1640 lines
58 KiB
Python
1640 lines
58 KiB
Python
import numpy as np
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from numpy.testing import assert_equal, assert_allclose, assert_
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from pytest import raises as assert_raises
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import pytest
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from scipy.interpolate import (BSpline, BPoly, PPoly, make_interp_spline,
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make_lsq_spline, _bspl, splev, splrep, splprep,
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splder, splantider, sproot, splint, insert,
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CubicSpline, make_smoothing_spline)
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import scipy.linalg as sl
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from scipy.interpolate._bsplines import (_not_a_knot, _augknt,
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_woodbury_algorithm, _periodic_knots,
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_make_interp_per_full_matr)
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import scipy.interpolate._fitpack_impl as _impl
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import os
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class TestBSpline:
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def test_ctor(self):
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# knots should be an ordered 1-D array of finite real numbers
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assert_raises((TypeError, ValueError), BSpline,
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**dict(t=[1, 1.j], c=[1.], k=0))
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with np.errstate(invalid='ignore'):
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assert_raises(ValueError, BSpline, **dict(t=[1, np.nan], c=[1.], k=0))
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assert_raises(ValueError, BSpline, **dict(t=[1, np.inf], c=[1.], k=0))
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assert_raises(ValueError, BSpline, **dict(t=[1, -1], c=[1.], k=0))
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assert_raises(ValueError, BSpline, **dict(t=[[1], [1]], c=[1.], k=0))
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# for n+k+1 knots and degree k need at least n coefficients
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assert_raises(ValueError, BSpline, **dict(t=[0, 1, 2], c=[1], k=0))
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assert_raises(ValueError, BSpline,
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**dict(t=[0, 1, 2, 3, 4], c=[1., 1.], k=2))
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# non-integer orders
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assert_raises(TypeError, BSpline,
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**dict(t=[0., 0., 1., 2., 3., 4.], c=[1., 1., 1.], k="cubic"))
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assert_raises(TypeError, BSpline,
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**dict(t=[0., 0., 1., 2., 3., 4.], c=[1., 1., 1.], k=2.5))
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# basic interval cannot have measure zero (here: [1..1])
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assert_raises(ValueError, BSpline,
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**dict(t=[0., 0, 1, 1, 2, 3], c=[1., 1, 1], k=2))
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# tck vs self.tck
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n, k = 11, 3
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t = np.arange(n+k+1)
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c = np.random.random(n)
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b = BSpline(t, c, k)
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assert_allclose(t, b.t)
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assert_allclose(c, b.c)
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assert_equal(k, b.k)
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def test_tck(self):
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b = _make_random_spline()
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tck = b.tck
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assert_allclose(b.t, tck[0], atol=1e-15, rtol=1e-15)
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assert_allclose(b.c, tck[1], atol=1e-15, rtol=1e-15)
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assert_equal(b.k, tck[2])
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# b.tck is read-only
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with pytest.raises(AttributeError):
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b.tck = 'foo'
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def test_degree_0(self):
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xx = np.linspace(0, 1, 10)
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b = BSpline(t=[0, 1], c=[3.], k=0)
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assert_allclose(b(xx), 3)
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b = BSpline(t=[0, 0.35, 1], c=[3, 4], k=0)
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assert_allclose(b(xx), np.where(xx < 0.35, 3, 4))
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def test_degree_1(self):
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t = [0, 1, 2, 3, 4]
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c = [1, 2, 3]
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k = 1
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b = BSpline(t, c, k)
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x = np.linspace(1, 3, 50)
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assert_allclose(c[0]*B_012(x) + c[1]*B_012(x-1) + c[2]*B_012(x-2),
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b(x), atol=1e-14)
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assert_allclose(splev(x, (t, c, k)), b(x), atol=1e-14)
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def test_bernstein(self):
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# a special knot vector: Bernstein polynomials
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k = 3
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t = np.asarray([0]*(k+1) + [1]*(k+1))
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c = np.asarray([1., 2., 3., 4.])
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bp = BPoly(c.reshape(-1, 1), [0, 1])
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bspl = BSpline(t, c, k)
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xx = np.linspace(-1., 2., 10)
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assert_allclose(bp(xx, extrapolate=True),
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bspl(xx, extrapolate=True), atol=1e-14)
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assert_allclose(splev(xx, (t, c, k)),
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bspl(xx), atol=1e-14)
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def test_rndm_naive_eval(self):
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# test random coefficient spline *on the base interval*,
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# t[k] <= x < t[-k-1]
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b = _make_random_spline()
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t, c, k = b.tck
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xx = np.linspace(t[k], t[-k-1], 50)
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y_b = b(xx)
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y_n = [_naive_eval(x, t, c, k) for x in xx]
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assert_allclose(y_b, y_n, atol=1e-14)
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y_n2 = [_naive_eval_2(x, t, c, k) for x in xx]
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assert_allclose(y_b, y_n2, atol=1e-14)
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def test_rndm_splev(self):
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b = _make_random_spline()
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t, c, k = b.tck
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xx = np.linspace(t[k], t[-k-1], 50)
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assert_allclose(b(xx), splev(xx, (t, c, k)), atol=1e-14)
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def test_rndm_splrep(self):
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np.random.seed(1234)
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x = np.sort(np.random.random(20))
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y = np.random.random(20)
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tck = splrep(x, y)
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b = BSpline(*tck)
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t, k = b.t, b.k
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xx = np.linspace(t[k], t[-k-1], 80)
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assert_allclose(b(xx), splev(xx, tck), atol=1e-14)
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def test_rndm_unity(self):
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b = _make_random_spline()
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b.c = np.ones_like(b.c)
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xx = np.linspace(b.t[b.k], b.t[-b.k-1], 100)
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assert_allclose(b(xx), 1.)
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def test_vectorization(self):
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n, k = 22, 3
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t = np.sort(np.random.random(n))
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c = np.random.random(size=(n, 6, 7))
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b = BSpline(t, c, k)
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tm, tp = t[k], t[-k-1]
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xx = tm + (tp - tm) * np.random.random((3, 4, 5))
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assert_equal(b(xx).shape, (3, 4, 5, 6, 7))
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def test_len_c(self):
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# for n+k+1 knots, only first n coefs are used.
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# and BTW this is consistent with FITPACK
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n, k = 33, 3
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t = np.sort(np.random.random(n+k+1))
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c = np.random.random(n)
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# pad coefficients with random garbage
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c_pad = np.r_[c, np.random.random(k+1)]
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b, b_pad = BSpline(t, c, k), BSpline(t, c_pad, k)
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dt = t[-1] - t[0]
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xx = np.linspace(t[0] - dt, t[-1] + dt, 50)
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assert_allclose(b(xx), b_pad(xx), atol=1e-14)
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assert_allclose(b(xx), splev(xx, (t, c, k)), atol=1e-14)
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assert_allclose(b(xx), splev(xx, (t, c_pad, k)), atol=1e-14)
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def test_endpoints(self):
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# base interval is closed
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b = _make_random_spline()
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t, _, k = b.tck
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tm, tp = t[k], t[-k-1]
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for extrap in (True, False):
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assert_allclose(b([tm, tp], extrap),
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b([tm + 1e-10, tp - 1e-10], extrap), atol=1e-9)
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def test_continuity(self):
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# assert continuity at internal knots
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b = _make_random_spline()
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t, _, k = b.tck
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assert_allclose(b(t[k+1:-k-1] - 1e-10), b(t[k+1:-k-1] + 1e-10),
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atol=1e-9)
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def test_extrap(self):
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b = _make_random_spline()
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t, c, k = b.tck
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dt = t[-1] - t[0]
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xx = np.linspace(t[k] - dt, t[-k-1] + dt, 50)
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mask = (t[k] < xx) & (xx < t[-k-1])
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# extrap has no effect within the base interval
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assert_allclose(b(xx[mask], extrapolate=True),
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b(xx[mask], extrapolate=False))
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# extrapolated values agree with FITPACK
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assert_allclose(b(xx, extrapolate=True),
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splev(xx, (t, c, k), ext=0))
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def test_default_extrap(self):
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# BSpline defaults to extrapolate=True
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b = _make_random_spline()
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t, _, k = b.tck
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xx = [t[0] - 1, t[-1] + 1]
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yy = b(xx)
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assert_(not np.all(np.isnan(yy)))
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def test_periodic_extrap(self):
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np.random.seed(1234)
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t = np.sort(np.random.random(8))
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c = np.random.random(4)
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k = 3
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b = BSpline(t, c, k, extrapolate='periodic')
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n = t.size - (k + 1)
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dt = t[-1] - t[0]
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xx = np.linspace(t[k] - dt, t[n] + dt, 50)
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xy = t[k] + (xx - t[k]) % (t[n] - t[k])
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assert_allclose(b(xx), splev(xy, (t, c, k)))
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# Direct check
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xx = [-1, 0, 0.5, 1]
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xy = t[k] + (xx - t[k]) % (t[n] - t[k])
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assert_equal(b(xx, extrapolate='periodic'), b(xy, extrapolate=True))
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def test_ppoly(self):
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b = _make_random_spline()
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t, c, k = b.tck
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pp = PPoly.from_spline((t, c, k))
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xx = np.linspace(t[k], t[-k], 100)
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assert_allclose(b(xx), pp(xx), atol=1e-14, rtol=1e-14)
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def test_derivative_rndm(self):
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b = _make_random_spline()
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t, c, k = b.tck
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xx = np.linspace(t[0], t[-1], 50)
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xx = np.r_[xx, t]
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for der in range(1, k+1):
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yd = splev(xx, (t, c, k), der=der)
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assert_allclose(yd, b(xx, nu=der), atol=1e-14)
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# higher derivatives all vanish
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assert_allclose(b(xx, nu=k+1), 0, atol=1e-14)
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def test_derivative_jumps(self):
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# example from de Boor, Chap IX, example (24)
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# NB: knots augmented & corresp coefs are zeroed out
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# in agreement with the convention (29)
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k = 2
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t = [-1, -1, 0, 1, 1, 3, 4, 6, 6, 6, 7, 7]
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np.random.seed(1234)
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c = np.r_[0, 0, np.random.random(5), 0, 0]
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b = BSpline(t, c, k)
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# b is continuous at x != 6 (triple knot)
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x = np.asarray([1, 3, 4, 6])
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assert_allclose(b(x[x != 6] - 1e-10),
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b(x[x != 6] + 1e-10))
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assert_(not np.allclose(b(6.-1e-10), b(6+1e-10)))
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# 1st derivative jumps at double knots, 1 & 6:
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x0 = np.asarray([3, 4])
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assert_allclose(b(x0 - 1e-10, nu=1),
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b(x0 + 1e-10, nu=1))
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x1 = np.asarray([1, 6])
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assert_(not np.all(np.allclose(b(x1 - 1e-10, nu=1),
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b(x1 + 1e-10, nu=1))))
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# 2nd derivative is not guaranteed to be continuous either
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assert_(not np.all(np.allclose(b(x - 1e-10, nu=2),
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b(x + 1e-10, nu=2))))
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def test_basis_element_quadratic(self):
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xx = np.linspace(-1, 4, 20)
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b = BSpline.basis_element(t=[0, 1, 2, 3])
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assert_allclose(b(xx),
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splev(xx, (b.t, b.c, b.k)), atol=1e-14)
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assert_allclose(b(xx),
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B_0123(xx), atol=1e-14)
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b = BSpline.basis_element(t=[0, 1, 1, 2])
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xx = np.linspace(0, 2, 10)
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assert_allclose(b(xx),
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np.where(xx < 1, xx*xx, (2.-xx)**2), atol=1e-14)
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def test_basis_element_rndm(self):
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b = _make_random_spline()
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t, c, k = b.tck
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xx = np.linspace(t[k], t[-k-1], 20)
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assert_allclose(b(xx), _sum_basis_elements(xx, t, c, k), atol=1e-14)
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def test_cmplx(self):
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b = _make_random_spline()
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t, c, k = b.tck
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cc = c * (1. + 3.j)
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b = BSpline(t, cc, k)
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b_re = BSpline(t, b.c.real, k)
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b_im = BSpline(t, b.c.imag, k)
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xx = np.linspace(t[k], t[-k-1], 20)
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assert_allclose(b(xx).real, b_re(xx), atol=1e-14)
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assert_allclose(b(xx).imag, b_im(xx), atol=1e-14)
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def test_nan(self):
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# nan in, nan out.
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b = BSpline.basis_element([0, 1, 1, 2])
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assert_(np.isnan(b(np.nan)))
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def test_derivative_method(self):
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b = _make_random_spline(k=5)
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t, c, k = b.tck
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b0 = BSpline(t, c, k)
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xx = np.linspace(t[k], t[-k-1], 20)
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for j in range(1, k):
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b = b.derivative()
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assert_allclose(b0(xx, j), b(xx), atol=1e-12, rtol=1e-12)
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def test_antiderivative_method(self):
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b = _make_random_spline()
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t, c, k = b.tck
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xx = np.linspace(t[k], t[-k-1], 20)
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assert_allclose(b.antiderivative().derivative()(xx),
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b(xx), atol=1e-14, rtol=1e-14)
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# repeat with N-D array for c
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c = np.c_[c, c, c]
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c = np.dstack((c, c))
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b = BSpline(t, c, k)
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assert_allclose(b.antiderivative().derivative()(xx),
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b(xx), atol=1e-14, rtol=1e-14)
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def test_integral(self):
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b = BSpline.basis_element([0, 1, 2]) # x for x < 1 else 2 - x
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assert_allclose(b.integrate(0, 1), 0.5)
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assert_allclose(b.integrate(1, 0), -1 * 0.5)
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assert_allclose(b.integrate(1, 0), -0.5)
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# extrapolate or zeros outside of [0, 2]; default is yes
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assert_allclose(b.integrate(-1, 1), 0)
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assert_allclose(b.integrate(-1, 1, extrapolate=True), 0)
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assert_allclose(b.integrate(-1, 1, extrapolate=False), 0.5)
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assert_allclose(b.integrate(1, -1, extrapolate=False), -1 * 0.5)
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# Test ``_fitpack._splint()``
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assert_allclose(b.integrate(1, -1, extrapolate=False),
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_impl.splint(1, -1, b.tck))
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# Test ``extrapolate='periodic'``.
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b.extrapolate = 'periodic'
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i = b.antiderivative()
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period_int = i(2) - i(0)
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assert_allclose(b.integrate(0, 2), period_int)
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assert_allclose(b.integrate(2, 0), -1 * period_int)
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assert_allclose(b.integrate(-9, -7), period_int)
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assert_allclose(b.integrate(-8, -4), 2 * period_int)
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assert_allclose(b.integrate(0.5, 1.5), i(1.5) - i(0.5))
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assert_allclose(b.integrate(1.5, 3), i(1) - i(0) + i(2) - i(1.5))
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assert_allclose(b.integrate(1.5 + 12, 3 + 12),
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i(1) - i(0) + i(2) - i(1.5))
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assert_allclose(b.integrate(1.5, 3 + 12),
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i(1) - i(0) + i(2) - i(1.5) + 6 * period_int)
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assert_allclose(b.integrate(0, -1), i(0) - i(1))
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assert_allclose(b.integrate(-9, -10), i(0) - i(1))
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assert_allclose(b.integrate(0, -9), i(1) - i(2) - 4 * period_int)
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def test_integrate_ppoly(self):
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# test .integrate method to be consistent with PPoly.integrate
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x = [0, 1, 2, 3, 4]
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b = make_interp_spline(x, x)
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b.extrapolate = 'periodic'
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p = PPoly.from_spline(b)
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for x0, x1 in [(-5, 0.5), (0.5, 5), (-4, 13)]:
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assert_allclose(b.integrate(x0, x1),
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p.integrate(x0, x1))
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def test_subclassing(self):
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# classmethods should not decay to the base class
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class B(BSpline):
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pass
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b = B.basis_element([0, 1, 2, 2])
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assert_equal(b.__class__, B)
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assert_equal(b.derivative().__class__, B)
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assert_equal(b.antiderivative().__class__, B)
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@pytest.mark.parametrize('axis', range(-4, 4))
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def test_axis(self, axis):
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n, k = 22, 3
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t = np.linspace(0, 1, n + k + 1)
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sh = [6, 7, 8]
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# We need the positive axis for some of the indexing and slices used
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# in this test.
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pos_axis = axis % 4
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sh.insert(pos_axis, n) # [22, 6, 7, 8] etc
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c = np.random.random(size=sh)
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b = BSpline(t, c, k, axis=axis)
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assert_equal(b.c.shape,
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[sh[pos_axis],] + sh[:pos_axis] + sh[pos_axis+1:])
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xp = np.random.random((3, 4, 5))
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assert_equal(b(xp).shape,
|
|
sh[:pos_axis] + list(xp.shape) + sh[pos_axis+1:])
|
|
|
|
# -c.ndim <= axis < c.ndim
|
|
for ax in [-c.ndim - 1, c.ndim]:
|
|
assert_raises(np.AxisError, BSpline,
|
|
**dict(t=t, c=c, k=k, axis=ax))
|
|
|
|
# derivative, antiderivative keeps the axis
|
|
for b1 in [BSpline(t, c, k, axis=axis).derivative(),
|
|
BSpline(t, c, k, axis=axis).derivative(2),
|
|
BSpline(t, c, k, axis=axis).antiderivative(),
|
|
BSpline(t, c, k, axis=axis).antiderivative(2)]:
|
|
assert_equal(b1.axis, b.axis)
|
|
|
|
def test_neg_axis(self):
|
|
k = 2
|
|
t = [0, 1, 2, 3, 4, 5, 6]
|
|
c = np.array([[-1, 2, 0, -1], [2, 0, -3, 1]])
|
|
|
|
spl = BSpline(t, c, k, axis=-1)
|
|
spl0 = BSpline(t, c[0], k)
|
|
spl1 = BSpline(t, c[1], k)
|
|
assert_equal(spl(2.5), [spl0(2.5), spl1(2.5)])
|
|
|
|
def test_design_matrix_bc_types(self):
|
|
'''
|
|
Splines with different boundary conditions are built on different
|
|
types of vectors of knots. As far as design matrix depends only on
|
|
vector of knots, `k` and `x` it is useful to make tests for different
|
|
boundary conditions (and as following different vectors of knots).
|
|
'''
|
|
def run_design_matrix_tests(n, k, bc_type):
|
|
'''
|
|
To avoid repetition of code the following function is provided.
