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**A comparison of three methods for principal component analysis of fuzzy interval data.**
*(English)*
Zbl 1157.62426

Summary: Vertices Principal Component Analysis (V-PCA), and Centers Principal Component Analysis (C-PCA) generalize Principal Component Analysis (PCA) in order to summarize interval valued data. Neural Network Principal Component Analysis (NN-PCA) represents an extension of PCA for fuzzy interval data. However, also the first two methods can be used for analyzing fuzzy interval data, but they then ignore the spread information. In the literature, the V-PCA method is usually considered computationally cumbersome because it requires the transformation of the interval valued data matrix into a single valued data matrix the number of rows of which depends exponentially on the number of variables and linearly on the number of observation units. However, it has been shown that this problem can be overcome by considering the cross-products matrix which is easy to compute. A review of C-PCA and V-PCA (which hence also includes the computational short-cut to V-PCA) and NN-PCA is provided. Furthermore, a comparison is given of the three methods by means of a simulation study and by an application to an empirical data set. In the simulation study, fuzzy interval data are generated according to various models, and it is reported in which conditions each method performs best.

### MSC:

62H25 | Factor analysis and principal components; correspondence analysis |

62M45 | Neural nets and related approaches to inference from stochastic processes |

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\textit{P. Giordani} and \textit{H. A. L. Kiers}, Comput. Stat. Data Anal. 51, No. 1, 379--397 (2006; Zbl 1157.62426)

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### References:

[1] | Baldi, P.; Hornik, K., Neural networks and principal component analysis: learning from examples without local minima, Neural networks, 2, 53-58, (1989) |

[2] | Bock, H.H.; Diday, E., Analysis of symbolic data: exploratory methods for extracting statistical information from complex data, (2000), Springer Heidelberg · Zbl 1039.62501 |

[3] | Boulard, H.; Kamp, Y., Auto-association by multilayer perceptrons and singular value decomposition, Biological cybernetics, 59, 291-294, (1988) · Zbl 0651.92006 |

[4] | Cazes, P., Analyse factorielle d’un tableau de lois de probabilité, Revue de statistique appliquée, 50, 5-24, (2002) |

[5] | Cazes, P.; Chouakria, A.; Diday, E.; Schektman, Y., Extension de l’analyse en composantes principales à des données de type intervalle, Revue de statistique appliquée, 45, 5-24, (1997) |

[6] | Coppi, R., D’Urso, P., Giordani, P., 2004. Component models for fuzzy data, Technical report n.1, Department of Statistics, Probability and Applied Statistics, University of Rome “La Sapienza”. Submitted for publication. |

[7] | Denœux, T.; Masson, M.H., Principal component analysis of fuzzy data using autoassociative neural networks, IEEE transactions on fuzzy systems, 12, 336-349, (2004) |

[8] | D’Urso, P.; Giordani, P., A least square approach to principal component analysis for interval valued data, Chemometrics and intelligent laboratory systems, 70, 179-192, (2004) |

[9] | D’Urso, P.; Giordani, P., A possibilistic approach to latent component analysis for fuzzy data, Fuzzy sets and systems, 150, 285-305, (2005) · Zbl 1058.62050 |

[10] | Giordani, P.; Kiers, H.A.L., Principal component analysis of symmetric fuzzy data, Computational statist. and data analysis, 45, 519-548, (2004) · Zbl 1429.62338 |

[11] | Giordani, P.; Kiers, H.A.L., Three-way component analysis of interval-valued data, Journal of chemometrics, 18, 253-264, (2004) |

[12] | Kiers, H.A.L., Some procedures for displaying results from three-way methods, Journal of chemometrics, 14, 151-170, (2000) |

[13] | Lauro, C.; Palumbo, F., Principal component analysis of interval data: a symbolic data analysis approach, Computational statistics, 15, 73-87, (2000) · Zbl 0953.62058 |

[14] | Lodwick, W.A.; Jamison, K.D., Special issue: interfaces between fuzzy set theory and interval analysis, Fuzzy sets and systems, 135, 1-3, (2003) |

[15] | Palumbo, F.; Lauro, C., A PCA for interval-valued data based on midpoints and radii, (), 641-648 |

[16] | Watada, J.; Yabuuchi, Y., Fuzzy principal component analysis and its application, Biomedical fuzzy human sciences, 3, 83-92, (1997) |

[17] | Zadeh, L.A., Fuzzy sets, Information and control, 8, 338-353, (1965) · Zbl 0139.24606 |

[18] | Zadeh, L.A., The concept of a linguistic variable and its application to approximate reasoning I, Information sciences, 8, 199-249, (1975) · Zbl 0397.68071 |

[19] | Zimmermann, H.J., Fuzzy set theory and its applications, (2001), Kluwer Academic Press Dordrecht |

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