EM-based Fast Uncertainty Quantification for Bayesian Multi-setup Operational Modal Analysis

Wei Zhu; Binbin Li; Zuo Zhu

Home/Archive/arXiv

arXiv · 2412.00685

EM-based Fast Uncertainty Quantification for Bayesian Multi-setup Operational Modal Analysis

Wei Zhu·Binbin Li·Zuo Zhu

View original source ↗Read source document ↗Export metadata ↓

Abstract

The current Bayesian FFT algorithm relies on direct differentiation to obtain the posterior covariance matrix (PCM), which is time-consuming, memory-intensive, and hard to code, especially for the multi-setup operational modal analysis (OMA). Aiming at accelerating the uncertainty quantification in multi-setup OMA, an expectation-maximization (EM)-based algorithm is proposed by reformulating the Hessian matrix of the negative log-likelihood function (NLLF) as a sum of simplified components corresponding to the complete-data NLLF. Matrix calculus is employed to derive these components in a compact manner, resulting in expressions similar to those in the single-setup case. This similarity allows for the reuse of existing Bayesian single-setup OMA codes, simplifying implementation. The singularity caused by mode shape norm constraints is addressed through null space projection, eliminating potential numerical errors from the conventional pseudoinverse operation. A sparse assembly strategy is further adopted, avoiding unnecessary calculations and storage of predominant zero elements in the Hessian matrix. The proposed method is then validated through a comprehensive parametric study and applied to a multi-setup OMA of a high-rise building. Results demonstrate that the proposed method efficiently calculates the PCM within seconds, even for cases with hundreds of parameters. This represents an efficiency improvement of at least one order of magnitude over the state-of-the-art method. Such performance paves the way for a real-time modal identification of large-scale structures, including those with closely-spaced modes.

Explore related subjects

stat.CO

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Wei Zhu, Binbin Li, Zuo Zhu. 2024-12-01. EM-based Fast Uncertainty Quantification for Bayesian Multi-setup Operational Modal Analysis. https://arxiv.org/abs/2412.00685

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Extended One-Liners for the Gamma, Poisson, and Binomial Distributions

We demonstrate explicit transformations of three independent uniforms with exact Gamma$(a,1)$, $a>0$, Poisson$(\lambda)$, $\lambda>0$, and Binomial$(N,p)$, $N\geq 1$, $0\leq p \leq 1$ laws using only elementary operations.

stat.CO↗

arXiv2026

Estimating Hierarchically Rank Structured Covariance Matrices

We consider the problem of estimating a high-dimensional covariance matrix from a very limited number of samples. This problem is ubiquitous in computational fluid dynamics, where a small number of fluid snapshots must be used to construct a Gramian matrix determining a reduced-order model, as well as in computational geoscience, where a small ensemble of Earth system forecasts must be used to estimate the covariance matrix associated with the forecast uncertainty. It is common practice to regularize the small-sample covariance by imposing a "localization" structure that enforces a physically realistic correlation length scale, imposing a sparsity constraint, "shrinking" towards a prescribed target, or attenuating small correlations. We propose an alternate technique that regularizes the small-sample covariance by imposing hierarchical rank structure. Compared to regularization methods that assume sparsity such as spatial localization, hierarchical rank structure accommodates a wider range of covariance matrices, roughly corresponding to situations where long-range correlations vary more smoothly than short-range ones. It also results in a data-sparse matrix format that permits highly efficient matrix-vector products. We present theory and algorithms which show how to efficiently estimate a high-dimensional, hierarchically rank structured covariance matrix from limited samples. Through an error analysis and numerical experiments with a variety of model problems, we demonstrate that these techniques are effective at reducing sampling errors, and that in many cases they achieve smaller estimation error than conventional techniques.

stat.CO↗

arXiv2026

Optimal Slice-Adaptive Tuning of Hybrid Slice Sampling

Slice sampling is a Markov chain Monte Carlo algorithm that draws its next state uniformly from a "slice"---a super-level set of the target density function---at each iteration, thereby providing automatic local adaptivity to the scale of the target. In practice the exact slice is not known, so general-purpose implementations use an approximate slice that is grown from a starting interval of length $w>0$, with a computational cost that depends on $w$. This work presents an analysis of the average per-iteration number of target density evaluations, as a function of $w$, of hybrid slice sampling with various slice-finding schemes for targets with contiguous slices. The paper uses the results of the analysis to develop automated, slice-adaptive tuning schemes along with suboptimality bounds and asymptotic convergence guarantees. Simulations demonstrate that the tuning schemes reliably yield near-optimal slice-adaptive tuning with essentially no dependence on the initial setting of $w$.

stat.CO↗