def init_model_gaussian1d(observations, nstates, reversible=True):
    """Generate an initial model with 1D-Gaussian output densities

    Parameters
    ----------
    observations : list of ndarray((T_i), dtype=float)
        list of arrays of length T_i with observation data
    nstates : int
        The number of states.

    Examples
    --------

    Generate initial model for a gaussian output model.

    >>> from bhmm import testsystems
    >>> [model, observations, states] = testsystems.generate_synthetic_observations(output='gaussian')
    >>> initial_model = init_model_gaussian1d(observations, model.nstates)

    """
    ntrajectories = len(observations)

    # Concatenate all observations.
    collected_observations = np.array([], dtype=config.dtype)
    for o_t in observations:
        collected_observations = np.append(collected_observations, o_t)

    # Fit a Gaussian mixture model to obtain emission distributions and state stationary probabilities.
    from bhmm._external.sklearn import mixture
    gmm = mixture.GMM(n_components=nstates)
    gmm.fit(collected_observations[:,None])
    from bhmm import GaussianOutputModel
    output_model = GaussianOutputModel(nstates, means=gmm.means_[:,0], sigmas=np.sqrt(gmm.covars_[:,0]))

    logger().info("Gaussian output model:\n"+str(output_model))

    # Extract stationary distributions.
    Pi = np.zeros([nstates], np.float64)
    Pi[:] = gmm.weights_[:]

    logger().info("GMM weights: %s" % str(gmm.weights_))

    # Compute fractional state memberships.
    Nij = np.zeros([nstates, nstates], np.float64)
    for o_t in observations:
        # length of trajectory
        T = o_t.shape[0]
        # output probability
        pobs = output_model.p_obs(o_t)
        # normalize
        pobs /= pobs.sum(axis=1)[:,None]
        # Accumulate fractional transition counts from this trajectory.
        for t in range(T-1):
            Nij[:,:] = Nij[:,:] + np.outer(pobs[t,:], pobs[t+1,:])

        logger().info("Nij\n"+str(Nij))

    # Compute transition matrix maximum likelihood estimate.
    import msmtools.estimation as msmest
    import msmtools.analysis as msmana
    Tij = msmest.transition_matrix(Nij, reversible=reversible)
    pi = msmana.stationary_distribution(Tij)

    # Update model.
    model = HMM(pi, Tij, output_model)

    return model
Generate an initial model with 1D-Gaussian output densities

    Parameters
    ----------
    observations : list of ndarray((T_i), dtype=float)
        list of arrays of length T_i with observation data
    nstates : int
        The number of states.

    Examples
    --------

    Generate initial model for a gaussian output model.

    >>> from bhmm import testsystems
    >>> [model, observations, states] = testsystems.generate_synthetic_observations(output='gaussian')
    >>> initial_model = init_model_gaussian1d(observations, model.nstates)