1224 lines
42 KiB
Python
1224 lines
42 KiB
Python
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from __future__ import division, absolute_import, print_function
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import numpy as np
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from numpy.testing import assert_equal, assert_allclose, assert_
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from scipy._lib._numpy_compat import suppress_warnings
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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, splder, splantider,
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sproot, splint, insert)
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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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import scipy.interpolate._fitpack_impl as _impl
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from scipy.interpolate._fitpack import _splint
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class TestBSpline(object):
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def test_ctor(self):
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# knots should be an ordered 1D 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(ValueError, 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(ValueError, 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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try:
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b.tck = 'foo'
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except AttributeError:
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pass
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except:
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raise AssertionError("AttributeError not raised.")
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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)
|
||
|
|
for j in range(1, k):
|
||
|
|
b = b.derivative()
|
||
|
|
assert_allclose(b0(xx, j), b(xx), atol=1e-12, rtol=1e-12)
|
||
|
|
|
||
|
|
def test_antiderivative_method(self):
|
||
|
|
b = _make_random_spline()
|
||
|
|
t, c, k = b.tck
|
||
|
|
xx = np.linspace(t[k], t[-k-1], 20)
|
||
|
|
assert_allclose(b.antiderivative().derivative()(xx),
|
||
|
|
b(xx), atol=1e-14, rtol=1e-14)
|
||
|
|
|
||
|
|
# repeat with n-D array for c
|
||
|
|
c = np.c_[c, c, c]
|
||
|
|
c = np.dstack((c, c))
|
||
|
|
b = BSpline(t, c, k)
|
||
|
|
assert_allclose(b.antiderivative().derivative()(xx),
|
||
|
|
b(xx), atol=1e-14, rtol=1e-14)
|
||
|
|
|
||
|
|
def test_integral(self):
|
||
|
|
b = BSpline.basis_element([0, 1, 2]) # x for x < 1 else 2 - x
|
||
|
|
assert_allclose(b.integrate(0, 1), 0.5)
|
||
|
|
assert_allclose(b.integrate(1, 0), -1 * 0.5)
|
||
|
|
assert_allclose(b.integrate(1, 0), -0.5)
|
||
|
|
|
||
|
|
# extrapolate or zeros outside of [0, 2]; default is yes
|
||
|
|
assert_allclose(b.integrate(-1, 1), 0)
|
||
|
|
assert_allclose(b.integrate(-1, 1, extrapolate=True), 0)
|
||
|
|
assert_allclose(b.integrate(-1, 1, extrapolate=False), 0.5)
|
||
|
|
assert_allclose(b.integrate(1, -1, extrapolate=False), -1 * 0.5)
|
||
|
|
|
||
|
|
# Test ``_fitpack._splint()``
|
||
|
|
t, c, k = b.tck
|
||
|
|
assert_allclose(b.integrate(1, -1, extrapolate=False),
|
||
|
|
_splint(t, c, k, 1, -1)[0])
