def f_updatef2_pw(f2_curr, q2_curr, y2, q1, v_coef_curr, v_basis, SSE_curr,
K_f2, K_f2prop, sigma_curr, sigma2_curr):
time = np.linspace(0, 1, y2.shape[0])
v = uf.f_basistofunction(v_basis["x"], 0, v_coef_curr, v_basis)
f2_prop = multivariate_normal(f2_curr, K_f2prop)
q2_prop = uf.f_to_srsf(f2_prop, time)
SSE_prop = f_SSEv_pw(v, q1, q2_prop)
postlog_curr = f_f2postlogl_pw(f2_curr, y2, SSE_curr, K_f2, sigma_curr,
sigma2_curr)
postlog_prop = f_f2postlogl_pw(f2_prop, y2, SSE_prop, K_f2, sigma_curr,
sigma2_curr)
ratio = np.minimum(1, np.exp(postlog_prop - postlog_curr))
u = rand()
if (u <= ratio):
f2_curr = f2_prop
q2_curr = q2_prop
f2_accept = True
else:
f2_accept = False
return f2_curr, q2_curr, f2_accept
def f_updatef1_pw(f1_curr, q1_curr, y1, q2, v_coef_curr, v_basis, SSE_curr,
K_f1, K_f1prop, sigma_curr, sigma1_curr):
time = np.linspace(0, 1, y1.shape[0])
v = uf.f_basistofunction(v_basis["x"], 0, v_coef_curr, v_basis)
f1_prop = multivariate_normal(f1_curr, K_f1prop)
q1_prop = uf.f_to_srsf(f1_prop, time)
SSE_prop = f_SSEv_pw(v, q1_prop, q2)
postlog_curr = f_f1postlogl_pw(f1_curr, y1, SSE_curr, K_f1, sigma_curr,
sigma1_curr)
postlog_prop = f_f1postlogl_pw(f1_prop, y1, SSE_prop, K_f1, sigma_curr,
sigma1_curr)
ratio = np.minimum(1, np.exp(postlog_prop - postlog_curr))
u = rand()
if (u <= ratio):
f1_curr = f1_prop
q1_curr = q1_prop
f1_accept = True
else:
f1_accept = False
return f1_curr, q1_curr, f1_accept
def map_driver(q1, f, bet, t, dt):
q2 = uf.f_to_srsf(f, t)
gam = uf.optimum_reparam(q1, t, q2)
fn = uf.warp_f_gamma(t, f, gam)
tmp = bet * fn
y = tmp.sum() * dt
return y
def predict(self, newdata=None):
"""
This function performs prediction on regression model on new data if available or current stored data in object
Usage: obj.predict()
obj.predict(newdata)
:param newdata: dict containing new data for prediction (needs the keys below, if None predicts on training data)
:type newdata: dict
:param f: (M,N) matrix of functions
:param time: vector of time points
:param y: truth if available
:param smooth: smooth data if needed
:param sparam: number of times to run filter
"""
if newdata != None:
f = newdata['f']
time = newdata['time']
y = newdata['y']
q = uf.f_to_srsf(f, time, newdata['smooth'])
n = f.shape[1]
yhat = np.zeros(n)
for ii in range(0, n):
diff = self.q - q[:, ii][:, np.newaxis]
dist = np.sum(np.abs(diff)**2, axis=0)**(1. / 2)
q_tmp = uf.warp_q_gamma(time, q[:, ii],
self.gamma[:, dist.argmin()])
yhat[ii] = self.alpha + trapz(q_tmp * self.beta, time)
if y is None:
self.SSE = np.nan
else:
self.SSE = np.sum((y - yhat)**2)
self.y_pred = yhat
else:
n = self.f.shape[1]
yhat = np.zeros(n)
for ii in range(0, n):
diff = self.q - self.q[:, ii][:, np.newaxis]
dist = np.sum(np.abs(diff)**2, axis=0)**(1. / 2)
q_tmp = uf.warp_q_gamma(self.time, self.q[:, ii],
self.gamma[:, dist.argmin()])
yhat[ii] = self.alpha + trapz(q_tmp * self.beta, self.time)
self.SSE = np.sum((self.y - yhat)**2)
self.y_pred = yhat
return
def MapC_to_y(n, c, B, t, f, parallel):
dt = np.diff(t)
dt = dt.mean()
y = np.zeros(n)
if parallel:
bet = np.dot(B, c)
q1 = uf.f_to_srsf(bet, t)
y = Parallel(n_jobs=-1)(delayed(map_driver)(q1, f[:, k], bet, t, dt)
for k in range(n))
else:
for k in range(0, n):
bet = np.dot(B, c)
q1 = uf.f_to_srsf(bet, t)
q2 = uf.f_to_srsf(f[:, k], t)
gam = uf.optimum_reparam(q1, t, q2)
fn = uf.warp_f_gamma(t, f[:, k], gam)
tmp = bet * fn
y[k] = tmp.sum() * dt
