73 research outputs found
Distribution on Warp Maps for Alignment of Open and Closed Curves
Alignment of curve data is an integral part of their statistical analysis,
and can be achieved using model- or optimization-based approaches. The
parameter space is usually the set of monotone, continuous warp maps of a
domain. Infinite-dimensional nature of the parameter space encourages sampling
based approaches, which require a distribution on the set of warp maps.
Moreover, the distribution should also enable sampling in the presence of
important landmark information on the curves which constrain the warp maps. For
alignment of closed and open curves in , possibly with
landmark information, we provide a constructive, point-process based definition
of a distribution on the set of warp maps of and the unit circle
that is (1) simple to sample from, and (2) possesses the
desiderata for decomposition of the alignment problem with landmark constraints
into multiple unconstrained ones. For warp maps on , the distribution is
related to the Dirichlet process. We demonstrate its utility by using it as a
prior distribution on warp maps in a Bayesian model for alignment of two
univariate curves, and as a proposal distribution in a stochastic algorithm
that optimizes a suitable alignment functional for higher-dimensional curves.
Several examples from simulated and real datasets are provided
Asymptotics of a Clustering Criterion for Smooth Distributions
We develop a clustering framework for observations from a population with a
smooth probability distribution function and derive its asymptotic properties.
A clustering criterion based on a linear combination of order statistics is
proposed. The asymptotic behavior of the point at which the observations are
split into two clusters is examined. The results obtained can then be utilized
to construct an interval estimate of the point which splits the data and
develop tests for bimodality and presence of clusters
Bayesian sensitivity analysis with the Fisher–Rao metric
We propose a geometric framework to assess sensitivity of Bayesian procedures to modelling assumptions based on the nonparametric Fisher–Rao metric. While the framework is general, the focus of this article is on assessing local and global robustness in Bayesian procedures with respect to perturbations of the likelihood and prior, and on the identification of influential observations. The approach is based on a square-root representation of densities, which enables analytical computation of geodesic paths and distances, facilitating the definition of naturally calibrated local and global discrepancy measures. An important feature of our approach is the definition of a geometric ϵ-contamination class of sampling distributions and priors via intrinsic analysis on the space of probability density functions. We demonstrate the applicability of our framework to generalized mixed-effects models and to directional and shape data
Spatially Penalised Registration of Multivariate Functional Data
Registration of multivariate functional data involves handling of both
cross-component and cross-observation phase variations. Allowing for the two
phase variations to be modelled as general diffeomorphic time warpings, in this
work we focus on the hitherto unconsidered setting where phase variation of the
component functions are spatially correlated. We propose an algorithm to
optimize a metric-based objective function for registration with a novel
penalty term that incorporates the spatial correlation between the component
phase variations through a kriging estimate of an appropriate phase random
field. The penalty term encourages the overall phase at a particular location
to be similar to the spatially weighted average phase in its neighbourhood, and
thus engenders a regularization that prevents over-alignment. Utility of the
registration method, and its superior performance compared to methods that fail
to account for the spatial correlation, is demonstrated through performance on
simulated examples and two multivariate functional datasets pertaining to EEG
signals and ozone concentrations. The generality of the framework opens up the
possibility for extension to settings involving different forms of correlation
between the component functions and their phases
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