Quasi-likelihood inference for modulated non-stationary time series

In this thesis we propose a new class of non-stationary time series models and a quasi-likelihood inference method that is computationally efficient and consistent for that class of processes. A standard class of non-stationary processes is that of locally-stationary processes, where a smooth time-varying spectral representation extends the spectral representation of stationary time series. This allows us to apply stationary estimation methods when analysing slowly-varying non-stationary processes. However, stationary inference methods may lead to large biases for more rapidly-varying non-stationary processes. We present a class of such processes based on the framework of modulated processes. A modulated process is formed by pointwise multiplying a stationary process, called the latent process, by a sequence, called the modulation sequence. Our interest lies in estimating a parametric model for the latent stationary process from observing the modulated process in parallel with the modulation sequence. Very often exact likelihood is not computationally viable when analysing large time series datasets. The Whittle likelihood is a stan- dard quasi-likelihood for stationary time series. Our inference method adapts this function by specifying the expected periodogram of the modulated process for a given parameter vector of the latent time series model, and then fits this quantity to the sample periodogram. We prove that this approach conserves the computational efficiency and convergence rate of the Whittle likelihood under increasing sample size. Finally, our real-data application is concerned with the study of ocean surface currents. We analyse bivariate non-stationary velocities obtained from instruments following the ocean surface currents, and infer key physical quantities from this dataset. Our simulations show the benefit of our modelling and estimation method

Diffusion-Based smoothers for spatial filtering of gridded geophysical data

© The Author(s), 2021. This article is distributed under the terms of the Creative Commons Attribution License. The definitive version was published in Grooms, I., Loose, N., Abernathey, R., Steinberg, J. M., Bachman, S. D., Marques, G., Guillaumin, A. P., & Yankovsky, E. Diffusion-Based smoothers for spatial filtering of gridded geophysical data. Journal of Advances in Modeling Earth Systems, 13(9), (2021): e2021MS002552, https://doi.org/10.1029/2021MS002552.We describe a new way to apply a spatial filter to gridded data from models or observations, focusing on low-pass filters. The new method is analogous to smoothing via diffusion, and its implementation requires only a discrete Laplacian operator appropriate to the data. The new method can approximate arbitrary filter shapes, including Gaussian filters, and can be extended to spatially varying and anisotropic filters. The new diffusion-based smoother's properties are illustrated with examples from ocean model data and ocean observational products. An open-source Python package implementing this algorithm, called gcm-filters, is currently under development.I.G. and N.L. are supported by NSF OCE 1912332. R.A. is supported by NSF OCE 1912325. J.S. is supported by NSF OCE 1912302. S.B. and G.M. are supported by NSF OCE 1912420. A.G. and E.Y. are supported by NSF GEO 1912357 and NOAA CVP NA19OAR4310364

The debiased Whittle likelihood

The Whittle likelihood is a widely used and computationally efficient pseudolikelihood. However, it is known to produce biased parameter estimates with finite sample sizes for large classes of models. We propose a method for debiasing Whittle estimates for second-order stationary stochastic processes. The debiased Whittle likelihood can be computed in the same O(n log n) operations as the standard Whittle approach. We demonstrate the superior performance of our method in simulation studies and in application to a large-scale oceanographic dataset, where in both cases the debiased approach reduces bias by up to two orders of magnitude, achieving estimates that are close to those of the exact maximum likelihood, at a fraction of the computational cost. We prove that the method yields estimates that are consistent at an optimal convergence rate of n(-1/2) for Gaussian processes and for certain classes of non-Gaussian or nonlinear processes. This is established under weaker assumptions than in the standard theory, and in particular the power spectral density is not required to be continuous in frequency. We describe how the method can be readily combined with standard methods of bias reduction, such as tapering and differencing, to further reduce bias in parameter estimates

The Debiased Spatial Whittle Likelihood

We provide a computationally and statistically efficient method for estimating the parameters of a stochastic covariance model observed on a regular spatial grid in any number of dimensions. Our proposed method, which we call the Debiased Spatial Whittle likelihood, makes important corrections to the well-known Whittle likelihood to account for large sources of bias caused by boundary effects and aliasing. We generalise the approach to flexibly allow for significant volumes of missing data including those with lower-dimensional substructure, and for irregular sampling boundaries. We build a theoretical framework under relatively weak assumptions which ensures consistency and asymptotic normality in numerous practical settings including missing data and non-Gaussian processes. We also extend our consistency results to multivariate processes. We provide detailed implementation guidelines which ensure the estimation procedure can be conducted in O(n log n) operations, where n is the number of points of the encapsulating rectangular grid, thus keeping the computational scalability of Fourier and Whittle-based methods for large data sets. We validate our procedure over a range of simulated and real-world settings, and compare with state-of-the-art alternatives, demonstrating the enduring practical appeal of Fourier-based methods, provided they are corrected by the procedures developed in this paper

Efficient Parameter Estimation of Sampled Random Fields

We provide a computationally and statistically efficient method for estimating the parameters of a stochastic Gaussian model observed on a spatial grid, which need not be rectangular. Standard methods are plagued by computational intractability, where designing methods that can be implemented for realistically sized problems has been an issue for a long time. This has motivated the use of the Fourier Transform and the Whittle likelihood approximation. The challenge of frequency-domain methods is to determine and account for observational boundary effects, missing data, and the shape of the observed spatial grid. In this paper we address these effects explicitly by proposing a new quasi-likelihood estimator. We prove consistency and asymptotic normality of our estimator in settings that include irregularly shaped grids. Our simulations show that the proposed method solves boundary issues with Whittle estimation for random fields, yielding parameter estimates with significantly reduced bias and error. We demonstrate the effectiveness of our method for incomplete lattices, in comparison to other recent methods. Finally, we apply our method to estimate the parameters of a Mat\'ern process used to model data from Venus' topography