|Spatial Processes with Stochastic Heteroscadesticity
| Wenying Huang
Ph.D. candidate, Colorado State University, Department of Statistics
Wednesday, February 28, 2007
Spatial data modeling and analysis aim at the description, explanation and prediction of a spatial process based on a sample of observations. Spatial modeling and analysis have been widely used in areas such as meteorology, ecology, environmental health, computer experiment modeling and so on. For example, the Northeast Fisheries Center of the National Marine Fisheries Service in Woods Hole, Massachusetts, samples on the continental shelf oﬀ the Northeastern United States to estimate the abundance of sea scallops and other shellﬁsh. The scallops data have been studied by many researchers to explore spatial association and make predictions of abundance at unsampled locations.
Let Y (s) denotes a spatial process over a domain D. For most cases, D will be a subset of Rd with positive volume. Stationary Gaussian processes are widely used in modeling spatial data. The Gaussian process is completely characterized by its mean and covariance structure. The covariance function explores the spatial dependence of deviations from the mean. A Gaussian process Y (·) is stationary if its mean is constant and the covariance between Y (s) and Y (t) only depends on s − t. A stationary process is isotropic if the covariance between Y (s)and Y (t) is a function solely of s−t. Parametric covariance/correlation functions such as exponential, Gaussian and Mat´ern are commonly used in practice. For ﬁtting Gaussian processes to spatial data, these covariance functions may be modiﬁed through the addition of a nugget, which corresponds to the height of the jump discontinuity of the covariance at s = 0. The nugget can be interpreted as the variance component due to measurement error and microscale variation.
The classical approach to spatial prediction is kriging. Kriging is an approach for computing Best Linear Unbiased Predictor (BLUP) or empirical BLUP. Hierarchical modeling and Bayesian analysis have also been developed in recent years and are well suited for nonlinear modeling problems.
Recognizing that isotropy and stationarity are assumptions regarding spatial association that will rarely hold in practice, some anisotropic covariance functions and a variety of non-stationary spatial process models have been developed. But most of these approaches have severe limitations. There is no universally accepted approach. It is desired to develop new methods that are capable of modeling a wide variety of spatial processes and are attractively interpretable.
In ﬁnancial time series data analysis, stochastic volatility (SV) models have been successfully applied to capture the conditional heteroscedasticity or volatility clustering. SV models the conditional variance as a log-normal latent process. Yan (2006) proposes a spatial stochastic volatility (SSV) model for lattice data, in which the spatial domain D is ﬁnite
or countable. In analogy, by introducing stochastic volatility into a Gaussian process, we propose a stochastic heteroscedastic process (SHP). Conditional on a latent Gaussian process, the SHP is a Gaussian process with non-stationary covariance structure. Unconditionally, the SHP is a stationary non-Gaussian process. The realizations from SHP are versatile and can represent spatial inhomogeneity. The unconditional correlation of SHP oﬀers a very rich class of correlation functions which can also allow for a smoothed nugget eﬀect. This smoothed nugget explains the microscale variation in a natural way. The unconditional correlation function can be used independently as a ﬂexible isotropic correlation class.
There are diﬃculties in estimating parameters for SHP model due to the high-dimensional latent vector. The existence of correlation in the latent process leads to slow convergence in a Bayesian approach. For maximum likelihood estimation, we propose to apply importance sampling in the likelihood calculation and latent process estimation. From our preliminary simulations, the importance density we proposed is computationally eﬃcient and maximum likelihood estimates perform well. Empirical BLUP can then be used for prediction.
In part of this dissertation research, we will improve the estimation and prediction strategies. Some potential modiﬁcations and extensions of the SHP model will also be developed. We will apply the SHP model and its unconditional correlation function to real data. Multivariate responses and gradient prediction will also be considered.