为类别型输入模型提供精确的可解释性分解，无需采样。
Exact Functional ANOVA Decomposition for Categorical Inputs Models
- 结合函数分析与离散傅里叶变换，推导出类别输入的闭式分解公式。
- 计算高效，能处理任意依赖结构，包括非矩形支撑分布。
- 自然推广了SHAP值，适用于有依赖关系的类别变量场景。
函数ANOVA为可解释性提供了严谨框架，将模型预测分解为主效应和高阶交互效应。对于独立特征，该分解定义明确，与SHAP值紧密关联，是加性解释的核心。然而，由于缺乏一般依赖分布的显式闭式表达，实践者不得不依赖昂贵的采样近似。本文针对类别型输入完全解决了这一局限。通过连接函数分析与离散傅里叶分析的扩展，我们推导出无需任何假设的闭式分解公式。该公式计算高效，无缝恢复经典独立情形，并可推广至任意依赖结构，包括非矩形支撑分布。此外，借助独立情形下SHAP与ANOVA的内在联系，本框架自然推广了SHAP值，适用于一般的类别型设定。
原文摘要 · Abstract (English)
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.
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