def plot(self):
"""
plot plot elastic vertical fPCA result
Usage: obj.plot()
"""
no = self.no
M = self.time.shape[0]
Nstd = self.stds.shape[0]
num_plot = int(np.ceil(no / 3))
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())
k = 0
for ii in range(0, num_plot):
if k > (no - 1):
break
fig, ax = plt.subplots(2, 3)
for k1 in range(0, 3):
k = k1 + (ii) * 3
axt = ax[0, k1]
if k > (no - 1):
break
for l in range(0, Nstd):
axt.plot(self.time,
self.q_pca[0:M, l, k],
color=CBcdict[cl[l]])
axt.set_title('q domain: PD %d' % (k + 1))
axt = ax[1, k1]
for l in range(0, Nstd):
axt.plot(self.time,
self.f_pca[:, l, k],
color=CBcdict[cl[l]])
axt.set_title('f domain: PD %d' % (k + 1))
fig.set_tight_layout(True)
cumm_coef = 100 * np.cumsum(self.latent) / sum(self.latent)
idx = np.arange(0, self.latent.shape[0]) + 1
plot.f_plot(idx, cumm_coef, "Coefficient Cumulative Percentage")
plt.ylabel("Percentage")
plt.xlabel("Index")
plt.show()
return
def plot(self):
"""
plot plot elastic horizontal fPCA results
Usage: obj.plot()
"""
no = self.no
TT = self.warp_data.time.shape[0]
num_plot = int(np.ceil(no / 3))
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),
}
k = 0
for ii in range(0, num_plot):
if k > (no - 1):
break
fig, ax = plt.subplots(1, 3)
for k1 in range(0, 3):
k = k1 + (ii) * 3
axt = ax[k1]
if k > (no - 1):
break
tmp = self.gam_pca[:, :, k]
axt.plot(np.linspace(0, 1, TT), tmp.transpose())
axt.set_title('PD %d' % (k + 1))
axt.set_aspect('equal')
fig.set_tight_layout(True)
cumm_coef = 100 * np.cumsum(self.latent[0:no]) / sum(self.latent[0:no])
idx = np.arange(0, no) + 1
plot.f_plot(idx, cumm_coef, "Coefficient Cumulative Percentage")
plt.ylabel("Percentage")
plt.xlabel("Index")
plt.show()
return
def plot(self):
"""
plot curve mean results
"""
fig, ax = plt.subplots()
n, T, K = self.beta.shape
for ii in range(0, K):
ax.plot(self.beta[0, :, ii], self.beta[1, :, ii])
plt.title('Curves')
ax.set_aspect('equal')
plt.axis('off')
plt.gca().invert_yaxis()
if hasattr(self, 'gams'):
M = self.gams.shape[0]
fig, ax = plot.f_plot(arange(0, M) / float(M - 1),
self.gams,
title="Warping Functions")
if hasattr(self, 'beta_mean'):
fig, ax = plt.subplots()
ax.plot(self.beta_mean[0, :], self.beta_mean[1, :])
plt.title('Karcher Mean')
ax.set_aspect('equal')
plt.axis('off')
plt.gca().invert_yaxis()
plt.show()
def align_fPLS(f,
g,
time,
comps=3,
showplot=True,
smoothdata=False,
delta=0.01,
max_itr=100):
"""
This function aligns a collection of functions while performing
principal least squares
: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 comps: number of fPLS components
:param showplot: Shows plots of results using matplotlib (default = T)
:param smooth_data: Smooth the data using a box filter (default = F)
:param delta: gradient step size
:param max_itr: maximum number of iterations
:type smooth_data: bool
:type f: np.ndarray
:type g: 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 gam: warping functions - similar structure to fn
:return wqf: srsf principal weight functions
:return wqg: srsf principal weight functions
:return wf: srsf principal weight functions
:return wg: srsf principal weight functions
:return cost: cost function value
"""
print("Initializing...")
