def test_B():
import pspline
import numpy as np
import matplotlib.pyplot as plt
lam_M = np.matrix((0, 2, 4))
lam_C = np.matrix((0, 1, 2, 3, 4))
A_M = 1/120.0 * np.power(lam_M, 5)
A_M_4d = np.power(lam_C, 1) #* 1/24.
N_k = lam_C.shape[1]
N_D = lam_M.shape[1]
# Prep arrays to be of correct shape
A_M, lam_M, lam_C, N_k = pspline.prep_spectrum(lam_M, A_M, N_k=N_k)
lam_0 = lam_M[0, 0]
Delta = lam_M[1, 0] - lam_M[0, 0]
B = pspline.construct_B(lam_C, lam_M)
B_true = Delta / 12.0 * np.array((
(0, 0, 0, 0, 0),
(0, 8, 0, 0, 0),
(0, 16, 64, 0, 0),
))
from pprint import pprint
if 0:
print 'B calculated'
print B / np.min(B[B != 0])
print 'B True'
print B_true / Delta * 12.0
#assert np.array_equal(B, B_true)
# Integrate to original function
A_C = B* A_M_4d.T
A_M = np.squeeze(np.asarray(A_M))
A_M_4d = np.squeeze(np.asarray(A_M_4d))
A_C = np.squeeze(np.asarray(A_C))
if 0:
print 'A_C = '
print A_C
print ''
print 'A_M = '
print A_M
print ''
if 1:
print 'A_C / A_M = '
print A_C / A_M
print ''
x_C = np.squeeze(np.array(x_C))
#y_4d = np.squeeze(np.array(y_4d_calc[:-4]))
y_4d = np.squeeze(np.array(y_4d))
y_3d = cumtrapz(y_4d, x_C, initial=0)
y_2d = cumtrapz(y_3d, x_C, initial=0)
y_1d = cumtrapz(y_2d, x_C, initial=0)
y_0d = cumtrapz(y_1d, x_C, initial=0)
# Plot
plt.clf(); plt.close()
plt.plot(lam_M, A_M, label='f(x)',)
plt.plot(lam_C, A_M_4d, label="f''''(x)")
plt.plot(lam_M, A_C, label="Integrated f''''(x)")
#plt.plot(lam_M, A_C - A_M, label="Integrated f''''(x) - f(x)")
#plt.xlim(-30, 30)
#plt.ylim(-15, max(As)*1.1)
plt.legend(loc='best')
plt.savefig('figures/linear_integration_test.png')
def test_B():
import pspline
import numpy as np
import matplotlib.pyplot as plt
lam_M = np.matrix((0, 2, 4))
lam_C = np.matrix((0, 1, 2, 3, 4))
A_M = 1 / 120.0 * np.power(lam_M, 5)
A_M_4d = np.power(lam_C, 1) #* 1/24.
N_k = lam_C.shape[1]
N_D = lam_M.shape[1]
# Prep arrays to be of correct shape
A_M, lam_M, lam_C, N_k = pspline.prep_spectrum(lam_M, A_M, N_k=N_k)
lam_0 = lam_M[0, 0]
Delta = lam_M[1, 0] - lam_M[0, 0]
B = pspline.construct_B(lam_C, lam_M)
B_true = Delta / 12.0 * np.array((
(0, 0, 0, 0, 0),
(0, 8, 0, 0, 0),
(0, 16, 64, 0, 0),
))
from pprint import pprint
if 0:
print 'B calculated'
print B / np.min(B[B != 0])
print 'B True'
print B_true / Delta * 12.0
#assert np.array_equal(B, B_true)
# Integrate to original function
A_C = B * A_M_4d.T
A_M = np.squeeze(np.asarray(A_M))
A_M_4d = np.squeeze(np.asarray(A_M_4d))
A_C = np.squeeze(np.asarray(A_C))
if 0:
print 'A_C = '
print A_C
print ''
print 'A_M = '
print A_M
print ''
if 1:
print 'A_C / A_M = '
print A_C / A_M
print ''
x_C = np.squeeze(np.array(x_C))
#y_4d = np.squeeze(np.array(y_4d_calc[:-4]))
