, Structural Health Monitoring of deterministic vectors. Lower-case letters q, w, and x are deterministic vectors. Upper-case letters such as X or H are real-valued random variables. Boldface upper-case letters such as X or H are vector-valued random variables. Upper-case letters Q, U, W, and X are vector-valued random variables
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, In practice, the initial dataset can be made up of heterogeneous numerical values and must be scaled for performing computational statistics
, Consequently, quantities q j , x j , and x j are assumed to be scaled quantities (see Soize and Ghanem (2016) for the scaling)
, X Nd ] be the random matrix with values in M n,Nd , whose columns are N d independent copies of random vector X. The normalization of random matrix [X] is attained with random matrix, Data normalization. Let X be the R n -valued second-order random vector defined by Equation (A3) for which the N d independent realizations are N d data points in R n , represented by the matrix
× ?) diagonal matrix of the ? positive eigenvalues of the empirical estimate of the covariance matrix of X (computed using x 1 ,
,
, When n is small, ? can be chosen as n. If some eigenvalues are zero, they must be eliminated and then ? < n. When n is high, a statistical reduction can be done as usual and therefore ? < n in such a case
For 1 < m ? N d , let, 2005. ,
, the positive-definite diagonal real matrix in M Nd such that [b] ij = ? ij Nd j =1 [K] ij in which [K] ij = exp(? 1 4? diff ? i ?? j 2 )