30 Wireless Acoustic Emission Monitoring of Structural Behavior 235 As it is known, the MLE finds the particular values of the model parameters which make the observed results the most probable or, in other words, which maximize the likelihood functionL. Working equivalently with the logarithm of Eq. 30.4 and searching for the MLE of the model parameters we get: @lnL.‚1;‚2;k;Mjx/ @ i D0 i D1;2; (30.5) which has the solution: 2 i;max D 1 ni qi XjDpi xj M XmD1 ai mxj m! 2 i D1;2: (30.6) Inserting Eq. 30.6 into Eq. 30.4 we get the maximized logarithmic likelihood function [29–31]: lnL.‚1;‚2;k;Mjx/ D k M 2 ln 2 1;max n k M 2 ln 2 2;max CC1 (30.7) where C1 is a constant. The expression in Eq. 30.7 is the basis for the Akaike Information Criterion (AIC), in which the AICfunction is defined as AICD2P 2ln(maximized likelihood function), where Pis the number of parameters in the statistical model. Generally, a model with minimumAICvalue is thought to be most suitable one among the competing models. Originally this function was designed to determine the optimal order for an AR process fitting a time series. In the current application, the order Mof the AR process is fixed, and therefore the AICfunction is a measure for the model fit. The point k where AICis minimized, or Lis maximized, determines the optimal separation of the two time series—the first representing noise and the second containing the signal—in the least square sense, and is interpreted as the onset time of the signal. In this sense, the AICas a function of k is knownas AICpicker [29]: AIC.k/ D.k M/ln 2 1;max CŒn k M ln 2 2;max CC2; (30.8) where C2 is a constant. Alternatively, the AIC value can be directly calculated from the signal without dealing with the AR coefficients. As M<<n, Eq. 30.8 can be simplified [29]: AIC.k/ Dkln.var.xŒ1;k // C.n k 1/ln.var.x Œ1Ck;n //; (30.9) where k goes through all the signal trace and var is the sample variance. As AICpicker finds the onset point as the global minimum, it is necessary to choose a time window that includes only the segment of interest of the signal. If the time window is chosen properly, AICpicker can find the first arrival of the signal (P-wave arrival for AE) accurately. In case of low S/N ratios (as for noisy EM signals) or more seismic phases (as P-wave and S-wave for AE signals) in a time window, global minimum cannot guarantee to indicate the first arrival of the signal. For this reason a pre-selection of this window is necessary to apply the procedure. Here, the onset time is firstly pre-determined using a threshold amplitude level: 10 XkDiC1 jxkj!=10 4 i XkD1 jxkj!=i; (30.10) The first value for the index k that makes relation (30.10) fulfilled is named k0 and it is the first estimation for the onset time. This first estimation is always localized after the actual onset time. Thus, we applyAICpicker to the interval [1,k0] for a rough determination of the onset time, k1. Then, the application of AICpicker to the time window with center in k1 and width2(k1 k0) gives the value kmin, which is regarded as the actual onset time of the analyzed signal.
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