|
|
'''
|
|
np.random.seed(1234)
|
|
x = np.sort(np.random.random_sample(n) * 40 - 20)
|
|
y = np.random.random_sample(n) * 40 - 20
|
|
if bc_type == "periodic":
|
|
y[0] = y[-1]
|
|
|
|
bspl = make_interp_spline(x, y, k=k, bc_type=bc_type)
|
|
|
|
c = np.eye(len(bspl.t) - k - 1)
|
|
des_matr_def = BSpline(bspl.t, c, k)(x)
|
|
des_matr_csr = BSpline.design_matrix(x,
|
|
bspl.t,
|
|
k).toarray()
|
|
assert_allclose(des_matr_csr @ bspl.c, y, atol=1e-14)
|
|
assert_allclose(des_matr_def, des_matr_csr, atol=1e-14)
|
|
|
|
# "clamped" and "natural" work only with `k = 3`
|
|
n = 11
|
|
k = 3
|
|
for bc in ["clamped", "natural"]:
|
|
run_design_matrix_tests(n, k, bc)
|
|
|
|
# "not-a-knot" works with odd `k`
|
|
for k in range(3, 8, 2):
|
|
run_design_matrix_tests(n, k, "not-a-knot")
|
|
|
|
# "periodic" works with any `k` (even more than `n`)
|
|
n = 5 # smaller `n` to test `k > n` case
|
|
for k in range(2, 7):
|
|
run_design_matrix_tests(n, k, "periodic")
|
|
|
|
@pytest.mark.parametrize('extrapolate', [False, True, 'periodic'])
|
|
@pytest.mark.parametrize('degree', range(5))
|
|
def test_design_matrix_same_as_BSpline_call(self, extrapolate, degree):
|
|
"""Test that design_matrix(x) is equivalent to BSpline(..)(x)."""
|
|
np.random.seed(1234)
|
|
x = np.random.random_sample(10 * (degree + 1))
|
|
xmin, xmax = np.amin(x), np.amax(x)
|
|
k = degree
|
|
t = np.r_[np.linspace(xmin - 2, xmin - 1, degree),
|
|
np.linspace(xmin, xmax, 2 * (degree + 1)),
|
|
np.linspace(xmax + 1, xmax + 2, degree)]
|
|
c = np.eye(len(t) - k - 1)
|
|
bspline = BSpline(t, c, k, extrapolate)
|
|
assert_allclose(
|
|
bspline(x), BSpline.design_matrix(x, t, k, extrapolate).toarray()
|
|
)
|
|
|
|
# extrapolation regime
|
|
x = np.array([xmin - 10, xmin - 1, xmax + 1.5, xmax + 10])
|
|
if not extrapolate:
|
|
with pytest.raises(ValueError):
|
|
BSpline.design_matrix(x, t, k, extrapolate)
|
|
else:
|
|
assert_allclose(
|
|
bspline(x),
|
|
BSpline.design_matrix(x, t, k, extrapolate).toarray()
|
|
)
|
|
|
|
def test_design_matrix_x_shapes(self):
|
|
# test for different `x` shapes
|
|
np.random.seed(1234)
|
|
n = 10
|
|
k = 3
|
|
x = np.sort(np.random.random_sample(n) * 40 - 20)
|
|
y = np.random.random_sample(n) * 40 - 20
|
|
|
|
bspl = make_interp_spline(x, y, k=k)
|
|
for i in range(1, 4):
|
|
xc = x[:i]
|
|
yc = y[:i]
|
|
des_matr_csr = BSpline.design_matrix(xc,
|
|
bspl.t,
|
|
k).toarray()
|
|
assert_allclose(des_matr_csr @ bspl.c, yc, atol=1e-14)
|
|
|
|
def test_design_matrix_t_shapes(self):
|
|
# test for minimal possible `t` shape
|
|
t = [1., 1., 1., 2., 3., 4., 4., 4.]
|
|
des_matr = BSpline.design_matrix(2., t, 3).toarray()
|
|
assert_allclose(des_matr,
|
|
[[0.25, 0.58333333, 0.16666667, 0.]],
|
|
atol=1e-14)
|
|
|
|
def test_design_matrix_asserts(self):
|
|
np.random.seed(1234)
|
|
n = 10
|
|
k = 3
|
|
x = np.sort(np.random.random_sample(n) * 40 - 20)
|
|
y = np.random.random_sample(n) * 40 - 20
|
|
bspl = make_interp_spline(x, y, k=k)
|
|
# invalid vector of knots (should be a 1D non-descending array)
|
|
# here the actual vector of knots is reversed, so it is invalid
|
|
with assert_raises(ValueError):
|
|
BSpline.design_matrix(x, bspl.t[::-1], k)
|
|
k = 2
|
|
t = [0., 1., 2., 3., 4., 5.]
|
|
x = [1., 2., 3., 4.]
|
|
# out of bounds
|
|
with assert_raises(ValueError):
|
|
BSpline.design_matrix(x, t, k)
|
|
|
|
@pytest.mark.parametrize('bc_type', ['natural', 'clamped',
|
|
'periodic', 'not-a-knot'])
|
|
def test_from_power_basis(self, bc_type):
|
|
np.random.seed(1234)
|
|
x = np.sort(np.random.random(20))
|
|
y = np.random.random(20)
|
|
if bc_type == 'periodic':
|
|
y[-1] = y[0]
|
|
cb = CubicSpline(x, y, bc_type=bc_type)
|
|
bspl = BSpline.from_power_basis(cb, bc_type=bc_type)
|
|
xx = np.linspace(0, 1, 20)
|
|
assert_allclose(cb(xx), bspl(xx), atol=1e-15)
|
|
bspl_new = make_interp_spline(x, y, bc_type=bc_type)
|
|
assert_allclose(bspl.c, bspl_new.c, atol=1e-15)
|
|
|
|
@pytest.mark.parametrize('bc_type', ['natural', 'clamped',
|
|
'periodic', 'not-a-knot'])
|
|
def test_from_power_basis_complex(self, bc_type):
|
|
np.random.seed(1234)
|
|
x = np.sort(np.random.random(20))
|
|
y = np.random.random(20) + np.random.random(20) * 1j
|
|
if bc_type == 'periodic':
|
|
y[-1] = y[0]
|
|
cb = CubicSpline(x, y, bc_type=bc_type)
|
|
bspl = BSpline.from_power_basis(cb, bc_type=bc_type)
|
|
bspl_new_real = make_interp_spline(x, y.real, bc_type=bc_type)
|
|
bspl_new_imag = make_interp_spline(x, y.imag, bc_type=bc_type)
|
|
assert_equal(bspl.c.dtype, (bspl_new_real.c
|
|
+ 1j * bspl_new_imag.c).dtype)
|
|
assert_allclose(bspl.c, bspl_new_real.c
|
|
+ 1j * bspl_new_imag.c, atol=1e-15)
|
|
|
|
def test_from_power_basis_exmp(self):
|
|
'''
|
|
For x = [0, 1, 2, 3, 4] and y = [1, 1, 1, 1, 1]
|
|
the coefficients of Cubic Spline in the power basis:
|
|
|
|
$[[0, 0, 0, 0, 0],\\$
|
|
$[0, 0, 0, 0, 0],\\$
|
|
$[0, 0, 0, 0, 0],\\$
|
|
$[1, 1, 1, 1, 1]]$
|
|
|
|
It could be shown explicitly that coefficients of the interpolating
|
|
function in B-spline basis are c = [1, 1, 1, 1, 1, 1, 1]
|
|
'''
|
|
x = np.array([0, 1, 2, 3, 4])
|
|
y = np.array([1, 1, 1, 1, 1])
|
|
bspl = BSpline.from_power_basis(CubicSpline(x, y, bc_type='natural'),
|
|
bc_type='natural')
|
|
assert_allclose(bspl.c, [1, 1, 1, 1, 1, 1, 1], atol=1e-15)
|
|
|
|
|
|
def test_knots_multiplicity():
|
|
# Take a spline w/ random coefficients, throw in knots of varying
|
|
# multiplicity.
|
|
|
|
def check_splev(b, j, der=0, atol=1e-14, rtol=1e-14):
|
|
# check evaluations against FITPACK, incl extrapolations
|
|
t, c, k = b.tck
|
|
x = np.unique(t)
|
|
x = np.r_[t[0]-0.1, 0.5*(x[1:] + x[:1]), t[-1]+0.1]
|
|
assert_allclose(splev(x, (t, c, k), der), b(x, der),
|
|
atol=atol, rtol=rtol, err_msg='der = %s k = %s' % (der, b.k))
|
|
|
|
# test loop itself
|
|
# [the index `j` is for interpreting the traceback in case of a failure]
|
|
for k in [1, 2, 3, 4, 5]:
|
|
b = _make_random_spline(k=k)
|
|
for j, b1 in enumerate(_make_multiples(b)):
|
|
check_splev(b1, j)
|
|
for der in range(1, k+1):
|
|
check_splev(b1, j, der, 1e-12, 1e-12)
|
|
|
|
|
|
### stolen from @pv, verbatim
|
|
def _naive_B(x, k, i, t):
|
|
"""
|
|
Naive way to compute B-spline basis functions. Useful only for testing!
|
|
computes B(x; t[i],..., t[i+k+1])
|
|
"""
|
|
if k == 0:
|
|
return 1.0 if t[i] <= x < t[i+1] else 0.0
|
|
if t[i+k] == t[i]:
|
|
c1 = 0.0
|
|
else:
|
|
c1 = (x - t[i])/(t[i+k] - t[i]) * _naive_B(x, k-1, i, t)
|
|
if t[i+k+1] == t[i+1]:
|
|
c2 = 0.0
|
|
else:
|
|
c2 = (t[i+k+1] - x)/(t[i+k+1] - t[i+1]) * _naive_B(x, k-1, i+1, t)
|
|
return (c1 + c2)
|
|
|
|
|
|
### stolen from @pv, verbatim
|
|
def _naive_eval(x, t, c, k):
|
|
"""
|
|
Naive B-spline evaluation. Useful only for testing!