return (y)
def predict(self, newdata=None):
"""
This function performs prediction on regression model on new data if available or current stored data in object
Usage: obj.predict()
obj.predict(newdata)
:param newdata: dict containing new data for prediction (needs the keys below, if None predicts on training data)
:type newdata: dict
:param f: (M,N) matrix of functions
:param time: vector of time points
:param y: truth if available
:param smooth: smooth data if needed
:param sparam: number of times to run filter
"""
omethod = self.warp_data.method
lam = self.warp_data.lam
m = self.n_classes
M = self.time.shape[0]
if newdata != None:
f = newdata['f']
time = newdata['time']
y = newdata['y']
sparam = newdata['sparam']
if newdata['smooth']:
f = fs.smooth_data(f, sparam)
q1 = fs.f_to_srsf(f, time)
n = q1.shape[1]
self.y_pred = np.zeros((n, m))
mq = self.warp_data.mqn
fn = np.zeros((M, n))
qn = np.zeros((M, n))
gam = np.zeros((M, n))
for ii in range(0, n):
gam[:, ii] = uf.optimum_reparam(mq, time, q1[:, ii], omethod)
fn[:, ii] = uf.warp_f_gamma(time, f[:, ii], gam[:, ii])
qn[:, ii] = uf.f_to_srsf(fn[:, ii], time)
m_new = np.sign(fn[self.pca.id, :]) * np.sqrt(
np.abs(fn[self.pca.id, :]))
qn1 = np.vstack((qn, m_new))
U = self.pca.U
no = U.shape[1]
if self.pca.__class__.__name__ == 'fdajpca':
C = self.pca.C
TT = self.time.shape[0]
mu_g = self.pca.mu_g
mu_psi = self.pca.mu_psi
vec = np.zeros((M, n))
psi = np.zeros((TT, n))
binsize = np.mean(np.diff(self.time))
for i in range(0, n):
psi[:, i] = np.sqrt(np.gradient(gam[:, i], binsize))
vec[:, i] = geo.inv_exp_map(mu_psi, psi[:, i])
g = np.vstack((qn1, C * vec))
a = np.zeros((n, no))
for i in range(0, n):
for j in range(0, no):
tmp = (g[:, i] - mu_g)
a[i, j] = dot(tmp.T, U[:, j])
elif self.pca.__class__.__name__ == 'fdavpca':
a = np.zeros((n, no))
for i in range(0, n):
for j in range(0, no):
tmp = (qn1[:, i] - self.pca.mqn)
a[i, j] = dot(tmp.T, U[:, j])
elif self.pca.__class__.__name__ == 'fdahpca':
a = np.zeros((n, no))
mu_psi = self.pca.psi_mu
vec = np.zeros((M, n))
TT = self.time.shape[0]
psi = np.zeros((TT, n))
binsize = np.mean(np.diff(self.time))
for i in range(0, n):
psi[:, i] = np.sqrt(np.gradient(gam[:, i], binsize))
vec[:, i] = geo.inv_exp_map(mu_psi, psi[:, i])
vm = self.pca.vec.mean(axis=1)
for i in range(0, n):
for j in range(0, no):
a[i, j] = np.sum(dot(vec[:, i] - vm, U[:, j]))
else:
raise Exception('Invalid fPCA Method')
for ii in range(0, n):
for jj in range(0, m):
self.y_pred[ii, jj] = self.alpha[jj] + np.sum(
a[ii, :] * self.b[:, jj])
if y == None:
self.y_pred = rg.phi(self.y_pred.reshape((1, n * m)))
self.y_pred = self.y_pred.reshape((n, m))
self.y_labels = np.argmax(self.y_pred, axis=1)
self.PC = np.nan
else:
self.y_pred = rg.phi(self.y_pred.reshape((1, n * m)))
self.y_pred = self.y_pred.reshape((n, m))
self.y_labels = np.argmax(self.y_pred, axis=1)
self.PC = np.zeros(m)
cls_set = np.arange(0, m)
for ii in range(0, m):
cls_sub = np.setdiff1d(cls_set, ii)
TP = np.sum(y[self.y_labels == ii] == ii)
FP = np.sum(y[np.in1d(self.y_labels, cls_sub)] == ii)
TN = np.sum(y[np.in1d(self.y_labels, cls_sub)] ==
self.y_labels[np.in1d(self.y_labels, cls_sub)])
FN = np.sum(np.in1d(y[self.y_labels == ii], cls_sub))
self.PC[ii] = (TP + TN) / (TP + FP + FN + TN)
self.PCo = np.sum(y == self.y_labels) / self.y_labels.shape[0]
else:
n = self.pca.coef.shape[1]
self.y_pred = np.zeros((n, m))
for ii in range(0, n):
