Statistical Methods: Latent Trajectory Modeling with ESF (The LTM-ESF Approach)

Landscape modelers often have to deal with changes both over space and over time—Given a time series of spatial data (maps), how can we elegantly embrace changes on space (a strength of traditional GIS) without losing insights into changes over time? On the one hand, social science literature shows that traditional latent trajectory modeling (LTM, or latent growth curve modeling) is powerful in handling temporal change patterns, e.g., providing insight into the trajectory of each study unit over time, with a reasonable assumption that the study units are uncorrelated over space (Preacher et al. 2008). When applying LTM to landscape or spatial analysis, this assumption often fails due to the "curse" of spatial autocorrelation: knowing the status of one unit automatically reveals, often to a large extent, the status of other adjacent or nearby units.

On the other hand, the eigenvector spatial filtering (ESF) has been developed by geographers to handle spatial autocorrelation in many cross-sectional or spatial regression applications (e.g., Griffith 2000, Tiefelsdorf and Griffith 2007). Any chance to take advantage of both LTM, due to its strength in handling temporal changes, and ESF, due to its power in dealing with spatial autocorrelation?

Our answer is YES! CHES member Dr. Li An has been extending the traditional LTM approach to applications in landscape analysis/modeling, geography, and spatial sciences. Creatively connecting ESF to LTM, the LTM-ESF approach empowers modelers to simultaneously take into accounts both spatial and temporal autocorrelations, making landscape modelers capable of elegantly handling variabilities over both time and space (An et al. 2016, Crook et al. 2016). A couple of exemplar references are listed below.

Readings and References:

An, L., M. Tsou, B. Spitzberg, J.M. Gawron, and D.K. Gupta (2016). Latent trajectory models for space-time analysis: An application in deciphering spatial panel data. Geographical Analysis 48 (3): 314–336 (see the link below for "The LTM-ESF Paper").

Crook, S.E.S., L. An, D.A. Stow, and J.R. Weeks (2016). Latent trajectory modeling of spatiotemporal relationships between land cover and land use, socioeconomics, and obesity in Ghana. Spatial Demography 4(3):221-244.

Griffith, D.A. (2000). A linear regression solution to the spatial autocorrelation problem. Journal of Geographical Systems 2 (2):141–156.

Preacher, K. J., A. L. Wichman, R. C. MacCallum, and N. E. Briggs (2008). Latent growth curve modeling. Los Angeles: SAGE Publications.

Tiefelsdorf, M., and D. A. Griffith (2007). Semiparametric filtering of spatial autocorrelation: The eigenvector approach. Environment and Planning A 39 (5):1193 – 1221.

Examples, Models, and/or Documents:

Presentation about the LTM-ESF approach

A step-by-steb instruction about how to create eigenvectors

The LTM-ESF Paper (Published at Geographical Analysis)

Steps for creating and using eigenvectors for space time analysis

Alternative steps for creating and using eigenvectors for space time analysis