64 research outputs found

    Robust Estimation and Wavelet Thresholding in Partial Linear Models

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    This paper is concerned with a semiparametric partially linear regression model with unknown regression coefficients, an unknown nonparametric function for the non-linear component, and unobservable Gaussian distributed random errors. We present a wavelet thresholding based estimation procedure to estimate the components of the partial linear model by establishing a connection between an l1l_1-penalty based wavelet estimator of the nonparametric component and Huber's M-estimation of a standard linear model with outliers. Some general results on the large sample properties of the estimates of both the parametric and the nonparametric part of the model are established. Simulations and a real example are used to illustrate the general results and to compare the proposed methodology with other methods available in the recent literature

    Wavelet penalized likelihood estimation in generalized functional models

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    The paper deals with generalized functional regression. The aim is to estimate the influence of covariates on observations, drawn from an exponential distribution. The link considered has a semiparametric expression: if we are interested in a functional influence of some covariates, we authorize others to be modeled linearly. We thus consider a generalized partially linear regression model with unknown regression coefficients and an unknown nonparametric function. We present a maximum penalized likelihood procedure to estimate the components of the model introducing penalty based wavelet estimators. Asymptotic rates of the estimates of both the parametric and the nonparametric part of the model are given and quasi-minimax optimality is obtained under usual conditions in literature. We establish in particular that the LASSO penalty leads to an adaptive estimation with respect to the regularity of the estimated function. An algorithm based on backfitting and Fisher-scoring is also proposed for implementation. Simulations are used to illustrate the finite sample behaviour, including a comparison with kernel and splines based methods

    Wavelet methods in statistics: Some recent developments and their applications

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    The development of wavelet theory has in recent years spawned applications in signal processing, in fast algorithms for integral transforms, and in image and function representation methods. This last application has stimulated interest in wavelet applications to statistics and to the analysis of experimental data, with many successes in the efficient analysis, processing, and compression of noisy signals and images. This is a selective review article that attempts to synthesize some recent work on ``nonlinear'' wavelet methods in nonparametric curve estimation and their role on a variety of applications. After a short introduction to wavelet theory, we discuss in detail several wavelet shrinkage and wavelet thresholding estimators, scattered in the literature and developed, under more or less standard settings, for density estimation from i.i.d. observations or to denoise data modeled as observations of a signal with additive noise. Most of these methods are fitted into the general concept of regularization with appropriately chosen penalty functions. A narrow range of applications in major areas of statistics is also discussed such as partial linear regression models and functional index models. The usefulness of all these methods are illustrated by means of simulations and practical examples.Comment: Published in at http://dx.doi.org/10.1214/07-SS014 the Statistics Surveys (http://www.i-journals.org/ss/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Statistical methods for topology inference, denoising, and bootstrapping in networks

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    Quite often, the data we observe can be effectively represented using graphs. The underlying structure of the resulting graph, however, might contain noise and does not always hold constant across scales. With the right tools, we could possibly address these two problems. This thesis focuses on developing the right tools and provides insights in looking at them. Specifically, I study several problems that incorporate network data within the multi-scale framework, aiming at identifying common patterns and differences, of signals over networks across different scales. Additional topics in network denoising and network bootstrapping will also be discussed. The first problem we consider is the connectivity changes in dynamic networks constructed from multiple time series data. Multivariate time series data is often non-stationary. Furthermore, it is not uncommon to expect changes in a system across multiple time scales. Motivated by these observations, we in-corporate the traditional Granger-causal type of modeling within the multi-scale framework and propose a new method to detect the connectivity changes and recover the dynamic network structure. The second problem we consider is how to denoise and approximate signals over a network adjacency matrix. We propose an adaptive unbalanced Haar wavelet based transformation of the network data, and show that it is efficient in approximation and denoising of the graph signals over a network adjacency matrix. We focus on the exact decompositions of the network, the corresponding approximation theory, and denoising signals over graphs, particularly from the perspective of compression of the networks. We also provide a real data application on denoising EEG signals over a DTI network. The third problem we consider is in network denoising and network inference. Network representation is popular in characterizing complex systems. However, errors observed in the original measurements will propagate to network statistics and hence induce uncertainties to the summaries of the networks. We propose a spectral-denoising based resampling method to produce confidence intervals that propagate the inferential errors for a number of Lipschitz continuous net- work statistics. We illustrate the effectiveness of the method through a series of simulation studies

