Kernel Methods For Estimating Covariance Functions From Curves

We propose kernel methods for estimating covariance functions, when the data consists of a collection of curves. Every curve is modelled as an independent realization of a stochastic process with unknown mean and covariance structure. We consider a kernel density estimator, which has the positive semi-definiteness property on the “time” points and also in the continuum. We describe a cross-validation procedure, which leaves out an entire curve at a time, to choose the bandwidth (smoothing parameter) automatically from the observed collection of curves.