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Solve the following problem using Python, implementing the functions described below, one line at a time
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Description:
 def estimate_hidden_markov_model(dtrajs, nstates, lag, reversible=True, stationary=False, connectivity=None, mincount_connectivity='1/n', separate=None, observe_nonempty=True, stride=1, dt_traj='1 step', accuracy=1e-3, maxit=1000):
 r""" Estimates a Hidden Markov state model from discrete trajectories Returns a :class:`MaximumLikelihoodHMSM` that contains a transition matrix between a few (hidden) metastable states. Each metastable state has a probability distribution of visiting the discrete 'microstates' contained in the input trajectories. The resulting object is a hidden Markov model that allows to compute a large number of quantities. Parameters dtrajs : list containing ndarrays(dtype=int) or ndarray(n, dtype=int) discrete trajectories, stored as integer ndarrays (arbitrary size) or a single ndarray for only one trajectory. lag : int lagtime for the MSM estimation in multiples of trajectory steps nstates : int the number of metastable states in the resulting HMM reversible : bool, optional, default = True If true compute reversible MSM, else non-reversible MSM stationary : bool, optional, default=False If True, the initial distribution of hidden states is self-consistently computed as the stationary distribution of the transition matrix. If False, it will be estimated from the starting states. Only set this to true if you're sure that the observation trajectories are initiated from a global equilibrium distribution. connectivity : str, optional, default = None Defines if the resulting HMM will be defined on all hidden states or on a connected subset. Connectivity is defined by counting only transitions with at least mincount_connectivity counts. If a subset of states is used, all estimated quantities (transition matrix, stationary distribution, etc) are only defined on this subset and are correspondingly smaller than nstates. Following modes are available: * None or 'all' : The active set is the full set of states. Estimation is done on all weakly connected subsets separately. The resulting transition matrix may be disconnected. * 'largest' : The active set is the largest reversibly connected set. * 'populous' : The active set is the reversibly connected set with most counts. mincount_connectivity : float or '1/n' minimum number of counts to consider a connection between two states. Counts lower than that will count zero in the connectivity check and may thus separate the resulting transition matrix. The default evaluates to 1/nstates. separate : None or iterable of int Force the given set of observed states to stay in a separate hidden state. The remaining nstates-1 states will be assigned by a metastable decomposition. observe_nonempty : bool If True, will restricted the observed states to the states that have at least one observation in the lagged input trajectories. dt_traj : str, optional, default='1 step' Description of the physical time corresponding to the trajectory time step. May be used by analysis algorithms such as plotting tools to pretty-print the axes. By default '1 step', i.e. there is no physical time unit. Specify by a number, whitespace and unit. Permitted units are (* is an arbitrary string):
 | 'fs', 'femtosecond*' | 'ps', 'picosecond*' | 'ns', 'nanosecond*' | 'us', 'microsecond*' | 'ms', 'millisecond*' | 's', 'second*' accuracy : float convergence threshold for EM iteration. When two the likelihood does not increase by more than accuracy, the iteration is stopped successfully. maxit : int stopping criterion for EM iteration. When so many iterations are performed without reaching the requested accuracy, the iteration is stopped without convergence (a warning is given) Returns ------- hmsm : :class:`MaximumLikelihoodHMSM <pyemma.msm.MaximumLikelihoodHMSM>` Estimator object containing the HMSM and estimation information. Example ------- We have estimated a 2x2 hidden transition matrix between the metastable states: [[ 0.684 0.316] [ 0.242 0.758]] With the equilibrium distribution: The observed states are the three discrete clusters that we have in our discrete trajectory: [0 1 2] The metastable distributions (mm.metastable_distributions), or equivalently the observation probabilities are the probability to be in a given cluster ('microstate') if we are in one of the hidden metastable states. So it's a 2 x 3 matrix: [[ 0.9620883 0.0379117 0. ] [ 0. 0.28014352 0.71985648]] The first metastable state ist mostly in cluster 0, and a little bit in the transition state cluster 1. The second metastable state is less well defined, but mostly in cluster 2 and less prominently in the transition state cluster 1. We can print the lifetimes of the metastable states: And the timescale of the hidden transition matrix - now we only have one relaxation timescale: The mean first passage times can also be computed between metastable states: .. autoclass:: pyemma.msm.estimators.maximum_likelihood_hmsm.MaximumLikelihoodHMSM :members: :undoc-members: .. rubric:: Methods .. autoautosummary:: pyemma.msm.estimators.maximum_likelihood_hmsm.MaximumLikelihoodHMSM :methods: .. rubric:: Attributes .. autoautosummary:: pyemma.msm.estimators.maximum_likelihood_hmsm.MaximumLikelihoodHMSM :attributes: References [1]_ is an excellent review of estimation algorithms for discrete Hidden Markov Models. This function estimates a discrete HMM on the discrete input states using the Baum-Welch algorithm [2]_. We use a maximum-likelihood Markov state model to initialize the HMM estimation as described in [3]_. .. [1] L. R. Rabiner: A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition. Proc. IEEE 77, 257-286 (1989) .. [2] L. Baum, T. Petrie, G. Soules and N. Weiss N: A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains. Ann. Math. Statist. 41, 164-171 (1970) .. [3] F. Noe, H. Wu, J.-H. Prinz and N. Plattner: Projected and hidden Markov models for calculating kinetics and metastable states of complex molecules. J. Chem. Phys. 139, 184114 (2013) """

    # initialize HMSM estimator
    hmsm_estimator = _ML_HMSM(lag=lag, nstates=nstates, reversible=reversible, stationary=stationary,
                              msm_init='largest-strong',
                              connectivity=connectivity, mincount_connectivity=mincount_connectivity, separate=separate,
                              observe_nonempty=observe_nonempty, stride=stride, dt_traj=dt_traj,
                              accuracy=accuracy, maxit=maxit)
    # run estimation
    return hmsm_estimator.estimate(dtrajs)