|
|
"""
|
|
if x == t[k]:
|
|
i = k
|
|
else:
|
|
i = np.searchsorted(t, x) - 1
|
|
assert t[i] <= x <= t[i+1]
|
|
assert i >= k and i < len(t) - k
|
|
return sum(c[i-j] * _naive_B(x, k, i-j, t) for j in range(0, k+1))
|
|
|
|
|
|
def _naive_eval_2(x, t, c, k):
|
|
"""Naive B-spline evaluation, another way."""
|
|
n = len(t) - (k+1)
|
|
assert n >= k+1
|
|
assert len(c) >= n
|
|
assert t[k] <= x <= t[n]
|
|
return sum(c[i] * _naive_B(x, k, i, t) for i in range(n))
|
|
|
|
|
|
def _sum_basis_elements(x, t, c, k):
|
|
n = len(t) - (k+1)
|
|
assert n >= k+1
|
|
assert len(c) >= n
|
|
s = 0.
|
|
for i in range(n):
|
|
b = BSpline.basis_element(t[i:i+k+2], extrapolate=False)(x)
|
|
s += c[i] * np.nan_to_num(b) # zero out out-of-bounds elements
|
|
return s
|
|
|
|
|
|
def B_012(x):
|
|
""" A linear B-spline function B(x | 0, 1, 2)."""
|
|
x = np.atleast_1d(x)
|
|
return np.piecewise(x, [(x < 0) | (x > 2),
|
|
(x >= 0) & (x < 1),
|
|
(x >= 1) & (x <= 2)],
|
|
[lambda x: 0., lambda x: x, lambda x: 2.-x])
|
|
|
|
|
|
def B_0123(x, der=0):
|
|
"""A quadratic B-spline function B(x | 0, 1, 2, 3)."""
|
|
x = np.atleast_1d(x)
|
|
conds = [x < 1, (x > 1) & (x < 2), x > 2]
|
|
if der == 0:
|
|
funcs = [lambda x: x*x/2.,
|
|
lambda x: 3./4 - (x-3./2)**2,
|
|
lambda x: (3.-x)**2 / 2]
|
|
elif der == 2:
|
|
funcs = [lambda x: 1.,
|
|
lambda x: -2.,
|
|
lambda x: 1.]
|
|
else:
|
|
raise ValueError('never be here: der=%s' % der)
|
|
pieces = np.piecewise(x, conds, funcs)
|
|
return pieces
|
|
|
|
|
|
def _make_random_spline(n=35, k=3):
|
|
np.random.seed(123)
|
|
t = np.sort(np.random.random(n+k+1))
|
|
c = np.random.random(n)
|
|
return BSpline.construct_fast(t, c, k)
|
|
|
|
|
|
def _make_multiples(b):
|
|
"""Increase knot multiplicity."""
|
|
c, k = b.c, b.k
|
|
|
|
t1 = b.t.copy()
|
|
t1[17:19] = t1[17]
|
|
t1[22] = t1[21]
|
|
yield BSpline(t1, c, k)
|
|
|
|
t1 = b.t.copy()
|
|
t1[:k+1] = t1[0]
|
|
yield BSpline(t1, c, k)
|
|
|
|
t1 = b.t.copy()
|
|
t1[-k-1:] = t1[-1]
|
|
yield BSpline(t1, c, k)
|
|
|
|
|
|
class TestInterop:
|
|
#
|
|
# Test that FITPACK-based spl* functions can deal with BSpline objects
|
|
#
|
|
def setup_method(self):
|
|
xx = np.linspace(0, 4.*np.pi, 41)
|
|
yy = np.cos(xx)
|
|
b = make_interp_spline(xx, yy)
|
|
self.tck = (b.t, b.c, b.k)
|
|
self.xx, self.yy, self.b = xx, yy, b
|
|
|
|
self.xnew = np.linspace(0, 4.*np.pi, 21)
|
|
|
|
c2 = np.c_[b.c, b.c, b.c]
|
|
self.c2 = np.dstack((c2, c2))
|
|
self.b2 = BSpline(b.t, self.c2, b.k)
|
|
|
|
def test_splev(self):
|
|
xnew, b, b2 = self.xnew, self.b, self.b2
|
|
|
|
# check that splev works with 1-D array of coefficients
|
|
# for array and scalar `x`
|
|
assert_allclose(splev(xnew, b),
|
|
b(xnew), atol=1e-15, rtol=1e-15)
|
|
assert_allclose(splev(xnew, b.tck),
|
|
b(xnew), atol=1e-15, rtol=1e-15)
|
|
assert_allclose([splev(x, b) for x in xnew],
|
|
b(xnew), atol=1e-15, rtol=1e-15)
|
|
|
|
# With N-D coefficients, there's a quirck:
|
|
# splev(x, BSpline) is equivalent to BSpline(x)
|
|
with assert_raises(ValueError, match="Calling splev.. with BSpline"):
|
|
splev(xnew, b2)
|
|
|
|
# However, splev(x, BSpline.tck) needs some transposes. This is because
|
|
# BSpline interpolates along the first axis, while the legacy FITPACK
|
|
# wrapper does list(map(...)) which effectively interpolates along the
|
|
# last axis. Like so:
|
|
sh = tuple(range(1, b2.c.ndim)) + (0,) # sh = (1, 2, 0)
|
|
cc = b2.c.transpose(sh)
|
|
tck = (b2.t, cc, b2.k)
|
|
assert_allclose(splev(xnew, tck),
|
|
b2(xnew).transpose(sh), atol=1e-15, rtol=1e-15)
|
|
|
|
def test_splrep(self):
|
|
x, y = self.xx, self.yy
|
|
# test that "new" splrep is equivalent to _impl.splrep
|
|
tck = splrep(x, y)
|
|
t, c, k = _impl.splrep(x, y)
|
|
assert_allclose(tck[0], t, atol=1e-15)
|
|
assert_allclose(tck[1], c, atol=1e-15)
|
|
assert_equal(tck[2], k)
|
|
|
|
# also cover the `full_output=True` branch
|
|
tck_f, _, _, _ = splrep(x, y, full_output=True)
|
|
assert_allclose(tck_f[0], t, atol=1e-15)
|
|
assert_allclose(tck_f[1], c, atol=1e-15)
|
|
assert_equal(tck_f[2], k)
|
|
|
|
# test that the result of splrep roundtrips with splev:
|
|
# evaluate the spline on the original `x` points
|
|
yy = splev(x, tck)
|
|
assert_allclose(y, yy, atol=1e-15)
|
|
|
|
# ... and also it roundtrips if wrapped in a BSpline
|
|
b = BSpline(*tck)
|
|
assert_allclose(y, b(x), atol=1e-15)
|
|
|
|
def test_splrep_errors(self):
|
|
# test that both "old" and "new" splrep raise for an N-D ``y`` array
|
|
# with n > 1
|
|
x, y = self.xx, self.yy
|
|
y2 = np.c_[y, y]
|
|
with assert_raises(ValueError):
|
|
splrep(x, y2)
|
|
with assert_raises(ValueError):
|
|
_impl.splrep(x, y2)
|
|
|
|
# input below minimum size
|
|
with assert_raises(TypeError, match="m > k must hold"):
|
|
splrep(x[:3], y[:3])
|
|
with assert_raises(TypeError, match="m > k must hold"):
|
|
_impl.splrep(x[:3], y[:3])
|
|
|
|
def test_splprep(self):
|
|
x = np.arange(15).reshape((3, 5))
|
|
b, u = splprep(x)
|
|
tck, u1 = _impl.splprep(x)
|
|
|
|
# test the roundtrip with splev for both "old" and "new" output
|
|
assert_allclose(u, u1, atol=1e-15)
|
|
assert_allclose(splev(u, b), x, atol=1e-15)
|
|
assert_allclose(splev(u, tck), x, atol=1e-15)
|
|
|
|
# cover the ``full_output=True`` branch
|
|
(b_f, u_f), _, _, _ = splprep(x, s=0, full_output=True)
|
|
assert_allclose(u, u_f, atol=1e-15)
|
|
assert_allclose(splev(u_f, b_f), x, atol=1e-15)
|
|
|
|
def test_splprep_errors(self):
|
|
# test that both "old" and "new" code paths raise for x.ndim > 2
|
|
x = np.arange(3*4*5).reshape((3, 4, 5))
|
|
with assert_raises(ValueError, match="too many values to unpack"):
|
|
splprep(x)
|
|
with assert_raises(ValueError, match="too many values to unpack"):
|
|
_impl.splprep(x)
|
|
|
|
# input below minimum size
|
|
x = np.linspace(0, 40, num=3)
|
|
with assert_raises(TypeError, match="m > k must hold"):
|
|
splprep([x])
|
|
with assert_raises(TypeError, match="m > k must hold"):
|
|
_impl.splprep([x])
|
|
|
|
# automatically calculated parameters are non-increasing
|
|
# see gh-7589
|
|
x = [-50.49072266, -50.49072266, -54.49072266, -54.49072266]
|
|
with assert_raises(ValueError, match="Invalid inputs"):
|
|
splprep([x])
|
|
with assert_raises(ValueError, match="Invalid inputs"):
|
|
_impl.splprep([x])
|
|
|
|
# given non-increasing parameter values u
|
|
x = [1, 3, 2, 4]
|
|
u = [0, 0.3, 0.2, 1]
|
|
with assert_raises(ValueError, match="Invalid inputs"):
|
