def test_roots_discont(self):
# Check that a discontinuity across zero is reported as root
c = np.array([[1], [-1]]).T
x = np.array([0, 0.5, 1])
pp = PPoly(c, x)
assert_array_equal(pp.roots(), [0.5])
assert_array_equal(pp.roots(discontinuity=False), [])
def test_roots_discont(self):
# Check that a discontinuity across zero is reported as root
c = np.array([[1], [-1]]).T
x = np.array([0, 0.5, 1])
pp = PPoly(c, x)
assert_array_equal(pp.roots(), [0.5])
assert_array_equal(pp.roots(discontinuity=False), [])
def test_roots_random(self):
# Check high-order polynomials with random coefficients
np.random.seed(1234)
num = 0
for extrapolate in (True, False):
for order in range(0, 20):
x = np.unique(np.r_[0, 10 * np.random.rand(30), 10])
c = 2 * np.random.rand(order + 1, len(x) - 1, 2, 3) - 1
pp = PPoly(c, x)
r = pp.roots(discontinuity=False, extrapolate=extrapolate)
for i in range(2):
for j in range(3):
rr = r[i, j]
if rr.size > 0:
# Check that the reported roots indeed are roots
num += rr.size
val = pp(rr, extrapolate=extrapolate)[:, i, j]
cmpval = pp(rr, nu=1,
extrapolate=extrapolate)[:, i, j]
assert_allclose(val / cmpval,
0,
atol=1e-7,
err_msg="(%r) r = %s" % (
extrapolate,
repr(rr),
))
# Check that we checked a number of roots
assert_(num > 100, repr(num))
def test_roots_random(self):
# Check high-order polynomials with random coefficients
np.random.seed(1234)
num = 0
for extrapolate in (True, False):
for order in range(0, 20):
x = np.unique(np.r_[0, 10 * np.random.rand(30), 10])
c = 2*np.random.rand(order+1, len(x)-1, 2, 3) - 1
pp = PPoly(c, x)
r = pp.roots(discontinuity=False, extrapolate=extrapolate)
for i in range(2):
for j in range(3):
rr = r[i,j]
if rr.size > 0:
# Check that the reported roots indeed are roots
num += rr.size
val = pp(rr, extrapolate=extrapolate)[:,i,j]
cmpval = pp(rr, nu=1, extrapolate=extrapolate)[:,i,j]
assert_allclose(val/cmpval, 0, atol=1e-7,
err_msg="(%r) r = %s" % (extrapolate,
repr(rr),))
# Check that we checked a number of roots
assert_(num > 100, repr(num))
def test_roots_idzero(self):
# Roots for piecewise polynomials with identically zero
# sections.
c = np.array([[-1, 0.25], [0, 0], [-1, 0.25]]).T
x = np.array([0, 0.4, 0.6, 1.0])
pp = PPoly(c, x)
assert_array_equal(pp.roots(), [0.25, 0.4, np.nan, 0.6 + 0.25])
def test_roots_idzero(self):
# Roots for piecewise polynomials with identically zero
# sections.
c = np.array([[-1, 0.25], [0, 0], [-1, 0.25]]).T
x = np.array([0, 0.4, 0.6, 1.0])
pp = PPoly(c, x)
assert_array_equal(pp.roots(),
[0.25, 0.4, np.nan, 0.6 + 0.25])
def _get_piecewise_mean_price_vs_size_from_orderbook_entry(orders):
""" orders is just asks or just orders """
cm = [0] + [x['cm'] for x in orders]
# integral (price times qty) d_qty / qty
# represent this as integral of piecewise polynomial with coeff [0, price]
price = np.zeros((2, len(cm) - 1))
price[1, :] = [x['price'] for x in orders]
f = PPoly(price, cm, extrapolate=False)
F = f.antiderivative()
return lambda x: F(x) / x
def test_bp_from_pp(self):
x = [0, 1, 3]
c = [[3, 2], [1, 8], [4, 3]]
pp = PPoly(c, x)
bp = BPoly.from_power_basis(pp)
pp1 = PPoly.from_bernstein_basis(bp)
xp = [0.1, 1.4]
assert_allclose(pp(xp), bp(xp))
assert_allclose(pp(xp), pp1(xp))
def _make_ppoly():
if n_waypoints == 1:
waypoints = _extract_waypoints(0)
pp_coeffs = np.zeros((1, 1, self.dof))
for idof in range(self.dof):
pp_coeffs[0, 0, idof] = waypoints[0, idof]
