  The growing field of large-scale time domain astronomy requires methods for
probabilistic data analysis that are computationally tractable, even with large
datasets. Gaussian Processes are a popular class of models used for this
purpose but, since the computational cost scales, in general, as the cube of
the number of data points, their application has been limited to small
datasets. In this paper, we present a novel method for Gaussian Process
modeling in one-dimension where the computational requirements scale linearly
with the size of the dataset. We demonstrate the method by applying it to
simulated and real astronomical time series datasets. These demonstrations are
examples of probabilistic inference of stellar rotation periods, asteroseismic
oscillation spectra, and transiting planet parameters. The method exploits
structure in the problem when the covariance function is expressed as a mixture
of complex exponentials, without requiring evenly spaced observations or
uniform noise. This form of covariance arises naturally when the process is a
mixture of stochastically-driven damped harmonic oscillators -- providing a
physical motivation for and interpretation of this choice -- but we also
demonstrate that it can be a useful effective model in some other cases. We
present a mathematical description of the method and compare it to existing
scalable Gaussian Process methods. The method is fast and interpretable, with a
range of potential applications within astronomical data analysis and beyond.
We provide well-tested and documented open-source implementations of this
method in C++, Python, and Julia.
