def inverse_exp_coord(beta1, beta2):
"""
Calculate the inverse exponential to obtain a shooting vector from
beta1 to beta2 in shape space of open curves
:param beta1: numpy ndarray of shape (2,M) of M samples
:param beta2: numpy ndarray of shape (2,M) of M samples
:rtype: numpy ndarray
:return v: shooting vectors
:return dist: distance
"""
T = beta1.shape[1]
centroid1 = calculatecentroid(beta1)
beta1 = beta1 - tile(centroid1, [T, 1]).T
centroid2 = calculatecentroid(beta2)
beta2 = beta2 - tile(centroid2, [T, 1]).T
q1 = curve_to_q(beta1)
# Iteratively optimize over SO(n) x Gamma
for i in range(0, 1):
# Optimize over SO(n)
beta2, O_hat, tau = find_rotation_and_seed_coord(beta1, beta2)
q2 = curve_to_q(beta2)
# Optimize over Gamma
gam = optimum_reparam_curve(q2, q1, 0.0)
gamI = uf.invertGamma(gam)
# Applying optimal re-parameterization to the second curve
beta2 = group_action_by_gamma_coord(beta2, gamI)
# Optimize over SO(n)
beta2, O_hat, tau = find_rotation_and_seed_coord(beta1, beta2)
q2n = curve_to_q(beta2)
# Compute geodesic distance
q1dotq2 = innerprod_q2(q1, q2n)
dist = arccos(q1dotq2)
# Compute shooting vector
if q1dotq2 > 1:
q1dotq2 = 1
u = q2n - q1dotq2 * q1
normu = sqrt(innerprod_q2(u, u))
if normu > 1e-4:
v = u * arccos(q1dotq2) / normu
else:
v = zeros((2, T))
return (v, dist)
def find_C(C, qn, vec, q0, m, mu_psi):
qhat, gamhat, a, U, s, mu_g = jointfPCAd(qn, vec, C, m, mu_psi)
(M,N) = qn.shape
time = np.linspace(0,1,M-1)
d = np.zeros(N)
for i in range(0,N):
tmp = uf.warp_q_gamma(time, qhat[0:(M-1),i], uf.invertGamma(gamhat[:,i]))
d[i] = trapz((tmp-q0[:,i])*(tmp-q0[:,i]), time)
out = sum(d*d)/N
return out
def oc_srvf_align(beta, mode='O'):
"""
This claculates the mean of a set of curves and aligns them
:param beta: numpy ndarray of shape (n, M, N) describing N curves
in R^M
:param mode: Open ('O') or closed curve ('C') (default 'O')
:rtype: tuple of numpy array
:return betan: aligned curves
:return qn: aligned srvf
:return betamean: mean curve
:return mu: mean srvf
"""
n, T, N = beta.shape
# find mean
mu, betamean, v, q = curve_karcher_mean(beta, mode=mode)
qn = zeros((n, T, N))
betan = zeros((n, T, N))
centroid2 = cf.calculatecentroid(betamean)
betamean = betamean - tile(centroid2, [T, 1]).T
q_mu = cf.curve_to_q(betamean)
# align to mean
for ii in range(0, N):
beta1 = beta[:, :, ii]
centroid1 = cf.calculatecentroid(beta1)
beta1 = beta1 - tile(centroid1, [T, 1]).T
# Iteratively optimize over SO(n) x Gamma
for i in range(0, 1):
# Optimize over SO(n)
beta1, O_hat, tau = cf.find_rotation_and_seed_coord(betamean,
