Canonical variables based on robust sacales

STELLA MARIS DONATO; JORGE GABRIEL ADROVER

Punta del Este

Congreso; CIMPA-UNESCO-MESR-MICINN research School 2012 "New trends in Mathematical Statistics"; 2012

CIMPA

Canonical analysis is a dimension-reduction technique in which two random vectors x ∈ Rp and y ∈ Rq are reduced to r−dimensional vectors u = Cx =(u1, ..., ur) and v = By =(v1, ..., vr) , with C ∈ Rr×p and B ∈ Rr×q, r ≤ min (p, q) . The elements of u and v are uncorrelated respectively and the squares of the correlations between uj and vj , j = 1, ..., r, are maximized subject to CΣ11Ct = I and BΣ22Bt = I, where Σ11 and Σ22 are the covariance matrices of x and y respectively. This approach amounts to find 2r linear combinations u = Cx and v = By which minimize E
u − v − a2  , with a ∈ Rr. This measure of closeness is very sensitive to outlying observations. Rather, the proposal is to minimize a robust univariate dispersion measure based on u − v − a2 to derive robust canonical vectors and correlations. Canonical analysis is closely related to perform principal component analysis (PCA) with the block vector 
Σ−1/2 11 x t ,
Σ−1/2 22 y t t . By these means an iterative and efficient algorithm implemented by Maronna (2005) to perform robust PCA based on robust univariate scales is used. A simulation study and a real data analysis is conduced to compare the new procedure with some other competitors, showing a remarkable performance.