Principal Component Analysis (PCA) for Interval-Valued Symbolic Data: A Comparison of the Center and Vertex (TOPS) Methods
DOI:
https://doi.org/10.30871/jaic.v10i2.12381Keywords:
Symbolic Data Analysis (SDA), Principal Component Analysis (PCA), Centers, Vertices (TOPS), Dimensionality ReductionAbstract
Classical dimensionality reduction techniques, such as Principal Component Analysis (PCA), are widely used to explore the structure of multivariate datasets. However, these methods are traditionally restricted to situations in which each variable is represented by a single numerical value per individual. The emergence of symbolic data, particularly interval-valued data, has introduced new challenges in the field of data science. In this framework, a single variable may take multiple possible values, reflecting either measurement uncertainty or intrinsic variability of the observation. Such data therefore provide a more faithful representation of the complexity of observed phenomena, but they require specifically adapted analytical methodologies. This paper aims to compare two PCA variants applied to interval-valued symbolic data: The Center Method, in which each interval is represented by its midpoint, and the Vertex Method (TOPS), in which the lower and upper bounds of each interval are jointly exploited. We formally define interval-valued variables, present the algorithmic steps of both the Center and TOPS methods, analyze their computational complexity, and introduce evaluation metrics including explained variance, reconstruction error, and sensitivity analysis with respect to interval width. The objective is to assess the extent to which these approaches preserve the information contained within intervals and to determine which method proves more appropriate for a given dataset. Using a biomedical dataset (n = 1021 individuals, p = 7 interval-valued variables), we show that while the Center method provides strong dimensional condensation and interpretability, the TOPS method more faithfully preserves the geometry of intervals in the presence of high variability. This study clarifies the theoretical differences between the two approaches and proposes a systematic evaluation framework for interval-valued symbolic PCA methods.
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Copyright (c) 2026 Benjamin Boono, Mabela Rostin Makengo , Kikomba Kahungu Michael, Mbuyi Lunkondo Patience, Kipulu Ngimbi Serge
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