MingJie Zhao, Herbert Jaeger

• Observable operator models (OOMs), a recently developed matrix model class of stochastic processes [6], possesses several advantages over hidden Markov models (HMMs). Nevertheless, there is a critical issue, the negative probability problem (NPP), which remains unsolved in OOMs theory; and which has heavily prevented it from being an alternative to HMMs in practice. To avoid the NPP we introduce in this report a variation of OOM, the norm observable operator models (norm-OOMs). Like OOMs, norm-OOMs describe stochastic processes also using linear observable operators. But norm-OOMs differ from OOMs in that they employ a nonlinear function acting on the state vectors, instead of the linear one used by OOMs, to compute probabilities. Under this nonlinear map, the family of all probability distributions can be embedded into a special inner product space. This provides novel insights into the relationship between the stochastic processes theory and linear algebra; and enables us to study stochastic processes by concepts and methods from linear algebra, a more convenient field of mathematics. In this report the basic theory of norm-OOMs is set up; an iterative method for learning norm-OOMs is developed based upon the maximum likelihood (ML) principle; the advantages and limitations of norm-OOMs are discussed; and some problems for future investigation are outlined.