HJ-Biplot como metodología exploratoria para el análisis multidimensional de los objetivos de desarrollo sostenible (ODS) a nivel municipios en Bolivia
Palabras clave:
Estadística multivariante, métodos Biplot, HJ-Biplot, ODS, ClústeresResumen
La Agenda 2030 y los Objetivos de Desarrollo Sostenible (ODS) se presentan como oportunidades clave para reconsiderar prácticas y abordar los desafíos de desarrollo, medir y analizar el cumplimiento de estos objetivos es de vital importancia para una planificación objetiva. Por tanto, se destaca la importancia de un análisis estadístico multidimensional descriptivo, tales como, las técnicas de análisis Biplot, metodologías usadas por ciencia de datos, inteligencia artificial y machine learning, como un nuevo paradigma para comprender la información de manera más profunda y como base para el diseño de políticas con alto impacto social. La medición continua y evaluación de indicadores ODS se consideran esenciales para ajustar y mejorar las políticas públicas a lo largo del tiempo. En este sentido, el artículo presenta el HJ-Biplot como una técnica de análisis multivariante, una herramienta analítica avanzada para interpretar grandes volúmenes de información, en este caso, el cumplimiento a los ODS a nivel municipio, logrando clústeres a nivel departamental y nacional de estos de acuerdo a la similiaridad y disimilaridad como insumo para el diseño de políticas publicas objetivas y de alto impacto.
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Aitchison, J. (1982). The Statistical Analysis of Compositional Data. Journal of the Royal Statistical Society: Series B (Methodological), 44(2), 139–160. https://doi.org/10.1111/j.2517-6161.1982.tb01195.x
Aitchison, J. (1997). The one-hour course in compositional data analysis or compositional data analysis is simple. In in: V. Pawlowsky-Glahn (Ed.), Proceedings of IAMG’97 – The III Annual Conference of the International Association for Mathematical Geology. (Vols. I, II and addendum, p. 3-35). Barcelona: International Center for Numerical Methods in Engineering (CIMNE).
Aitchison, J., & Greenacre, M. (2002). Biplots of compositional data. Journal of the Royal Statistical Society: Series C (Applied Statistics), 51(4), 375–392. https://doi.org/10.1111/1467-9876.00275
Aitchison, J., Barceló-Vidal, C., Egozcue, J. J., & Pawlowsky-Glahn, V. (2002). A concise guide for the algebraic-geometric structure of the simplex, the sample space for compositional data analysis. In Proceedings of Eighth Annual Conference of the International Association for Mathematical Geology. (Vol. 2, pp. 387-392).
Alcaraz Lamana, A.; Alonso Torres, P. (2019). La contribución de las universidades a la Agenda 2030. Unitat de Cooperació, Servei de Relacions Internacionals i Cooperació Universitat de València
Amaratunga, D., & Cabrera, J. (2016). High-dimensional data. Journal of the National Science Foundation of Sri Lanka, 44(1).
Amaro, I. R. (2001). Manova biplot para diseños con varios factores, basado en modelos lineales generales multivariantes. [Tesis doctoral]. Universidad de Salamanca, España.
Amaro, I. R., Vicente-Villardón, J. L. y, & Galindo-Villardón, P. (2004). MANOVA BIPLOT para arreglos de tratamientos con dos factores basado en modelos lineales generales multivariantes. Interciencia, 29(1), 26–32.
Baccalá, N. (2004). Contribuciones al Análisis de Matrices de Datos Multivía: tipología de las variables. [Tesis doctoral]. Universidad de Salamanca, España, Spain.
Becker, H. C. (1981). Correlations among some statistical measures of phenotypic stability. Euphytica, 30(3), 835–840.
Benzécri, J. P. (1973). L’analyse des données (Vol. 2). Paris: Dunod.
Berman, J. J. (2013). Principles of big data: preparing, sharing, and analyzing complex information. Newnes.
Billheimer, D., Guttorp, P., & Fagan, W. F. (2001). Statistical Interpretation of Species Composition. Journal of the American Statistical Association, 96(456), 1205–1214. https://doi.org/10.1198/016214501753381850
Blázquez, A. (1998). Análisis biplot basado en modelos lineales generalizados. [Tesis doctoral]. Universidad de Salamanca, España.
