10,890 research outputs found
Bayesian Lattice Filters for Time-Varying Autoregression and Time-Frequency Analysis
Modeling nonstationary processes is of paramount importance to many
scientific disciplines including environmental science, ecology, and finance,
among others. Consequently, flexible methodology that provides accurate
estimation across a wide range of processes is a subject of ongoing interest.
We propose a novel approach to model-based time-frequency estimation using
time-varying autoregressive models. In this context, we take a fully Bayesian
approach and allow both the autoregressive coefficients and innovation variance
to vary over time. Importantly, our estimation method uses the lattice filter
and is cast within the partial autocorrelation domain. The marginal posterior
distributions are of standard form and, as a convenient by-product of our
estimation method, our approach avoids undesirable matrix inversions. As such,
estimation is extremely computationally efficient and stable. To illustrate the
effectiveness of our approach, we conduct a comprehensive simulation study that
compares our method with other competing methods and find that, in most cases,
our approach performs superior in terms of average squared error between the
estimated and true time-varying spectral density. Lastly, we demonstrate our
methodology through three modeling applications; namely, insect communication
signals, environmental data (wind components), and macroeconomic data (US gross
domestic product (GDP) and consumption).Comment: 49 pages, 16 figure
Bayesian Nonstationary Spatial Modeling for Very Large Datasets
With the proliferation of modern high-resolution measuring instruments
mounted on satellites, planes, ground-based vehicles and monitoring stations, a
need has arisen for statistical methods suitable for the analysis of large
spatial datasets observed on large spatial domains. Statistical analyses of
such datasets provide two main challenges: First, traditional
spatial-statistical techniques are often unable to handle large numbers of
observations in a computationally feasible way. Second, for large and
heterogeneous spatial domains, it is often not appropriate to assume that a
process of interest is stationary over the entire domain.
We address the first challenge by using a model combining a low-rank
component, which allows for flexible modeling of medium-to-long-range
dependence via a set of spatial basis functions, with a tapered remainder
component, which allows for modeling of local dependence using a compactly
supported covariance function. Addressing the second challenge, we propose two
extensions to this model that result in increased flexibility: First, the model
is parameterized based on a nonstationary Matern covariance, where the
parameters vary smoothly across space. Second, in our fully Bayesian model, all
components and parameters are considered random, including the number,
locations, and shapes of the basis functions used in the low-rank component.
Using simulated data and a real-world dataset of high-resolution soil
measurements, we show that both extensions can result in substantial
improvements over the current state-of-the-art.Comment: 16 pages, 2 color figure
Modeling Non-Stationary Processes Through Dimension Expansion
In this paper, we propose a novel approach to modeling nonstationary spatial
fields. The proposed method works by expanding the geographic plane over which
these processes evolve into higher dimensional spaces, transforming and
clarifying complex patterns in the physical plane. By combining aspects of
multi-dimensional scaling, group lasso, and latent variables models, a
dimensionally sparse projection is found in which the originally nonstationary
field exhibits stationarity. Following a comparison with existing methods in a
simulated environment, dimension expansion is studied on a classic test-bed
data set historically used to study nonstationary models. Following this, we
explore the use of dimension expansion in modeling air pollution in the United
Kingdom, a process known to be strongly influenced by rural/urban effects,
amongst others, which gives rise to a nonstationary field
Compression and Conditional Emulation of Climate Model Output
Numerical climate model simulations run at high spatial and temporal
resolutions generate massive quantities of data. As our computing capabilities
continue to increase, storing all of the data is not sustainable, and thus it
is important to develop methods for representing the full datasets by smaller
compressed versions. We propose a statistical compression and decompression
algorithm based on storing a set of summary statistics as well as a statistical
model describing the conditional distribution of the full dataset given the
summary statistics. The statistical model can be used to generate realizations
representing the full dataset, along with characterizations of the
uncertainties in the generated data. Thus, the methods are capable of both
compression and conditional emulation of the climate models. Considerable
attention is paid to accurately modeling the original dataset--one year of
daily mean temperature data--particularly with regard to the inherent spatial
nonstationarity in global fields, and to determining the statistics to be
stored, so that the variation in the original data can be closely captured,
while allowing for fast decompression and conditional emulation on modest
computers
A dynamic nonstationary spatio-temporal model for short term prediction of precipitation
Precipitation is a complex physical process that varies in space and time.
