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Strategies for Multidimensional Data Visualization

  • Gintautas Dzemyda
  • Olga Kurasova
  • Julius Žilinskas
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 75)

Abstract

In this chapter, an analytical review of methods for multidimensional data visualization is presented. The methods based on direct visualization and projections are described. Some quantitative criteria of the visualization quality are also introduced.

Keywords

Linear Discriminant Analysis Geodesic Distance Multidimensional Data Locally Linear Embedding Laplacian Eigenmaps 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 2.
    Andrews, D.F.: Plots of high dimensional data. Biometrics 28, 125–136 (1972). DOI 10.2307/2528964CrossRefGoogle Scholar
  2. 5.
    Becker, R.A., Cleveland, W.S., Shyu, M.J.: The design and control of trellis display. J. Comput. Stat. Graph. 5, 123–155 (1996)Google Scholar
  3. 6.
    Belkin, M., Niyogi, P.: Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput. 15(6), 1373–1396 (2003). DOI  10.1162/089976603321780317 CrossRefMATHGoogle Scholar
  4. 7.
    Bernatavičienė, J., Dzemyda, G., Kurasova, O., Marcinkevičius, V.: Optimal decisions in combining the SOM with nonlinear projection methods. Eur. J. Oper. Res. 173(3), 729–745 (2006). DOI 10.1016/j. ejor.2005.05.030CrossRefMATHGoogle Scholar
  5. 8.
    Bernatavičienė, J., Dzemyda, G., Kurasova, O., Marcinkevičius, V.: Strategies of selecting the basis vector set in the relative MDS. Technol. Econ. Dev. Econ. 12(4), 283–288 (2006). DOI 10.1080/13928619.2006. 9637755Google Scholar
  6. 10.
    Bernatavičienė, J., Dzemyda, G., Marcinkevičius, V.: Conditions for optimal efficiency of relative MDS. Informatica 18(2), 187–202 (2007)MATHGoogle Scholar
  7. 11.
    Bernatavičienė, J., Dzemyda, G., Marcinkevičius, V.: Diagonal majorization algorithm: Properties and efficiency. Inform. Tech. Contr. 36(4), 353–358 (2007). URL http://itc.ktu.lt/itc364/Bernat364.pdf
  8. 12.
    Bezdek, J.C., Pal, N.R.: An index of topological preservation for feature extraction. Pattern Recogn. 28(3), 381–391 (1995). DOI 10. 1016/0031-3203(94)00111-XCrossRefGoogle Scholar
  9. 14.
    Borg, I., Groenen, P.J.F.: Modern Multidimensional Scaling: Theory and Applications, 2nd edn. Springer, New York (2005)Google Scholar
  10. 16.
    Bray, J.R., Curtis, J.T.: An ordination of the upland forest communities of southern wisconsin. Ecol. Monogr. 27(4), 325–349 (1957). URL http://www.jstor.org/stable/1942268
  11. 26.
    Chambers, J.M.: Graphical Methods for Data Analysis (Statistics). Chapman & Hall/CRC, Boca Raton (1983)Google Scholar
  12. 29.
    Chernoff, H.: The use of faces to represent points in k-dimensional space graphically. J. Am. Stat. Assoc. 68(342), 361–368 (1973)CrossRefGoogle Scholar
  13. 31.
    Cox, T.F., Cox, M.A.A.: Multidimensional Scaling, 2nd edn. Chapman & Hall/CRC, Boca Raton (2001)MATHGoogle Scholar
  14. 32.
    Cristianini, N., Shawe-Taylor, J.: An Introduction to Support Vector Machines and Other Kernel-based Learning Methods. Cambridge University Press, Cambridge (2000)CrossRefGoogle Scholar
  15. 34.
    Decoste, D.: Visualizing Mercer kernel feature spaces via kernelized locally-linear embeddings. In: Proceedings of the Eighth International Conference on Neural Information Processing (2001)Google Scholar
  16. 36.
    Delicado, P.: Another look at principal curves and surfaces. J. Multivariate Anal. 77(1), 84–116 (2001). URL http://ideas.repec.org/a/eee/jmvana/v77y2001i1p84-116.html
  17. 40.
    Dijkstra E., W.: A note on two problems in connexion with graphs. Numer. Math. 1, 269–271 (1959)Google Scholar
  18. 41.
