# negative penalised HSMM log-likelihood function Evaluates the negative penalised log-likelihood function of the HSMM (internal function, called by the function `pmleHSMM`). ```r npllHSMM(parvect ,N, y, R_vec, lambda, order_diff, y_dist=c("norm","gamma","pois","bern"), stationary=TRUE, T_y, p_ref=2) ``` ## Arguments - `parvect`: vector of unconstraint working parameter as returned by the function `n2wHSMM`. - `N`: number of states of the HSMM, integer greater than 1. - `y`: vector containing the observed time series. - `R_vec`: vector of length `N` containing the lengths of the unstructured starts of the dwell-time distributions. - `lambda`: vector of length `N` containing the smoothing parameter values to weight the penalty term. - `order_diff`: order of the differences used for the penalty term, positive integer which does not exceed the length of the unstructured starts. - `y_dist`: character determining the class of state-dependent distributions. Supported values are `"norm"` (normal distribution), `"gamma"` (gamma distribution), `"pois"` (Poisson distribution) and `"bern"` (Bernoulli distribution). - `stationary`: Logical, if `TRUE` (default), stationarity is assumed, if `FALSE`, the underlying state-sequence is assumed to enter a new state at time t=1. - `T_y`: length of the observed time series. - `p_ref`: positive integer determining the reference dwell-time probability used for the multinomial logit parameter transformation. Default value is 2. Only needs to be changed if the dwell-time probability for dwell time r=2 is estimated very close to zero in order to avoid numerical problems. ## Returns Returns the value of the negative penalised HSMM log-likelihood function for the given parameters and time series. ## Details The penalised log-likelihood function relies on the exact HMM representation of the HSMM and is evaluated using the forward algorithm which is implemented in `C++` to speed up the calculation. ## References Pohle, J., Adam, T. and Beumer, L.T. (2021): Flexible estimation of the state dwell-time distribution in hidden semi-Markov models. arXiv:[https://arxiv.org/abs/2101.09197](https://arxiv.org/abs/2101.09197). Zucchini, W., MacDonald, I.L. and Langrock, R. (2016): **Hidden Markov models for time series: An introduction using R**. 2nd edition. Chapman & Hall/CRC, Boca Raton. ## Examples ```r # 3-state gamma HSMM and hourly muskox step length # natural parameters p_list0<-list() p_list0[[1]]<-c(dgeom(0:9,0.2),1-pgeom(9,0.2)) p_list0[[2]]<-c(dgeom(0:9,0.2),1-pgeom(9,0.2)) p_list0[[3]]<-c(dgeom(0:9,0.2),1-pgeom(9,0.2)) omega0<-matrix(0.5,3,3) diag(omega0)<-0 mu0<-c(5,100,350) sigma0<-c(3,90,300) R_vec<-sapply(p_list0,length)-1 # lengths of the unstructured starts # working parameter vector parvect<-n2wHSMM(N=3,p_list=p_list0,mu=mu0,sigma=sigma0, omega=omega0,y_dist='gamma') # evaluate the negative penalised log-likelihood function npllHSMM(parvect,N=3,muskox$step,R_vec=R_vec,lambda=c(1000,1000,1000), order_diff=2,y_dist="gamma",T_y=nrow(muskox)) ```

npllHSMM function