Geurt Jongbloed

The Iterative Convex Minorant Algorithm for Nonparametric Estimation

Abstract The problem of minimizing a smooth convex function over a specific cone in IRn is frequently encountered in nonparametric statistics. For that type of problem we suggest an algorithm and show that this algorithm converges to the solution of the minimization problem. Moreover, a simulation study is presented, showing the superiority of our algorithm compared to the EM algorithm in the interval censoring case 2 setting.

A regional peaks-over-threshold model in a nonstationary climate

Regional frequency analysis is often used to reduce the uncertainty in the estimation of distribution parameters and quantiles. In this paper a regional peaks-over-threshold model is introduced that can be used to analyze precipitation extremes in a changing climate. We use a temporally varying threshold, which is determined by quantile regression for each site separately. The marginal distributions of the excesses are described by generalized Pareto distributions (GPD). The parameters of these distributions may vary over time and their spatial variation is modeled by the index flood (IF) approach. We consider different models for the temporal dependence of the GPD parameters. Parameter estimation is based on the framework of composite likelihood. Composite likelihood ratio tests that account for spatial dependence are used to test the significance of temporal trends in the model parameters and to test the IF assumption. We apply the method to gridded, observed daily precipitation data from the Netherlands for the winter season. A general increase of the threshold is observed, especially along the west coast and northern parts of the country. Moreover, there is no indication that the ratio between the GPD scale parameter and the threshold has changed over time, which implies that the scale parameter increases by the same percentage as the threshold. These positive trends lead to an increase of rare extremes of on average 22% over the country during the observed period.

Maximum smoothed likelihood estimation and smoothed maximum likelihood estimation in the current status model

We consider the problem of estimating the distribution function, the density and the hazard rate of the (unobservable) event time in the current status model. A well studied and natural nonparametric estimator for the distribution function in this model is the nonparametric maximum likelihood estimator (MLE). We study two alternative methods for the estimation of the distribution function, assuming some smoothness of the event time distribution. The first estimator is based on a maximum smoothed likelihood approach. The second method is based on smoothing the (discrete) MLE of the distribution function. These estimators can be used to estimate the density and hazard rate of the event time distribution based on the plug-in principle.

Density estimation in the uniform deconvolution model

We consider the problem of estimating a probability density function based on data that are corrupted by noise from a uniform distribution. The (nonparametric) maximum likelihood estimator for the corresponding distribution function is well defined. For the density function this is not the case. We study two nonparametric estimators for this density. The first is a type of kernel density estimate based on the empirical distribution function of the observable data. The second is a kernel density estimate based on the MLE of the distribution function of the unobservable (uncorrupted) data.

Exponential deconvolution: two asymptotically equivalent estimators

Two isotonic estimators for the distribution function in a specific deconvolution model, the exponential deconvolution model, are considered. The first estimator is a least squares projection of a naive estimator for the distribution function on the set of distribution functions. The second estimator is the well known maximum likelihood estimator. The two estimators are shown to be first order asymptotically equivalent at a fixed point.

Evaluation of O2PLS in Omics data integration.

BackgroundRapid computational and technological developments made large amounts of omics data available in different biological levels. It is becoming clear that simultaneous data analysis methods are needed for better interpretation and understanding of the underlying systems biology. Different methods have been proposed for this task, among them Partial Least Squares (PLS) related methods. To also deal with orthogonal variation, systematic variation in the data unrelated to one another, we consider the Two-way Orthogonal PLS (O2PLS): an integrative data analysis method which is capable of modeling systematic variation, while providing more parsimonious models aiding interpretation.ResultsA simulation study to assess the performance of O2PLS showed positive results in both low and higher dimensions. More noise (50 % of the data) only affected the systematic part estimates. A data analysis was conducted using data on metabolomics and transcriptomics from a large Finnish cohort (DILGOM). A previous sequential study, using the same data, showed significant correlations between the Lipo-Leukocyte (LL) module and lipoprotein metabolites. The O2PLS results were in agreement with these findings, identifying almost the same set of co-varying variables. Moreover, our integrative approach identified other associative genes and metabolites, while taking into account systematic variation in the data. Including orthogonal components enhanced overall fit, but the orthogonal variation was difficult to interpret.ConclusionsSimulations showed that the O2PLS estimates were close to the true parameters in both low and higher dimensions. In the presence of more noise (50 %), the orthogonal part estimates could not distinguish well between joint and unique variation. The joint estimates were not systematically affected. Simultaneous analysis with O2PLS on metabolome and transcriptome data showed that the LL module, together with VLDL and HDL metabolites, were important for the metabolomic and transcriptomic relation. This is in agreement with an earlier study. In addition more gene expression and metabolites are identified being important for the joint covariation.

