28 2. BackgroundFigure 2.11 The general state space model is the base for the discrete or continuousrepresentation of the sequential nature for a stochastic process. The progress of thestochastic process, a hidden state sequence sT1, can only be observed by measuringa sequence of noisy feature vectors x1T.2.5.6 TrainingFor learning the parameters of the transition and emission probabilities of a HMM,a corpus of state-annotated measurement sequences is needed. If such a corpus isavailable, inference of the parameters of the stochastic models can be performedwith the Baum-Welch algorithm. This algorithm represents the adoption of thebasic assumptions of the estimate-maximize algorithm to sequential data.But during the development of the presented system, no such training corpus for theTracking problem was available. Therefore, the complex intrinsics of the inferencealgorithm should not be presented here(see[4] for details).2.6 Continuous ModelsContinuous models are the generalization of models with a discrete state space, likethe HMM, to a continuous state space. Therefore, the once discrete PDF for p( s| s)is now modelled as a continuous PDF, often as a Gaussian. But that is the onlyconceptual difference to the HMM model, as both are based on the state spacemodel of figure 2.11. The memorylessnes, analogue to the Markov property, and theconcept of observable emission probabilities are preserved. So the same factorizationfor the joint probability p( s1T, x1T) under the first order Markov assumptions can bederived:Tp( sT1, xT1)= p( st| st- 1) · p( xt| st)t=1with the difference of a now continuously distributed p( st| st- 1).2.6.1 Linear Dynamic SystemA Linear Dynamic System(LDS), also known under the name Kalman Filter , issuch a continuous state space model that is briefly motivated before continuing withmore flexible concept of the Particle Filter.The motivation for this approach arises from the following practical problem. Anunknown quantity s should be measured by a noisy sensor. The measured observa-tion x represents the underlying s distorted by zero-mean Gaussian. If a sequence