binsize = np.diff(time)
binsize = binsize.mean()
eps = np.finfo(np.double).eps
M = f.shape[0]
N = f.shape[1]
f0 = f
g0 = g
if showplot:
plot.f_plot(time, f, title="f Original Data")
plot.f_plot(time, g, title="g Original Data")
# Compute q-function of f and g
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("Calculating fPLS weight functions for %d Warped Functions..." % N)
itr = 0
fi = np.zeros((M, N, max_itr + 1))
fi[:, :, itr] = f
gi = np.zeros((M, N, max_itr + 1))
gi[:, :, itr] = g
qfi = np.zeros((M, N, max_itr + 1))
qfi[:, :, itr] = qf
qgi = np.zeros((M, N, max_itr + 1))
qgi[:, :, itr] = qg
wqf1, wqg1, alpha, values, costmp = pls_svd(time, qfi[:, :, itr],
qgi[:, :, itr], 2, 0)
wqf = np.zeros((M, max_itr + 1))
wqf[:, itr] = wqf1[:, 0]
wqg = np.zeros((M, max_itr + 1))
wqg[:, itr] = wqg1[:, 0]
gam = np.zeros((M, N, max_itr + 1))
tmp = np.tile(np.linspace(0, 1, M), (N, 1))
gam[:, :, itr] = tmp.transpose()
wqf_diff = np.zeros(max_itr + 1)
cost = np.zeros(max_itr + 1)
cost_diff = 1
while itr <= max_itr:
# warping
gamtmp = np.ascontiguousarray(gam[:, :, 0])
qftmp = np.ascontiguousarray(qfi[:, :, 0])
qgtmp = np.ascontiguousarray(qgi[:, :, 0])
wqftmp = np.ascontiguousarray(wqf[:, itr])
wqgtmp = np.ascontiguousarray(wqg[:, itr])
gam[:, :, itr + 1] = fpls.fpls_warp(time,
gamtmp,
qftmp,
qgtmp,
wqftmp,
wqgtmp,
display=0,
delta=delta,
tol=1e-6,
max_iter=4000)
for k in range(0, N):
gam_k = gam[:, k, itr + 1]
time0 = (time[-1] - time[0]) * gam_k + time[0]
fi[:, k, itr + 1] = np.interp(time0, time, fi[:, k, 0])
gi[:, k, itr + 1] = np.interp(time0, time, gi[:, k, 0])
qfi[:, k, itr + 1] = uf.warp_q_gamma(time, qfi[:, k, 0], gam_k)
qgi[:, k, itr + 1] = uf.warp_q_gamma(time, qgi[:, k, 0], gam_k)
# PLS
wqfi, wqgi, alpha, values, costmp = pls_svd(time, qfi[:, :, itr + 1],
qgi[:, :, itr + 1], 2, 0)
wqf[:, itr + 1] = wqfi[:, 0]
wqg[:, itr + 1] = wqgi[:, 0]
wqf_diff[itr] = np.sqrt(sum(wqf[:, itr + 1] - wqf[:, itr])**2)
rfi = np.zeros(N)
rgi = np.zeros(N)
for l in range(0, N):
rfi[l] = uf.innerprod_q(time, qfi[:, l, itr + 1], wqf[:, itr + 1])
rgi[l] = uf.innerprod_q(time, qgi[:, l, itr + 1], wqg[:, itr + 1])
cost[itr] = np.cov(rfi, rgi)[1, 0]
if itr > 1:
cost_diff = cost[itr] - cost[itr - 1]
print("Iteration: %d - Diff Value: %f - %f" %
(itr + 1, wqf_diff[itr], cost[itr]))
if wqf_diff[itr] < 1e-1 or abs(cost_diff) < 1e-3:
break
itr += 1
cost = cost[0:(itr + 1)]
# Aligned data & stats
fn = fi[:, :, itr + 1]
gn = gi[:, :, itr + 1]
qfn = qfi[:, :, itr + 1]
qf0 = qfi[:, :, 0]
qgn = qgi[:, :, itr + 1]
qg0 = qgi[:, :, 0]
wqfn, wqgn, alpha, values, costmp = pls_svd(time, qfn, qgn, comps, 0)
wf = np.zeros((M, comps))
wg = np.zeros((M, comps))
for ii in range(0, comps):
wf[:, ii] = cumtrapz(wqfn[:, ii] * np.abs(wqfn[:, ii]),
time,
initial=0)
wg[:, ii] = cumtrapz(wqgn[:, ii] * np.abs(wqgn[:, ii]),
time,
initial=0)
gam_f = gam[:, :, itr + 1]
if showplot:
# Align Plots
fig, ax = plot.f_plot(np.arange(0, M) / float(M - 1),
gam_f,
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")
plot.f_plot(time, wf, title="wf")
plot.f_plot(time, wg, title="wg")
plt.show()
align_fPLSresults = collections.namedtuple('align_fPLS', [
'wf', 'wg', 'fn', 'gn', 'qfn', 'qgn', 'qf0', 'qg0', 'wqf', 'wqg',
'gam', 'values', 'cost'
])
out = align_fPLSresults(wf, wg, fn, gn, qfn, qgn, qf0, qg0, wqfn, wqgn,
gam_f, values, cost)
return out
def align_fPLS(f, g, time, comps=3, showplot=True, smoothdata=False,
delta=0.01, max_itr=100):
"""
This function aligns a collection of functions while performing
principal least squares
: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 comps: number of fPLS components
:param showplot: Shows plots of results using matplotlib (default = T)
:param smooth_data: Smooth the data using a box filter (default = F)
:param delta: gradient step size
:param max_itr: maximum number of iterations
:type smooth_data: bool
:type f: np.ndarray
:type g: 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 gam: warping functions - similar structure to fn
:return wqf: srsf principal weight functions
:return wqg: srsf principal weight functions
:return wf: srsf principal weight functions
:return wg: srsf principal weight functions
:return cost: cost function value
"""
print ("Initializing...")