|
||
|
|
|
||
|
|
# Test ``extrapolate='periodic'``.
|
||
|
|
b.extrapolate = 'periodic'
|
||
|
|
i = b.antiderivative()
|
||
|
|
period_int = i(2) - i(0)
|
||
|
|
|
||
|
|
assert_allclose(b.integrate(0, 2), period_int)
|
||
|
|
assert_allclose(b.integrate(2, 0), -1 * period_int)
|
||
|
|
assert_allclose(b.integrate(-9, -7), period_int)
|
||
|
|
assert_allclose(b.integrate(-8, -4), 2 * period_int)
|
||
|
|
|
||
|
|
assert_allclose(b.integrate(0.5, 1.5), i(1.5) - i(0.5))
|
||
|
|
assert_allclose(b.integrate(1.5, 3), i(1) - i(0) + i(2) - i(1.5))
|
||
|
|
assert_allclose(b.integrate(1.5 + 12, 3 + 12),
|
||
|
|
i(1) - i(0) + i(2) - i(1.5))
|
||
|
|
assert_allclose(b.integrate(1.5, 3 + 12),
|
||
|
|
i(1) - i(0) + i(2) - i(1.5) + 6 * period_int)
|
||
|
|
|
||
|
|
assert_allclose(b.integrate(0, -1), i(0) - i(1))
|
||
|
|
assert_allclose(b.integrate(-9, -10), i(0) - i(1))
|
||
|
|
assert_allclose(b.integrate(0, -9), i(1) - i(2) - 4 * period_int)
|
||
|
|
|
||
|
|
def test_integrate_ppoly(self):
|
||
|
|
# test .integrate method to be consistent with PPoly.integrate
|
||
|
|
x = [0, 1, 2, 3, 4]
|
||
|
|
b = make_interp_spline(x, x)
|
||
|
|
b.extrapolate = 'periodic'
|
||
|
|
p = PPoly.from_spline(b)
|
||
|
|
|
||
|
|
for x0, x1 in [(-5, 0.5), (0.5, 5), (-4, 13)]:
|
||
|
|
assert_allclose(b.integrate(x0, x1),
|
||
|
|
p.integrate(x0, x1))
|
||
|
|
|
||
|
|
def test_subclassing(self):
|
||
|
|
# classmethods should not decay to the base class
|
||
|
|
class B(BSpline):
|
||
|
|
pass
|
||
|
|
|
||
|
|
b = B.basis_element([0, 1, 2, 2])
|
||
|
|
assert_equal(b.__class__, B)
|
||
|
|
assert_equal(b.derivative().__class__, B)
|
||
|
|
assert_equal(b.antiderivative().__class__, B)
|
||
|
|
|
||
|
|
def test_axis(self):
|
||
|
|
n, k = 22, 3
|
||
|
|
t = np.linspace(0, 1, n + k + 1)
|
||
|
|
sh0 = [6, 7, 8]
|
||
|
|
for axis in range(4):
|
||
|
|
sh = sh0[:]
|
||
|
|
sh.insert(axis, n) # [22, 6, 7, 8] etc
|
||
|
|
c = np.random.random(size=sh)
|
||
|
|
b = BSpline(t, c, k, axis=axis)
|
||
|
|
assert_equal(b.c.shape,
|
||
|
|
[sh[axis],] + sh[:axis] + sh[axis+1:])
|
||
|
|
|
||
|
|
xp = np.random.random((3, 4, 5))
|
||
|
|
assert_equal(b(xp).shape,
|
||
|
|
sh[:axis] + list(xp.shape) + sh[axis+1:])
|
||
|
|
|
||
|
|
#0 <= axis < c.ndim
|
||
|
|
for ax in [-1, len(sh)+1]:
|
||
|
|
assert_raises(ValueError, 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_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(object):
|
||
|
|
#
|
||
|
|
# 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 1D 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 suppress_warnings() as sup:
|
||
|
|
sup.filter(DeprecationWarning,
|
||
|
|
"Calling splev.. with BSpline objects with c.ndim > 1 is not recommended.")
|
||
|
|
assert_allclose(splev(xnew, b2), b2(xnew), atol=1e-15, rtol=1e-15)
|
||
|
|
|
||
|
|
# 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]
|
||
|
|
msg = "failed in converting 3rd argument `y' of dfitpack.curfit to C/Fortran array"
|
||
|
|
with assert_raises(Exception, message=msg):
|
||
|
|
splrep(x, y2)
|
||
|
|
with assert_raises(Exception, message=msg):
|
||
|
|
_impl.splrep(x, y2)
|
||
|
|
|
||
|
|
# input below minimum size
|
||
|
|
with assert_raises(TypeError, message="m > k must hold"):
|
||
|
|
splrep(x[:3], y[:3])
|
||
|
|
with assert_raises(TypeError, message="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, message="too many values to unpack"):
|
||
|
|
splprep(x)
|
||
|
|
with assert_raises(ValueError, message="too many values to unpack"):
|
||
|
|
_impl.splprep(x)
|
||
|
|
|
||
|
|
# input below minimum size
|
||
|
|
x = np.linspace(0, 40, num=3)
|
||
|
|
with assert_raises(TypeError, message="m > k must hold"):
|
||
|
|
splprep([x])
|
||
|
|
with assert_raises(TypeError, message="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, message="Invalid inputs"):
|
||
|
|
splprep([x])
|
||
|
|
with assert_raises(ValueError, message="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, message="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/ 1D 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 suppress_warnings() as sup:
|
||
|
|
sup.filter(DeprecationWarning,
|
||
|
|
"Calling sproot.. with BSpline objects with c.ndim > 1 is not recommended.")