Bodor, A., Csabai, I., Mahoney, M. W., & Solymosi, N. (2012). rCUR: an R package for CUR matrix decomposition. BMC Bioinformatics, 13(1), 103. https://doi.org/10.1186/1471-2105-13-103
Braak, C. J. Ter, & Looman, C. W. (1994). Biplots in reduced-rank regression. Biometrical Journal, 36(8), 983–1003.
Bradu, D., & Gabriel, K. R. (1974). Simultaneous statistical inference on interactions in two-way analysis of variance. Journal of the American Statistical Association, 69(346), 428–436.
Bradu, D., & Gabriel, K. R. (1978). The biplot as a diagnostic tool for models of two-way tables. Technometrics, 20(1), 47–68.
Breiman, L. (1995). Better Subset Regression Using the Nonnegative Garrote. Technometrics, 37(4), 373–384. https://doi.org/10.1080/00401706.1995.10484371
Cadima, J., & Jolliffe, I. T. (1995). Loading and correlations in the interpretation of principle compenents. Journal of Applied Statistics, 22(2), 203–214. https://doi.org/10.1080/757584614
Cárdenas, O. C., & Galindo, M. P. (2004). Biplot con información externa basado en modelos bilineales generalizados.
Cárdenas, O., Noguera, C., Galindo, M. P., & Vicente-Villardón, J. L. (2003). El uso de información externa en aproximaciones Biplot. Revista Venezolana de Análisis de Coyuntura, 9(2), 257–276.
Cárdenas, O., Noguera, C., Galindo, P., & Vicente-Villardón, J. L. (2006). An alternative to principal components regression based on Regression Biplot. INTERCIENCIA, 31(3), 160–167.
Carlier, A., & Kroonenberg, P. M. (1996). Decompositions and biplots in threeway correspondence analysis. Psychometrika, 61(2), 355–373.
Chessel, D., Dufour, A., & Thioulouse, J. (2004). The ade4 package-I-One-table methods. Pdfs.Semanticscholar.Org.
Chessel, D., Dufour, A.B., Dray, S., Jombart, T., Lobry, J.R., Ollier, S. y, & Thioulouse, J. (2013). ade4. R package version 1.5-2: analysis of ecological data: exploratory and Euclidean methods in environmental sciences. cran.rproject.org/package=ade4, 2013.
Choulakian, V. (1996). Generalized bilinear models. Psychometrika, 61(2), 271– 283.
Cortés-Rodríguez, M., & Sánchez-Barba, M. (2013). Biplot de datos composicionales: una herramienta útil en el estudio de test psicológicos. Universidad de Salamanca, España.
Crossa, J., Cornelius, P. L., & Yan, W. (2002). Biplots of linear-bilinear models for
studying crossover genotype× environment interaction. Crop Science, 42(2), 619– 633.
Crossa, J., Gauch, H. G., & Zobel, R. W. (1990). Additive main effects and multiplicative interaction analysis of two international maize cultivar trials. Crop Science, 30, 493–500.
Cubilla-Montilla, M., Nieto-Librero, A.-B., Galindo-Villardón, M. P., Vicente Galindo, M. P., & Garcia-Sanchez, I.-M. (2019). Are cultural values sufficient to improve stakeholder engagement human and labour rights issues? Corporate Social Responsibility and Environmental Management, 26(4), 938–955. https://doi.org/10.1002/csr.1733
d’Aspremont, A., El Ghaoui, L., Jordan, M. I., & Lanckriet, G. R. G. (2007). A Direct Formulation for Sparse PCA Using Semidefinite Programming. SIAM Review, 49(3), 434–448. https://doi.org/10.1137/050645506
De Falguerolles, A. (1996). Generalized bilinear models and generalized biplots: some examples. Publications Du Laboratoire de Statistique et Probabilités.
Demey, J. R. (2008). Diversidad genética en bancos de Germoplasma: un enfoque Biplot.
Denis, J. B. (1991). Ajustements de modèles linéaires et bilinéaires sous contraintes linéaires avec données manquantes. Revue de Statistique Appliquée, 39(2), 5–24.
Donoho, D. L., & Johnstone, I. M. (1994). Ideal spatial adaptation by wavelet shrinkage. Biometrika, 81(3), 425–455. https://doi.org/10.1093/biomet/81.3.425
Dray, S., Dufour, A.B., Chessel, D. (2007). The ade4 package-II: Two-table and Ktable methods. R Journal, 7(2), 47–52.
