The Analysis of Noisy Signals by Nonparametric Smoothing and Differentiation

While smoothing methods are commonly applied to noisy signals, this is not true for differentiation. Derivatives are often of intrinsic interest when analyzing biological dynamics, and as will be illustrated, they are useful for determining characteristic points (local extrema, inflection, and saddle points) in the curve, in the presence of noise. There are inherent difficulties in computing derivatives which might have inhibited wider usage. Kernel estimation is a statistical approach to nonparametric regression (i.e., without specifying a functional model for the signal), which allows detertining the signal itself and its derivatives from noisy data. This method is presented, together with its properties. The influence and the choice of the weight function (kernel) of the smoothing parameter and the treatment of boundary points deserve particular attention.