Developing Statistical Methods for Spatial and Spatio-Temporal Prediction with Applications to Air Quality Data

Yang, Lifeng (2021) Developing Statistical Methods for Spatial and Spatio-Temporal Prediction with Applications to Air Quality Data. University of Southampton, Doctoral Thesis, 130pp.

Record type: Thesis (Doctoral)

Abstract

Prediction at an unobserved location for spatial and spatial time-series data, also known as Kriging, with complex structures is flourishing in a wide range of disciplines lately. It acts as a powerful tool capable of revealing meaningful insights by studying, seemingly isolated, spatial information on the subjects of
interest. Despite the vast demand, methods for exploring spatial data collected at irregularly spaced sampling locations remain limited to mostly parametric linear techniques owning to the multi-lateral nature of space. We aim to provide semiparametric nonlinear alternatives to these applications.

The current linear spatial prediction methods for spatial data, conventionally are based on an assumption that the underlying spatial data-generating process can be decomposed into two components: a deterministic linear trend function1 and a Gaussian stochastic process2. In practice, such an assumption may not be
reasonable as the linear-structured spatial trend function and the Gaussian stochastic process may not be true. We hence develop new ideas in this thesis. Firstly, a nonparametric-trend universal Kriging (NTUK) method is proposed by replacing the deterministic linear component1 with a nonparametric local linear
fitting regression function, as such the solution space of the trend function is vastly enlarged. Secondly, we adopt a semiparametric model structure, i.e., the model averaging marginal regression approximation for Kriging. Through a nonparametric estimation of spatial probability density functions, an affine combination of one-dimensional conditional marginal regression functions is used for approximation in Kriging.

By suggesting a K-radius averaging function to the Kriging, the stochastic process2 part which is not assumed Gaussian is also predicted. A complete semiparametric spatial nonlinear prediction procedure is thus developed.
In spatial time-series setting, we further extend our developed methods above to the prediction of the future data at an unobserved location. We integrate the above semiparametric spatial nonlinear prediction procedure with a semiparametric spatio-temporal nonlinear regression model, which allows the spatiotemporal random field to be non-stationary over space (but stationary along time; for time series, say, through differencing) while the sampling spatial grids can be irregular. Hence the proposed model uses a two-phase framework performing firstly a spatio-temporal forecasting for a future time at the observed locations, followed by our spatial nonlinear prediction procedure stated above.

Empirical applications to air quality data are demonstrated. The performances of the proposed models are evaluated against those obtained from linear methods with significant improvement.