Authors: Christopher K. Wikle*, Scott H. Holan* and Noel Cressie*+
Spatio-temporal statistical modeling has seen tremendous growth as a subdiscipline in recent years. The associated methodological and theoretical developments have been driven primarily by problems emerging from the environmental sciences. These include issues related to climate change, ocean/atmosphere forecasting and data assimilation, ecological prediction of invasive species, and animal movement, to name a few. Two components have proven to be exceptionally useful in this development: the hierarchical spatio-temporal modeling framework, and the use of dynamical spatio-temporal models (DSTMs). These ideas and methodologies may prove to be equally useful in analyzing data associated with Federal surveys. We discuss a few cases where such methodologies appear to be naturally applicable.
Before we provide specific examples, it is instructive to consider the hierarchical modeling perspective. Although the notion of hierarchical modeling has been around for nearly 50 years, it is only relatively recently, with the advent of associated computational methods, that this framework has made a dramatic impact. In the context of environmental science, as reviewed in Cressie and Wikle (2011), hierarchical modeling in the environmental sciences is based on the simple, yet powerful notion that complicated real-world processes can be modeled in a conditional and probabilistically coherent framework that clearly distinguishes the data, process, and parameter portions of models. For example, within the modeling hierarchy, let “D” correspond to “data,” “Y” to “process,” and “P” correspond to parameters. The total law of probability allows one to express the joint distribution of D, Y, and P (i.e., [D,Y,P], where we are using the brackets to denote a generic probability distribution as is customary in hierarchical modeling) in terms of the product of conditional distributions: [D,Y,P] = [D|Y,P][Y|P][P]. In this expression, the vertical line means “conditional on”; that is, for two generic random variables Z and X, [Z|X] denotes the conditional distribution of Z, conditional on X. In the Bayesian context, one can then consider learning about the process and parameters given the data through: [Y,P | D] α [D|Y,P][Y|P][P], where α denotes “proportional to,” and the constant of proportionality (i.e., the normalizing constant) ensures that the integral of the left-hand side over the process and parameters is equal to 1. Such a model is referred to as a Bayesian hierarchical model (BHM). In some cases, we do not put distributions on the parameters, but we estimate them from the data, so that [Y| D, P] α [D|Y,P][Y|P], which is sometimes called an empirical hierarchical model (EHM).
The key to both the BHM and EHM approaches is that we have specified a data model (i.e., [D|Y, P]) that has, conditional upon the process, a greatly simplified error structure. It also allows one to easily account for multiple sources of information informing the same process. Similarly, this framework allows one to focus on the process of interest (Y) separately from the data, which often makes it possible to incorporate scientific information into this model specification. This permits one to incorporate temporal and/or spatial dependencies simply (e.g., Markovian structures in DSTMs) at the process-modeling stage of the hierarchy. Finally, in the BHM framework, one can also specify distributions for the parameters, allowing complex dependence structures such as temporally and/or spatially dependent parameters. Critically, each of these primary stages (data, process, and parameters) can be decomposed into sub-stages in more complex modeling situations.
We have recently been involved in transformative research that brings several of the hierarchical spatio-temporal modeling approaches to various problems associated with the analysis of US Federal surveys.
In the environmental sciences, much effort has been devoted to developing models for all three components of the hierarchical framework. These include issues of change-of- support in the data model (accounting for variations in scale, resolution, and alignment between the observations and the process), dynamical models for the process (in some cases including complex linear and non-linear dynamics), and parameters that may be restructured to ensure physically realistic process behavior, or they may be allowed to be random with dependence structures that allow realistic adaptation to the process and data. These models can be quite complicated, and typically advanced computational methodologies must be used for their implementation (e.g., Markov chain Monte Carlo for the BHM and the E-M algorithm for the EHM).
We have recently been involved in transformative research that brings several of the hierarchical spatio-temporal modeling approaches to various problems associated with the analysis of US Federal surveys. These data are inherently spatio-temporal, with the added complications associated with sample surveys and their uncertainty. For example, such spatio-temporal methodology is well suited for addressing the problem of small area estimation, where salient examples include change-of-support issues, spatially dependent errors, and the incorporation of functional covariates.
For motivating our research, we consider questions associated with the American Community Survey (ACS). The ACS contains many demographic, social, and economic variables. The publicly released ACS data have a unique structure comprised of rolling multiyear estimates (MYEs). The survey’s broad coverage and timely dissemination make the ACS an ideal resource for investigating a broad range of research and policy questions across a range of spatial and temporal scales. Importantly, the shift from the decennial census long-form data to using MYEs from the ACS offers many statistical challenges.
In the ACS (as in other Federal surveys), there are many situations in socio-economic- demographic data analysis in which the data come aggregated at various levels of support across space and time (e.g., census tracts and Public Use Microdata Areas or PUMAs). It is often advantageous to consider a common level of support for the associated “true” dynamical process in order to facilitate inference and prediction (subject to the usual caveats related to the modifiable areal unit problem). One can envision using the hierarchical modeling structure to accommodate such change-of-support for DSTMs, allowing for efficient borrowing of information across space and time. It should be noted that although we use the terminology, “process,” in this context, we also include as process other “true” latent quantities of interest such as the superpopulation mean.
In an age where one has available a wide variety of non-standard auxiliary information such as social-media search loads or satellite observations of various land-surface characteristics, it may prove worthwhile to include such “big data” covariates in small area estimation.
As another example, consider the Fay-Herriot model, which is widely used in small area estimation. The power of this model is that it uses auxiliary information to model undersampled locations so as to reduce estimation variance. In an age where one has available a wide variety of non-standard auxiliary information such as social-media search loads or satellite observations of various land-surface characteristics, it may prove worthwhile to include such “big data” covariates in small area estimation. Clearly, such information comes spatially and temporally referenced, and they can be quite voluminous. Thus, it would be useful to extend the type of covariate information used in the FH model to include such “functional” covariates. The inclusion of these functional covariates may be facilitated through dimension reduction methods that include stochastic search variable selection.
Additionally, the fact that there is inherent spatial and temporal autocorrelation in the underlying small area responses suggests that Fay-Herriot models should include explicit spatial and temporal error structures (e.g., conditional autoregressive spatial models, autoregressive temporal models, and DSTMs). In particular, given the relationship between various demographic processes and underlying quantities of interest that are measured from Federal-survey data, much of the historical development in human and ecological population demographics, as well as spatial epidemiology, can be used within the hierarchical framework to inform the underlying data and process distributions.
In conclusion, although much of the methodological and theoretical development of spatial and spatio-temporal statistics has been in the environmental sciences, the hierarchical-modeling framework is ideal for providing “technology transfer” to inform relevant data and process models in the Federal-survey-statistics realm. To be sure, the transfer of these ideas to this new realm is not without its unique challenges (e.g., sampling errors, data confidentiality, multi-year estimates), but the framework is there. The next decade looks to be an exciting time for advancing the spatio-temporal complexity of the analysis of Federal survey data.
Cressie, N. and C.K. Wikle (2011) Statistics for Spatio-Temporal Data. John Wiley & Sons, Hoboken, NJ.
Professor Noel Cressie is Adjunct Professor in the Department of Statistics, University of Missouri-Columbia, and he is Distinguished Professor in the National Institute for Applied Statistics Research Australia (NIASRA), School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia.