Data fusion by T3-PCA: A global model for the simultaneous analysis of coupled three-way and two-way real-valued data

In various areas of science, researchers try to gain insight into important processes by jointly analysing different datasets containing information regarding common aspects of these processes. For example, to explain individual differences in personality, researchers collect, for the same set of persons, data regarding behavioural signatures (i.e., the reaction profile of a person across different situations), on the one hand, and traits or dispositions, on the other hand. To uncover the processes underlying such coupled data, to all N-way $N$ -mode data blocks simultaneously a global model is fitted, in which each data block is represented by an $N$ -way $N$ -mode decomposition model (e.g., principal component analysis [PCA], Parafac, Tucker3) and the parameters underlying the common mode are required to be the same for all data blocks this mode belongs to. To estimate the parameters underlying the common mode, a simultaneous strategy is used that pools the information present in all data blocks (i.e., data fusion). In this paper, we propose the T3-PCA model, which represents three- and two-way data with Tucker3 and PCA respectively. This model is less restrictive than the already proposed LMPCA model in which the three-way data block is decomposed according to a Parafac model. To estimate the T3-PCA model parameters, an alternating least-squares algorithm is proposed. The superior performance of the simultaneous T3-PCA strategy over a sequential strategy (i.e., estimating common parameters using information from the three-way data block only) is demonstrated in an extensive simulation study and an application to empirical coupled anxiety data.