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My research focuses on statistical models and methods
for spatial-temporal processes and their applications to environmental
The key problem, in my view, is to find models that allow for
spatial-temporal interactions and methods for estimating and assessing
fit of such models.
My present focus is on models for Gaussian processes (or processes that
can be made Gaussian after a transformation), so that it suffices to
specify the mean and covariance structure.
The following are the main topics of my personal research;
- Simply stated, the ultimate
goal of any model for space-time covariance
functions is to capture the variance of any
linear combination of observations of a process
To do this, it is essential to consider the
nature of space-time interactions of
the process and not just the
spatial and temporal variations separately.
I have been studying how these interactions
relate to the behavior of
space-time covariance functions away from the origin
and, in turn, how this behavior
depends on the space-time spectral density
- Most atmospheric spatial-temporal
processes are not time reversible.
In particular, the space-time covariance
functions are generally not fully symmetric, in that
covariance between site x
at time s and
site y at time t is
not the same as that between
at time t and
the goals of this work are to have flexible
asymmetries, explicit expressions for the
resulting covariance functions,
and useful diagnostics for assessing asymmetry.
- A continuing interest of mine is the development of
covariance structures for
when the spatial domain is
a sphere, which has obvious relevance
to the atmospheric sciences and climatology. In joint
work with a number of students over the years, including
Marcin Hitczenko, Stefano Castruccio and currently Michael Horrell, I have been
developing approaches to generate
explicit covariance functions
for this setting, including ways of producing
rich classes of space-time asymmetries.
Specific challenges that arise when modeling atmospheric processes on a global
scale include the variation of dependence structures with latitude and on
whether one is over land or water.
- While new and better models
are critical to advancing
spatial-temporal statistics, one
also needs computational and
graphical techniques for analyzing and summarizing
large space-time datasets.
One setting I have been working on for some time
is regularly collected monitoring data,
for which a combination
of ideas from multiple time
series and spatial statistics is appropriate.
Models and analyses that work in the spectral
domain in time and
the spatial domain in space are natural
for regular monitoring data and lead
to interesting classes of partially nonparametric
models that can be fitted reasonably
easily using spectral methods.
With Joseph Guinness, I have worked on extending these ideas to processes
that are nonstationary in time, which is critical for modeling high frequency
More recently, I have been focusing on computational methods that can
be applied when the observations are not regularly spaced in space or time,
which is commonly the case for many environmental datasets and for which
spectral methods are not so useful. This research makes heavy use of
iterative methods for solving large systems of linear equations.
Collaborators on this work include Mihai Anitescu, Jie Chen and Ying Sun.