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

机译：具有O（n）计算成本的近似分数高斯噪声模型

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摘要

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(n2), 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(n3). 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.
机译：分数高斯噪声（FGN）是一种静止时间序列模型，具有在各种领域应用的长内存属性，如经济学，水文和气候学。使用基于似然的方法拟合长度N的FGN模型的计算成本是O（n2），利用协方差矩阵的Toeplitz结构。在最具现实的情况下，我们不会直接观察FGN过程，而只能通过间接高斯观察，因此陷阱结构很容易丢失，并且计算成本增加到O（N3）。本文介绍了o（n）计算成本的近似FGN模型，包括直接和间接高斯观察，有或没有调理。这是通过用具有独立的独立阶权自由额（AR）处理的加权之和的FGN来实现，拟合近似的参数以匹配FGN模型的自相关函数。尽管是Markov，所产生的近似是静止的，并且仅使用四个AR部件提供非常精确的拟合。具体地，给定的近似FGN模型结合在潜伏的高斯模型中，其中使用集成嵌套拉普拉斯近似的方法获得了贝叶斯推断。仿真和两个真实数据示例中展示了近似FGN模型的性能。

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