def make_knots(size=30, num_knots=4, order=7):
knots = np.round(-1 +
np.logspace(np.log10(1), np.log10(size - 1), num_knots))
knots = np.round(np.sqrt(knots) * (size - 1) / max(np.sqrt(knots)))
knots = np.unique(knots)
knots = splinelab.augknt(knots, order)
return knots
def create_bspline_basis(xmin, xmax, p=3, nknots=5):
""" compute a Bspline basis set where:
:param p: order of spline (3 = cubic)
:param nknots: number of knots (endpoints only counted once)
"""
knots = np.linspace(xmin, xmax, nknots)
k = splinelab.augknt(knots, p) # pad the knot vector
B = bspline.Bspline(k, p)
return B
def bspline_fit(X, order, nknots):
X = X.squeeze()
if len(X.shape) > 1:
raise ValueError('Bspline method only works for a single covariate.')
knots = np.linspace(X.min(), X.max(), nknots)
k = splinelab.augknt(knots, order)
bsp_basis = bspline.Bspline(k, order)
return bsp_basis
def test():
########################
# config
########################
p = 3 # order of spline basis (as-is! 3 = cubic)
nknots = 5 # for testing: number of knots to generate (here endpoints count only once)
tau = [0.1, 0.33] # collocation sites (i.e. where to evaluate)
########################
# usage example
########################
knots = np.linspace(
0, 1, nknots) # create a knot vector without endpoint repeats
k = splinelab.augknt(
knots, p) # add endpoint repeats as appropriate for spline order p
B = bspline.Bspline(k, p) # create spline basis of order p on knots k
# build some collocation matrices:
#
A0 = B.collmat(tau) # function value at sites tau
A2 = B.collmat(tau, deriv_order=2) # second derivative at sites tau
print(A0)
print(A2)
########################
# tests
########################
# number of basis functions
n_interior_knots = len(knots) - 2
n_basis_functions_expected = n_interior_knots + (p + 1)
n_basis_functions_actual = len(
B(0.)) # perform dummy evaluation to get number of basis functions
assert n_basis_functions_actual == n_basis_functions_expected, "something went wrong, number of basis functions is incorrect"
# partition-of-unity property of the spline basis
assert np.allclose(
np.sum(A0, axis=1), 1.0
), "something went wrong, the basis functions do not form a partition of unity"
Phips = np.zeros((Xs.shape[0], dimpoly))
colid = np.arange(0, 1)
for d in range(1, dimpoly + 1):
Phips[:, colid] = np.vstack(Xs**d)
colid += 1
# generative model
b = [0.5, -0.1, 0.005] # true regression coefficients
s2 = 0.05 # noise variance
y = Phip.dot(b) + np.random.normal(0, s2, N)
# cubic B-spline basis (used for regression)
p = 3 # order of spline (3 = cubic)
nknots = 5 # number of knots (endpoints only counted once)
knots = np.linspace(0, 10, nknots)
k = splinelab.augknt(knots, p) # pad the knot vector
B = bspline.Bspline(k, p)
Phi = np.array([B(i) for i in X])
Phis = np.array([B(i) for i in Xs])
hyp0 = np.zeros(2)
#hyp0 = np.zeros(4) # use ARD
B = BLR(hyp0, Phi, y)
hyp = B.estimate(hyp0, Phi, y, optimizer='powell')
yhat, s2 = B.predict(hyp, Phi, y, Phis)
plt.fill_between(Xs,
yhat - 1.96 * np.sqrt(s2),
yhat + 1.96 * np.sqrt(s2),
alpha=0.2)
plt.scatter(X, y)
start_n = 1000 # sample size at first analysis
typ_list = [2, 3] # phi function type list
nk_list = [(200, 5), (20, 50)] #
delta_list = [0.0, 0.10, 0.15, 0.20, 0.25, 0.30] # effect size typ three
alpha = 0.05
e = 0.5 # error variance
ms = [1, 2, 3] # alpha spending functions
lm = len(ms)
lb, ub = -2, 2
order = 3 # order of spline (as-is; 3 = cubic)
nknots = 4 # number of knots to generate
knots = np.linspace(lb, ub,
nknots) # create a knot vector without endpoint repeats
knots = splinelab.augknt(
knots, order) # add endpoint repeats as appropriate for spline order
bases = bspline.Bspline(knots,
order) # create spline basis of order p on knots k
nfeatures = 3
p = nfeatures * (order + nknots - 1) + 1
rho = 0.5
cov_mat = (1 - rho) * np.eye(nfeatures) + rho * np.ones((nfeatures, nfeatures))
L = np.linalg.cholesky(cov_mat).T
for (typ, (n, k), delta) in product(typ_list, nk_list, delta_list):
name = 'result/HTE/' + 'typ' + str(typ) + 'n' + str(n) + 'k' + str(
k) + 'delta' + str(delta) + '_nonlinear_adaptive_HTE.npz'
# if os.path.exists(name):
def main():
"""Demonstration: plot a B-spline basis and its first three derivatives."""
