# 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. """ This module 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. """ 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, 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 xrange(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()

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