    Donoho, D., Grimes, C.: Hessian eigenmaps: New locally linear embedding techniques for high-dimensional data. Proc. Natl. Acad. Sci. 100(10), 5591–5596 (2003)CrossRefMATHMathSciNetGoogle Scholar
  19. 43.
    Duda, R., Hart, P.: Pattern Recognition and Scene Analysis. Wiley, New York (1973)Google Scholar
  20. 44.
    Duda, R.O., Hart, P.E., Stork, D.G.: Pattern Classification, 2nd edn. Wiley, New York (2000)Google Scholar
  21. 45.
    Dunham, M.H.: Data Mining: Introductory and Advanced Topics. Prentice Hall PTR, Upper Saddle River, NJ (2002)Google Scholar
  22. 48.
    Dzemyda, G.: Visualization of correlation-based environmental data. Environmetrics 15(8), 827–836 (2004). DOI 10.1002/env.672CrossRefGoogle Scholar
  23. 49.
    Dzemyda, G.: Multidimensional data visualization in the statistical analysis of curricula. Comput. Stat. Data Anal. 49(1), 265–281 (2005). DOI 10.1016/j.csda.2004.05.001CrossRefMATHMathSciNetGoogle Scholar
  24. 50.
    Dzemyda, G., Bernatavičienė, J., Kurasova, O., Marcinkevičius, V.: Minimization of the mapping error using coordinate descent. In: Proceedings of the 13th International Conference in Central Europe on Computer Graphics, Visualization and Computer Vision, WSCG’2005 (Short Papers), pp. 169–172 (2005)Google Scholar
  25. 53.
    Dzemyda, G., Kurasova, O.: Heuristic approach for minimizing the projection error in the integrated mapping. Eur. J. Oper. Res. 171(3), 859–878 (2006). DOI  10.1016/j.ejor.2004.09.011 CrossRefMATHMathSciNetGoogle Scholar
  26. 58.
    Ebert, D.S., Rohrer, R.M., Shaw, C.D., Panda, P., Kukla, J.M., Roberts, D.A.: Procedural shape generation for multi-dimensional data visualization. Comput. Graph. 24(3), 375–384 (2000)CrossRefGoogle Scholar
  27. 59.
    Estévez, P.A., Figueroa, C.J., Saito, K.: Cross-entropy embedding of high-dimensional data using the neural gas model. Neural Network 18(5–6), 727–737 (2005). DOI 10.1016/j.neunet.2005.06.010CrossRefGoogle Scholar
  28. 63.
    Fisher, R.A.: The use of multiple measurements in taxonomic problems. Ann. Eug. 7, 179–188 (1936)CrossRefGoogle Scholar
  29. 67.
    Fua, Y.H., Ward, M.O., Rundensteiner, E.A.: Hierarchical parallel coordinates for exploration of large datasets. In: VIS’99: Proceedings of the Conference on Visualization, pp. 43–50. IEEE Computer Society Press, Los Alamitos, CA (1999)Google Scholar
  30. 68.
    Fukunaga, K.: Introduction to Statistical Pattern Recognition, 2nd edn. Academic, London (1990)MATHGoogle Scholar
  31. 69.
    Ge, S.S., Yang, Y., Lee, T.H.: Hand gesture recognition and tracking based on distributed locally linear embedding. Image Vis. Comput. 26(12), 1607–1620 (2008). DOI 10.1016/j.imavis.2008.03.004CrossRefGoogle Scholar
  32. 71.
    Goodhill, G., Sejnowski, T.: Quantifying neighbourhood preservation in topographic mappings. In: Proceedings of the 3rd Joint Symposium on Neural Computation, pp. 61–82 (1996)Google Scholar
  33. 73.
    Grinstein, G., Trutschl, M., Cvek, U.: High-dimensional visualizations. In: Proceedings of Workshop on Visual Data Mining, ACM Conference on Knowledge Discovery and Data Mining, pp. 1–14. ACM, New York (2001)Google Scholar
  34. 74.
    Grinstein, G.G., Ward, M.O.: Introduction to data visualization. In: Fayyad, U., Grinstein, G.G., Wierse, A. (eds.) Information visualization in data mining and knowledge discovery, pp. 21–45. Morgan Kaufmann, San Francisco, CA (2002)Google Scholar
  35. 81.
    Guttman, L.: A general nonmetric technique for finding the smallest coordinate space for a configuration of points. Psychometrika 33(4), 469–506 (1968)CrossRefMATHGoogle Scholar
  36. 82.
