Hidden Markov models (HMMs) are one of the most popular and successful statistical models for time series. Observable operator models (OOMs) are generalizations of HMMs that exhibit several attractive advantages. In particular, a variety of highly efficient, constructive, and asymptotically correct learning algorithms are available for OOMs. However, the OOM theory suffers from the negative probability problem (NPP): a given, learned OOM may sometimes predict negative probabilities for certain events. It was recently shown that it is undecidable whether a given OOM will eventually produce such negative values. We propose a novel variant of OOMs, called norm-observable operator models (NOOMs), which avoid the NPP by design. Like OOMs, NOOMs use a set of linear operators to update system states. But differing from OOMs, they represent probabilities by the square of the norm of system states, thus precluding negative probability values. While being free of the NPP, NOOMs retain most advantages of OOMs. For example, NOOMs also capture (some) processes that cannot be modeled by HMMs. More importantly, in principle, NOOMs can be learned from data in a constructive way, and the learned models are asymptotically correct. We also prove that NOOMs capture all Markov chain (MC) describable processes. This letter presents the mathematical foundations of NOOMs, discusses the expressiveness of the model class, and explains how a NOOM can be estimated from data constructively.

Observable operator models (OOMs) are a class of models for stochastic processes that properly subsumes the class that can be modeled by finite-dimensional hidden Markov models (HMMs). One of the main advantages of OOMs over HMMs is that they admit asymptotically correct learning algorithms. A series of learning algorithms has been developed, with increasing computational and statistical efficiency, whose recent culmination was the error-controlling (EC) algorithm developed by the first author. The EC algorithm is an iterative, asymptotically correct algorithm that yields (and minimizes) an assured upper bound on the modeling error. The run time is faster by at least one order of magnitude than EM-based HMM learning algorithms and yields significantly more accurate models than the latter. Here we present a significant improvement of the EC algorithm: the constructive error-controlling (CEC) algorithm. CEC inherits from EC the main idea of minimizing an upper bound on the modeling error but is constructive where EC needs iterations. As a consequence, we obtain further gains in learning speed without loss in modeling accuracy.

Observable operator models (OOMs) generalize hidden Markov models (HMMs) and can be represented in a structurally similar matrix formalism. The mathematical theory of OOMs gives rise to a family of constructive, fast, and asymptotically correct learning algorithms, whose statistical efficiency, however, depends crucially on the optimization of two auxiliary transformation matrices. This optimization task is nontrivial; indeed, even formulating computationally accessible optimality criteria is not easy. Here we derive how a bound on the modeling error of an OOM can be expressed in terms of these auxiliary matrices, which in turn yields an optimization procedure for them and finally affords us with a complete learning algorithm: the error-controlling algorithm. Models learned by this algorithm have an assured error bound on their parameters. The performance of this algorithm is illuminated by comparisons with two types of HMMs trained by the expectation-maximization algorithm, with the efficiency-sharpening algorithm, another recently found learning algorithm for OOMs, and with predictive state representations (Littman & Sutton, 2001 ) trained by methods representing the state of the art in that field.