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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 fingerprint_correlation(T, obs1, obs2=None, tau=1, k=None, ncv=None):
 r"""Dynamical fingerprint for equilibrium correlation experiment. Parameters T : (M, M) ndarray or scipy.sparse matrix Transition matrix obs1 : (M,) ndarray Observable, represented as vector on state space obs2 : (M,) ndarray (optional) Second observable, for cross-correlations k : int (optional) Number of time-scales and amplitudes to compute tau : int (optional) Lag time of given transition matrix, for correct time-scales ncv : int (optional) The number of Lanczos vectors generated, `ncv` must be greater than k; it is recommended that ncv > 2*k Returns ------- timescales : (N,) ndarray Time-scales of the transition matrix amplitudes : (N,) ndarray Amplitudes for the correlation experiment See also -------- correlation, fingerprint_relaxation References .. [1] Noe, F, S Doose, I Daidone, M Loellmann, M Sauer, J D Chodera and J Smith. 2010. Dynamical fingerprints for probing individual relaxation processes in biomolecular dynamics with simulations and kinetic experiments. PNAS 108 (12):
 4822-4827. Notes ----- Fingerprints are a combination of time-scale and amplitude spectrum for a equilibrium correlation or a non-equilibrium relaxation experiment. **Auto-correlation** The auto-correlation of an observable :math:`a(x)` for a system in equilibrium is .. math:: \mathbb{E}_{\mu}[a(x,0)a(x,t)]=\sum_x \mu(x) a(x, 0) a(x, t) :math:`a(x,0)=a(x)` is the observable at time :math:`t=0`. It can be propagated forward in time using the t-step transition matrix :math:`p^{t}(x, y)`. The propagated observable at time :math:`t` is :math:`a(x, t)=\sum_y p^t(x, y)a(y, 0)`. Using the eigenvlaues and eigenvectors of the transition matrix the autocorrelation can be written as .. math:: \mathbb{E}_{\mu}[a(x,0)a(x,t)]=\sum_i \lambda_i^t \langle a, r_i\rangle_{\mu} \langle l_i, a \rangle. The fingerprint amplitudes :math:`\gamma_i` are given by .. math:: \gamma_i=\langle a, r_i\rangle_{\mu} \langle l_i, a \rangle. And the fingerprint time scales :math:`t_i` are given by .. math:: t_i=-\frac{\tau}{\log \lvert \lambda_i \rvert}. **Cross-correlation** The cross-correlation of two observables :math:`a(x)`, :math:`b(x)` is similarly given .. math:: \mathbb{E}_{\mu}[a(x,0)b(x,t)]=\sum_x \mu(x) a(x, 0) b(x, t) The fingerprint amplitudes :math:`\gamma_i` are similarly given in terms of the eigenvectors .. math:: \gamma_i=\langle a, r_i\rangle_{\mu} \langle l_i, b \rangle. Examples -------- array([ inf, 9.49122158, 0.43429448]) array([ 0.20661157, 0.22727273, 0.02066116]) """

    # check if square matrix and remember size
    T = _types.ensure_ndarray_or_sparse(T, ndim=2, uniform=True, kind='numeric')
    n = T.shape[0]
    # will not do fingerprint analysis for nonreversible matrices
    if not is_reversible(T):
        raise ValueError('Fingerprint calculation is not supported for nonreversible transition matrices. ')
    obs1 = _types.ensure_ndarray(obs1, ndim=1, size=n, kind='numeric')
    obs1 = _types.ensure_ndarray_or_None(obs1, ndim=1, size=n, kind='numeric')
    # go
    if _issparse(T):
        return sparse.fingerprints.fingerprint_correlation(T, obs1, obs2=obs2, tau=tau, k=k, ncv=ncv)
    else:
        return dense.fingerprints.fingerprint_correlation(T, obs1, obs2, tau=tau, k=k)