def srsf_align(self,
method="mean",
omethod="DP",
smoothdata=False,
parallel=False,
lam=0.0,
cores=-1):
"""
This function aligns a collection of functions using the elastic
square-root slope (srsf) framework.
:param method: (string) warp calculate Karcher Mean or Median (options = "mean" or "median") (default="mean")
:param omethod: optimization method (DP, DP2) (default = DP)
:param smoothdata: Smooth the data using a box filter (default = F)
:param parallel: run in parallel (default = F)
:param lam: controls the elasticity (default = 0)
:param cores: number of cores for parallel (default = -1 (all))
:type lam: double
:type smoothdata: bool
Examples
>>> import tables
>>> fun=tables.open_file("../Data/simu_data.h5")
>>> f = fun.root.f[:]
>>> f = f.transpose()
>>> time = fun.root.time[:]
>>> obj = fs.fdawarp(f,time)
>>> obj.srsf_align()
"""
M = self.f.shape[0]
N = self.f.shape[1]
self.lam = lam
if M > 500:
parallel = True
elif N > 100:
parallel = True
eps = np.finfo(np.double).eps
f0 = self.f
self.method = omethod
methods = ["mean", "median"]
self.type = method
# 0 mean, 1-median
method = [i for i, x in enumerate(methods) if x == method]
if len(method) == 0:
method = 0
else:
method = method[0]
# Compute SRSF function from data
f, g, g2 = uf.gradient_spline(self.time, self.f, smoothdata)
q = g / np.sqrt(abs(g) + eps)
print("Initializing...")
mnq = q.mean(axis=1)
a = mnq.repeat(N)
d1 = a.reshape(M, N)
d = (q - d1)**2
dqq = np.sqrt(d.sum(axis=0))
min_ind = dqq.argmin()
mq = q[:, min_ind]
mf = f[:, min_ind]
if parallel:
out = Parallel(n_jobs=cores)(delayed(uf.optimum_reparam)(
mq, self.time, q[:, n], omethod, lam, mf[0], f[0, n])
for n in range(N))
gam = np.array(out)
gam = gam.transpose()
else:
gam = np.zeros((M, N))
for k in range(0, N):
gam[:, k] = uf.optimum_reparam(mq, self.time, q[:, k], omethod,
lam, mf[0], f[0, k])
gamI = uf.SqrtMeanInverse(gam)
mf = np.interp((self.time[-1] - self.time[0]) * gamI + self.time[0],
self.time, mf)
mq = uf.f_to_srsf(mf, self.time)
# Compute Karcher Mean
if method == 0:
print("Compute Karcher Mean of %d function in SRSF space..." % N)
if method == 1:
print("Compute Karcher Median of %d function in SRSF space..." % N)
MaxItr = 20
ds = np.repeat(0.0, MaxItr + 2)
ds[0] = np.inf
qun = np.repeat(0.0, MaxItr + 1)
tmp = np.zeros((M, MaxItr + 2))
tmp[:, 0] = mq
mq = tmp
tmp = np.zeros((M, MaxItr + 2))
tmp[:, 0] = mf
mf = tmp
tmp = np.zeros((M, N, MaxItr + 2))
tmp[:, :, 0] = self.f
f = tmp
tmp = np.zeros((M, N, MaxItr + 2))
tmp[:, :, 0] = q
q = tmp
for r in range(0, MaxItr):
print("updating step: r=%d" % (r + 1))
if r == (MaxItr - 1):
print("maximal number of iterations is reached")
# Matching Step
if parallel:
out = Parallel(n_jobs=cores)(delayed(uf.optimum_reparam)(
mq[:, r], self.time, q[:, n,
0], omethod, lam, mf[0, r], f[0, n,
0])
for n in range(N))
gam = np.array(out)
gam = gam.transpose()
else:
for k in range(0, N):
gam[:, k] = uf.optimum_reparam(mq[:, r], self.time,
q[:, k, 0], omethod, lam,
mf[0, r], f[0, k, 0])
gam_dev = np.zeros((M, N))
vtil = np.zeros((M, N))
dtil = np.zeros(N)
for k in range(0, N):
f[:, k, r + 1] = np.interp(
(self.time[-1] - self.time[0]) * gam[:, k] + self.time[0],
self.time, f[:, k, 0])
q[:, k, r + 1] = uf.f_to_srsf(f[:, k, r + 1], self.time)
gam_dev[:, k] = np.gradient(gam[:, k], 1 / float(M - 1))
v = q[:, k, r + 1] - mq[:, r]
d = np.sqrt(trapz(v * v, self.time))
vtil[:, k] = v / d
dtil[k] = 1.0 / d
mqt = mq[:, r]
a = mqt.repeat(N)
d1 = a.reshape(M, N)
d = (q[:, :, r + 1] - d1)**2
if method == 0:
d1 = sum(trapz(d, self.time, axis=0))
d2 = sum(trapz((1 - np.sqrt(gam_dev))**2, self.time, axis=0))
ds_tmp = d1 + lam * d2
ds[r + 1] = ds_tmp
# Minimization Step
# compute the mean of the matched function
qtemp = q[:, :, r + 1]
ftemp = f[:, :, r + 1]
mq[:, r + 1] = qtemp.mean(axis=1)
mf[:, r + 1] = ftemp.mean(axis=1)
qun[r] = norm(mq[:, r + 1] - mq[:, r]) / norm(mq[:, r])
if method == 1:
d1 = np.sqrt(sum(trapz(d, self.time, axis=0)))
d2 = sum(trapz((1 - np.sqrt(gam_dev))**2, self.time, axis=0))
ds_tmp = d1 + lam * d2
ds[r + 1] = ds_tmp
# Minimization Step
# compute the mean of the matched function
stp = .3
vbar = vtil.sum(axis=1) * (1 / dtil.sum())
qtemp = q[:, :, r + 1]
ftemp = f[:, :, r + 1]
mq[:, r + 1] = mq[:, r] + stp * vbar
tmp = np.zeros(M)
tmp[1:] = cumtrapz(mq[:, r + 1] * np.abs(mq[:, r + 1]),
self.time)
mf[:, r + 1] = np.median(f0[1, :]) + tmp
qun[r] = norm(mq[:, r + 1] - mq[:, r]) / norm(mq[:, r])
if qun[r] < 1e-2 or r >= MaxItr:
break
# Last Step with centering of gam
r += 1
if parallel:
out = Parallel(n_jobs=cores)(delayed(uf.optimum_reparam)(
mq[:, r], self.time, q[:, n,
0], omethod, lam, mf[0, r], f[0, n, 0])
for n in range(N))
gam = np.array(out)
gam = gam.transpose()
else:
for k in range(0, N):
gam[:, k] = uf.optimum_reparam(mq[:, r], self.time, q[:, k, 0],
omethod, lam, mf[0, r], f[0, k,
0])
gam_dev = np.zeros((M, N))
for k in range(0, N):
gam_dev[:, k] = np.gradient(gam[:, k], 1 / float(M - 1))
gamI = uf.SqrtMeanInverse(gam)
gamI_dev = np.gradient(gamI, 1 / float(M - 1))
time0 = (self.time[-1] - self.time[0]) * gamI + self.time[0]
mq[:,
r + 1] = np.interp(time0, self.time, mq[:, r]) * np.sqrt(gamI_dev)
for k in range(0, N):
q[:, k, r +
1] = np.interp(time0, self.time, q[:, k, r]) * np.sqrt(gamI_dev)
f[:, k, r + 1] = np.interp(time0, self.time, f[:, k, r])
gam[:, k] = np.interp(time0, self.time, gam[:, k])
# Aligned data & stats
self.fn = f[:, :, r + 1]
self.qn = q[:, :, r + 1]
self.q0 = q[:, :, 0]
mean_f0 = f0.mean(axis=1)
std_f0 = f0.std(axis=1)
mean_fn = self.fn.mean(axis=1)
std_fn = self.fn.std(axis=1)
self.gam = gam
self.mqn = mq[:, r + 1]
tmp = np.zeros(M)
tmp[1:] = cumtrapz(self.mqn * np.abs(self.mqn), self.time)
self.fmean = np.mean(f0[1, :]) + tmp
fgam = np.zeros((M, N))
for k in range(0, N):
time0 = (self.time[-1] - self.time[0]) * gam[:, k] + self.time[0]
fgam[:, k] = np.interp(time0, self.time, self.fmean)
var_fgam = fgam.var(axis=1)
self.orig_var = trapz(std_f0**2, self.time)
self.amp_var = trapz(std_fn**2, self.time)
self.phase_var = trapz(var_fgam, self.time)
return
def align_fPCA(f, time, num_comp=3, showplot=True, smoothdata=False, cores=-1):
"""
aligns a collection of functions while extracting principal components.
The functions are aligned to the principal components
:param f: numpy ndarray of shape (M,N) of N functions with M samples
:param time: vector of size M describing the sample points
:param num_comp: number of fPCA components
:param showplot: Shows plots of results using matplotlib (default = T)
:param smooth_data: Smooth the data using a box filter (default = F)
:param cores: number of cores for parallel (default = -1 (all))
:type sparam: double
:type smooth_data: bool
:type f: np.ndarray
:type time: np.ndarray
:rtype: tuple of numpy array
:return fn: aligned functions - numpy ndarray of shape (M,N) of N
functions with M samples
:return qn: aligned srvfs - similar structure to fn
:return q0: original srvf - similar structure to fn
:return mqn: srvf mean or median - vector of length M
:return gam: warping functions - similar structure to fn
:return q_pca: srsf principal directions
:return f_pca: functional principal directions
:return latent: latent values
:return coef: coefficients
:return U: eigenvectors
:return orig_var: Original Variance of Functions
:return amp_var: Amplitude Variance
:return phase_var: Phase Variance
"""
lam = 0.0
MaxItr = 50
coef = np.arange(-2., 3.)
Nstd = coef.shape[0]
M = f.shape[0]
N = f.shape[1]
if M > 500:
parallel = True
elif N > 100:
parallel = True
else:
parallel = False
eps = np.finfo(np.double).eps
f0 = f
if showplot:
plot.f_plot(time, f, title="Original Data")
# Compute SRSF function from data
f, g, g2 = uf.gradient_spline(time, f, smoothdata)
q = g / np.sqrt(abs(g) + eps)
print("Initializing...")
mnq = q.mean(axis=1)
a = mnq.repeat(N)
d1 = a.reshape(M, N)
d = (q - d1)**2
dqq = np.sqrt(d.sum(axis=0))
min_ind = dqq.argmin()
print("Aligning %d functions in SRVF space to %d fPCA components..." %
(N, num_comp))
itr = 0
mq = np.zeros((M, MaxItr + 1))
mq[:, itr] = q[:, min_ind]
fi = np.zeros((M, N, MaxItr + 1))
fi[:, :, 0] = f
qi = np.zeros((M, N, MaxItr + 1))
qi[:, :, 0] = q
gam = np.zeros((M, N, MaxItr + 1))
cost = np.zeros(MaxItr + 1)
while itr < MaxItr:
print("updating step: r=%d" % (itr + 1))
if itr == MaxItr:
print("maximal number of iterations is reached")
# PCA Step
a = mq[:, itr].repeat(N)
d1 = a.reshape(M, N)
qhat_cent = qi[:, :, itr] - d1
K = np.cov(qi[:, :, itr])
U, s, V = svd(K)
alpha_i = np.zeros((num_comp, N))
for ii in range(0, num_comp):
for jj in range(0, N):
alpha_i[ii, jj] = trapz(qhat_cent[:, jj] * U[:, ii], time)
U1 = U[:, 0:num_comp]
tmp = U1.dot(alpha_i)
qhat = d1 + tmp
# Matching Step
if parallel:
out = Parallel(n_jobs=cores)(
delayed(uf.optimum_reparam)(qhat[:,
n], time, qi[:, n,
itr], "DP", lam)
for n in range(N))
gam_t = np.array(out)
gam[:, :, itr] = gam_t.transpose()
else:
gam[:, :, itr] = uf.optimum_reparam(qhat, time, qi[:, :, itr],
"DP", lam)
for k in range(0, N):
time0 = (time[-1] - time[0]) * gam[:, k, itr] + time[0]
fi[:, k, itr + 1] = np.interp(time0, time, fi[:, k, itr])
