  Partial Least Squares (PLS) methods have been heavily exploited to analyse
the association between two blocs of data. These powerful approaches can be
applied to data sets where the number of variables is greater than the number
of observations and in presence of high collinearity between variables.
Different sparse versions of PLS have been developed to integrate multiple data
sets while simultaneously selecting the contributing variables. Sparse
modelling is a key factor in obtaining better estimators and identifying
associations between multiple data sets. The cornerstone of the sparsity
version of PLS methods is the link between the SVD of a matrix (constructed
from deflated versions of the original matrices of data) and least squares
minimisation in linear regression. We present here an accurate description of
the most popular PLS methods, alongside their mathematical proofs. A unified
algorithm is proposed to perform all four types of PLS including their
regularised versions. Various approaches to decrease the computation time are
offered, and we show how the whole procedure can be scalable to big data sets.
