2.8. Summary 31distance for the comparison and decides for the location with the least distance. TheLMSE decision rule rlmse: x s is thus defined as:rlmse( x)= argmins1DD( xd- ysd)2d=1D= argmin( xd- ysd)2sd=1The method is derived from the Bayes Decision Rule that has been introduced insection 2.4:rbayes( x)= argmax[ p( x| s) p( s)]sAll locations are equally probable, and the calculation is done in logspace:rbayes( x)= argmax[ log( p( x| s))]sThe location-conditional p( x| s) is modelled as a Gaussian with independent compo-nents: p( x| s) N( µ,) . Furthermore, by assuming a constant pooled variance overall locations and removing the constant coefficients of the Gaussian5, this leads to:Drbayes( x)= argmax-( xd- µsd)2sd=1D= argmin( xd- µsd)2sd=1Therefore, under the given assumptions for p( x| s) and if ys= µsthe followingequality holds:rlmse( x)= rbayes( x)2.8 SummaryA thorough introduction into the general principles that are employed to approachthe localization problem and the radio propagation modelling was given. TheBayesian approach to the pattern recognition problem was introduced and threealgorithms, the HMM , the PF and the LMSE technique were derived. For the topicof radio propagation, the basic background information needed for understanding theprimary physical effects that influence the RSSI signal distribution were presented.Building on that foundations, a brief overview over the mechanics of employed PHO-TON system was given.5A similar transformation into logspace has been done with the HMM emission probabilitiesp( x| s) in section 2.5.4. A pooled variance, has been used there as well.