Laplace Prior-Based Bayesian Compressive Sensing Using K-SVD for Vibration Signal Transmission and Fault Detection

Vibration signal transmission plays a fundamental role in equipment prognostics and health management. However, long-term condition monitoring requires signal compression before transmission because of the high sampling frequency. In this paper, an efficient Bayesian compressive sensing algorithm is proposed. The contribution is explicitly decomposed into two components: a multitask scenario and a Laplace prior-based hierarchical model. This combination makes full use of the sparse promotion under Laplace priors and the correlation between sparse blocks to improve the efficiency. Moreover, a K-singular value decomposition (K-SVD) dictionary learning method is used to find the best sparse representation of the signal. Simulation results show that the Laplace prior-based reconstruction performs better than typical algorithms. The comparison between a fixed dictionary and learning dictionary also illustrates the advantage of the K-SVD method. Finally, a fault detection case of a reconstructed signal is analyzed. The effectiveness of the proposed method is validated by simulation and experimental tests.

Weight Smoothing for Generalized Linear Models Using a Laplace Prior

When analyzing data sampled with unequal inclusion probabilities, correlations between the probability of selection and the sampled data can induce bias if the inclusion probabilities are ignored in the analysis. Weights equal to the inverse of the probability of inclusion are commonly used to correct possible bias. When weights are uncorrelated with the descriptive or model estimators of interest, highly disproportional sample designs resulting in large weights can introduce unnecessary variability, leading to an overall larger mean square error compared to unweighted methods. We describe an approach we term ‘weight smoothing’ that models the interactions between the weights and the estimators as random effects, reducing the root mean square error (RMSE) by shrinking interactions toward zero when such shrinkage is allowed by the data. This article adapts a flexible Laplace prior distribution for the hierarchical Bayesian model to gain a more robust bias-variance tradeoff than previous approaches using normal priors. Simulation and application suggest that under a linear model setting, weight-smoothing models with Laplace priors yield robust results when weighting is necessary, and provide considerable reduction in RMSE otherwise. In logistic regression models, estimates using weight-smoothing models with Laplace priors are robust, but with less gain in efficiency than in linear regression settings.