return PPoly(pp_coeffs, [0, 1])
if self._interpolation == "quadratic":
waypoints = _extract_waypoints(0)
waypoints_d = _extract_waypoints(1)
waypoints_dd = []
for i in range(n_waypoints - 1):
qdd = (waypoints_d[i + 1] - waypoints_d[i]) / (
self.ss_waypoints[i + 1] - self.ss_waypoints[i]
)
waypoints_dd.append(qdd)
waypoints_dd = np.array(waypoints_dd)
# Fill the coefficient matrix for scipy.PPoly class
pp_coeffs = np.zeros((3, n_waypoints - 1, self.dof))
for idof in range(self.dof):
for iseg in range(n_waypoints - 1):
pp_coeffs[:, iseg, idof] = [
waypoints_dd[iseg, idof] / 2,
waypoints_d[iseg, idof],
waypoints[iseg, idof],
]
return PPoly(pp_coeffs, self.ss_waypoints)
if self._interpolation == "cubic":
waypoints = _extract_waypoints(0)
waypoints_d = _extract_waypoints(1)
waypoints_dd = _extract_waypoints(2)
waypoints_ddd = []
for i in range(n_waypoints - 1):
qddd = (waypoints_dd[i + 1] - waypoints_dd[i]) / (
self.ss_waypoints[i + 1] - self.ss_waypoints[i]
)
waypoints_ddd.append(qddd)
waypoints_ddd = np.array(waypoints_ddd)
# Fill the coefficient matrix for scipy.PPoly class
pp_coeffs = np.zeros((4, n_waypoints - 1, self.dof))
for idof in range(self.dof):
for iseg in range(n_waypoints - 1):
pp_coeffs[:, iseg, idof] = [
waypoints_ddd[iseg, idof] / 6,
waypoints_dd[iseg, idof] / 2,
waypoints_d[iseg, idof],
waypoints[iseg, idof],
]
return PPoly(pp_coeffs, self.ss_waypoints)
raise ValueError("An error has occured. Unable to form PPoly.")
def test_roots_repeated(self):
# Check roots repeated in multiple sections are reported only
# once.
# [(x + 1)**2 - 1, -x**2] ; x == 0 is a repeated root
c = np.array([[1, 0, -1], [-1, 0, 0]]).T
x = np.array([-1, 0, 1])
pp = PPoly(c, x)
assert_array_equal(pp.roots(), [-2, 0])
assert_array_equal(pp.roots(extrapolate=False), [0])
def test_roots_repeated(self):
# Check roots repeated in multiple sections are reported only
# once.
# [(x + 1)**2 - 1, -x**2] ; x == 0 is a repeated root
c = np.array([[1, 0, -1], [-1, 0, 0]]).T
x = np.array([-1, 0, 1])
pp = PPoly(c, x)
assert_array_equal(pp.roots(), [-2, 0])
assert_array_equal(pp.roots(extrapolate=False), [0])
def test_bp_from_pp_random(self):
np.random.seed(1234)
m, k = 5, 8 # number of intervals, order
x = np.sort(np.random.random(m))
c = np.random.random((k, m - 1))
pp = PPoly(c, x)
bp = BPoly.from_power_basis(pp)
pp1 = PPoly.from_bernstein_basis(bp)
xp = np.linspace(x[0], x[-1], 21)
assert_allclose(pp(xp), bp(xp))
assert_allclose(pp(xp), pp1(xp))
def test_derivative_simple(self):
np.random.seed(1234)
c = np.array([[4, 3, 2, 1]]).T
dc = np.array([[3*4, 2*3, 2]]).T
ddc = np.array([[2*3*4, 1*2*3]]).T
x = np.array([0, 1])
pp = PPoly(c, x)
dpp = PPoly(dc, x)
ddpp = PPoly(ddc, x)
assert_allclose(pp.derivative().c, dpp.c)
assert_allclose(pp.derivative(2).c, ddpp.c)
def test_multi_shape(self):
c = np.random.rand(6, 2, 1, 2, 3)
x = np.array([0, 0.5, 1])
p = PPoly(c, x)
assert_equal(p.x.shape, x.shape)
assert_equal(p.c.shape, c.shape)
assert_equal(p(0.3).shape, c.shape[2:])
assert_equal(p(np.random.rand(5, 6)).shape, (5, 6) + c.shape[2:])
dp = p.derivative()
assert_equal(dp.c.shape, (5, 2, 1, 2, 3))
ip = p.antiderivative()
assert_equal(ip.c.shape, (7, 2, 1, 2, 3))
def __init__(self, poly_coeff_set):
self.poly_coeff_set = np.array(poly_coeff_set)
self.shape = np.array(self.poly_coeff_set.shape)
self.amount_segment = self.shape[1]
self.dof = self.shape[2]