Drineas, P., Kannan, R., & Mahoney, M. (2006). Fast Monte Carlo algorithms for matrices III: Computing a compressed approximate matrix decomposition. SIAM, Journal on Computing, 36(1), 184–206.
Drineas, P., Mahoney, M. W., & Muthukrishnan, S. (2008). Relative-Error CUR Matrix Decompositions. SIAM Journal on Matrix Analysis and Applications, 30(2), 844–881. https://doi.org/10.1137/07070471X
Eckart, C., & Young, G. (1936). The approximation of one matrix by another of lower rank. Psychometrika, 1(3), 211–218.
Eckart, C., & Young, G. (1939). A principal axis transformation for non-Hermitian matrices. Bulletin of the American Mathematical Society, 45(2), 118–121.
Efron, B, & Tibshirani, R. (1994). An introduction to the bootstrap.
Efron, B., & Gong, G. (1983). A leisurely look at the bootstrap, the jackknife, and cross-validation. The American Statistician, 37(1), 36–48.
Efron, B., Hastie, T., Johnstone, I., & Tibshirani, R. (2004). Least angle regression. The Annals of Statistics, 32(2), 407–499.173
Efron, Bradley. (1979). Computers and the Theory of Statistics: Thinking the Unthinkable. SIAM Review, 21(4), 460–480. https://doi.org/10.1137/1021092
Egido, J. (2014). dynBiplotGUI. R package versión 1.0.1. cran.rproject.org/web/packages/dynBiplotGUI.
Erichson, N., Zheng, P., Manohar, K., Brunton, S., Kuetz, J., & Aravkin, A. (2018). Sparse Principal Component Analysis via Variable Projection. ArXiv Preprint ArXiv:1804.00341.
Fan, J., & Li, R. (2001). Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties. Journal of the American Statistical Association, 96(456), 1348–1360. https://doi.org/10.1198/016214501753382273
Farcomeni, A. (2009). An exact approach to sparse principal component analysis. Computational Statistics, 24(4), 583–604. https://doi.org/10.1007/s00180-008-0147-3
Faria, J. C., & Demetrio, C. G. B. (2012). Biplot of multivariate data based on principal components analysis. R package version 1.02. URL http://cran.rproject.org/package=bpca.
Fernández-Gómez, M. J. (1995). Contribuciones al análisis multivariante directo del gradiente mediante estudio combinado de configuraciones espaciales. [Tesis doctoral]. Universidad de Salamanca, España.
Fisher, R. A. (1936). The use of mutiple measurements in taxonomic problems. Annals of Eugenics, 7(2), 179–188. https://doi.org/10.1111/j.1469-1809.1936.tb02137
Friedman, J., Hastie, T., & Tibshirani, R. (2010). Regularization paths for generalized linear models via coordinate descent. Journal of Statistical Software, 33(1), 1.
Frieze, A., Kannan, R., & Vempala, S. (2004). Fast Monte-Carlo algorithms for finding low-rank approximations. Journal of the ACM (JACM), 51(6), 1025–1041.
Frutos, E., & Galindo, M. P. (2013). GGEBiplotGUI. R package version 1.0-6: interactive GGE biplots in R. cran.r-project.org/package=GGEBiplotGUI,.
Gabriel, K. R. (1971). The biplot graphic display of matrices with application to principal component analysis. Biometrika, 58(3), 453–467.
Gabriel, K. R. (1972). Analysis of Meteorological Data by Means of Canonical Decomposition and Biplots. Journal of Applied Meteorology, 11(7), 1071–1077. https://journals.ametsoc.org/view/journals/apme/11/7/1520-0450_1972_011_1071_aomdbm_2_0_co_2.xml
Gabriel, K. R. (1995). “MANOVA biplots for two-way contingency tables.” In: W.J. Krzanowski (Ed.), Recent Advances in Descriptive Multivariate Analysis, Oxford: Clarendon Press.
