Distance measures for nonparametric weak process models

Nonparametric versions of hidden Markov models, what we call weak models, are robust for process detection and easy to construct, as the assumption of knowing precise probabilities in HMMs is weakened to {0,1}-values of reachabilities. Weak models are shown to be equivalent to DFAs/ NFAs. The concept of minimal unifilar weak model (μ-WM) is introduced. The spectral radius of the transition matrix of μ-WM determines the growth rate of acceptable observation sequences. An absolute weak model distance is defined for model clustering purpose, while a relative distance is a measure of how fast the performance of detection gets improved as more observations arrive. Convergence of the distance measures is proved.