self.degree_poly = self.shape[0]
self.s_start = 0
self.s_end = self.amount_segment
self.duration = self.amount_segment
self.s_waypoint = np.arange(0, self.duration + 1, 1)
self.picecwise_poly = PPoly(self.poly_coeff_set, self.s_waypoint)
self.picecwise_poly_s = self.picecwise_poly.derivative()
self.picecwise_poly_ss = self.picecwise_poly_s.derivative()
import ipdb
ipdb.set_trace()
def test_multi_shape(self):
c = np.random.rand(6, 2, 1, 2, 3)
x = np.array([0, 0.5, 1])
p = PPoly(c, x)
assert_equal(p.x.shape, x.shape)
assert_equal(p.c.shape, c.shape)
assert_equal(p(0.3).shape, c.shape[2:])
assert_equal(p(np.random.rand(5,6)).shape,
(5,6) + c.shape[2:])
dp = p.derivative()
assert_equal(dp.c.shape, (5, 2, 1, 2, 3))
ip = p.antiderivative()
assert_equal(ip.c.shape, (7, 2, 1, 2, 3))
def test_construct_fast(self):
np.random.seed(1234)
c = np.array([[1, 4], [2, 5], [3, 6]], dtype=float)
x = np.array([0, 0.5, 1])
p = PPoly.construct_fast(c, x)
assert_allclose(p(0.3), 1 * 0.3**2 + 2 * 0.3 + 3)
assert_allclose(p(0.7), 4 * (0.7 - 0.5)**2 + 5 * (0.7 - 0.5) + 6)
def test_antiderivative_vs_derivative(self):
np.random.seed(1234)
x = np.linspace(0, 1, 30)**2
y = np.random.rand(len(x))
spl = splrep(x, y, s=0, k=5)
pp = PPoly.from_spline(spl)
for dx in range(0, 10):
ipp = pp.antiderivative(dx)
# check that derivative is inverse op
pp2 = ipp.derivative(dx)
assert_allclose(pp.c, pp2.c)
# check continuity
for k in range(dx):
pp2 = ipp.derivative(k)
r = 1e-13
endpoint = r * pp2.x[:-1] + (1 - r) * pp2.x[1:]
assert_allclose(pp2(pp2.x[1:]),
pp2(endpoint),
rtol=1e-7,
err_msg="dx=%d k=%d" % (dx, k))
def get_piecewise_price_vs_size_from_orderbook_entry(orders, mean=True):
""" orders is just asks or just orders. takes maybe 300 micros though timeit reports much less """
if not orders:
return None
cm = [0] + [x['cm'] for x in orders]
# integral (price times qty) d_qty / qty
# represent this as integral of piecewise polynomial with coeff [0, price]
price = np.zeros((2, len(cm)-1))
price[1,:] = [x['price'] for x in orders]
f = PPoly(price, cm, extrapolate=False)
F = f.antiderivative()
if mean:
# generally you want mean price if you took out the stack up to some size
return lambda x: F(x) / x
else:
return F
def test_size_history_R(rnd_eta):
p = rnd_eta
q = PPoly(x=p.t, c=[1.0 / p.Ne]).antiderivative()
for t in np.random.rand(100) * len(p.t):
assert abs(p.R(t) - q(t)) < 1e-4
for t1, t2 in np.random.rand(100, 2) * len(p.t):
assert abs((p.R(t1 + t2) - p.R(t1)) - (q(t1 + t2) - q(t1))) < 1e-4
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_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 cubic_interp(obs_t, cum_obs):
"""
Construct a cubic count interpolant
(which for monotonic counts is a quadratic rate)
"""
# extend with null counts
# so that it extrapolates, but conservatively
obs_t = np.concatenate([
[obs_t[0] - 2, obs_t[0] - 1],
obs_t,
[obs_t[-1] + 1, obs_t[-1] + 2]
])
cum_obs = np.concatenate([
[cum_obs[0], cum_obs[0]],
cum_obs,
[cum_obs[-1], cum_obs[-1]]
])
big_n_hat = PPoly.from_bernstein_basis(
PchipInterpolator(
obs_t,
cum_obs,
extrapolate=True
)
)
return big_n_hat
def curve_from_controls(durations, accels, t0=0., x0=0., v0=0.):
assert len(durations) == len(accels)
#from numpy.polynomial import Polynomial
#t = Polynomial.identity()
times = [t0]
positions = [x0]
velocities = [v0]
coeffs = []
for duration, accel in zip(durations, accels):
assert duration >= 0.