for jj in range(0, m):
self.y_pred[ii, jj] = self.alpha[jj] + np.sum(
self.pca.coef[ii, :] * self.b[:, jj])
self.y_pred = rg.phi(self.y_pred.reshape((1, n * m)))
self.y_pred = self.y_pred.reshape((n, m))
self.y_labels = np.argmax(self.y_pred, axis=1)
self.PC = np.zeros(m)
cls_set = np.arange(0, m)
for ii in range(0, m):
cls_sub = np.setdiff1d(cls_set, ii)
TP = np.sum(self.y[self.y_labels == ii] == ii)
FP = np.sum(self.y[np.in1d(self.y_labels, cls_sub)] == ii)
TN = np.sum(self.y[np.in1d(self.y_labels, cls_sub)] ==
self.y_labels[np.in1d(self.y_labels, cls_sub)])
FN = np.sum(np.in1d(y[self.y_labels == ii], cls_sub))
self.PC[ii] = (TP + TN) / (TP + FP + FN + TN)
self.PCo = np.sum(y == self.y_labels) / self.y_labels.shape[0]
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 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 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 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 calc_model(self,
B=None,
lam=0,
df=20,
max_itr=20,
delta=.01,
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]
m = self.y.max()
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
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.LL = 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])
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)
# Find alpha and beta using l_bfgs
b0 = np.zeros(m * (Nb + 1))
out = fmin_l_bfgs_b(mlogit_loss,
b0,
fprime=mlogit_gradient,
args=(Phi, self.Y),
pgtol=1e-10,
maxiter=200,
maxfun=250,
factr=1e-30)
b = out[0]
B0 = b.reshape(Nb + 1, m)
alpha = B0[0, :]
beta = np.zeros((M, m))
for i in range(0, m):
beta[:, i] = B.dot(B0[1:Nb + 1, i])
# compute the logistic loss
self.LL[itr - 1] = mlogit_loss(b, Phi, self.Y)
# find gamma
gamma_new = np.zeros((M, N))
if parallel:
out = Parallel(n_jobs=cores)(
delayed(mlogit_warp_grad)(alpha,
beta,
self.time,
self.q[:, n],
self.Y[n, :],
delta=delta) 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] = mlogit_warp_grad(alpha,
beta,
self.time,
self.q[:, ii],
self.Y[ii, :],
delta=delta)
if norm(gamma - gamma_new) < 1e-5:
break
else:
gamma = gamma_new
itr += 1
self.qn = qn
self.fn = fn
self.gamma = gamma
self.alpha = alpha
self.beta = beta
self.b = b[1:-1]
self.n_classes = m
self.LL = self.LL[0:itr]
return
def elastic_mlogistic(f, y, time, B=None, df=20, max_itr=20, cores=-1,
delta=.01, parallel=True, smooth=False):
"""
This function identifies a multinomial logistic 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 labels {1,2,...,m} for m classes
:param time: vector of size M describing the sample points
:param B: optional matrix describing Basis elements
: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 N
functions with M 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 Loss: logistic loss
"""
M = f.shape[0]
N = f.shape[1]
# Code labels
m = y.max()
Y = np.zeros((N, m), dtype=int)
for ii in range(0, N):
Y[ii, y[ii]-1] = 1
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]
q = uf.f_to_srsf(f, time, smooth)
gamma = np.tile(np.linspace(0, 1, M), (N, 1))
gamma = gamma.transpose()
itr = 1
LL = 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])
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)
# Find alpha and beta using l_bfgs
b0 = np.zeros(m * (Nb+1))
out = fmin_l_bfgs_b(mlogit_loss, b0, fprime=mlogit_gradient,
args=(Phi, Y), pgtol=1e-10, maxiter=200,
maxfun=250, factr=1e-30)
b = out[0]
B0 = b.reshape(Nb+1, m)