    Hadid, A., Kouropteva, O., Pietikinen, M.: Unsupervised learning using locally linear embedding: Experiments with face pose analysis. Proc. Int. Conf. Pattern Recogn. 1, 111–114 (2002). DOI 10.1109/ICPR.2002. 1044625Google Scholar
  37. 83.
    Han, J., Kamber, M.: Data Mining: Concepts and Techniques, 2nd edn. Morgan Kaufmann, San Francisco, CA (2006)Google Scholar
  38. 86.
    Hassoun, M.H.: Fundamentals of Artificial Neural Networks. MIT, Cambridge, MA (1995)MATHGoogle Scholar
  39. 87.
    Hastie, T.: Principal curves and surfaces. Ph.D. thesis, Stanford Linear Accelerator Center, Stanford University (1984)Google Scholar
  40. 88.
    Hastie, T., Stuetzle, W.: Principal curves. J. Am. Stat. Assoc. 84(406), 502–516 (1989). URL http://www.jstor.org/stable/2289936
  41. 93.
    Hoffman, P., Grinstein, G., Pinkney, D.: Dimensional anchors: a graphic primitive for multidimensional multivariate information visualizations. In: NPIVM’99: Proceedings of the 1999 workshop on new paradigms in information visualization and manipulation in conjunction with the eighth ACM international conference on Information and knowledge management, pp. 9–16. ACM, New York (1999). DOI 10.1145/331770. 331775Google Scholar
  42. 94.
    Hoffman, P.E., Grinstein, G.G.: A survey of visualizations for high-dimensional data mining. In: Fayyad, U., Grinstein, G.G., Wierse, A. (eds.) Information Visualization in Data Mining and Knowledge Discovery, pp. 47–82. Morgan Kaufmann, San Francisco, CA (2002)Google Scholar
  43. 95.
    Honggui, L., Xingguo, L.: Gait analysis using LLE. In: ICSP’04: Proceedings of I7th International Conference on Signal Processing, vol. 2, pp. 1423–1426 (2004). DOI 10.1109/ICOSP.2004.1441593Google Scholar
  44. 103.
    Inselberg, A.: The plane with parallel coordinates. Vis. Comput. 1(2), 69–91 (1985)CrossRefMATHGoogle Scholar
  45. 108.
    Jain, V., Saul, L.K.: Exploratory analysis and visualization of speech and music by locally linear embedding. In: ICASSP’04: Proceedings of IEEE International Conference of Speech, Acoustics, and Signal Processing, vol. 3, pp. 984–987 (2004)Google Scholar
  46. 110.
    Jolliffe, I.: Principal Component Analysis. Springer, Berlin (1986)CrossRefGoogle Scholar
  47. 111.
    Karbauskaitė, R., Dzemyda, G.: Topology preservation measures in the visualization of manifold-type multidimensional data. Informatica 20(2), 235–254 (2009)MATHMathSciNetGoogle Scholar
  48. 112.
    Karbauskaitė, R., Dzemyda, G., Marcinkevičius, V.: Dependence of locally linear embedding on the regularization parameter. TOP Offic. J. Spanish Soc. Stat. Oper. Res. 18(2), 354–376 (2010). DOI 10.1007/ s11750-010-0151-yMATHGoogle Scholar
  49. 113.
    Karbauskaitė, R., Kurasova, O., Dzemyda, G.: Selection of the number of neighbours of each data point for the Locally Linear Embedding algorithm. Inform. Tech. Contr. 36(4), 359–364 (2007). URL http://itc.ktu.lt/itc364/Karbausk364.pdf
  50. 114.
    Karbowski, A.: Direct method of hierarchical nonlinear optimization - reassessment after 30 years. In: DSTIS 2003: Proceedings of III International Conference on Decision Support for Telecommunications and Information Society, pp. 15–30. Warsaw (2003)Google Scholar
  51. 115.
    Kaski, S.: Data exploration using self-organizing maps. Ph.D. thesis, Helsinki University of Technology, Department of Computer Science and Engineering (1997)Google Scholar
  52. 120.
    Kohonen, T.: Self-Organizing Maps, 3rd edn. Springer Series in Information Science. Springer, Berlin (2001)CrossRefMATHGoogle Scholar
  53. 122.
    Konig, A.: Interactive visualization and analysis of hierarchical neural projections for data mining. IEEE Trans. Neural Network 11(3), 615–624 (2000). DOI 10.1109/72.846733CrossRefGoogle Scholar
  54. 125.
