# Multiple Factor Analysis Based on NIPALS Algorithm to Solve Missing Data Problems
## 2 Methods
\begin{pmatrix}X_{1}\\ NA\\ X_{3}\\ \vdots\\ X_{p}\end{pmatrix} ,\qquad\mathbf{Y}= \begin{pmatrix}NA\\ Y_{2}\\ Y_{3}\\ \vdots\\ Y_{p}\end{pmatrix} ,\]

the inner product between \(\mathbf{X}\) and \(\mathbf{Y}\) using the available data principle is:

\[\langle\mathbf{X},\mathbf{Y}\rangle=\sum_{X_{i}Y_{i}\
eq NA}X_{i}Y_{i}=X_{3}Y_{3}+X_{4 }Y_{4}+\ldots+X_{p}Y_{p}. \tag{17}\]

### Nonlinear Iterative Partial Least Squares Based on Multiple Factor Analysis

As mentioned in Section 2.1, classic MFA does a PCA for each \(k\)th table in step 1. For this reason, it is proposed to implement NIPALS in this step and obtain the eigenvalues to weigh the tables of quantitative variables. After weighing the tables with \(\lambda_{1}^{(k)}\), a global NIPALS operation is performed in the last step for \(\mathbf{Z}\) which contains NAs. Thus, MFA-NIPALS involves the following steps:

1. A NIPALS operation on each \(k\)th table of quantitative variables;
2. Weigh the variables with \(1/\lambda_{1}^{(k)}\) obtained by NIPALS in the \(k\)th table;
3. A global NIPALS of the juxtaposed table \(\mathbf{Z}\). MFA-NIPALS has the following properties: 1. Eigenvalues are decreasing [35; 36], i.e., \(\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}\geq\ldots\geq\lambda_{q}\); 2. Components \(\mathbf{t}_{h}\) are orthogonal (\(\mathbf{t}_{h}^{\top}\mathbf{t}_{h}=0\)); 3. Eigenvectors \(\mathbf{p}_{h}\) are orthonormal (\(\mathbf{p}_{h}^{\top}\mathbf{p}_{h}=1\)). ## 3 Applications

In this section, we present two real-world datasets to illustrate the method's performance, the simulation scenarios, and implementation of methodologies. ### Qualification Dataset

In this dataset, the rows represent students and columns are the qualifications obtained in the subjects of Mathematics, Spanish, and Natural Sciences. We analyzed the student qualifications in a longitudinal way. A similar example can be found in [37; 38], where [PERSON] adapted an MFA. Table 1 shows the first rows of the data, where \(p=9\) columns (quantitative variables) illustrate three academic periods in a longitudinal way. Using this dataset, we generated random matrices with \(n=50\) observations, each containing a random number of NAs of 5%, 10%, \(\ldots\), and 30% of \(n\) observations. As a first step, students' factorial coordinates \(\mathbf{\psi}\) and eigenvalues \(\lambda_{\alpha}\) were obtained with the classic MFA, RIMFA, and MFA-NIPALS. In the second step, the methods were compared via coordinate correlations of classic MFA versus RIMFA, \(cor(\mathbf{\psi}_{MFA},\mathbf{\psi}_{RIMFA})\), and of classic MFA versus MFA-NIPALS, \(cor(\mathbf{\psi}_{MFA},\mathbf{\psi}_{MFA-NIPALS})\). We used version 4.1.0 of R software for all computations, the FactoMineR library for classic MFA [39; 40], missMDA for RIMFA [14], and ade4 for the NIPALS algorithm [41]. In the next sections, we present the descriptive results, the MFA with complete data, the MFA-NIPALS with 10% of NAs, and simulation results for 5% to 30% of NAs. Figure 3 presents a correlation circle of MFA with complete data, where 41.26% of the variance percentage explained was obtained in the first factorial plane. A high correlation between the qualifications of Mathematics and Natural Sciences in the first academic period can be observed, as well as a moderate correlation between Spanish of periods 1 and 2 and a low correlation between subjects in the third period. Figure 4 presents the individual factor map where students 11, 21, and 42 highly contribute to the axis, according to their position in the plane far from the average individual or gravity center. Figure 2: Correlation matrix of qualifications for the three academic periods. #### 3.1.1 MFA-NIPALS with 10% of NAs

In this section, the main results of the proposed MFA-NIPALS algorithm are presented. Figure 5 illustrates the correlation circle of MFA-NIPALS with 10% of NA.