4 Structural Damage Detection Through Vibrational Feature Analysis with Missing Data 25 'T 1 .k/ D z T .k 1/ zT .k n/ 'T 2 .k/ D u.k 1/ u.k n/ 1 Dha 1 b1 an bn i T 2 D c1 cn T So, Eqs. (4.1) and (4.2) may be written as below. y.k/ D y C' T 1 .k/ 1 Cv.k/ u.k/ D u C' T 2 .k/ 2 Cw.k/ (4.3) The complete data loglikelihood criterion is derived from the joint density of the N observations, given in Eq. (4.4). f D N Y kD1 ( 1 p2 1 exp " y.k/ 'T 1 .k/ 1 y 2 2 1 #) ( 1 p2 2 exp " u.k/ 'T 2 .k/ 2 u 2 2 2 #) (4.4) The loglikelihood criterion is then derived by taking the natural logarithm of this density: L. ;ƒ/ DC N 2 log . 1/ N 2 log . 2/ 1 2 1 N X kD1 y.k/ ' T 1 .k/ 1 y 2 1 2 2 N X kD1 u.k/ ' T 2 .k/ 2 u 2 (4.5) Differentiating with respect to yields the following criteria to maximize the likelihood with respect to the variances: 1 D 1 N N X kD1 Eh y.k/ ' T 1 .k/ 1 y 2i (4.6) 2 D 1 N N X kD1 Eh u.k/ ' T 2 .k/ 2 u 2i (4.7) To define the other unknown parameters y and u and parameter vectors 1 and 2, it is simplest to back substitute Eqs. (4.6) and (4.7) into Eq. (4.5): L. ;ƒ/ DC N 2 log 1 N N X kD1 Eh y.k/ ' T 1 .k/ 1 y 2i ! N 2 log 1 N N X kD1 Eh u.k/ ' T 2 .k/ 2 u 2i ! (4.8) Maximizing this likelihood is equivalent to minimizing the terms within the logarithms, so differentiating those terms with respect to the unknown parameters and parameter vectors yields: 1 D N X kD1 E '1.k/' T 1 .k/ ! 1 N X kD1 EŒ'1.k/y.k/ (4.9) 2 D N X kD1 E '2.k/' T 2 .k/ ! 1 N X kD1 EŒ'2.k/u.k/ (4.10) y D 1 N N X kD1 y.k/ ' T 1 .k/ 1 (4.11) u D 1 N N X kD1 u.k/ ' T 2 .k/ 2 (4.12)
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