Nonlinear Principal and Canonical Directions from Continuous Extensions of Multidimensional Scaling
Abstract: A continuous random variable is expanded as a sum of a sequence of uncorrelated random variables. These variables are principal dimensions in continuous scaling on a distance function, as an extension of classic scaling on a distance matrix. For a particular distance, these dimensions are principal components. Then some properties are studied and an inequality is obtained. Diagonal expansions are considered from the same continuous scaling point of view, by means of the chi-square distance. The geometric dimension of a bivariate distribution is defined and illustrated with copulas. It is shown that the dimension can have the power of continuum.
Cite this paper: C. Cuadras, "Nonlinear Principal and Canonical Directions from Continuous Extensions of Multidimensional Scaling," Open Journal of Statistics, Vol. 4 No. 2, 2014, pp. 154-171. doi: 10.4236/ojs.2014.42015.
References

[1]   T. W. Anderson and M. A. Stephens, “The Continuous and Discrete Brownian Bridges: Representation and Applications,” Linear Algebra and Its Applications, Vol. 264, 1996, pp. 145-171. http://dx.doi.org/10.1016/S0024-3795(97)00015-3

[2]   J. Durbin and M. Knott, “Components of Cramér-von Mises Statistics. I,” Journal of the Royal Statistical Society: Series B, Vol. 34, 1972, pp. 290-307.

[3]   J. Fortiana and C. M. Cuadras, “A Family of Matrices, the Discretized Brownian Bridges and Distance-Based Regression,” Linear Algebra and Its Applications, Vol. 264, 1997, pp. 173-188. http://dx.doi.org/10.1016/S0024-3795(97)00051-7

[4]   J. L. Doob, “Stochastic Processes,” Wiley, New York, 1953.

[5]   M. Loève, “Probability Theory,” 3rd Edition, Van Nostrand, Princeton, 1963.

[6]   G. K. Eagleson, “Orthogonal Expansions and U-Statistics,” Australian Journal of Statistics, Vol. 21, No. 3, 1979, pp. 221-237.
http://dx.doi.org/10.1111/j.1467-842X.1979.tb01141.x

[7]   C. M. Cuadras and D. Cuadras, “Orthogonal Expansions and Distinction between Logistic and Normal,” In: C. Huber-Carol, N. Balakrishnan, M. S. Nikulin and M. Mesbah, Eds., Goodness-Of-Fit Tests and Model Validity, Birkhauser, Boston, 2002, pp. 327-339. http://dx.doi.org/10.1007/978-1-4612-0103-8_24

[8]   J. Fortiana and A. Grané, “Goodness-Of-Fit Tests Based on Maximum Correlations and Their Orthogonal Decompositions,” Journal of the Royal Statistical Society: Series B, Vol. 65, No. 1, 2003, pp. 115-126.
http://dx.doi.org/10.1111/1467-9868.00375

[9]   C. M. Cuadras, “Diagonal Distributions via Orthogonal Expansions and Tests of Independence,” In: C. M. Cuadras, J. Fortiana and J. A. Rodriguez-Lallena, Eds., Distributions with Given Marginals and Statistical Modelling, Kluwer Academic Press, Dordrecht, 2002, pp. 35-42.
http://dx.doi.org/10.1007/978-94-017-0061-0_5

[10]   C. M. Cuadras, “First Principal Component Characterization of a Continuous Random Variable,” In: N. Balakrishnan, I. G. Bairamov and O. L. Gebizlioglu, Eds., Advances on Models, Characterizations and Applications, Chapman and Hall/CRC, London, 2005, pp. 189-199.
http://dx.doi.org/10.1201/9781420028690.ch12

[11]   C. M. Cuadras and J. Fortiana, “Continuous Metric Scaling and Prediction,” In: C. M. Cuadras and C. R. Rao, Eds., Multivariate Analysis, Future Directions 2, Elsevier Science Publishers B. V. (North-Holland), Amsterdam, 1993, pp. 47-66.

[12]   C. M. Cuadras and J. Fortiana, “A Continuous Metric Scaling Solution for a Random Variable,” Journal of Multivariate Analysis, Vol. 52, No. 1, 1995, pp. 1-14.
http://www.sciencedirect.com/science/article/pii/S0047259X85710019
http://dx.doi.org/10.1006/jmva.1995.1001

[13]   C. M. Cuadras and J. Fortiana, “Weighted Continuous Metric Scaling,” In: A. K. Gupta and V. L. Girko, Eds., Multidimensional Statistical Analysis and Theory of Random Matrices, VSP, Zeist, 1996, pp. 27-40.

[14]   C. M. Cuadras and J. Fortiana, “The Importance of Geometry in Multivariate Analysis and Some Applications,” In: C. R. Rao and G. Szekely, Eds., Statistics for the 21st Century, Marcel Dekker, New York, 2000, pp. 93-108.

[15]   C. M. Cuadras and D. Cuadras, “A Parametric Approach to Correspondence Analysis,” Linear Algebra and Its Applications, Vol. 417, No. 1, 2006, pp. 64-74.
http://www.sciencedirect.com/science/article/pii/S0024379505005203
http://dx.doi.org/10.1016/j.laa.2005.10.029

[16]   C. M. Cuadras and D. Cuadras, “Eigenanalysis on a Bivariate Covariance Kernel,” Journal of Multivariate Analysis, Vol. 99, No. 10, 2008, pp. 2497-2507.
http://www.sciencedirect.com/science/article/pii/S0047259X08000754
http://dx.doi.org/10.1016/j.jmva.2008.02.039

[17]   H. O. Lancaster, “The Chi-Squared Distribution,” Wiley, New York, 1969.

