A unified scheme to solving arbitrary complex-valued ratio distribution with application to statistical inference for raw frequency response functions and transmissibility functions. (November 2020) - Yan, Wang-Ji; Zhao, Meng-Yun; Beer, Michael; Ren, Wei-Xin; Chronopoulos, Dimitrios
Highlights: A unified scheme is proposed to solve complex-valued ratio distribution without closed-form solution. A unified formula is derived by reducing the problem into multi-dimensional integrals. Fast sparse-grid quadrature scheme is utilized to improve the efficiency of numerical integrals. The unified method enables uncertainty quantification for FRFs and TFs. The unified scheme is applicable to statistical inference for arbitrary ratio random variables. Abstract: Complex-valued ratio distributions arises in many real applications such as statistical inference for frequency response functions (FRFs) and transmissibility functions (TFs) in structural health monitoring. As a sequel to our previous study, a unified scheme to solving complex ratio random variables is proposed in this study for the case when it is highly non-trivial or impossible to discover a closed-form solution such as the complex-valued t ratio distribution. Based on the probability transformation principle in the complex-valued domain, a unified formula is derived by reducing the concerned problem into multi-dimensional integrals, which can be solved by advanced numerical techniques. A fast sparse-grid quadrature (SGQ) scheme by constructing multivariate quadrature formulas using the combinations of tensor products of suitable one-dimensional formulas is utilized to improve the computational efficiency by avoiding the problem of curse of integral dimensionality. The unified methodology enables the efficient calculation of the probability density function (PDF) of a ratio random variable with its denominator and nominator specified by arbitrary probability distributions including Gaussian or non-Gaussian ratio random variables, correlated or independent random variables, bounded or unbounded ratio random variables. The unified scheme is applied to uncertainty quantification for raw FRFs and TFs without any post-processing such as averaging, smoothing and windowing, and the efficiency of the proposed scheme is verified by using the vibration test field data from a simply supported beam and from the Alamosa Canyon Bridge.