beta1)
q1 = cf.curve_to_q(beta1)
# Optimize over Gamma
gam = cf.optimum_reparam_curve(q1, q_mu, 0.0)
gamI = uf.invertGamma(gam)
# Applying optimal re-parameterization to the second curve
beta1 = cf.group_action_by_gamma_coord(beta1, gamI)
# Optimize over SO(n)
beta1, O_hat, tau = cf.find_rotation_and_seed_coord(betamean, beta1)
qn[:, :, ii] = cf.curve_to_q(beta1)
betan[:, :, ii] = beta1
align_results = collections.namedtuple('align', ['betan', 'qn', 'betamean', 'mu'])
out = align_results(betan, qn, betamean, q_mu)
return out
def geod_sphere(beta1, beta2, k=5):
"""
This function caluclates the geodecis between open curves beta1 and
beta2 with k steps along path
:param beta1: numpy ndarray of shape (2,M) of M samples
:param beta2: numpy ndarray of shape (2,M) of M samples
:param k: number of samples along path (Default = 5)
:rtype: numpy ndarray
:return dist: geodesic distance
:return path: geodesic path
:return O: rotation matrix
"""
lam = 0.0
elastic = 1
rotation = 1
returnpath = 1
n, T = beta1.shape
beta1 = cf.resamplecurve(beta1, T)
beta2 = cf.resamplecurve(beta2, T)
centroid1 = cf.calculatecentroid(beta1)
beta1 = beta1 - tile(centroid1, [T, 1]).T
centroid2 = cf.calculatecentroid(beta2)
beta2 = beta2 - tile(centroid2, [T, 1]).T
q1 = cf.curve_to_q(beta1)
if rotation:
beta2, O1, tau = cf.find_rotation_and_seed_coord(beta1, beta2)
q2 = cf.curve_to_q(beta2)
else:
O1 = eye(2)
q2 = cf.curve_to_q(beta2)
if elastic:
# Find the optimal coorespondence
gam = cf.optimum_reparam_curve(q2, q1, lam)
gamI = uf.invertGamma(gam)
# Applying optimal re-parameterization to the second curve
beta2n = cf.group_action_by_gamma_coord(beta2, gamI)
q2n = cf.curve_to_q(beta2n)
if rotation:
beta2n, O2, tau = cf.find_rotation_and_seed_coord(beta1, beta2n)
centroid2 = cf.calculatecentroid(beta2n)
beta2n = beta2n - tile(centroid2, [T, 1]).T
q2n = cf.curve_to_q(beta2n)
O = O1.dot(O2)
else:
q2n = q2
O = O1
# Forming geodesic between the registered curves
dist = arccos(cf.innerprod_q2(q1, q2n))
if returnpath:
PsiQ = zeros((n, T, k))
PsiX = zeros((n, T, k))
for tau in range(0, k):
s = dist * tau / (k - 1.)
PsiQ[:, :, tau] = (sin(dist-s)*q1+sin(s)*q2n)/sin(dist)
PsiX[:, :, tau] = cf.q_to_curve(PsiQ[:, :, tau])
path = PsiQ
else:
path = 0
return(dist, path, O)
def init_path_rand(beta1, beta_mid, beta2, T=100, k=5):
"""
Initializes a path in \cal{C}. beta1, beta_mid beta2 are already
standardized curves. Creates a path from beta1 to beta_mid to beta2 in
shape space, then projects to the closed shape manifold.
