In contemporary data-driven environments, the generation and processing of multivariate time series data is an omnipresent challenge, often complicated by time delays between different time series. These delays, originating from a multitude of sources like varying data transmission dynamics, sensor interferences, and environmental changes, introduce significant complexities. Traditional Time Delay Estimation methods, which typically assume a fixed constant time delay, may not fully capture these variabilities, compromising the precision of predictive models in diverse settings. To address this issue, we introduce the Time Series Model Bootstrap (TSMB), a versatile framework designed to handle potentially varying or even nondeterministic time delays in time series modeling. Contrary to traditional approaches that hinge on the assumption of a single, consistent time delay, TSMB adopts a nonparametric stance, acknowledging and incorporating time delay uncertainties. TSMB significantly bolsters the performance of models that are trained and make predictions using this framework, making it highly suitable for a wide range of dynamic and interconnected data environments.

Bayesian Regularization and Nonnegative Deconvolution (BRAND) is proposed for estimating time delays of acoustic signals in reverberant environments. Sparsity of the nonnegative filter coefficients is enforced using an L1-norm regularization. A probabilistic generative model is used to simultaneously estimate the regularization parameters and filter coefficients from the signal data. Iterative update rules are derived under a Bayesian framework using the Expectation-Maximization procedure. The resulting time delay estimation algorithm is demonstrated on noisy acoustic data.