An approximate fractional Gaussian noise model with $$\mathcal {O}(n)$$ O ( n ) computational cost

Abstract

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 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 O(n 3). This paper presents an approximate fGn model of O(n) computational cost, both with direct or indirect Gaussian observations, with or without conditioning. This is achieved Sigrunn H. Sørbye

The variance inflation factor to account for correlations in likelihood ratio tests: deformation analysis with terrestrial laser scanners

“…Clearly, there is a slight discrepancy at high frequencies between the two spectra. Indeed, a fGn can only be approximated as a sum of k AR processes of first order (Sørbye et al 2019). This sum is itself modellable as an autoregressive process with moving average (ARMA) of order (k, k − 1) for the AR and MA components, respectively.…”

Warming Trends and Long-Range Dependent Climate Variability Since Year 1900: A Bayesian Approach

“…This is not the case when the noise term ǫ is fGn having an LRD structure, but the precision matrix of an AR1 process is tridiagonal. In Sørbye et al (2019), the fGn process is approximated using a weighted sum of AR1 processes where the weights and the first-lag autocorrelation coefficients of the approximation are optimized such that the covariance function of the approximation matches the exact covariance function of fGn defined in Equation 4. The latent field x is extended to include the AR(1) components that make up the approximation.…”

“…Second, due to the LRD assumption, the precision matrix, defined as the inverse covariance matrix, is dense and therefore unsuited for the computationally efficient algorithms that INLA applies. In order to retain sparsity we have to introduce an approximation such as that introduced by Sørbye et al (2019). The model with these modifications are available in the R-package INLA.climate which can be downloaded from the GitHub repository eirikmn/INLA.climate.…”

Emergent Scale Invariance and Climate Sensitivity

“…First, the LRD properties of the fGn process make the precision matrix of the latent field dense. To ensure computational efficiency, this is circumvented by approximating the fGn as a weighted sum of four AR(1) processes as introduced in [27]. Second, the mean of the observation vector, E(∆T) = σ f G(β) (F 0 + F) has a non-standard form.…”