# Multiple Factor Analysis Based on NIPALS Algorithm to Solve Missing Data Problems
## 2 Methods
### Nonlinear Iterative Partial Least Squares
### Wine Dataset

The wine dataset of the \(\mathtt{FactoMineR}\) library has \(n=21\) observations related to Valle de Loira (France) wines and \(p=29\) sensory variables related to wine quality. This dataset contains qualitative and quantitative variables described in [4; 39]. For this illustration, we considered quantitative variables. First, we deployed the MFA algorithm for a complete dataset. Additionally, a subset with 7% of NAs of total observations was randomly selected to deploy the RIMFA and MFA-NIPALS algorithms. The results were compared in terms of percentage of explained variance and coordinate correlation of the complete dataset versus RIMFA and MFA-NIPALS algorithms. Figure 9 shows the correlation of each table's variables. The percentage of explained variance in the first two components is 70.94% of the total variability. Figure 10 represents each wine in the factorial plane, where wines T1 and T2 stand out due to similarities of their sensory variables and atypical features. In addition, the \(1\mathtt{VAU}\) and \(2\mathtt{ING}\) wines are highlighted on the left of the factorial plane, as they presented similarities and atypical features because they differ greatly from the original wines with average features. Considering Figures 11 and 12, obtained from the MFA-NIPALS algorithm with 7% of NAs, it is evident that the results feature similarities with Figures 9 and 10. Moreover, more similarities for wines in the factorial plane are highlighted. In fact, the correlation between the coordinates of wines from MFA and MFA-NIPALS on the first factorial axis is \(cor(\boldsymbol{\psi_{1 MFA}},\boldsymbol{\psi_{1 MFA-NIPALS}})=0.9816\) and \(cor(\boldsymbol{\psi_{2 MFA}},\boldsymbol{\psi_{2 MFA-NIPALS}})=0.9704\) for the second factorial axis. The percentage of explained variance of 72.35%, which is similar to the MFA case with complete dataset, is also highlighted. For RIMFA, 72.53% of the explained variance was found in the first two factorial axes. Comparing their coordinates versus MFA coordinates using a complete dataset, correlations \(cor(\boldsymbol{\psi_{1 MFA}},\boldsymbol{\psi_{1 RIMFA}})=0.9985\) and \(cor(\boldsymbol{\psi_{2 MFA}},\boldsymbol{\psi_{2 RIMFA}})=0.9930\) were found. Thus, the RIMFA algorithm performed better than the MFA-NIPALS one. Nevertheless, MFA-NIPALS performed better than the MFA algorithm in the estimation of coordinates, where \(cor(\boldsymbol{\psi_{MFA}},\boldsymbol{\psi_{MFA-NIPALS}})\) is close to 1. Figure 11: Correlation circle of MFA-NIPALS algorithm with 7% of NAs in wine dataset. Figure 10: Individual factor map of MFA algorithm in wine dataset. ## 4 Conclusions and Further Works

We successfully coupled the NIPALS algorithm with MFA for missing data, called the MFA-NIPALS algorithm. The proposed algorithm was implemented with R software and an alternative method for data imputation with RIMFA was configured. MFA-NIPALS was adapted using the nipals function of the ade4 library. Other options such as the nipals function of the plsdepot library [42] could be explored. [PERSON] et al. [43] considered phase space reconstruction techniques mainly oriented toward classification tasks, where the integrate-and-differentiate approach is focused on improving identification accuracy through the elimination of classification errors needed for the parameter estimation of nonlinear equations. Further studies could explore the integrate-and-differentiate approach regarding the missing data problem. Further works could analyze if Gram-Schmidt orthogonalization helps to find better properties for MFA-NIPALS. The literature contains multiple factorial methods, in which the missing data problem has not been addressed with NIPALS, for example, the STATISTACT method and canonical correlation analysis (ACC) [44; 45]. The NIPALS algorithm seems suitable to address missing data in STATIS-ACT and ACC. Another important concept related to MFS is the RV coefficient of Escoufier [46; 47], which is a matrix version of the Pearson coefficient of correlation.