code: def corr_nearest_factor(
      corr, rank, ctol=1e-6, lam_min=1e-30, lam_max=1e30, maxiter=1000
  ):
      """
      Find the nearest correlation matrix with factor structure to a
      given square matrix.
  
      Parameters
      ----------
      corr : square array
          The target matrix (to which the nearest correlation matrix is
          sought).  Must be square, but need not be positive
          semidefinite.
      rank : int
          The rank of the factor structure of the solution, i.e., the
          number of linearly independent columns of X.
      ctol : positive real
          Convergence criterion.
      lam_min : float
          Tuning parameter for spectral projected gradient optimization
          (smallest allowed step in the search direction).
      lam_max : float
          Tuning parameter for spectral projected gradient optimization
          (largest allowed step in the search direction).
      maxiter : int
          Maximum number of iterations in spectral projected gradient
          optimization.
  
      Returns
      -------
      rslt : Bunch
          rslt.corr is a FactoredPSDMatrix defining the estimated
          correlation structure.  Other fields of `rslt` contain
          returned values from spg_optim.
  
      Notes
      -----
      A correlation matrix has factor structure if it can be written in
      the form I + XX' - diag(XX'), where X is n x k with linearly
      independent columns, and with each row having sum of squares at
      most equal to 1.  The approximation is made in terms of the
      Frobenius norm.
  
      This routine is useful when one has an approximate correlation
      matrix that is not positive semidefinite, and there is need to
      estimate the inverse, square root, or inverse square root of the
      population correlation matrix.  The factor structure allows these
      tasks to be done without constructing any n x n matrices.
  
      This is a non-convex problem with no known guaranteed globally
      convergent algorithm for computing the solution.  Borsdof, Higham
      and Raydan (2010) compared several methods for this problem and
      found the spectral projected gradient (SPG) method (used here) to
      perform best.
  
      The input matrix `corr` can be a dense numpy array or any scipy
      sparse matrix.  The latter is useful if the input matrix is
      obtained by thresholding a very large sample correlation matrix.
      If `corr` is sparse, the calculations are optimized to save
      memory, so no working matrix with more than 10^6 elements is
      constructed.
  
      References
      ----------
      .. [*] R Borsdof, N Higham, M Raydan (2010).  Computing a nearest
         correlation matrix with factor structure. SIAM J Matrix Anal Appl,
         31:5, 2603-2622.
         http://eprints.ma.man.ac.uk/1523/01/covered/MIMS_ep2009_87.pdf
  
      Examples
      --------
      Hard thresholding a correlation matrix may result in a matrix that
      is not positive semidefinite.  We can approximate a hard
      thresholded correlation matrix with a PSD matrix as follows, where
      `corr` is the input correlation matrix.
  
      >>> import numpy as np
      >>> from statsmodels.stats.correlation_tools import corr_nearest_factor
      >>> np.random.seed(1234)
      >>> b = 1.5 - np.random.rand(10, 1)
      >>> x = np.random.randn(100,1).dot(b.T) + np.random.randn(100,10)
      >>> corr = np.corrcoef(x.T)
      >>> corr = corr * (np.abs(corr) >= 0.3)
      >>> rslt = corr_nearest_factor(corr, 3)
      """
  
      p, _ = corr.shape
  
      # Starting values (following the PCA method in BHR).
      u, s, vt = svds(corr, rank)
      X = u * np.sqrt(s)
      nm = np.sqrt((X**2).sum(1))
      ii = np.flatnonzero(nm > 1e-5)
      X[ii, :] /= nm[ii][:, None]
  
      # Zero the diagonal
      corr1 = corr.copy()
      if type(corr1) is np.ndarray:
          np.fill_diagonal(corr1, 0)
      elif sparse.issparse(corr1):
          corr1.setdiag(np.zeros(corr1.shape[0]))
          corr1.eliminate_zeros()
          corr1.sort_indices()
      else:
          raise ValueError("Matrix type not supported")
  
      # The gradient, from lemma 4.1 of BHR.
      def grad(X):
          gr = np.dot(X, np.dot(X.T, X))
          if type(corr1) is np.ndarray:
              gr -= np.dot(corr1, X)
          else:
              gr -= corr1.dot(X)
          gr -= (X * X).sum(1)[:, None] * X
          return 4 * gr
  
      # The objective function (sum of squared deviations between fitted
      # and observed arrays).
      def func(X):
          if type(corr1) is np.ndarray:
              M = np.dot(X, X.T)
              np.fill_diagonal(M, 0)
              M -= corr1
              fval = (M * M).sum()
              return fval
          else:
              fval = 0.0
              # Control the size of intermediates
              max_ws = 1e6
              bs = int(max_ws / X.shape[0])
              ir = 0
              while ir < X.shape[0]:
                  ir2 = min(ir + bs, X.shape[0])
                  u = np.dot(X[ir:ir2, :], X.T)
                  ii = np.arange(u.shape[0])
                  u[ii, ir + ii] = 0
                  u -= np.asarray(corr1[ir:ir2, :].todense())
                  fval += (u * u).sum()
                  ir += bs
              return fval
  
      rslt = _spg_optim(
          func,
          grad,
          X,
          _project_correlation_factors,
          ctol=ctol,
          lam_min=lam_min,
          lam_max=lam_max,
          maxiter=maxiter,
      )
      root = rslt.params
      diag = 1 - (root**2).sum(1)
      soln = FactoredPSDMatrix(diag, root)
      rslt.corr = soln
      del rslt.params
      return rslt
  
imports: import numpy
  import scipy.sparse
  from scipy.sparse.linalg import svds
  from scipy.optimize import fminbound
  import warnings
  from statsmodels.tools.tools import Bunch
  from statsmodels.tools.sm_exceptions import IterationLimitWarning, iteration_limit_doc
label_desc: Use image node_id 8 for calling a global function with example usage: corr_nearest_factor(corr, rank, ctol, lam_min, lam_max, maxiter) and returns: rslt, unknown, fval, fval