Bayesian analysis of spatially-dependent functional responses with spatially-dependent multi-dimensional functional predictors

Authors: YANG, WEN-HSI - Commonwealth Scientific And Industrial Research Organisation (CSIRO); WIKLE, CHRISTOPHER - University Of Missouri; HOLAN, SCOTT - University Of Missouri; MYERS, DAIVD - University Of Missouri; Sudduth, Kenneth - Ken

Submitted to: STATISTICA SINICA
Publication Type: Peer Reviewed Journal
Publication Acceptance Date: 2/25/2014
Publication Date: 12/21/2014
Citation: Yang, W., Wikle, C.K., Holan, S.H., Myers, D.B., Sudduth, K.A. 2015. Bayesian analysis of spatially-dependent functional responses with spatially-dependent multi-dimensional functional predictors. Statistica Sinica. 25:205-223.

Interpretive Summary: Scientists in many fields are often confronted with analyzing and interpreting datasets containing a large number of related variables. An example is in proximal soil sensing, where spectrometers may be used to collect soil reflectance data at hundreds or even thousands of wavelengths. Relating those large datasets to parameters of interest requires advanced multivariate statistical techniques. A further complication is that the data are often collected at multiple locations or over time, leading to the need to account for spatial and/or temporal dependence in the analysis. In the past, this type of analysis has often been done in a two-step procedure, which leads to inefficiencies. This research developed a new, merged approach called multi-dimensional spatial functional models. The approach was tested with soil spectroscopy data collected over multiple depths at multiple locations in an experimental site. Using multi-dimensional spatial functional models, reasonable maps of soil properties were obtained over the site. This new approach has potential for improved interpretation of large datasets such as those obtained in proximal soil sensing. This could enhance the utilization of these data by scientists and practitioners in the fields of precision agriculture and digital soil mapping.

Technical Abstract: Recent advances in technology have led to the collection of high-dimensional data not previously encountered in many scientific environments. As a result, scientists are often faced with the challenging task of including these high-dimensional data into statistical models. For example, data from sensor networks, near infrared spectroscopy, and satellite imagery, among others, produce high-dimensional data in the form of functional-curves and/or multi-dimensional functional objects. Modeling functional observations utilizing multi-dimensional functional covariates is often complicated by spatial and/or temporal dependence in the observations and high-dimensional predictors. To utilize these rich sources of information we develop multi-dimensional spatial functional models. These models utilize hierarchical Bayesian methodology and can take advantage of stochastic search variable selection. Importantly, we employ low-rank basis function expansions to facilitate model development. In this context, our model accounts for several sources of error, including the truncation error that arises from truncating the several infinite-dimensional basis function expansions. We illustrate this methodology through spatial models of soil electrical conductivity depth profiles using spatially dependent near infrared spectral images of electrical conductivity covariates.