A probabilistic and statistical view of fuzzy methods
Technometrics
Fuzzy set theory—and its applications (3rd ed.)
Fuzzy set theory—and its applications (3rd ed.)
Three-way component analysis with smoothness constraints
Computational Statistics & Data Analysis
Uncertainty and Information: Foundations of Generalized Information Theory
Uncertainty and Information: Foundations of Generalized Information Theory
Strong consistency of least-squares estimation in linear regression models with vague concepts
Journal of Multivariate Analysis
Limit distributions of least squares estimators in linear regression models with vague concepts
Journal of Multivariate Analysis
A comparison of three methods for principal component analysis of fuzzy interval data
Computational Statistics & Data Analysis
Principal component analysis of fuzzy data using autoassociative neural networks
IEEE Transactions on Fuzzy Systems
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Data are often affected by uncertainty. Uncertainty is usually referred to as randomness. Nonetheless, other sources of uncertainty may occur. In particular, the empirical information may also be affected by imprecision. Also in these cases it can be fruitful to analyze the underlying structure of the data. In this paper we address the problem of summarizing a sample of three-way imprecise data. In order to manage the different sources of uncertainty a twofold strategy is adopted. On the one hand, imprecise data are transformed into fuzzy sets by means of the so-called fuzzification process. The so-obtained fuzzy data are then analyzed by suitable generalizations of the Tucker3 and CANDECOMP/PARAFAC models, which are the two most popular three-way extensions of Principal Component Analysis. On the other hand, the statistical validity of the obtained underlying structure is evaluated by (nonparametric) bootstrapping. A simulation experiment is performed for assessing whether the use of fuzzy data is helpful in order to summarize three-way uncertain data. Finally, to show how our models work in practice, an application to real data is discussed.