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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 tram( ttrajs, dtrajs, bias, lag, unbiased_state=None, count_mode='sliding', connectivity='post_hoc_RE', maxiter=10000, maxerr=1.0E-15, save_convergence_info=0, dt_traj='1 step', connectivity_factor=1.0, nn=None, direct_space=False, N_dtram_accelerations=0, callback=None, init='mbar', init_maxiter=10000, init_maxerr=1e-8, equilibrium=None, overcounting_factor=1.0):
 r""" Transition-based reweighting analysis method Parameters ttrajs : numpy.ndarray(T), or list of numpy.ndarray(T_i) 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(T, num_therm_states), or list of numpy.ndarray(T_i, num_therm_states) A single reduced bias energy trajectory or a list of reduced bias energy trajectories. For every simulation frame seen in trajectory i and time step t, btrajs[i][t, k] is the reduced bias energy of that frame evaluated in the k'th thermodynamic state (i.e. at the k'th umbrella/Hamiltonian/temperature) 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. 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*' connectivity : str, optional, default='post_hoc_RE' One of 'post_hoc_RE', 'BAR_variance', 'reversible_pathways' or 'summed_count_matrix'. 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. * 'post_hoc_RE' : similar to 'reversible_pathways' but with a more strict requirement for the overlap between thermodynamic states. It is required that every state in the connected set can be reached by following a pathway of reversible transitions or jumping between overlapping thermodynamic states while staying in the same Markov state. 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). Two thermodynamic states k and l are defined to overlap at Markov state n if a replica exchange simulation [2]_ restricted to state n would show at least one transition from k to l or one transition from from l to k. The expected number of replica exchanges is estimated from the simulation data. The minimal number required of replica exchanges per Markov state can be increased by decreasing `connectivity_factor`. * 'BAR_variance' : like 'post_hoc_RE' but with a different condition to define the thermodynamic overlap based on the variance of the BAR estimator [3]_. Two thermodynamic states k and l are defined to overlap at Markov state n if the variance of the free energy difference Delta f_{kl} computed with BAR (and restricted to conformations form Markov state n) is less or equal than one. The minimally required variance can be controlled with `connectivity_factor`. * '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! For more details see :func:`pyemma.thermo.extensions.cset.compute_csets_TRAM`. connectivity_factor : float, optional, default=1.0 Only needed if connectivity='post_hoc_RE' or 'BAR_variance'. Values greater than 1.0 weaken the connectivity conditions. For 'post_hoc_RE' this multiplies the number of hypothetically observed transitions. For 'BAR_variance' this scales the threshold for the minimal allowed variance of free energy differences. direct_space : bool, optional, default=False Whether to perform the self-consistent iteration with Boltzmann factors (direct space) or free energies (log-space). When analyzing data from multi-temperature simulations, direct-space is not recommended. N_dtram_accelerations : int, optional, default=0 Convergence of TRAM can be speeded up by interleaving the updates in the self-consistent iteration with a dTRAM-like update step. N_dtram_accelerations says how many times the dTRAM-like update step should be applied in every iteration of the TRAM equations. Currently this is only effective if direct_space=True. 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. The bias would be a list of :math:`T \times K` arrays which specify each frame's bias energy in all thermodynamic states: Let us try the above example: array([[[1 1] [0 4]] [[0 3] [2 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 2016 Multiensemble Markov models of molecular thermodynamics and kinetics Proc. Natl. Acad. Sci. USA 113 E3221--E3230 .. [2]_ Hukushima et al, Exchange Monte Carlo method and application to spin glass simulations, J. Phys. Soc. Jan. 65, 1604 (1996) .. [3]_ Shirts and Chodera, Statistically optimal analysis of samples from multiple equilibrium states, J. Chem. Phys. 129, 124105 (2008) """

    # 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)))
    if len(ttrajs) != len(bias):
        raise ValueError("Unmatching number of ttraj/bias elements: %d!=%d" % (
            len(ttrajs), len(bias)))
    for ttraj, dtraj, btraj in zip(ttrajs, dtrajs, bias):
        if len(ttraj) != len(dtraj):
            raise ValueError("Unmatching number of data points in ttraj/dtraj: %d!=%d" % (
                len(ttraj), len(dtraj)))
        if len(ttraj) != btraj.shape[0]:
            raise ValueError("Unmatching number of data points in ttraj/bias trajectory: %d!=%d" % (
                len(ttraj), len(btraj)))
    # check lag time(s)
    lags = _np.asarray(lag, dtype=_np.intc).reshape((-1,)).tolist()
    # build TRAM and run estimation
    from pyemma.thermo import TRAM as _TRAM
    tram_estimators = []
    from pyemma._base.progress import ProgressReporter
    pg = ProgressReporter()
    pg.register(amount_of_work=len(lags), description='Estimating TRAM for lags')
    with pg.context():
        for lag in lags:
            t = _TRAM(
                lag, count_mode=count_mode, connectivity=connectivity,
                maxiter=maxiter, maxerr=maxerr, save_convergence_info=save_convergence_info,
                dt_traj=dt_traj, connectivity_factor=connectivity_factor, nn=nn,
                direct_space=direct_space, N_dtram_accelerations=N_dtram_accelerations,
                callback=callback, init=init, init_maxiter=init_maxiter, init_maxerr=init_maxerr,
                equilibrium=equilibrium, overcounting_factor=overcounting_factor).estimate((ttrajs, dtrajs, bias))
            tram_estimators.append(t)
            pg.update(1)
    _assign_unbiased_state_label(tram_estimators, unbiased_state)
    # return
    if len(tram_estimators) == 1:
        return tram_estimators[0]
    return tram_estimators