Least-squares best fit of a deterministic model to experimental data using linear prediction: estimating confidence intervals of the rms with prediction and application to dimensional metrology of mirrors

Abstract

We present a novel method for estimating the root mean square (rms) of a mirror. Firstly, a least-squares method using linear prediction is applied, for fitting a reference surface to the data. At any point of the mirror, the probability density function of the mirror, conditioned by the data, is considered. The knowledge of this density makes it possible to take into account all the points of the surface, in a stochastic sense, instead of the sole data. The conditional expectation and variance can be estimated, yielding confidence intervals. This approach provides a dramatic increase in accuracy for the estimations, a lesser sensitivity to random sampling fluctuations, and confidence intervals of the estimated parameters. This leads to higher quality and reliability in measurement procedures. Tests on both simulated and experimental data validate the theoretical results together with the approximation formulae.