:param beta1: numpy ndarray of shape (2,M) of M samples (first curve)
:param betamid: numpy ndarray of shape (2,M) of M samples (mid curve)
:param beta2: numpy ndarray of shape (2,M) of M samples (end curve)
:param T: Number of samples of curve (Default = 100)
:param k: number of samples along path (Default = 5)
:rtype: numpy ndarray
:return alpha: a path between two q-functions
:return beta: a path between two curves
:return O: rotation matrix
"""
alpha = zeros((2, T, k))
beta = zeros((2, T, k))
q1 = cf.curve_to_q(beta1)
q_mid = cf.curve_to_q(beta_mid)
# find optimal rotation of q2
beta2, O1, tau1 = cf.find_rotation_and_seed_coord(beta1, beta2)
q2 = cf.curve_to_q(beta2)
# find the optimal coorespondence
gam = cf.optimum_reparam_curve(q2, q1)
gamI = uf.invertGamma(gam)
# apply optimal reparametrization
beta2n = cf.group_action_by_gamma_coord(beta2, gamI)
# find optimal rotation of q2
beta2n, O2, tau1 = cf.find_rotation_and_seed_coord(beta1, beta2n)
centroid2 = cf.calculatecentroid(beta2n)
beta2n = beta2n - tile(centroid2, [T, 1]).T
q2n = cf.curve_to_q(beta2n)
O = O1.dot(O2)
# Initialize a path as a geodesic through q1 --- q_mid --- q2
theta1 = arccos(cf.innerprod_q2(q1, q_mid))
theta2 = arccos(cf.innerprod_q2(q_mid, q2n))
tmp = arange(2, int((k - 1) / 2) + 1)
t = zeros(tmp.size)
alpha[:, :, 0] = q1
beta[:, :, 0] = beta1
i = 0
for tau in range(2, int((k - 1) / 2) + 1):
t[i] = (tau - 1.0) / ((k - 1) / 2.0)
qnew = (1 / sin(theta1)) * (sin((1 - t[i]) * theta1) * q1 + sin(t[i] * theta1) * q_mid)
alpha[:, :, tau - 1] = cf.project_curve(qnew)
x = cf.q_to_curve(alpha[:, :, tau - 1])
a = -1 * cf.calculatecentroid(x)
beta[:, :, tau - 1] = x + tile(a, [T, 1]).T
i += 1
alpha[:, :, int((k - 1) / 2)] = q_mid
beta[:, :, int((k - 1) / 2)] = beta_mid
i = 0
for tau in range(int((k - 1) / 2) + 1, k - 1):
qnew = (1 / sin(theta2)) * (sin((1 - t[i]) * theta2) * q_mid + sin(t[i] * theta2) * q2n)
alpha[:, :, tau] = cf.project_curve(qnew)
x = cf.q_to_curve(alpha[:, :, tau])
a = -1 * cf.calculatecentroid(x)
beta[:, :, tau] = x + tile(a, [T, 1]).T
i += 1
alpha[:, :, k - 1] = q2n
beta[:, :, k - 1] = beta2n
return (alpha, beta, O)
def geod_sphere(beta1, beta2, k=5):
"""
This function caluclates the geodecis between open curves beta1 and
beta2 with k steps along path
:param beta1: numpy ndarray of shape (2,M) of M samples
:param beta2: numpy ndarray of shape (2,M) of M samples
:param k: number of samples along path (Default = 5)
:rtype: numpy ndarray
:return dist: geodesic distance
:return path: geodesic path
:return O: rotation matrix
"""
lam = 0.0
elastic = 1
rotation = 1
returnpath = 1
n, T = beta1.shape
beta1 = cf.resamplecurve(beta1, T)
beta2 = cf.resamplecurve(beta2, T)
centroid1 = cf.calculatecentroid(beta1)
beta1 = beta1 - tile(centroid1, [T, 1]).T
centroid2 = cf.calculatecentroid(beta2)
beta2 = beta2 - tile(centroid2, [T, 1]).T
q1 = cf.curve_to_q(beta1)
if rotation:
beta2, O1, tau = cf.find_rotation_and_seed_coord(beta1, beta2)
q2 = cf.curve_to_q(beta2)
else:
O1 = eye(2)
q2 = cf.curve_to_q(beta2)
if elastic:
# Find the optimal coorespondence
gam = cf.optimum_reparam_curve(q2, q1, lam)
gamI = uf.invertGamma(gam)
# Applying optimal re-parameterization to the second curve
beta2n = cf.group_action_by_gamma_coord(beta2, gamI)
q2n = cf.curve_to_q(beta2n)
if rotation:
beta2n, O2, tau = cf.find_rotation_and_seed_coord(beta1, beta2n)
centroid2 = cf.calculatecentroid(beta2n)
beta2n = beta2n - tile(centroid2, [T, 1]).T
q2n = cf.curve_to_q(beta2n)
O = O1.dot(O2)
else:
q2n = q2
O = O1
