We introduce Graphical Quadratic Algebra (GQA), a string diagrammatic calculus extending the language of Graphical Affine Algebra with a new generator characterised by invariance under rotation matrices. We show that GQA is a sound and complete axiomatisation for three different models: quadratic relations, which are a compositional formalism for least-squares problems, Gaussian stochastic processes, and Gaussian stochastic processes extended with non-determinisms. The equational theory of GQA sheds light on the connections between these perspectives, giving an algebraic interpretation to the interplay of stochastic behaviour, relational behaviour, non-determinism, and conditioning. As applications, we discuss various case studies, including linear regression, probabilistic programming, and electrical circuits with realistic (noisy) components.

翻译：我们引入图形二次代数（Graphical Quadratic Algebra，GQA），这是一种字符串图演算，通过添加一种由旋转矩阵不变性刻画的新生成元，扩展了图形仿射代数的语言。我们证明，GQA 对于三种不同的模型是可靠且完备的公理化体系：二次关系（一种用于最小二乘问题的组合形式）、高斯随机过程，以及扩展了非确定性的高斯随机过程。GQA 的等式理论揭示了这些视角之间的联系，为随机行为、关系行为、非确定性以及条件化之间的相互作用提供了代数解释。作为应用，我们讨论了多个案例研究，包括线性回归、概率编程以及具有实际（含噪声）元件的电路。