Unlike CA and MCA, which are primarily designed for categorical data and contingency tables, the HJ-Biplot can handle continuous, categorical, and mixed data. This flexibility makes it particularly useful in studies where variables of different types must be analyzed simultaneously [14]. The HJ-Biplot stands out for its ability to provide a symmetric representation of rows and columns, ensuring that the quality of representation is optimal for both clouds. This contrasts with other Biplots where the quality of representation is not the same for rows and columns. The technique maximizes the quality of the representation using appropriate metrics in the row and column spaces, allowing for the accurate interpretation of relative positions and interrelations. Additionally, the HJ-Biplot can incorporate external information, such as additional categorical variables, to enrich the analysis and provide deeper context. This capability makes it a powerful tool for exploring and visualizing complex data, facilitating the identification of hidden patterns and trends. Given a data matrix \(X\), with dimensions \(n\times p\), where \(n\) is the number of individuals and \(p\) is the number of variables, this matrix contains the observations of each individual for each variable. It is common to center the matrix \(X\) to eliminate the effect of different average levels of the variables. This is accomplished by subtracting the mean of each column (variable) from each of the elements in that column, thus obtaining the _centered matrix_\(\mathbf{X}=X-\mathbf{1}\mathbf{m}^{\mathsf{T}}\), where \(\mathbf{1}\) is a column vector of ones of length \(n\) and \(\mathbf{m}\) is a vector of means of length \(p\). The centered matrix \(\mathbf{X}\) is decomposed using SVD. This decomposition is a generalization of _eigenvalue_ and _eigenvector_ decomposition for non-square matrices. The SVD of \(\mathbf{X}\) is expressed as \(\mathbf{X}=\mathbf{U}\mathbf{D}\mathbf{V}^{\mathsf{T}}\), where \(\mathbf{U}\) is an orthogonal matrix with dimensions \(n\times r\), whose column vectors are the left singular vectors; \(\mathbf{D}\) is a diagonal matrix of dimensions \(r\times r\) with the singular values of \(\mathbf{X}\) on its diagonal; and \(\mathbf{V}\) is an orthogonal matrix with dimensions \(p\times r\), whose column vectors are the right singular vectors. In practice, the first \(k\) singular values that explain the majority of the variability in the data are selected. Thus, the first \(k\) columns of \(\mathbf{U}\) and \(\mathbf{V}\) and the first \(k\) singular values of \(\mathbf{D}\) are selected. Figure 1:3 Techniques used by [PERSON] to develop the HJ-Biplot technique [12; 14; 13]. The _coordinates of the individuals_ in the reduced space are obtained by multiplying \(\mathbf{U}_{k}\) by \(\mathbf{D}_{k}\), resulting in \(\mathbf{G}=\mathbf{U}_{k}\mathbf{D}_{k}\). Similarly, the coordinates of the variables are obtained by multiplying \(\mathbf{V}_{k}\) by \(\mathbf{D}_{k}\), obtaining \(\mathbf{H}=\mathbf{V}_{k}\mathbf{D}_{k}\). In the HJ-Biplot, both individuals (rows) and variables (columns) are represented in the same two-dimensional or three-dimensional space, if applicable. The coordinates \(\mathbf{G}\) and \(\mathbf{H}\) are used to plot these points on a graph. The interpretation of the relative positions of the points is key in the HJ-Biplot: the proximity between points of individuals indicates _similarities_ between them, while the orientation and length of the vectors of the variables indicate the direction and magnitude of their influence on the data. The orthogonal projection of an individual's point onto a variable vector approximates the value of the individual on that variable. The _quality of the representation_ in the HJ-Biplot can be evaluated using the proportion of the variability explained by the selected singular values. The sum of the squares of the selected singular values, divided by the sum of the squares of all singular values, provides a measure of the quality of the representation.