    Kraus, M., Ertl, T.: Interactive data exploration with customized glyphs. In: WSCG01: Proceedings of the 9-th International Conference in Central Europe on Computer Graphics, Visualization and Computer Vision, pp. 20–23 (2001)Google Scholar
  55. 126.
    Kruskal, J.: Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika 29, 1–27 (1964)CrossRefMATHMathSciNetGoogle Scholar
  56. 128.
    Kurasova, O., Molytė, A.: Combination of vector quantization and visualization. In: MLDM’09: Proceedings of the 6th International Conference on Machine Learning and Data Mining in Pattern Recognition, pp. 29–43. Springer, Berlin (2009). DOI http://dx.doi.org/10.1007/978-3-642-03070-3_3
  57. 132.
    Lance, G.N., Williams, W.T.: Computer programs for hierarchical polythetic classification (‘similarity analyses’). Comput. J. 9(1), 60–64 (1966)CrossRefMATHGoogle Scholar
  58. 141.
    Li, W., Pardalos, P.M., Han, C.G.: Gauss-Seidel method for least-distance problems. J. Optim. Theor Appl. 75(3), 487–500 (1992). DOI 10.1007/BF00940488CrossRefMATHMathSciNetGoogle Scholar
  59. 142.
    Liou, C.Y., Kuo, Y.T.: Economic states on neuronic maps. In: ICONIP’02: Proceedings of the 9th International Conference on Neural Information Processing, vol. 2, pp. 787–791 (2002)Google Scholar
  60. 143.
    Liu, K., Weissenfeld, A., Ostermann, J.: Parameterization of mouth images by LLE and PCA for image-based facial animation. In: ICASSP06: IEEE Proceedings of International Conference on Acoustics, Speech and Signal Processing, vol. 5, pp. 461–464 (2006). URL ftp://ftp.tnt.uni-hannover.de/pub/papers/2006/ICASSP-KLAWJO.pdf
  61. 158.
    Mekuz, N., Bauckhage, C., Tsotsos, J.K.: Face recognition with weighted locally linear embedding. Comput. Robot Vis Can. Conf. 0, 290–296 (2005). DOI 10.1109/CRV.2005.42CrossRefGoogle Scholar
  62. 162.
    Murtagh, F.: Multivariate data analysis software and resources. URL http://www.classification-society.org/csna/mda-sw/
  63. 163.
    Naud, A.: Visualization of high-dimensional data using an association of multidimensional scaling to clustering. In: IEEE Conference on Cybernetics and Intelligent Systems, pp. 252–255 (2004). DOI 10. 1109/ICCIS.2004.1460421Google Scholar
  64. 164.
    Naud, A., Duch, W.: Interactive data exploration using mds mapping. In: Proceedings of the Fifth Conference: Neural Networks and Soft Computing, pp. 255–260 (2000)Google Scholar
  65. 166.
    Nene, S.A., Nayar, S.K., Murase, H.: Columbia object image library (COIL-20. Tech. Rep. CUCS-005-96, Columbia University (1996)Google Scholar
  66. 168.
    Oja, E.: Principal components, minor components, and linear neural networks. Neural Network 5(6), 927–935 (1992). DOI 10.1016/ S0893-6080(05)80089-9CrossRefGoogle Scholar
  67. 169.
    Opitz, O., Hilbert, A.: Visualization of multivariate data by scaling and property fitting. In: Gaul, W., Opitz, O., Schader, M. (eds.) Data Analysis: Scientific Modeling and Practical Applications, pp. 505–514. Springer, New York (2000)CrossRefGoogle Scholar
  68. 171.
    Pearson, K.: On lines and planes of closest fit to systems of points in space. Phil. Mag. 2(6), 559–572 (1901)CrossRefGoogle Scholar
  69. 176.
    R. S., M.: A planar geometric model for representing multidimensional discrete spaces and multiple-valued logic functions. Tech. Rep. UIUCDCSR-78-897, University of Illinois at Urbaba-Champaign (1978)Google Scholar
  70. 179.
    Ribarsky, W., Ayers, E., Eble, J., Mukherjea, S.: Glyphmaker: creating customized visualization of complex data. IEEE Comput. 27(7), 57–64 (1994)CrossRefGoogle Scholar
  71. 180.
    de Ridder, D., Kouropteva, O., Okun, O., Pietikinen, M., Duin, R.: Supervised locally linear embedding. In: ICANN/ICONIP’2003: Proceedings of the International Conference on Artificial Neural Networks and Neural Information Processing. Lecture Notes in Computer Science, vol. 2714, pp. 333–341. Springer, New York (2003)Google Scholar
  72. 184.