    Statistical Diffusion Tensor Imaging

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    Magnetic resonance diffusion tensor imaging (DTI) allows to infere the ultrastructure of living tissue. In brain mapping, neural fiber trajectories can be identified by exploiting the anisotropy of diffusion processes. Manifold statistical methods may be linked into the comprehensive processing chain that is spanned between DTI raw images and the reliable visualization of fibers. In this work, a space varying coefficients model (SVCM) using penalized B-splines was developed to integrate diffusion tensor estimation, regularization and interpolation into a unified framework. The implementation challenges originating in multiple 3d space varying coefficient surfaces and the large dimensions of realistic datasets were met by incorporating matrix sparsity and efficient model approximation. Superiority of B-spline based SVCM to the standard approach was demonstrable from simulation studies in terms of the precision and accuracy of the individual tensor elements. The integration with a probabilistic fiber tractography algorithm and application on real brain data revealed that the unified approach is at least equivalent to the serial application of voxelwise estimation, smoothing and interpolation. From the error analysis using boxplots and visual inspection the conclusion was drawn that both the standard approach and the B-spline based SVCM may suffer from low local adaptivity. Therefore, wavelet basis functions were employed for filtering diffusion tensor fields. While excellent local smoothing was indeed achieved by combining voxelwise tensor estimation with wavelet filtering, no immediate improvement was gained for fiber tracking. However, the thresholding strategy needs to be refined and the proposed model of an incorporation of wavelets into an SVCM needs to be implemented to finally assess their utility for DTI data processing. In summary, an SVCM with specific consideration of the demands of human brain DTI data was developed and implemented, eventually representing a unified postprocessing framework. This represents an experimental and statistical platform to further improve the reliability of tractography

    Paradigm free mapping: detection and characterization of single trial fMRI BOLD responses without prior stimulus information

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    The increased contrast to noise ratio available at Ultrahigh (7T) Magnetic Resonance Imaging (MRI) allows mapping in space and time the brain's response to single trial events with functional MRI (fMRI) based on the Blood Oxygenation Level Dependent (BOLD) contrast. This thesis primarily concerns with the development of techniques to detect and characterize single trial event-related BOLD responses without prior paradigm information, Paradigm Free Mapping, and assess variations in BOLD sensitivity across brain regions at high field fMRI. Based on a linear haemodynamic response model, Paradigm Free Mapping (PFM) techniques rely on the deconvolution of the neuronal-related signal driving the BOLD effect using regularized least squares estimators. The first approach, named PFM, builds on the ridge regression estimator and spatio-temporal t-statistics to detect statistically significant changes in the deconvolved fMRI signal. The second method, Sparse PFM, benefits from subset selection features of the LASSO and Dantzig Selector estimators that automatically detect the single trial BOLD responses by promoting a sparse deconvolution of the signal. The third technique, Multicomponent PFM, exploits further the benefits of sparse estimation to decompose the fMRI signal into a haemodynamical component and a baseline component using the morphological component analysis algorithm. These techniques were evaluated in simulations and experimental fMRI datasets, and the results were compared with well-established fMRI analysis methods. In particular, the methods developed here enabled the detection of single trial BOLD responses to visually-cued and self-paced finger tapping responses without prior information of the events. The potential application of Sparse PFM to identify interictal discharges in idiopathic generalized epilepsy was also investigated. Furthermore, Multicomponent PFM allowed us to extract cardiac and respiratory fluctuations of the signal without the need of physiological monitoring. To sum up, this work demonstrates the feasibility to do single trial fMRI analysis without prior stimulus or physiological information using PFM techniques

    Paradigm free mapping: detection and characterization of single trial fMRI BOLD responses without prior stimulus information

    Get PDF
    The increased contrast to noise ratio available at Ultrahigh (7T) Magnetic Resonance Imaging (MRI) allows mapping in space and time the brain's response to single trial events with functional MRI (fMRI) based on the Blood Oxygenation Level Dependent (BOLD) contrast. This thesis primarily concerns with the development of techniques to detect and characterize single trial event-related BOLD responses without prior paradigm information, Paradigm Free Mapping, and assess variations in BOLD sensitivity across brain regions at high field fMRI. Based on a linear haemodynamic response model, Paradigm Free Mapping (PFM) techniques rely on the deconvolution of the neuronal-related signal driving the BOLD effect using regularized least squares estimators. The first approach, named PFM, builds on the ridge regression estimator and spatio-temporal t-statistics to detect statistically significant changes in the deconvolved fMRI signal. The second method, Sparse PFM, benefits from subset selection features of the LASSO and Dantzig Selector estimators that automatically detect the single trial BOLD responses by promoting a sparse deconvolution of the signal. The third technique, Multicomponent PFM, exploits further the benefits of sparse estimation to decompose the fMRI signal into a haemodynamical component and a baseline component using the morphological component analysis algorithm. These techniques were evaluated in simulations and experimental fMRI datasets, and the results were compared with well-established fMRI analysis methods. In particular, the methods developed here enabled the detection of single trial BOLD responses to visually-cued and self-paced finger tapping responses without prior information of the events. The potential application of Sparse PFM to identify interictal discharges in idiopathic generalized epilepsy was also investigated. Furthermore, Multicomponent PFM allowed us to extract cardiac and respiratory fluctuations of the signal without the need of physiological monitoring. To sum up, this work demonstrates the feasibility to do single trial fMRI analysis without prior stimulus or physiological information using PFM techniques
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