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.