qi[:, k, itr + 1] = uf.f_to_srsf(fi[:, k, itr + 1], time)
qtemp = qi[:, :, itr + 1]
mq[:, itr + 1] = qtemp.mean(axis=1)
cost_temp = np.zeros(N)
for ii in range(0, N):
cost_temp[ii] = norm(qtemp[:, ii] - qhat[:, ii])**2
cost[itr + 1] = cost_temp.mean()
if abs(cost[itr + 1] - cost[itr]) < 1e-06:
break
itr += 1
if itr >= MaxItr:
itrf = MaxItr
else:
itrf = itr + 1
cost = cost[1:(itrf + 1)]
# Aligned data & stats
fn = fi[:, :, itrf]
qn = qi[:, :, itrf]
q0 = qi[:, :, 0]
mean_f0 = f0.mean(axis=1)
std_f0 = f0.std(axis=1)
mqn = mq[:, itrf]
gamf = gam[:, :, 0]
for k in range(1, itr):
gam_k = gam[:, :, k]
for l in range(0, N):
time0 = (time[-1] - time[0]) * gam_k[:, l] + time[0]
gamf[:, l] = np.interp(time0, time, gamf[:, l])
# Center Mean
gamI = uf.SqrtMeanInverse(gamf)
gamI_dev = np.gradient(gamI, 1 / float(M - 1))
time0 = (time[-1] - time[0]) * gamI + time[0]
mqn = np.interp(time0, time, mqn) * np.sqrt(gamI_dev)
for k in range(0, N):
qn[:, k] = np.interp(time0, time, qn[:, k]) * np.sqrt(gamI_dev)
fn[:, k] = np.interp(time0, time, fn[:, k])
gamf[:, k] = np.interp(time0, time, gamf[:, k])
mean_fn = fn.mean(axis=1)
std_fn = fn.std(axis=1)
# Get Final PCA
mididx = int(np.round(time.shape[0] / 2))
m_new = np.sign(fn[mididx, :]) * np.sqrt(np.abs(fn[mididx, :]))
mqn2 = np.append(mqn, m_new.mean())
qn2 = np.vstack((qn, m_new))
K = np.cov(qn2)
U, s, V = svd(K)
stdS = np.sqrt(s)
# compute the PCA in the q domain
q_pca = np.ndarray(shape=(M + 1, Nstd, num_comp), dtype=float)
for k in range(0, num_comp):
for l in range(0, Nstd):
q_pca[:, l, k] = mqn2 + coef[l] * stdS[k] * U[:, k]
# compute the correspondence in the f domain
f_pca = np.ndarray(shape=(M, Nstd, num_comp), dtype=float)
for k in range(0, num_comp):
for l in range(0, Nstd):
q_pca_tmp = q_pca[0:M, l, k] * np.abs(q_pca[0:M, l, k])
q_pca_tmp2 = np.sign(q_pca[M, l, k]) * (q_pca[M, l, k]**2)
f_pca[:, l, k] = uf.cumtrapzmid(time, q_pca_tmp, q_pca_tmp2,
np.floor(time.shape[0] / 2),
mididx)
N2 = qn.shape[1]
c = np.zeros((N2, num_comp))
for k in range(0, num_comp):
for l in range(0, N2):
c[l, k] = sum((np.append(qn[:, l], m_new[l]) - mqn2) * U[:, k])
if showplot:
CBcdict = {
'Bl': (0, 0, 0),
'Or': (.9, .6, 0),
'SB': (.35, .7, .9),
'bG': (0, .6, .5),
'Ye': (.95, .9, .25),
'Bu': (0, .45, .7),
'Ve': (.8, .4, 0),
'rP': (.8, .6, .7),
}
cl = sorted(CBcdict.keys())
# Align Plots
fig, ax = plot.f_plot(np.arange(0, M) / float(M - 1),
gamf,
title="Warping Functions")
ax.set_aspect('equal')
plot.f_plot(time, fn, title="Warped Data")
tmp = np.array([mean_f0, mean_f0 + std_f0, mean_f0 - std_f0])
tmp = tmp.transpose()
plot.f_plot(time, tmp, title=r"Original Data: Mean $\pm$ STD")
tmp = np.array([mean_fn, mean_fn + std_fn, mean_fn - std_fn])
tmp = tmp.transpose()
plot.f_plot(time, tmp, title=r"Warped Data: Mean $\pm$ STD")
# PCA Plots
fig, ax = plt.subplots(2, num_comp)
for k in range(0, num_comp):
axt = ax[0, k]
for l in range(0, Nstd):
axt.plot(time, q_pca[0:M, l, k], color=CBcdict[cl[l]])
axt.hold(True)
axt.set_title('q domain: PD %d' % (k + 1))
plot.rstyle(axt)
axt = ax[1, k]
for l in range(0, Nstd):
axt.plot(time, f_pca[:, l, k], color=CBcdict[cl[l]])
axt.hold(True)
axt.set_title('f domain: PD %d' % (k + 1))
plot.rstyle(axt)
fig.set_tight_layout(True)
cumm_coef = 100 * np.cumsum(s) / sum(s)
idx = np.arange(0, M + 1) + 1
plot.f_plot(idx, cumm_coef, "Coefficient Cumulative Percentage")
plt.xlabel("Percentage")
plt.ylabel("Index")
plt.show()
mean_f0 = f0.mean(axis=1)
std_f0 = f0.std(axis=1)
mean_fn = fn.mean(axis=1)
std_fn = fn.std(axis=1)
tmp = np.zeros(M)
tmp[1:] = cumtrapz(mqn * np.abs(mqn), time)
fmean = np.mean(f0[1, :]) + tmp
fgam = np.zeros((M, N))
for k in range(0, N):
time0 = (time[-1] - time[0]) * gamf[:, k] + time[0]
fgam[:, k] = np.interp(time0, time, fmean)
var_fgam = fgam.var(axis=1)
orig_var = trapz(std_f0**2, time)
amp_var = trapz(std_fn**2, time)
phase_var = trapz(var_fgam, time)
K = np.cov(fn)
U, s, V = svd(K)
align_fPCAresults = collections.namedtuple('align_fPCA', [
'fn', 'qn', 'q0', 'mqn', 'gam', 'q_pca', 'f_pca', 'latent', 'coef',
'U', 'orig_var', 'amp_var', 'phase_var', 'cost'
])
out = align_fPCAresults(fn, qn, q0, mqn, gamf, q_pca, f_pca, s, c, U,
orig_var, amp_var, phase_var, cost)
return out
def multiple_align_functions(self,
mu,
omethod="DP",
smoothdata=False,
parallel=False,
lam=0.0,
cores=-1):
"""
This function aligns a collection of functions using the elastic square-root
slope (srsf) framework.
Usage: obj.multiple_align_functions(mu)
obj.multiple_align_functions(lambda)
obj.multiple_align_functions(lambda, ...)