Gabriel, K. R. (1998). Generalised bilinear regression. Biometrika, 85(3), 689–700. Gabriel, K. R., & Odoroff, C. L. (1990). Biplots in biomedical research. Statistics in Medicine, 9(5), 469–485. https://doi.org/10.1002/sim.4780090502174
Galindo, M. P. (1986). Una alternativa de representacion simultanea: HJ-Biplot. Qüestiió: Quaderns d’estadística i Investigación Operativa, 10(1), 13–23.
Galindo, M. P., & Cuadras, C. M. (1986). Una extensión del método Biplot y su relación con otras técnicas. Publicaciones de Bioestadística y Biomatemática, 17.
Gandomi, A., & Haider, M. (2015). Beyond the hype: Big data concepts, methods, and analytics. International Journal of Information Management, 35(2), 137–144.
Gauch, H. G. (1988). Model selection and validation for yield trials with interaction. Biometrics, 705–715.
Gauch, H. G., & Zobel, R. W. (1989). Accuracy and selection success in yield trial analyses. Theoretical and Applied Genetics, 77(4), 473–481. https://doi.org/10.1007/BF00274266
Gollob, H. F. (1968). A statistical model which combines features of factor analytic and analysis of variance techniques. Psychometrika, 33(1), 73–115.
Goreinov, S. ., Tyrtyshnikov, E. E., & Zamarashkin, N. L. (1997). A theory of pseudoskeleton approximations. Linear Algebra and Its Applications, 261(1–3), 1– 21.
Goreinov, S. A., & Tyrtyshnikov, E. E. (2001). The maximal-volume concept in approximation by low-rank matrices. Contemporary Mathematics, 280, 47–52.
Gower, J. C. (1992). Generalized biplots. Biometrika, 79(3), 475–493.
Gower, J. C., & Hand, D. J. (1995). Biplots (Vol. 54). CRC Press.
Gower, J. C., & Hand, D. J. (1996). Biplots. Chapman&Hall, London UK.
Gower, J. C., & Harding, S. A. (1988). Nonlinear biplots. Biometrika, 75(3), 445–455.
Graffelman, J. (2012). Calibrate: calibration of scatterplot and biplot axes. R package version 1(1). URL cran.r-project.org/package=calibrate, 2012.
Greenacre, M. J. (1984). Correspondence analysis. London: Academic Press.
Greenacre, M. J. (1993). Biplots in correspondence analysis. Journal of Applied Statistics, 20(2), 251–269. https://doi.org/10.1080/02664769300000021
Greenacre, M., & Nenadic, O. (2012). The ca R package version 0.53: simple, multiple and joint correspondence analysis. cran.r-project.org/package=ca.
Hashem, I. A. T., Yaqoob, I., Anuar, N. B., Mokhtar, S., Gani, A., & Khan, S. U. (2015). The rise of “big data” on cloud computing: Review and open research issues. Information Systems, 47, 98–115.
Hausman, R. E. (1982). Constrained multivariate analysis. In: Zanakis SH, Rustagi JS (eds). Optimisation in Statistics. North-Holland, Amsterdam, 137–151.
Hernández Sampieri, R., Fernández Collado, C., & Baptista Lucio, P. (2014). Selección de la muestra en Metodología de la Investigación. México: McGrawHill.175
Hernández Sánchez, J. C. (2016). Biplot logístico para datos nominales y ordinales. [Tesis Doctoral]. Universidad de Salamanca, España.
Hernández Suárez, M., Molina Pérez, D., Rodríguez-Rodríguez, E., Díaz Romero, C., Espinosa Borreguero, F., & Galindo-Villardón, P. (2016). The Compositional HJ-Biplot—A New Approach to Identifying the Links among Bioactive Compounds of Tomatoes. International Journal of Molecular Sciences, 17(11), 1828.
Hernández, J. C., & Vicente-Villardón, J. L. (2013a). NominalLogisticBiplot. R package version 0.1: biplot representations of categorical data. cran.rproject.org/web/packages/NominalLogisticBiplot/index.html,.
Hernández, J. C., & Vicente-Villardón, J. L. (2013b). OrdinalLogisticBiplot. R package version 0.2: ordinal logistic biplots. cran.rproject.org/web/packages/OrdinalLogisticBiplot/index.html.
Hernández, S. (2005). Biplots robustos. [Tesis doctoral]. Universidad de Salamanca, España.
