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Classical Least Squares (CLS) Methods

Course Description

Principal components analysis PCA) and inverse least squares (ILS) methods such as partial least squares (PLS) are ubiquitous to chemometrics. However, classical least squares (CLS or forward least squares) techniques are seeing a resurgence in popularity. Two major reasons are better interpretability and the ability to control aspects of the regression modeling. As with ILS, CLS methods can be used for exploratory analysis, detection, classification and quantification. This half-day course will start by covering CLS regression methods including classical, extended, weighted and generalized least squares. It will be shown how these methods can be used to account for interferents (i.e. analytes other than the one of interest) in spectroscopic systems. CLS also provides a natural framework for the development of popular de-cluttering methods such as External Parameter Orthogonalization (EPO) and Generalized Least Squares (GLS) weighting. These methods can be combined with CLS to form Gray CLS models which have greatly improved prediction performance compared to conventional CLS. The course includes hands-on computer time using PLS_Toolbox or Solo to work through example problems. It will also be shown how constraints can be easily employed with these methods to allow greater control over the modeling.


Linear Algebra for Machine Learning and ChemometricsChemometrics I –– PCA, and Chemometrics II – Regression and PLS or equivalent experience.

Course Outline

1.0 Introduction
1.1 The CLS Model
1.2 Inherent Challenges

2.0 Practical use of CLS Models
2.1 Model identification
2.2 Model application
2.3 Examples

3.0 Extended CLS
3.1 Why?
3.2 Creating extended factors
3.3 Application
3.4 Examples

4.0 Weighted CLS
4.1 Introduction
4.2 Suitable scenarios
4.3 Application
4.4 Examples

5.0 Generalized Least Squares
5.1 Origins
5.2 Applications

6.0 Gray-CLS
6.1 Combining CLS with EPO and GLS
6.2 Cross validating to determined optimal parameters

7.0 Conclusions