|
||
|
|
r = sproot(b2, mest=50)
|
||
|
|
r = np.asarray(r)
|
||
|
|
|
||
|
|
assert_equal(r.shape, (3, 2, 4))
|
||
|
|
assert_allclose(r - roots, 0, atol=1e-12)
|
||
|
|
|
||
|
|
# 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 suppress_warnings() as sup:
|
||
|
|
sup.filter(DeprecationWarning,
|
||
|
|
"Calling splint.. with BSpline objects with c.ndim > 1 is not recommended.")
|
||
|
|
assert_allclose(splint(0, 1, b2), b2.integrate(0, 1), atol=1e-14)
|
||
|
|
|
||
|
|
# 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(object):
|
||
|
|
#
|
||
|
|
# Test basic ways of constructing interpolating splines.
|
||
|
|
#
|
||
|
|
xx = np.linspace(0., 2.*np.pi)
|
||
|
|
yy = np.sin(xx)
|
||
|
|
|
||
|
|
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)
|
||
|
|
|
||
|
|
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)
|
||
|
|
|
||
|
|
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_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 derivs 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)
|
||
|
|
|
||
|
|
# If one of the derivatives is omitted, the spline definition is
|
||
|
|
# incomplete:
|
||
|
|
assert_raises(ValueError, make_interp_spline, x, y, k,
|
||
|
|
**dict(bc_type=([(1, 0.)], None)))
|
||
|
|
|
||
|
|
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 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
|
||
|
|
|
||
|
|
|
||
|
|
### XXX: 'periodic' interp spline using full matrices
|
||
|
|
def make_interp_per_full_matr(x, y, t, k):
|
||
|
|
x, y, t = map(np.asarray, (x, y, t))
|
||
|
|
|
||
|
|
n = x.size
|
||
|
|
nt = t.size - k - 1
|
||
|
|
|
||
|
|
# have `n` conditions for `nt` coefficients; need nt-n derivatives
|
||
|
|
assert nt - n == k - 1
|
||
|
|
|
||
|
|
# LHS: the collocation matrix + derivatives at edges
|
||
|
|
A = np.zeros((nt, nt), dtype=np.float_)
|
||
|
|
|
||
|
|
# derivatives at x[0]:
|
||
|
|
offset = 0
|
||
|
|
|
||
|
|
if x[0] == t[k]:
|
||
|
|
left = k
|
||
|
|
else:
|
||
|
|
left = np.searchsorted(t, x[0]) - 1
|
||
|
|
|
||
|
|
if x[-1] == t[k]:
|
||
|
|
left2 = k
|
||
|
|
else:
|
||
|
|
left2 = np.searchsorted(t, x[-1]) - 1
|
||
|
|
|
||
|
|
for i in range(k-1):
|
||
|
|
bb = _bspl.evaluate_all_bspl(t, k, x[0], left, nu=i+1)
|
||
|
|
A[i, left-k:left+1] = bb
|
||
|
|
bb = _bspl.evaluate_all_bspl(t, k, x[-1], left2, nu=i+1)
|
||
|
|
A[i, left2-k:left2+1] = -bb
|
||
|
|
offset += 1
|
||
|
|
|
||
|
|
# RHS
|
||
|
|
y = np.r_[[0]*(k-1), y]
|
||
|
|
|
||
|
|
# collocation matrix
|
||
|
|
for j in range(n):
|
||
|
|
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 + offset, 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(object):
|
||
|
|
#
|
||
|
|
# 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)
|
||
|
|
|