160 M. Sheibani et al. with the predictive models, either fragilities curves or ANN trained models. In this paper, we propose to predict the damage of similar structures in the whole area based on a limited number of visually or instrumentally inspected structures. Visual inspections can be made shortly after the hazard and the information will be sent to the data center for further analysis. Moreover, there may be some structures which are pre-equipped with monitoring equipment, such as accelerometers, and this could be another source of information. Combining these two sources, the data center will train a model to predict the responses for other uninspected structures. In this paper, we aim to train our machine learning model from some limited posthazard observations, and predict the responses for the larger test set. Data from the 2011 Fukushima earthquake is utilized in the form of 592 ground motions recorded at different locations of the country of Japan. The training set is consisted of 100 randomly selected records and the rest is used for verification of the method. Gaussian Process Regression learning method is used to that end. Four different building types are considered in this study, one of which is randomly selected and assigned to each ground motion record. Finally, the predicted responses are compared to the responses obtained by nonlinear time history analysis. 15.2 Gaussian Process Regression To implement the Machine Learning (ML) to predict the structure responses, GPR algorithm has been used in this paper. GPR works basically like linear regression but for non-parametric functions. In this case, we have a multi-dimensional vector as the independent variable xi and observations yi are made at different intervals of x. Assuming that all dependent variables, i.e. y1, y2, . . . , yn, are different elements of one point sampled from a multivariate Gaussian distribution, one can relate observations to each other using the covariance matrixKas follows. Considering the exponential kernel, each element of the matrixKcan be expressed as [10], Ki,j =k xi, xj =σ 2 f exp − 1 2 xi −xj Tdiag(m) xi −xj +σ 2 nδij (15.1) where θ = m,σ 2 f ,σ 2 n is the vector of the hyperparameters. The maximum allowable covariance is represented by σ 2 f, the variance of the Gaussian distributed noise affecting the observations is denoted byσ 2 n, andδij is the Kronecker delta function. Different choices for the matrixmcan be found in [10]. Using the above-mentionednobservations, the prediction of y ∗ given x ∗ is achieved by GPR. In order to predict, two more Matrices need to be defined, k∗ =[k(x∗,x1)k(x∗,x2) . . .k(x∗,xn)] (15.2) k∗∗ =k(x∗,x∗) (15.3) As mentioned before, the observations are considered as a sample from a multivariate Gaussian distribution. Based on this assumption, and considering the conditional probability, we have, y∗ | y ∼N k∗K−1y, k∗∗ −k∗K−1kT ∗ (15.4) Therefore, the mean and variance of the predicted instance can be expressed as E(y∗) =k∗K−1y (15.5) var (y∗) =k∗∗ −k∗K−1kT ∗ (15.6) It is clear that in order to have predictions with lower variances, the testing instances should be well covered with the training data.
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