#########################################################################################
# config
#########################################################################################
# Order of spline basis.
#
p = 3
# Knot vector, including the endpoints.
#
# For convenience, endpoints are specified only once, regardless of the value of p.
#
# Duplicate knots *in the interior* are allowed, with the standard meaning for B-splines.
#
knots = [0., 0.25, 0.5, 0.75, 1.]
# How many plotting points to use on each subinterval [knots[i], knots[i+1]).
#
# Only intervals with length > 0 are actually plotted.
#
nt_per_interval = 101
#########################################################################################
# the demo itself
#########################################################################################
# The evaluation algorithm used in bspline.py uses half-open intervals t_i <= x < t_{i+1}.
#
# This causes the right endpoint of each interval to actually be the start point of the next interval.
#
# Especially, the right endpoint of the last interval is the start point of the next (nonexistent) interval,
# so the basis will return a value of zero there.
#
# We work around this by using a small epsilon to avoid evaluation exactly at t_{i+1} (for each interval).
#
epsrel = 1e-10
epsabs = epsrel * (knots[-1] - knots[0])
original_knots = knots
knots = splinelab.augknt(
knots, p) # add repeated endpoint knots for splines of order p
# treat each interval separately to preserve discontinuities
#
# (useful especially when plotting the highest-order nonzero derivative)
#
B = bspline.Bspline(knots, p)
xxs = []
for I in zip(knots[:-1], knots[1:]):
t_i = I[0]
t_ip1 = I[1] - epsabs
if t_ip1 - t_i > 0.: # accept only intervals of length > 0 (to skip higher-multiplicity knots in the interior)
xxs.append(np.linspace(t_i, t_ip1, nt_per_interval))
# common settings for all plotted lines
settings = {"linestyle": 'solid', "linewidth": 1.0}
# create a list of unique colors for plotting
#
# http://stackoverflow.com/questions/8389636/creating-over-20-unique-legend-colors-using-matplotlib
#
NUM_COLORS = nbasis = len(
B(0.)) # perform dummy evaluation to get number of basis functions
cm = plt.get_cmap('gist_rainbow')
cNorm = matplotlib.colors.Normalize(vmin=0, vmax=NUM_COLORS - 1)
scalarMap = matplotlib.cm.ScalarMappable(norm=cNorm, cmap=cm)
colors = [scalarMap.to_rgba(i) for i in range(NUM_COLORS)]
labels = [
r"$B$", r"$\mathrm{d}B\,/\,\mathrm{d}x$",
r"$\mathrm{d}^2B\,/\,\mathrm{d}x^2$",
r"$\mathrm{d}^3B\,/\,\mathrm{d}x^3$"
]
# for plotting the knot positions:
unique_knots_xx = np.unique(original_knots)
unique_knots_yy = np.zeros_like(unique_knots_xx)
# plot the basis functions B(x) and their first three derivatives
plt.figure(1)
plt.clf()
for k in range(4):
ax = plt.subplot(2, 2, k + 1)
# place the axis label where it fits
if k % 2 == 0:
ax.yaxis.set_label_position("left")
else:
ax.yaxis.set_label_position("right")
# plot the kth derivative; each basis function gets a unique color
f = B.diff(order=k) # order=0 is a passthrough
for xx in xxs:
yy = np.array([
f(x) for x in xx
]) # f(scalar) -> rank-1 array, one element per basis function
for i in range(nbasis):
settings["color"] = colors[i]
plt.plot(xx, yy[:, i], **settings)
plt.ylabel(labels[k])
# show knot positions
plt.plot(unique_knots_xx, unique_knots_yy, "kx")
plt.suptitle(r"$B$-spline basis functions, $p=%d$" % (p))
