def main():
# Testing the get points
num_splines = 10
num_dados = 1000
x_min = 0
x_max = 10
lmbda = 100
# Generate a random spline to get samples from it
weights = np.random.rand(num_splines)
spl = spline(weights, x_min=x_min, x_max=x_max)
# Get samples with noise
t, x_t, spl_t = get_points(num_dados, num_splines, spl, x_min, x_max)
if num_dados < 1000:
plt.scatter(t, x_t, label="Simulated with noise", color=np.random.rand(3,))
plt.plot(t, spl_t, label="Original data")
plt.xlabel("t")
plt.ylabel("spl(t)")
plt.title("Dados")
# Interpolate the samples
res = interpolate(t, x_t, num_splines, lmbda)
new_spline = spline(res, x_min=x_min, x_max=x_max)
plt.plot(t, new_spline(t), label="Simulated curve")
plt.legend(loc="upper right")
plt.show()
def interpolate(x, y, num_splines, lmbda):
default_spl = spline(np.ones(num_splines), x_min=x[0], x_max=x[-1])
mu = np.zeros((len(x), num_splines))
for i in range(len(x)):
for j in range(num_splines):
mu[i][j] = default_spl.beta_j(j, x[i])
m_1 = mu.T@mu
b = [email protected]
m_2 = matrix_m2(num_splines)
M = m_1 + lmbda*m_2
res = np.linalg.solve(M, b)
return res
def points_random_noise(timeline, n_data_points, n_splines, x_min, x_max):
# Gera pesos aleatórios e cria uma spline a partir deles
rand_weights = np.array(np.random.rand(n_splines))
rand_spline = spl.spline(rand_weights,
x_min=timeline[0],
x_max=timeline[-1])(timeline)
# Adiciona ruído aos pontos da spline gerada
sigma = np.std(rand_spline) / 5
noise = np.array(np.random.normal(0, sigma, n_data_points))
points_generated = rand_spline + noise
return points_generated
def main(n_data_points=350, n_splines=14, lmb=0.1, x_min=0, x_max=50):
timeline = np.linspace(x_min, x_max, n_data_points)
points = points_random_noise(timeline, n_data_points, n_splines, x_min,
x_max)
adjusted = adjust_spline(timeline, points, n_splines, lmb)
points_with_noise = plt.scatter(timeline, points, s=3.0, color="green")
generated_spline = plt.plot(timeline,
spl.spline(adjusted, timeline[0],
timeline[-1])(timeline),
color='red')
plt.show()
def adjust_spline(timeline, x_t, num_splines, lmb):
# Cria uma spline "base", com coeficientes iguais a 1 para usar os beta_j's
base_spline = spl.spline(np.ones(num_splines),
x_min=timeline[0],
x_max=timeline[-1])
mu_aux = []
for j in range(num_splines):
mu_aux.append(base_spline.beta_j(j, timeline))
mu = np.array(mu_aux).T
m1 = np.dot(mu.T, mu)
y = np.dot(mu.T, x_t.T)
m2 = spl.matrix_m2(num_splines)
M = m1 + lmb * m2
# Resolve o sistema linear e retorna a solução
adjusted = np.linalg.solve(M, y)
return adjusted
def sphere_best_mobility_known(location, ETA, A):
'''Best mobility known for a single sphere close to a wall. This function uses the mobilities
translational-perpendicular: P. Huang and K. S. Breuer PRE 76, 046307 (2007).
translational-parallel: for distances very close to the wall
A. J. Goldman, R. G. Cox and H. Brenner, Chemical engineering science, 22, 637 (1967).
For other distances
L. P. Fauxcheux and A. J. Libchaber PRE 49, 5158 (1994).
rotational-perpendicular: Cubic spline fit to the mobility of a sphere discretize with 162 markers.
rotational-parallel: Cubic spline fit to the mobility of a sphere discretize with 162 markers.
rotation-translation-coupling: Cubic spline fit to the mobility of a sphere discretize with 162 markers.
'''
# Init the function the first time
try:
# Check if the spline function has been called
sphere_best_mobility_known.init += 1e-01
except:
# Exception to call the spline function only once per mobility component
sphere_best_mobility_known.init = 1e-01
# Create variables to store mobilities:
# distance to the wall
sphere_best_mobility_known.mX = []
# Mobilities
sphere_best_mobility_known.mRotationRotationParallel = []
sphere_best_mobility_known.mRotationRotationPerpendicular = []
sphere_best_mobility_known.mRotationTranslationCoupling = []
