具有${\mathcal O}(n)$计算代价的近似分数高斯噪声模型
An approximate fractional Gaussian noise model with ${\mathcal O}(n)$ computational cost
摘要：
分数阶高斯噪声（fGn）是一种具有长记忆特性的平稳时间序列模型，在计量经济学、水文学和气候学等领域有着广泛的应用。利用协方差阵的Toeplitz结构，使用基于似然的方法拟合长度为$n$的fGn模型的计算成本为${\mathcal O}(n^{2})$。在大多数现实情况下，我们不直接观察fGn过程，而只是通过间接的高斯观测，所以Toeplitz结构很容易丢失，计算成本增加到${\mathcal O}(n^{3})$。本文提出了一个近似的fGn模型的${\mathcal O}(n)$计算成本，无论是直接或间接的高斯观测，有或没有条件。这是通过用独立的一阶自回归过程的加权和近似fGn来实现的，拟合近似的参数以匹配fGn模型的自相关函数。由此产生的近似是平稳的，尽管是马尔可夫的，并给出了一个非常准确的拟合只使用四个组件。近似fGn模型的性能在模拟和两个实际数据的例子中证明。
英文摘要：
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 ${\mathcal O}(n^{2})$, 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.
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