Hernández, S., & Galindo-Villardón, M. P. (2006). BIPROB: UN MÉTODO PARA OBTENER UN BIPLOT ROBUSTO. Investigación Operacional, 27(3), 287–299.
Hoerl, A. E., & Kennard, R. W. (1970). Ridge Regression: Biased Estimation for Nonorthogonal Problems. Technometrics, 12(1), 55–67. https://doi.org/10.1080/00401706.1970.10488634
Hofstede, G. (2011). Dimensionalizing cultures: The Hofstede model in context. Readings in Psychology and Culture, 2(1), 8.
Hotelling. H. (1933). Analysis of a complex of statistical variables into principal components. Journal of Educational Psychology., 24(6), 417.
Jolliffe, I. T. (1995). Rotation of principal components: choice of normalization constraints. Journal of Applied Statistics, 22(1), 29–35. https://doi.org/10.1080/757584395
Jolliffe, I. T., & Uddin, M. (2000). The Simplified Component Technique: An Alternative to Rotated Principal Components. Journal of Computational and Graphical Statistics, 9(4), 689–710. https://doi.org/10.1080/10618600.2000.10474908
Jolliffe, I. T., Trendafilov, N. T., & Uddin, M. (2003). A Modified Principal Component Technique Based on the LASSO. Journal of Computational and Graphical Statistics, 12(3), 531–547. https://doi.org/10.1198/1061860032148
Kempton, R. A. (1984). The use of biplots in interpreting variety by environment interactions. The Journal of Agricultural Science, 103(1), 123–135.
La Grange, A. M., Le Roux, N. J., Rousseeuw, I. R., & Tukey, J. W. (2009). BiplotGUI: interactive biplots in R. R package version 0.0-7. cran.rproject.org/package=BiplotGUI.
Lafortune, G., Fuller, G., Moreno, J., Schmidt-Traub, G., Kroll, C. (2018). SDG Index and Dashboards: Detailed Methodological paper. Paris: Sustainable Development Solutions Network (SDSN).
Mahoney, M. W., & Drineas, P. (2009). CUR matrix decompositions for improved data analysis. Proceedings of the National Academy of Sciences of the United States of America, 106(3), 697–702. https://doi.org/10.1073/pnas.0803205106
Mahoney, M. W., Maggioni, M., & Drineas, P. (2008). Tensor-CUR Decompositions for Tensor-Based Data. SIAM Journal on Matrix Analysis and Applications, 30(3), 957–987. https://doi.org/10.1137/060665336
Mandel, J. (1971). A New Analysis of Variance Model for Non-additive Data. Technometrics, 13(1), 1–18. https://doi.org/10.1080/00401706.1971.10488751
Mardia, K. V., Kent, J. T., & Bibby, J. M. (1979). Multivariate Analysis. Press Inc. London.
Markos, A. (2012). Package caGUI. The GUI ca R package version 0.1-4: a Tcl/Tk GUI for the functions. cran.r-project.org/package=caGUI, 2012.
Martín-Rodríguez, J. (1996). Contribuciones a la integración de subespacios desde una perspectiva biplot. [Tesis doctoral]. Universidad de Salamanca, España.
McCabe, G. P. (1984). Principal Variables. Technometrics, 26(2), 137–144.
https://doi.org/10.1080/00401706.1984.10487939
Moghaddam, B., Weiss, Y., & Avidan, S. (2006). Spectral bounds for sparse PCA: Exact and greedy algorithms. Advances in Neural Information Processing Systems.
Nenadic, O., & Greenacre, M. (2007). Correspondence analysis in R, with two-and three-dimensional graphics: The ca package. Journal of Statistical Software, 20(3), 1–13.
Nieto-Librero, A. B. (2015). Versión inferencial de los métodos Biplot basada en remuestreo Bootstrap y su aplicación a tablas de tres vías. [Tesis doctoral]. Universidad de Salamanca, España.
Nieto-Librero, A. B., & Galindo-Villardón, P. (2015). biplotbootGUI. R package version 1.0: Bootstrap on Classical Biplots and Clustering Disjoint Biplot. cran.rproject.org/web/packages/biplotbootGUI/.
Nieto-Librero, A. B., Baccalá, N., & Galindo, M. P. (2012). MultibiplotGUI. R package version 0.0-1: Multibiplot Analysis. cran.rproject.org/package=multibiplotGUI.
