Analysis and comparison of an improved unreconstructed variance criterion to other criteria for estimating the dimension of PCA model

• Various selection criteria are analysed and compared to each other.
• Mathematical proofs showing the limitations of some selection criteria are provided.
• New selection criterion called improved unreconstructed variance (IUV) is proposed.
• Training data are deeply investigated and modelled using the studied criteria.

This paper provides a new criterion to select the significant components of an empirical process model using the principal component analysis approach. The proposed criterion is an improved unreconstructed variance (IUV) applied to a changing of process data representation. Four other criteria are studied to perform fundamental analyses and comparisons to each other. They are well known in the literature as the minimum description length (MDL), the imbedded error (IE), the equality of the eigenvalue (EOE) and the variance of reconstruction error (VRE). The selection of the significant components is usually constrained by three main difficulties such as the noise included in data, the presence of independent and quasi-independent process variables and the size of training samples. This paper presents two fundamental proofs that clarify the limitations of both criteria which are IE and VRE. The consistency of the MDL and EOE criteria improves by increasing the number of training observations. The purpose of the IUV criterion is to enhance the VRE in order to remedy the encountered limitations. The proposed criterion shows a promising consistency as well as a highly robustness versus the mentioned difficulties. Its potential and the limitations of the other criteria are illustrated using two numerical examples and the CSTR process.