Bayesian analysis of hierarchical random fields for material modeling

The spatial averaging method for non-homogeneous random fields with application to reliability analysis

“…,38]. Consider a Student's 𝑡-RF 𝑌 (𝒛), in which 𝐹 𝑌 at any 𝒛 ∈ 𝒁 is a Student's 𝑡-distribution, with location parameter 𝜇 𝑌 (𝒛), scale parameter 𝜎 𝑌 (𝒛) and degrees of freedom 𝜈 𝑌 [30,38,39]:…”

“…The log-Student's 𝑡-distribution combines the lognormal and the Student's 𝑡-distribution and thus, can be used to model non-negative quantities accounting for parameter uncertainty [30]. Consider a log-Student's 𝑡-RF 𝑉 (𝒛), i.e., 𝐹 𝑉 at any 𝒛 ∈ 𝒁 is a log-Student's 𝑡distribution [30,40]:…”

“…The parametrization of 𝐹 𝑉 is done by means of the parameters of the underlying Student's 𝑡-distribution. At any 𝒛 ∈ 𝒁 it holds that 𝑌 (𝒛) = ln(𝑉 (𝒛)) follows a Student's 𝑡-distribution with location parameter 𝜇 𝑌 (𝒛) = 𝜇 ln𝑉 (𝒛), scale parameter 𝜎 𝑌 (𝒛) = 𝜎 ln𝑉 (𝒛) and degrees of freedom 𝜈 𝑌 = 𝜈 𝑉 [30]. By combining Eqs.…”

“…Furthermore, we extend SA to a special class of non-Gaussian RFs, so-called translation RFs [29], and present application for RF models with lognormal, Student's 𝑡-and log-Student's 𝑡-marginal distribution. Student's 𝑡-and log-Student's 𝑡-RFs appear as predictive RFs when learning is performed with spatial data [30]. RFs with lognormal or log-Student's 𝑡-marginal distribution are advantageous for modeling non-negative quantities, such as strength parameter of materials, as the support of these distributions is limited to the positive axis.…”