TY - JOUR

T1 - Using image and curve registration for measuring the goodness of fit of spatial and temporal predictions

AU - Reilly, Cavan

AU - Price, Phillip

AU - Gelman, Andrew

AU - Sandgathe, Scott A.

PY - 2004/12

Y1 - 2004/12

N2 - 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, λ, controls the tradeoff between (a) and (b), so that as λ → ∞ no deformation is allowed, whereas for λ = 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.

AB - 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, λ, controls the tradeoff between (a) and (b), so that as λ → ∞ no deformation is allowed, whereas for λ = 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.

KW - Calculus of variations

KW - Deformation

KW - Dynamic programming

KW - Errors-in-variables regression

KW - Goodness of fit

KW - Image registration

KW - Morphometrics

KW - Spatial distribution

KW - Time series

KW - Variance components

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U2 - 10.1111/j.0006-341X.2004.00251.x

DO - 10.1111/j.0006-341X.2004.00251.x

M3 - Article

C2 - 15606416

AN - SCOPUS:10944263500

VL - 60

SP - 954

EP - 964

JO - Biometrics

JF - Biometrics

SN - 0006-341X

IS - 4

ER -