Due to its simplicity and explicit algebraic and geometric meanings, latent dimensions, and identification structures associated with these dimensions, reliability of the latent dimensions obtained by orthoblique transformation of principal components can be determined in a clear and unambiguous manner.
Let G = (gij); i = 1, ..., n; j = 1, ..., m is an acceptably unknown matrix of measurement errors in the description of a set E on a set V. Then the matrix of true results of entities from E on the variables from V will be Y = Z - G.
Assume, in accordance with the classical theory of measurement (Gulliksen, 1950, Lord - Novick, 1968; Pfanzagl, 1968), that matrix G is such that YtG = 0 and
GtGn-1 = E2 = (ejj2)
where E2 is a diagonal matrix, the covariance matrix of true results will be H = YtYn-1 = R - E2
if R = ZtZn-1
is an intercorrelation matrix of variables from V defined on set E.
Suppose that the reliability coefficients of variables from V are known; let P be a diagonal matrix whose elements j are these reliability coefficients. Then the variances of measurement errors for the standardized results on variables from V will be just elements of the matrix E2 = I - .
Now the true values on the latent dimensions will be elements of the matrix
= (Z - G)Q
with the covariance matrix = tn-1 = QtHQ = QtRQ - QtE2Q = (pq).
Therefore, the true variances of the latent dimensions will be the diagonal elements of matrix ; denote those elements with p2. Based on the formal definition of the reliability coefficient of some variable = t2 / where t2 is a true variance of the variable and is the total variance of the variable, or the variance that also includes the error variance, the reliability coefficients of the latent dimensions, if the reliability coefficients of the variables from which these dimensions have been derived are known, will be
p = p2 / sp2 = 1 - (qptE2qp )(qptRq )-1 p = 1,...,k
Evaluation criteria of landslide stability
