Comparison of separate and joint modeling of bivariate response with emphasis on PLS

Abstract

In this thesis we have tried to find if or when multiresponse Partial Least Squares Regression(PLS2)predicts better than uniresponse PLS(PLS1).

With a simulation study and analysis of variance we have investigated how PLS1 predicts with different simulation parameter settings. The result showed that if we had small relevant eigenvalues, the predictor based on PLS1 does not predict well. We have also compared the estimated values with the true values of parameters, with focus on eigenvalues and covariances. Then we found that if we had small relevant eigenvalues, the estimated values was often very different from the true parameters.

The estimated regression coefficients found by PLS1 and PLS2 differ. We found empirical that for one component the PLS2 estimator is a linear combination of the two PLS1 estimators, one for each response.

For prediction the two PLS1 predictors and PLS2 predictor provide very similar result. The results showed that with some simulation parameter settings PLS2 was a better predictor than PLS1. This happened if we had only one common relevant component with a small relevant eigenvalue. Based on analysis of variance we found that the difference in prediction error between the two methods was larger, when the number of observations were few and there was high degree of collinearity simultaneous. However the variation between replications was found to be large. We have also tested the methods on real data sets, but PLS2 did not predict better than PLS1 on these. Therefore we concluded with that as far as we have seen PLS1 is a better choice as a predictor than PLS2.