  Fractional Gaussian noise (fGn) is a stationary time series model with long
memory properties applied in various fields like econometrics, hydrology and
climatology. The computational cost in fitting an fGn model of length $n$ using
a likelihood-based approach is ${\mathcal O}(n^{2})$, exploiting the Toeplitz
structure of the covariance matrix. In most realistic cases, we do not observe
the fGn process directly but only through indirect Gaussian observations, so
the Toeplitz structure is easily lost and the computational cost increases to
${\mathcal O}(n^{3})$. This paper presents an approximate fGn model of
${\mathcal O}(n)$ computational cost, both with direct or indirect Gaussian
observations, with or without conditioning. This is achieved by approximating
fGn with a weighted sum of independent first-order autoregressive processes,
fitting the parameters of the approximation to match the autocorrelation
function of the fGn model. The resulting approximation is stationary despite
being Markov and gives a remarkably accurate fit using only four components.
The performance of the approximate fGn model is demonstrated in simulations and
two real data examples.
