This quality is expressed as follows:  
Figure 1:4  
where \(\lambda_{i}\) represents the singular eigenvalues. The HJ-Biplot also allows the incorporation of external information to enrich the analysis. This can be accomplished by including additional categorical variables that were not in the original dataset. These additional variables are projected into the same space, providing additional context for interpretation [15]. Interpretation of the HJ-Biplot involves examining the relative positions of the individuals and variables on the graph. Some key questions include the following. Which individuals are close to each other, and what does this mean in terms of the variables? Which variables are strongly correlated as indicated by nearby vectors? Which individuals are most influenced by specific variables as reflected in the proximity of their projections on the graph? Table 1 below evaluates three Biplot techniques: GH-Biplot, JK-Biplot, and HJ-Biplot, in terms of their _global goodness of fit_, for rows and columns. Each technique is analyzed using specific mathematical formulas that measure the proportion of the total variability explained. For the GH-Biplot, the global goodness of fit is calculated as the sum of the squares of the first \(k\) singular values (\(\lambda_{i}\)) divided by the sum of the squares of all singular values. This global measure provides an overview of the representation quality. The goodness of fit for rows in the GH-Biplot is simplified as \(2/r\), where \(r\) is the number of retained dimensions, while for the columns, the proportion of variability explained by the first two singular values is used over the total. The JK-Biplot, on the other hand, uses the same formula for global goodness of fit as the GH-Biplot--that is, the proportion of total variability explained by the first  
\begin{table}
\begin{tabular}{l l l l} \hline
**Technique** & **Global Fit** & **Rows Fit** & **Columns Fit** \\ \hline GH–Biplot & \(\sum_{i=1}^{k}\lambda_{i}^{2}\) & \(\dfrac{2}{r}\) & \(\dfrac{\lambda_{1}^{2}+\lambda_{2}^{2}}{\sum_{i=1}^{r}\lambda_{i}^{2}}\) \\ JK–Biplot & \(\dfrac{\sum_{i=1}^{r}\lambda_{i}^{2}}{\sum_{i=1}^{r}\lambda_{i}^{2}}\) & \(\dfrac{\lambda  
singular values. However, it differs in how it calculates the goodness of fit for rows and columns. For the rows, the JK-Biplot employs the following formula:  
Figure 1:5  
which implies a more accurate representation based on the first two singular values. For the columns, the technique simplifies the goodness of fit to \(2/r\), which is similar to what the GH-Biplot does for the rows. This difference in the formulas reflects an alternative approach to data representation and adjustment. The HJ-Biplot is notable for its symmetry in the representation of rows and columns. It uses the same formula for global _goodness of fit_ as the other two techniques:  
Figure 1:6  
ensuring a consistent measure of explained variability. For both rows and columns, the HJ-Biplot uses (1), which ensures high-quality representation for both rows and columns. This symmetry is a significant advantage of the HJ-Biplot, providing a balanced and accurate representation of multivariate data, making it a robust and versatile technique for various types of analyses. To address the practical implementation of the HJ-Biplot technique, it is relevant to mention that its development and application have been supported by the creation of various software tools. These tools have enabled deeper analysis and interactive visualizations, facilitating the interpretation of multivariate data. In this context, Table 2 presents some of the programs and packages specifically developed for the study and application of the HJ-Biplot, highlighting their evolution and adaptation to different platforms and research needs. This technological advancement has been key to the dissemination and use of this technique in various fields, allowing researchers to effectively explore and communicate the complex structures present in their data.