DESCRIPTION OF IV-PCA METHOD

Historic: PCA with Instrumental Variables appear for the first time in Rao (1964). The method has been mainly developed by Robert Sabatier in France (in the biometry sector).

Aim: IV-PCA can be seen either as a causal model, either as a predictive model. It is employed in both cases in the case there are much correlated variables for X (nxp)and Y(nxq). There are 2 parameters to choose, one for X and one for Y, pcx and pcy.

Causal model: It is in this case a PCA of Y, with linear constraints (related to X) on the mode of the objects, in order to underline the variations of Y due to X (see criterion).

Predictive model: it can be a robust predictive model too.Suppose X is of rank p and Y of rank q:

• if pcy=q, then IV-PCA is a PCR model (Principal Components Regression) .this case occurs notably when there is only one response variable y (pcy=q=1).
• if pcx=p, then IV-PCA is a RRR model (Reduced Rank Regression, called Redundancy Analysis too).
  

Criterion:

Algorithm: The algorithm uses projection tools associate to Singular Values Decomposition algorithms, and classical regression tools.

Code: The code comes from CReS, using built-in Matlab function "svd" and "svds", and chemo m-file "svd2".

Applications: Biometry,...

Dataset reference: bat.mat (X), viscosity.mat (Y)

References:

Rao C.R., The use and interpretation of principal components analysis in applied research, Sankhya, serie A, Vol 26, 1964, 329-357

Sabatier, R., Lebreton, J.D. and Chessel D., Principal component analysis with instrumental variables as a tool for modelling composition data, 1989, 341-352 in COPPI, R. & BOLASCO, S. (Eds) Multiway data analysis. North Holland, Amsterdam

Franc A., Etude algébrique des multitableaux: apports de l'analyse tensorielle, PhD thesis, 1992, Université de Montpellier.