# 6.7. Introduction to Projection to Latent Structures (PLS)¶

Projection to Latent Structures (PLS) is the first step we will take to extending latent variable methods to using more than one block of data. In the PLS method we divide our variables (columns) into two blocks: called \(\mathbf{X}\) and \(\mathbf{Y}\).

Learning how to choose which variables go in each block will become apparent later, but for now you may use the rule of thumb that says \(\mathbf{X}\) takes the variables which are always available when using the model, while \(\mathbf{Y}\) takes the variables that are *not always available*. Both \(\mathbf{X}\) and \(\mathbf{Y}\) must be available when building the model, but later, when using the model, only \(\mathbf{X}\) is required. As you can guess, one of the major uses of PLS is for predicting variables in \(\mathbf{Y}\) using variables in \(\mathbf{X}\), but this is not its only purpose as a model. It is a very good model for process understanding and troubleshooting.

PLS can be used for process monitoring and for optimizing the performance of a process. It is also widely used for new product development, or for improving existing products. In all these cases the \(\mathbf{Y}\) block most often contains the outcome, or quality properties.

However, PLS is most commonly used for prediction. And this is also a good way to introduce PLS. In (chemical) engineering processes we use it to develop software sensors (also known as inferential sensors) that predict time-consuming lab measurement in real-time, using the on-line data from our processes. In laboratories we use spectral data (e.g. NIR spectra) to predict the composition of a liquid; this is known as the calibration problem; once calibrated with samples of known composition we can predict the composition of future samples.

But why use the PLS method at all?

- 6.7.1. Advantages of the projection to latent structures (PLS) method
- 6.7.2. A conceptual explanation of PLS
- 6.7.3. A mathematical/statistical interpretation of PLS
- 6.7.4. A geometric interpretation of PLS
- 6.7.5. Interpreting the scores in PLS
- 6.7.6. Interpreting the loadings in PLS
- 6.7.7. How the PLS model is calculated
- 6.7.8. Variability explained with each component
- 6.7.9. Coefficient plots in PLS
- 6.7.10. Analysis of designed experiments using PLS models
- 6.7.11. PLS Exercises