coeff = [0.5 * accel, 1. * velocities[-1], positions[-1]] # 0. jerk
coeffs.append(coeff)
times.append(times[-1] + duration)
p_curve = np.poly1d(coeff) # Not centered
positions.append(p_curve(duration))
v_curve = p_curve.deriv() # Not centered
velocities.append(v_curve(duration))
# print(positions)
# print(velocities)
#np.piecewise
# max_order = max(p_curve.order for p_curve in p_curves)
# coeffs = np.zeros([max_order + 1, len(p_curves), 1])
# for i, p_curve in enumerate(p_curves):
# # TODO: need to center
# for k, c in iterate_poly1d(p_curve):
# coeffs[max_order - k, i] = c
from scipy.interpolate import PPoly
# TODO: check continuity
#from scipy.interpolate import CubicHermiteSpline
return PPoly(c=np.array(coeffs).T, x=times) # TODO: spline.extend
def test_construct_fast(self):
np.random.seed(1234)
c = np.array([[1, 4], [2, 5], [3, 6]], dtype=float)
x = np.array([0, 0.5, 1])
p = PPoly.construct_fast(c, x)
assert_allclose(p(0.3), 1*0.3**2 + 2*0.3 + 3)
assert_allclose(p(0.7), 4*(0.7-0.5)**2 + 5*(0.7-0.5) + 6)
def from_poly(poly):
from scipy.interpolate import PPoly
if isinstance(poly, MultiPPoly):
return poly
assert len(poly.c.shape) == 3
d = poly.c.shape[2]
return MultiPPoly(
[PPoly(c=poly.c[..., k], x=poly.x) for k in range(d)])
def add_anchor_point(x, y):
"""Auxilliary function to create stepping potential used by energy_model.peq_models.jump_spline2 module
Find additional anchoring point (x_add, y_add), such that x[0] < x_add < x[1], and y_add = y_spline (x_add_mirror),
where y_spline is spline between x[1] and x[2], and x_add_mirror is a mirror point of x_add w.r.t. x[1], e.g. |x_add-x[1]|=|x[1]-x_add_mirror|.
This additional point will force the barriers to have symmetrical shape.
Params:
x: x - values at three right or three left boundary points, e.g. interp_x[0:3], or interp_x[-3:][::-1] (reverse order on the right boundary)
y: corresponding y values
Returns:
x_add, y_add - additional point in between x[0] < x_add < x[1] that can be used for spline interpolation
"""
# Start with the middle point
x_add_mirror = 0.5 * (x[1] + x[2])
not_found = True
cpoly = PPoly(CubicSpline(np.sort(x[1:3]), np.sort(
y[1:3]), bc_type=[(1, 0), (1, 0)]).c, np.sort([x[1], x[2]]))
first_peak_is_maximum = True if y[0] <= y[1] else False
while not_found:
# Check if y-value at anchor point exceeds value at the boundary and that x-value is within an interval
if first_peak_is_maximum:
if cpoly(x_add_mirror) > y[0] and np.abs(x_add_mirror - x[1]) < np.abs(x[1] - x[0]):
x_add = 2 * x[1] - x_add_mirror
if x[1] > x[0]:
poly2 = PPoly(CubicSpline([x[0], x_add, x[1]], [y[0], cpoly(
x_add_mirror), y[1]], bc_type=[(1, 0), (1, 0)]).c, [x[0], x_add, x[1]])
else:
poly2 = PPoly(CubicSpline([x[1], x_add, x[0]], [y[1], cpoly(
x_add_mirror), y[0]], bc_type=[(1, 0), (1, 0)]).c, [x[1], x_add, x[0]])
x_dense = np.linspace(x[0], x[1], 100)
if all(poly2(x_dense) <= y[1]):
not_found = False
else:
