Package Bio :: Package Statistics :: Module lowess
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Source Code for Module Bio.Statistics.lowess

  1  # Copyright 2004-2008 by M de Hoon. 
  2  # All rights reserved. 
  3  # This code is part of the Biopython distribution and governed by its 
  4  # license.  Please see the LICENSE file that should have been included 
  5  # as part of this package. 
  6  """Implements the Lowess function for nonparametric regression. 
  7   
  8  Functions: 
  9  lowess        Fit a smooth nonparametric regression curve to a scatterplot. 
 10   
 11  For more information, see 
 12   
 13  William S. Cleveland: "Robust locally weighted regression and smoothing 
 14  scatterplots", Journal of the American Statistical Association, December 1979, 
 15  volume 74, number 368, pp. 829-836. 
 16   
 17  William S. Cleveland and Susan J. Devlin: "Locally weighted regression: An 
 18  approach to regression analysis by local fitting", Journal of the American 
 19  Statistical Association, September 1988, volume 83, number 403, pp. 596-610. 
 20  """ 
 21   
 22  from __future__ import print_function 
 23   
 24  from Bio._py3k import range 
 25   
 26  import numpy 
 27   
 28  try: 
 29      from Bio.Cluster import median 
 30      # The function median in Bio.Cluster is faster than the function median 
 31      # in NumPy, as it does not require a full sort. 
 32  except ImportError as x: 
 33      # Use the median function in NumPy if Bio.Cluster is not available 
 34      from numpy import median 
 35   
 36   
37 -def lowess(x, y, f=2. / 3., iter=3):
38 """lowess(x, y, f=2./3., iter=3) -> yest 39 40 Lowess smoother: Robust locally weighted regression. 41 The lowess function fits a nonparametric regression curve to a scatterplot. 42 The arrays x and y contain an equal number of elements; each pair 43 (x[i], y[i]) defines a data point in the scatterplot. The function returns 44 the estimated (smooth) values of y. 45 46 The smoothing span is given by f. A larger value for f will result in a 47 smoother curve. The number of robustifying iterations is given by iter. The 48 function will run faster with a smaller number of iterations. 49 50 x and y should be numpy float arrays of equal length. The return value is 51 also a numpy float array of that length. 52 53 e.g. 54 >>> import numpy 55 >>> x = numpy.array([4, 4, 7, 7, 8, 9, 10, 10, 10, 11, 11, 12, 12, 12, 56 ... 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 16, 16, 57 ... 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 20, 20, 20, 20, 58 ... 20, 22, 23, 24, 24, 24, 24, 25], numpy.float) 59 >>> y = numpy.array([2, 10, 4, 22, 16, 10, 18, 26, 34, 17, 28, 14, 20, 24, 60 ... 28, 26, 34, 34, 46, 26, 36, 60, 80, 20, 26, 54, 32, 40, 61 ... 32, 40, 50, 42, 56, 76, 84, 36, 46, 68, 32, 48, 52, 56, 62 ... 64, 66, 54, 70, 92, 93, 120, 85], numpy.float) 63 >>> result = lowess(x, y) 64 >>> len(result) 65 50 66 >>> print("[%0.2f, ..., %0.2f]" % (result[0], result[-1])) 67 [4.85, ..., 84.98] 68 """ 69 n = len(x) 70 r = int(numpy.ceil(f * n)) 71 h = [numpy.sort(abs(x - x[i]))[r] for i in range(n)] 72 w = numpy.clip(abs(([x] - numpy.transpose([x])) / h), 0.0, 1.0) 73 w = 1 - w * w * w 74 w = w * w * w 75 yest = numpy.zeros(n) 76 delta = numpy.ones(n) 77 for iteration in range(iter): 78 for i in range(n): 79 weights = delta * w[:, i] 80 weights_mul_x = weights * x 81 b1 = numpy.dot(weights, y) 82 b2 = numpy.dot(weights_mul_x, y) 83 A11 = sum(weights) 84 A12 = sum(weights_mul_x) 85 A21 = A12 86 A22 = numpy.dot(weights_mul_x, x) 87 determinant = A11 * A22 - A12 * A21 88 beta1 = (A22 * b1 - A12 * b2) / determinant 89 beta2 = (A11 * b2 - A21 * b1) / determinant 90 yest[i] = beta1 + beta2 * x[i] 91 residuals = y - yest 92 s = median(abs(residuals)) 93 delta[:] = numpy.clip(residuals / (6 * s), -1, 1) 94 delta[:] = 1 - delta * delta 95 delta[:] = delta * delta 96 return yest
97 98
99 -def _test():
100 """Run the Bio.Statistics.lowess module's doctests.""" 101 print("Running doctests...") 102 import doctest 103 doctest.testmod() 104 print("Done")
105 106 if __name__ == "__main__": 107 _test() 108