# Read 162-blobs sphere mobility.
# rotational-roational mobility is normalize by (8*pi*ETA*A**3)
# rotational-translation mobility is normalize by (6*pi*ETA*A**2)
f = open('mobility.162-blob.dat', 'r')
next(f)
for line in f:
data = line.split()
# distance to the wall
sphere_best_mobility_known.mX.append(float(data[0]))
# Mobilities
sphere_best_mobility_known.mRotationRotationParallel.append(
float(data[3]))
sphere_best_mobility_known.mRotationRotationPerpendicular.append(
float(data[4]))
sphere_best_mobility_known.mRotationTranslationCoupling.append(
float(data[5]))
# Create second derivative of mobility finctions
# Call spline function
sphere_best_mobility_known.mRotationRotationParallel_y2 = splines.spline(
sphere_best_mobility_known.mX,
sphere_best_mobility_known.mRotationRotationParallel,
len(sphere_best_mobility_known.mX), 1e+30, 1e+30)
sphere_best_mobility_known.mRotationRotationPerpendicular_y2 = splines.spline(
sphere_best_mobility_known.mX,
sphere_best_mobility_known.mRotationRotationPerpendicular,
len(sphere_best_mobility_known.mX), 1e+30, 1e+30)
sphere_best_mobility_known.mRotationTranslationCoupling_y2 = splines.spline(
sphere_best_mobility_known.mX,
sphere_best_mobility_known.mRotationTranslationCoupling,
len(sphere_best_mobility_known.mX), 1e+30, 1e+30)
# Compute mobilities
# define threshold, at this distance the parallel mobilities
# of Goldman and Feucheux cross
threshold = 1.02979 * A
# distance to the wall
h = location[2]
# dimensional factors
factor_tt = 1.0 / (6.0 * np.pi * ETA * A)
factor_rr = 1.0 / (8.0 * np.pi * ETA * A**3)
factor_tr = 1.0 / (6.0 * np.pi * ETA * A**2)
fluid_mobility = np.zeros([6, 6])
# translation-translation perpendicular to the wall
fluid_mobility[2, 2] = factor_tt * Huang.selfMobilityHuang(A, h)[1]
# translation-translation parallel to the wall
if (h < threshold):
mobility = factor_tt * Goldman.selfMobilityGoldman(A, h)[0, 0]
fluid_mobility[0, 0] = mobility
fluid_mobility[1, 1] = mobility
else:
mobility = factor_tt * Faucheux.selfMobilityFaucheux(A, h)
fluid_mobility[0, 0] = mobility
fluid_mobility[1, 1] = mobility
# Rescale distance to the wall, splines are for a sphere of radius 1
h = h / A
# rotation-rotation parallel to the wall
mobility = factor_rr * splines.splint(
sphere_best_mobility_known.mX,
sphere_best_mobility_known.mRotationRotationParallel,
sphere_best_mobility_known.mRotationRotationParallel_y2,
len(sphere_best_mobility_known.mX), h)
fluid_mobility[3, 3] = mobility
fluid_mobility[4, 4] = mobility
# rotation-rotation perpendicular to the wall
mobility = factor_rr * splines.splint(
sphere_best_mobility_known.mX,
sphere_best_mobility_known.mRotationRotationPerpendicular,
sphere_best_mobility_known.mRotationRotationPerpendicular_y2,
len(sphere_best_mobility_known.mX), h)
fluid_mobility[5, 5] = mobility
# rotation-translation coupling
mobility = factor_tr * splines.splint(
sphere_best_mobility_known.mX,
sphere_best_mobility_known.mRotationTranslationCoupling,
sphere_best_mobility_known.mRotationTranslationCoupling_y2,
len(sphere_best_mobility_known.mX), h)
fluid_mobility[0, 4] = mobility
fluid_mobility[1, 3] = mobility
fluid_mobility[3, 1] = mobility
fluid_mobility[4, 0] = mobility
return fluid_mobility
def sphere_best_mobility_known(location, ETA, A):
'''Best mobility known for a single sphere close to a wall. This function uses the mobilities
translational-perpendicular: P. Huang and K. S. Breuer PRE 76, 046307 (2007).
translational-parallel: for distances very close to the wall
A. J. Goldman, R. G. Cox and H. Brenner, Chemical engineering science, 22, 637 (1967).
For other distances
L. P. Fauxcheux and A. J. Libchaber PRE 49, 5158 (1994).
rotational-perpendicular: Cubic spline fit to the mobility of a sphere discretize with 162 markers.
rotational-parallel: Cubic spline fit to the mobility of a sphere discretize with 162 markers.
rotation-translation-coupling: Cubic spline fit to the mobility of a sphere discretize with 162 markers.