[18]   C. M. Cuadras, J. Fortiana and M. J. Greenacre, “Continuous Extensions of Matrix Formulations in Correspondence Analysis, with Applications to the FGM Family of Distributions,” In: R. D. H. Heijmans, D. S. G. Pollock and A. Satorra, Eds., Innovations in Multivariate Statistical Analysis, Kluwer Academic Publisher, Dordrecht, 2000, pp. 101-116.
http://dx.doi.org/10.1007/978-1-4615-4603-0_7

[19]   C. M. Cuadras, J. Fortiana and F. Oliva, “The Proximity of an Individual to a Population with Applications in Discriminant Analysis,” Journal of Classification, Vol. 14, No. 1, 1997, pp. 117-136. http://dx.doi.org/10.1007/s003579900006

[20]   K. V. Mardia, J. T. Kent and J. M. Bibby, “Multivariate Analysis,” Academic Press, London, 1979.

[21]   T. F. Cox and M. A. Cox, “Multidimensional Scaling,” Chapman and Hall, London, 1994.

[22]   I. J. Schoenberg, “Remarks to Maurice Fréchet’s Article ‘Sur la définition axiomtique d’une classe d’espaces vectoriels distanciés applicables vectoriellment sur l’espace de Hilbert’,” Annals of Mathematics, Vol. 36, No. 3, 1935, pp. 724-732.
http://dx.doi.org/10.2307/1968654

[23]   C. M. Cuadras, “Distance Analysis in Discrimination and Classification Using Both Continuous and Categorical Variables,” In: Y. Dodge, Ed., Statistical Data Analysis and Inference, Elsevier Science Publishers B. V. (North-Holland), Amsterdam, 1989, pp. 459-473.

[24]   C. M. Cuadras, E. A. Atkinson and J. Fortiana, “Probability Densities from Distances and Discriminant Analysis,” Statistics and Probability Letters, Vol. 33, No. 4, 1997, pp. 405-411. http://dx.doi.org/10.1016/S0167-7152(96)00154-X

[25]   C. R. Rao, “Diversity: Its Measurement, Decomposition, Apportionment and Analysis,” Sankhyā: The Indian Journal of Statistics, Series A, Vol. 44, No. 1, 1982, pp. 1-21.

[26]   Z. Liu and C. R. Rao, “Asymptotic Distribution of Statistics Based on Quadratic Entropy and Bootstrapping,” Journal of Statistical Planning and Inference, Vol. 43, No. 1-2, 1995, pp. 1-18. http://dx.doi.org/10.1016/0378-3758(94)00005-G

[27]   C. M. Cuadras and Y. Lahlou, “Some Orthogonal Expansions for the Logistic Distribution,” Communications in Statistics— Theory and Methods, Vol. 29, No. 12, 2000, pp. 2643-2663. http://dx.doi.org/10.1080/03610920008832629

[28]   C. M. Cuadras, “On the Covariance between Functions,” Journal of Multivariate Analysis, Vol. 81, No. 1, 2002, pp. 19-27.
http://www.sciencedirect.com/science/article/pii/S0047259X01920007
http://dx.doi.org/10.1006/jmva.2001.2000

[29]   C. M. Cuadras and Y. Lahlou, “Principal Components of the Pareto Distribution,” In: C. M. Cuadras, J. Fortiana and J. A. Rodriguez-Lallena, Eds., Distributions with Given Marginals and Statistical Modelling, Kluwer Academic Press, Dordrecht, 2002, pp. 43-50. http://dx.doi.org/10.1007/978-94-017-0061-0_6

[30]   E. Salinelli, “Nonlinear Principal Components, II: Characterization of Normal Distributions,” Journal of Multivariate Analysis, Vol. 100, No. 4, 2009, pp. 652-660.
http://dx.doi.org/10.1016/j.jmva.2008.07.001

[31]   H. Chernoff, “A Note on an Inequality Involving the Normal Distribution,” Annals of Probability, Vol. 9, No. 3, 1981, pp. 533-535. http://dx.doi.org/10.1214/aop/1176994428

[32]   T. Cacoullos, “On Upper and Lower Bounds for the Variance of a Function of a Random Variable,” Annals of Probability, Vol. 10, No. 3, 1982. pp. 799-809. http://dx.doi.org/10.1214/aop/1176993788

[33]   C. A. J. Klaassen, “On an Inequality of Chernoff,” Annals of Probability, Vol. 13, No. 3, 1985, pp. 966-974.
http://dx.doi.org/10.1214/aop/1176992917

[34]   M. J. Greenacre, “Theory and Applications of Correspondence Analysis,” Academic Press, London, 1984.

[35]   C. M. Cuadras, “Correspondence Analysis and Diagonal Expansions in Terms of Distribution Functions,” Journal of Statistical Planning and Inference, Vol. 103, No. 1-2, 2002, pp. 137-150. http://dx.doi.org/10.1016/S0378-3758(01)00216-6

[36]   R. B. Nelsen, “An Introduction to Copulas,” 2nd Edition, Springer, New York, 2006.

[37]   R. B. Nelsen, J. J. Quesada-Molina and J. A. Rodriguez-Lallena, “Bivariate Copulas with Cubic Sections,” Journal of Nonparametric Statistics, Vol. 7, No. 3, 1997, pp. 205-220. http://dx.doi.org/10.1080/10485259708832700

[38]   C. M. Cuadras and W. Daz, “Another Generalization of the Bivariate FGM Distribution with Two-Dimensional Extensions,” Acta et Commentationes Universitatis Tartuensis de Mathematica, Vol. 16, No. 1, 2012, pp. 3-12.