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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 dtram( ttrajs, dtrajs, bias, lag, unbiased_state=None, count_mode='sliding', connectivity='reversible_pathways', maxiter=10000, maxerr=1.0E-15, save_convergence_info=0, dt_traj='1 step', init=None, init_maxiter=10000, init_maxerr=1.0E-8):
 r""" Discrete transition-based reweighting analysis method Parameters ttrajs : numpy.ndarray(T) of int, or list of numpy.ndarray(T_i) of int A single discrete trajectory or a list of discrete trajectories. The integers are in at any time. dtrajs : numpy.ndarray(T) of int, or list of numpy.ndarray(T_i) of int A single discrete trajectory or a list of discrete trajectories. The integers are indexes trajectory is in at any time. bias : numpy.ndarray(shape=(num_therm_states, num_conf_states)) object bias_energies_full[j, i] is the bias energy in units of kT for each discrete state i at thermodynamic state j. lag : int or list of int, optional, default=1 Integer lag time at which transitions are counted. Providing a list of lag times will trigger one estimation per lag time. unbiased_state : int, optional, default=None Index of the unbiased thermodynamic state or None if there is no unbiased data available. count_mode : str, optional, default='sliding' Mode to obtain count matrices from discrete trajectories. Should be one of: * 'sliding' : a trajectory of length T will have :math:`T-\tau` counts at time indexes .. math:: * 'sample' : a trajectory of length T will have :math:`T/\tau` counts at time indexes .. math:: Currently only 'sliding' is supported. connectivity : str, optional, default='reversible_pathways' One of 'reversible_pathways', 'summed_count_matrix' or None. Defines what should be considered a connected set in the joint (product) space of conformations and thermodynamic ensembles. * 'reversible_pathways' : requires that every state in the connected set can be reached by following a pathway of reversible transitions. A reversible transition between two Markov states (within the same thermodynamic state k) is a pair of Markov states that belong to the same strongly connected component of the count matrix (from thermodynamic state k). A pathway of reversible transitions is a list of (i_(N-1), i_N)]. The thermodynamic state where the reversible transitions happen, is ignored in constructing the reversible pathways. This is equivalent to assuming that two ensembles overlap at some Markov state whenever there exist frames from both ensembles in that Markov state. * 'summed_count_matrix' : all thermodynamic states are assumed to overlap. The connected set is then computed by summing the count matrices over all thermodynamic states and taking it's largest strongly connected set. Not recommended! * None : assume that everything is connected. For debugging. For more details see :func:`pyemma.thermo.extensions.cset.compute_csets_dTRAM`. maxiter : int, optional, default=10000 The maximum number of dTRAM iterations before the estimator exits unsuccessfully. maxerr : float, optional, default=1e-15 Convergence criterion based on the maximal free energy change in a self-consistent iteration step. save_convergence_info : int, optional, default=0 Every save_convergence_info iteration steps, store the actual increment and the actual loglikelihood; 0 means no storage. dt_traj : str, optional, default='1 step' Description of the physical time corresponding to the lag. 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*' init : str, optional, default=None Use a specific initialization for self-consistent iteration: | None: use a hard-coded guess for free energies and Lagrangian multipliers | 'wham': perform a short WHAM estimate to initialize the free energies init_maxiter : int, optional, default=10000 The maximum number of self-consistent iterations during the initialization. init_maxerr : float, optional, default=1.0E-8 Convergence criterion for the initialization. Returns ------- A :class:`MEMM <pyemma.thermo.models.memm.MEMM>` object or list thereof A multi-ensemble Markov state model (for each given lag time) which consists of stationary and kinetic quantities at all temperatures/thermodynamic states. Example ------- **Umbrella sampling**: Suppose we simulate in K umbrellas, centered at .. math:: b_k(x) = \frac{c_k}{2 \textrm{kT}} \cdot (x - y_k)^2 Suppose we have one simulation of length T in each umbrella, and they are ordered from 0 to K-1. We have discretized the x-coordinate into 100 bins. Then dtrajs and ttrajs should each be a list of :math:`K` arrays. dtrajs would look for example like this:: where each array has length T, and is the sequence of bins (in the range 0 to 99) visited along the trajectory. ttrajs would look like this:: Because trajectory 1 stays in umbrella 1 (index 0), trajectory 2 stays in umbrella 2 (index 1), and so forth. bias is a :math:`K \times n` matrix with all reduced bias energies evaluated at all centers: .. math:: \left(\begin{array}{cccc} \end{array}\right) Let us try the above example: array([[[5, 1], [1, 2]], [[1, 4], [3, 1]]], dtype=int32) See :class:`MEMM <pyemma.thermo.models.memm.MEMM>` for a full documentation. .. autoclass:: pyemma.thermo.models.memm.MEMM :members: :undoc-members: .. rubric:: Methods .. autoautosummary:: pyemma.thermo.models.memm.MEMM :methods: .. rubric:: Attributes .. autoautosummary:: pyemma.thermo.models.memm.MEMM :attributes: References .. [1] Wu, H. et al 2014 Statistically optimal analysis of state-discretized trajectory data from multiple thermodynamic states J. Chem. Phys. 141, 214106 """

    # prepare trajectories
    ttrajs = _types.ensure_dtraj_list(ttrajs)
    dtrajs = _types.ensure_dtraj_list(dtrajs)
    if len(ttrajs) != len(dtrajs):
        raise ValueError("Unmatching number of dtraj/ttraj elements: %d!=%d" % (
            len(dtrajs), len(ttrajs)) )
    for ttraj, dtraj in zip(ttrajs, dtrajs):
        if len(ttraj) != len(dtraj):
            raise ValueError("Unmatching number of data points in ttraj/dtraj: %d!=%d" % (
                len(ttraj), len(dtraj)))
    # check lag time(s)
    lags = _np.asarray(lag, dtype=_np.intc).reshape((-1,)).tolist()
    # build DTRAM and run estimation
    from pyemma.thermo import DTRAM
    from pyemma._base.progress import ProgressReporter
    pg = ProgressReporter()
    pg.register(len(lags), description='Estimating DTRAM for lags')
    dtram_estimators = []
    with pg.context():
        for _lag in lags:
            d = DTRAM(
                bias, _lag,
                count_mode=count_mode, connectivity=connectivity,
                maxiter=maxiter, maxerr=maxerr, save_convergence_info=save_convergence_info,
                dt_traj=dt_traj, init=init, init_maxiter=init_maxiter,
                init_maxerr=init_maxerr).estimate((ttrajs, dtrajs))
            dtram_estimators.append(d)
            pg.update(1)
    _assign_unbiased_state_label(dtram_estimators, unbiased_state)
    # return
    if len(dtram_estimators) == 1:
        return dtram_estimators[0]
    return dtram_estimators