41 An Exploratory Study on Removing Environmental and Operational Effects Using a Regime-Switching Cointegration Method 333 Fig. 41.2 Upper panel: the time series of the four natural frequency series of the system in Fig. 41.1 plotted as a function of time; Lower panel: temperature series plotted against time. Red dashed line imposes damage introduction Following the conventional cointegration procedures proposed in [4], one can obtain a residual series as shown in Fig. 41.3. Because the underlying cointegration has not been accurately modelled, only the first part of the residual series stays largely stationary, the high variance from the cold zone has been consequently left in the residual, damage indication has therefore failed. By further investigating the mutual relationship of the four natural frequency series in Fig. 41.1, one can see a clear bilinear relationship as plotted in Fig. 41.4. The knee points in the figure correspond to the breakpoint where the cointegrating relationship changes. Thus, a two-regime cointegration model should be appropriate to model this four-DOF system. Following the procedure in Section 3 to validate the performance of the regime-switching cointegration method: 1. Not all data points are needed for estimating the model, only the alternative points from point 5000 to 8000 are chosen as a training set. In order to find the relationship between the frequency series and temperature, the training set is rearranged according to the magnitude of the temperature series, as displayed in Fig. 41.5 along with the temperature series. Denote the shuffled series as ft D.f1t; f2t; f3t; f4t/; t D1;2; : : :N, where N is the sample size. Even though the stiffness is set to have a bilinear relationship with temperature, Fig. 41.5 does not show any clear sign of a breakpoint. 2. Next, the ADF test statistic is utilised as a tool to ascertain the position of the breakpoint. As the breakpoint can be anywhere in the series from the beginning to the end, the following step is to evaluate the ADF statistic at every possible breakpoint. However, in practice, it needs a minimum amount of data to calculate the ADF statistic, so only the data in the middle are used to evaluate it, that is the data set in the interval .Œ0:15N ; Œ0:85N /. Assume a breakpoint is inserted at position , then the original ft.1 W N/ is split into two sets: f1 .1 W /, f2 . C1 W N/. One then uses the Johansen procedure presented earlier to estimate the cointegrating vector of each set, sayˇ1 andˇ2 , and to construct the residual series at this breakpoint, e D.ˇ1 f1 I ˇ2 f2 /, where “;” is used to concatenate these two vector series; the subscript denotes the fact the residual series depends on the position of the breakpoint. 3. Repeat the procedure above to evaluate all the points in .Œ0:15N ; Œ0:85N / as breakpoints, and calculate the respective ADF statistics, which are plotted in Fig. 41.6; the horizontal axis represents the number index of the training set. The blank space at the beginning and the end of the figure indicates the fact that ADF statistics are only evaluated in the interval .Œ0:15N ; Œ0:85N /. The smallest value of the curve is at data point 976, corresponding to the temperature 0.4767ıC, which is quite close to the simulation assumption.
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