|
splprep(*[[x], None, u])
|
|
|
|
def test_sproot(self):
|
|
b, b2 = self.b, self.b2
|
|
roots = np.array([0.5, 1.5, 2.5, 3.5])*np.pi
|
|
# sproot accepts a BSpline obj w/ 1-D coef array
|
|
assert_allclose(sproot(b), roots, atol=1e-7, rtol=1e-7)
|
|
assert_allclose(sproot((b.t, b.c, b.k)), roots, atol=1e-7, rtol=1e-7)
|
|
|
|
# ... and deals with trailing dimensions if coef array is N-D
|
|
with assert_raises(ValueError, match="Calling sproot.. with BSpline"):
|
|
sproot(b2, mest=50)
|
|
|
|
# and legacy behavior is preserved for a tck tuple w/ N-D coef
|
|
c2r = b2.c.transpose(1, 2, 0)
|
|
rr = np.asarray(sproot((b2.t, c2r, b2.k), mest=50))
|
|
assert_equal(rr.shape, (3, 2, 4))
|
|
assert_allclose(rr - roots, 0, atol=1e-12)
|
|
|
|
def test_splint(self):
|
|
# test that splint accepts BSpline objects
|
|
b, b2 = self.b, self.b2
|
|
assert_allclose(splint(0, 1, b),
|
|
splint(0, 1, b.tck), atol=1e-14)
|
|
assert_allclose(splint(0, 1, b),
|
|
b.integrate(0, 1), atol=1e-14)
|
|
|
|
# ... and deals with N-D arrays of coefficients
|
|
with assert_raises(ValueError, match="Calling splint.. with BSpline"):
|
|
splint(0, 1, b2)
|
|
|
|
# and the legacy behavior is preserved for a tck tuple w/ N-D coef
|
|
c2r = b2.c.transpose(1, 2, 0)
|
|
integr = np.asarray(splint(0, 1, (b2.t, c2r, b2.k)))
|
|
assert_equal(integr.shape, (3, 2))
|
|
assert_allclose(integr,
|
|
splint(0, 1, b), atol=1e-14)
|
|
|
|
def test_splder(self):
|
|
for b in [self.b, self.b2]:
|
|
# pad the c array (FITPACK convention)
|
|
ct = len(b.t) - len(b.c)
|
|
if ct > 0:
|
|
b.c = np.r_[b.c, np.zeros((ct,) + b.c.shape[1:])]
|
|
|
|
for n in [1, 2, 3]:
|
|
bd = splder(b)
|
|
tck_d = _impl.splder((b.t, b.c, b.k))
|
|
assert_allclose(bd.t, tck_d[0], atol=1e-15)
|
|
assert_allclose(bd.c, tck_d[1], atol=1e-15)
|
|
assert_equal(bd.k, tck_d[2])
|
|
assert_(isinstance(bd, BSpline))
|
|
assert_(isinstance(tck_d, tuple)) # back-compat: tck in and out
|
|
|
|
def test_splantider(self):
|
|
for b in [self.b, self.b2]:
|
|
# pad the c array (FITPACK convention)
|
|
ct = len(b.t) - len(b.c)
|
|
if ct > 0:
|
|
b.c = np.r_[b.c, np.zeros((ct,) + b.c.shape[1:])]
|
|
|
|
for n in [1, 2, 3]:
|
|
bd = splantider(b)
|
|
tck_d = _impl.splantider((b.t, b.c, b.k))
|
|
assert_allclose(bd.t, tck_d[0], atol=1e-15)
|
|
assert_allclose(bd.c, tck_d[1], atol=1e-15)
|
|
assert_equal(bd.k, tck_d[2])
|
|
assert_(isinstance(bd, BSpline))
|
|
assert_(isinstance(tck_d, tuple)) # back-compat: tck in and out
|
|
|
|
def test_insert(self):
|
|
b, b2, xx = self.b, self.b2, self.xx
|
|
|
|
j = b.t.size // 2
|
|
tn = 0.5*(b.t[j] + b.t[j+1])
|
|
|
|
bn, tck_n = insert(tn, b), insert(tn, (b.t, b.c, b.k))
|
|
assert_allclose(splev(xx, bn),
|
|
splev(xx, tck_n), atol=1e-15)
|
|
assert_(isinstance(bn, BSpline))
|
|
assert_(isinstance(tck_n, tuple)) # back-compat: tck in, tck out
|
|
|
|
# for N-D array of coefficients, BSpline.c needs to be transposed
|
|
# after that, the results are equivalent.
|
|
sh = tuple(range(b2.c.ndim))
|
|
c_ = b2.c.transpose(sh[1:] + (0,))
|
|
tck_n2 = insert(tn, (b2.t, c_, b2.k))
|
|
|
|
bn2 = insert(tn, b2)
|
|
|
|
# need a transpose for comparing the results, cf test_splev
|
|
assert_allclose(np.asarray(splev(xx, tck_n2)).transpose(2, 0, 1),
|
|
bn2(xx), atol=1e-15)
|
|
assert_(isinstance(bn2, BSpline))
|
|
assert_(isinstance(tck_n2, tuple)) # back-compat: tck in, tck out
|
|
|
|
|
|
class TestInterp:
|
|
#
|
|
# Test basic ways of constructing interpolating splines.
|
|
#
|
|
xx = np.linspace(0., 2.*np.pi)
|
|
yy = np.sin(xx)
|
|
|
|
def test_non_int_order(self):
|
|
with assert_raises(TypeError):
|
|
make_interp_spline(self.xx, self.yy, k=2.5)
|
|
|
|
def test_order_0(self):
|
|
b = make_interp_spline(self.xx, self.yy, k=0)
|
|
assert_allclose(b(self.xx), self.yy, atol=1e-14, rtol=1e-14)
|
|
b = make_interp_spline(self.xx, self.yy, k=0, axis=-1)
|
|
assert_allclose(b(self.xx), self.yy, atol=1e-14, rtol=1e-14)
|
|
|
|
def test_linear(self):
|
|
b = make_interp_spline(self.xx, self.yy, k=1)
|
|
assert_allclose(b(self.xx), self.yy, atol=1e-14, rtol=1e-14)
|
|
b = make_interp_spline(self.xx, self.yy, k=1, axis=-1)
|
|
assert_allclose(b(self.xx), self.yy, atol=1e-14, rtol=1e-14)
|
|
|
|
@pytest.mark.parametrize('k', [0, 1, 2, 3])
|
|
def test_incompatible_x_y(self, k):
|
|
x = [0, 1, 2, 3, 4, 5]
|
|
y = [0, 1, 2, 3, 4, 5, 6, 7]
|
|
with assert_raises(ValueError, match="Shapes of x"):
|
|
make_interp_spline(x, y, k=k)
|
|
|
|
@pytest.mark.parametrize('k', [0, 1, 2, 3])
|
|
def test_broken_x(self, k):
|
|
x = [0, 1, 1, 2, 3, 4] # duplicates
|
|
y = [0, 1, 2, 3, 4, 5]
|
|
with assert_raises(ValueError, match="x to not have duplicates"):
|
|
make_interp_spline(x, y, k=k)
|
|
|
|
x = [0, 2, 1, 3, 4, 5] # unsorted
|
|
with assert_raises(ValueError, match="Expect x to be a 1D strictly"):
|
|
make_interp_spline(x, y, k=k)
|
|
|
|
x = [0, 1, 2, 3, 4, 5]
|
|
x = np.asarray(x).reshape((1, -1)) # 1D
|
|
with assert_raises(ValueError, match="Expect x to be a 1D strictly"):
|
|
make_interp_spline(x, y, k=k)
|
|
|
|
def test_not_a_knot(self):
|
|
for k in [3, 5]:
|
|
b = make_interp_spline(self.xx, self.yy, k)
|
|
assert_allclose(b(self.xx), self.yy, atol=1e-14, rtol=1e-14)
|
|
|
|
def test_periodic(self):
|
|
# k = 5 here for more derivatives
|
|
b = make_interp_spline(self.xx, self.yy, k=5, bc_type='periodic')
|
|
assert_allclose(b(self.xx), self.yy, atol=1e-14, rtol=1e-14)
|
|
# in periodic case it is expected equality of k-1 first
|
|
# derivatives at the boundaries
|
|
for i in range(1, 5):
|
|
assert_allclose(b(self.xx[0], nu=i), b(self.xx[-1], nu=i), atol=1e-11)
|
|
# tests for axis=-1
|
|
b = make_interp_spline(self.xx, self.yy, k=5, bc_type='periodic', axis=-1)
|
|
assert_allclose(b(self.xx), self.yy, atol=1e-14, rtol=1e-14)
|
|
for i in range(1, 5):
|
|
assert_allclose(b(self.xx[0], nu=i), b(self.xx[-1], nu=i), atol=1e-11)
|
|
|
|
@pytest.mark.parametrize('k', [2, 3, 4, 5, 6, 7])
|
|
def test_periodic_random(self, k):
|
|
# tests for both cases (k > n and k <= n)
|
|
n = 5
|
|
np.random.seed(1234)
|
|
x = np.sort(np.random.random_sample(n) * 10)
|
|
y = np.random.random_sample(n) * 100
|
|
y[0] = y[-1]
|
|
b = make_interp_spline(x, y, k=k, bc_type='periodic')
|
|
assert_allclose(b(x), y, atol=1e-14)
|
|
|
|
def test_periodic_axis(self):
|
|
n = self.xx.shape[0]
|
|
np.random.seed(1234)
|
|
x = np.random.random_sample(n) * 2 * np.pi
|
|
x = np.sort(x)
|
|
x[0] = 0.
