Partial Least-Squares Method for Three-Mode Three-Way Datasets Based on Tucker Model

When analyzing two three-mode three-way datasets (object × variable × condition), the objective is to obtain common factors that show the relationships between the two datasets. The partial least-squares (PLS) method has been applied to such datasets to investigate the common factors. However, the PLS method was proposed for two-mode two-way datasets, such as multivariate datasets. Therefore, this method does not consider the condition when searching for relationships between datasets; that is, it tends to regard the same variable under different conditions as different variables. To address this problem, we extended the PLS method to three-mode three-way datasets by using the Tucker model so that the same variable under different conditions is regarded as the same. Moreover, we can apply the proposed method to three-mode three-way datasets with different dimensions for the conditions and variables, and the output is obtained in the form of three-mode three-way datasets. We show the advantage of the proposed method by applying it to a multicollinearity case as a numerical example.