  We consider ECNoise, a practical tool for estimating the magnitude of noise
in evaluations of a black-box function. Recent developments in numerical
optimization algorithms have seen increased usage of ECNoise as a subroutine to
provide a solver with noise level estimates, so that the solver might somehow
proportionally adjust for noise. Particularly motivated by problems in
computationally expensive derivative-free optimization, we question a
fundamental assumption made in the original development of ECNoise,
particularly the assumption that the set of points provided to ECNoise must
satisfy fairly restrictive geometric conditions (in particular, that the points
be collinear and equally spaced). Driven by prior practical experience, we show
that in many situations, noise estimates obtained from providing an arbitrary
(that is, not collinear) geometry of points as input to ECNoise are often
indistinguishable from noise estimates obtained from using the standard
(collinear and equally spaced) geometry. We analyze this via parametric curves
that interpolate the arbitrary input points. The analysis provides insight into
the circumstances in which one can expect arbitrary point selection to cause
significant degradation of ECNoise. Moreover, the analysis suggests a practical
means (the solution of a small mixed integer linear program) by which one can
gradually adjust an initial arbitrary point selection to yield better noise
estimates with higher probability.
