Current File : //lib/python3/dist-packages/scipy/interpolate/tests/test_bsplines.py
import numpy as np
from numpy.testing import (assert_equal, assert_allclose, assert_,
suppress_warnings)
from pytest import raises as assert_raises
import pytest
from scipy.interpolate import (BSpline, BPoly, PPoly, make_interp_spline,
make_lsq_spline, _bspl, splev, splrep, splprep, splder, splantider,
sproot, splint, insert, CubicSpline)
import scipy.linalg as sl
from scipy._lib import _pep440
from scipy.interpolate._bsplines import (_not_a_knot, _augknt,
_woodbury_algorithm, _periodic_knots,
_make_interp_per_full_matr)
import scipy.interpolate._fitpack_impl as _impl
from scipy.interpolate._fitpack import _splint
class TestBSpline:
def test_ctor(self):
# knots should be an ordered 1-D array of finite real numbers
assert_raises((TypeError, ValueError), BSpline,
**dict(t=[1, 1.j], c=[1.], k=0))
with np.errstate(invalid='ignore'):
assert_raises(ValueError, BSpline, **dict(t=[1, np.nan], c=[1.], k=0))
assert_raises(ValueError, BSpline, **dict(t=[1, np.inf], c=[1.], k=0))
assert_raises(ValueError, BSpline, **dict(t=[1, -1], c=[1.], k=0))
assert_raises(ValueError, BSpline, **dict(t=[[1], [1]], c=[1.], k=0))
# for n+k+1 knots and degree k need at least n coefficients
assert_raises(ValueError, BSpline, **dict(t=[0, 1, 2], c=[1], k=0))
assert_raises(ValueError, BSpline,
**dict(t=[0, 1, 2, 3, 4], c=[1., 1.], k=2))
# non-integer orders
assert_raises(TypeError, BSpline,
**dict(t=[0., 0., 1., 2., 3., 4.], c=[1., 1., 1.], k="cubic"))
assert_raises(TypeError, BSpline,
**dict(t=[0., 0., 1., 2., 3., 4.], c=[1., 1., 1.], k=2.5))
# basic interval cannot have measure zero (here: [1..1])
assert_raises(ValueError, BSpline,
**dict(t=[0., 0, 1, 1, 2, 3], c=[1., 1, 1], k=2))
# tck vs self.tck
n, k = 11, 3
t = np.arange(n+k+1)
c = np.random.random(n)
b = BSpline(t, c, k)
assert_allclose(t, b.t)
assert_allclose(c, b.c)
assert_equal(k, b.k)
def test_tck(self):
b = _make_random_spline()
tck = b.tck
assert_allclose(b.t, tck[0], atol=1e-15, rtol=1e-15)
assert_allclose(b.c, tck[1], atol=1e-15, rtol=1e-15)
assert_equal(b.k, tck[2])
# b.tck is read-only
with pytest.raises(AttributeError):
b.tck = 'foo'
def test_degree_0(self):
xx = np.linspace(0, 1, 10)
b = BSpline(t=[0, 1], c=[3.], k=0)
assert_allclose(b(xx), 3)
b = BSpline(t=[0, 0.35, 1], c=[3, 4], k=0)
assert_allclose(b(xx), np.where(xx < 0.35, 3, 4))
def test_degree_1(self):
t = [0, 1, 2, 3, 4]
c = [1, 2, 3]
k = 1
b = BSpline(t, c, k)
x = np.linspace(1, 3, 50)
assert_allclose(c[0]*B_012(x) + c[1]*B_012(x-1) + c[2]*B_012(x-2),
b(x), atol=1e-14)
assert_allclose(splev(x, (t, c, k)), b(x), atol=1e-14)
def test_bernstein(self):
# a special knot vector: Bernstein polynomials
k = 3
t = np.asarray([0]*(k+1) + [1]*(k+1))
c = np.asarray([1., 2., 3., 4.])
