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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 transition_matrix(C, reversible=False, mu=None, method='auto', **kwargs):
 r"""Estimate the transition matrix from the given countmatrix. Parameters C : numpy ndarray or scipy.sparse matrix Count matrix reversible : bool (optional) If True restrict the ensemble of transition matrices to those having a detailed balance symmetry otherwise the likelihood optimization is carried out over the whole space of stochastic matrices. mu : array_like The stationary distribution of the MLE transition matrix. method : str Select which implementation to use for the estimation. One of 'auto', 'dense' and 'sparse', optional, default='auto'. 'dense' always selects the dense implementation, 'sparse' always selects the sparse one. 'auto' selects the most efficient implementation according to the sparsity structure of the matrix: if the occupation of the C matrix is less then one third, select sparse. Else select dense. The type of the T matrix returned always matches the type of the C matrix, irrespective of the method that was used to compute it. **kwargs: Optional algorithm-specific parameters. See below for special cases Xinit : (M, M) ndarray Optional parameter with reversible = True. initial value for the matrix of absolute transition probabilities. Unless set otherwise, will use X = diag(pi) t, where T is a nonreversible transition matrix estimated from C, i.e. T_ij = c_ij / sum_k c_ik, and pi is its stationary distribution. maxiter : 1000000 : int Optional parameter with reversible = True. maximum number of iterations before the method exits maxerr : 1e-8 : float 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. rev_pisym : bool, default=False Fast computation of reversible transition matrix by normalizing :math:`x_{ij} = \pi_i p_{ij} + \pi_j p_{ji}`. :math:`p_{ij}` is the direct (nonreversible) estimate and :math:`pi_i` is its stationary distribution. This estimator is asympotically unbiased but not maximum likelihood. return_statdist : bool, default=False Optional parameter with reversible = True. If set to true, the stationary distribution is also returned return_conv : bool, default=False Optional parameter with reversible = True. If set to true, the likelihood history and the pi_change history is returned. warn_not_converged : bool, default=True Prints a warning if not converged. sparse_newton : bool, default=False If True, use the experimental primal-dual interior-point solver for sparse input/computation method. Returns ------- P : (M, M) ndarray or scipy.sparse matrix The MLE transition matrix. P has the same data type (dense or sparse) as the input matrix C. The reversible estimator returns by default only P, but may also return (P,pi) or (P,lhist,pi_changes) or (P,pi,lhist,pi_changes) depending on the return settings P : ndarray (n,n) transition matrix. This is the only return for return_statdist = False, return_conv = False (pi) : ndarray (n) stationary distribution. Only returned if return_statdist = True (lhist) : ndarray (k) likelihood history. Has the length of the number of iterations needed. Only returned if return_conv = True (pi_changes) : ndarray (k) history of likelihood history. Has the length of the number of iterations needed. Only returned if return_conv = True Notes ----- The transition matrix is a maximum likelihood estimate (MLE) of the probability distribution of transition matrices with parameters given by the count matrix. References .. [1] Prinz, J H, H Wu, M Sarich, B Keller, M Senne, M Held, J D Chodera, C Schuette and F Noe. 2011. Markov models of molecular kinetics: Generation and validation. J Chem Phys 134: 174105 .. [2] Bowman, G R, K A Beauchamp, G Boxer and V S Pande. 2009. Progress and challenges in the automated construction of Markov state models for full protein systems. J. Chem. Phys. 131: 124101 .. [3] Trendelkamp-Schroer, B, H Wu, F Paul and F. Noe. 2015 Estimation and uncertainty of reversible Markov models. J. Chem. Phys. 143: 174101 Examples -------- Non-reversible estimate array([[ 0.83333333, 0.08333333, 0.08333333], [ 0.4 , 0. , 0.6 ], [ 0. , 0.2 , 0.8 ]]) Reversible estimate array([[ 0.83333333, 0.10385551, 0.06281115], [ 0.35074677, 0. , 0.64925323], [ 0.04925323, 0.15074677, 0.8 ]]) Reversible estimate with given stationary vector array([[ 0.94771371, 0.00612645, 0.04615984], [ 0.42885157, 0. , 0.57114843], [ 0.11142031, 0.01969477, 0.86888491]]) """

    if issparse(C):
        sparse_input_type = True
    elif isdense(C):
        sparse_input_type = False
    else:
        raise NotImplementedError('C has an unknown type.')

    if method == 'dense':
        sparse_computation = False
    elif method == 'sparse':
        sparse_computation = True
    elif method == 'auto':
        # heuristically determine whether is't more efficient to do a dense of sparse computation
        if sparse_input_type:
            dof = C.getnnz()
        else:
            dof = np.count_nonzero(C)
        dimension = C.shape[0]
        if dimension*dimension < 3*dof:
            sparse_computation = False
        else:
            sparse_computation = True
    else:
        raise ValueError(('method="%s" is no valid choice. It should be one of'
                          '"dense", "sparse" or "auto".') % method)

    # convert input type
    if sparse_computation and not sparse_input_type:
        C = coo_matrix(C)
    if not sparse_computation and sparse_input_type:
        C = C.toarray()

    return_statdist = 'return_statdist' in kwargs

    if not return_statdist:
        kwargs['return_statdist'] = False

    sparse_newton = kwargs.pop('sparse_newton', False)

    if reversible:
        rev_pisym = kwargs.pop('rev_pisym', False)

        if mu is None:
            if sparse_computation:
                if rev_pisym:
                    result = sparse.transition_matrix.transition_matrix_reversible_pisym(C, **kwargs)
                elif sparse_newton:
                    from msmtools.estimation.sparse.newton.mle_rev import solve_mle_rev
                    result = solve_mle_rev(C, **kwargs)
                else:
                    result = sparse.mle_trev.mle_trev(C, **kwargs)
            else:
                if rev_pisym:
                    result = dense.transition_matrix.transition_matrix_reversible_pisym(C, **kwargs)
                else:
                    result = dense.mle_trev.mle_trev(C, **kwargs)
        else:
            kwargs.pop('return_statdist') # pi given, keyword unknown by estimators.
            if sparse_computation:
                # Sparse, reversible, fixed pi (currently using dense with sparse conversion)
                result = sparse.mle_trev_given_pi.mle_trev_given_pi(C, mu, **kwargs)
            else:
                result = dense.mle_trev_given_pi.mle_trev_given_pi(C, mu, **kwargs)
    else:  # nonreversible estimation
        if mu is None:
            if sparse_computation:
                # Sparse,  nonreversible
                result = sparse.transition_matrix.transition_matrix_non_reversible(C)
            else:
                # Dense,  nonreversible
                result = dense.transition_matrix.transition_matrix_non_reversible(C)
            # Both methods currently do not have an iterate of pi, so we compute it here for consistency.
            if return_statdist:
                from msmtools.analysis import stationary_distribution
                mu = stationary_distribution(result)
        else:
            raise NotImplementedError('nonreversible mle with fixed stationary distribution not implemented.')

    if return_statdist and isinstance(result, tuple):
        T, mu = result
    else:
        T = result

    # convert return type
    if sparse_computation and not sparse_input_type:
        T = T.toarray()
    elif not sparse_computation and sparse_input_type:
        T = csr_matrix(T)

    if return_statdist:
        return T, mu
    return T