30 2. Backgroundanalogue to the remaining unpruned states of the pruning HMM. The unprunedstates at a time frame t determine the new set of unpruned states at t+ 1 byevaluation of a new xt. In the Particle Filter, the set of samples at a time frame tis equivalently used to generate the new set of samples for t+ 1 with respect to thenew observations of xt.The sampling process is defined recursively4. Let{ st( l)} be set of samples withcardinality L for a timeframe t and{ s0(l)} an initial set of samples that are drawnfrom a p( s0) distribution. Then, the sampling weights{ wt( l)} are defined by:wt( l)=p( xt| st( l))Lm=1p( xt| tt( m))The weight for a sample is therefore computed from the emission probabilitiesp( xt| st) that are obtained from the radio propagation models. The weights sat-isfy the normalization constraintlwt( l)= 1 and are in the range 0 wt( l) 1.They can be understood as a way to explain the importance of each sample underthe current observation xt. The posterior p( st| xt) is represented by the combinationof the samples and the corresponding weight for each sample. The next posteriorp( st+1| xt) is defined by combining the weights of t and the transition probabilitiesp( s| s) to the following mixture:Lp( st+1| xt)=wt( l)p( st+1| st)l=1By drawing new samples{ wt(+ l)1} from this distribution, the recursive step is com-pleted. Therefore, the algorithm can be interpreted as to switch between two modes.At first, the weight of the samples under the observation x is determined by usingthe stored p( x| s). The weight defines the number of new samples that are spawnedfrom this sample. In the context of this thesis: If the sample represents a very prob-able location for the RSSI readings, the sample location is used as a major sourcefor the next set of samples.The following set of samples is then generated by spawning the appropriate numberof samples at each source into the most probable direction/distance given by thetransition probability p( s| s).2.7 Least Mean Squared ErrorAnother localization method, that has been prominently used due to it's simplisticprinciples, is given by the so called Least Mean Squared Error(LMSE). Contrary tothe HMM and the PF based approaches, the method ignores the sequential natureof the measurements and is therefore not based on the state space model of figure2.11. The basic idea is given by comparing the RSSI reading of the D-dimensionalmeasurement vector x with all location s annotated measurements ysof a databaserepresenting the prior knowledge of the distribution. The LMSE uses the euclidian4A more detailed presentation of the PF can be found in[4].