A kernel-based mixed effect regression model for earthquake ground motions

Abstract This paper presents a semi-parametric mixed-effect regression approach for analyzing and modeling earthquake ground motions, taking into account the correlations between records. Using kernels, the proposed method extends the classical mixed model equations to complicated relationships. The predictive equation is composed of parametric and nonparametric parts. The parametric part incorporates known relationships into the model, while the nonparametric part captures the relationships which cannot be cast into a simple parametric form. A least squares kernel machine is used to infer the nonparametric part of the model. The resulting semi-parametric model combines the strengths of parametric and nonparametric approaches, allowing incorporation of prior, well-justified knowledge into the model while retaining flexibility with respect to the explanatory variables for which the functional form is uncertain. Equations for pointwise confidence and prediction intervals around the conditional mean are provided. The validity of the proposed method is demonstrated through numerical simulations and using recorded ground motions.

[1]  W. B. Joyner,et al.  Methods for regression analysis of strong-motion data , 1993, Bulletin of the Seismological Society of America.

[2]  Alexander Gammerman,et al.  Ridge Regression Learning Algorithm in Dual Variables , 1998, ICML.

[3]  Jale Tezcan,et al.  A Probabilistic Nonparametric Model for the Vertical-to-Horizontal Spectral Ratio , 2012 .

[4]  J. Mercer.  Functions of Positive and Negative Type, and their Connection with the Theory of Integral Equations , 1909 .

[5]  J. Berger.  Minimax estimation of a multivariate normal mean under arbitrary quadratic loss , 1976 .

[6]  Qiang Cheng,et al.  A nonparametric characterization of vertical ground motion effects , 2012 .

[7]  D. Harville.  Maximum Likelihood Approaches to Variance Component Estimation and to Related Problems , 1977 .

[8]  Paul L. Speckman,et al.  Confidence bands in nonparametric regression , 1993 .

[9]  N. A. Abrahamson,et al.  A stable algorithm for regression analyses using the random effects model , 1992, Bulletin of the Seismological Society of America.

[10]  Johan A. K. Suykens,et al.  Approximate Confidence and Prediction Intervals for Least Squares Support Vector Regression , 2011, IEEE Transactions on Neural Networks.

[11]  David R. Brillinger,et al.  An exploratory analysis of the Joyner-Boore attenuation data , 1984 .

[12]  C. R. Henderson.  Statistical methods in animal improvement: historical overview. , 1990 .

[13]  Chih-Jen Lin,et al.  Asymptotic Behaviors of Support Vector Machines with Gaussian Kernel , 2003, Neural Computation.

[14]  Johan A. K. Suykens,et al.  Least Squares Support Vector Machines , 2002 .

[15]  Qiang Cheng,et al.  Confidence and prediction intervals for semiparametric mixed-effect least squares support vector machine , 2014, Pattern Recognit. Lett..

[16]  Vladimir Vapnik,et al.  Statistical learning theory , 1998 .

[17]  Johan A. K. Suykens,et al.  Least Squares Support Vector Machine Classifiers , 1999, Neural Processing Letters.

[18]  David R. Brillinger,et al.  Further analysis of the Joyner-Boore attenuation data , 1985 .

[19]  J. Suykens,et al.  Primal and dual model representations in kernel-based learning , 2010 .

[20]  M. Aizerman,et al.  Theoretical Foundations of the Potential Function Method in Pattern Recognition Learning , 1964 .

[21]  Xihong Lin,et al.  Semiparametric Regression of Multidimensional Genetic Pathway Data: Least‐Squares Kernel Machines and Linear Mixed Models , 2007, Biometrics.

[22]  David M. Boore,et al.  Peak horizontal acceleration and velocity from strong motion records including records from the 1979 Imperial Valley, California, earthquake , 1981 .