bp = BPoly(c.reshape(-1, 1), [0, 1])
bspl = BSpline(t, c, k)
xx = np.linspace(-1., 2., 10)
assert_allclose(bp(xx, extrapolate=True),
bspl(xx, extrapolate=True), atol=1e-14)
assert_allclose(splev(xx, (t, c, k)),
bspl(xx), atol=1e-14)
def test_rndm_naive_eval(self):
# test random coefficient spline *on the base interval*,
# t[k] <= x < t[-k-1]
b = _make_random_spline()
t, c, k = b.tck
xx = np.linspace(t[k], t[-k-1], 50)
y_b = b(xx)
y_n = [_naive_eval(x, t, c, k) for x in xx]
assert_allclose(y_b, y_n, atol=1e-14)
y_n2 = [_naive_eval_2(x, t, c, k) for x in xx]
assert_allclose(y_b, y_n2, atol=1e-14)
def test_rndm_splev(self):
b = _make_random_spline()
t, c, k = b.tck
xx = np.linspace(t[k], t[-k-1], 50)
assert_allclose(b(xx), splev(xx, (t, c, k)), atol=1e-14)
def test_rndm_splrep(self):
np.random.seed(1234)
x = np.sort(np.random.random(20))
y = np.random.random(20)
tck = splrep(x, y)
b = BSpline(*tck)
t, k = b.t, b.k
xx = np.linspace(t[k], t[-k-1], 80)
assert_allclose(b(xx), splev(xx, tck), atol=1e-14)
def test_rndm_unity(self):
b = _make_random_spline()
b.c = np.ones_like(b.c)
xx = np.linspace(b.t[b.k], b.t[-b.k-1], 100)
assert_allclose(b(xx), 1.)
def test_vectorization(self):
n, k = 22, 3
t = np.sort(np.random.random(n))
c = np.random.random(size=(n, 6, 7))
b = BSpline(t, c, k)
tm, tp = t[k], t[-k-1]
xx = tm + (tp - tm) * np.random.random((3, 4, 5))
assert_equal(b(xx).shape, (3, 4, 5, 6, 7))
def test_len_c(self):
# for n+k+1 knots, only first n coefs are used.
# and BTW this is consistent with FITPACK
n, k = 33, 3
t = np.sort(np.random.random(n+k+1))
c = np.random.random(n)
# pad coefficients with random garbage
c_pad = np.r_[c, np.random.random(k+1)]
b, b_pad = BSpline(t, c, k), BSpline(t, c_pad, k)
dt = t[-1] - t[0]
xx = np.linspace(t[0] - dt, t[-1] + dt, 50)
assert_allclose(b(xx), b_pad(xx), atol=1e-14)
assert_allclose(b(xx), splev(xx, (t, c, k)), atol=1e-14)
assert_allclose(b(xx), splev(xx, (t, c_pad, k)), atol=1e-14)
def test_endpoints(self):
# base interval is closed
b = _make_random_spline()
t, _, k = b.tck
tm, tp = t[k], t[-k-1]
for extrap in (True, False):
assert_allclose(b([tm, tp], extrap),
b([tm + 1e-10, tp - 1e-10], extrap), atol=1e-9)
def test_continuity(self):
# assert continuity at internal knots
b = _make_random_spline()
t, _, k = b.tck
assert_allclose(b(t[k+1:-k-1] - 1e-10), b(t[k+1:-k-1] + 1e-10),
atol=1e-9)
def test_extrap(self):
b = _make_random_spline()
t, c, k = b.tck
dt = t[-1] - t[0]
xx = np.linspace(t[k] - dt, t[-k-1] + dt, 50)
mask = (t[k] < xx) & (xx < t[-k-1])
# extrap has no effect within the base interval
assert_allclose(b(xx[mask], extrapolate=True),
b(xx[mask], extrapolate=False))
# extrapolated values agree with FITPACK
assert_allclose(b(xx, extrapolate=True),
splev(xx, (t, c, k), ext=0))
def test_default_extrap(self):
# BSpline defaults to extrapolate=True
b = _make_random_spline()
t, _, k = b.tck
xx = [t[0] - 1, t[-1] + 1]
yy = b(xx)
assert_(not np.all(np.isnan(yy)))
def test_periodic_extrap(self):
np.random.seed(1234)
t = np.sort(np.random.random(8))
c = np.random.random(4)
k = 3
b = BSpline(t, c, k, extrapolate='periodic')
n = t.size - (k + 1)
dt = t[-1] - t[0]
xx = np.linspace(t[k] - dt, t[n] + dt, 50)
xy = t[k] + (xx - t[k]) % (t[n] - t[k])
assert_allclose(b(xx), splev(xy, (t, c, k)))
# Direct check
xx = [-1, 0, 0.5, 1]
xy = t[k] + (xx - t[k]) % (t[n] - t[k])
assert_equal(b(xx, extrapolate='periodic'), b(xy, extrapolate=True))
def test_ppoly(self):