binsize = np.diff(time)
binsize = binsize.mean()
eps = np.finfo(np.double).eps
M = f.shape[0]
N = f.shape[1]
f0 = f
g0 = g
if showplot:
plot.f_plot(time, f, title="f Original Data")
plot.f_plot(time, g, title="g Original Data")
# Compute q-function of f and g
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("Calculating fPLS weight functions for %d Warped Functions..." % N)
itr = 0
fi = np.zeros((M, N, max_itr + 1))
fi[:, :, itr] = f
gi = np.zeros((M, N, max_itr + 1))
gi[:, :, itr] = g
qfi = np.zeros((M, N, max_itr + 1))
qfi[:, :, itr] = qf
qgi = np.zeros((M, N, max_itr + 1))
qgi[:, :, itr] = qg
wqf1, wqg1, alpha, values, costmp = pls_svd(time, qfi[:, :, itr],
qgi[:, :, itr], 2, 0)
wqf = np.zeros((M, max_itr + 1))
wqf[:, itr] = wqf1[:, 0]
wqg = np.zeros((M, max_itr + 1))
wqg[:, itr] = wqg1[:, 0]
gam = np.zeros((M, N, max_itr + 1))
tmp = np.tile(np.linspace(0, 1, M), (N, 1))
gam[:, :, itr] = tmp.transpose()
wqf_diff = np.zeros(max_itr + 1)
cost = np.zeros(max_itr + 1)
cost_diff = 1
while itr <= max_itr:
# warping
gamtmp = np.ascontiguousarray(gam[:, :, 0])
qftmp = np.ascontiguousarray(qfi[:, :, 0])
qgtmp = np.ascontiguousarray(qgi[:, :, 0])
wqftmp = np.ascontiguousarray(wqf[:, itr])
wqgtmp = np.ascontiguousarray(wqg[:, itr])
gam[:, :, itr + 1] = fpls.fpls_warp(time, gamtmp, qftmp, qgtmp,
wqftmp, wqgtmp, display=0,
delta=delta, tol=1e-6,
max_iter=4000)
for k in range(0, N):
gam_k = gam[:, k, itr + 1]
time0 = (time[-1] - time[0]) * gam_k + time[0]
fi[:, k, itr + 1] = np.interp(time0, time, fi[:, k, 0])
gi[:, k, itr + 1] = np.interp(time0, time, gi[:, k, 0])
qfi[:, k, itr + 1] = uf.warp_q_gamma(time, qfi[:, k, 0], gam_k)
qgi[:, k, itr + 1] = uf.warp_q_gamma(time, qgi[:, k, 0], gam_k)
# PLS
wqfi, wqgi, alpha, values, costmp = pls_svd(time, qfi[:, :, itr + 1],
qgi[:, :, itr + 1], 2, 0)
wqf[:, itr + 1] = wqfi[:, 0]
wqg[:, itr + 1] = wqgi[:, 0]
wqf_diff[itr] = np.sqrt(sum(wqf[:, itr + 1] - wqf[:, itr]) ** 2)
rfi = np.zeros(N)
rgi = np.zeros(N)
for l in range(0, N):
rfi[l] = uf.innerprod_q(time, qfi[:, l, itr + 1], wqf[:, itr + 1])
rgi[l] = uf.innerprod_q(time, qgi[:, l, itr + 1], wqg[:, itr + 1])
cost[itr] = np.cov(rfi, rgi)[1, 0]
if itr > 1:
cost_diff = cost[itr] - cost[itr - 1]
print("Iteration: %d - Diff Value: %f - %f" % (itr + 1, wqf_diff[itr],
cost[itr]))
if wqf_diff[itr] < 1e-1 or abs(cost_diff) < 1e-3:
break
itr += 1
cost = cost[0:(itr + 1)]
# Aligned data & stats
fn = fi[:, :, itr + 1]