# Forming geodesic between the registered curves
dist = arccos(cf.innerprod_q2(q1, q2n))
if returnpath:
PsiQ = zeros((n, T, k))
PsiX = zeros((n, T, k))
for tau in range(0, k):
s = dist * tau / (k - 1.0)
PsiQ[:, :, tau] = (sin(dist - s) * q1 + sin(s) * q2n) / sin(dist)
PsiX[:, :, tau] = cf.q_to_curve(PsiQ[:, :, tau])
path = PsiQ
else:
path = 0
return (dist, path, O)
def find_C_sub(time, qhat, gamhat, q0):
tmp = uf.warp_q_gamma(time, qhat, uf.invertGamma(gamhat))
d = trapz((tmp - q0) * (tmp - q0), time)
return d
def gauss_model(fn, time, qn, gam, n=1, sort_samples=False):
"""
This function models the functional data using a Gaussian model
extracted from the principal components of the srvfs
:param fn: numpy ndarray of shape (M,N) of N aligned functions with
M samples
:param time: vector of size M describing the sample points
:param qn: numpy ndarray of shape (M,N) of N aligned srvfs with M samples
:param gam: warping functions
:param n: number of random samples
:param sort_samples: sort samples (default = T)
:type n: integer
:type sort_samples: bool
:type fn: np.ndarray
:type qn: np.ndarray
:type gam: np.ndarray
:type time: np.ndarray
:rtype: tuple of numpy array
:return fs: random aligned samples
:return gams: random warping functions
:return ft: random samples
"""
# Parameters
eps = np.finfo(np.double).eps
binsize = np.diff(time)
binsize = binsize.mean()
M = time.size
# compute mean and covariance in q-domain
mq_new = qn.mean(axis=1)
mididx = np.round(time.shape[0] / 2)
m_new = np.sign(fn[mididx, :]) * np.sqrt(np.abs(fn[mididx, :]))
mqn = np.append(mq_new, m_new.mean())
qn2 = np.vstack((qn, m_new))
C = np.cov(qn2)
q_s = np.random.multivariate_normal(mqn, C, n)
q_s = q_s.transpose()
# compute the correspondence to the original function domain
fs = np.zeros((M, n))
for k in range(0, n):
fs[:, k] = uf.cumtrapzmid(time, q_s[0:M, k] * np.abs(q_s[0:M, k]),
np.sign(q_s[M, k]) * (q_s[M, k] ** 2),
mididx)
fbar = fn.mean(axis=1)
fsbar = fs.mean(axis=1)
err = np.transpose(np.tile(fbar-fsbar, (n,1)))
fs += err
# random warping generation
rgam = uf.randomGamma(gam, n)
gams = np.zeros((M, n))
for k in range(0, n):
gams[:, k] = uf.invertGamma(rgam[:, k])
# sort functions and warping
if sort_samples:
mx = fs.max(axis=0)
seq1 = mx.argsort()
# compute the psi-function
fy = np.gradient(rgam, binsize)
psi = fy / np.sqrt(abs(fy) + eps)
ip = np.zeros(n)
len = np.zeros(n)
for i in range(0, n):
tmp = np.ones(M)
ip[i] = tmp.dot(psi[:, i] / M)
len[i] = np.acos(tmp.dot(psi[:, i] / M))
seq2 = len.argsort()
# combine x-variability and y-variability
ft = np.zeros((M, n))
for k in range(0, n):
ft[:, k] = np.interp(gams[:, seq2[k]], np.arange(0, M) /
np.double(M - 1), fs[:, seq1[k]])
tmp = np.isnan(ft[:, k])
while tmp.any():
rgam2 = uf.randomGamma(gam, 1)
ft[:, k] = np.interp(gams[:, seq2[k]], np.arange(0, M) /
np.double(M - 1), uf.invertGamma(rgam2))
else:
# combine x-variability and y-variability
ft = np.zeros((M, n))
for k in range(0, n):
ft[:, k] = np.interp(gams[:, k], np.arange(0, M) /
np.double(M - 1), fs[:, k])
tmp = np.isnan(ft[:, k])
while tmp.any():
rgam2 = uf.randomGamma(gam, 1)
ft[:, k] = np.interp(gams[:, k], np.arange(0, M) /
np.double(M - 1), uf.invertGamma(rgam2))
samples = collections.namedtuple('samples', ['fs', 'gams', 'ft'])
out = samples(fs, rgam, ft)
return out
def init_path_rand(beta1, beta_mid, beta2, T=100, k=5):
r"""
Initializes a path in :math:`\cal{C}`. beta1, beta_mid beta2 are already
standardized curves. Creates a path from beta1 to beta_mid to beta2 in
shape space, then projects to the closed shape manifold.
