Eirik Myrvoll-Nilsen

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 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.

Layer-counted proxy records from paleoclimatic archives are subject to considerable dating uncertainties. These uncertainties originate from irregularities in the archive’s deposition process that result, in turn, in errors during the layer counting process. Dating uncertainties can be quantified by assuming a probabilistic model for the relationship between the depth of a sample in the proxy archive and the age of that sample. However, systematic biases in counting or depositional processes can cause the counted chronology to deviate substantially from the true age and possibly corrupt the age model. By synchronizing a given chronology with other, independently dated archives, one can constrain the dating uncertainties and correct potential biases. This can be done by matching the chronology to tie-points obtained by identifying characteristic events which were recorded simultaneously by different archives or with independent methods. However, updating the counted age–depth relationship under the consideration of tie-points is not straightforward and no generally accepted method is presently available for layer-counted archives. A key requirement for such a method is that it should include an appropriate uncertainty-sensitive interpolation between tie-points. Using a Gaussian model to represent a potential bias, we show how tie-points and their uncertainties can be incorporated into a previously suggested Bayesian modeling framework to reflect the general uncertainties of a counted chronology. Both the uncertainty inherent to the tie-points and the age-correlation between the data from different depths in the archive are consistently represented in this approach. We demonstrate the methodology in two applications: first, using synthetic data, and second, applying the methodology to data from the NGRIP ice core, an iconic paleoclimate proxy archive.

A tipping point is defined by the IPCC as a critical threshold beyond which a system reorganizes, often abruptly and/or irreversibly. Tipping points can be crossed solely by internal variation in the system or by approaching a bifurcation point where the current state loses stability, which forces the system to move to another stable state. It can be shown that before a bifurcation point is reached there are observable changes in the statistical properties of the state variable. These are known as early warning signals and include increased fluctuation and autocorrelation time. It is currently debated whether or not Dansgaard–Oeschger (DO) events, which are abrupt warmings of the North Atlantic region which occurred during the last glacial period, are preceded by early warning signals. To express the changes in statistical behavior we propose a model based on the well-known first-order autoregressive (AR) process, with modifications to the autocorrelation parameter such that it depends linearly on time. In order to estimate the time evolution of the autocorrelation parameter we adopt a hierarchical Bayesian modeling framework, from which Bayesian analysis can be performed using the methodology of integrated nested Laplace approximations. We then apply the model to segments of the oxygen isotope ratios from the Northern Greenland Ice Core Project record corresponding to 17 DO events. Statistically significant early warning signals are detected for a number of DO events, which suggests that such events could indeed exhibit signs of ongoing destabilization and may have been caused by approaching a bifurcation point. The methodology developed to perform the given early warning analyses can be applied more generally and is publicly available as the R package INLA.ews.