Sparse HJ Biplot: A New Methodology via Elastic Net

STATIS multivariate three-way method for evaluating quality of life after corneal surgery: Methodology and case study in Costa Rica

Vision-related quality of life (QoL) analyzes the visual function concerning individual well-being based on activity and social participation. Because QoL is a multivariate construct, a multivariate statistical method must be used to analyze this construct. In this paper, we present a methodology based on STATIS multivariate three-way methods to assess the real change in vision-related QoL for myopic patients by comparing their conditions before and after corneal surgery. We conduct a case study in Costa Rica to detect the outcomes of patients referred for myopia that underwent refractive surgery. We consider a descriptive, observational and prospective study. We utilize the NEI VFQ-25 instrument to measure the vision-related QoL in five different stages over three months. After applying this instrument/questionnaire, a statistically significant difference was detected between the perceived QoL levels. In addition, strong correlations were identified with highly similar structures ranging from 0.857 to 0.940. The application of the dual STATIS method found the non-existence of reconceptualization in myopic patients, but a statistically significant recalibration was identified. Furthermore, a real change was observed in all patients after surgery. This finding has not been stated previously due to the limitations of the existing statistical tools. We demonstrated that dual STATIS is a multivariate method capable of evaluating vision-related QoL data and detecting changes in recalibration and reconceptualization.

Amaro I, Fabricio Salinas J. Statistical Analysis of Mortality by Non-Communicable Diseases (NCDs) and food supply in Ecuador, 1990-2017

An unhealthy diet is a risk factor for non-communicable diseases (NCDs). Data on the average protein supply and meat products between 1990-2017 consumed by the Ecuadorian population are collected. The HJ-Biplot method is used to identify clusters with MultBiplot software to obtain a two-dimensional representation of the data in the plane along with the quality of representation of the variables and the total variance explained. This work aims to determine how much the source of protein, animal or vegetal, influences the amount of death caused by non-communicable diseases such as Cardiovascular diseases and Diabetes Mellitus. Finally, a Hierarchical Cluster Analysis is carried out with the food supply related to those diseases. It can be concluded that NCD mortality is primarily explained by the average protein supply of animal and meat products in Ecuador. Keywords: Ecuador, Non-Communicable Diseases, Cardiovascular disease, Diabetes, HJ-Biplot, food, proteins.]

CenetBiplot: a new proposal of sparse and orthogonal biplots methods by means of elastic net CSVD

In this work, a new mathematical algorithm for sparse and orthogonal constrained biplots, called CenetBiplots, is proposed. Biplots provide a joint representation of observations and variables of a multidimensional matrix in the same reference system. In this subspace the relationships between them can be interpreted in terms of geometric elements. CenetBiplots projects a matrix onto a low-dimensional space generated simultaneously by sparse and orthogonal principal components. Sparsity is desired to select variables automatically, and orthogonality is necessary to keep the geometrical properties that ensure the biplots graphical interpretation. To this purpose, the present study focuses on two different objectives: 1) the extension of constrained singular value decomposition to incorporate an elastic net sparse constraint (CenetSVD), and 2) the implementation of CenetBiplots using CenetSVD. The usefulness of the proposed methodologies for analysing high-dimensional and low-dimensional matrices is shown. Our method is implemented in R software and available for download from https://github.com/ananieto/SparseCenetMA.

Using HJ-Biplot and External Logistic Biplot as Machine Learning Methods for Corporate Social Responsibility Practices for Sustainable Development

In recent years, social responsibility has been revolutionizing sustainable development. After the development of new mathematical techniques, the improvement of computers’ processing capacity and the greater availability of possible explanatory variables, the analysis of these topics is moving towards the use of different machine learning techniques. However, within the field of machine learning, the use of Biplot techniques is little known for these analyses. For this reason, in this paper we explore the performance of two of the most popular techniques in multivariate statistics: External Logistic Biplot and the HJ-Biplot, to analyse the data structure in social responsibility studies. The results obtained from the sample of companies representing the Fortune Global 500 list indicate that the most frequently reported indicators are related to the social aspects are labour practices and decent work and society. On the contrary, the disclosure of indicators is less frequently related to human rights and product responsibility. Additionally, we have identified the countries and sectors with the highest CSR in social matters. We discovered that both machine learning algorithms are extremely competitive and practical to apply in CSR since they are simple to implement and work well with relatively big datasets.

Logistic Biplot by Conjugate Gradient Algorithms and Iterated SVD

Multivariate binary data are increasingly frequent in practice. Although some adaptations of principal component analysis are used to reduce dimensionality for this kind of data, none of them provide a simultaneous representation of rows and columns (biplot). Recently, a technique named logistic biplot (LB) has been developed to represent the rows and columns of a binary data matrix simultaneously, even though the algorithm used to fit the parameters is too computationally demanding to be useful in the presence of sparsity or when the matrix is large. We propose the fitting of an LB model using nonlinear conjugate gradient (CG) or majorization–minimization (MM) algorithms, and a cross-validation procedure is introduced to select the hyperparameter that represents the number of dimensions in the model. A Monte Carlo study that considers scenarios with several sparsity levels and different dimensions of the binary data set shows that the procedure based on cross-validation is successful in the selection of the model for all algorithms studied. The comparison of the running times shows that the CG algorithm is more efficient in the presence of sparsity and when the matrix is not very large, while the performance of the MM algorithm is better when the binary matrix is balanced or large. As a complement to the proposed methods and to give practical support, a package has been written in the R language called BiplotML. To complete the study, real binary data on gene expression methylation are used to illustrate the proposed methods.