docs: def correlation(T, obs1, obs2=None, times=(1), maxtime=None, k=None, ncv=None, return_times=False):
      r"""Time-correlation for equilibrium 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
      times : array-like of int (optional), default=(1)
          List of times (in tau) at which to compute correlation
      maxtime : int, optional, default=None
          Maximum time step to use. Equivalent to . Alternative to times.
      k : int (optional)
          Number of eigenvalues and eigenvectors to use for computation
      ncv : int (optional)
          The number of Lanczos vectors generated, `ncv` must be greater than k;
          it is recommended that ncv > 2*k
  
      Returns
      -------
      correlations : ndarray
          Correlation values at given times
      times : ndarray, optional
          time points at which the correlation was computed (if return_times=True)
  
      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
      -----
  
      **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.
  
      **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)
  
      Examples
      --------
  
      >>> import numpy as np
      >>> from msmtools.analysis import 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])
      >>> times = np.array([1, 5, 10, 20])
  
      >>> corr = correlation(T, a, times=times)
      >>> corr
      array([ 0.40909091,  0.34081364,  0.28585667,  0.23424263])
  
      """
      # check if square matrix and remember size
      T = _types.ensure_ndarray_or_sparse(T, ndim=2, uniform=True, kind='numeric')
      n = T.shape[0]
      obs1 = _types.ensure_ndarray(obs1, ndim=1, size=n, kind='numeric')
      obs2 = _types.ensure_ndarray_or_None(obs2, ndim=1, size=n, kind='numeric')
      times = _types.ensure_int_vector(times, require_order=True)
  
      # check input
      # go
      if _issparse(T):
          return sparse.fingerprints.correlation(T, obs1, obs2=obs2, times=times, k=k, ncv=ncv)
      else:
          return dense.fingerprints.correlation(T, obs1, obs2=obs2, times=times, k=k)