Patterns of interest in dynamical systems are often represented by a number of semantic features such as probabilistic finite state automata (PFSA) and cross machines over possibly different alphabets. Previous publications have reported a Hilbert space formulation of PFSA over the same alphabet. This paper introduces an isomorphism between the Hilbert space of PFSA and the Euclidean space to improve the computational efficiency of algebraic operations. Furthermore, this formulation is extended to cross machines and it shows that these semantic features can be structured in a unified mathematical framework. In this framework, an algorithm of supervised learning is formulated for generating semantic features in the setting of linear discriminant analysis (LDA). The proposed algorithm has the flexibility for adaptation under different environments by tuning a set of parameters that can be updated autonomously or be specified by the human user. The proposed algorithm has been validated on real-life data for target detection as applied to border control.