Using Image and Curve Registration for Measuring the Goodness of Fit of Spatial and Temporal Predictions
Conventional measures of model fit for indexed data (e.g., time series or spatial data) summarize errors in y, for instance by integrating (or summing) the squared difference between predicted and measured values over a range of x. We propose an approach which recognizes that errors can occur in the x-direction as well. Instead of just measuring the difference between the predictions and observations at each site (or time), we first “deform” the predictions, stretching or compressing along the x-direction or directions, so as to improve the agreement between the observations and the deformed predictions. Error is then summarized by (a) the amount of deformation in x, and (b) the remaining difference in y between the data and the deformed predictions (i.e., the residual error in y after the deformation). A parameter, , controls the tradeoff between (a) and (b), so that as →∞ no deformation is allowed, whereas for = 0 the deformation minimizes the errors in y. In some applications, the deformation itself is of interest because it characterizes the (temporal or spatial) structure of the errors. The optimal deformation can be computed by solving a system of nonlinear partial differential equations, or, for a unidimensional index, by using a dynamic programming algorithm. We illustrate the procedure with examples from nonlinear time series and fluid dynamics.
Keywords: Calculus of variations; Deformation; Dynamic programming; Errors-in-variables regression; Goodness of fit; Image registration; Morphometrics; Spatial distribution; Time series; Variance components
Document Type: Research Article
Affiliations: 1: Indoor Environment Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, U.S.A. 2: Department of Statistics, Columbia University, New York, New York 10027, U.S.A. 3: Applied Physics Laboratory, University of Washington, 1013 NE 40th Street, Seattle, Washington 98105, U.S.A.
Publication date: December 1, 2004