gn = gi[:, :, itr + 1]
qfn = qfi[:, :, itr + 1]
qf0 = qfi[:, :, 0]
qgn = qgi[:, :, itr + 1]
qg0 = qgi[:, :, 0]
wqfn, wqgn, alpha, values, costmp = pls_svd(time, qfn, qgn, comps, 0)
wf = np.zeros((M, comps))
wg = np.zeros((M, comps))
for ii in range(0, comps):
wf[:, ii] = cumtrapz(wqfn[:, ii] * np.abs(wqfn[:, ii]), time, initial=0)
wg[:, ii] = cumtrapz(wqgn[:, ii] * np.abs(wqgn[:, ii]), time, initial=0)
gam_f = gam[:, :, itr + 1]
if showplot:
# Align Plots
fig, ax = plot.f_plot(np.arange(0, M) / float(M - 1), gam_f,
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")
plot.f_plot(time, wf, title="wf")
plot.f_plot(time, wg, title="wg")
plt.show()
align_fPLSresults = collections.namedtuple('align_fPLS', ['wf', 'wg', 'fn',
'gn', 'qfn', 'qgn', 'qf0',
'qg0', 'wqf', 'wqg', 'gam',
'values', 'cost'])
out = align_fPLSresults(wf, wg, fn, gn, qfn, qgn, qf0, qg0, wqfn,
wqgn, gam_f, values, cost)
return out
def align_fPCA(f, time, num_comp=3, showplot=True, smoothdata=False):
"""
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 sparam: Number of times to run box filter (default = 25)
: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=-1)(
delayed(uf.optimum_reparam)(qhat[:, n], time, qi[:, n, itr],
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], 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 = 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)
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="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")
# 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 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 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 jointfPCA(fn, time, qn, q0, gam, no=2, showplot=True):
"""
This function calculates joint functional principal component analysis
on aligned data
:param fn: numpy ndarray of shape (M,N) of N aligned functions with M
samples
:param time: vector of size N describing the sample points
:param qn: numpy ndarray of shape (M,N) of N aligned SRSF with M samples
:param no: number of components to extract (default = 2)
:param showplot: Shows plots of results using matplotlib (default = T)
:type showplot: bool
:type no: int
:rtype: tuple of numpy ndarray
:return q_pca: srsf principal directions
:return f_pca: functional principal directions
:return latent: latent values
:return coef: coefficients
:return U: eigenvectors
"""
coef = np.arange(-1., 2.)
