1
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5
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
31
32 except ImportError as x:
33
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
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