:param mu: vector of function to align to
:param omethod: optimization method (DP, DP2) (default = DP)
:param smoothdata: Smooth the data using a box filter (default = F)
:param parallel: run in parallel (default = F)
:param lam: controls the elasticity (default = 0)
:param cores: number of cores for parallel (default = -1 (all))
:type lam: double
:type smoothdata: bool
"""
M = self.f.shape[0]
N = self.f.shape[1]
self.lam = lam
if M > 500:
parallel = True
elif N > 100:
parallel = True
eps = np.finfo(np.double).eps
self.method = omethod
self.type = "multiple"
# Compute SRSF function from data
f, g, g2 = uf.gradient_spline(self.time, self.f, smoothdata)
q = g / np.sqrt(abs(g) + eps)
mq = uf.f_to_srsf(mu, self.time)
if parallel:
out = Parallel(n_jobs=cores)(delayed(uf.optimum_reparam)(
mq, self.time, q[:, n], omethod, lam, mu[0], f[0, n])
for n in range(N))
gam = np.array(out)
gam = gam.transpose()
else:
gam = np.zeros((M, N))
for k in range(0, N):
gam[:, k] = uf.optimum_reparam(mq, self.time, q[:, k], omethod,
lam, mu[0], f[0, k])
self.gamI = uf.SqrtMeanInverse(gam)
fn = np.zeros((M, N))
qn = np.zeros((M, N))
for k in range(0, N):
fn[:, k] = np.interp(
(self.time[-1] - self.time[0]) * gam[:, k] + self.time[0],
self.time, f[:, k])
qn[:, k] = uf.f_to_srsf(f[:, k], self.time)
# Aligned data & stats
self.fn = fn
self.qn = qn
self.q0 = q
mean_f0 = f.mean(axis=1)
std_f0 = f.std(axis=1)
mean_fn = self.fn.mean(axis=1)
std_fn = self.fn.std(axis=1)
self.gam = gam
self.mqn = mq
self.fmean = mu
fgam = np.zeros((M, N))
for k in range(0, N):
time0 = (self.time[-1] - self.time[0]) * gam[:, k] + self.time[0]
fgam[:, k] = np.interp(time0, self.time, self.fmean)
var_fgam = fgam.var(axis=1)
self.orig_var = trapz(std_f0**2, self.time)
self.amp_var = trapz(std_fn**2, self.time)
self.phase_var = trapz(var_fgam, self.time)
return
def srsf_align_pair(f, g, time, method="mean", showplot=True,
smoothdata=False, lam=0.0):
"""
This function aligns a collection of functions using the elastic square-
root slope (srsf) framework.
:param f: numpy ndarray of shape (M,N) of N functions with M samples
:param g: numpy ndarray of shape (M,N) of N functions with M samples
:param time: vector of size M describing the sample points
:param method: (string) warp calculate Karcher Mean or Median (options =
"mean" or "median") (default="mean")
:param showplot: Shows plots of results using matplotlib (default = T)
:param smoothdata: Smooth the data using a box filter (default = F)
:param lam: controls the elasticity (default = 0)
:type lam: double
:type smoothdata: bool
:type f: np.ndarray
:type time: np.ndarray
:rtype: tuple of numpy array
:return fn: aligned functions - numpy ndarray of shape (M,N) of N
functions with M samples
:return gn: aligned functions - numpy ndarray of shape (M,N) of N
functions with M samples
:return qfn: aligned srvfs - similar structure to fn
:return qgn: aligned srvfs - similar structure to fn
:return qf0: original srvf - similar structure to fn
:return qg0: original srvf - similar structure to fn
:return fmean: f function mean or median - vector of length N
:return gmean: g function mean or median - vector of length N
:return mqfn: srvf mean or median - vector of length N
:return mqgn: srvf mean or median - vector of length N
:return gam: warping functions - similar structure to fn
"""
M = f.shape[0]
N = f.shape[1]
if M > 500:
parallel = True
elif N > 100:
parallel = True
else:
parallel = False
eps = np.finfo(np.double).eps
f0 = f
g0 = g
methods = ["mean", "median"]
# 0 mean, 1-median
method = [i for i, x in enumerate(methods) if x == method]
if method != 0 or method != 1:
method = 0
if showplot:
plot.f_plot(time, f, title="Original Data")
plot.f_plot(time, g, title="g Original Data")
# Compute SRSF function from data
f, g1, g2 = uf.gradient_spline(time, f, smoothdata)
qf = g1 / np.sqrt(abs(g1) + eps)
g, g1, g2 = uf.gradient_spline(time, g, smoothdata)
qg = g1 / np.sqrt(abs(g1) + eps)
print ("Initializing...")
