Nowadays, data analysis applied to high dimension has arisen. The edification of high-dimensional data can be achieved by the gathering of different independent data. However, each independent set can introduce its own bias. We can cope with this bias introducing the observation set structure into our model. The goal of this article is to build theoretical background for the dimension reduction method sparse Partial Least Square (sPLS) in the context of data presenting such an observation set structure. The innovation consists in building different sPLS models and linking them through a common-Lasso penalization. This theory could be applied to any field, where observation present this kind of structure and, therefore, improve the sPLS in domains, where it is competitive. Furthermore, it can be extended to the particular case, where variables can be gathered in given a priori groups, where sPLS is defined as a sparse group Partial Least Square.
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