alpha = B0[0, :]
beta = np.zeros((M, m))
for i in range(0, m):
beta[:, i] = B.dot(B0[1:Nb+1, i])
# compute the logistic loss
LL[itr - 1] = mlogit_loss(b, Phi, Y)
# find gamma
gamma_new = np.zeros((M, N))
if parallel:
out = Parallel(n_jobs=cores)(delayed(mlogit_warp_grad)(alpha, beta,
time, q[:, n], Y[n, :], delta=delta) 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] = mlogit_warp_grad(alpha, beta, time,
q[:, ii], Y[ii, :], delta=delta)
if norm(gamma - gamma_new) < 1e-5:
break
else:
gamma = gamma_new
itr += 1
# Last Step with centering of gam
gamma = gamma_new
# gamI = uf.SqrtMeanInverse(gamma)
# 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[:, ii])
model = collections.namedtuple('model', ['alpha', 'beta', 'fn',
'qn', 'gamma', 'q', 'B', 'b',
'Loss', 'n_classes', 'type'])
out = model(alpha, beta, fn, qn, gamma, q, B, b[1:-1], LL[0:itr],
m, 'mlogistic')
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
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 predict(self, newdata=None):
"""
This function performs prediction on regression model on new data if available or current stored data in object
Usage: obj.predict()
obj.predict(newdata)
:param newdata: dict containing new data for prediction (needs the keys below, if None predicts on training data)
:type newdata: dict
:param f: (M,N) matrix of functions
:param time: vector of time points
:param y: truth if available
:param smooth: smooth data if needed
:param sparam: number of times to run filter
"""
omethod = self.warp_data.method
lam = self.warp_data.lam
M = self.time.shape[0]
if newdata != None:
f = newdata['f']
time = newdata['time']
y = newdata['y']
if newdata['smooth']:
sparam = newdata['sparam']
f = fs.smooth_data(f,sparam)
q1 = fs.f_to_srsf(f,time)
n = q1.shape[1]
self.y_pred = np.zeros(n)
mq = self.warp_data.mqn
fn = np.zeros((M,n))
qn = np.zeros((M,n))
gam = np.zeros((M,n))
for ii in range(0,n):
gam[:,ii] = uf.optimum_reparam(mq,time,q1[:,ii],omethod,lam)
fn[:,ii] = uf.warp_f_gamma(time,f[:,ii],gam[:,ii])
qn[:,ii] = uf.f_to_srsf(fn[:,ii],time)
U = self.pca.U
no = U.shape[1]
if self.pca.__class__.__name__ == 'fdajpca':
m_new = np.sign(fn[self.pca.id,:])*np.sqrt(np.abs(fn[self.pca.id,:]))
qn1 = np.vstack((qn, m_new))
C = self.pca.C
TT = self.time.shape[0]
mu_g = self.pca.mu_g
mu_psi = self.pca.mu_psi
vec = np.zeros((M,n))
psi = np.zeros((TT,n))
binsize = np.mean(np.diff(self.time))
for i in range(0,n):
psi[:,i] = np.sqrt(np.gradient(gam[:,i],binsize))
out, theta = geo.inv_exp_map(mu_psi, psi[:,i])
vec[:,i] = out
g = np.vstack((qn1, C*vec))
a = np.zeros((n,no))
for i in range(0,n):
for j in range(0,no):
tmp = (g[:,i]-mu_g)
a[i,j] = np.dot(tmp.T, U[:,j])
elif self.pca.__class__.__name__ == 'fdavpca':
m_new = np.sign(fn[self.pca.id,:])*np.sqrt(np.abs(fn[self.pca.id,:]))
qn1 = np.vstack((qn, m_new))
a = np.zeros((n,no))
for i in range(0,n):
for j in range(0,no):
tmp = (qn1[:,i]-self.pca.mqn)
a[i,j] = np.dot(tmp.T, U[:,j])
elif self.pca.__class__.__name__ == 'fdahpca':
a = np.zeros((n,no))
mu_psi = self.pca.psi_mu
vec = np.zeros((M,n))
TT = self.time.shape[0]
psi = np.zeros((TT,n))
binsize = np.mean(np.diff(self.time))
for i in range(0,n):
psi[:,i] = np.sqrt(np.gradient(gam[:,i],binsize))
out, theta = geo.inv_exp_map(mu_psi, psi[:,i])
vec[:,i] = out
vm = self.pca.vec.mean(axis=1)
for i in range(0,n):
for j in range(0,no):
a[i,j] = np.sum(np.dot(vec[:,i]-vm,U[:,j]))
else:
raise Exception('Invalid fPCA Method')
for ii in range(0,n):
self.y_pred[ii] = self.alpha + np.dot(a[ii,:],self.b)
if y is None:
self.SSE = np.nan
else:
self.SSE = np.sum((y-self.y_pred)**2)
else:
n = self.pca.coef.shape[0]
self.y_pred = np.zeros(n)
for ii in range(0,n):
self.y_pred[ii] = self.alpha + np.dot(self.pca.coef[ii,:],self.b)
self.SSE = np.sum((self.y-self.y_pred)**2)
return
def elastic_prediction(f, time, model, y=None, smooth=False):
"""
This function performs prediction from an elastic regression model
with phase-variablity
: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 model: indentified model from elastic_regression
:param y: truth, optional used to calculate SSE
: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 N
functions with M 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
"""
q = uf.f_to_srsf(f, time, smooth)
n = q.shape[1]
if model.type == 'linear' or model.type == 'logistic':
y_pred = np.zeros(n)
elif model.type == 'mlogistic':
m = model.n_classes
y_pred = np.zeros((n, m))
for ii in range(0, n):
diff = model.q - q[:, ii][:, np.newaxis]
dist = np.sum(np.abs(diff)**2, axis=0)**(1. / 2)
q_tmp = uf.warp_q_gamma(time, q[:, ii], model.gamma[:, dist.argmin()])
if model.type == 'linear':
y_pred[ii] = model.alpha + trapz(q_tmp * model.beta, time)
elif model.type == 'logistic':
y_pred[ii] = model.alpha + trapz(q_tmp * model.beta, time)
elif model.type == 'mlogistic':
for jj in range(0, m):
y_pred[ii, jj] = model.alpha[jj] + trapz(
q_tmp * model.beta[:, jj], time)
if y is None:
if model.type == 'linear':
SSE = None
elif model.type == 'logistic':
y_pred = phi(y_pred)
y_labels = np.ones(n)
y_labels[y_pred < 0.5] = -1
PC = None
elif model.type == 'mlogistic':
y_pred = phi(y_pred.ravel())
y_pred = y_pred.reshape(n, m)
y_labels = y_pred.argmax(axis=1) + 1
PC = None
else:
if model.type == 'linear':
SSE = sum((y - y_pred)**2)
elif model.type == 'logistic':
y_pred = phi(y_pred)
y_labels = np.ones(n)
y_labels[y_pred < 0.5] = -1
TP = sum(y[y_labels == 1] == 1)
FP = sum(y[y_labels == -1] == 1)
TN = sum(y[y_labels == -1] == -1)
FN = sum(y[y_labels == 1] == -1)
PC = (TP + TN) / float(TP + FP + FN + TN)
elif model.type == 'mlogistic':
y_pred = phi(y_pred.ravel())
y_pred = y_pred.reshape(n, m)
y_labels = y_pred.argmax(axis=1) + 1
PC = np.zeros(m)
cls_set = np.arange(1, m + 1)
for ii in range(0, m):
cls_sub = np.delete(cls_set, ii)
TP = sum(y[y_labels == (ii + 1)] == (ii + 1))
FP = sum(y[np.in1d(y_labels, cls_sub)] == (ii + 1))
TN = sum(y[np.in1d(y_labels, cls_sub)] == y_labels[np.in1d(
y_labels, cls_sub)])
FN = sum(np.in1d(y[y_labels == (ii + 1)], cls_sub))
PC[ii] = (TP + TN) / float(TP + FP + FN + TN)
PC = sum(y == y_labels) / float(y_labels.size)
if model.type == 'linear':
prediction = collections.namedtuple('prediction', ['y_pred', 'SSE'])
out = prediction(y_pred, SSE)
elif model.type == 'logistic':
prediction = collections.namedtuple('prediction',
['y_prob', 'y_labels', 'PC'])
out = prediction(y_pred, y_labels, PC)
elif model.type == 'mlogistic':
prediction = collections.namedtuple('prediction',
['y_prob', 'y_labels', 'PC'])
out = prediction(y_pred, y_labels, PC)
return out
def elastic_mlogistic(f,
y,
time,
B=None,
df=20,
max_itr=20,
cores=-1,
delta=.01,
parallel=True,
smooth=False):
"""
This function identifies a multinomial logistic 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 labels {1,2,...,m} for m classes