Nieto-Librero, A. B., Sierra, C., Vicente-Galindo, M. P., Ruíz-Barzola, O., & Galindo-Villardón, M. P. (2017). Clustering Disjoint HJ-Biplot: A new tool for identifying pollution patterns in geochemical studies. Chemosphere, 176, 389–396.
Oksanen, J., Blanchet, F. G., Friendly, M., Kindt, R., Legendre, P., Minchin, P. R., … Wagner, H. (2013). Package vegan. Comunity Ecology Package, Version 2.9. R package version 2.0-8. cran.r-project.org/package=vegan.177
Pawlowsky-Glahn, V., & Egozcue, J. J.-S. (2001). Geometric approach to statistical analysis on the simplex. Stochastic Environmental Research and Risk Assessment, 15(5), 384–398.
Pearson, K. (1901). On lines and planes of closest fit to systems of points in space. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 2(11), 559–572.
Qi, X., Luo, R., & Zhao, H. (2013). Sparse principal component analysis by choice of norm. Journal of Multivariate Analysis, 114:127-150. https://doi.org/10.1016/j.jmva.2012.07.004
Quenouille, M. H. (1950). An application of least squares to family diet surveys. Econometrica: Journal of the Econometric SocietY, 27–44.
Resolución 70/1 [Asamblea General de las Naciones Unidas]. Por la cual se aprobó la Agenda 2030 para el Desarrollo Sostenible. 25 de septiembre de 2015.
Rodríguez-Mazahua, L., Rodríguez-Enríquez, C.-A., Sánchez-Cervantes, J. L., Cervantes, J., García-Alcaraz, J. L., & Alor-Hernández, G. (2016). A general perspective of Big Data: applications, tools, challenges and trends. The Journal of Supercomputing, 72(8), 3073–3113.
R-TEAM. (2014). A language and environment for statistical computing. R foundation for statistical computing. Vienna, Austria, 2014. Retrieved from www.R-project. org.
Shen, H., & Huang, J. (2008). Sparse principal component analysis via regularized low rank matrix approximation. Journal of Multivariate Analysis, 99(6), 1015–1034.
Stewart, G. W. (1999). Four algorithms for the the efficient computation of truncated pivoted QR approximations to a sparse matrix. Numerische Mathematik, 83(2), 313–323. https://doi.org/10.1007/s002110050451
Talia, D. (2013). Clouds for scalable big data analytics. Computer, 46(5), 98–101.
Ter Braak, C. J. (1986). Canonical Correspondence Analysis: a new eigenvector technique for Multivariate Direct Gradient Analysis. Ecology, 67(5), 1167–1179.
Ter Braak, C. J. (1990). Interpreting canonical correlation analysis through biplots of structure correlations and weights. Psychometrika, 55(3), 519–531.
Thioulouse, J., & Dray, S. (2007). Interactive multivariate data analysis in R with the ade4 and ade4TkGUI packages. Journal of Statistical Software, 22(5), 1–14.
Thioulouse, J., & Dray, S. (2012). ade4TkGUI. R package version 0.2-6: ade4 Tcl/Tk graphical user interface. cran.r-project.org/package=ade4TkGUI, 2012.
Tibshirani, R. (1996). Regression Shrinkage and Selection Via the Lasso. Journal of the Royal Statistical Society: Series B (Methodological), 58(1), 267–288. https://doi.org/10.1111/j.2517-6161.1996.tb02080.x
Tibshirani, R. (2011). Regression shrinkage and selection via the lasso: a retrospective. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73(3), 273–282.178
Torgerson, W. S. (1952). Multidimensional scaling: I. Theory and method. Psychometrika, 17(4), 401–419. https://doi.org/10.1007/BF02288916
Trendafilov, N. T. (2014). From simple structure to sparse components: A review. Computational Statistics, 29(3–4), 431–454.
Tukey, J. (1958). Bias and confidence in not quite large samples. Ann. Math. Statis, 29, 614.
Vairinhos, V. M. (2003). Desarrollo de un sistema para minería de datos basado en los métodos Biplot.
Van Eeuwijk, F. A. (1995). Multiplicative interaction in generalized linear models. Biometrics, 1017–1032.
