Characterizing the variability of system kernel and input estimates

The identification of the input to, or kernels of, a system using nonparametric representations and least-squares estimation is becoming increasingly popular. Nonparametric representations avoid making a priori assumptions about the input or having detailed knowledge about the system, and only need to guarantee known general characteristics (for example, positivity), which are obtained through the imposition of constraints on the estimates. An often overlooked problem is how to characterize the variability of the estimates so obtained. This problem is caused by the presence of constraints--and/or the nonlinearities of the estimates, or the complexity of the (regression based) estimation algorithms used--which make standard methods of estimating variability incorrect. In this article we investigate the use of a resampling technique called the "bootstrap" to obtain the desired estimates of variability. We present real data analysis demonstrating the approach, and through simulations we test the performance of a novel bootstrap technique obtaining confidence bands for the estimated functions.