2.6. Continuous Models 29of measurements is available in the form of xT1, the hidden value of s can be easilyestimated by averaging of the measurements.The problem will again become more complex if the quantity s is allowed to changeover time. An obvious approach would be to use some form of historical averageof the last measurements to estimate the current true value of s. But how farshould that evaluated window of measurements reach back? If the value for s ischanging slowly over time, a large window capturing a lot of valuable informationis appropriate. But if the value for s is changing fast, this leads to averaging overhigh variety information and therefore introduce a new form of error. Here, a shortmoving average window is better. This approach can be made more complicated byintroducing a weighed average of the history, probably giving more recent ones moreweight. This reasoning relates to the need for state-conditional probabilities p( x| s),now the transitions will be investigated.The other source of information, the transition probability p( s| s) is modelled by anon-stationary multivariate linear-Gaussian distribution:p( s| s)= N( s| As, B)where A is the linear transformation representing the movement vector and B theinitial mean and variance. The requirement of a linearly changing state sequence isgiven by the need for integration instead of summing during finding the marginalsof p( x1T| s1T). A similar restriction is enforced on the emission probabilities. Onlysimple PDFs like Gaussians are allowed for p( x| s). Mixtures densities lead also toproblems during integration due to their contained summations.These limitations were addressed by the introduction variants of that approach. Forexample, the extended Kalman filter[14] or the also preseneted Particle Filter, dropsthe restriction of linearity in the state changes.For a more detailed introduction of the LDS technique, it should be referred to[21]and[4].2.6.2 Particle FilterOne of the more recent approaches to the tracking problem, that was also evaluatedin this thesis, is given by the Particle Filter(PF)[7]. The model is a responseto the problem that estimating the non-linear movement of a person in an indoorlocalization scenario with the linear Kalman Filter is prone to fail. The concepts ofthe PF are based on the idea to sample the posterior distribution p( st| xt) directlyfor a given time frame t. In the HMM formalism, this posterior distribution isapproximated by the recursion equation Q( s, t). And due to the enumerability ofthe discrete states, the Viterbi Algorithm is able to compute p( st| xt) and can thusdecide locally for the state stwith maximum probability. The goal of this procedurecan also be described as a marking of the most probable states or locations given anobserved measurement sequence.The same marking of probable states in a continuous space is conducted if p( st| xt)is sampled under the conditions of the observations. By generating enough samples,the best state hypothesis will get the most probability mass. This is in a way an