Functional data analysis in an operator-based mixed-model framework

Functional data analysis in a mixed-effects model framework is done using operator calculus. In this approach the functional parameters are treated as serially correlated effects giving an alternative to the penalized likelihood approach, where the functional parameters are treated as fixed effects. Operator approximations for the necessary matrix computations are proposed, and semi-explicit and numerically stable formulae of linear computational complexity are derived for likelihood analysis. The operator approach renders the usage of a functional basis unnecessary and clarifies the role of the boundary conditions.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ389 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Laplace approximation of transition densities posed as Brownian expectations

AbstractWe construct the Laplace approximation of the Lebesgue density for a discrete partial observation of a multi-dimensional stochastic differential equation. This approximation may be computed integrating systems of ordinary differential equations. The construction of the Laplace approximation begins with the definition of the point of minimum energy. We show how such a point can be defined in the Cameron–Martin space as a maximum a posteriori estimate of the underlying Brownian motion given the observation of a finite-dimensional functional. The definition of the MAP estimator is possible via a renormalization of the densities of piecewise linear approximations of the Brownian motion. Using the renormalized Brownian density the Laplace approximation of the integral over all Brownian paths can be defined. The developed theory provides a method for performing approximate maximum likelihood estimation

Simultaneous inference for misaligned multivariate functional data

We consider inference for misaligned multivariate functional data that represents the same underlying curve, but where the functional samples have systematic differences in shape. In this paper we introduce a new class of generally applicable models where warping effects are modeled through nonlinear transformation of latent Gaussian variables and systematic shape differences are modeled by Gaussian processes. To model cross-covariance between sample coordinates we introduce a class of low-dimensional cross-covariance structures suitable for modeling multivariate functional data. We present a method for doing maximum-likelihood estimation in the models and apply the method to three data sets. The first data set is from a motion tracking system where the spatial positions of a large number of body-markers are tracked in three-dimensions over time. The second data set consists of height and weight measurements for Danish boys. The third data set consists of three-dimensional spatial hand paths from a controlled obstacle-avoidance experiment. We use the developed method to estimate the cross-covariance structure, and use a classification setup to demonstrate that the method outperforms state-of-the-art methods for handling misaligned curve data.Comment: 44 pages in total including tables and figures. Additional 9 pages of supplementary material and reference

Statistical modelling of conidial discharge of entomophthoralean fungi using a newly discovered Pandora species

Entomophthoralean fungi are insect pathogenic fungi and are characterized by their active discharge of infective conidia that infect insects. Our aim was to study the effects of temperature on the discharge and to characterize the variation in the associated temporal pattern of a newly discovered Pandora species with focus on peak location and shape of the discharge. Mycelia were incubated at various temperatures in darkness, and conidial discharge was measured over time. We used a novel modification of a statistical model (pavpop), that simultaneously estimates phase and amplitude effects, into a setting of generalized linear models. This model is used to test hypotheses of peak location and discharge of conidia. The statistical analysis showed that high temperature leads to an early and fast decreasing peak, whereas there were no significant differences in total number of discharged conidia. Using the proposed model we also quantified the biological variation in the timing of the peak location at a fixed temperature.Comment: 23 pages including supplementary materia

Gaussian counter models for visual identification of briefly presented, mutually confusable single stimuli in pure accuracy tasks

Highlights: • Visual identification of single stimuli in pure accuracy task is investigated. • Multivariate Wiener and Ornstein–Uhlenbeck counter models are proposed and tested. • Two classes of models, race and first passage time models, are proposed and analyzed. • The models are compared with the Poisson counter model from the literature. • Model selection favors Gaussian race models over Poisson or first passage time models. Abstract: When identifying confusable visual stimuli, accumulation of information over time is an obvious strategy of the observer. However, the nature of the accumulation process is unresolved: for example it may be discrete or continuous in terms of the information encoded. Another unanswered question is whether or not stimulus sampling continues after the stimulus offset. In the present paper we propose various continuous Gaussian counter models of the time course of visual identification of briefly presented, mutually confusable single stimuli in a pure accuracy task. During stimulus analysis, tentative categorizations that stimulus belongs to category are made until a maximum time after the stimulus disappears. Two classes of models are proposed. First, the overt response is based on the categorization that had the highest value at the time the stimulus disappears (race models). Second, the overt response is based on the categorization that made the minimum first passage time through a constant boundary (first passage time models). Within this framework, multivariate Wiener and Ornstein–Uhlenbeck counter models are considered under different parameter regimes, assuming either that the stimulus sampling stops immediately or that it continues for some time after the stimulus offset. Each type of model was evaluated by Monte Carlo tests of goodness of fit against observed probability distributions of responses in two extensive experiments. A comparison of these continuous models with a simple discrete Poisson counter model proposed by Kyllingsbæk, Markussen, and Bundesen (2012) was carried out, together with model selection among the competing candidates. Both the Wiener and the Ornstein–Uhlenbeck race models provide a close fit to individual data on identification of both digits and Landolt rings, outperforming the first passage time model and the Poisson counter race model