y_4d = np.squeeze(np.array(y_4d))
y_3d = cumtrapz(y_4d, x_C, initial=0)
y_2d = cumtrapz(y_3d, x_C, initial=0)
y_1d = cumtrapz(y_2d, x_C, initial=0)
y_0d = cumtrapz(y_1d, x_C, initial=0)
# Plot
plt.clf()
plt.close()
plt.plot(
lam_M,
A_M,
label='f(x)',
)
plt.plot(lam_C, A_M_4d, label="f''''(x)")
plt.plot(lam_M, A_C, label="Integrated f''''(x)")
#plt.plot(lam_M, A_C - A_M, label="Integrated f''''(x) - f(x)")
#plt.xlim(-30, 30)
#plt.ylim(-15, max(As)*1.1)
plt.legend(loc='best')
plt.savefig('figures/linear_integration_test.png')
def test_gauss_integration(ngauss=1):
import pspline
import numpy as np
from scipy import linalg
import matplotlib.pyplot as plt
# Test integration of fourth derivative of gaussians
Delta = 0.5
x0s = (0, 15)
sigmas = (5, 10)
As = (20, 10)
x0s = (-5, 0)
sigmas = (5, 5)
As = (10, 20)
x = np.arange(-30, 30, Delta)
x = np.linspace(-30, 30, 71)
if ngauss == 2:
y = gauss(x, sigmas[0], x0s[0], As[0]) \
+ gauss(x, sigmas[1], x0s[1], As[1]) \
+ np.random.normal(0,1, len(x))
elif ngauss == 1:
y = gauss(x, sigmas[0], x0s[0], As[0]) \
+ np.random.normal(0,0.1, len(x))
A_M, lam_M, lam_C, N_k = pspline.prep_spectrum(x, y)
x_C = np.squeeze(np.asarray(lam_C))
if ngauss == 2:
y_4d = gauss_4th_deriv(x_C, sigmas[0], x0s[0], As[0]) \
+ gauss_4th_deriv(x_C, sigmas[1], x0s[1], As[1])
elif ngauss == 1:
y_4d = gauss_4th_deriv(x_C, sigmas[0], x0s[0], As[0])
y_4d = np.matrix(y_4d).T
#y_1d = np.matrix(y_1d).T
lam_0 = lam_M[0, 0]
Delta = x[1] - x[0]
N_D = len(x)
#print 'lam_0', lam_0
#print 'Delta', Delta
B = pspline.construct_B(lam_C, lam_C)
trap_integ = pspline.construct_trap_integ(lam_C, lam_C)
#print B.shape, y_4d.shape
#print B
# Integrate to original function
y_calc = B[:-2, :] * y_4d[:, 0]
y = np.squeeze(np.asarray(y))
#y_1d = np.squeeze(np.asarray(y_1d))
y_calc = np.squeeze(np.asarray(y_calc))
lam_0 = lam_M[0, 0]
Delta = abs(lam_C[-1, 0] - lam_C[0, 0]) / (N_k - 1)
ones = np.matrix(np.ones(A_M.shape[0])).T
lam_M0 = lam_M - ones * lam_0
B = pspline.construct_B(lam_C, lam_M)
B_prime = np.hstack((B, ones, lam_M0, np.power(lam_M0, 2) / 2.0,
np.power(lam_M0, 3) / 6.0))
beta = pspline.construct_beta(lam_C)
chi = 10
y_4d_calc = linalg.inv(B_prime.T * B_prime + chi / Delta**4 * \
beta.T*beta) * B_prime.T * A_M
#print y_calc.shape
#print y_4d.shape
#x_C = x_[1:-1]
# Integrate 4th derive to 0th deriv
from scipy.integrate import cumtrapz
x_C = np.squeeze(np.array(x_C))
y_4d = np.squeeze(np.array(y_4d_calc[:-4]))
#y_4d = np.squeeze(np.array(y_4d))
if 1:
y_3d = cumtrapz(y_4d, x_C, initial=y_4d_calc[-4, 0])
y_2d = cumtrapz(y_3d, x_C, initial=y_4d_calc[-3, 0])
y_1d = cumtrapz(y_2d, x_C, initial=y_4d_calc[-2, 0])
y_0d = cumtrapz(y_1d, x_C, initial=y_4d_calc[-1, 0])
