PSI - Issue 78

A. Romanazzi et al. / Procedia Structural Integrity 78 (2026) 777–784 781 time instants of the section [ , ] , initial minimum AR-order, X-order and time delay order are set as [ , , ] and ARX fitting is conducted. To identify the ARX model that best describes each segment, a minimum normalized root mean squared error, , on the estimation data is fixed as fit requirement. In case the requirement is not satisfied, the AR-order, X-order, and the delay order are increased of the order of the iteration, = + 1 , as [ , , ]+ , and the fitting is performed again considering all the new combinations given by: [ [ , + ] [ , + ] [ , + ] ] (10) When the requirement is met, the iteration over the orders stops and the ARX model that provides the highest NRMSE among all the orders combinations is selected as: ̂ = [ [ , + ] [ , + ] [ , + ] ] { } (11) In order to prevent infinite loops, the maximum AR-order, X-order, and time delay order, can be imposed as [ , , ] . Then, frequency response analysis is performed considering the inferred ARX model that best describes the pre-seismic scenario ̂ , and post-seismic scenario ̂ , as transfer function ( ) . As for the procedure of the segmentation of the remaining data points, the algorithm is explained as follows and assuming that the post-seismic scenario is a priori selected. The segment =1 is initialised, and for the first iteration on the segment length =1 , the initial and final instants, [ , ] . Therefore, the ARX fitting is performed on the selected time interval after setting for the iteration orders =0 the minimum AR-order, X-order and time delay order as [ , , m in ] . To individuate the ARX model that best describes each segment, a minimum normalized root mean squared error, , on the estimation data is fixed as fit requirement. In case the requirement is not satisfied, the AR-order, X-order, and the delay order are increased of the order of the iteration, = + 1 , as [ , , ]+ , and the fitting is performed again considering all the new combinations. When the requirement is met, the iteration over the orders stops and the ARX model that provides the highest NRMSE among all the orders combinations is selected as: ̂ = [ [ , + ] [ , + ] [ , + ] ] { } (12) In order to prevent infinite loops, the maximum AR-order, X-order, and time delay order, can be imposed as [ , , m ax ] . However, in case the requirement is not satisfied and the maximum model order [ , , ] is attained, the iteration over the length of the segment is increased of = + 1 and the data points are halved as: [ , ] − 1 2 (13)