Tracking a smooth fault line in a response surface

We suggest a sequential, or "tracking," algorithm for estimating a smooth fault line in a response surface. The method starts with an approximation to a point on the line, and from there the line is tracked as it meanders through the plane. The technique differs from recent approaches in that it does not require a large part of the plane to be searched for evidence of a fault line. This offers potential computational savings, and produces a method that is invariant under rotations of coordinate axes (except insofar as a rotation might affect the estimated starting point, and the relative orientation of the grid on which calculations are done). That feature is important if design points are not located on a regular grid. We investigate properties of the method under very general conditions on the design, allowing Poisson cluster processes, jiggled grid processes and deterministic, regular lattices. Uniform rates of convergence are derived in all these settings, for the case of noisy data, and shown to be within logarithmic factors of optimal pointwise convergence rates in the no-noise setting.