336 H. Shi et al. Fig. 41.7 Residual series of the cointegration model, the vertical red dashed line indicates damage introduction, the two horizontal red lines represent the three standard error bars; the grey shaded areas show where cointegration switches regimes It is clear that the residual series is stationary before damage introduction, any effect from temperature is effectively eliminated, and the nonlinear behaviour of the frequency response is precisely captured. After 5000 data points, the magnitude of the residual series exceeds the confidence interval immediately, which indicates strongly the occurrence of damage, the overlaid grey areas show where cointegration switches from one regime to the other. The result can be interpreted by the fact that the regime-switching cointegration is estimated with training data under normal condition, and the health state of the system has been accurately modelled. Whenever damage occurs, the long term relationship of the variables no longer holds, thus the residual series turns nonstationary immediately. 41.5 Discussions and Conclusions Despite the fact that the method suggests very good results, one may still argue that shuffling the original series may break the underlying cointegrating relationship, therefore the estimation procedure might be ill-conditioned. This argument is partly true, that rearranging the order of series will surely break the underlying error correction mechanism [as expressed in Eq. (41.5)], but the long term relationship stays the same, or in other words, the rearranged series have the same cointegrating vectors as the original series, because the cointegrating relationships are stacked pointwise in time. One should bear in mind that the final goal here is fundamentally different from the aim of the econometricians, the concern is more about the long term relationship between variables, the short term adjustments are less of interest for the moment. Therefore, it is legitimate to use temperature as a reference series to rearrange the original series, and estimate the cointegrating vectors of the yielded series. This paper is concerned with exploring a new nonlinear cointegration method aiming to address the issue of nonlinear effects of EOVs in SHM data. The proposed method is based on a breakpoint model from econometrics to build a piecewise linear cointegration model. The proposed method is validated with a synthetic case, the results suggest that environmental effects on systems can be successfully removed; it needs the authors to further investigate this approach to solve real engineering problems.
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