text: # Copyright 2004-2008 by M de Hoon.
  # All rights reserved.
  # This code is part of the Biopython distribution and governed by its
  # license.  Please see the LICENSE file that should have been included
  # as part of this package.
  """Implements the Lowess function for nonparametric regression.
  
  Functions:
  lowess        Fit a smooth nonparametric regression curve to a scatterplot.
  
  For more information, see
  
  William S. Cleveland: "Robust locally weighted regression and smoothing
  scatterplots", Journal of the American Statistical Association, December 1979,
  volume 74, number 368, pp. 829-836.
  
  William S. Cleveland and Susan J. Devlin: "Locally weighted regression: An
  approach to regression analysis by local fitting", Journal of the American
  Statistical Association, September 1988, volume 83, number 403, pp. 596-610.
  """
  
  from __future__ import print_function
  
  from Bio._py3k import range
  
  import numpy
  
  try:
      from Bio.Cluster import median
      # The function median in Bio.Cluster is faster than the function median
      # in NumPy, as it does not require a full sort.
  except ImportError as x:
      # Use the median function in NumPy if Bio.Cluster is not available
      from numpy import median
  
  
  def lowess(x, y, f=2. / 3., iter=3):
      """lowess(x, y, f=2./3., iter=3) -> yest
  
      Lowess smoother: Robust locally weighted regression.
      The lowess function fits a nonparametric regression curve to a scatterplot.
      The arrays x and y contain an equal number of elements; each pair
      (x[i], y[i]) defines a data point in the scatterplot. The function returns
      the estimated (smooth) values of y.
  
      The smoothing span is given by f. A larger value for f will result in a
      smoother curve. The number of robustifying iterations is given by iter. The
      function will run faster with a smaller number of iterations.
  
      x and y should be numpy float arrays of equal length.  The return value is
      also a numpy float array of that length.
  
      e.g.
      >>> import numpy
      >>> x = numpy.array([4,  4,  7,  7,  8,  9, 10, 10, 10, 11, 11, 12, 12, 12,
      ...                 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 16, 16,
      ...                 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 20, 20, 20, 20,
      ...                 20, 22, 23, 24, 24, 24, 24, 25], numpy.float)
      >>> y = numpy.array([2, 10,  4, 22, 16, 10, 18, 26, 34, 17, 28, 14, 20, 24,
      ...                 28, 26, 34, 34, 46, 26, 36, 60, 80, 20, 26, 54, 32, 40,
      ...                 32, 40, 50, 42, 56, 76, 84, 36, 46, 68, 32, 48, 52, 56,
      ...                 64, 66, 54, 70, 92, 93, 120, 85], numpy.float)
      >>> result = lowess(x, y)
      >>> len(result)
      50
      >>> print("[%0.2f, ..., %0.2f]" % (result[0], result[-1]))
      [4.85, ..., 84.98]
      """
      n = len(x)
      r = int(numpy.ceil(f * n))
      h = [numpy.sort(abs(x - x[i]))[r] for i in range(n)]
      w = numpy.clip(abs(([x] - numpy.transpose([x])) / h), 0.0, 1.0)
      w = 1 - w * w * w
      w = w * w * w
      yest = numpy.zeros(n)
      delta = numpy.ones(n)
      for iteration in range(iter):
          for i in range(n):
              weights = delta * w[:, i]
              weights_mul_x = weights * x
              b1 = numpy.dot(weights, y)
              b2 = numpy.dot(weights_mul_x, y)
              A11 = sum(weights)
              A12 = sum(weights_mul_x)
              A21 = A12
              A22 = numpy.dot(weights_mul_x, x)
              determinant = A11 * A22 - A12 * A21
              beta1 = (A22 * b1 - A12 * b2) / determinant
              beta2 = (A11 * b2 - A21 * b1) / determinant
              yest[i] = beta1 + beta2 * x[i]
          residuals = y - yest
          s = median(abs(residuals))
          delta[:] = numpy.clip(residuals / (6 * s), -1, 1)
          delta[:] = 1 - delta * delta
          delta[:] = delta * delta
      return yest
  
  
  def _test():
      """Run the Bio.Statistics.lowess module's doctests."""
      print("Running doctests...")
      import doctest
      doctest.testmod()
      print("Done")
  
  if __name__ == "__main__":
      _test()
  