12,720 research outputs found
Semiparametric estimation of spectral density function for irregular spatial data
Estimation of the covariance structure of spatial processes is of fundamental
importance in spatial statistics. In the literature, several non-parametric and
semi-parametric methods have been developed to estimate the covariance
structure based on the spectral representation of covariance functions.
However,they either ignore the high frequency properties of the spectral
density, which are essential to determine the performance of interpolation
procedures such as Kriging, or lack of theoretical justification. We propose a
new semi-parametric method to estimate spectral densities of isotropic spatial
processes with irregular observations. The spectral density function at low
frequencies is estimated using smoothing spline, while a parametric model is
used for the spectral density at high frequencies, and the parameters are
estimated by a method-of-moment approach based on empirical variograms at small
lags. We derive the asymptotic bounds for bias and variance of the proposed
estimator. The simulation study shows that our method outperforms the existing
non-parametric estimator by several performance criteria.Comment: 29 pages, 2 figure
Half-tapering strategy for conditional simulation with large datasets
Gaussian conditional realizations are routinely used for risk assessment and
planning in a variety of Earth sciences applications. Conditional realizations
can be obtained by first creating unconditional realizations that are then
post-conditioned by kriging. Many efficient algorithms are available for the
first step, so the bottleneck resides in the second step. Instead of doing the
conditional simulations with the desired covariance (F approach) or with a
tapered covariance (T approach), we propose to use the taper covariance only in
the conditioning step (Half-Taper or HT approach). This enables to speed up the
computations and to reduce memory requirements for the conditioning step but
also to keep the right short scale variations in the realizations. A criterion
based on mean square error of the simulation is derived to help anticipate the
similarity of HT to F. Moreover, an index is used to predict the sparsity of
the kriging matrix for the conditioning step. Some guides for the choice of the
taper function are discussed. The distributions of a series of 1D, 2D and 3D
scalar response functions are compared for F, T and HT approaches. The
distributions obtained indicate a much better similarity to F with HT than with
T.Comment: 39 pages, 2 Tables and 11 Figure
Covariance matrix estimation for stationary time series
We obtain a sharp convergence rate for banded covariance matrix estimates of
stationary processes. A precise order of magnitude is derived for spectral
radius of sample covariance matrices. We also consider a thresholded covariance
matrix estimator that can better characterize sparsity if the true covariance
matrix is sparse. As our main tool, we implement Toeplitz [Math. Ann. 70 (1911)
351-376] idea and relate eigenvalues of covariance matrices to the spectral
densities or Fourier transforms of the covariances. We develop a large
deviation result for quadratic forms of stationary processes using m-dependence
approximation, under the framework of causal representation and physical
dependence measures.Comment: Published in at http://dx.doi.org/10.1214/11-AOS967 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Aggregation and long memory: recent developments
It is well-known that the aggregated time series might have very different
properties from those of the individual series, in particular, long memory. At
the present time, aggregation has become one of the main tools for modelling of
long memory processes. We review recent work on contemporaneous aggregation of
random-coefficient AR(1) and related models, with particular focus on various
long memory properties of the aggregated process
Pseudo-nonstationarity in the scaling exponents of finite-interval time series
The accurate estimation of scaling exponents is central in the observational study of scale-invariant phenomena. Natural systems unavoidably provide observations over restricted intervals; consequently, a stationary stochastic process (time series) can yield anomalous time variation in the scaling exponents, suggestive of nonstationarity. The variance in the estimates of scaling exponents computed from an interval of N observations is known for finite variance processes to vary as ~1/N as N for certain statistical estimators; however, the convergence to this behavior will depend on the details of the process, and may be slow. We study the variation in the scaling of second-order moments of the time-series increments with N for a variety of synthetic and “real world” time series, and we find that in particular for heavy tailed processes, for realizable N, one is far from this ~1/N limiting behavior. We propose a semiempirical estimate for the minimum N needed to make a meaningful estimate of the scaling exponents for model stochastic processes and compare these with some “real world” time series
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