OPLS

OPLS stands for orthogonal projections to latent structures. It is a multivariate statistical method used to model the relationship between two data matrices, X (predictors) and Y (responses). It was introduced by Trygg and Wold in 2002 as an extension of partial least squares (PLS) regression. The key idea is to partition the variation in X into a predictive part that is correlated with Y and one or more orthogonal parts that are uncorrelated with Y. This separation isolates the signal of interest from unrelated variation, improving model interpretability and often predictive performance.

In OPLS, the model aims to maximize the covariance between the predictive X scores and Y, while orthogonal components capture systematic variation in X orthogonal to Y. The result is a model with typically one predictive component and several orthogonal components. The method can be used for regression (OPLS) and discriminant analysis (OPLS-DA) where Y is a binary or categorical variable. Outputs include score plots, loading plots, and VIP scores to identify influential variables.

Applications are widespread in chemometrics, especially with metabolomics, spectroscopy, and other omics datasets, where high collinearity and structured noise are common. Preprocessing steps such as centering, scaling, and possibly log transformation are important for robust results.

Advantages include improved interpretability, clearer separation of predictive versus nonpredictive variation, and often better cross-validated performance. Limitations include reliance on linear relationships, potential overfitting if component numbers are not properly chosen, and sensitivity to outliers. Proper cross-validation and permutation testing are recommended to assess model validity.