Robust Factor Analysis for Compositional Data
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Factor analysis as frequent technique for multivariate data inspection is widely used also for compositional data analysis. The usual way is to use a centered logratio (clr)
transformation to obtain the random vector y of dimension D. The factor model is
then
y = Λf + e (1)
with the factors f of dimension k < D, the error term e, and the loadings matrix Λ.
Using the usual model assumptions (see, e.g., Basilevsky, 1994), the factor analysis
model (1) can be written as
Cov(y) = ΛΛT + ψ (2)
where ψ = Cov(e) has a diagonal form. The diagonal elements of ψ as well as the
loadings matrix Λ are estimated from an estimation of Cov(y).
Given observed clr transformed data Y as realizations of the random vector
y. Outliers or deviations from the idealized model assumptions of factor analysis
can severely effect the parameter estimation. As a way out, robust estimation of
the covariance matrix of Y will lead to robust estimates of Λ and ψ in (2), see
Pison et al. (2003). Well known robust covariance estimators with good statistical
properties, like the MCD or the S-estimators (see, e.g. Maronna et al., 2006), rely
on a full-rank data matrix Y which is not the case for clr transformed data (see,
e.g., Aitchison, 1986).
The isometric logratio (ilr) transformation (Egozcue et al., 2003) solves this
singularity problem. The data matrix Y is transformed to a matrix Z by using
an orthonormal basis of lower dimension. Using the ilr transformed data, a robust
covariance matrix C(Z) can be estimated. The result can be back-transformed to
the clr space by
C(Y ) = V C(Z)V T
where the matrix V with orthonormal columns comes from the relation between
the clr and the ilr transformation. Now the parameters in the model (2) can be
estimated (Basilevsky, 1994) and the results have a direct interpretation since the
links to the original variables are still preserved.
The above procedure will be applied to data from geochemistry. Our special
interest is on comparing the results with those of Reimann et al. (2002) for the Kola
project data
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