partial least squares

PLS (Partial Least Squares Regression) is a multivariate regression technique that finds latent variables (components) in the predictor space that are maximally correlated with the response variables — superior to PCA regression when the goal is prediction rather than variance explanation.

How Does PLS Work?

• Latent Variables: Find directions in $X$ space that explain maximum covariance with $Y$ (not just variance in $X$).
• Decomposition: $X = TP^T + E$, $Y = UQ^T + F$ with maximum correlation between $T$ and $U$.
• Prediction: New $X$ values are projected onto latent variables to predict $Y$.
• Variable Importance (VIP): PLS provides Variable Importance in Projection scores for feature ranking.

Why It Matters

• Few Samples, Many Variables: Works when $p >> n$ (more variables than observations) — common in semiconductor data.
• Correlated Predictors: Handles multicollinearity that breaks ordinary least squares regression.
• Virtual Metrology: PLS is a standard algorithm for virtual metrology models in semiconductor fabs.

PLS is regression designed for correlated, high-dimensional data — finding the process variations that actually matter for predicting output quality.