Presence Function as a Mechanism for Correct Handling of Missing Values in Fuzzy Data Mining

When data are incomplete, standard fuzzy data mining methods face an unacceptable trade-off: deleting objects with missing values loses up to a third of the sample and introduces systematic bias under the MAR mechanism, while mean imputation destroys the covariance structure and produces rules with artificially inflated support. This paper proposes a different solution – a presence function φ: U × A → [0, 1] that embeds the handling of missing values directly into the fuzzy support formula. An object with a missing attribute is neither discarded nor «repaired»: it participates in the analysis with a reduced weight φ₀, and the rest is handled by the t-norm. A number of theoretical properties are proved: anti-monotonicity of support for an arbitrary t-norm, robustness to attribute noise via the Lipschitz constant, invariance to monotone scale transformations, and the limiting cases φ₀ → 0 and φ₀ → 1/kₐ. Under 30 % MCAR missingness with φ₀ = 0,3, the share of recovered patterns is 21–38 percentage points higher than with listwise deletion. Three strategies for constructing fuzzy partitions are proposed: uniform, quantile-based and optimization-based. The dependence of the quality of extracted patterns on the parameter φ₀ under different missingness mechanisms is studied in detail. An experimental comparison on synthetic data confirmed that φ₀ = 0,3 under MCAR missingness provides a 21–38 percentage point higher share of recovered patterns than listwise deletion at a 30 % missingness level.