else:
y_3d = cumtrapz(y_4d, x_C, initial=0)
y_2d = cumtrapz(y_3d, x_C, initial=0)
y_1d = cumtrapz(y_2d, x_C, initial=0)
y_0d = cumtrapz(y_1d, x_C, initial=0)
# Calculate spectrum to compare with computed
if ngauss == 2:
y_C = gauss(x_C, sigmas[0], x0s[0], As[0]) \
+ gauss(x_C, sigmas[1], x0s[1], As[1])
elif ngauss == 1:
y_C = gauss(x_C, sigmas[0], x0s[0], As[0])
norm = np.sum((y_0d - y_C)**2)
print('L2 norm of calc y and true y = {0:.2f}'.format(norm))
print('\ny_0d[0] =', y_0d[0])
# Plot
scale = np.max(y) / np.max(y_4d)
plt.clf()
plt.close()
plt.plot(
x,
y,
label='f(x)',
)
plt.plot(x_C, y_4d * scale, label="f''''(x) x " + str(scale))
plt.plot(x_C, y_3d, label="Calc. f'''(x)")
plt.plot(x_C, y_2d, label="Calc. f''(x)")
plt.plot(x_C, y_1d, label="Calc. f'(x)")
plt.plot(
x_C,
y_0d,
label="Calc. f(x)",
) #marker='o')
plt.plot(
x_C[:-2],
y_calc,
label="Integrated f''''(x)",
) #marker='+')
#plt.plot(x, y_calc - y, label="Integrated f''''(x) - f(x)")
plt.xlim(-30, 30)
#plt.ylim(-15, max(As)*1.1)
plt.legend(loc='best')
plt.savefig('figures/integration_test_ngauss' + str(ngauss) + '.png')
plt.show()
plt.clf()
plt.close()
plt.plot(x_C, y_4d, label="f''''(x)")
plt.plot(x_C, y_4d_calc[:-4], label="f''''(x) Calc")
#plt.plot(x_C, y_3d, label="Calc. f'''(x)")
#plt.plot(x_C, y_2d, label="Calc. f''(x)")
#plt.plot(x_C, y_1d, label="Calc. f'(x)")
#plt.plot(x_C, y_0d, label="Calc. f(x)")
#plt.plot(x, y_calc - y, label="Integrated f''''(x) - f(x)")
plt.xlim(-30, 30)
#plt.ylim(-15, max(As)*1.1)
plt.legend(loc='best')
plt.savefig('figures/integration_test_derivs_ngauss_' + str(ngauss) +
'.png')
def test_gauss_integration(ngauss=1):
import pspline
import numpy as np
from scipy import linalg
import matplotlib.pyplot as plt
# Test integration of fourth derivative of gaussians
Delta = 0.5
x0s = (0, 15)
sigmas = (5, 10)
As = (20, 10)
x0s = (-5, 0)
sigmas = (5, 5)
As = (10, 20)
x = np.arange(-30, 30, Delta)
x = np.linspace(-30, 30, 71)
if ngauss == 2:
y = gauss(x, sigmas[0], x0s[0], As[0]) \
+ gauss(x, sigmas[1], x0s[1], As[1]) \
+ np.random.normal(0,1, len(x))
elif ngauss == 1:
y = gauss(x, sigmas[0], x0s[0], As[0]) \
+ np.random.normal(0,0.1, len(x))
A_M, lam_M, lam_C, N_k = pspline.prep_spectrum(x, y)
x_C = np.squeeze(np.asarray(lam_C))
if ngauss == 2:
y_4d = gauss_4th_deriv(x_C, sigmas[0], x0s[0], As[0]) \
+ gauss_4th_deriv(x_C, sigmas[1], x0s[1], As[1])
elif ngauss == 1:
y_4d = gauss_4th_deriv(x_C, sigmas[0], x0s[0], As[0])
y_4d = np.matrix(y_4d).T
#y_1d = np.matrix(y_1d).T
lam_0 = lam_M[0, 0]
Delta = x[1] - x[0]
N_D = len(x)
#print 'lam_0', lam_0
#print 'Delta', Delta
B = pspline.construct_B(lam_C, lam_C)
trap_integ = pspline.construct_trap_integ(lam_C, lam_C)