if cpoly(x_add_mirror) < y[0] and np.abs(x_add_mirror - x[1]) < np.abs(x[1] - x[0]):
x_add = 2 * x[1] - x_add_mirror
if x[1] > x[0]:
poly2 = PPoly(CubicSpline([x[0], x_add, x[1]], [y[0], cpoly(
x_add_mirror), y[1]], bc_type=[(1, 0), (1, 0)]).c, [x[0], x_add, x[1]])
else:
poly2 = PPoly(CubicSpline([x[1], x_add, x[0]], [y[1], cpoly(
x_add_mirror), y[0]], bc_type=[(1, 0), (1, 0)]).c, [x[1], x_add, x[0]])
x_dense = np.linspace(x[0], x[1], 100)
if all(poly2(x_dense) >= y[1]):
not_found = False
x_add_mirror = 0.5 * (x[1] + x_add_mirror)
return x_add, cpoly(2 * x[1] - x_add)
def setup(self):
np.random.seed(1234)
m, k = 55, 3
x = np.sort(np.random.random(m + 1))
c = np.random.random((3, m))
self.pp = PPoly(c, x)
npts = 100
self.xp = np.linspace(0, 1, npts)
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_roots(self):
x = np.linspace(0, 1, 31)**2
y = np.sin(30 * x)
spl = splrep(x, y, s=0, k=3)
pp = PPoly.from_spline(spl)
r = pp.roots()
r = r[(r >= 0) & (r <= 1)]
assert_allclose(r, sproot(spl), atol=1e-15)
def test_from_spline(self):
np.random.seed(1234)
x = np.sort(np.r_[0, np.random.rand(11), 1])
y = np.random.rand(len(x))
spl = splrep(x, y, s=0)
pp = PPoly.from_spline(spl)
xi = np.linspace(0, 1, 200)
assert_allclose(pp(xi), splev(xi, spl))
def smooth(self):
"Return an C2 interpolant of self, suitable for optimization"
pc = scipy.interpolate.PchipInterpolator(x=self.t[:-1],
y=1.0 / self.Ne)
# the base class of PchipInterpolator seems to have switched from a BPoly to a PPoly at some point
assert isinstance(pc, scipy.interpolate.PPoly)
# add a final piece at the back that is constant
cinf = np.eye(pc.c.shape[0])[:, -1:] * pc.c[-1, -1]
ret = PPoly(c=np.append(pc.c, cinf, axis=1), x=np.append(pc.x, np.inf))
return ret
def test_from_spline(self):
np.random.seed(1234)
x = np.sort(np.r_[0, np.random.rand(11), 1])
y = np.random.rand(len(x))
spl = splrep(x, y, s=0)
pp = PPoly.from_spline(spl)
xi = np.linspace(0, 1, 200)
assert_allclose(pp(xi), splev(xi, spl))
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_pp_from_bp(self):
x = [0, 1, 3]
c = [[3, 3], [1, 1], [4, 2]]
bp = BPoly(c, x)
pp = PPoly.from_bernstein_basis(bp)
bp1 = BPoly.from_power_basis(pp)
xp = [0.1, 1.4]
assert_allclose(bp(xp), pp(xp))
assert_allclose(bp(xp), bp1(xp))
def test_roots(self):
x = np.linspace(0, 1, 31)**2
y = np.sin(30*x)
spl = splrep(x, y, s=0, k=3)
pp = PPoly.from_spline(spl)
r = pp.roots()
r = r[(r >= 0) & (r <= 1)]
assert_allclose(r, sproot(spl), atol=1e-15)
def trim_end(poly, end):
from scipy.interpolate import PPoly
times = poly.x
if end >= times[-1]:
return poly
if end <= times[0]:
return None
last = find(lambda i: times[i] <= end, reversed(range(len(times))))
times = list(times[:last + 1]) + [end]
c = poly.c[:, :last + 1, ...]
return PPoly(c=c, x=times)
def to_pp(self) -> PPoly:
"""Return a piecewise polynomial representation of this segmentation.
Notes:
Only represents segments heights. Identity of haplotype panel is currently ignored.
