mmpp              package:HiddenMarkov              R Documentation

_M_a_r_k_o_v _M_o_d_u_l_a_t_e_d _P_o_i_s_s_o_n _P_r_o_c_e_s_s _O_b_j_e_c_t

_D_e_s_c_r_i_p_t_i_o_n:

     Creates a Markov modulated Poisson process model object with class
     '"mmpp"'.

_U_s_a_g_e:

     mmpp(tau, Q, delta, lambda, nonstat = TRUE)

_A_r_g_u_m_e_n_t_s:

     tau: vector containing the _event times_. Note that the first
          event is at time zero. Alternatively, 'tau' could be
          specified as 'NULL', meaning that the data will be added
          later (e.g. simulated).

       Q: the infinitesimal generator matrix of the Markov process.

   delta: is the marginal probability distribution of the m hidden
          states at time zero.

  lambda: a vector containing the Poisson rates.

 nonstat: is logical, 'TRUE' if the homogeneous Markov process is
          assumed to be non-stationary, default.

_D_e_t_a_i_l_s:

     The Markov modulated Poisson process is based on a hidden Markov
     process in continuous time. The initial state probabilities (at
     time zero) are specified by 'delta' and the transition rates by
     the 'Q' matrix. The rate parameter of the Poisson process
     ('lambda') is determined by the current state of the hidden Markov
     process. Within each state, the Poisson process is homogeneous
     (constant rate parameter). A Poisson event is assumed to occur at
     time zero and at the end of the observation period, however, state
     transitions of the Markov process do not necessarily coincide with
     Poisson events. For more details, see Ryden (1996).

_V_a_l_u_e:

     A 'list' object with class '"mmpp"', containing the above
     arguments as named components.

_R_e_f_e_r_e_n_c_e_s:

     Klemm, A.; Lindemann, C. & Lohmann, M. (2003). Modeling IP traffic
     using the batch Markovian arrival process. _Performance
     Evaluation_ *54(2)*, 149-173.

     Roberts, W.J.J.; Ephraim, Y. & Dieguez, E. (2006). On Ryden's EM
     algorithm for estimating MMPPs. _IEEE Signal Processing Letters_
     *13(6)*, 373-376.

     Ryden, T. (1994). Parameter estimation for Markov modulated
     Poisson processes. _Stochastic Models_ *10(4)*, 795-829.

     Ryden, T. (1996). An EM algorithm for estimation in
     Markov-modulated Poisson processes. _Computational Statistics &
     Data Analysis_ *21(4)*, 431-447.

_E_x_a_m_p_l_e_s:

     Q <- matrix(c(-2,  2,
                    1, -1),
                 byrow=TRUE, nrow=2)/10

     #    NULL indicates that we have no data at this point
     x <- mmpp(NULL, Q, delta=c(0, 1), lambda=c(5, 1))

     x <- simulate(x, nsim=1000, seed=5)

     x$delta <- c(0.5, 0.5)
     y <- BaumWelch(x)

     print(summary(y))

     #    log-likelihood using initial parameter values
     print(logLik(x))

     #    log-likelihood using estimated parameter values
     print(logLik(y))