Predictions and interpolations at unobserved times and/or locations help to
solve important problems in many areas. In this paper, we present a
hierarchical Bayesian model for spatio-temporal data and apply it to obtain
short term predictions of rainfall. The model incorporates physical knowledge
about the underlying processes that determine rainfall, such as advection,
diffusion and convection. It is based on a temporal autoregressive convolution
with spatially colored and temporally white innovations. By linking the
advection parameter of the convolution kernel to an external wind vector, the
model is temporally nonstationary. Further, it allows for nonseparable and
anisotropic covariance structures. With the help of the Voronoi tessellation,
we construct a natural parametrization, that is, space as well as time
resolution consistent, for data lying on irregular grid points. In the
application, the statistical model combines forecasts of three other
meteorological variables obtained from a numerical weather prediction model
with past precipitation observations. The model is then used to predict
three-hourly precipitation over 24 hours. It performs better than a separable,
stationary and isotropic version, and it performs comparably to a deterministic
numerical weather prediction model for precipitation and has the advantage that
it quantifies prediction uncertainty.Comment: Published in at http://dx.doi.org/10.1214/12-AOAS564 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Markov Processes, Hurst Exponents, and Nonlinear Diffusion Equations with application to finance
We show by explicit closed form calculations that a Hurst exponent H that is
not 1/2 does not necessarily imply long time correlations like those found in
fractional Brownian motion. We construct a large set of scaling solutions of
Fokker-Planck partial differential equations where H is not 1/2. Thus Markov
processes, which by construction have no long time correlations, can have H not
equal to 1/2. If a Markov process scales with Hurst exponent H then it simply
means that the process has nonstationary increments. For the scaling solutions,
we show how to reduce the calculation of the probability density to a single
integration once the diffusion coefficient D(x,t) is specified. As an example,
we generate a class of student-t-like densities from the class of quadratic
diffusion coefficients. Notably, the Tsallis density is one member of that
large class. The Tsallis density is usually thought to result from a nonlinear
diffusion equation, but instead we explicitly show that it follows from a
Markov process generated by a linear Fokker-Planck equation, and therefore from
a corresponding Langevin equation. Having a Tsallis density with H not equal to
1/2 therefore does not imply dynamics with correlated signals, e.g., like those
of fractional Brownian motion. A short review of the requirements for
fractional Brownian motion is given for clarity, and we explain why the usual
simple argument that H unequal to 1/2 implies correlations fails for Markov
processes with scaling solutions. Finally, we discuss the question of scaling
of the full Green function g(x,t;x',t') of the Fokker-Planck pde.Comment: to appear in Physica
Smoothing and mean-covariance estimation of functional data with a Bayesian hierarchical model
Functional data, with basic observational units being functions (e.g.,
curves, surfaces) varying over a continuum, are frequently encountered in
various applications. While many statistical tools have been developed for
functional data analysis, the issue of smoothing all functional observations
simultaneously is less studied. Existing methods often focus on smoothing each
individual function separately, at the risk of removing important systematic
patterns common across functions. We propose a nonparametric Bayesian approach
to smooth all functional observations simultaneously and nonparametrically. In
the proposed approach, we assume that the functional observations are
independent Gaussian processes subject to a common level of measurement errors,
enabling the borrowing of strength across all observations. Unlike most
Gaussian process regression models that rely on pre-specified structures for
the covariance kernel, we adopt a hierarchical framework by assuming a Gaussian
process prior for the mean function and an Inverse-Wishart process prior for
the covariance function. These prior assumptions induce an automatic
mean-covariance estimation in the posterior inference in addition to the
simultaneous smoothing of all observations. Such a hierarchical framework is
flexible enough to incorporate functional data with different characteristics,
including data measured on either common or uncommon grids, and data with
either stationary or nonstationary covariance structures. Simulations and real
data analysis demonstrate that, in comparison with alternative methods, the
proposed Bayesian approach achieves better smoothing accuracy and comparable
mean-covariance estimation results. Furthermore, it can successfully retain the
systematic patterns in the functional observations that are usually neglected
by the existing functional data analyses based on individual-curve smoothing.Comment: Submitted to Bayesian Analysi
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