mnq = qf.mean(axis=1)
a = mnq.repeat(N)
d1 = a.reshape(M, N)
d = (qf - d1) ** 2
dqq = np.sqrt(d.sum(axis=0))
min_ind = dqq.argmin()
mq = np.column_stack((qf[:, min_ind], qg[:, min_ind]))
mf = np.column_stack((f[:, min_ind], g[:, min_ind]))
if parallel:
out = Parallel(n_jobs=-1)(delayed(uf.optimum_reparam_pair)(mq, time, qf[:, n], qg[:, n], lam) for n in range(N))
gam = np.array(out)
gam = gam.transpose()
else:
gam = uf.optimum_reparam_pair(mq, time, qf, qg, lam)
gamI = uf.SqrtMeanInverse(gam)
time0 = (time[-1] - time[0]) * gamI + time[0]
for k in range(0, 2):
mf[:, k] = np.interp(time0, time, mf[:, k])
mq[:, k] = uf.f_to_srsf(mf[:, k], time)
# Compute Karcher Mean
if method == 0:
print("Compute Karcher Mean of %d function in SRSF space..." % N)
if method == 1:
print("Compute Karcher Median of %d function in SRSF space..." % N)
MaxItr = 20
ds = np.repeat(0.0, MaxItr + 2)
ds[0] = np.inf
qfun = np.repeat(0.0, MaxItr + 1)
qgun = np.repeat(0.0, MaxItr + 1)
tmp = np.zeros((M, 2, MaxItr + 2))
tmp[:, :, 0] = mq
mq = tmp
tmp = np.zeros((M, N, MaxItr + 2))
tmp[:, :, 0] = f
f = tmp
tmp = np.zeros((M, N, MaxItr + 2))
tmp[:, :, 0] = g
g = tmp
tmp = np.zeros((M, N, MaxItr + 2))
tmp[:, :, 0] = qf
qf = tmp
tmp = np.zeros((M, N, MaxItr + 2))
tmp[:, :, 0] = qg
qg = tmp
for r in range(0, MaxItr):
print("updating step: r=%d" % (r + 1))
if r == (MaxItr - 1):
print("maximal number of iterations is reached")
# Matching Step
if parallel:
out = Parallel(n_jobs=-1)(
delayed(uf.optimum_reparam_pair)(mq[:, :, r], time, qf[:, n, 0], qg[:, n, 0], lam) for n in range(N))
gam = np.array(out)
gam = gam.transpose()
else:
gam = uf.optimum_reparam_pair(mq[:, :, r], time, qf[:, :, 0],
qg[:, :, 0], lam)
gam_dev = np.zeros((M, N))
for k in range(0, N):
time0 = (time[-1] - time[0]) * gam[:, k] + time[0]
f[:, k, r + 1] = np.interp(time0, time, f[:, k, 0])
g[:, k, r + 1] = np.interp(time0, time, g[:, k, 0])
qf[:, k, r + 1] = uf.f_to_srsf(f[:, k, r + 1], time)
qg[:, k, r + 1] = uf.f_to_srsf(g[:, k, r + 1], time)
gam_dev[:, k] = np.gradient(gam[:, k], 1 / float(M - 1))
mqt = mq[:, 0, r]
a = mqt.repeat(N)
d1 = a.reshape(M, N)
df = (qf[:, :, r + 1] - d1) ** 2
mqt = mq[:, 1, r]
a = mqt.repeat(N)
d1 = a.reshape(M, N)
dg = (qg[:, :, r + 1] - d1) ** 2
if method == 0:
d1 = sum(trapz(df, time, axis=0))
d2 = sum(trapz((1 - np.sqrt(gam_dev)) ** 2, time, axis=0))
ds_tmp = d1 + lam * d2
d1 = sum(trapz(dg, time, axis=0))
d2 = sum(trapz((1 - np.sqrt(gam_dev)) ** 2, time, axis=0))
ds_tmp1 = d1 + lam * d2
ds[r + 1] = (ds_tmp + ds_tmp1) / 2
# Minimization Step
# compute the mean of the matched function
qtemp = qf[:, :, r + 1]
mq[:, 0, r + 1] = qtemp.mean(axis=1)
qtemp = qg[:, :, r + 1]
mq[:, 1, r + 1] = qtemp.mean(axis=1)
qfun[r] = norm(mq[:, 0, r + 1] - mq[:, 0, r]) / norm(mq[:, 0, r])
qgun[r] = norm(mq[:, 1, r + 1] - mq[:, 1, r]) / norm(mq[:, 1, r])
if method == 1:
d1 = sum(trapz(df, time, axis=0))
d2 = sum(trapz((1 - np.sqrt(gam_dev)) ** 2, time, axis=0))
ds_tmp = np.sqrt(d1) + lam * d2
ds_tmp1 = np.sqrt(sum(trapz(dg, time, axis=0))) + lam * sum(
trapz((1 - np.sqrt(gam_dev)) ** 2, time, axis=0))
ds[r + 1] = (ds_tmp + ds_tmp1) / 2
# Minimization Step
# compute the mean of the matched function
dist_iinv = ds[r + 1] ** (-1)
qtemp = qf[:, :, r + 1] / ds[r + 1]
mq[:, 0, r + 1] = qtemp.sum(axis=1) * dist_iinv
qtemp = qg[:, :, r + 1] / ds[r + 1]
mq[:, 1, r + 1] = qtemp.sum(axis=1) * dist_iinv
qfun[r] = norm(mq[:, 0, r + 1] - mq[:, 0, r]) / norm(mq[:, 0, r])
qgun[r] = norm(mq[:, 1, r + 1] - mq[:, 1, r]) / norm(mq[:, 1, r])
if (qfun[r] < 1e-2 and qgun[r] < 1e-2) or r >= MaxItr:
break
# Last Step with centering of gam
r += 1
if parallel:
out = Parallel(n_jobs=-1)(
delayed(uf.optimum_reparam_pair)(mq[:, :, r], time, qf[:, n, 0],
qg[:, n, 0], lam) for n in range(N))
gam = np.array(out)
gam = gam.transpose()
else:
gam = uf.optimum_reparam_pair(mq[:, :, r], time, qf[:, :, 0],
qg[:, :, 0], lam)
gam_dev = np.zeros((M, N))
for k in range(0, N):
gam_dev[:, k] = np.gradient(gam[:, k], 1 / float(M - 1))
gamI = uf.SqrtMeanInverse(gam)
gamI_dev = np.gradient(gamI, 1 / float(M - 1))
time0 = (time[-1] - time[0]) * gamI + time[0]
for k in range(0, 2):
mq[:, k, r + 1] = np.interp(time0, time,
mq[:, k, r]) * np.sqrt(gamI_dev)
for k in range(0, N):
qf[:, k, r + 1] = np.interp(time0, time,
qf[:, k, r]) * np.sqrt(gamI_dev)
f[:, k, r + 1] = np.interp(time0, time, f[:, k, r])
qg[:, k, r + 1] = np.interp(time0, time,
qg[:, k, r]) * np.sqrt(gamI_dev)
g[:, k, r + 1] = np.interp(time0, time, g[:, k, r])
gam[:, k] = np.interp(time0, time, gam[:, k])
# Aligned data & stats
fn = f[:, :, r + 1]
gn = g[:, :, r + 1]
qfn = qf[:, :, r + 1]
qf0 = qf[:, :, 0]
qgn = qg[:, :, r + 1]
qg0 = qg[:, :, 0]
mean_f0 = f0.mean(axis=1)
std_f0 = f0.std(axis=1)
mean_fn = fn.mean(axis=1)
std_fn = fn.std(axis=1)
mean_g0 = g0.mean(axis=1)
std_g0 = g0.std(axis=1)
mean_gn = gn.mean(axis=1)
std_gn = gn.std(axis=1)
mqfn = mq[:, 0, r + 1]
mqgn = mq[:, 1, r + 1]
tmp = np.zeros(M)
tmp[1:] = cumtrapz(mqfn * np.abs(mqfn), time)
fmean = np.mean(f0[1, :]) + tmp
tmp = np.zeros(M)
tmp[1:] = cumtrapz(mqgn * np.abs(mqgn), time)
gmean = np.mean(g0[1, :]) + tmp
if showplot:
fig, ax = plot.f_plot(np.arange(0, M) / float(M - 1), gam,
title="Warping Functions")
ax.set_aspect('equal')
plot.f_plot(time, fn, title="fn Warped Data")
plot.f_plot(time, gn, title="gn Warped Data")
tmp = np.array([mean_f0, mean_f0 + std_f0, mean_f0 - std_f0])
tmp = tmp.transpose()
plot.f_plot(time, tmp, title="f Original Data: Mean $\pm$ STD")
tmp = np.array([mean_fn, mean_fn + std_fn, mean_fn - std_fn])
tmp = tmp.transpose()
plot.f_plot(time, tmp, title="fn Warped Data: Mean $\pm$ STD")
tmp = np.array([mean_g0, mean_g0 + std_g0, mean_g0 - std_g0])
tmp = tmp.transpose()
plot.f_plot(time, tmp, title="g Original Data: Mean $\pm$ STD")
tmp = np.array([mean_gn, mean_gn + std_gn, mean_gn - std_gn])
tmp = tmp.transpose()
plot.f_plot(time, tmp, title="gn Warped Data: Mean $\pm$ STD")
plot.f_plot(time, fmean, title="$f_{mean}$")
plot.f_plot(time, gmean, title="$g_{mean}$")
plt.show()
align_results = collections.namedtuple('align', ['fn', 'gn', 'qfn', 'qf0',
'qgn', 'qg0', 'fmean',
'gmean', 'mqfn', 'mqgn',
'gam'])
out = align_results(fn, gn, qfn, qf0, qgn, qg0, fmean, gmean, mqfn,
mqgn, gam)
return out
def srsf_align(f, time, method="mean", showplot=True, smoothdata=False,
lam=0.0):
"""
This function aligns a collection of functions using the elastic
square-root slope (srsf) framework.
