98 G. Busca et al. The layout of the next paper steps is the following: Sect. 10.2 provides a brief description of the theory adopted in this paper, through the adopted autoregressive model (10.2.1) and the estimation of the optimal model order (10.2.2); in the end a summary about the Mahalanobis distance method will be given (10.2.3). Section 10.3 describes the monitored structure, the Meazza stadium based in Milan, with a structural description of the stand under examination (10.3) and of its dynamical behaviour (10.3.1). Section 10.4 then deals with data analysis, through the basic statistics on the available vibration data (10.4.1), the implementation and results of the autoregressive model (10.4.2). Finally, Sect. 10.5 gives the final comments. 10.2 Theory and Description of the Adopted Models In this section, the AR modelling, the choice of optimal model order and Mahalanobis distance are briefly explained. 10.2.1 Autoregressive Models: AR In time series analysis, one of the most useful representation to express a time series process is the autoregressive (AR) model. It is a stochastic finite linear model that will be briefly explained in the following. For a complete theory of this topic, please refer to [3]. If Z is a generic stationary process, we can estimate the value of Z at time t just basing the evaluation on its past values plus white noise. Let us fix the value of a process at equally spaced times t, t 1, t 2, : : : by zt, zt 1, zt 2, : : : . Also let Qzt , Qzt 1, Qzt 2, : : : be the deviation from the process mean value (assumed stationary); for example, Qzt Dzt . Then the value Qzt can be written as: Qzt Dˆ1Qzt 1 Cˆ2Qzt 2 C CˆpQzt p Cat (10.1) where p is the order of the model, ˚i is the constant coefficient of the autoregressive model and at is white noise. at is a sequence of uncorrelated random values from a fixed distribution with constant mean E(at), (usually assumed to be 0) and constant variance Var.at / D 2 a). If we define an autoregressive operator of order p by the following equation: ˆ.B/ D1 ˆ1B ˆ2B2 ˆpB2 (10.2) Then the autoregressive model may be written as: ˆ.B/Qzt Dat (10.3) The model contains pC2 unknown parameters , ˆ1, ˆ2, : : : , ˆp, 2 a, which in practice have to be estimated from the data. 10.2.2 Optimal Order Estimation In time series analysis or, more generally, in any data analysis, there may be several adequate models that can be used to represent a given data set. Sometimes, the best choice is easy, other times the choice can be very difficult. Thus, plenty of criteria for model order estimation have been introduced and are described in literature for a proper model selection. Model identification techniques such as Partial Autocorrelation Function (PACF), Akaike’s Information Criterion (AIC) and Bayesian Information Criterion (BIC) [4] are used to identify the most adequate model order. These three models are those most used in literature. A way to assess an adequate model is to check if residuals are white noise. For a given data set, when there are multiple adequate models, the selection is normally the model with the minimum order. In the following, we briefly introduce the formulations of some model selection criteria.
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