Функция присутствия как механизм корректной обработки пропущенных значений в нечётком интеллектуальном анализе данных

Научно-аналитический журнал "Вестник Санкт-Петербургского университета Государственной противопожарной службы МЧС России". 2026. № 2. С. 92–103.

Starodubov K., Максимова Е. А., Утешева Г. Ш., Громов Ю. Ю., Карасев П. И.

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.

Keywords: нечеткая логика, интеллектуальный анализ данных, Неполные данные