Spatial interpolation of high-frequency monitoring data

Climate modelers generally require meteorological information on regular grids, but monitoring stations are, in practice, sited irregularly. Thus, there is a need to produce public data records that interpolate available data to a high density grid, which can then be used to generate meteorological maps at a broad range of spatial and temporal scales. In addition to point predictions, quantifications of uncertainty are also needed. One way to accomplish this is to provide multiple simulations of the relevant meteorological quantities conditional on the observed data taking into account the various uncertainties in predicting a space-time process at locations with no monitoring data. Using a high-quality dataset of minute-by-minute measurements of atmospheric pressure in north-central Oklahoma, this work describes a statistical approach to carrying out these conditional simulations. Based on observations at 11 stations, conditional simulations were produced at two other sites with monitoring stations. The resulting point predictions are very accurate and the multiple simulations produce well-calibrated prediction uncertainties for temporal changes in atmospheric pressure but are substantially overconservative for the uncertainties in the predictions of (undifferenced) pressure.Comment: Published in at http://dx.doi.org/10.1214/08-AOAS208 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Spatial variation of total column ozone on a global scale

The spatial dependence of total column ozone varies strongly with latitude, so that homogeneous models (invariant to all rotations) are clearly unsuitable. However, an assumption of axial symmetry, which means that the process model is invariant to rotations about the Earth's axis, is much more plausible and considerably simplifies the modeling. Using TOMS (Total Ozone Mapping Spectrometer) measurements of total column ozone over a six-day period, this work investigates the modeling of axially symmetric processes on the sphere using expansions in spherical harmonics. It turns out that one can capture many of the large scale features of the spatial covariance structure using a relatively small number of terms in such an expansion, but the resulting fitted model provides a horrible fit to the data when evaluated via its likelihood because of its inability to describe accurately the process's local behavior. Thus, there remains the challenge of developing computationally tractable models that capture both the large and small scale structure of these data.Comment: Published at http://dx.doi.org/10.1214/07-AOAS106 in the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Interpolation of nonstationary high frequency spatial-temporal temperature data

The Atmospheric Radiation Measurement program is a U.S. Department of Energy project that collects meteorological observations at several locations around the world in order to study how weather processes affect global climate change. As one of its initiatives, it operates a set of fixed but irregularly-spaced monitoring facilities in the Southern Great Plains region of the U.S. We describe methods for interpolating temperature records from these fixed facilities to locations at which no observations were made, which can be useful when values are required on a spatial grid. We interpolate by conditionally simulating from a fitted nonstationary Gaussian process model that accounts for the time-varying statistical characteristics of the temperatures, as well as the dependence on solar radiation. The model is fit by maximizing an approximate likelihood, and the conditional simulations result in well-calibrated confidence intervals for the predicted temperatures. We also describe methods for handling spatial-temporal jumps in the data to interpolate a slow-moving cold front.Comment: Published in at http://dx.doi.org/10.1214/13-AOAS633 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

A New Measure of the Clustering of QSO Heavy-Element Absorption-Line Systems

We examine the line-of-sight clustering of QSO heavy-element absorption-line systems, using a new measure of clustering, called the reduced second moment measure, that directly measures the mean over-density of absorbers. While closely related to other second-order measures such as the correlation function or the power spectrum, this measure has a number of distinct statistical properties which make possible a continuous exploration of clustering as a function of scale. From a sample of 352 C IV absorbers with median redshift 2.2, drawn from the spectra of 274 QSOs, we find that the absorbers are strongly clustered on scales from 1 to 20 Mpc. Furthermore, there appears to be a sharp break at 20 Mpc, with significant clustering on scales up to 100 Mpc in excess of that which would be expected from a smooth transition to homogeneity. There is no evidence of clustering on scales greater than 100 Mpc. These results suggest that strong C IV absorbers along a line of sight are indicators of clusters and possibly superclusters, a relationship that is supported by recent observations of ``Lyman break'' galaxies.Comment: 13 pages (LaTex, uses aaspp4.sty and psfig.sty), with 3 encapsulated PostScript figures. To appear in The Astrophysical Journal. Extended new discussion of the statistical properties of the reduced second moment measure, and a new figure highlighting the excess clustering on comoving scales greater than 20 Mp

Stochastic approximation of score functions for Gaussian processes

We discuss the statistical properties of a recently introduced unbiased stochastic approximation to the score equations for maximum likelihood calculation for Gaussian processes. Under certain conditions, including bounded condition number of the covariance matrix, the approach achieves $O(n)$ storage and nearly $O(n)$ computational effort per optimization step, where $n$ is the number of data sites. Here, we prove that if the condition number of the covariance matrix is bounded, then the approximate score equations are nearly optimal in a well-defined sense. Therefore, not only is the approximation efficient to compute, but it also has comparable statistical properties to the exact maximum likelihood estimates. We discuss a modification of the stochastic approximation in which design elements of the stochastic terms mimic patterns from a $2^n$ factorial design. We prove these designs are always at least as good as the unstructured design, and we demonstrate through simulation that they can produce a substantial improvement over random designs. Our findings are validated by numerical experiments on simulated data sets of up to 1 million observations. We apply the approach to fit a space-time model to over 80,000 observations of total column ozone contained in the latitude band $40^{\circ}$-$50^{\circ}$N during April 2012.Comment: Published in at http://dx.doi.org/10.1214/13-AOAS627 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org