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<p>Directional data, i.e., data collected in the form of angles or natural directions
arise in many scientific fields, such as oceanography, climatology, geology, meteorology
and biology to name a few. The non-Euclidean nature of such data poses difficulties
in applying ordinary statistical methods developed for inline data, motivating the
need for specialized modeling framework for directional data. Motivated in particular
by a marine application of modeling spatial association of wave directions and additionally
association between spatial wave directions and spatial wave heights, this dissertation
focuses on providing general frameworks of modeling spatial and spatio-temporal directional
data, while also studying the theoretical properties of the proposed methods. In particular,
the projected normal family of circular distributions is proposed as a default parametric
family of distributions for directional data. Operating in a Bayesian framework and
exploiting standard data augmentation techniques, the projected normal family is shown
to have straightforward extensions to the regression and process setting. </p><p>
</p><p>A fully model-based approach is developed to capture structured spatial dependence
for modeling directional data at different spatial locations. A stochastic process
taking values on the circle, a projected Gaussian spatial process, is introduced.
This spatial angular process is induced from an inline bivariate Gaussian process.
The properties of the projected Gaussian process is discussed with special emphasis
on the ``covariance'' structure. We show how to fit this process as a model for data,
using suitable latent variables with Markov chain Monte Carlo methods. We also show
how to implement spatial interpolation and conduct model comparison in this setting.
Simulated examples are provided as proof of concept. A real data application arises
for modeling the aforementioned wave direction data in the Adriatic sea, off the coast
of Italy. This directional data being available dynamically, naturally motivated extension
to a space-time setting. </p><p>As the basis of the projected Gaussian process, the
properties of the general projected normal distribution is first clarified. The general
projected normal distribution on a circle is defined to be the distribution of a bivariate
normal random variable with arbitrary mean and covariance, projected on the unit circle.
The projected normal distribution is an under-utilized model for explaining directional
data. In particular, the general version with non-identity covariance provides flexibility,
e.g., bimodality, asymmetry, and convenient regression specification. </p><p>For analyzing
non-spatial circular data, fully Bayesian hierarchical models using the general projected
normal distribution are developed and fitting using Markov chain Monte Carlo methods
with suitable latent variables is illustrated. The posterior inference for distributional
features such as the angular mean direction and concentration can be implemented as
well as how prediction within the regression setting can be handled. For analyzing
spatial directional data, latent variables are also introduced to facilitate the model
fitting with MCMC methods. The implementation of spatial interpolation and conduction
of model comparison are demonstrated. With regard to model comparison, an out-of-sample
approach using both a predictive likelihood scoring loss criterion and a cumulative
rank probability score criterion is utilized.</p><p>This dissertation later focuses
on building model extensions based on the framework of the projected Gaussian process.
The wave directions data studied in the previous chapters also include wave height
information at the same space and time resolution. Motivated by joint modeling of
these important attributes of wave (wave directions and wave heights), a hierarchical
framework is developed for jointly modeling spatial directional and ordinary linear
observations. We show that the Bayesian model fitting under our model specification
is straightforward using suitable latent variable augmentation via Markov chain Monte
Carlo (MCMC). This joint model framework can easily incorporate space-time covariate
information, enabling both spatial interpolation and temporal forecast. </p><p>The
spatial projected Gaussian process also provides a natural application in geosciences
as aspect processes for the elevation maps. Compared to conventional calculations,
a fully process model for aspects is provided, allowing full inference and arbitrary
interpolation. The aspect processes can directly be inferred from a sample from the
surface of elevations, providing the estimate and its uncertainties of the aspect
at any new location over the region.</p>
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