# Multiple Factor Analysis Based on NIPALS Algorithm to Solve Missing Data Problems
## 2 Methods
### Nonlinear Iterative Partial Least Squares
The RV coefficient between tables \(X_{n\times p}\) and \(Y_{n\times q}\) is:

\[RV(\mathbf{X},\mathbf{Y})=\frac{tr\Big{\{}\mathbf{X}\mathbf{X}^{\top}\mathbf{Y}\mathbf{Y}^{\top}\Big{\}} }{tr\Big{\{}\mathbf{X}\mathbf{X}^{\top}\mathbf{X}\mathbf{X}^{\top}\Big{\}}tr\Big{\{}\mathbf{Y}\mathbf{ Y}^{\top}\mathbf{Y}\mathbf{Y}^{\top}\Big{\}}}, \tag{18}\]

where \(tr\{\mathbf{A}\}\) denotes the trace of matrix \(\mathbf{A}\)[36]. The RV coefficient is often used to study the correlation between tables or groups of variables. The coefficients related to the covariance of a pair of tables appear in the numerator of (18). If \(X\) and \(Y\) include NAs and MFA-NIPALS is used, further work could focus on RV coefficient computation using the available data principle (see Section 2.4). Another proposal is the use of coordinates between individuals \(\mathbf{\psi}^{(k)}\) in the \(k\)th table as an estimation of \(X\) and \(Y\), since coordinates \(\mathbf{\psi}^{(k)}\) do not have NAs when MFA-NIPALS is deployed. Figure 12: Individual factor map of MFA algorithm with 7% of NAs in wine dataset. In this paper, MFA-NIPALS was used for longitudinal quantitative variables with the presence of NAs. This approach could be extended to multiple \(k\) qualitative tables using MCA under the available data principle [19]. This idea allows working with MFA-NIPALS with mixed data using NIPALS for quantitative tables and MCA for qualitative ones. Given that the available data principle reduces computational cost when MFA-NIPALS is used compared to RIMFA, the MFA-NIPALS algorithm is a novel approach to addressing missing data problems in multiple quantitative tables. Nevertheless, further studies are needed to compare MFA-NIPALS with RIMFA across datasets with higher dimensions, analyzing the computational performance of both methods by comparing the estimated coordinates of \(\mathbf{\psi}\) (in presence of NAs) with MFA coordinates (of a complete dataset). It is expected that MFA-NIPALS performs better with datasets with more variables than observations (\(p>n\)), where PLS methods have advantages over classic methods [48]. Based on simulation scenarios, it is recommended to work the MFA-NIPALS proposal until 15% of NAs of the total number of observations. This result is in line with the results by [24], highlighting the NAs percentage recommended for NIPALS. Moreover, it is recommended to use MFA-NIPALS when data imputation is not feasible. Though the RIMFA algorithm performed well when coordinates are compared to the MFS one, this study showed that the MFA-NIPALS algorithm was a good alternative for NA handling in the MFA. Moreover, our proposal is promising, as it yielded favorable results regarding the percentage of explained variance. It is highly probable that other studies generate even better results by mixing the MFA-NIPALS and RIMFA approaches. Research data and R codes are available online at [[https://www.mdpi.com/article/10.3390/a16100457/s1](https://www.mdpi.com/article/10.3390/a16100457/s1)]([https://www.mdpi.com/article/10.3390/a16100457/s1](https://www.mdpi.com/article/10.3390/a16100457/s1)) in the Supplementary Materials. Conceptualization, A.F.O.-M.; data curation, A.F.O.-M.; formal analysis, A.F.O.-M. and J.E.C.-R.; investigation, A.F.O.-M.; methodology, A.F.O.-M. and J.E.C.-R.; project administration, A.F.O.-M.; resources, A.F.O.-M.; software, A.F.O.-M.; supervision, J.E.C.-R.; validation, A.F.O.-M.; visualization, A.F.O.-M.; writing--original draft, A.F.O.-M. and J.E.C.-R.; writing--review and editing, J.E.C.-R. All authors have read and agreed to the published version of the manuscript. [PERSON]'s research was funded by FIB-UV grant Res. Ex. Nro. 2286, from Universidad de Valparaiso, Chile. Research data and R codes are available in the Supplementary Materials. The authors thank the editor and three anonymous referees for their helpful comments and suggestions. The authors declare that there are no conflict of interests in the publication of this paper. The following abbreviations are used in this manuscript: