categorymoments

Categorymoments is a data-analytic concept used to describe and extract distributional properties of a numerical feature within the segments defined by a categorical variable. It fits within feature engineering and exploratory data analysis and is used to capture how a variable behaves differently across categories.

Definition: Given a dataset with a numerical feature X and a categorical feature C with categories c in a set, categorymoments computes the moments of X within each category. For order k, the conditional moment m_k(c) can be defined as E[(X - μ_c)^k | C=c], where μ_c is the mean of X in category c. In practice, implementations often include the first four moments (mean, variance, skewness, kurtosis) or use raw moments such as E[X^k | C=c].

Computation: In offline settings, moments are computed by grouping data by C and calculating sample moments for each group. In streaming or online settings, incremental updates (for example, per-category Welford’s algorithm) allow real-time calculation. Storage and computation scale with the number of categories and the number of moments tracked, which can be a consideration for high-cardinality categoricals.

Applications: Category-specific moments are used as features for predictive models to capture heterogeneity across categories. They support anomaly detection, pricing and segmentation analyses, and monitoring distribution shifts across groups. They can also inform smoothing or hierarchical modeling to improve stability.

Limitations: Sparse categories can yield unstable estimates, necessitating smoothing or shrinkage. Higher-order moments can be noisy and may require regularization. Privacy or fairness concerns may arise if category definitions reveal sensitive attributes.