A Semi-Parametric Tool for Seismic Risk Management Applications

This study presents a semi-parametric mixed-effect regression approach for analysis and modeling of clustered observations. The proposed approach allows incorporation of known relationships with respect to a subset of the predictors into the model while retaining flexibility in the absence of a well-justified functional form. The model is separated into parametric and non-parametric parts. The predictors of the non-parametric part are then mapped into a high dimensional space via a kernel function, thereby transforming the model into one which can be solved using standard linear mixed model equations. The proposed method can be used as part of a seismic loss estimation framework to analyze and model data containing correlated observations. Introduction Losses caused by earthquakes and other natural disasters are of major concern to various stakeholders. As seismic loss estimates are used in a number of applications, such as code provisions, insurance calculations, and public policy decisions, the manner uncertainties are quantified and used has major political, economic and social implications. Regression analysis is a statistical procedure used in various stages of seismic loss estimation. To date, numerous regression approaches with varying levels of sophistication have been proposed. The majority of the existing regression algorithms are of parametric nature, relying on a fixed mathematical form. While non-parametric models are advantageous when a well-justified mathematical form does not exist, they do not allow incorporation of knowledge regarding the functional form with respect to a subset of the predictors. This study presents a semi-parametric regression approach which allows incorporation of known relationships with respect to a subset of the predictors into the model while retaining flexibility with respect to other predictors. A demonstrative example, based on a sample of earthquake records, is presented. The Semi-Parametric Model Consider a dataset containing observations from N events with a total of Nn = ∑ ni i=1 observations, where ni is the number of observations from the ith event. Let yij, the jth 1 Associate Professor, Dept. of Civil and Env. Engineering, SIUC, Carbondale, IL 62901, Email: jale@siu.edu 2 Graduate Student, Dept. of Civil and Env. Engineering, SIUC, Carbondale, IL 62901, Email: yildiz.dak@siu.edu 3 Associate Professor, Computer Science Dept., SIUC, Carbondale, IL 62901, Email: qcheng@cs.siu.edu Tezcan J, Dak Hazirbaba Y and Cheng Q. A semi-parametric tool for seismic risk management applications. Proceedings of the 3 International Conference on Urban Disaster Reduction, Earthquake Engineering Research Institute, Boulder, CO, USA, 2014. observation from the ith event, be modeled as yij = βxij + h�sij� + ui + eij , i = 1:N, j = 1:ni (1) where xij and sij denote the vector of predictors of the parametric part and the non-parametric part, respectively; β is a coefficient vector; h is an unknown function; ui is a random intercept; and eij is a random error term associated with yij . For notational convenience, the first entry of β is reserved for the constant term, and the first entry of vector xij is set to 1. Using the index map (1,1) . (1,n1) (2,1) . (2,n2) . (N, 1) . (N,nN) ↓ . ↓ ↓ . ↓ . ↓ . ↓ 1 . n1 n1 + 1 . n1 + n2 . �nj + 1 N−1