Functional ANOVA offers a principled framework for interpretability by decomposing a model's prediction into main effects and higher-order interactions. For independent features, this decomposition is well-defined, strongly linked with SHAP values, and serves as a cornerstone of additive explainability. However, the lack of an explicit closed-form expression for general dependent distributions has forced practitioners to rely on costly sampling-based approximations. We completely resolve this limitation for categorical inputs. By bridging functional analysis with the extension of discrete Fourier analysis, we derive a closed-form decomposition without any assumption. Our formulation is computationally very efficient. It seamlessly recovers the classical independent case and extends to arbitrary dependence structures, including distributions with non-rectangular support. Furthermore, leveraging the intrinsic link between SHAP and ANOVA under independence, our framework yields a natural generalization of SHAP values for the general categorical setting.

翻译：函数方差分析通过将模型预测分解为主效应和高阶交互作用，为可解释性提供了一个原则性框架。对于独立特征，该分解具有明确的定义，与SHAP值紧密关联，并构成可加性解释方法的基石。然而，由于缺乏适用于一般相依分布的显式闭式表达式，实践者不得不依赖计算成本高昂的基于采样的近似方法。我们针对分类输入完全解决了这一局限性。通过将泛函分析与离散傅里叶分析的扩展相结合，我们推导出了无需任何假设的闭式分解表达式。该公式在计算上具有很高的效率。它能够无缝恢复经典的独立情形，并可推广至任意相依结构，包括具有非矩形支撑的分布。此外，借助独立条件下SHAP与方差分析的内在联系，我们的框架为一般分类场景下的SHAP值提供了一种自然的推广形式。