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302 lines
10 KiB
Python
302 lines
10 KiB
Python
import numpy as np
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from numpy.testing import (assert_allclose, assert_equal,
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assert_almost_equal, assert_array_equal,
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assert_array_almost_equal)
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from scipy.ndimage import convolve1d
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from scipy.signal import savgol_coeffs, savgol_filter
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from scipy.signal._savitzky_golay import _polyder
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def check_polyder(p, m, expected):
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dp = _polyder(p, m)
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assert_array_equal(dp, expected)
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def test_polyder():
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cases = [
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([5], 0, [5]),
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([5], 1, [0]),
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([3, 2, 1], 0, [3, 2, 1]),
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([3, 2, 1], 1, [6, 2]),
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([3, 2, 1], 2, [6]),
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([3, 2, 1], 3, [0]),
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([[3, 2, 1], [5, 6, 7]], 0, [[3, 2, 1], [5, 6, 7]]),
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([[3, 2, 1], [5, 6, 7]], 1, [[6, 2], [10, 6]]),
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([[3, 2, 1], [5, 6, 7]], 2, [[6], [10]]),
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([[3, 2, 1], [5, 6, 7]], 3, [[0], [0]]),
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]
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for p, m, expected in cases:
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check_polyder(np.array(p).T, m, np.array(expected).T)
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#--------------------------------------------------------------------
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# savgol_coeffs tests
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#--------------------------------------------------------------------
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def alt_sg_coeffs(window_length, polyorder, pos):
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"""This is an alternative implementation of the SG coefficients.
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It uses numpy.polyfit and numpy.polyval. The results should be
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equivalent to those of savgol_coeffs(), but this implementation
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is slower.
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window_length should be odd.
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"""
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if pos is None:
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pos = window_length // 2
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t = np.arange(window_length)
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unit = (t == pos).astype(int)
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h = np.polyval(np.polyfit(t, unit, polyorder), t)
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return h
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def test_sg_coeffs_trivial():
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# Test a trivial case of savgol_coeffs: polyorder = window_length - 1
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h = savgol_coeffs(1, 0)
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assert_allclose(h, [1])
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h = savgol_coeffs(3, 2)
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assert_allclose(h, [0, 1, 0], atol=1e-10)
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h = savgol_coeffs(5, 4)
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assert_allclose(h, [0, 0, 1, 0, 0], atol=1e-10)
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h = savgol_coeffs(5, 4, pos=1)
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assert_allclose(h, [0, 0, 0, 1, 0], atol=1e-10)
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h = savgol_coeffs(5, 4, pos=1, use='dot')
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assert_allclose(h, [0, 1, 0, 0, 0], atol=1e-10)
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def compare_coeffs_to_alt(window_length, order):
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# For the given window_length and order, compare the results
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# of savgol_coeffs and alt_sg_coeffs for pos from 0 to window_length - 1.
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# Also include pos=None.
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for pos in [None] + list(range(window_length)):
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h1 = savgol_coeffs(window_length, order, pos=pos, use='dot')
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h2 = alt_sg_coeffs(window_length, order, pos=pos)
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assert_allclose(h1, h2, atol=1e-10,
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err_msg=("window_length = %d, order = %d, pos = %s" %
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(window_length, order, pos)))
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def test_sg_coeffs_compare():
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# Compare savgol_coeffs() to alt_sg_coeffs().
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for window_length in range(1, 8, 2):
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for order in range(window_length):
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compare_coeffs_to_alt(window_length, order)
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def test_sg_coeffs_exact():
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polyorder = 4
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window_length = 9
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halflen = window_length // 2
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x = np.linspace(0, 21, 43)
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delta = x[1] - x[0]
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# The data is a cubic polynomial. We'll use an order 4
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# SG filter, so the filtered values should equal the input data
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# (except within half window_length of the edges).
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y = 0.5 * x ** 3 - x
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h = savgol_coeffs(window_length, polyorder)
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y0 = convolve1d(y, h)
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assert_allclose(y0[halflen:-halflen], y[halflen:-halflen])
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# Check the same input, but use deriv=1. dy is the exact result.
