Simultaneous Inference for Misaligned Multivariate Functional Data

Abstract

Summary We consider inference for misaligned multivariate functional data that represents the same underlying curve, but where the functional samples have systematic differences in shape. We introduce a class of generally applicable models where warping effects are modelled through non‐linear transformation of latent Gaussian variables and systematic shape differences are modelled by Gaussian processes. To model cross‐covariance between sample co‐ordinates we propose a class of low dimensional cross‐covariance structures that are suitable for modelling 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 longitudinal 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 method to estimate the cross‐covariance structure and use a classification set‐up to demonstrate that the method outperforms state of the art methods for handling misaligned curve data.

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

“…Examples of statistical analyses (some using the pavpop model) of other biological systems, where a model of the temporal variation was essential for the data analysis and interpretation of results, include electrophoretic spectra of cheese [28], growth of boys [23] and hand movements [23, 24].…”

“…Estimation is a major challenge, as direct estimation is not feasible due to the large number of latent variables. Furthermore, unlike [23], the response is not Gaussian, which require additional considerations. We propose to use a twofold Laplace approximation for doing approximate maximum likelihood estimation; details on the Laplace approximation are found in the appendix.…”

“…The main difference from the estimation procedure of [23] is the non-trivial addition of a second layer of latent variables, u .…”

“…We intend to follow the novel methodology of [22], which we will refer to as the pavpop model ( Phase and Amplitude Variation of POPulation means ). This methodology has been used in different applications with great success [23, 24]. The main idea of [22] is simultaneous modelling of amplitude and temporal variation, where temporal variation is modelled as a spline interpolation of a latent Gaussian variable that represents temporal deviation from the idealised system.…”

“…The main idea of [22] is simultaneous modelling of amplitude and temporal variation, where temporal variation is modelled as a spline interpolation of a latent Gaussian variable that represents temporal deviation from the idealised system. For a review of methods for handling misaligned functional data, we refer to [22, 23].…”