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1. Regularization and tempering for a moment‐matching localized particle filter

   https://doi.org/10.1002/qj.4328  

   Poterjoy, Jonathan (June 2022, Quarterly Journal of the Royal Meteorological Society)

   Abstract Iterative ensemble filters and smoothers are now commonly used for geophysical models. Some of these methods rely on a factorization of the observation likelihood function to sample from a posterior density through a set of “tempered” transitions to ensemble members. For Gaussian‐based data assimilation methods, tangent linear versions of nonlinear operators can be relinearized between iterations, thus leading to a solution that is less biased than a single‐step approach. This study adopts similar iterative strategies for a localized particle filter (PF) that relies on the estimation of moments to adjust unobserved variables based on importance weights. This approach builds off a “regularization” of the local PF, which forces weights to be more uniform through heuristic means. The regularization then leads to an adaptive tempering, which can also be combined with filter updates from parametric methods, such as ensemble Kalman filters. The role of iterations is analyzed by deriving the localized posterior probability density assumed by current local PF formulations and then examining how single‐step and tempered PFs sample from this density. From experiments performed with a low‐dimensional nonlinear system, the iterative and hybrid strategies show the largest benefits in observation‐sparse regimes, where only a few particles contain high likelihoods and prior errors are non‐Gaussian. This regime mimics specific applications in numerical weather prediction, where small ensemble sizes, unresolved model error, and highly nonlinear dynamics lead to prior uncertainty that is larger than measurement uncertainty. 

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2. A Statistical Hypothesis Testing Strategy for Adaptively Blending Particle Filters and Ensemble Kalman Filters for Data Assimilation

   https://doi.org/10.1175/MWR-D-22-0108.1  

   Kurosawa, Kenta; Poterjoy, Jonathan (January 2023, Monthly Weather Review)

   Abstract Particle filters avoid parametric estimates for Bayesian posterior densities, which alleviates Gaussian assumptions in nonlinear regimes. These methods, however, are more sensitive to sampling errors than Gaussian-based techniques such as ensemble Kalman filters. A recent study by the authors introduced an iterative strategy for particle filters that match posterior moments—where iterations improve the filter’s ability to draw samples from non-Gaussian posterior densities. The iterations follow from a factorization of particle weights, providing a natural framework for combining particle filters with alternative filters to mitigate the impact of sampling errors. The current study introduces a novel approach to forming an adaptive hybrid data assimilation methodology, exploiting the theoretical strengths of nonparametric and parametric filters. At each data assimilation cycle, the iterative particle filter performs a sequence of updates while the prior sample distribution is non-Gaussian, then an ensemble Kalman filter provides the final adjustment when Gaussian distributions for marginal quantities are detected. The method employs the Shapiro–Wilk test to determine when to make the transition between filter algorithms, which has outstanding power for detecting departures from normality. Experiments using low-dimensional models demonstrate that the approach has a significant value, especially for nonhomogeneous observation networks and unknown model process errors. Moreover, hybrid factors are extended to consider marginals of more than one collocated variables using a test for multivariate normality. Findings from this study motivate the use of the proposed method for geophysical problems characterized by diverse observation networks and various dynamic instabilities, such as numerical weather prediction models. Significance Statement Data assimilation statistically processes observation errors and model forecast errors to provide optimal initial conditions for the forecast, playing a critical role in numerical weather forecasting. The ensemble Kalman filter, which has been widely adopted and developed in many operational centers, assumes Gaussianity of the prior distribution and solves a linear system of equations, leading to bias in strong nonlinear regimes. On the other hand, particle filters avoid many of those assumptions but are sensitive to sampling errors and are computationally expensive. We propose an adaptive hybrid strategy that combines their advantages and minimizes the disadvantages of the two methods. The hybrid particle filter–ensemble Kalman filter is achieved with the Shapiro–Wilk test to detect the Gaussianity of the ensemble members and determine the timing of the transition between these filter updates. Demonstrations in this study show that the proposed method is advantageous when observations are heterogeneous and when the model has an unknown bias. Furthermore, by extending the statistical hypothesis test to the test for multivariate normality, we consider marginals of more than one collocated variable. These results encourage further testing for real geophysical problems characterized by various dynamic instabilities, such as real numerical weather prediction models. 

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3. Scalable Gaussian Process Inference with Finite-data Mean and Variance Guarantees

   Huggins, Jonathan H.; Campbell, Trevor; Kasprzak, Mikolaj; Broderick, Tamara (April 2019, Proceedings of Machine Learning Research)

   Gaussian processes (GPs) offer a flexible class of priors for nonparametric Bayesian regression, but popular GP posterior inference methods are typically prohibitively slow or lack desirable finite-data guarantees on quality. We develop a scalable approach to approximate GP regression, with finite-data guarantees on the accuracy of our pointwise posterior mean and variance estimates. Our main contribution is a novel objective for approximate inference in the nonparametric setting: the preconditioned Fisher (pF) divergence. We show that unlike the Kullback–Leibler divergence (used in variational inference), the pF divergence bounds bounds the 2-Wasserstein distance, which in turn provides tight bounds on the pointwise error of mean and variance estimates. We demonstrate that, for sparse GP likelihood approximations, we can minimize the pF divergence bounds efficiently. Our experiments show that optimizing the pF divergence bounds has the same computational requirements as variational sparse GPs while providing comparable empirical performance—in addition to our novel finite-data quality guarantees. 

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4. Challenges for Inline Observation Error Estimation in the Presence of Misspecified Background Uncertainty

   https://doi.org/10.1175/MWR-D-22-0298.1  

   Walsworth, Andrew; Poterjoy, Jonathan; Satterfield, Elizabeth (September 2023, Monthly Weather Review)

   Abstract For data assimilation to provide faithful state estimates for dynamical models, specifications of observation uncertainty need to be as accurate as possible. Innovation-based methods based on Desroziers diagnostics, are commonly used to estimate observation uncertainty, but such methods can depend greatly on the prescribed background uncertainty. For ensemble data assimilation, this uncertainty comes from statistics calculated from ensemble forecasts, which require inflation and localization to address under sampling. In this work, we use an ensemble Kalman filter (EnKF) with a low-dimensional Lorenz model to investigate the interplay between the Desroziers method and inflation. Two inflation techniques are used for this purpose: 1) a rigorously tuned fixed multiplicative scheme and 2) an adaptive state-space scheme. We document how inaccuracies in observation uncertainty affect errors in EnKF posteriors and study the combined impacts of misspecified initial observation uncertainty, sampling error, and model error on Desroziers estimates. We find that whether observation uncertainty is over- or underestimated greatly affects the stability of data assimilation and the accuracy of Desroziers estimates and that preference should be given to initial overestimates. Inline estimates of Desroziers tend to remove the dependence between ensemble spread–skill and the initially prescribed observation error. In addition, we find that the inclusion of model error introduces spurious correlations in observation uncertainty estimates. Further, we note that the adaptive inflation scheme is less robust than fixed inflation at mitigating multiple sources of error. Last, sampling error strongly exacerbates existing sources of error and greatly degrades EnKF estimates, which translates into biased Desroziers estimates of observation error covariance. Significance StatementTo generate accurate predictions of various components of the Earth system, numerical models require an accurate specification of state variables at our current time. This step adopts a probabilistic consideration of our current state estimate versus information provided from environmental measurements of the true state. Various strategies exist for estimating uncertainty in observations within this framework, but are sensitive to a host of assumptions, which are investigated in this study. 

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5. On global normal linear approximations for nonlinear Bayesian inverse problems

   https://doi.org/10.1088/1361-6420/acc129  

   Nicholson, Ruanui; Petra, Noémi; Villa, Umberto; Kaipio, Jari P (March 2023, Inverse Problems)

   Abstract The replacement of a nonlinear parameter-to-observable mapping with a linear (affine) approximation is often carried out to reduce the computational costs associated with solving large-scale inverse problems governed by partial differential equations (PDEs). In the case of a linear parameter-to-observable mapping with normally distributed additive noise and a Gaussian prior measure on the parameters, the posterior is Gaussian. However, substituting an accurate model for a (possibly well justified) linear surrogate model can give misleading results if the induced model approximation error is not accounted for. To account for the errors, the Bayesian approximation error (BAE) approach can be utilised, in which the first and second order statistics of the errors are computed via sampling. The most common linear approximation is carried out via linear Taylor expansion, which requires the computation of (Fréchet) derivatives of the parameter-to-observable mapping with respect to the parameters of interest. In this paper, we prove that the (approximate) posterior measure obtained by replacing the nonlinear parameter-to-observable mapping with a linear approximation is in fact independent of the choice of the linear approximation when the BAE approach is employed. Thus, somewhat non-intuitively, employing the zero-model as the linear approximation gives the same approximate posterior as any other choice of linear approximations of the parameter-to-observable model. The independence of the linear approximation is demonstrated mathematically and illustrated with two numerical PDE-based problems: an inverse scattering type problem and an inverse conductivity type problem. 

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