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dy = 1.5 * x ** 2 - 1
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h = savgol_coeffs(window_length, polyorder, deriv=1, delta=delta)
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y1 = convolve1d(y, h)
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assert_allclose(y1[halflen:-halflen], dy[halflen:-halflen])
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# Check the same input, but use deriv=2. d2y is the exact result.
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d2y = 3.0 * x
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h = savgol_coeffs(window_length, polyorder, deriv=2, delta=delta)
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y2 = convolve1d(y, h)
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assert_allclose(y2[halflen:-halflen], d2y[halflen:-halflen])
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def test_sg_coeffs_deriv():
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# The data in `x` is a sampled parabola, so using savgol_coeffs with an
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# order 2 or higher polynomial should give exact results.
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i = np.array([-2.0, 0.0, 2.0, 4.0, 6.0])
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x = i ** 2 / 4
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dx = i / 2
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d2x = np.full_like(i, 0.5)
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for pos in range(x.size):
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coeffs0 = savgol_coeffs(5, 3, pos=pos, delta=2.0, use='dot')
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assert_allclose(coeffs0.dot(x), x[pos], atol=1e-10)
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coeffs1 = savgol_coeffs(5, 3, pos=pos, delta=2.0, use='dot', deriv=1)
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assert_allclose(coeffs1.dot(x), dx[pos], atol=1e-10)
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coeffs2 = savgol_coeffs(5, 3, pos=pos, delta=2.0, use='dot', deriv=2)
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assert_allclose(coeffs2.dot(x), d2x[pos], atol=1e-10)
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def test_sg_coeffs_deriv_gt_polyorder():
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"""
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If deriv > polyorder, the coefficients should be all 0.
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This is a regression test for a bug where, e.g.,
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savgol_coeffs(5, polyorder=1, deriv=2)
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raised an error.
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"""
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coeffs = savgol_coeffs(5, polyorder=1, deriv=2)
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assert_array_equal(coeffs, np.zeros(5))
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coeffs = savgol_coeffs(7, polyorder=4, deriv=6)
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assert_array_equal(coeffs, np.zeros(7))
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def test_sg_coeffs_large():
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# Test that for large values of window_length and polyorder the array of
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# coefficients returned is symmetric. The aim is to ensure that
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# no potential numeric overflow occurs.
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coeffs0 = savgol_coeffs(31, 9)
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assert_array_almost_equal(coeffs0, coeffs0[::-1])
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coeffs1 = savgol_coeffs(31, 9, deriv=1)
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assert_array_almost_equal(coeffs1, -coeffs1[::-1])
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#--------------------------------------------------------------------
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# savgol_filter tests
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#--------------------------------------------------------------------
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def test_sg_filter_trivial():
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""" Test some trivial edge cases for savgol_filter()."""
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x = np.array([1.0])
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y = savgol_filter(x, 1, 0)
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assert_equal(y, [1.0])
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# Input is a single value. With a window length of 3 and polyorder 1,
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# the value in y is from the straight-line fit of (-1,0), (0,3) and
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# (1, 0) at 0. This is just the average of the three values, hence 1.0.
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x = np.array([3.0])
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y = savgol_filter(x, 3, 1, mode='constant')
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assert_almost_equal(y, [1.0], decimal=15)
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x = np.array([3.0])
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y = savgol_filter(x, 3, 1, mode='nearest')
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assert_almost_equal(y, [3.0], decimal=15)
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x = np.array([1.0] * 3)
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y = savgol_filter(x, 3, 1, mode='wrap')
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assert_almost_equal(y, [1.0, 1.0, 1.0], decimal=15)
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def test_sg_filter_basic():
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# Some basic test cases for savgol_filter().