'''
# Init the function the first time
try:
# Check if the spline function has been called
sphere_best_mobility_known.init += 1e-01
except:
# Exception to call the spline function only once per mobility component
sphere_best_mobility_known.init = 1e-01
# Create variables to store mobilities:
# distance to the wall
sphere_best_mobility_known.mX = []
# Mobilities
sphere_best_mobility_known.mRotationRotationParallel = []
sphere_best_mobility_known.mRotationRotationPerpendicular = []
sphere_best_mobility_known.mRotationTranslationCoupling = []
# Read 162-blobs sphere mobility.
# rotational-roational mobility is normalize by (8*pi*ETA*A**3)
# rotational-translation mobility is normalize by (6*pi*ETA*A**2)
f = open('mobility.162-blob.dat', 'r')
next(f)
for line in f:
data = line.split()
# distance to the wall
sphere_best_mobility_known.mX.append(float(data[0]))
# Mobilities
sphere_best_mobility_known.mRotationRotationParallel.append(float(data[3]))
sphere_best_mobility_known.mRotationRotationPerpendicular.append(float(data[4]))
sphere_best_mobility_known.mRotationTranslationCoupling.append(float(data[5]))
# Create second derivative of mobility finctions
# Call spline function
sphere_best_mobility_known.mRotationRotationParallel_y2 = splines.spline(sphere_best_mobility_known.mX,
sphere_best_mobility_known.mRotationRotationParallel,
len(sphere_best_mobility_known.mX),
1e+30,
1e+30)
sphere_best_mobility_known.mRotationRotationPerpendicular_y2 = splines.spline(sphere_best_mobility_known.mX,
sphere_best_mobility_known.mRotationRotationPerpendicular,
len(sphere_best_mobility_known.mX),
1e+30,
1e+30)
sphere_best_mobility_known.mRotationTranslationCoupling_y2 = splines.spline(sphere_best_mobility_known.mX,
sphere_best_mobility_known.mRotationTranslationCoupling,
len(sphere_best_mobility_known.mX),
1e+30,
1e+30)
# Compute mobilities
# define threshold, at this distance the parallel mobilities
# of Goldman and Feucheux cross
threshold = 1.02979 * A
# distance to the wall
h = location[2]
# dimensional factors
factor_tt = 1.0 / (6.0*np.pi*ETA*A)
factor_rr = 1.0 / (8.0*np.pi*ETA*A**3)
factor_tr = 1.0 / (6.0*np.pi*ETA*A**2)
fluid_mobility = np.zeros( [6, 6] )
# translation-translation perpendicular to the wall
fluid_mobility[2,2] = factor_tt * Huang.selfMobilityHuang(A,h)[1]
# translation-translation parallel to the wall
if(h < threshold):
mobility = factor_tt * Goldman.selfMobilityGoldman(A, h)[0,0]
fluid_mobility[0,0] = mobility
fluid_mobility[1,1] = mobility
else:
mobility = factor_tt * Faucheux.selfMobilityFaucheux(A, h)
fluid_mobility[0,0] = mobility
fluid_mobility[1,1] = mobility
# Rescale distance to the wall, splines are for a sphere of radius 1
h = h / A
# rotation-rotation parallel to the wall
mobility = factor_rr * splines.splint(sphere_best_mobility_known.mX,
sphere_best_mobility_known.mRotationRotationParallel,
sphere_best_mobility_known.mRotationRotationParallel_y2,
len(sphere_best_mobility_known.mX),
h)
fluid_mobility[3,3] = mobility
fluid_mobility[4,4] = mobility
# rotation-rotation perpendicular to the wall
mobility = factor_rr * splines.splint(sphere_best_mobility_known.mX,
sphere_best_mobility_known.mRotationRotationPerpendicular,
sphere_best_mobility_known.mRotationRotationPerpendicular_y2,
len(sphere_best_mobility_known.mX),
h)
fluid_mobility[5,5] = mobility
# rotation-translation coupling
mobility = factor_tr * splines.splint(sphere_best_mobility_known.mX,
sphere_best_mobility_known.mRotationTranslationCoupling,
sphere_best_mobility_known.mRotationTranslationCoupling_y2,
len(sphere_best_mobility_known.mX),
h)
fluid_mobility[0,4] = mobility
fluid_mobility[1,3] = mobility
fluid_mobility[3,1] = mobility
fluid_mobility[4,0] = mobility
return fluid_mobility
import splines as sp
import numpy as np
import matplotlib.pyplot as plt
# a função original
n_weights = 10
w = np.random.randn(n_weights)
x = np.arange(0, 15.01, 0.01)
s = sp.spline(w, x.min(), x.max())
y = s(x)
label1, = plt.plot(x, y, c='g')
# criação de amostras com ruídos
x = np.arange(0, 15.5, 0.5)
n_samples = x.shape[0]
noise = np.random.normal(loc=0.0, scale=20, size=n_samples)
y = s(x) + noise
label2 = plt.scatter(x, y, c='r', marker='x')
# matriz B
B = np.zeros((n_samples, n_weights))
for i in range(n_samples):
for j in range(n_weights):
B[i, j] = s.beta_j(j, x[i])
# suavizador
M2 = sp.matrix_m2(n_weights)
b = B.T.dot(y)
M1 = B.T.dot(B)