26 2. BackgroundFigure 2.10 The states of a second order HMM carry the history of one predecessorstate(first index) and the current state(second index). Due to this contextualizationthe number of states is increased by the factor 2, the number of transitions. Equalcolors represent equal current states.multiplications leads to numerical underflows. The classical approach to retain nu-merical stability in the recursions is to transform the probabilities into the negativelogspace. Doing so, leads to a reinterpretation of the probabilities to costs. Insteadof searching for the state sequence with the maximum probability, the task is nowtransformed to a search for the minimum costs. It should be noted, that applyingthe logarithm to a function is an invariant operation with respect to the maximum.Therefore, the decision for the resulting sequence will not be influenced.So starting with the dynamic programming recursion for the local decisions:Q( t, s):= max[ p( xt, s| s) · Q( t- 1, s)]sand applying the logarithm leads to:Q( t, s):= max[log( p( xt, s| s))+ Q( t- 1, s)]swith further interpreting as minimum costs:Q( t, s):= min[- log( p( xt, s| s))+ Q( t- 1, s)]s:= min[ Q( t- 1, s)- log( p( xt| s))- log( p( s| s))]sIt can be seen, that the evaluation of Q( t, s) is performed very efficient in logspace.The evaluation of the costs for a candidate state s is a simple summing of the storedtransition probability- log( p( s| s)) and the calculation of the costs of the emissionprobability.In the presented system, the emission probabilities are modelled as a multivariateGaussian distribution N( µ, ) with independent components. An input feature vec-tor x for a time frame t is D-dimensional. Therefore, the vector represents a list