). We use Hamiltonian Monte Carlo to approximate the posterior distribution of model parameters because this method provides superior performance in the presence of the high dimensionality and strong parameter correlations exhibited by our models. We compare the performance of three prior formulations using simulated data and find the horseshoe prior provides the best compromise between bias and precision. We apply SPMRF models to two benchmark data examples frequently used to test nonparametric methods. We find that this method is flexible enough to accommodate a variety of data generating models and offers the adaptive properties and computational tractability to make it a useful addition to the Bayesian nonparametric toolbox.Comment: 38 pages, to appear in Bayesian Analysi