ifa: Inverse of a matrix based on its factor decomposition.
In ERP: Significance Analysis of Event-Related Potentials Data

Description Usage Arguments Details Value Author(s) References See Also Examples

This function is not supposed to be called directly by users. It is needed in function emfa implementing an EM algorithm for a factor model.

1	ifa(Psi, B)

`Psi`	m-vector of specific variances, where m is the number of rows and columns of the matrix.
`B`	m x q matrix of loadings, where q is the number of factors

If Sigma has the q-factor decomposition Sigma=diag(Psi)+BB', then Sigma^-1 has the corresponding q-factor decomposition Sigma^-1=diag(phi)(I-theta theta')diag(phi), where theta is a mxq matrix and phi the vector of inverse specific standard deviations.

`iS`	m x m inverse of diag(Psi)+BB'.
`iSB`	m x q matrix (diag(Psi)+BB')^-1B.
`Phi`	m-vector of inverse specific standard deviations.
`Theta`	mxq matrix of loadings for the inverse factor model (see the details Section).

David Causeur, IRMAR, UMR 6625 CNRS, Agrocampus Ouest, Rennes, France.

Woodbury, M.A. (1949) The Stability of Out-Input Matrices. Chicago, Ill., 5 pp

emfa, isqrtfa

data(impulsivity)
erpdta = as.matrix(impulsivity[,5:505])   # erpdta contains the whole set of ERP curves     
fa = emfa(erpdta,nbf=20)          # 20-factor modelling of the ERP curves in erpdta
Sfa = diag(fa$Psi)+tcrossprod(fa$B) # Factorial estimation of the variance 
iSfa = ifa(fa$Psi,fa$B)$iS        # Matrix inversion    
max(abs(crossprod(Sfa,iSfa)-diag(ncol(erpdta)))) # Checks that Sfa x iSfa = diag(ncol(erpdta))