Vicente-Tavera, S. (1992). Las técnicas de representación de datos multidimensionales en el estudio del Indice de Producción Industrial en la CEE (Tesis Doctoral). Universidad de Salamanca, España.
Vicente-Villardón, J. L. (1992). Una alternativa a las técnicas factoriales clásicas basada en una generalización de los métodos BIPLOT. [Tesis doctoral]. Universidad de Salamanca, España.
Vicente-Villardón, J. L. (2001). Biplot for binary data based on logistic response surfaces. In Salamanca Statistics Seminar IV: Advances in Multivariate Analysis. Salamanca, Spain.
Vicente-Villardón, J. L. (2010). MULTBIPLOT: package for multivariate análisis using biplots. Departamento de Estadística. Universidad de Salamanca. Retrieved from http://biplot.usal.es/multbiplot/introduction.html
Vicente-Villardón, J. L. (2017). MultBiplotR: Multivariate Analysis using Biplot. R package version 0.1.0. http://biplot.usal.es/classicalbiplot/multbiplot-in-r/.
Vicente-Villardón, J. L., Galindo-Villardón, M. P., & Blázquez-Zavallos, A. (2004). Constrained Logistic Biplots. In SALAMANCA STATISTICS SEMINAR V. Advances in Descriptive Multivariate Analysis . Universidad de Salamanca, España.
Vicente-Villardón, J. L., Galindo-Villardón, M. P., & Blázquez-Zavallos, A. (2006). Logistic biplots. Multiple Correspondence Analysis and Related Methods. London: Chapman & Hall. 503-521.
Vichi, M., & Saporta, G. (2009). Clustering and disjoint principal component analysis. Computational Statistics & Data Analysis, 53(8), 3194–3208.
Vines, S. K. (2000). Simple principal components. Journal of the Royal Statistical Society: Series C (Applied Statistics), 49(4), 441–451. https://doi.org/10.1111/1467-9876.00204
Witten, D. M., Tibshirani, R., & Hastie, T. (2009). A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis. Biostatistics, 10(3), 515–534.179
Yan, W. (2001). GGEbiplot—a Windows application for graphical analysis of multienvironment trial data and other types of two-way data. Agronomy, 93(5),1111–1118.
Yan, W. (2002). Singular-value partitioning in biplot analysis of multienvironment trial data. Agronomy Journal, 94(5), 990–996.
Yan, W., & Hunt, L. A. (2002). Biplot analysis of diallel data. Crop Science, 42(1), 21–30.
Yan, W., & Kang, M. S. (2002). GGE biplot analysis: A graphical tool for breeders, geneticists, and agronomists. CRC Press.
Yan, W., & Tinker, N. A. (2006). Biplot analysis of multi-environment trial data: Principles and applications. Canadian Journal of Plant Science, 86(3), 623–645.
Yan, W., Cornelius, P. L., Crossa, J., & Hunt, L. A. (2001). Two types of GGE biplots for analyzing multi-environment trial data. Crop Science, 41, 656–663.
Yan, W., Hunt, L. A., Sheng, Q., & Szlavnics, Z. (2000). Cultivar evaluation and mega-environment investigation based on the GGE biplot. Crop Science, 40, 597– 605.
Yang, J., Rübel, O., Prabhat, Mahoney, M. W., & Bowen, B. P. (2015). Identifying Important Ions and Positions in Mass Spectrometry Imaging Data Using CUR Matrix Decompositions. Analytical Chemistry, 87(9), 4658–4666. https://doi.org/10.1021/ac5040264
Young, G., & Householder, A. S. (1938). Discussion of a set of points in terms of their mutual distances. Psychometrika, 3(1), 19–22.
Zhang, Z., Xu, Y., Yang, J., Li, X., & Zhang, D. (2015). A survey of sparse representation: algorithms and applications. IEEE, 490–530.
Zobel, R. W., Wright, M. J., & Gauch, H. G. (1988). Statistical analysis of a yield trial. Agronomy Journal, 80, 388–393.
Zou, H., & Hastie, T. (2005). Regularization and variable selection via the elastic net. Journal of the Royal Statistical Society:, 67(2), 301–320.
Zou, H., Hastie, T., & Tibshirani, R. (2006). Sparse Principal Component Analysis. Journal of Computational and Graphical Statistics, 15(2), 265–286. https://doi.org/10.1198/106186006X113430.
