NIPALS

NIPALS stands for Nonlinear Iterative Partial Least Squares. It is an iterative algorithm used to compute latent variables in principal component analysis (PCA) and partial least squares (PLS) regression, particularly for small data sets or data with missing values. The method was introduced by Herman Wold in the 1960s and has become a standard technique in chemometrics and related fields.

The method begins with a data matrix X, typically centered and scaled, and, for PLS, a response matrix Y. Components are extracted one at a time by alternating updates of score vectors and loading vectors. In the PCA setting, the algorithm seeks a weight vector w that maximizes the variance of X projected onto t = X w, with the loading p = X^T t / (t^T t) and X deflated as X := X - t p^T. In PLS, the goal is to maximize the covariance between X and Y; the algorithm updates t and a corresponding response score u until convergence, then deflates both X and Y.

Advantages of NIPALS include its simplicity, ability to handle missing data, and suitability for small data sets and noisy measurements. It does not require the full singular value decomposition and can produce results with a lower computational burden in certain contexts. The method relies on centering and scaling and on deflation to separate successive latent variables.

Limitations include potential sensitivity to outliers, dependence on initialization, and possible convergence to local optima rather than a global solution. It has several variants and has been superseded in some applications by more direct methods such as SIMPLS, but it remains widely taught and implemented in chemometric software.