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

TL;DR

This paper introduces a computationally efficient approximate fractional Gaussian noise model that reduces complexity from quadratic or cubic to linear, enabling faster analysis of long-memory processes in various scientific fields.

Contribution

The paper proposes a novel approximate fGn model using a weighted sum of AR(1) processes, achieving linear computational cost while maintaining high accuracy.

Findings

The approximation fits the autocorrelation function well with only four components.

The model performs accurately in simulations and real data examples.

It significantly reduces computational complexity for large datasets.

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 ${\mathcal O}(n^{2})$, 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.