Function Estimation Using Data Adaptive Kernel Smoothers - How Much Smoothing?

We consider a common problem in physics: How to estimate a smooth function given noisy measurements. We assume that the unknown signal is measured at N different times, {ti: i = 1, . . . N} and that the measurements, {yi}, have been contaminated by additive noise. Thus the measurements satisfy yi = g(ti) + i, where g(t) is the unknown signal and i are random errors. For simplicity, we assume that the errors are independent and have zero mean and uniform variance, σ. As an example, we consider a chirp signal: g(t) = sin(4πt). This signal is called a chirp because its “frequency” is growing linearly: d dt {phase}= 8πt, which corresponds to the changing pitch in a bird’s chirp. Figure 1 plots the chirp over two periods. Superimposed on the chirp is a point random realization of the noisy signal with σ = 0.5. A simple estimator of the unknown signal is a local average: