Thomas Bengtsson

Obstacles to High-Dimensional Particle Filtering

Abstract Particle filters are ensemble-based assimilation schemes that, unlike the ensemble Kalman filter, employ a fully nonlinear and non-Gaussian analysis step to compute the probability distribution function (pdf) of a system’s state conditioned on a set of observations. Evidence is provided that the ensemble size required for a successful particle filter scales exponentially with the problem size. For the simple example in which each component of the state vector is independent, Gaussian, and of unit variance and the observations are of each state component separately with independent, Gaussian errors, simulations indicate that the required ensemble size scales exponentially with the state dimension. In this example, the particle filter requires at least 1011 members when applied to a 200-dimensional state. Asymptotic results, following the work of Bengtsson, Bickel, and collaborators, are provided for two cases: one in which each prior state component is independent and identically distributed, and ...

Curse-of-dimensionality revisited: Collapse of the particle filter in very large scale systems

It has been widely realized that Monte Carlo methods (approximation via a sample ensemble) may fail in large scale systems. This work offers some theoretical insight into this phenomenon in the context of the particle filter. We demonstrate that the maximum of the weights associated with the sample ensemble converges to one as both the sample size and the system dimension tends to infinity. Specifically, under fairly weak assumptions, if the ensemble size grows sub-exponentially in the cube root of the system dimension, the convergence holds for a single update step in state-space models with independent and identically distributed kernels. Further, in an important special case, more refined arguments show (and our simulations suggest) that the convergence to unity occurs unless the ensemble grows super-exponentially in the system dimension. The weight singularity is also established in models with more general multivariate likelihoods, e.g. Gaussian and Cauchy. Although presented in the context of atmospheric data assimilation for numerical weather prediction, our results are generally valid for high-dimensional particle filters.

Sharp failure rates for the bootstrap particle filter in high dimensions

We prove that the maximum of the sample importance weights in a high-dimensional Gaussian particle filter converges to unity unless the ensemble size grows exponentially in the system dimension. Our work is motivated by and parallels the derivations of Bengtsson, Bickel and Li (2007); however, we weaken their assumptions on the eigenvalues of the covariance matrix of the prior distribution and establish rigorously their strong conjecture on when weight collapse occurs. Specifically, we remove the assumption that the nonzero eigenvalues are bounded away from zero, which, although the dimension of the involved vectors grow to infinity, essentially permits the effective system dimension to be bounded. Moreover, with some restrictions on the rate of growth of the maximum eigenvalue, we relax their assumption that the eigenvalues are bounded from above, allowing the system to be dominated by a single mode.

Performance Bounds for Particle Filters Using the Optimal Proposal

AbstractParticle filters may suffer from degeneracy of the particle weights. For the simplest “bootstrap” filter, it is known that avoiding degeneracy in large systems requires that the ensemble size must increase exponentially with the variance of the observation log-likelihood. The present article shows first that a similar result applies to particle filters using sequential importance sampling and the optimal proposal distribution and, second, that the optimal proposal yields minimal degeneracy when compared to any other proposal distribution that depends only on the previous state and the most recent observations. Thus, the optimal proposal provides performance bounds for filters using sequential importance sampling and any such proposal. An example with independent and identically distributed degrees of freedom illustrates both the need for exponentially large ensemble size with the optimal proposal as the system dimension increases and the potentially dramatic advantages of the optimal proposal rela...

A Regression Paradox for Linear Models: Sufficient Conditions and Relation to Simpson's Paradox

An analysis of customer survey data using direct and reverse linear regression leads to inconsistent conclusions with respect to the effect of a group variable. This counterintuitive phenomenon, called the “regression paradox,” causes seemingly contradictory group effects when the predictor and regressand are interchanged. Using analytical developments as well as geometric arguments, we describe sufficient conditions under which the regression paradox will appear in linear Gaussian models. The results show that the phenomenon depends on a distribution shift between the groups relative to the predictability of the model. As a consequence, the paradox can appear naturally in certain distributions, and may not be caused by sampling error or incorrectly specified models. Simulations verify that the paradox may appear in more general, non-Gaussian settings. An interesting, geometric connection to Simpson’s paradox is provided.

Monitoring and Diagnostics of Power Anomalies in Transparent Optical Networks

Challenges in monitoring optically-transparent networks are highlighted for dynamically-controlled Raman amplification systems. We use models of amplifier physics together with statistical estimation to automatically discriminate between measurement errors, anomalous losses, and pump failures.

Model-Based Anomaly Detection for a Transparent Optical Transmission System

In this chapter, we present an approach for anomaly detection at the physical layer of networks where detailed knowledge about the devices and their operations is available. The approach combines physics-based process models with observational data models to characterize the uncertainties and derive the alarm decision rules. We formulate and apply three different methods based on this approach for a well-defined problem in optical network monitoring that features many typical challenges for this methodology. Specifically, we address the problem of monitoring optically transparent transmission systems that use dynamically controlled Raman amplification systems. We use models of amplifier physics together with statistical estimation to derive alarm decision rules and use these rules to automatically discriminate between measurement errors, anomalous losses, and pump failures. Our approach has led to an efficient tool for systematically detecting anomalies in the system behavior of a deployed network, where pro-active measures to address such anomalies are key to preventing unnecessary disturbances to the system’s continuous operation.