    Roweis, S.T., Saul, L.K.: Nonlinear dimensionality reduction by locally linear embedding. Science 290(5500), 2323–2326 (2000)CrossRefGoogle Scholar
  73. 185.
    Rubner, J., Tavan, P.: A self-organizing network for principal-component analysis. EPL (Europhys. Lett.) 10(7), 693–698 (1989). URL http://stacks.iop.org/0295-5075/10/693
  74. 187.
    Sachinopoulou, A.: Multidimensional visualization. Tech. rep., Technical Research Centre of Finland, VTT Tiedotteita, Meddelanden, Research Notes 2114 (2001)Google Scholar
  75. 188.
    Sammon, J.W.: A nonlinear mapping for data structure analysis. IEEE Trans. Comput. 18, 401–409 (1969)CrossRefGoogle Scholar
  76. 190.
    Saul, L., Roweis, S.: Think globally, fit locally: Unsupervised learning of low dimensional manifolds. J. Mach. Learn. Res. 4, 119–155 (2003)MathSciNetGoogle Scholar
  77. 193.
    de Silva, V., Tenenbaum, J.B.: Global versus local methods for nonlinear dimensionality reduction. In: Becker, S., Thrun, S., Obermayer, K. (eds.) Advances in Neural Information Processing Systems, 15, pp. 721–728. MIT, Cambridge, MA (2003)Google Scholar
  78. 198.
    Taylor, P.: Statistical methods. In: Berthold, M., Hand, D.J. (eds.) Intelligent Data Analysis: An Introduction, pp. 69–129. Springer, New York (2003)CrossRefGoogle Scholar
  79. 200.
    Tenenbaum, J.B., de Silva, V., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290(5500), 2319–2323 (2000). DOI 10.1126/science.290.5500.2319CrossRefGoogle Scholar
  80. 206.
    Trosset, M.W., Groenen, P.J.F.: Multidimensional scaling algorithms for large data sets. CD-ROM (2005)Google Scholar
  81. 209.
    Varini, C., Nattkemper T., Degenhard, A., Wismuller, A.: Breast MRI data analysis by LLE. In: Proceedings of 2004 IEEE International Joint Conference on Neural Networks, vol. 3, pp. 2449–2454 (2004)Google Scholar
  82. 213.
    Ward, M.O.: XmdvTool: integrating multiple methods for visualizing multivariate data. In: VIS’94: Proceedings of the Conference on Visualization, pp. 326–333. IEEE Computer Society Press, Los Alamitos, CA (1994)Google Scholar
  83. 215.
    Williams, M., Munzner, T.: Steerable, progressive multidimensional scaling. In: INFOVIS’04: Proceedings of the IEEE Symposium on Information Visualization, pp. 57–64. IEEE Computer Society, Washington, DC (2004). DOI 10.1109/INFOVIS.2004.60Google Scholar
  84. 216.
    Wittenbrink, C.M., Pang, A.T., Lodha, S.K.: Glyphs for visualizing uncertainty in vector fields. IEEE Trans. Visual. Comput. Graph. 2(3), 266–279 (1996). DOI 10.1109/2945.537309CrossRefGoogle Scholar
  85. 218.
    Yang, L.: Sammon’s nonlinear mapping using geodesic distances. In: ICPR’04: Proceedings of 17th International Conference on the Pattern Recognition, vol. 2, pp. 303–306. Washington (2004)Google Scholar
  86. 219.
    Zhao, Q., Zhang, D., Lu, H.: Supervised LLE in ICA space for facial expression recognition. In: ICNNB’05: Proceedings of International Conference on Neural Networks and Brain, vol. 3, pp. 1970–1975 (2005). DOI 10.1109/ICIEA.2006.257259Google Scholar
  87. 220.
    Zhu, L., Zhu, S.A.: Face recognition based on extended locally linear embedding. In: Proceedings of 1st IEEE Conference on Industrial Electronics and Applications, pp. 1–4 (2006). DOI 10.1109/ICIEA. 2006.257259Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2013

Authors and Affiliations

  • Gintautas Dzemyda
    • 1
  • Olga Kurasova
    • 1
  • Julius Žilinskas
    • 1
  1. 1.Institute of Mathematics and InformaticsVilnius UniversityVilniusLithuania

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