Nstd = coef.shape[0]
# set up for fPCA in q-space
mq_new = qn.mean(axis=1)
M = time.shape[0]
mididx = int(np.round(M / 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))
# calculate vector space of warping functions
mu_psi, gam_mu, psi, vec = uf.SqrtMean(gam)
# joint fPCA
C = fminbound(find_C,0,1e4,(qn2,vec,q0,no,mu_psi))
qhat, gamhat, a, U, s, mu_g = jointfPCAd(qn2, vec, C, no, mu_psi)
# geodesic paths
q_pca = np.ndarray(shape=(M, Nstd, no), dtype=float)
f_pca = np.ndarray(shape=(M, Nstd, no), dtype=float)
for k in range(0, no):
for l in range(0, Nstd):
qhat = mqn + dot(U[0:(M+1),k],coef[l]*np.sqrt(s[k]))
vechat = dot(U[(M+1):,k],(coef[l]*np.sqrt(s[k]))/C)
psihat = geo.exp_map(mu_psi,vechat)
gamhat = cumtrapz(psihat*psihat,np.linspace(0,1,M),initial=0)
gamhat = (gamhat - gamhat.min()) / (gamhat.max() - gamhat.min())
if (sum(vechat)==0):
gamhat = np.linspace(0,1,M)
fhat = uf.cumtrapzmid(time, qhat[0:M]*np.fabs(qhat[0:M]), np.sign(qhat[M])*(qhat[M]*qhat[M]), mididx)
f_pca[:,l,k] = uf.warp_f_gamma(np.linspace(0,1,M), fhat, gamhat)
q_pca[:,l,k] = uf.warp_q_gamma(np.linspace(0,1,M), qhat[0:M], gamhat)
jfpca_results = collections.namedtuple('jfpca', ['q_pca', 'f_pca', 'latent', 'coef', 'U'])
jfpca = jfpca_results(q_pca, f_pca, s, a, U)
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())
fig, ax = plt.subplots(2, no)
for k in range(0, no):
axt = ax[0, k]
for l in range(0, Nstd):
axt.plot(time, q_pca[0:M, l, k], color=CBcdict[cl[l]])
axt.set_title('q domain: PD %d' % (k + 1))
axt = ax[1, k]
for l in range(0, Nstd):
axt.plot(time, f_pca[:, l, k], color=CBcdict[cl[l]])
axt.set_title('f domain: PD %d' % (k + 1))
fig.set_tight_layout(True)
cumm_coef = 100 * np.cumsum(s) / sum(s)
idx = np.arange(0, s.shape[0]) + 1
plot.f_plot(idx, cumm_coef, "Coefficient Cumulative Percentage")
plt.xlabel("Percentage")
plt.ylabel("Index")
plt.show()
return jfpca
def vertfPCA(fn, time, qn, no=2, showplot=True):
"""
This function calculates vertical functional principal component analysis
on aligned data
:param fn: numpy ndarray of shape (M,N) of N aligned functions with M
samples
:param time: vector of size N describing the sample points
:param qn: numpy ndarray of shape (M,N) of N aligned SRSF with M samples
:param no: number of components to extract (default = 2)
:param showplot: Shows plots of results using matplotlib (default = T)
:type showplot: bool
:type no: int
:rtype: tuple of numpy ndarray
:return q_pca: srsf principal directions
:return f_pca: functional principal directions
:return latent: latent values
:return coef: coefficients
:return U: eigenvectors
"""
coef = np.arange(-2., 3.)
Nstd = coef.shape[0]
# FPCA
mq_new = qn.mean(axis=1)
N = mq_new.shape[0]
mididx = int(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))
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=(N + 1, Nstd, no), dtype=float)
for k in range(0, no):
for l in range(0, Nstd):
q_pca[:, l, k] = mqn + coef[l] * stdS[k] * U[:, k]
# compute the correspondence in the f domain
f_pca = np.ndarray(shape=(N, Nstd, no), dtype=float)
for k in range(0, no):
for l in range(0, Nstd):
f_pca[:, l, k] = uf.cumtrapzmid(time, q_pca[0:N, l, k] * np.abs(q_pca[0:N, l, k]),
np.sign(q_pca[N, l, k]) * (q_pca[N, l, k] ** 2),
mididx)
fbar = fn.mean(axis=1)
fsbar = f_pca[:, :, k].mean(axis=1)
err = np.transpose(np.tile(fbar-fsbar, (Nstd,1)))
f_pca[:, :, k] += err
N2 = qn.shape[1]
c = np.zeros((N2, no))
for k in range(0, no):
for l in range(0, N2):