:param f: numpy ndarray of shape (M,N) of N functions with M samples
:param time: vector of size M describing the sample points
:param method: (string) warp calculate Karcher Mean or Median
(options = "mean" or "median") (default="mean")
:param showplot: Shows plots of results using matplotlib (default = T)
:param smoothdata: Smooth the data using a box filter (default = F)
:param lam: controls the elasticity (default = 0)
:type lam: double
:type smoothdata: bool
:type f: np.ndarray
:type time: np.ndarray
:rtype: tuple of numpy array
:return fn: aligned functions - numpy ndarray of shape (M,N) of N
functions with M samples
:return qn: aligned srvfs - similar structure to fn
:return q0: original srvf - similar structure to fn
:return fmean: function mean or median - vector of length M
:return mqn: srvf mean or median - vector of length M
:return gam: warping functions - similar structure to fn
:return orig_var: Original Variance of Functions
:return amp_var: Amplitude Variance
:return phase_var: Phase Variance
Examples
>>> import tables
>>> fun=tables.open_file("../Data/simu_data.h5")
>>> f = fun.root.f[:]
>>> f = f.transpose()
>>> time = fun.root.time[:]
>>> out = srsf_align(f,time)
"""
M = f.shape[0]
N = f.shape[1]
if M > 500:
parallel = True
elif N > 100:
parallel = True
else:
parallel = False
eps = np.finfo(np.double).eps
f0 = f
methods = ["mean", "median"]
# 0 mean, 1-median
method = [i for i, x in enumerate(methods) if x == method]
if len(method) == 0:
method = 0
else:
method = method[0]
if showplot:
plot.f_plot(time, f, title="f Original Data")
# Compute SRSF function from data
f, g, g2 = uf.gradient_spline(time, f, smoothdata)
q = g / np.sqrt(abs(g) + eps)
print("Initializing...")
mnq = q.mean(axis=1)
a = mnq.repeat(N)
d1 = a.reshape(M, N)
d = (q - d1) ** 2
dqq = np.sqrt(d.sum(axis=0))
min_ind = dqq.argmin()
mq = q[:, min_ind]
mf = f[:, min_ind]
if parallel:
out = Parallel(n_jobs=-1)(delayed(uf.optimum_reparam)(mq, time,
q[:, n], lam) for n in range(N))
gam = np.array(out)
gam = gam.transpose()
else:
gam = uf.optimum_reparam(mq, time, q, lam)
gamI = uf.SqrtMeanInverse(gam)
mf = np.interp((time[-1] - time[0]) * gamI + time[0], time, mf)
mq = uf.f_to_srsf(mf, time)
# Compute Karcher Mean
if method == 0:
print("Compute Karcher Mean of %d function in SRSF space..." % N)
if method == 1:
print("Compute Karcher Median of %d function in SRSF space..." % N)
MaxItr = 20
ds = np.repeat(0.0, MaxItr + 2)
ds[0] = np.inf
qun = np.repeat(0.0, MaxItr + 1)
tmp = np.zeros((M, MaxItr + 2))
tmp[:, 0] = mq
mq = tmp
tmp = np.zeros((M, N, MaxItr + 2))
tmp[:, :, 0] = f
f = tmp
tmp = np.zeros((M, N, MaxItr + 2))
tmp[:, :, 0] = q
q = tmp
for r in range(0, MaxItr):
print("updating step: r=%d" % (r + 1))
if r == (MaxItr - 1):
print("maximal number of iterations is reached")
# Matching Step
if parallel:
out = Parallel(n_jobs=-1)(delayed(uf.optimum_reparam)(mq[:, r],
time, q[:, n, 0], lam) for n in range(N))
gam = np.array(out)
gam = gam.transpose()
else:
gam = uf.optimum_reparam(mq[:, r], time, q[:, :, 0], lam)
gam_dev = np.zeros((M, N))
for k in range(0, N):
f[:, k, r + 1] = np.interp((time[-1] - time[0]) * gam[:, k]
+ time[0], time, f[:, k, 0])
q[:, k, r + 1] = uf.f_to_srsf(f[:, k, r + 1], time)
gam_dev[:, k] = np.gradient(gam[:, k], 1 / float(M - 1))
mqt = mq[:, r]
a = mqt.repeat(N)
d1 = a.reshape(M, N)
d = (q[:, :, r + 1] - d1) ** 2
if method == 0:
d1 = sum(trapz(d, time, axis=0))
d2 = sum(trapz((1 - np.sqrt(gam_dev)) ** 2, time, axis=0))
ds_tmp = d1 + lam * d2
ds[r + 1] = ds_tmp
# Minimization Step
# compute the mean of the matched function
qtemp = q[:, :, r + 1]
mq[:, r + 1] = qtemp.mean(axis=1)
qun[r] = norm(mq[:, r + 1] - mq[:, r]) / norm(mq[:, r])
if method == 1:
d1 = np.sqrt(sum(trapz(d, time, axis=0)))
d2 = sum(trapz((1 - np.sqrt(gam_dev)) ** 2, time, axis=0))
ds_tmp = d1 + lam * d2
ds[r + 1] = ds_tmp
# Minimization Step
# compute the mean of the matched function
dist_iinv = ds[r + 1] ** (-1)
qtemp = q[:, :, r + 1] / ds[r + 1]
mq[:, r + 1] = qtemp.sum(axis=1) * dist_iinv
qun[r] = norm(mq[:, r + 1] - mq[:, r]) / norm(mq[:, r])
if qun[r] < 1e-2 or r >= MaxItr:
break
# Last Step with centering of gam
r += 1
if parallel:
out = Parallel(n_jobs=-1)(delayed(uf.optimum_reparam)(mq[:, r], time, q[:, n, 0], lam) for n in range(N))
gam = np.array(out)
gam = gam.transpose()
else:
gam = uf.optimum_reparam(mq[:, r], time, q[:, :, 0], lam)
gam_dev = np.zeros((M, N))
for k in range(0, N):
gam_dev[:, k] = np.gradient(gam[:, k], 1 / float(M - 1))
gamI = uf.SqrtMeanInverse(gam)
gamI_dev = np.gradient(gamI, 1 / float(M - 1))
time0 = (time[-1] - time[0]) * gamI + time[0]
mq[:, r + 1] = np.interp(time0, time, mq[:, r]) * np.sqrt(gamI_dev)
for k in range(0, N):
q[:, k, r + 1] = np.interp(time0, time, q[:, k, r]) * np.sqrt(gamI_dev)
f[:, k, r + 1] = np.interp(time0, time, f[:, k, r])
gam[:, k] = np.interp(time0, time, gam[:, k])
# Aligned data & stats
fn = f[:, :, r + 1]
qn = q[:, :, r + 1]
q0 = q[:, :, 0]
mean_f0 = f0.mean(axis=1)
std_f0 = f0.std(axis=1)
mean_fn = fn.mean(axis=1)
std_fn = fn.std(axis=1)
mqn = mq[:, r + 1]
tmp = np.zeros((1, M))
tmp = tmp.flatten()
tmp[1:] = cumtrapz(mqn * np.abs(mqn), time)
fmean = np.mean(f0[1, :]) + tmp
fgam = np.zeros((M, N))
for k in range(0, N):
time0 = (time[-1] - time[0]) * gam[:, k] + time[0]
fgam[:, k] = np.interp(time0, time, fmean)
var_fgam = fgam.var(axis=1)
orig_var = trapz(std_f0 ** 2, time)
amp_var = trapz(std_fn ** 2, time)
phase_var = trapz(var_fgam, time)
if showplot:
fig, ax = plot.f_plot(np.arange(0, M) / float(M - 1), gam,
title="Warping Functions")
ax.set_aspect('equal')
plot.f_plot(time, fn, title="Warped Data")
tmp = np.array([mean_f0, mean_f0 + std_f0, mean_f0 - std_f0])
tmp = tmp.transpose()
plot.f_plot(time, tmp, title="Original Data: Mean $\pm$ STD")
tmp = np.array([mean_fn, mean_fn + std_fn, mean_fn - std_fn])
tmp = tmp.transpose()
plot.f_plot(time, tmp, title="Warped Data: Mean $\pm$ STD")
plot.f_plot(time, fmean, title="$f_{mean}$")
plt.show()
align_results = collections.namedtuple('align', ['fn', 'qn', 'q0', 'fmean',
'mqn', 'gam', 'orig_var',
'amp_var', 'phase_var'])
out = align_results(fn, qn, q0, fmean, mqn, gam, orig_var, amp_var,
phase_var)
return out
def elastic_regression(f,
y,
time,
B=None,
lam=0,
df=20,
max_itr=20,
cores=-1,
smooth=False):
"""
This function identifies a regression model with phase-variablity
using elastic methods
:param f: numpy ndarray of shape (M,N) of N functions with M samples
:param y: numpy array of N responses
:param time: vector of size M describing the sample points
:param B: optional matrix describing Basis elements
:param lam: regularization parameter (default 0)
:param df: number of degrees of freedom B-spline (default 20)
:param max_itr: maximum number of iterations (default 20)
:param cores: number of cores for parallel processing (default all)
:type f: np.ndarray
:type time: np.ndarray
:rtype: tuple of numpy array
:return alpha: alpha parameter of model
:return beta: beta(t) of model
:return fn: aligned functions - numpy ndarray of shape (M,N) of M
functions with N samples
:return qn: aligned srvfs - similar structure to fn
:return gamma: calculated warping functions
:return q: original training SRSFs
:return B: basis matrix
:return b: basis coefficients
:return SSE: sum of squared error
"""
M = f.shape[0]
N = f.shape[1]
if M > 500:
parallel = True
elif N > 100:
parallel = True
else:
parallel = False
binsize = np.diff(time)
binsize = binsize.mean()
# Create B-Spline Basis if none provided
if B is None:
B = bs(time, df=df, degree=4, include_intercept=True)
Nb = B.shape[1]
# second derivative for regularization
Bdiff = np.zeros((M, Nb))
for ii in range(0, Nb):
Bdiff[:, ii] = np.gradient(np.gradient(B[:, ii], binsize), binsize)
q = uf.f_to_srsf(f, time, smooth)
gamma = np.tile(np.linspace(0, 1, M), (N, 1))
gamma = gamma.transpose()
itr = 1
SSE = np.zeros(max_itr)
while itr <= max_itr:
print("Iteration: %d" % itr)
# align data
fn = np.zeros((M, N))
qn = np.zeros((M, N))
for ii in range(0, N):
fn[:, ii] = np.interp(
(time[-1] - time[0]) * gamma[:, ii] + time[0], time, f[:, ii])
qn[:, ii] = uf.warp_q_gamma(time, q[:, ii], gamma[:, ii])
# OLS using basis
Phi = np.ones((N, Nb + 1))
for ii in range(0, N):
for jj in range(1, Nb + 1):
Phi[ii, jj] = trapz(qn[:, ii] * B[:, jj - 1], time)
R = np.zeros((Nb + 1, Nb + 1))
for ii in range(1, Nb + 1):
for jj in range(1, Nb + 1):
R[ii, jj] = trapz(Bdiff[:, ii - 1] * Bdiff[:, jj - 1], time)
xx = dot(Phi.T, Phi)
inv_xx = inv(xx + lam * R)
xy = dot(Phi.T, y)
b = dot(inv_xx, xy)
alpha = b[0]
beta = B.dot(b[1:Nb + 1])
beta = beta.reshape(M)
# compute the SSE
int_X = np.zeros(N)
for ii in range(0, N):
int_X[ii] = trapz(qn[:, ii] * beta, time)