"""
x = np.array(
[self.segments[0].interval[0]] + [s.interval[1] for s in self.segments]
)
c = np.array([s.height for s in self.segments])[None]
return PPoly(x=x, c=c)
def test_derivative_eval(self):
np.random.seed(1234)
x = np.sort(np.r_[0, np.random.rand(11), 1])
y = np.random.rand(len(x))
spl = splrep(x, y, s=0)
pp = PPoly.from_spline(spl)
xi = np.linspace(0, 1, 200)
for dx in range(0, 3):
assert_allclose(pp(xi, dx), splev(xi, spl, dx))
def test_derivative_eval(self):
np.random.seed(1234)
x = np.sort(np.r_[0, np.random.rand(11), 1])
y = np.random.rand(len(x))
spl = splrep(x, y, s=0)
pp = PPoly.from_spline(spl)
xi = np.linspace(0, 1, 200)
for dx in range(0, 3):
assert_allclose(pp(xi, dx), splev(xi, spl, dx))
def test_extrapolate_attr(self):
# [ 1 - x**2 ]
c = np.array([[-1, 0, 1]]).T
x = np.array([0, 1])
for extrapolate in [True, False, None]:
pp = PPoly(c, x, extrapolate=extrapolate)
pp_d = pp.derivative()
pp_i = pp.antiderivative()
if extrapolate is False:
assert_(np.isnan(pp([-0.1, 1.1])).all())
assert_(np.isnan(pp_i([-0.1, 1.1])).all())
assert_(np.isnan(pp_d([-0.1, 1.1])).all())
assert_equal(pp.roots(), [1])
else:
assert_allclose(pp([-0.1, 1.1]), [1-0.1**2, 1-1.1**2])
assert_(not np.isnan(pp_i([-0.1, 1.1])).any())
assert_(not np.isnan(pp_d([-0.1, 1.1])).any())
assert_allclose(pp.roots(), [1, -1])
def test_derivative(self):
np.random.seed(1234)
x = np.sort(np.r_[0, np.random.rand(11), 1])
y = np.random.rand(len(x))
spl = splrep(x, y, s=0, k=5)
pp = PPoly.from_spline(spl)
xi = np.linspace(0, 1, 200)
for dx in range(0, 10):
assert_allclose(pp(xi, dx), pp.derivative(dx)(xi),
err_msg="dx=%d" % (dx,))
def test_bp_from_pp_random(self):
np.random.seed(1234)
m, k = 5, 8 # number of intervals, order
x = np.sort(np.random.random(m))
c = np.random.random((k, m-1))
pp = PPoly(c, x)
bp = BPoly.from_power_basis(pp)
pp1 = PPoly.from_bernstein_basis(bp)
xp = np.linspace(x[0], x[-1], 21)
assert_allclose(pp(xp), bp(xp))
assert_allclose(pp(xp), pp1(xp))
def test_vs_alternative_implementations(self):
np.random.seed(1234)
c = np.random.rand(3, 12, 22)
x = np.sort(np.r_[0, np.random.rand(11), 1])
p = PPoly(c, x)
xp = np.r_[0.3, 0.5, 0.33, 0.6]
expected = _ppoly_eval_1(c, x, xp)
assert_allclose(p(xp), expected)
expected = _ppoly_eval_2(c[:, :, 0], x, xp)
assert_allclose(p(xp)[:, 0], expected)
def test_derivative(self):
np.random.seed(1234)
x = np.sort(np.r_[0, np.random.rand(11), 1])
y = np.random.rand(len(x))
spl = splrep(x, y, s=0, k=5)
pp = PPoly.from_spline(spl)
xi = np.linspace(0, 1, 200)
for dx in range(0, 10):
assert_allclose(pp(xi, dx),
pp.derivative(dx)(xi),
err_msg="dx=%d" % (dx, ))
def test_antiderivative_simple(self):
np.random.seed(1234)
# [ p1(x) = 3*x**2 + 2*x + 1,
# p2(x) = 1.6875]
c = np.array([[3, 2, 1], [0, 0, 1.6875]]).T
# [ pp1(x) = x**3 + x**2 + x,
# pp2(x) = 1.6875*(x - 0.25) + pp1(0.25)]
ic = np.array([[1, 1, 1, 0], [0, 0, 1.6875, 0.328125]]).T
# [ ppp1(x) = (1/4)*x**4 + (1/3)*x**3 + (1/2)*x**2,
# ppp2(x) = (1.6875/2)*(x - 0.25)**2 + pp1(0.25)*x + ppp1(0.25)]
iic = np.array([[1/4, 1/3, 1/2, 0, 0],