|
|
x[-1] = 2 * np.pi
|
|
y = np.zeros((2, n))
|
|
y[0] = np.sin(x)
|
|
y[1] = np.cos(x)
|
|
b = make_interp_spline(x, y, k=5, bc_type='periodic', axis=1)
|
|
for i in range(n):
|
|
assert_allclose(b(x[i]), y[:, i], atol=1e-14)
|
|
assert_allclose(b(x[0]), b(x[-1]), atol=1e-14)
|
|
|
|
def test_periodic_points_exception(self):
|
|
# first and last points should match when periodic case expected
|
|
np.random.seed(1234)
|
|
k = 5
|
|
n = 8
|
|
x = np.sort(np.random.random_sample(n))
|
|
y = np.random.random_sample(n)
|
|
y[0] = y[-1] - 1 # to be sure that they are not equal
|
|
with assert_raises(ValueError):
|
|
make_interp_spline(x, y, k=k, bc_type='periodic')
|
|
|
|
def test_periodic_knots_exception(self):
|
|
# `periodic` case does not work with passed vector of knots
|
|
np.random.seed(1234)
|
|
k = 3
|
|
n = 7
|
|
x = np.sort(np.random.random_sample(n))
|
|
y = np.random.random_sample(n)
|
|
t = np.zeros(n + 2 * k)
|
|
with assert_raises(ValueError):
|
|
make_interp_spline(x, y, k, t, 'periodic')
|
|
|
|
@pytest.mark.parametrize('k', [2, 3, 4, 5])
|
|
def test_periodic_splev(self, k):
|
|
# comparision values of periodic b-spline with splev
|
|
b = make_interp_spline(self.xx, self.yy, k=k, bc_type='periodic')
|
|
tck = splrep(self.xx, self.yy, per=True, k=k)
|
|
spl = splev(self.xx, tck)
|
|
assert_allclose(spl, b(self.xx), atol=1e-14)
|
|
|
|
# comparison derivatives of periodic b-spline with splev
|
|
for i in range(1, k):
|
|
spl = splev(self.xx, tck, der=i)
|
|
assert_allclose(spl, b(self.xx, nu=i), atol=1e-10)
|
|
|
|
def test_periodic_cubic(self):
|
|
# comparison values of cubic periodic b-spline with CubicSpline
|
|
b = make_interp_spline(self.xx, self.yy, k=3, bc_type='periodic')
|
|
cub = CubicSpline(self.xx, self.yy, bc_type='periodic')
|
|
assert_allclose(b(self.xx), cub(self.xx), atol=1e-14)
|
|
|
|
# edge case: Cubic interpolation on 3 points
|
|
n = 3
|
|
x = np.sort(np.random.random_sample(n) * 10)
|
|
y = np.random.random_sample(n) * 100
|
|
y[0] = y[-1]
|
|
b = make_interp_spline(x, y, k=3, bc_type='periodic')
|
|
cub = CubicSpline(x, y, bc_type='periodic')
|
|
assert_allclose(b(x), cub(x), atol=1e-14)
|
|
|
|
def test_periodic_full_matrix(self):
|
|
# comparison values of cubic periodic b-spline with
|
|
# solution of the system with full matrix
|
|
k = 3
|
|
b = make_interp_spline(self.xx, self.yy, k=k, bc_type='periodic')
|
|
t = _periodic_knots(self.xx, k)
|
|
c = _make_interp_per_full_matr(self.xx, self.yy, t, k)
|
|
b1 = np.vectorize(lambda x: _naive_eval(x, t, c, k))
|
|
assert_allclose(b(self.xx), b1(self.xx), atol=1e-14)
|
|
|
|
def test_quadratic_deriv(self):
|
|
der = [(1, 8.)] # order, value: f'(x) = 8.
|
|
|
|
# derivative at right-hand edge
|
|
b = make_interp_spline(self.xx, self.yy, k=2, bc_type=(None, der))
|
|
assert_allclose(b(self.xx), self.yy, atol=1e-14, rtol=1e-14)
|
|
assert_allclose(b(self.xx[-1], 1), der[0][1], atol=1e-14, rtol=1e-14)
|
|
|
|
# derivative at left-hand edge
|
|
b = make_interp_spline(self.xx, self.yy, k=2, bc_type=(der, None))
|
|
assert_allclose(b(self.xx), self.yy, atol=1e-14, rtol=1e-14)
|
|
assert_allclose(b(self.xx[0], 1), der[0][1], atol=1e-14, rtol=1e-14)
|
|
|
|
def test_cubic_deriv(self):
|
|
k = 3
|
|
|
|
# first derivatives at left & right edges:
|
|
der_l, der_r = [(1, 3.)], [(1, 4.)]
|
|
b = make_interp_spline(self.xx, self.yy, k, bc_type=(der_l, der_r))
|
|
assert_allclose(b(self.xx), self.yy, atol=1e-14, rtol=1e-14)
|
|
assert_allclose([b(self.xx[0], 1), b(self.xx[-1], 1)],
|
|
[der_l[0][1], der_r[0][1]], atol=1e-14, rtol=1e-14)
|
|
|
|
# 'natural' cubic spline, zero out 2nd derivatives at the boundaries
|
|
der_l, der_r = [(2, 0)], [(2, 0)]
|
|
b = make_interp_spline(self.xx, self.yy, k, bc_type=(der_l, der_r))
|
|
assert_allclose(b(self.xx), self.yy, atol=1e-14, rtol=1e-14)
|
|
|
|
def test_quintic_derivs(self):
|
|
k, n = 5, 7
|
|
x = np.arange(n).astype(np.float_)
|
|
y = np.sin(x)
|
|
der_l = [(1, -12.), (2, 1)]
|
|
der_r = [(1, 8.), (2, 3.)]
|
|
b = make_interp_spline(x, y, k=k, bc_type=(der_l, der_r))
|
|
assert_allclose(b(x), y, atol=1e-14, rtol=1e-14)
|
|
assert_allclose([b(x[0], 1), b(x[0], 2)],
|
|
[val for (nu, val) in der_l])
|
|
assert_allclose([b(x[-1], 1), b(x[-1], 2)],
|
|
[val for (nu, val) in der_r])
|
|
|
|
@pytest.mark.xfail(reason='unstable')
|
|
def test_cubic_deriv_unstable(self):
|
|
# 1st and 2nd derivative at x[0], no derivative information at x[-1]
|
|
# The problem is not that it fails [who would use this anyway],
|
|
# the problem is that it fails *silently*, and I've no idea
|
|
# how to detect this sort of instability.
|
|
# In this particular case: it's OK for len(t) < 20, goes haywire
|
|
# at larger `len(t)`.
|
|
k = 3
|
|
t = _augknt(self.xx, k)
|
|
|
|
der_l = [(1, 3.), (2, 4.)]
|
|
b = make_interp_spline(self.xx, self.yy, k, t, bc_type=(der_l, None))
|
|
assert_allclose(b(self.xx), self.yy, atol=1e-14, rtol=1e-14)
|
|
|
|
def test_knots_not_data_sites(self):
|
|
# Knots need not coincide with the data sites.
|
|
# use a quadratic spline, knots are at data averages,
|
|
# two additional constraints are zero 2nd derivatives at edges
|
|
k = 2
|
|
t = np.r_[(self.xx[0],)*(k+1),
|
|
(self.xx[1:] + self.xx[:-1]) / 2.,
|
|
(self.xx[-1],)*(k+1)]
|
|
b = make_interp_spline(self.xx, self.yy, k, t,
|
|
bc_type=([(2, 0)], [(2, 0)]))
|
|
|
|
assert_allclose(b(self.xx), self.yy, atol=1e-14, rtol=1e-14)
|
|
assert_allclose([b(self.xx[0], 2), b(self.xx[-1], 2)], [0., 0.],
|
|
atol=1e-14)
|
|
|
|
def test_minimum_points_and_deriv(self):
|
|
# interpolation of f(x) = x**3 between 0 and 1. f'(x) = 3 * xx**2 and
|
|
# f'(0) = 0, f'(1) = 3.
|
|
k = 3
|
|
x = [0., 1.]
|
|
y = [0., 1.]
|
|
b = make_interp_spline(x, y, k, bc_type=([(1, 0.)], [(1, 3.)]))
|
|
|
|
xx = np.linspace(0., 1.)
|
|
yy = xx**3
|
|
assert_allclose(b(xx), yy, atol=1e-14, rtol=1e-14)
|
|
|
|
def test_deriv_spec(self):
|
|
# If one of the derivatives is omitted, the spline definition is
|
|
# incomplete.
|
|
x = y = [1.0, 2, 3, 4, 5, 6]
|
|
|
|
with assert_raises(ValueError):
|
|
make_interp_spline(x, y, bc_type=([(1, 0.)], None))
|
|
|
|
with assert_raises(ValueError):
|
|
make_interp_spline(x, y, bc_type=(1, 0.))
|
|
|
|
with assert_raises(ValueError):
|
|
make_interp_spline(x, y, bc_type=[(1, 0.)])
