## 5 Data Fusion
### Multi-Block Data Exploration  
One of the first exploratory multi-block methods dates back to 1987. It was proposed by [PERSON] and [PERSON] and it is called Consensus PCA (CPCA) [177]. CPCA is the natural extension of PCA for the analysis of multiple datasets. Basically, the algorithm searches for directions of maximum variance common to all the data blocks at hand. CPCA encompasses four different steps: (i) one measured variable is selected as super-score; (ii) block loadings and scores are estimated from such a super-score; (iii) block scores are gathered in an updated super-score block which is used to calculate  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{3}{c}{**Applications**} \\ \hline
**Aim** & **Method** & **Reference** \\ \hline Traceability of rice varieties & UNEQ/SIMCA/Neural networks-based & [152] \\  & class modelling & \\ Authentication of wine samples & UNEQ/SIMCA/MRM/Neural & [153, 155, 156] \\ Authentication of beer samples & UNEQ/SIMCA/POTFUN & [132, 157, 158] \\ Traceability of extra virgin olive oils & SIMCA/Non-parametric class modelling & [151, 159, 160] \\ Authentication of olive seeds & UNEQ/SIMCA/PLSDM & [154, 161] \\ Traceability of coffee & SIMCA/POTFUN & [150, 162] \\ Traceability of walnuts & SIMCA & [163] \\ Authentication of Avola diamonds & SIMCA & [164] \\ Authentication of PGI Cragnano pasta & SIMCA & [165] \\ Authentication of Italian PDO hazelnut & SIMCA & [166] \\ Authentication of Vallerano chestnut & SIMCA & [167] \\ Plant ripening monitoring & PRIMA & [174] \\ Wheat straw fermentation monitoring & SVDD & [175] \\ \hline \hline \end{tabular}
\end{table}
Table 2: _Cont._a set of super-weights; and (iv) super-weights are normalized and a new super-score is computed. The procedure is iterated until convergence.  
A few years later, in 1996, [PERSON] himself, together with [PERSON] and [PERSON], developed a novel version of multi-block PCA, called Hierarchical PCA (HPCA), sharing similar algorithmic features with CPCA [178]. Their main difference, though, relates to the fact that in HPCA the super-scores are normalized instead of the super-weights. It was observed that both CPCA and HPCA exhibit severe convergence issues, eventually overcome through the modifications proposed by [PERSON] and [PERSON] [179].  
Later on, in 2001, a new sequential exploratory approach (Generalized PCA--GPCA [180]) appeared in literature. GPCA constructs a PCA model on the matrix resulting from the concatenation of the datasets under study and extracts normalized super-scores. These super-scores are resorted to for the retrieval of block scores and loadings. The outcomes of a comprehensive comparison of different variants of CPCA, HPCA and GPCA are reported in [181].  
Almost concomitantly, in 2000, [PERSON] et al. implemented Common Components and Specific Weights Analysis (CCSWA) [182; 183]. CCSWA seeks the shared information underlying all the investigated datasets. Specifically, \(A_{\text{com}}\) orthogonal common components are extracted from a combination of the cross-product matrices corresponding to the \(I\) data blocks coped with \((\mathbf{X}_{i}\mathbf{X}_{i}^{\mathsf{T}}\forall i\in[1,I])\), which is weighted according to their so-called _salience_, i.e., the portion of variance they explain in each one of them. Over the years, different formulations of CCSWA have been designed. CCSWA has nowadays become very well-known with the name ComDim.  
More recently, many alternative multi-block techniques (thoroughly reviewed in [184; 185]), based on the extraction of common factors, but at the same time also aimed at the recognition of the distinctive information carried by each of the individual considered datasets, have been proposed (e.g., DIStinctive and COmmon Simultaneous Component Analysis--DISCO-SCA [186]--or 2-block Orthogonal Projections to Latent Structures--O2 PLS [187]).