Measures of fit in principal component and canonical variate analyses

Abstract

Treating principal component analysis (PCA) and canonical variate analysis (CVA) as methods for approximating tables, we develop measures, collectively termed predictivity, that assess the quality of fit independently for each variable and for all dimensionalities. We illustrate their use with data from aircraft development, the African timber industry and copper froth measurements from the mining industry. Similar measures are described for assessing the predictivity associated with the individual samples (in the case of PCA and CVA) or group means (in the case of CVA). For these measures to be meaningful, certain essential orthogonality conditions must hold that are shown to be satisfied by predictivity.

Keywords:

• biplots
• canonical variate analysis
• measures of fit
• prediction
• principal component analysis

Acknowledgements

We thank an anonymous referee for identifying obscurities, which we hope we have eliminated to make a more readable paper. This work is based upon research supported by the National Research Foundation of South Africa. Any opinion, findings and conclusions or recommendations expressed in this material are those of the authors and therefore the NRF does not accept any liability in regard thereto.