高阶矢量Potts模型对离散数据的建模
Modeling Discrete Data with High-Order Vector Potts Models
AI总结：
本文通过引入q态自旋模型，将最大熵框架从二元数据推广到离散数据，提出高阶矢量Potts模型，并利用配分函数的圈展开和规范变换揭示其统计性质，最后聚焦于最小复杂模型实现快速模型选择。
摘要：
对高维数据进行建模具有挑战性，但对于理解许多复杂系统至关重要。最大熵模型（如Ising模型和Potts模型）已被广泛用于从数据中的相关模式捕获成对相互作用，从而能够从观测（例如，从蛋白质序列或神经群体活动）中推断复杂系统的图形表示。最近，人们对涉及三个或更多变量的高阶相关模式建模的兴趣日益增长。虽然在高阶Ising模型的二元数据方面取得了进展，但我们将此框架扩展到更一般的离散数据情况。我们引入了q态自旋模型，这是一个完整的最大熵模型族，将矢量Potts模型推广到包含长程和任意高阶相互作用。在成对情况下，与标准矢量Potts模型相比，我们的模型允许更多样化的相互作用类型。我们通过示例讨论了它们的统计解释，并将其与离散傅里叶分析联系起来。利用配分函数的圈展开，我们证明了自旋模型的统计性质完全由其相互作用的代数结构所捕获。我们定义了规范变换，在此变换下该结构（以及配分函数）保持不变。规范变换下等价的模型可以被视为同一抽象统计模型的不同表示，尽管通常具有不同阶数的相互作用，这扩展了二元情况的结果。对于数据分析的实际应用，我们专注于二元情况下称为最小复杂模型的一个子集，并将其推广到离散数据。我们获得了这些模型边际似然的闭式表达式，从而能够快速进行模型选择。我们通过简单的真实世界示例说明了它们的用途。
英文摘要：
Modeling high-dimensional data is challenging, yet essential to understanding many complex systems. Maximum entropy models such as Ising and Potts models have been used extensively to capture pairwise interactions from correlation patterns in data, allowing to infer graphical representations of complex systems from observations (e.g., from protein sequences or neural population activity). Recently, there has been growing interest in modeling higher-order correlation patterns involving simultaneously three or more variables. While progress has been made in binary data with high-order Ising models, we extend this framework to the more general case of discrete data. We introduce q-state spin models, a complete family of maximum entropy models that generalize the vector Potts model to include long-range and arbitrary high-order interactions. In the pairwise case, our models allow for more diverse interaction types compared to the standard vector Potts model. We discuss their statistical interpretation with examples and relate them to discrete Fourier analysis. Using a loop expansion of the partition function, we show that the statistical properties of spin models are fully captured by the algebraic structure of their interactions. We define gauge transformations under which this structure, and thus the partition function, remains invariant. Models equivalent under gauge transformations can be seen as different representations of the same abstract statistical model, despite generally having interactions of different orders, extending results from the binary case. For practical application to data analysis, we focus on a subset of models known in the binary case as Minimally Complex Models, generalizing them to discrete data. We obtain a closed-form expression for the marginal likelihood of these models, enabling fast model selection. We illustrate their use with simple real-world examples.
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