[0, 0, 1.6875/2, 0.328125, 0.037434895833333336]]).T
x = np.array([0, 0.25, 1])
pp = PPoly(c, x)
ipp = pp.antiderivative()
iipp = pp.antiderivative(2)
iipp2 = ipp.antiderivative()
assert_allclose(ipp.x, x)
assert_allclose(ipp.c.T, ic.T)
assert_allclose(iipp.c.T, iic.T)
assert_allclose(iipp2.c.T, iic.T)
def test_derivative_ppoly(self):
# make sure it's consistent w/ power basis
np.random.seed(1234)
m, k = 5, 8 # number of intervals, order
x = np.sort(np.random.random(m))
c = np.random.random((k, m-1))
bp = BPoly(c, x)
pp = PPoly.from_bernstein_basis(bp)
for d in range(k):
bp = bp.derivative()
pp = pp.derivative()
xp = np.linspace(x[0], x[-1], 21)
assert_allclose(bp(xp), pp(xp))
def test_derivative_ppoly(self):
# make sure it's consistent w/ power basis
np.random.seed(1234)
m, k = 5, 8 # number of intervals, order
x = np.sort(np.random.random(m))
c = np.random.random((k, m - 1))
bp = BPoly(c, x)
pp = PPoly.from_bernstein_basis(bp)
for d in range(k):
bp = bp.derivative()
pp = pp.derivative()
xp = np.linspace(x[0], x[-1], 21)
assert_allclose(bp(xp), pp(xp))
def test_antiderivative_vs_spline(self):
np.random.seed(1234)
x = np.sort(np.r_[0, np.random.rand(11), 1])
y = np.random.rand(len(x))
spl = splrep(x, y, s=0, k=5)
pp = PPoly.from_spline(spl)
for dx in range(0, 10):
pp2 = pp.antiderivative(dx)
spl2 = splantider(spl, dx)
xi = np.linspace(0, 1, 200)
assert_allclose(pp2(xi), splev(xi, spl2),
rtol=1e-7)
def test_integrate(self):
np.random.seed(1234)
x = np.sort(np.r_[0, np.random.rand(11), 1])
y = np.random.rand(len(x))
spl = splrep(x, y, s=0, k=5)
pp = PPoly.from_spline(spl)
a, b = 0.3, 0.9
ig = pp.integrate(a, b)
ipp = pp.antiderivative()
assert_allclose(ig, ipp(b) - ipp(a))
assert_allclose(ig, splint(a, b, spl))
a, b = -0.3, 0.9
ig = pp.integrate(a, b, extrapolate=True)
assert_allclose(ig, ipp(b) - ipp(a))
assert_(np.isnan(pp.integrate(a, b, extrapolate=False)).all())
def test_antiderivative_vs_derivative(self):
np.random.seed(1234)
x = np.linspace(0, 1, 30)**2
y = np.random.rand(len(x))
spl = splrep(x, y, s=0, k=5)
pp = PPoly.from_spline(spl)
for dx in range(0, 10):
ipp = pp.antiderivative(dx)
# check that derivative is inverse op
pp2 = ipp.derivative(dx)
assert_allclose(pp.c, pp2.c)
# check continuity
for k in range(dx):
pp2 = ipp.derivative(k)
r = 1e-13
endpoint = r*pp2.x[:-1] + (1 - r)*pp2.x[1:]
assert_allclose(pp2(pp2.x[1:]), pp2(endpoint),
rtol=1e-7, err_msg="dx=%d k=%d" % (dx, k))
from scipy.interpolate import PPoly from scipy.interpolate import splev, splrep W = np.array([-2.0, -2.5, -3.7538003, -5.00760061, -5.50760061]) tvec = np.linspace(0, 1, W.shape[0]) W = np.hstack((W[0], W[0], W[:], W[-1] - 0.1, W[-1] - 0.2)) tvec = np.hstack((-0.1, -0.05, tvec, 1.05, 1.1)) tck = splrep(tvec, W, s=0, k=3) x2 = np.linspace(0, 1, 100) # x2 = np.linspace(tvec[0], tvec[-1], 100) y2 = splev(x2, tck) print splev(0, tck) print splev(0, tck, der=1) print splev(1, tck, der=1) poly = PPoly.from_spline(tck) dpoly = poly.derivative(1) ddpoly = poly.derivative(2) [B, idx] = np.unique(poly.x, return_index=True) coeff = poly.c[:, idx] plt.plot(tvec, W, "o", x2, y2) plt.show()