:param beta1: numpy ndarray of shape (2,M) of M samples (first curve)
:param betamid: numpy ndarray of shape (2,M) of M samples (mid curve)
:param beta2: numpy ndarray of shape (2,M) of M samples (end curve)
:param T: Number of samples of curve (Default = 100)
:param k: number of samples along path (Default = 5)
:rtype: numpy ndarray
:return alpha: a path between two q-functions
:return beta: a path between two curves
:return O: rotation matrix
"""
alpha = zeros((2, T, k))
beta = zeros((2, T, k))
q1 = cf.curve_to_q(beta1)[0]
q_mid = cf.curve_to_q(beta_mid)[0]
# find optimal rotation of q2
beta2, O1, tau1 = cf.find_rotation_and_seed_coord(beta1, beta2)
q2 = cf.curve_to_q(beta2)[0]
# find the optimal coorespondence
gam = cf.optimum_reparam_curve(q2, q1)
gamI = uf.invertGamma(gam)
# apply optimal reparametrization
beta2n = cf.group_action_by_gamma_coord(beta2, gamI)
# find optimal rotation of q2
beta2n, O2, tau1 = cf.find_rotation_and_seed_coord(beta1, beta2n)
centroid2 = cf.calculatecentroid(beta2n)
beta2n = beta2n - tile(centroid2, [T, 1]).T
q2n = cf.curve_to_q(beta2n)[0]
O = O1 @ O2
# Initialize a path as a geodesic through q1 --- q_mid --- q2
theta1 = arccos(cf.innerprod_q2(q1, q_mid))
theta2 = arccos(cf.innerprod_q2(q_mid, q2n))
tmp = arange(2, int((k - 1) / 2) + 1)
t = zeros(tmp.size)
alpha[:, :, 0] = q1
beta[:, :, 0] = beta1
i = 0
for tau in range(2, int((k - 1) / 2) + 1):
t[i] = (tau - 1.) / ((k - 1) / 2.)
qnew = (1 / sin(theta1)) * (sin(
(1 - t[i]) * theta1) * q1 + sin(t[i] * theta1) * q_mid)
alpha[:, :, tau - 1] = cf.project_curve(qnew)
x = cf.q_to_curve(alpha[:, :, tau - 1])
a = -1 * cf.calculatecentroid(x)
beta[:, :, tau - 1] = x + tile(a, [T, 1]).T
i += 1
alpha[:, :, int((k - 1) / 2)] = q_mid
beta[:, :, int((k - 1) / 2)] = beta_mid
i = 0
for tau in range(int((k - 1) / 2) + 1, k - 1):
qnew = (1 / sin(theta2)) * (sin(
(1 - t[i]) * theta2) * q_mid + sin(t[i] * theta2) * q2n)
alpha[:, :, tau] = cf.project_curve(qnew)
x = cf.q_to_curve(alpha[:, :, tau])
a = -1 * cf.calculatecentroid(x)
beta[:, :, tau] = x + tile(a, [T, 1]).T
i += 1
alpha[:, :, k - 1] = q2n
beta[:, :, k - 1] = beta2n
return (alpha, beta, O)
def find_rotation_and_seed_q(q1, q2, closed=0, rotation=True, method="DP"):
"""
This function returns a candidate list of optimally oriented and
registered (seed) srvs w.r.t. q1
:param q1: numpy ndarray of shape (2,M) of M samples
:param q2: numpy ndarray of shape (2,M) of M samples
:param closed: Open (0) or Closed (1)
:param rotation: find rotation (default=True)
:param method: method to apply optimization (default="DP") options are "DP" or "RBFGS"
:rtype: numpy ndarray
:return q2best: optimal aligned q2 to q1
:return Rbest: rotation matrix
:return gamIbest: warping function
"""
n, T = q1.shape
scl = 4.
