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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_markov_model(dtrajs, lag, reversible=True, statdist=None, count_mode='sliding', weights='empirical', sparse=False, connectivity='largest', dt_traj='1 step', maxiter=1000000, maxerr=1e-8, score_method='VAMP2', score_k=10, mincount_connectivity='1/n'):
 r""" Estimates a Markov model from discrete trajectories Returns a :class:`MaximumLikelihoodMSM` that contains the estimated transition matrix and allows to compute a large number of quantities related to Markov models. 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 lag time at which transitions are counted and the transition matrix is estimated. reversible : bool, optional If true compute reversible MSM, else non-reversible MSM statdist : (M,) ndarray, optional Stationary vector on the full state-space. Transition matrix will be estimated such that statdist is its equilibrium distribution. 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:: * 'effective' : Uses an estimate of the transition counts that are statistically uncorrelated. Recommended when used with a Bayesian MSM. * 'sample' : A trajectory of length T will have :math:`T/\tau` counts at time indexes .. math:: weights : str, optional can be used to re-weight non-equilibrium data to equilibrium. Must be one of the following: * 'empirical': Each trajectory frame counts as one. (default) * 'oom': Each transition is re-weighted using OOM theory, see [11]_. sparse : bool, optional If true compute count matrix, transition matrix and all derived quantities using sparse matrix algebra. In this case python sparse matrices will be returned by the corresponding functions instead of numpy arrays. This behavior is suggested for very large numbers of states (e.g. > 4000) because it is likely to be much more efficient. connectivity : str, optional Connectivity mode. Three methods are intended (currently only 'largest' is implemented) * 'largest' : The active set is the largest reversibly connected set. All estimation will be done on this subset and all quantities (transition matrix, stationary distribution, etc) are only defined on this subset and are correspondingly smaller than the full set of states * 'all' : The active set is the full set of states. Estimation will be conducted on each reversibly connected set separately. That means the transition matrix will decompose into disconnected submatrices, the stationary vector is only defined within subsets, etc. Currently not implemented. * 'none' : The active set is the full set of states. Estimation will be conducted on the full set of states without ensuring connectivity. This only permits nonreversible estimation. Currently not implemented. dt_traj : str, optional 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*' maxiter : int, optional Optional parameter with reversible = True. maximum number of iterations before the transition matrix estimation method exits maxerr : float, optional Optional parameter with reversible = True. convergence tolerance for transition matrix estimation. This specifies the maximum change of the Euclidean norm of relative stationary probabilities (:math:`x_i = \sum_k x_{ik}`). The relative stationary probability changes :math:`e_i = (x_i^{(1)} - x_i^{(2)})/(x_i^{(1)} + x_i^{(2)})` are used in order to track changes in small probabilities. The Euclidean norm of the change vector, :math:`|e_i|_2`, is compared to maxerr. score_method : str, optional, default='VAMP2' Score to be used with MSM score function. Available scores are based on the variational approach for Markov processes [13]_ [14]_: * 'VAMP1' Sum of singular values of the symmetrized transition matrix [14]_ . If the MSM is reversible, this is equal to the sum of transition matrix eigenvalues, also called Rayleigh quotient [13]_ [15]_ . * 'VAMP2' Sum of squared singular values of the symmetrized transition matrix [14]_ . If the MSM is reversible, this is equal to the kinetic variance [16]_ . score_k : int or None The maximum number of eigenvalues or singular values used in the score. If set to None, all available eigenvalues will be used. 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. Returns ------- msm : :class:`MaximumLikelihoodMSM <pyemma.msm.MaximumLikelihoodMSM>` Estimator object containing the MSM and estimation information. See also -------- MaximumLikelihoodMSM An MSM object that has been estimated from data .. autoclass:: pyemma.msm.estimators.maximum_likelihood_msm.MaximumLikelihoodMSM :members: :undoc-members: .. rubric:: Methods .. autoautosummary:: pyemma.msm.estimators.maximum_likelihood_msm.MaximumLikelihoodMSM :methods: .. rubric:: Attributes .. autoautosummary:: pyemma.msm.estimators.maximum_likelihood_msm.MaximumLikelihoodMSM :attributes: References The mathematical theory of Markov (state) model estimation was introduced in [1]_ . Further theoretical developments were made in [2]_ . The term Markov state model was coined in [3]_ . Continuous-time Markov models (Master equation models) were suggested in [4]_. Reversible Markov model estimation was