c[l, k] = sum((np.append(qn[:, l], m_new[l]) - mqn) * U[:, k])
vfpca_results = collections.namedtuple('vfpca', ['q_pca', 'f_pca', 'latent', 'coef', 'U'])
vfpca = vfpca_results(q_pca, f_pca, s, c, U)
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())
fig, ax = plt.subplots(2, no)
for k in range(0, no):
axt = ax[0, k]
for l in range(0, Nstd):
axt.plot(time, q_pca[0:N, l, k], color=CBcdict[cl[l]])
axt.set_title('q domain: PD %d' % (k + 1))
axt = ax[1, k]
for l in range(0, Nstd):
axt.plot(time, f_pca[:, l, k], color=CBcdict[cl[l]])
axt.set_title('f domain: PD %d' % (k + 1))
fig.set_tight_layout(True)
cumm_coef = 100 * np.cumsum(s) / sum(s)
idx = np.arange(0, N + 1) + 1
plot.f_plot(idx, cumm_coef, "Coefficient Cumulative Percentage")
plt.xlabel("Percentage")
plt.ylabel("Index")
plt.show()
return vfpca
def horizfPCA(gam, time, no=2, showplot=True):
"""
This function calculates horizontal functional principal component analysis on aligned data
:param gam: numpy ndarray of shape (M,N) of N warping functions
:param time: vector of size M describing the sample points
:param no: number of components to extract (default = 2)
:param showplot: Shows plots of results using matplotlib (default = T)
:type showplot: bool
:type no: int
:rtype: tuple of numpy ndarray
:return q_pca: srsf principal directions
:return f_pca: functional principal directions
:return latent: latent values
:return coef: coefficients
:return U: eigenvectors
"""
# Calculate Shooting Vectors
mu, gam_mu, psi, vec = uf.SqrtMean(gam)
tau = np.arange(1, 6)
TT = time.shape[0]
# TFPCA
K = np.cov(vec)
U, s, V = svd(K)
vm = vec.mean(axis=1)
gam_pca = np.ndarray(shape=(tau.shape[0], mu.shape[0] + 1, no), dtype=float)
psi_pca = np.ndarray(shape=(tau.shape[0], mu.shape[0], no), dtype=float)
for j in range(0, no):
for k in tau:
v = (k - 3) * np.sqrt(s[j]) * U[:, j]
vn = norm(v) / np.sqrt(TT)
if vn < 0.0001:
psi_pca[k-1, :, j] = mu
else:
psi_pca[k-1, :, j] = np.cos(vn) * mu + np.sin(vn) * v / vn
tmp = np.zeros(TT)
tmp[1:TT] = np.cumsum(psi_pca[k-1, :, j] * psi_pca[k-1, :, j])
gam_pca[k-1, :, j] = (tmp - tmp[0]) / (tmp[-1] - tmp[0])
hfpca_results = collections.namedtuple('hfpca', ['gam_pca', 'psi_pca', 'latent', 'U', 'gam_mu'])
hfpca = hfpca_results(gam_pca, psi_pca, s, U, gam_mu)
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),
}
fig, ax = plt.subplots(1, no)
for k in range(0, no):
axt = ax[k]
tmp = gam_pca[:, :, k]
axt.plot(np.linspace(0, 1, TT), tmp.transpose())
axt.set_title('PD %d' % (k + 1))
axt.set_aspect('equal')
fig.set_tight_layout(True)
cumm_coef = 100 * np.cumsum(s) / sum(s)
idx = np.arange(0, TT-1) + 1
plot.f_plot(idx, cumm_coef, "Coefficient Cumulative Percentage")
plt.xlabel("Percentage")
plt.ylabel("Index")
plt.show()
return hfpca
def vertfPCA(fn, time, qn, no=1, showplot=True):
"""
This function calculates vertical functional principal component analysis
on aligned data
:param fn: numpy ndarray of shape (M,N) of N aligned functions with M
samples
:param time: vector of size N describing the sample points
:param qn: numpy ndarray of shape (M,N) of N aligned SRSF with M samples
:param no: number of components to extract (default = 1)
:param showplot: Shows plots of results using matplotlib (default = T)
:type showplot: bool
:type no: int
:rtype: tuple of numpy ndarray
:return q_pca: srsf principal directions
:return f_pca: functional principal directions
:return latent: latent values
:return coef: coefficients
:return U: eigenvectors
"""
coef = np.arange(-2., 3.)
