docs: 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
      --------
  
      >>> import numpy as np
      >>> from msmtools.analysis import fingerprint_correlation
  
      >>> T = np.array([[0.9, 0.1, 0.0], [0.5, 0.0, 0.5], [0.0, 0.1, 0.9]])
      >>> a = np.array([1.0, 0.0, 0.0])
      >>> ts, amp = fingerprint_correlation(T, a)
  
      >>> ts
      array([        inf,  9.49122158,  0.43429448])
  
      >>> amp
      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)