4.3. Positioning and Tracking 554.3.1.5 Result SequenceThe Viterbi Decoder can return three different result sequences. The first is the bestsequence s1Tfor all given measurements x1T. This sequence represents the result ofan offline run. Furthermore, the algorithm returns the sequence containing the beststate hypothesis stfor a time frame t during the viterbi decoding and representsthe results of an online run. By averaging the position over the top N locationhypothesis for a time frame t, the third result sequence sT1is defined and called theaveraged online result. The offline s1Thas a higher probability of correctness thanthe online variants since the latter has less information to rely on, as the futuremeasurements are excluded.4.3.2 Particle FilterThe localization problem can also be approached by modelling the states as contin-uous variables in the state space model(Figure 2.11). Such an approach is given bythe Particle Filter. As in the HMM approach, the most likely sequence of states sT1for an observed measurement sequence x1Thas to be found. But instead of searchingthis sequence by efficiently enumerating and evaluating the possible hypotheses, thesequence is generated by an iterative sampling process.Although a basic assumption of the PF are continuous states, the system mapsthem into a discrete voxel space. Therefore, most of the data-structures from theHMM design can be reused. Voxels with 40 cm edge size have shown to represent aresolution for reaching optimal results.4.3.2.1 Emission ProbabilitiesThe parameters of the emission probabilities p( x| s) are similar to the HMM design.p( x| s) is a Gaussian with a mean vector that is extracted from the radio propagationmodels by averaging over the RSSI values of corresponding voxel groups. The PFdoes not use the simplified logspace approach of the HMM model, and does thereforenot reduce the emission probabilities to distance based cost values. The full gaussianis evaluated and a signal variance of 5 dBm has been empirically determined tolead to good results. Missing components of the input signal vector xthave beeninterpolated from neighbouring signals as described in the related HMM chapter4.3.1.2.4.3.2.2 Transition ProbabilitiesThe transition probability p( s| s) is modelled as a multivariate zero-mean Gaussianwith independent components. Three dimensions of the Gaussian represent the threeaxis in free space. The variance of the horizontal components has been set to 5 m andthe variance of the vertical axis to 2 m. The reduced variability in the vertical axiscan be justified by the lower probability to move in that direction. Both values havebeen empirically determined by testing the PF on location annotated measurements.