  Gaussian processes are rich distributions over functions, which provide a
Bayesian nonparametric approach to smoothing and interpolation. We introduce
simple closed form kernels that can be used with Gaussian processes to discover
patterns and enable extrapolation. These kernels are derived by modelling a
spectral density -- the Fourier transform of a kernel -- with a Gaussian
mixture. The proposed kernels support a broad class of stationary covariances,
but Gaussian process inference remains simple and analytic. We demonstrate the
proposed kernels by discovering patterns and performing long range
extrapolation on synthetic examples, as well as atmospheric CO2 trends and
airline passenger data. We also show that we can reconstruct standard
covariances within our framework.