minE = 4000
if closed == 1:
end_idx = int(floor(T/scl))
scl = 4
else:
end_idx = 0
for ctr in range(0, end_idx+1):
if closed == 1:
q2n = shift_f(q2, scl*ctr)
else:
q2n = q2
if rotation:
q2new, R = find_best_rotation(q1, q2n)
else:
q2new = q2n.copy()
R = eye(n)
# Reparam
if norm(q1-q2new,'fro') > 0.0001:
gam = optimum_reparam_curve(q2new, q1, 0.0, method)
gamI = uf.invertGamma(gam)
q2new = group_action_by_gamma(q2new,gamI)
if closed == 1:
q2new = project_curve(q2new)
else:
gamI = linspace(0,1,T)
tmp = innerprod_q2(q1,q2new)
if tmp > 1:
tmp = 1
if tmp < -1:
tmp = -1
Ec = arccos(tmp)
if Ec < minE:
Rbest = R
q2best = q2new
gamIbest = gamI
minE = Ec
return (q2best, Rbest, gamIbest)
def gauss_model(fn, time, qn, gam, n=1, sort_samples=False):
"""
This function models the functional data using a Gaussian model
extracted from the principal components of the srvfs
:param fn: numpy ndarray of shape (M,N) of N aligned functions with
M samples
:param time: vector of size M describing the sample points
:param qn: numpy ndarray of shape (M,N) of N aligned srvfs with M samples
:param gam: warping functions
:param n: number of random samples
:param sort_samples: sort samples (default = T)
:type n: integer
:type sort_samples: bool
:type fn: np.ndarray
:type qn: np.ndarray
:type gam: np.ndarray
:type time: np.ndarray
:rtype: tuple of numpy array
:return fs: random aligned samples
:return gams: random warping functions
:return ft: random samples
"""
# Parameters
eps = np.finfo(np.double).eps
binsize = np.diff(time)
binsize = binsize.mean()
M = time.size
# compute mean and covariance in q-domain
mq_new = qn.mean(axis=1)
mididx = np.round(time.shape[0] / 2)
m_new = np.sign(fn[mididx, :]) * np.sqrt(np.abs(fn[mididx, :]))
mqn = np.append(mq_new, m_new.mean())
qn2 = np.vstack((qn, m_new))
C = np.cov(qn2)
q_s = np.random.multivariate_normal(mqn, C, n)
q_s = q_s.transpose()
# compute the correspondence to the original function domain
fs = np.zeros((M, n))
for k in range(0, n):
fs[:, k] = uf.cumtrapzmid(time, q_s[0:M, k] * np.abs(q_s[0:M, k]),
np.sign(q_s[M, k]) * (q_s[M, k]**2))
# random warping generation
rgam = uf.randomGamma(gam, n)
gams = np.zeros((M, n))
for k in range(0, n):
gams[:, k] = uf.invertGamma(rgam[:, k])
# sort functions and warping
if sort_samples:
mx = fs.max(axis=0)
seq1 = mx.argsort()
# compute the psi-function
fy = np.gradient(rgam, binsize)
psi = fy / np.sqrt(abs(fy) + eps)
ip = np.zeros(n)
len = np.zeros(n)
for i in range(0, n):
tmp = np.ones(M)
ip[i] = tmp.dot(psi[:, i] / M)
len[i] = np.acos(tmp.dot(psi[:, i] / M))
seq2 = len.argsort()
# combine x-variability and y-variability
ft = np.zeros((M, n))
for k in range(0, n):
ft[:, k] = np.interp(gams[:, seq2[k]],
np.arange(0, M) / np.double(M - 1),
fs[:, seq1[k]])
tmp = np.isnan(ft[:, k])
while tmp.any():
rgam2 = uf.randomGamma(gam, 1)
ft[:, k] = np.interp(gams[:, seq2[k]],
