22 2. BackgroundFigure 2.7 In a Hidden Markov Model the states are not directly observable. Butthe output of the states can be measured as sequence of feature vectors. From theseobservations the best matching state sequence s1Tcan be recovered.compensating distortions in the speed of the stochastic process. If the process isonly developing slowly, for example a slowly moving localization target, more 0-transitions can be used. In a 0-transition the process remains in the present state.The opposite effect have 2-transitions. They are employed to model acceleratedphases of the stochastic process. Elevating the one dimensional(0, 1, 2)-model intothe three dimensional localization space with six degrees of freedom leads to therather clumsy notation of((0, 1, 2)1,(0, 1, 2)2,..,(0, 1, 2)6)-model. By combining the0-transitions and counting only the number of predecessors on each dimension thisshould be called(5, 5, 5)-model. The(5, 5, 5)-model is parametrized by 125 possibletransitions for each state.In a Hidden Markov Model(HMM ), the states of the Markov chain are not directlyobservable. But the output of the states, also called the emission of the states,is visible. The visible emission x is coupled to the hidden state s through theprobability distribution p( x| s). Therefore, by observing a sequence of measurementsand relating them to the emission probabilities, the HMM can be used for assigningprobabilities to different hypothesized hidden sequences which on their part are listsof locations.This emission probability p( x| s) represents the state-conditional probability in theBayesian approach and has to be learned from the environment. In the context oflocalization, the state s represents a position in space. Therefore, it is understoodas a model for the probability to receive a special signal constellation at a givenposition. Emission probabilities are often modelled as multivariate Gaussians ormore complex mixture densities. The chosen model properties for the emissions inthis thesis are discussed in the later section 2.5.4, where the simplification of thementioned Gaussians is of concern.