TY - JOUR

T1 - An approximate fractional Gaussian noise model with O(n) computational cost

AU - Sørbye, Sigrunn Holbek

AU - Myrvoll-Nilsen, Eirik

AU - Rue, Haavard

N1 - KAUST Repository Item: Exported on 2020-10-01

PY - 2018/11/16

Y1 - 2018/11/16

N2 - 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 O(n) , 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 O(n). This paper presents an approximate fGn model of O(n) computational cost, both with direct and indirect Gaussian observations, with or without conditioning. This is achieved by approximating fGn with a weighted sum of independent first-order autoregressive (AR) 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 AR components. Specifically, the given approximate fGn model is incorporated within the class of latent Gaussian models in which Bayesian inference is obtained using the methodology of integrated nested Laplace approximation. The performance of the approximate fGn model is demonstrated in simulations and two real data examples.

AB - 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 O(n) , 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 O(n). This paper presents an approximate fGn model of O(n) computational cost, both with direct and indirect Gaussian observations, with or without conditioning. This is achieved by approximating fGn with a weighted sum of independent first-order autoregressive (AR) 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 AR components. Specifically, the given approximate fGn model is incorporated within the class of latent Gaussian models in which Bayesian inference is obtained using the methodology of integrated nested Laplace approximation. The performance of the approximate fGn model is demonstrated in simulations and two real data examples.

UR - http://hdl.handle.net/10754/630196

UR - http://link.springer.com/article/10.1007/s11222-018-9843-1

UR - http://www.scopus.com/inward/record.url?scp=85056737238&partnerID=8YFLogxK

U2 - 10.1007/s11222-018-9843-1

DO - 10.1007/s11222-018-9843-1

M3 - Article

VL - 29

SP - 821

EP - 833

JO - Statistics and Computing

JF - Statistics and Computing

SN - 0960-3174

IS - 4

ER -