np.arange(0, M) / np.double(M - 1),
uf.invertGamma(rgam2))
else:
# combine x-variability and y-variability
ft = np.zeros((M, n))
for k in range(0, n):
ft[:, k] = np.interp(gams[:, k],
np.arange(0, M) / np.double(M - 1), fs[:, k])
tmp = np.isnan(ft[:, k])
while tmp.any():
rgam2 = uf.randomGamma(gam, 1)
ft[:, k] = np.interp(gams[:, k],
np.arange(0, M) / np.double(M - 1),
uf.invertGamma(rgam2))
samples = collections.namedtuple('samples', ['fs', 'gams', 'ft'])
out = samples(fs, rgam, ft)
return out
def gauss_model(self, n=1, sort_samples=False):
"""
This function models the functional data using a Gaussian model
extracted from the principal components of the srvfs
:param n: number of random samples
:param sort_samples: sort samples (default = T)
:type n: integer
:type sort_samples: bool
"""
fn = self.fn
time = self.time
qn = self.qn
gam = self.gam
# Parameters
eps = np.finfo(np.double).eps
binsize = np.diff(time)
binsize = binsize.mean()
M = time.size
# compute mean and covariance in q-domain
mq_new = qn.mean(axis=1)
mididx = np.round(time.shape[0] / 2)
m_new = np.sign(fn[mididx, :]) * np.sqrt(np.abs(fn[mididx, :]))
mqn = np.append(mq_new, m_new.mean())
qn2 = np.vstack((qn, m_new))
C = np.cov(qn2)
q_s = np.random.multivariate_normal(mqn, C, n)
q_s = q_s.transpose()
# compute the correspondence to the original function domain
fs = np.zeros((M, n))
for k in range(0, n):
fs[:, k] = uf.cumtrapzmid(time, q_s[0:M, k] * np.abs(q_s[0:M, k]),
np.sign(q_s[M, k]) * (q_s[M, k]**2),
mididx)
fbar = fn.mean(axis=1)
fsbar = fs.mean(axis=1)
err = np.transpose(np.tile(fbar - fsbar, (n, 1)))
fs += err
# random warping generation
rgam = uf.randomGamma(gam, n)
gams = np.zeros((M, n))
for k in range(0, n):
gams[:, k] = uf.invertGamma(rgam[:, k])
# sort functions and warping
if sort_samples:
mx = fs.max(axis=0)
seq1 = mx.argsort()
# compute the psi-function
fy = np.gradient(rgam, binsize)
psi = fy / np.sqrt(abs(fy) + eps)
ip = np.zeros(n)
len = np.zeros(n)
for i in range(0, n):
tmp = np.ones(M)
ip[i] = tmp.dot(psi[:, i] / M)
len[i] = np.arccos(tmp.dot(psi[:, i] / M))
seq2 = len.argsort()
# combine x-variability and y-variability
ft = np.zeros((M, n))
for k in range(0, n):
ft[:, k] = np.interp(gams[:, seq2[k]],
np.arange(0, M) / np.double(M - 1),
fs[:, seq1[k]])
tmp = np.isnan(ft[:, k])
while tmp.any():
rgam2 = uf.randomGamma(gam, 1)
ft[:, k] = np.interp(gams[:, seq2[k]],
np.arange(0, M) / np.double(M - 1),
uf.invertGamma(rgam2))
else:
# combine x-variability and y-variability
ft = np.zeros((M, n))
for k in range(0, n):
ft[:, k] = np.interp(gams[:, k],
np.arange(0, M) / np.double(M - 1), fs[:,
k])
tmp = np.isnan(ft[:, k])
while tmp.any():
rgam2 = uf.randomGamma(gam, 1)
ft[:, k] = np.interp(gams[:, k],
np.arange(0, M) / np.double(M - 1),
uf.invertGamma(rgam2))
self.rsamps = True
self.fs = fs
self.gams = rgam
self.ft = ft
self.qs = q_s[0:M, :]
return
