Sequential Bayesian inference for implicit hidden Markov models and current limitations

Hidden Markov models can describe time series arising in various fields of science, by treating the data as noisy measurements of an arbitrarily complex Markov process. Sequential Monte Carlo (SMC) methods have become standard tools to estimate the hidden Markov process given the observations and a fixed parameter value. We review some of the recent developments allowing the inclusion of parameter uncertainty as well as model uncertainty. The shortcomings of the currently available methodology are emphasised from an algorithmic complexity perspective. The statistical objects of interest for time series analysis are illustrated on a toy "Lotka-Volterra" model used in population ecology. Some open challenges are discussed regarding the scalability of the reviewed methodology to longer time series, higher-dimensional state spaces and more flexible models.Comment: Review article written for ESAIM: proceedings and surveys. 25 pages, 10 figure

Real-time Loss Estimation for Instrumented Buildings

Motivation. A growing number of buildings have been instrumented to measure and record earthquake motions and to transmit these records to seismic-network data centers to be archived and disseminated for research purposes. At the same time, sensors are growing smaller, less expensive to install, and capable of sensing and transmitting other environmental parameters in addition to acceleration. Finally, recently developed performance-based earthquake engineering methodologies employ structural-response information to estimate probabilistic repair costs, repair durations, and other metrics of seismic performance. The opportunity presents itself therefore to combine these developments into the capability to estimate automatically in near-real-time the probabilistic seismic performance of an instrumented building, shortly after the cessation of strong motion. We refer to this opportunity as (near-) real-time loss estimation (RTLE). Methodology. This report presents a methodology for RTLE for instrumented buildings. Seismic performance is to be measured in terms of probabilistic repair cost, precise location of likely physical damage, operability, and life-safety. The methodology uses the instrument recordings and a Bayesian state-estimation algorithm called a particle filter to estimate the probabilistic structural response of the system, in terms of member forces and deformations. The structural response estimate is then used as input to component fragility functions to estimate the probabilistic damage state of structural and nonstructural components. The probabilistic damage state can be used to direct structural engineers to likely locations of physical damage, even if they are concealed behind architectural finishes. The damage state is used with construction cost-estimation principles to estimate probabilistic repair cost. It is also used as input to a quantified, fuzzy-set version of the FEMA-356 performance-level descriptions to estimate probabilistic safety and operability levels. CUREE demonstration building. The procedure for estimating damage locations, repair costs, and post-earthquake safety and operability is illustrated in parallel demonstrations by CUREE and Kajima research teams. The CUREE demonstration is performed using a real 1960s-era, 7-story, nonductile reinforced-concrete moment-frame building located in Van Nuys, California. The building is instrumented with 16 channels at five levels: ground level, floors 2, 3, 6, and the roof. We used the records obtained after the 1994 Northridge earthquake to hindcast performance in that earthquake. The building is analyzed in its condition prior to the 1994 Northridge Earthquake. It is found that, while hindcasting of the overall system performance level was excellent, prediction of detailed damage locations was poor, implying that either actual conditions differed substantially from those shown on the structural drawings, or inappropriate fragility functions were employed, or both. We also found that Bayesian updating of the structural model using observed structural response above the base of the building adds little information to the performance prediction. The reason is probably that Real-Time Loss Estimation for Instrumented Buildings ii structural uncertainties have only secondary effect on performance uncertainty, compared with the uncertainty in assembly damageability as quantified by their fragility functions. The implication is that real-time loss estimation is not sensitive to structural uncertainties (saving costly multiple simulations of structural response), and that real-time loss estimation does not benefit significantly from installing measuring instruments other than those at the base of the building. Kajima demonstration building. The Kajima demonstration is performed using a real 1960s-era office building in Kobe, Japan. The building, a 7-story reinforced-concrete shearwall building, was not instrumented in the 1995 Kobe earthquake, so instrument recordings are simulated. The building is analyzed in its condition prior to the earthquake. It is found that, while hindcasting of the overall repair cost was excellent, prediction of detailed damage locations was poor, again implying either that as-built conditions differ substantially from those shown on structural drawings, or that inappropriate fragility functions were used, or both. We find that the parameters of the detailed particle filter needed significant tuning, which would be impractical in actual application. Work is needed to prescribe values of these parameters in general. Opportunities for implementation and further research. Because much of the cost of applying this RTLE algorithm results from the cost of instrumentation and the effort of setting up a structural model, the readiest application would be to instrumented buildings whose structural models are already available, and to apply the methodology to important facilities. It would be useful to study under what conditions RTLE would be economically justified. Two other interesting possibilities for further study are (1) to update performance using readily observable damage; and (2) to quantify the value of information for expensive inspections, e.g., if one inspects a connection with a modeled 50% failure probability and finds that the connect is undamaged, is it necessary to examine one with 10% failure probability

Accounting for model error in Tempered Ensemble Transform Particle Filter and its application to non-additive model error

In this paper, we trivially extend Tempered (Localized) Ensemble Transform Particle Filter---T(L)ETPF---to account for model error. We examine T(L)ETPF performance for non-additive model error in a low-dimensional and a high-dimensional test problem. The former one is a nonlinear toy model, where uncertain parameters are non-Gaussian distributed but model error is Gaussian distributed. The latter one is a steady-state single-phase Darcy flow model, where uncertain parameters are Gaussian distributed but model error is non-Gaussian distributed. The source of model error in the Darcy flow problem is uncertain boundary conditions. We comapare T(L)ETPF to a Regularized (Localized) Ensemble Kalman Filter---R(L)EnKF. We show that T(L)ETPF outperforms R(L)EnKF for both the low-dimensional and the high-dimensional problem. This holds even when ensemble size of TLETPF is 100 while ensemble size of R(L)EnKF is greater than 6000. As a side note, we show that TLETPF takes less iterations than TETPF, which decreases computational costs; while RLEnKF takes more iterations than REnKF, which incerases computational costs. This is due to an influence of localization on a tempering and a regularizing parameter

Transform-based particle filtering for elliptic Bayesian inverse problems

We introduce optimal transport based resampling in adaptive SMC. We consider elliptic inverse problems of inferring hydraulic conductivity from pressure measurements. We consider two parametrizations of hydraulic conductivity: by Gaussian random field, and by a set of scalar (non-)Gaussian distributed parameters and Gaussian random fields. We show that for scalar parameters optimal transport based SMC performs comparably to monomial based SMC but for Gaussian high-dimensional random fields optimal transport based SMC outperforms monomial based SMC. When comparing to ensemble Kalman inversion with mutation (EKI), we observe that for Gaussian random fields, optimal transport based SMC gives comparable or worse performance than EKI depending on the complexity of the parametrization. For non-Gaussian distributed parameters optimal transport based SMC outperforms EKI