SSE[itr - 1] = sum((y.reshape(N) - alpha - int_X)**2)
# find gamma
gamma_new = np.zeros((M, N))
if parallel:
out = Parallel(n_jobs=cores)(
delayed(regression_warp)(beta, time, q[:, n], y[n], alpha)
for n in range(N))
gamma_new = np.array(out)
gamma_new = gamma_new.transpose()
else:
for ii in range(0, N):
gamma_new[:, ii] = regression_warp(beta, time, q[:, ii], y[ii],
alpha)
if norm(gamma - gamma_new) < 1e-5:
break
else:
gamma = gamma_new
itr += 1
# Last Step with centering of gam
gamI = uf.SqrtMeanInverse(gamma_new)
gamI_dev = np.gradient(gamI, 1 / float(M - 1))
beta = np.interp(
(time[-1] - time[0]) * gamI + time[0], time, beta) * np.sqrt(gamI_dev)
for ii in range(0, N):
qn[:, ii] = np.interp((time[-1] - time[0]) * gamI + time[0], time,
qn[:, ii]) * np.sqrt(gamI_dev)
fn[:, ii] = np.interp((time[-1] - time[0]) * gamI + time[0], time,
fn[:, ii])
gamma[:, ii] = np.interp((time[-1] - time[0]) * gamI + time[0], time,
gamma_new[:, ii])
model = collections.namedtuple(
'model',
['alpha', 'beta', 'fn', 'qn', 'gamma', 'q', 'B', 'b', 'SSE', 'type'])
out = model(alpha, beta, fn, qn, gamma, q, B, b[1:-1], SSE[0:itr],
'linear')
return out
def calc_model(self,
B=None,
lam=0,
df=20,
max_itr=20,
cores=-1,
smooth=False):
"""
This function identifies a regression model with phase-variability
using elastic pca
:param B: optional matrix describing Basis elements
:param lam: regularization parameter (default 0)
:param df: number of degrees of freedom B-spline (default 20)
:param max_itr: maximum number of iterations (default 20)
:param cores: number of cores for parallel processing (default all)
"""
M = self.f.shape[0]
N = self.f.shape[1]
if M > 500:
parallel = True
elif N > 100:
parallel = True
else:
parallel = False
binsize = np.diff(self.time)
binsize = binsize.mean()
# Create B-Spline Basis if none provided
if B is None:
B = bs(self.time, df=df, degree=4, include_intercept=True)
Nb = B.shape[1]
self.B = B
# second derivative for regularization
Bdiff = np.zeros((M, Nb))
for ii in range(0, Nb):
Bdiff[:, ii] = np.gradient(np.gradient(B[:, ii], binsize), binsize)
self.Bdiff = Bdiff
self.q = uf.f_to_srsf(self.f, self.time, smooth)
gamma = np.tile(np.linspace(0, 1, M), (N, 1))
gamma = gamma.transpose()
itr = 1
self.SSE = np.zeros(max_itr)
while itr <= max_itr:
print("Iteration: %d" % itr)
# align data
fn = np.zeros((M, N))
qn = np.zeros((M, N))
for ii in range(0, N):
fn[:, ii] = np.interp(
(self.time[-1] - self.time[0]) * gamma[:, ii] +
self.time[0], self.time, self.f[:, ii])
qn[:, ii] = uf.warp_q_gamma(self.time, self.q[:, ii],
gamma[:, ii])
# OLS using basis
Phi = np.ones((N, Nb + 1))
for ii in range(0, N):
for jj in range(1, Nb + 1):
Phi[ii, jj] = trapz(qn[:, ii] * B[:, jj - 1], self.time)
R = np.zeros((Nb + 1, Nb + 1))
for ii in range(1, Nb + 1):
for jj in range(1, Nb + 1):
R[ii, jj] = trapz(Bdiff[:, ii - 1] * Bdiff[:, jj - 1],
self.time)
xx = np.dot(Phi.T, Phi)
inv_xx = inv(xx + lam * R)
xy = np.dot(Phi.T, self.y)
b = np.dot(inv_xx, xy)
alpha = b[0]
beta = B.dot(b[1:Nb + 1])
beta = beta.reshape(M)
# compute the SSE
int_X = np.zeros(N)
for ii in range(0, N):
int_X[ii] = trapz(qn[:, ii] * beta, self.time)
self.SSE[itr - 1] = sum((self.y.reshape(N) - alpha - int_X)**2)
# find gamma
gamma_new = np.zeros((M, N))
if parallel:
out = Parallel(n_jobs=cores)(delayed(regression_warp)(
beta, self.time, self.q[:, n], self.y[n], alpha)
for n in range(N))
gamma_new = np.array(out)
gamma_new = gamma_new.transpose()
else:
for ii in range(0, N):
gamma_new[:, ii] = regression_warp(beta, self.time,
self.q[:, ii],
self.y[ii], alpha)
if norm(gamma - gamma_new) < 1e-5:
break
else:
gamma = gamma_new
itr += 1
# Last Step with centering of gam
gamI = uf.SqrtMeanInverse(gamma_new)
gamI_dev = np.gradient(gamI, 1 / float(M - 1))
beta = np.interp((self.time[-1] - self.time[0]) * gamI + self.time[0],
self.time, beta) * np.sqrt(gamI_dev)
for ii in range(0, N):
qn[:, ii] = np.interp(
(self.time[-1] - self.time[0]) * gamI + self.time[0],
self.time, qn[:, ii]) * np.sqrt(gamI_dev)
fn[:, ii] = np.interp(
(self.time[-1] - self.time[0]) * gamI + self.time[0],
self.time, fn[:, ii])
gamma[:, ii] = np.interp(
(self.time[-1] - self.time[0]) * gamI + self.time[0],
self.time, gamma_new[:, ii])
self.qn = qn
self.fn = fn
self.gamma = gamma
self.alpha = alpha
self.beta = beta
self.b = b[1:-1]
self.SSE = self.SSE[0:itr]
return