10 Autoregressive Model Applied to the Meazza Stadium for Damage Detection 107 Table 10.4 Days considered with the corresponding orders predicted by the three selected methods Date AIC BIC PACF 11th January 2011 100 29 26 16th May 2011 100 46 36 2nd August 2011 100 33 36 30th September 2011 100 44 11 Mean value 100 38 27 where b xis the standardized signal, and µx and¢x are the mean and standard deviation of x, respectively. We use the parameters of the AR models (˚i) as damage indicators. If damage occurs, the probability distribution function of the parameters of the autoregressive model changes. The parameters of each time history have been estimated by means of a least square approach. The choice of the time range is based on simulations with different duration: 1, 5, 10, 30 and 60 min. After the model estimation, the root mean squares (RMS) of the residuals have been considered, and the 10 min approach appears as the minimum duration to obtain a good representation of the time histories. Then, to create the database, the empty stadium condition 8:00 am and 8:10 am has been considered, for the 96 days previously selected. The time band considered for the events (Table 10.2) has been between 9:30 pm to 9.40 pm (when the most important events take place). In the end, for the days used as a check for the procedure, both mornings and evenings have been considered. The first step necessary to use the autoregressive models is the choice of the model order. In the next section, we report the result of the application of the three algorithms presented in Sect. 10.2.2. 10.4.2.1 Model Order Estimation The records used in this work are 96, so, the optimal model order estimation for each time history is time consuming, computationally speaking. In fact, the procedure consists of an iterative method with a fixed number of steps. Thereafter it could possible to identify the optimal order, in which one minimizing the function of the method adopted (Sect. 10.2.2). Therefore, we select four dates having a temperature range compatible with that of the considered events. The purpose is to ensure that the AR model is able to describe different environmental conditions. It has been decided to fix the maximum order to 100, which is a maximum that usually literature accepts as a limit the overfitting. In Table 10.4 the list of the selected days is shown, together with the corresponding predicted order by the three different methods, which are the most used in literature. In literature, the Bayesian information criterion (BIC) is considered as the most stable method, so it has been decided to establish the model order as the mean value of the four values predicted by BIC for the four considered days. The mean value is 38, and then we decide to fix the model order at 40, as a factor of safety to taking into account the great variability of the time considered histories. As a next step the different parameters for all the time histories have been computed, considering an autoregressive model of order 40, AR(40). After this step it is possible to predict the new time histories by a proper autoregressive model. The goodness of the data interpolation can be seen by considering the residuals between the original time history and the predicted one. Figure 10.13 shows the root mean square (RMS) of the residuals for all the 96 days considered. The RMS value of the residuals is on the range between 0.1 and 0.7 (Fig. 10.13). 10.4.2.2 Mahalanobis Distance on Parameters Once the AR model has been estimated, the Mahalanobis distance has been adopted to distinguish between different conditions. To do that, it is necessary to characterize the normal behaviour of the stand using a database made of the 96 sets of AR parameters evaluated on the 96 records. Computing the mean vector and the covariance matrix of the 96 groups of parameters it is possible to define a threshold allowing one to discern if a time history, not considered in the database, is referred to the empty stadium or not. In Fig. 10.14 the result of the considered cases are shown, with the ID cases reported in the Tables 10.2 and 10.3. The obtained threshold is 79.04. It is possible to see that the concerts and the football match are over the value of 79.04. Conversely, the other days of empty stadium are below this threshold. The evening of 20th December 2012 (C evening), appear as a fault, but it was discovered this was an outlier in the data catalogue of the stadium. In fact, not listed in the event agenda, a football match was played in that occasion, so the Mahalanobis index is correct, and it is comparable to the other
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