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x = np.array([1.0, 2.0, 1.0])
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y = savgol_filter(x, 3, 1, mode='constant')
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assert_allclose(y, [1.0, 4.0 / 3, 1.0])
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y = savgol_filter(x, 3, 1, mode='mirror')
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assert_allclose(y, [5.0 / 3, 4.0 / 3, 5.0 / 3])
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y = savgol_filter(x, 3, 1, mode='wrap')
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assert_allclose(y, [4.0 / 3, 4.0 / 3, 4.0 / 3])
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def test_sg_filter_2d():
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x = np.array([[1.0, 2.0, 1.0],
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[2.0, 4.0, 2.0]])
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expected = np.array([[1.0, 4.0 / 3, 1.0],
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[2.0, 8.0 / 3, 2.0]])
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y = savgol_filter(x, 3, 1, mode='constant')
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assert_allclose(y, expected)
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y = savgol_filter(x.T, 3, 1, mode='constant', axis=0)
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assert_allclose(y, expected.T)
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def test_sg_filter_interp_edges():
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# Another test with low degree polynomial data, for which we can easily
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# give the exact results. In this test, we use mode='interp', so
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# savgol_filter should match the exact solution for the entire data set,
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# including the edges.
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t = np.linspace(-5, 5, 21)
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delta = t[1] - t[0]
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# Polynomial test data.
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x = np.array([t,
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3 * t ** 2,
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t ** 3 - t])
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dx = np.array([np.ones_like(t),
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6 * t,
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3 * t ** 2 - 1.0])
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d2x = np.array([np.zeros_like(t),
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np.full_like(t, 6),
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6 * t])
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window_length = 7
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y = savgol_filter(x, window_length, 3, axis=-1, mode='interp')
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assert_allclose(y, x, atol=1e-12)
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y1 = savgol_filter(x, window_length, 3, axis=-1, mode='interp',
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deriv=1, delta=delta)
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assert_allclose(y1, dx, atol=1e-12)
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y2 = savgol_filter(x, window_length, 3, axis=-1, mode='interp',
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deriv=2, delta=delta)
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assert_allclose(y2, d2x, atol=1e-12)
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# Transpose everything, and test again with axis=0.
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x = x.T
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dx = dx.T
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d2x = d2x.T
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y = savgol_filter(x, window_length, 3, axis=0, mode='interp')
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assert_allclose(y, x, atol=1e-12)
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y1 = savgol_filter(x, window_length, 3, axis=0, mode='interp',
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deriv=1, delta=delta)
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assert_allclose(y1, dx, atol=1e-12)
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y2 = savgol_filter(x, window_length, 3, axis=0, mode='interp',
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deriv=2, delta=delta)
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assert_allclose(y2, d2x, atol=1e-12)
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def test_sg_filter_interp_edges_3d():
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# Test mode='interp' with a 3-D array.
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t = np.linspace(-5, 5, 21)
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delta = t[1] - t[0]
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x1 = np.array([t, -t])
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x2 = np.array([t ** 2, 3 * t ** 2 + 5])
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x3 = np.array([t ** 3, 2 * t ** 3 + t ** 2 - 0.5 * t])
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dx1 = np.array([np.ones_like(t), -np.ones_like(t)])
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dx2 = np.array([2 * t, 6 * t])
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dx3 = np.array([3 * t ** 2, 6 * t ** 2 + 2 * t - 0.5])
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# z has shape (3, 2, 21)
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z = np.array([x1, x2, x3])
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dz = np.array([dx1, dx2, dx3])
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y = savgol_filter(z, 7, 3, axis=-1, mode='interp', delta=delta)
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assert_allclose(y, z, atol=1e-10)
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dy = savgol_filter(z, 7, 3, axis=-1, mode='interp', deriv=1, delta=delta)
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assert_allclose(dy, dz, atol=1e-10)
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# z has shape (3, 21, 2)
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z = np.array([x1.T, x2.T, x3.T])
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dz = np.array([dx1.T, dx2.T, dx3.T])
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y = savgol_filter(z, 7, 3, axis=1, mode='interp', delta=delta)
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assert_allclose(y, z, atol=1e-10)
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dy = savgol_filter(z, 7, 3, axis=1, mode='interp', deriv=1, delta=delta)
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assert_allclose(dy, dz, atol=1e-10)
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# z has shape (21, 3, 2)
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z = z.swapaxes(0, 1).copy()
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dz = dz.swapaxes(0, 1).copy()
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y = savgol_filter(z, 7, 3, axis=0, mode='interp', delta=delta)
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assert_allclose(y, z, atol=1e-10)
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dy = savgol_filter(z, 7, 3, axis=0, mode='interp', deriv=1, delta=delta)
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assert_allclose(dy, dz, atol=1e-10)
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