|
|
|
|
with assert_raises(ValueError):
|
|
make_interp_spline(x, y, bc_type=42)
|
|
|
|
# CubicSpline expects`bc_type=(left_pair, right_pair)`, while
|
|
# here we expect `bc_type=(iterable, iterable)`.
|
|
l, r = (1, 0.0), (1, 0.0)
|
|
with assert_raises(ValueError):
|
|
make_interp_spline(x, y, bc_type=(l, r))
|
|
|
|
def test_complex(self):
|
|
k = 3
|
|
xx = self.xx
|
|
yy = self.yy + 1.j*self.yy
|
|
|
|
# first derivatives at left & right edges:
|
|
der_l, der_r = [(1, 3.j)], [(1, 4.+2.j)]
|
|
b = make_interp_spline(xx, yy, k, bc_type=(der_l, der_r))
|
|
assert_allclose(b(xx), yy, atol=1e-14, rtol=1e-14)
|
|
assert_allclose([b(xx[0], 1), b(xx[-1], 1)],
|
|
[der_l[0][1], der_r[0][1]], atol=1e-14, rtol=1e-14)
|
|
|
|
# also test zero and first order
|
|
for k in (0, 1):
|
|
b = make_interp_spline(xx, yy, k=k)
|
|
assert_allclose(b(xx), yy, atol=1e-14, rtol=1e-14)
|
|
|
|
def test_int_xy(self):
|
|
x = np.arange(10).astype(np.int_)
|
|
y = np.arange(10).astype(np.int_)
|
|
|
|
# Cython chokes on "buffer type mismatch" (construction) or
|
|
# "no matching signature found" (evaluation)
|
|
for k in (0, 1, 2, 3):
|
|
b = make_interp_spline(x, y, k=k)
|
|
b(x)
|
|
|
|
def test_sliced_input(self):
|
|
# Cython code chokes on non C contiguous arrays
|
|
xx = np.linspace(-1, 1, 100)
|
|
|
|
x = xx[::5]
|
|
y = xx[::5]
|
|
|
|
for k in (0, 1, 2, 3):
|
|
make_interp_spline(x, y, k=k)
|
|
|
|
def test_check_finite(self):
|
|
# check_finite defaults to True; nans and such trigger a ValueError
|
|
x = np.arange(10).astype(float)
|
|
y = x**2
|
|
|
|
for z in [np.nan, np.inf, -np.inf]:
|
|
y[-1] = z
|
|
assert_raises(ValueError, make_interp_spline, x, y)
|
|
|
|
@pytest.mark.parametrize('k', [1, 2, 3, 5])
|
|
def test_list_input(self, k):
|
|
# regression test for gh-8714: TypeError for x, y being lists and k=2
|
|
x = list(range(10))
|
|
y = [a**2 for a in x]
|
|
make_interp_spline(x, y, k=k)
|
|
|
|
def test_multiple_rhs(self):
|
|
yy = np.c_[np.sin(self.xx), np.cos(self.xx)]
|
|
der_l = [(1, [1., 2.])]
|
|
der_r = [(1, [3., 4.])]
|
|
|
|
b = make_interp_spline(self.xx, yy, k=3, bc_type=(der_l, der_r))
|
|
assert_allclose(b(self.xx), yy, atol=1e-14, rtol=1e-14)
|
|
assert_allclose(b(self.xx[0], 1), der_l[0][1], atol=1e-14, rtol=1e-14)
|
|
assert_allclose(b(self.xx[-1], 1), der_r[0][1], atol=1e-14, rtol=1e-14)
|
|
|
|
def test_shapes(self):
|
|
np.random.seed(1234)
|
|
k, n = 3, 22
|
|
x = np.sort(np.random.random(size=n))
|
|
y = np.random.random(size=(n, 5, 6, 7))
|
|
|
|
b = make_interp_spline(x, y, k)
|
|
assert_equal(b.c.shape, (n, 5, 6, 7))
|
|
|
|
# now throw in some derivatives
|
|
d_l = [(1, np.random.random((5, 6, 7)))]
|
|
d_r = [(1, np.random.random((5, 6, 7)))]
|
|
b = make_interp_spline(x, y, k, bc_type=(d_l, d_r))
|
|
assert_equal(b.c.shape, (n + k - 1, 5, 6, 7))
|
|
|
|
def test_string_aliases(self):
|
|
yy = np.sin(self.xx)
|
|
|
|
# a single string is duplicated
|
|
b1 = make_interp_spline(self.xx, yy, k=3, bc_type='natural')
|
|
b2 = make_interp_spline(self.xx, yy, k=3, bc_type=([(2, 0)], [(2, 0)]))
|
|
assert_allclose(b1.c, b2.c, atol=1e-15)
|
|
|
|
# two strings are handled
|
|
b1 = make_interp_spline(self.xx, yy, k=3,
|
|
bc_type=('natural', 'clamped'))
|
|
b2 = make_interp_spline(self.xx, yy, k=3,
|
|
bc_type=([(2, 0)], [(1, 0)]))
|
|
assert_allclose(b1.c, b2.c, atol=1e-15)
|
|
|
|
# one-sided BCs are OK
|
|
b1 = make_interp_spline(self.xx, yy, k=2, bc_type=(None, 'clamped'))
|
|
b2 = make_interp_spline(self.xx, yy, k=2, bc_type=(None, [(1, 0.0)]))
|
|
assert_allclose(b1.c, b2.c, atol=1e-15)
|
|
|
|
# 'not-a-knot' is equivalent to None
|
|
b1 = make_interp_spline(self.xx, yy, k=3, bc_type='not-a-knot')
|
|
b2 = make_interp_spline(self.xx, yy, k=3, bc_type=None)
|
|
assert_allclose(b1.c, b2.c, atol=1e-15)
|
|
|
|
# unknown strings do not pass
|
|
with assert_raises(ValueError):
|
|
make_interp_spline(self.xx, yy, k=3, bc_type='typo')
|
|
|
|
# string aliases are handled for 2D values
|
|
yy = np.c_[np.sin(self.xx), np.cos(self.xx)]
|
|
der_l = [(1, [0., 0.])]
|
|
der_r = [(2, [0., 0.])]
|
|
b2 = make_interp_spline(self.xx, yy, k=3, bc_type=(der_l, der_r))
|
|
b1 = make_interp_spline(self.xx, yy, k=3,
|
|
bc_type=('clamped', 'natural'))
|
|
assert_allclose(b1.c, b2.c, atol=1e-15)
|
|
|
|
# ... and for N-D values:
|
|
np.random.seed(1234)
|
|
k, n = 3, 22
|
|
x = np.sort(np.random.random(size=n))
|
|
y = np.random.random(size=(n, 5, 6, 7))
|
|
|
|
# now throw in some derivatives
|
|
d_l = [(1, np.zeros((5, 6, 7)))]
|
|
d_r = [(1, np.zeros((5, 6, 7)))]
|
|
b1 = make_interp_spline(x, y, k, bc_type=(d_l, d_r))
|
|
b2 = make_interp_spline(x, y, k, bc_type='clamped')
|
|
assert_allclose(b1.c, b2.c, atol=1e-15)
|
|
|
|
def test_full_matrix(self):
|
|
np.random.seed(1234)
|
|
k, n = 3, 7
|
|
x = np.sort(np.random.random(size=n))
|
|
y = np.random.random(size=n)
|
|
t = _not_a_knot(x, k)
|
|
|
|
b = make_interp_spline(x, y, k, t)
|
|
cf = make_interp_full_matr(x, y, t, k)
|
|
assert_allclose(b.c, cf, atol=1e-14, rtol=1e-14)
|
|
|
|
def test_woodbury(self):
|
|
'''
|
|
Random elements in diagonal matrix with blocks in the
|
|
left lower and right upper corners checking the
|
|
implementation of Woodbury algorithm.
|
|
'''
|
|
np.random.seed(1234)
|
|
n = 201
|
|
for k in range(3, 32, 2):
|
|
offset = int((k - 1) / 2)
|
|
a = np.diagflat(np.random.random((1, n)))
|
|
for i in range(1, offset + 1):
|
|
a[:-i, i:] += np.diagflat(np.random.random((1, n - i)))
|
|
a[i:, :-i] += np.diagflat(np.random.random((1, n - i)))
|
|
ur = np.random.random((offset, offset))
|
|
a[:offset, -offset:] = ur
|
|
ll = np.random.random((offset, offset))
|
|
a[-offset:, :offset] = ll
|
|
d = np.zeros((k, n))
|
|
for i, j in enumerate(range(offset, -offset - 1, -1)):
|
|
if j < 0:
|
|
d[i, :j] = np.diagonal(a, offset=j)
|
|
else:
|
|
d[i, j:] = np.diagonal(a, offset=j)
|
|
b = np.random.random(n)
|
|
assert_allclose(_woodbury_algorithm(d, ur, ll, b, k),
|
|
np.linalg.solve(a, b), atol=1e-14)
|
|
|
|
|
|
def make_interp_full_matr(x, y, t, k):
|
|
"""Assemble an spline order k with knots t to interpolate
|
|
y(x) using full matrices.
|
|
Not-a-knot BC only.
|
|
|
|
This routine is here for testing only (even though it's functional).
|
|
"""
|
|
assert x.size == y.size
|
|
assert t.size == x.size + k + 1
|
|
n = x.size
|
|
|
|
A = np.zeros((n, n), dtype=np.float_)
|
|
|
|
for j in range(n):
|
|
xval = x[j]
|
|
if xval == t[k]:
|
|
left = k
|
|
else:
|
|
left = np.searchsorted(t, xval) - 1
|
|
|
|
# fill a row
|
|
bb = _bspl.evaluate_all_bspl(t, k, xval, left)
|
|
A[j, left-k:left+1] = bb
|
|
|
|
c = sl.solve(A, y)
|
|
return c
|
|
|
|
|
|
def make_lsq_full_matrix(x, y, t, k=3):
|
|
"""Make the least-square spline, full matrices."""
