1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
|
# 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()
|