Nstd = coef.shape[0]
# FPCA
mq_new = qn.mean(axis=1)
N = mq_new.shape[0]
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))
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=(N + 1, Nstd, no), dtype=float)
for k in range(0, no):
for l in range(0, Nstd):
q_pca[:, l, k] = mqn + coef[l] * stdS[k] * U[:, k]
# compute the correspondence in the f domain
f_pca = np.ndarray(shape=(N, Nstd, no), dtype=float)
for k in range(0, no):
for l in range(0, Nstd):
f_pca[:, l, k] = uf.cumtrapzmid(time, q_pca[0:N, l, k] * np.abs(q_pca[0:N, l, k]),
np.sign(q_pca[N, l, k]) * (q_pca[N, l, k] ** 2))
N2 = qn.shape[1]
c = np.zeros((N2, no))
for k in range(0, no):
for l in range(0, N2):
c[l, k] = sum((np.append(qn[:, l], m_new[l]) - mqn) * U[:, k])
vfpca_results = collections.namedtuple('vfpca', ['q_pca', 'f_pca', 'latent', 'coef', 'U'])
vfpca = vfpca_results(q_pca, f_pca, s, c, U)
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())
fig, ax = plt.subplots(2, no)
for k in range(0, no):
axt = ax[0, k]
for l in range(0, Nstd):
axt.plot(time, q_pca[0:N, 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, N + 1) + 1
plot.f_plot(idx, cumm_coef, "Coefficient Cumulative Percentage")
plt.xlabel("Percentage")
plt.ylabel("Index")
plt.show()
return vfpca
def horizfPCA(gam, time, no, showplot=True):
"""
This function calculates horizontal functional principal component analysis on aligned data
:param gam: numpy ndarray of shape (M,N) of N warping functions
:param time: vector of size M describing the sample points
:param no: number of components to extract (default = 1)
:param showplot: Shows plots of results using matplotlib (default = T)
:type showplot: bool
:type no: int
:rtype: tuple of numpy ndarray
:return q_pca: srsf principal directions
:return f_pca: functional principal directions
:return latent: latent values
:return coef: coefficients
:return U: eigenvectors
"""
# Calculate Shooting Vectors
mu, gam_mu, psi, vec = uf.SqrtMean(gam)
tau = np.arange(1, 6)
TT = time.shape[0]
# TFPCA
K = np.cov(vec)
U, s, V = svd(K)
vm = vec.mean(axis=1)
gam_pca = np.ndarray(shape=(tau.shape[0], mu.shape[0] + 1, no), dtype=float)
psi_pca = np.ndarray(shape=(tau.shape[0], mu.shape[0], no), dtype=float)
for j in range(0, no):
for k in tau:
v = (k - 3) * np.sqrt(s[j]) * U[:, j]
vn = norm(v) / np.sqrt(TT)
if vn < 0.0001:
psi_pca[k-1, :, j] = mu
else:
psi_pca[k-1, :, j] = np.cos(vn) * mu + np.sin(vn) * v / vn
tmp = np.zeros(TT)
tmp[1:TT] = np.cumsum(psi_pca[k-1, :, j] * psi_pca[k-1, :, j])
gam_pca[k-1, :, j] = (tmp - tmp[0]) / (tmp[-1] - tmp[0])
hfpca_results = collections.namedtuple('hfpca', ['gam_pca', 'psi_pca', 'latent', 'U', 'gam_mu'])
hfpca = hfpca_results(gam_pca, psi_pca, s, U, gam_mu)
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),
}
fig, ax = plt.subplots(1, no)
for k in range(0, no):
axt = ax[k]
axt.set_color_cycle(CBcdict[c] for c in sorted(CBcdict.keys()))
tmp = gam_pca[:, :, k]
axt.plot(np.linspace(0, 1, TT), tmp.transpose())
axt.set_title('PD %d' % (k + 1))
axt.set_aspect('equal')
plot.rstyle(axt)
fig.set_tight_layout(True)
cumm_coef = 100 * np.cumsum(s) / sum(s)
idx = np.arange(0, TT-1) + 1
plot.f_plot(idx, cumm_coef, "Coefficient Cumulative Percentage")
plt.xlabel("Percentage")
plt.ylabel("Index")
plt.show()
return hfpca
def plot(self):
"""
plot plot functional alignment results
Usage: obj.plot()
"""
M = self.f.shape[0]
plot.f_plot(self.time, self.f, title="f Original Data")
fig, ax = plot.f_plot(np.arange(0, M) / float(M - 1),
self.gam,
title="Warping Functions")
ax.set_aspect('equal')
plot.f_plot(self.time, self.fn, title="Warped Data")
mean_f0 = self.f.mean(axis=1)
std_f0 = self.f.std(axis=1)
mean_fn = self.fn.mean(axis=1)
std_fn = self.fn.std(axis=1)
tmp = np.array([mean_f0, mean_f0 + std_f0, mean_f0 - std_f0])
tmp = tmp.transpose()
plot.f_plot(self.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(self.time, tmp, title=r"Warped Data: Mean $\pm$ STD")
plot.f_plot(self.time, self.fmean, title="$f_{mean}$")
plt.show()
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 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