introduced in [5]_ , and further developed in [6]_ [7]_ [9]_ . It was shown in [8]_ that the quality of Markov state models does in fact not depend on memory loss, but rather on where the discretization is suitable to approximate the eigenfunctions of the Markov operator (the 'reaction coordinates'). With a suitable choice of discretization and lag time, MSMs can thus become very accurate. [9]_ introduced a number of methodological improvements and gives a good overview of the methodological basics of Markov state modeling today. [10]_ is a more extensive review book of theory, methods and applications. .. [1] Schuette, C. , A. Fischer, W. Huisinga and P. Deuflhard: A Direct Approach to Conformational Dynamics based on Hybrid Monte Carlo. J. Comput. Phys., 151, 146-168 (1999) .. [2] Swope, W. C., J. W. Pitera and F. Suits: Describing protein folding kinetics by molecular dynamics simulations: 1. Theory J. Phys. Chem. B 108, 6571-6581 (2004) .. [3] Singhal, N., C. D. Snow, V. S. Pande: Using path sampling to build better Markovian state models: Predicting the folding rate and mechanism of a tryptophan zipper beta hairpin. J. Chem. Phys. 121, 415 (2004). .. [4] Sriraman, S., I. G. Kevrekidis and G. Hummer, G. J. Phys. Chem. B 109, 6479-6484 (2005) .. [5] Noe, F.: Probability Distributions of Molecular Observables computed from Markov Models. J. Chem. Phys. 128, 244103 (2008) .. [6] Buchete, N.-V. and Hummer, G.: Coarse master equations for peptide folding dynamics. J. Phys. Chem. B 112, 6057--6069 (2008) .. [7] Bowman, G. R., K. A. Beauchamp, G. Boxer and V. S. Pande: Progress and challenges in the automated construction of Markov state models for full protein systems. J. Chem. Phys. 131, 124101 (2009) .. [8] Sarich, M., F. Noe and C. Schuette: On the approximation quality of Markov state models. SIAM Multiscale Model. Simul. 8, 1154-1177 (2010) .. [9] Prinz, J.-H., H. Wu, M. Sarich, B. Keller, M. Senne, M. Held, J. D. Chodera, C. Schuette and F. Noe: Markov models of molecular kinetics: Generation and Validation J. Chem. Phys. 134, 174105 (2011) .. [10] Bowman, G. R., V. S. Pande and F. Noe: An Introduction to Markov State Models and Their Application to Long Timescale Molecular Simulation. Advances in Experimental Medicine and Biology 797, Springer, Heidelberg (2014) .. [11] Nueske, F., Wu, H., Prinz, J.-H., Wehmeyer, C., Clementi, C. and Noe, F.: Markov State Models from short non-Equilibrium Simulations - Analysis and Correction of Estimation Bias J. Chem. Phys. (submitted) (2017) .. [12] H. Wu and F. Noe: Variational approach for learning Markov processes from time series data (in preparation) .. [13] Noe, F. and F. Nueske: A variational approach to modeling slow processes in stochastic dynamical systems. SIAM Multiscale Model. Simul. 11, 635-655 (2013). .. [14] Wu, H and F. Noe: Variational approach for learning Markov processes from time series data (in preparation) .. [15] McGibbon, R and V. S. Pande: Variational cross-validation of slow dynamical modes in molecular kinetics, J. Chem. Phys. 142, 124105 (2015) .. [16] Noe, F. and C. Clementi: Kinetic distance and kinetic maps from molecular dynamics simulation. J. Chem. Theory Comput. 11, 5002-5011 (2015) Example ------- Which is the active set of states we are working on? [0 1 2] Show the count matrix [[ 7. 2. 1.] [ 2. 0. 4.] [ 2. 3. 9.]] Show the estimated transition matrix [[ 0.7 0.167 0.133] [ 0.388 0. 0.612] [ 0.119 0.238 0.643]] Is this model reversible (i.e. does it fulfill detailed balance)? True What is the equilibrium distribution of states? [ 0.393 0.17 0.437] Relaxation timescales? [ 3.415 1.297] Mean first passage time from state 0 to 2: """

    # Catch invalid inputs for weights:
    if isinstance(weights, str):
        if weights not in ['empirical', 'oom']:
            raise ValueError("Weights must be either \'empirical\' or \'oom\'")
    else:
        raise ValueError("Weights must be either \'empirical\' or \'oom\'")

    # transition matrix estimator
    if weights == 'empirical':
        mlmsm = _ML_MSM(lag=lag, reversible=reversible, statdist_constraint=statdist,
                        count_mode=count_mode,
                        sparse=sparse, connectivity=connectivity,
                        dt_traj=dt_traj, maxiter=maxiter,
                        maxerr=maxerr, score_method=score_method, score_k=score_k,
                        mincount_connectivity=mincount_connectivity)
        # estimate and return
        return mlmsm.estimate(dtrajs)
    elif weights == 'oom':
        if (statdist is not None) or (maxiter != 1000000) or (maxerr != 1e-8):
            import warnings
            warnings.warn("Values for statdist, maxiter or maxerr are ignored if OOM-correction is used.")
        oom_msm = _OOM_MSM(lag=lag, reversible=reversible, count_mode=count_mode,
                           sparse=sparse, connectivity=connectivity, dt_traj=dt_traj,
                           score_method=score_method, score_k=score_k,
                           mincount_connectivity=mincount_connectivity)
        # estimate and return
        return oom_msm.estimate(dtrajs)