|
|
x, y, t = map(np.asarray, (x, y, t))
|
|
m = x.size
|
|
n = t.size - k - 1
|
|
|
|
A = np.zeros((m, n), dtype=np.float_)
|
|
|
|
for j in range(m):
|
|
xval = x[j]
|
|
# find interval
|
|
if xval == t[k]:
|
|
left = k
|
|
else:
|
|
left = np.searchsorted(t, xval) - 1
|
|
|
|
# fill a row
|
|
bb = _bspl.evaluate_all_bspl(t, k, xval, left)
|
|
A[j, left-k:left+1] = bb
|
|
|
|
# have observation matrix, can solve the LSQ problem
|
|
B = np.dot(A.T, A)
|
|
Y = np.dot(A.T, y)
|
|
c = sl.solve(B, Y)
|
|
|
|
return c, (A, Y)
|
|
|
|
|
|
class TestLSQ:
|
|
#
|
|
# Test make_lsq_spline
|
|
#
|
|
np.random.seed(1234)
|
|
n, k = 13, 3
|
|
x = np.sort(np.random.random(n))
|
|
y = np.random.random(n)
|
|
t = _augknt(np.linspace(x[0], x[-1], 7), k)
|
|
|
|
def test_lstsq(self):
|
|
# check LSQ construction vs a full matrix version
|
|
x, y, t, k = self.x, self.y, self.t, self.k
|
|
|
|
c0, AY = make_lsq_full_matrix(x, y, t, k)
|
|
b = make_lsq_spline(x, y, t, k)
|
|
|
|
assert_allclose(b.c, c0)
|
|
assert_equal(b.c.shape, (t.size - k - 1,))
|
|
|
|
# also check against numpy.lstsq
|
|
aa, yy = AY
|
|
c1, _, _, _ = np.linalg.lstsq(aa, y, rcond=-1)
|
|
assert_allclose(b.c, c1)
|
|
|
|
def test_weights(self):
|
|
# weights = 1 is same as None
|
|
x, y, t, k = self.x, self.y, self.t, self.k
|
|
w = np.ones_like(x)
|
|
|
|
b = make_lsq_spline(x, y, t, k)
|
|
b_w = make_lsq_spline(x, y, t, k, w=w)
|
|
|
|
assert_allclose(b.t, b_w.t, atol=1e-14)
|
|
assert_allclose(b.c, b_w.c, atol=1e-14)
|
|
assert_equal(b.k, b_w.k)
|
|
|
|
def test_multiple_rhs(self):
|
|
x, t, k, n = self.x, self.t, self.k, self.n
|
|
y = np.random.random(size=(n, 5, 6, 7))
|
|
|
|
b = make_lsq_spline(x, y, t, k)
|
|
assert_equal(b.c.shape, (t.size-k-1, 5, 6, 7))
|
|
|
|
def test_complex(self):
|
|
# cmplx-valued `y`
|
|
x, t, k = self.x, self.t, self.k
|
|
yc = self.y * (1. + 2.j)
|
|
|
|
b = make_lsq_spline(x, yc, t, k)
|
|
b_re = make_lsq_spline(x, yc.real, t, k)
|
|
b_im = make_lsq_spline(x, yc.imag, t, k)
|
|
|
|
assert_allclose(b(x), b_re(x) + 1.j*b_im(x), atol=1e-15, rtol=1e-15)
|
|
|
|
def test_int_xy(self):
|
|
x = np.arange(10).astype(np.int_)
|
|
y = np.arange(10).astype(np.int_)
|
|
t = _augknt(x, k=1)
|
|
# Cython chokes on "buffer type mismatch"
|
|
make_lsq_spline(x, y, t, k=1)
|
|
|
|
def test_sliced_input(self):
|
|
# Cython code chokes on non C contiguous arrays
|
|
xx = np.linspace(-1, 1, 100)
|
|
|
|
x = xx[::3]
|
|
y = xx[::3]
|
|
t = _augknt(x, 1)
|
|
make_lsq_spline(x, y, t, k=1)
|
|
|
|
def test_checkfinite(self):
|
|
# check_finite defaults to True; nans and such trigger a ValueError
|
|
x = np.arange(12).astype(float)
|
|
y = x**2
|
|
t = _augknt(x, 3)
|
|
|
|
for z in [np.nan, np.inf, -np.inf]:
|
|
y[-1] = z
|
|
assert_raises(ValueError, make_lsq_spline, x, y, t)
|
|
|
|
|
|
def data_file(basename):
|
|
return os.path.join(os.path.abspath(os.path.dirname(__file__)),
|
|
'data', basename)
|
|
|
|
|
|
class TestSmoothingSpline:
|
|
#
|
|
# test make_smoothing_spline
|
|
#
|
|
def test_invalid_input(self):
|
|
np.random.seed(1234)
|
|
n = 100
|
|
x = np.sort(np.random.random_sample(n) * 4 - 2)
|
|
y = x**2 * np.sin(4 * x) + x**3 + np.random.normal(0., 1.5, n)
|
|
|
|
# ``x`` and ``y`` should have same shapes (1-D array)
|
|
with assert_raises(ValueError):
|
|
make_smoothing_spline(x, y[1:])
|
|
with assert_raises(ValueError):
|
|
make_smoothing_spline(x[1:], y)
|
|
with assert_raises(ValueError):
|
|
make_smoothing_spline(x.reshape(1, n), y)
|
|
|
|
# ``x`` should be an ascending array
|
|
with assert_raises(ValueError):
|
|
make_smoothing_spline(x[::-1], y)
|
|
|
|
x_dupl = np.copy(x)
|
|
x_dupl[0] = x_dupl[1]
|
|
|
|
with assert_raises(ValueError):
|
|
make_smoothing_spline(x_dupl, y)
|
|
|
|
def test_compare_with_GCVSPL(self):
|
|
"""
|
|
Data is generated in the following way:
|
|
>>> np.random.seed(1234)
|
|
>>> n = 100
|
|
>>> x = np.sort(np.random.random_sample(n) * 4 - 2)
|
|
>>> y = np.sin(x) + np.random.normal(scale=.5, size=n)
|
|
>>> np.savetxt('x.csv', x)
|
|
>>> np.savetxt('y.csv', y)
|
|
|
|
We obtain the result of performing the GCV smoothing splines
|
|
package (by Woltring, gcvspl) on the sample data points
|
|
using its version for Octave (https://github.com/srkuberski/gcvspl).
|
|
In order to use this implementation, one should clone the repository
|
|
and open the folder in Octave.
|
|
In Octave, we load up ``x`` and ``y`` (generated from Python code
|
|
above):
|
|
|
|
>>> x = csvread('x.csv');
|
|
>>> y = csvread('y.csv');
|
|
|
|
Then, in order to access the implementation, we compile gcvspl files in
|
|
Octave:
|
|
|
|
>>> mex gcvsplmex.c gcvspl.c
|
|
>>> mex spldermex.c gcvspl.c
|
|
|
|
The first function computes the vector of unknowns from the dataset
|
|
(x, y) while the second one evaluates the spline in certain points
|
|
with known vector of coefficients.
|
|
|
|
>>> c = gcvsplmex( x, y, 2 );
|
|
>>> y0 = spldermex( x, c, 2, x, 0 );
|
|
|
|
If we want to compare the results of the gcvspl code, we can save
|
|
``y0`` in csv file:
|
|
|
|
>>> csvwrite('y0.csv', y0);
|
|
|
|
"""
|
|
# load the data sample
|
|
data = np.load(data_file('gcvspl.npz'))
|
|
# data points
|
|
x = data['x']
|
|
y = data['y']
|
|
|
|
y_GCVSPL = data['y_GCVSPL']
|
|
y_compr = make_smoothing_spline(x, y)(x)
|
|
|
|
# such tolerance is explained by the fact that the spline is built
|
|
# using an iterative algorithm for minimizing the GCV criteria. These
|
|
# algorithms may vary, so the tolerance should be rather low.
|
|
assert_allclose(y_compr, y_GCVSPL, atol=1e-4, rtol=1e-4)
|
|
|
|
def test_non_regularized_case(self):
|
|
"""
|
|
In case the regularization parameter is 0, the resulting spline
|
|
is an interpolation spline with natural boundary conditions.
|
|
"""
|
|
# create data sample
|
|
np.random.seed(1234)
|
|
n = 100
|
|
x = np.sort(np.random.random_sample(n) * 4 - 2)
|
|
y = x**2 * np.sin(4 * x) + x**3 + np.random.normal(0., 1.5, n)
|
|
|
|
spline_GCV = make_smoothing_spline(x, y, lam=0.)
|
|
spline_interp = make_interp_spline(x, y, 3, bc_type='natural')
|
|
|
|
grid = np.linspace(x[0], x[-1], 2 * n)
|
|
assert_allclose(spline_GCV(grid),
|
|
spline_interp(grid),
|
|
atol=1e-15)
|
|
|
|
def test_weighted_smoothing_spline(self):
|
|
# create data sample
|
|
np.random.seed(1234)
|
|
n = 100
|
|
x = np.sort(np.random.random_sample(n) * 4 - 2)
|
|
y = x**2 * np.sin(4 * x) + x**3 + np.random.normal(0., 1.5, n)
|
|
|
|
spl = make_smoothing_spline(x, y)
|
|
|
|
# in order not to iterate over all of the indices, we select 10 of
|
|
# them randomly
|
|
for ind in np.random.choice(range(100), size=10):
|
|
w = np.ones(n)
|
|
w[ind] = 30.
|
|
spl_w = make_smoothing_spline(x, y, w)
|
|
# check that spline with weight in a certain point is closer to the
|
|
# original point than the one without weights
|
|
orig = abs(spl(x[ind]) - y[ind])
|
|
weighted = abs(spl_w(x[ind]) - y[ind])
|
|
|
|
if orig < weighted:
|
|
raise ValueError(f'Spline with weights should be closer to the'
|
|
f' points than the original one: {orig:.4} < '
|
|
f'{weighted:.4}')
|