:param time: vector of size M describing the sample points
:param B: optional matrix describing Basis elements
: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 N
functions with M 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 Loss: logistic loss
"""
M = f.shape[0]
N = f.shape[1]
# Code labels
m = y.max()
Y = np.zeros((N, m), dtype=int)
for ii in range(0, N):
Y[ii, y[ii] - 1] = 1
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]
q = uf.f_to_srsf(f, time, smooth)
gamma = np.tile(np.linspace(0, 1, M), (N, 1))
gamma = gamma.transpose()
itr = 1
LL = 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])
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)
# Find alpha and beta using l_bfgs
b0 = np.zeros(m * (Nb + 1))
out = fmin_l_bfgs_b(mlogit_loss,
b0,
fprime=mlogit_gradient,
args=(Phi, Y),
pgtol=1e-10,
maxiter=200,
maxfun=250,
factr=1e-30)
b = out[0]
B0 = b.reshape(Nb + 1, m)
alpha = B0[0, :]
beta = np.zeros((M, m))
for i in range(0, m):
beta[:, i] = B.dot(B0[1:Nb + 1, i])
# compute the logistic loss
LL[itr - 1] = mlogit_loss(b, Phi, Y)
# find gamma
gamma_new = np.zeros((M, N))
if parallel:
out = Parallel(n_jobs=cores)(delayed(mlogit_warp_grad)(
alpha, beta, time, q[:, n], Y[n, :], delta=delta)
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] = mlogit_warp_grad(alpha,
beta,
time,
q[:, ii],
Y[ii, :],
delta=delta)
if norm(gamma - gamma_new) < 1e-5:
break
else:
gamma = gamma_new
itr += 1
# Last Step with centering of gam
gamma = gamma_new
# gamI = uf.SqrtMeanInverse(gamma)
# 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[:, ii])
model = collections.namedtuple('model', [
'alpha', 'beta', 'fn', 'qn', 'gamma', 'q', 'B', 'b', 'Loss',
'n_classes', 'type'
])
out = model(alpha, beta, fn, qn, gamma, q, B, b[1:-1], LL[0:itr], m,
'mlogistic')
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 elastic_prediction(f, time, model, y=None, smooth=False):
"""
This function performs prediction from an elastic regression model
with phase-variablity
: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 model: indentified model from elastic_regression
:param y: truth, optional used to calculate SSE
: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 N
functions with M 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
"""
q = uf.f_to_srsf(f, time, smooth)
n = q.shape[1]
if model.type == 'linear' or model.type == 'logistic':
y_pred = np.zeros(n)
elif model.type == 'mlogistic':
m = model.n_classes
y_pred = np.zeros((n, m))
for ii in range(0, n):
diff = model.q - q[:, ii][:, np.newaxis]
dist = np.sum(np.abs(diff) ** 2, axis=0) ** (1. / 2)
q_tmp = uf.warp_q_gamma(time, q[:, ii],
model.gamma[:, dist.argmin()])
if model.type == 'linear':
y_pred[ii] = model.alpha + trapz(q_tmp * model.beta, time)
elif model.type == 'logistic':
y_pred[ii] = model.alpha + trapz(q_tmp * model.beta, time)
elif model.type == 'mlogistic':
for jj in range(0, m):
y_pred[ii, jj] = model.alpha[jj] + trapz(q_tmp * model.beta[:, jj], time)
if y is None:
if model.type == 'linear':
SSE = None
elif model.type == 'logistic':
y_pred = phi(y_pred)
y_labels = np.ones(n)
y_labels[y_pred < 0.5] = -1
PC = None
elif model.type == 'mlogistic':
y_pred = phi(y_pred.ravel())
y_pred = y_pred.reshape(n, m)
y_labels = y_pred.argmax(axis=1)+1
PC = None
else:
if model.type == 'linear':
SSE = sum((y - y_pred) ** 2)
elif model.type == 'logistic':
y_pred = phi(y_pred)
y_labels = np.ones(n)
