Diffusion Processes with Hidden States from ... - FU Berlin, FB MI

3.11 Hidden Markov Model - Vector Auto Regression (HMM-VAR) as Generalization of HMM-SDEGiven these, we proceed the way we did in the last section and find ˆλ = argmaxQ(O|λ) to beuniquely given byλˆπ (i) = α 1(i)β 1 (i)∑ M i=1 α 1(i)β 1 (i) ,ˆ T i j=∑T−1t=0 α t(i)T (i, j)ρ(O t |q t ,O t−1 ,λ)β t+1 ( j)∑ T −1t=0 α t(i)β t (i)ˆB (i) = ∑T t=2 α t(i)β t (i)[−O t + O t−1 ] 2τ ∑t=0 T α . (3.85)t(i)β t (i),3.11 Hidden Markov Model - Vector Auto Regression(HMM-VAR) as Generalization of HMM-SDE(This section is from [21]).Since we have established in the last chapter the parameter estimation of a one-dimensional SDE,we have to investigate now the generalization of the HMM-SDE approach. This generalization isbased on a d-dimensional SDE of the formż = F(z − µ) + ΣẆ. (3.86)The parameter estimation is obtained by investigation of an appropriate likelihood function. Thisextension is performed in [33]. A more generalized approach is based on Vector Auto regression(VAR) processes ([24]), here the interpretation of a SDE as VAR process is given and a comprehensionis presented. A detailed analysis of the difficulties of parameter estimation of generalizedmultidimensional Langevin processes is given in [18].The trajectory z t with t ∈ {1, ... ,T ′ } is given at discrete points in time. According to the equation(3.86) and given an observation z t , the conditional probability density of z t+1 is a Gaussian withthe density functionf λ (z t+1 |z t ) =[1√ exp − 1 ]|2πR(τ)| 2 (z t+1 − µ t ) T R(τ) −1 (z t+1 − µ t ) , (3.87)with the mean and variance of the distribution given asandµ t := µ + (z t − µ)exp(τF)ˆR(τ) :=dsexp(sF)ΣΣ T exp ( sF T ) .43