#print B.shape, y_4d.shape
#print B
# Integrate to original function
y_calc = B[:-2,:] * y_4d[:, 0]
y = np.squeeze(np.asarray(y))
#y_1d = np.squeeze(np.asarray(y_1d))
y_calc = np.squeeze(np.asarray(y_calc))
lam_0 = lam_M[0, 0]
Delta = abs(lam_C[-1, 0] - lam_C[0, 0]) / (N_k - 1)
ones = np.matrix(np.ones(A_M.shape[0])).T
lam_M0 = lam_M - ones * lam_0
B = pspline.construct_B(lam_C, lam_M)
B_prime = np.hstack((B,
ones,
lam_M0,
np.power(lam_M0, 2) / 2.0,
np.power(lam_M0, 3) / 6.0))
beta = pspline.construct_beta(lam_C)
chi = 10
y_4d_calc = linalg.inv(B_prime.T * B_prime + chi / Delta**4 * \
beta.T*beta) * B_prime.T * A_M
#print y_calc.shape
#print y_4d.shape
#x_C = x_[1:-1]
# Integrate 4th derive to 0th deriv
from scipy.integrate import cumtrapz
x_C = np.squeeze(np.array(x_C))
y_4d = np.squeeze(np.array(y_4d_calc[:-4]))
#y_4d = np.squeeze(np.array(y_4d))
if 1:
y_3d = cumtrapz(y_4d, x_C, initial=y_4d_calc[-4, 0])
y_2d = cumtrapz(y_3d, x_C, initial=y_4d_calc[-3, 0])
y_1d = cumtrapz(y_2d, x_C, initial=y_4d_calc[-2, 0])
y_0d = cumtrapz(y_1d, x_C, initial=y_4d_calc[-1, 0])
else:
y_3d = cumtrapz(y_4d, x_C, initial=0)
y_2d = cumtrapz(y_3d, x_C, initial=0)
y_1d = cumtrapz(y_2d, x_C, initial=0)
y_0d = cumtrapz(y_1d, x_C, initial=0)
# Calculate spectrum to compare with computed
if ngauss == 2:
y_C = gauss(x_C, sigmas[0], x0s[0], As[0]) \
+ gauss(x_C, sigmas[1], x0s[1], As[1])
elif ngauss == 1:
y_C = gauss(x_C, sigmas[0], x0s[0], As[0])
norm = np.sum((y_0d - y_C)**2)
print('L2 norm of calc y and true y = {0:.2f}'.format(norm))
print('\ny_0d[0] =', y_0d[0])
# Plot
scale = np.max(y) / np.max(y_4d)
plt.clf(); plt.close()
plt.plot(x, y, label='f(x)',)
plt.plot(x_C, y_4d * scale, label="f''''(x) x " + str(scale))
plt.plot(x_C, y_3d, label="Calc. f'''(x)")
plt.plot(x_C, y_2d, label="Calc. f''(x)")
plt.plot(x_C, y_1d, label="Calc. f'(x)")
plt.plot(x_C, y_0d, label="Calc. f(x)",)#marker='o')
plt.plot(x_C[:-2], y_calc, label="Integrated f''''(x)",) #marker='+')
#plt.plot(x, y_calc - y, label="Integrated f''''(x) - f(x)")
plt.xlim(-30, 30)
#plt.ylim(-15, max(As)*1.1)
plt.legend(loc='best')
plt.savefig('figures/integration_test_ngauss' + str(ngauss) + '.png')
plt.show()
plt.clf(); plt.close()
plt.plot(x_C, y_4d, label="f''''(x)")
plt.plot(x_C, y_4d_calc[:-4], label="f''''(x) Calc")
#plt.plot(x_C, y_3d, label="Calc. f'''(x)")
#plt.plot(x_C, y_2d, label="Calc. f''(x)")
#plt.plot(x_C, y_1d, label="Calc. f'(x)")
#plt.plot(x_C, y_0d, label="Calc. f(x)")
#plt.plot(x, y_calc - y, label="Integrated f''''(x) - f(x)")
plt.xlim(-30, 30)
#plt.ylim(-15, max(As)*1.1)
plt.legend(loc='best')
plt.savefig('figures/integration_test_derivs_ngauss_' + str(ngauss) + '.png')