Estimating a Unimodal Distribution From Interval-Censored Data

In this article we consider three nonparametric maximum likelihood estimators based on mixed-case interval-censored data. Apart from the unrestricted estimator, we consider estimators under the assumption that the underlying distribution function of event times is concave or unimodal. Characterizations of the estimates are derived, and algorithms are proposed for their computation. The estimators are shown to be asymptotically consistent, and the benefits of additional constraints are illustrated through simulations. Finally, the estimators are used as an ingredient in a nonparametric comparison of two samples.

Effects of attenuation correction and reconstruction method on PET activation studies

The outcome of Statistical Parametric Mapping (SPM) analyses of PET activation studies depends among others, on the quality of reconstructed data. In general, filtered back-projection (FBP) is used for reconstruction in PET activation studies. There is, however, increasing interest in iterative reconstruction algorithms such as ordered subset expectation maximization (OSEM) algorithms. The aim of the present study was to investigate the effects of reconstruction techniques and attenuation correction (AC) on the detection of activation foci following statistical analysis with SPM. First, a replicate study was performed to assess the effects of the reconstruction method on pixel variance. Second, a phantom study was performed to evaluate the influence of both locations of an activated area and applied reconstruction method on SPM outcome. A volumetric method was used to compute the number of false positive voxels for all reconstructions. In addition, average t values within activation foci and for false positive voxels were calculated. For the assessment of the effects of reconstruction on clinical data, a group of 11 patients was studied. For all reconstructions SPM maps were created and compared. Both the clinical and the phantom data showed that use of iterative reconstruction methods reduced false positive results, while showing similar SPM results within activated areas as FBP. Reconstruction of data without attenuation correction reduced noise for FBP only, but did not affect the quality of SPM results for OSEM. It is concluded that OSEM is a good alternative for FBP reconstructions providing SPM results with less noise.

Mathematical Methods for Signal and Image Analysis and Representation

Mathematical Methods for Signal and Image Analysis and Representation presents the mathematical methodology for generic image analysis tasks. In the context of this book an image may be any m-dimensional empirical signal living on an n-dimensional smooth manifold (typically, but not necessarily, a subset of spacetime). The existing literature on image methodology is rather scattered and often limited to either a deterministic or a statistical point of view. In contrast, this book brings together these seemingly different points of view in order to stress their conceptual relations and formal analogies. Furthermore, it does not focus on specific applications, although some are detailed for the sake of illustration, but on the methodological frameworks on which such applications are built, making it an ideal companion for those seeking a rigorous methodological basis for specific algorithms as well as for those interested in the fundamental methodology per se. Covering many topics at the forefront of current research, including anisotropic diffusion filtering of tensor fields, this book will be of particular interest to graduate and postgraduate students and researchers in the fields of computer vision, medical imaging and visual perception.

A maximum smoothed likelihood estimator in the current status continuous mark model

We consider the problem of estimating the joint distribution function of the event time and a continuous mark variable based on censored data. More specifically, the event time is subject to current status censoring and the continuous mark is only observed in case inspection takes place after the event time. The nonparametric maximum likelihood estimator in this model is known to be inconsistent. We propose and study an alternative likelihood-based estimator, maximising a smoothed log-likelihood, hence called a maximum smoothed likelihood estimator (MSLE). This estimator is shown to be well defined and consistent, and a simple algorithm is described that can be used to compute it. The MSLE is compared with other estimators in a small simulation study.