b = _make_random_spline()
t, c, k = b.tck
pp = PPoly.from_spline((t, c, k))
xx = np.linspace(t[k], t[-k], 100)
assert_allclose(b(xx), pp(xx), atol=1e-14, rtol=1e-14)
def test_derivative_rndm(self):
b = _make_random_spline()
t, c, k = b.tck
xx = np.linspace(t[0], t[-1], 50)
xx = np.r_[xx, t]
for der in range(1, k+1):
yd = splev(xx, (t, c, k), der=der)
assert_allclose(yd, b(xx, nu=der), atol=1e-14)
# higher derivatives all vanish
assert_allclose(b(xx, nu=k+1), 0, atol=1e-14)
def test_derivative_jumps(self):
# example from de Boor, Chap IX, example (24)
# NB: knots augmented & corresp coefs are zeroed out
# in agreement with the convention (29)
k = 2
t = [-1, -1, 0, 1, 1, 3, 4, 6, 6, 6, 7, 7]
np.random.seed(1234)
c = np.r_[0, 0, np.random.random(5), 0, 0]
b = BSpline(t, c, k)
# b is continuous at x != 6 (triple knot)
x = np.asarray([1, 3, 4, 6])
assert_allclose(b(x[x != 6] - 1e-10),
b(x[x != 6] + 1e-10))
assert_(not np.allclose(b(6.-1e-10), b(6+1e-10)))
# 1st derivative jumps at double knots, 1 & 6:
x0 = np.asarray([3, 4])
assert_allclose(b(x0 - 1e-10, nu=1),
b(x0 + 1e-10, nu=1))
x1 = np.asarray([1, 6])
assert_(not np.all(np.allclose(b(x1 - 1e-10, nu=1),
b(x1 + 1e-10, nu=1))))
# 2nd derivative is not guaranteed to be continuous either
assert_(not np.all(np.allclose(b(x - 1e-10, nu=2),
b(x + 1e-10, nu=2))))
def test_basis_element_quadratic(self):
xx = np.linspace(-1, 4, 20)
b = BSpline.basis_element(t=[0, 1, 2, 3])
assert_allclose(b(xx),
splev(xx, (b.t, b.c, b.k)), atol=1e-14)
assert_allclose(b(xx),
B_0123(xx), atol=1e-14)
b = BSpline.basis_element(t=[0, 1, 1, 2])
xx = np.linspace(0, 2, 10)
assert_allclose(b(xx),
np.where(xx < 1, xx*xx, (2.-xx)**2), atol=1e-14)
def test_basis_element_rndm(self):
b = _make_random_spline()
t, c, k = b.tck
xx = np.linspace(t[k], t[-k-1], 20)
assert_allclose(b(xx), _sum_basis_elements(xx, t, c, k), atol=1e-14)
def test_cmplx(self):
b = _make_random_spline()
t, c, k = b.tck
cc = c * (1. + 3.j)
b = BSpline(t, cc, k)
b_re = BSpline(t, b.c.real, k)
b_im = BSpline(t, b.c.imag, k)
xx = np.linspace(t[k], t[-k-1], 20)
assert_allclose(b(xx).real, b_re(xx), atol=1e-14)
assert_allclose(b(xx).imag, b_im(xx), atol=1e-14)
def test_nan(self):
# nan in, nan out.
b = BSpline.basis_element([0, 1, 1, 2])
assert_(np.isnan(b(np.nan)))
def test_derivative_method(self):
b = _make_random_spline(k=5)
t, c, k = b.tck
b0 = BSpline(t, c, k)
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)
@pytest.mark.parametrize('axis', range(-4, 4))
def test_axis(self, axis):
n, k = 22, 3
t = np.linspace(0, 1, n + k + 1)
sh = [6, 7, 8]
# We need the positive axis for some of the indexing and slices used
# in this test.
pos_axis = axis % 4
sh.insert(pos_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[pos_axis],] + sh[:pos_axis] + sh[pos_axis+1:])
xp = np.random.random((3, 4, 5))
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 the 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")
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 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)
@pytest.mark.xfail(_pep440.parse(np.__version__) < _pep440.Version('1.14.0'),
reason='requires NumPy >= 1.14.0')
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 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:
#
# 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)
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)
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