y_labels[y_pred < 0.5] = -1
TP = sum(y[y_labels == 1] == 1)
FP = sum(y[y_labels == -1] == 1)
TN = sum(y[y_labels == -1] == -1)
FN = sum(y[y_labels == 1] == -1)
PC = (TP+TN)/float(TP+FP+FN+TN)
elif model.type == 'mlogistic':
y_pred = phi(y_pred.ravel())
y_pred = y_pred.reshape(n, m)
y_labels = y_pred.argmax(axis=1)+1
PC = np.zeros(m)
cls_set = np.arange(1, m+1)
for ii in range(0, m):
cls_sub = np.delete(cls_set, ii)
TP = sum(y[y_labels == (ii+1)] == (ii+1))
FP = sum(y[np.in1d(y_labels, cls_sub)] == (ii+1))
TN = sum(y[np.in1d(y_labels, cls_sub)] ==
y_labels[np.in1d(y_labels, cls_sub)])
FN = sum(np.in1d(y[y_labels == (ii+1)], cls_sub))
PC[ii] = (TP+TN)/float(TP+FP+FN+TN)
PC = sum(y == y_labels) / float(y_labels.size)
if model.type == 'linear':
prediction = collections.namedtuple('prediction', ['y_pred', 'SSE'])
out = prediction(y_pred, SSE)
elif model.type == 'logistic':
prediction = collections.namedtuple('prediction', ['y_prob',
'y_labels', 'PC'])
out = prediction(y_pred, y_labels, PC)
elif model.type == 'mlogistic':
prediction = collections.namedtuple('prediction', ['y_prob',
'y_labels', 'PC'])
out = prediction(y_pred, y_labels, PC)
return out
def pairwise_align_bayes(f1i, f2i, time, mcmcopts=None):
"""
This function aligns two functions using Bayesian framework. It will align
f2 to f1. It is based on mapping warping functions to a hypersphere, and a
subsequent exponential mapping to a tangent space. In the tangent space,
the Z-mixture pCN algorithm is used to explore both local and global
structure in the posterior distribution.
The Z-mixture pCN algorithm uses a mixture distribution for the proposal
distribution, controlled by input parameter zpcn. The zpcn$betas must be
between 0 and 1, and are the coefficients of the mixture components, with
larger coefficients corresponding to larger shifts in parameter space. The
zpcn["probs"] give the probability of each shift size.
Usage: out = pairwise_align_bayes(f1i, f2i, time)
out = pairwise_align_bayes(f1i, f2i, time, mcmcopts)
:param f1i: vector defining M samples of function 1
:param f2i: vector defining M samples of function 2
:param time: time vector of length M
:param mcmopts: dict of mcmc parameters
:type mcmcopts: dict
default mcmc options:
tmp = {"betas":np.array([0.5,0.5,0.005,0.0001]),"probs":np.array([0.1,0.1,0.7,0.1])}
mcmcopts = {"iter":2*(10**4) ,"burnin":np.minimum(5*(10**3),2*(10**4)//2),
"alpha0":0.1, "beta0":0.1,"zpcn":tmp,"propvar":1,
"initcoef":np.repeat(0,20), "npoints":200, "extrainfo":True}
:rtype collection containing
:return f2_warped: aligned f2
:return gamma: warping function
:return g_coef: final g_coef
:return psi: final psi
:return sigma1: final sigma
if extrainfo
:return accept: accept of psi samples
:return betas_ind
:return logl: log likelihood
:return gamma_mat: posterior gammas
:return gamma_stats: posterior gamma stats
:return xdist: phase distance posterior
:return ydist: amplitude distance posterior)
"""
if mcmcopts is None:
tmp = {
"betas": np.array([0.5, 0.5, 0.005, 0.0001]),
"probs": np.array([0.1, 0.1, 0.7, 0.1])
}
mcmcopts = {
"iter": 2 * (10**4),
"burnin": np.minimum(5 * (10**3), 2 * (10**4) // 2),
"alpha0": 0.1,
"beta0": 0.1,
"zpcn": tmp,
"propvar": 1,
"initcoef": np.repeat(0, 20),
"npoints": 200,
"extrainfo": True
}
if f1i.shape[0] != f2i.shape[0]:
raise Exception('Length of f1 and f2 must be equal')
if f1i.shape[0] != time.shape[0]:
raise Exception('Length of f1 and time must be equal')
if mcmcopts["zpcn"]["betas"].shape[0] != mcmcopts["zpcn"]["probs"].shape[0]:
raise Exception('In zpcn, betas must equal length of probs')
if np.mod(mcmcopts["initcoef"].shape[0], 2) != 0:
raise Exception('Length of mcmcopts.initcoef must be even')
# Number of sig figs to report in gamma_mat
SIG_GAM = 13
iter = mcmcopts["iter"]
# parameter settings
pw_sim_global_burnin = mcmcopts["burnin"]
valid_index = np.arange(pw_sim_global_burnin - 1, iter)
pw_sim_global_Mg = mcmcopts["initcoef"].shape[0] // 2
g_coef_ini = mcmcopts["initcoef"]
numSimPoints = mcmcopts["npoints"]
pw_sim_global_domain_par = np.linspace(0, 1, numSimPoints)
g_basis = uf.basis_fourier(pw_sim_global_domain_par, pw_sim_global_Mg, 1)
sigma1_ini = 1
zpcn = mcmcopts["zpcn"]
pw_sim_global_sigma_g = mcmcopts["propvar"]
def propose_g_coef(g_coef_curr):
pCN_beta = zpcn["betas"]
pCN_prob = zpcn["probs"]
probm = np.insert(np.cumsum(pCN_prob), 0, 0)
z = np.random.rand()
result = {"prop": g_coef_curr, "ind": 1}
for i in range(0, pCN_beta.shape[0]):
if z <= probm[i + 1] and z > probm[i]:
g_coef_new = normal(
0, pw_sim_global_sigma_g /
np.repeat(np.arange(1, pw_sim_global_Mg + 1), 2))
result["prop"] = np.sqrt(
1 -
pCN_beta[i]**2) * g_coef_curr + pCN_beta[i] * g_coef_new
result["ind"] = i
return result
# normalize time to [0,1]
time = (time - time.min()) / (time.max() - time.min())
timet = np.linspace(0, 1, numSimPoints)
f1 = uf.f_predictfunction(f1i, timet, 0)
f2 = uf.f_predictfunction(f2i, timet, 0)
# srsf transformation
q1 = uf.f_to_srsf(f1, timet)
q1i = uf.f_to_srsf(f1i, time)
q2 = uf.f_to_srsf(f2, timet)
tmp = uf.f_exp1(uf.f_basistofunction(g_basis["x"], 0, g_coef_ini, g_basis))
if tmp.min() < 0:
raise Exception("Invalid initial value of g")
# result vectors
g_coef = np.zeros((iter, g_coef_ini.shape[0]))
sigma1 = np.zeros(iter)
logl = np.zeros(iter)
SSE = np.zeros(iter)
accept = np.zeros(iter, dtype=bool)
accept_betas = np.zeros(iter)
# init
g_coef_curr = g_coef_ini
sigma1_curr = sigma1_ini
SSE_curr = f_SSEg_pw(
uf.f_basistofunction(g_basis["x"], 0, g_coef_ini, g_basis), q1, q2)
logl_curr = f_logl_pw(
uf.f_basistofunction(g_basis["x"], 0, g_coef_ini, g_basis), q1, q2,
sigma1_ini**2, SSE_curr)
g_coef[0, :] = g_coef_ini
sigma1[0] = sigma1_ini
SSE[0] = SSE_curr
logl[0] = logl_curr
# update the chain for iter-1 times
for m in tqdm(range(1, iter)):
# update g
g_coef_curr, tmp, SSE_curr, accepti, zpcnInd = f_updateg_pw(
g_coef_curr, g_basis, sigma1_curr**2, q1, q2, SSE_curr,
propose_g_coef)
# update sigma1
newshape = q1.shape[0] / 2 + mcmcopts["alpha0"]
newscale = 1 / 2 * SSE_curr + mcmcopts["beta0"]
sigma1_curr = np.sqrt(1 / np.random.gamma(newshape, 1 / newscale))
logl_curr = f_logl_pw(
uf.f_basistofunction(g_basis["x"], 0, g_coef_curr, g_basis), q1,
q2, sigma1_curr**2, SSE_curr)
# save updates to results
g_coef[m, :] = g_coef_curr
sigma1[m] = sigma1_curr
SSE[m] = SSE_curr
if mcmcopts["extrainfo"]:
logl[m] = logl_curr
accept[m] = accepti
accept_betas[m] = zpcnInd
# calculate posterior mean of psi
pw_sim_est_psi_matrix = np.zeros((numSimPoints, valid_index.shape[0]))
for k in range(0, valid_index.shape[0]):
g_temp = uf.f_basistofunction(g_basis["x"], 0,
g_coef[valid_index[k], :], g_basis)
psi_temp = uf.f_exp1(g_temp)
pw_sim_est_psi_matrix[:, k] = psi_temp
result_posterior_psi_simDomain = uf.f_psimean(pw_sim_global_domain_par,
pw_sim_est_psi_matrix)
# resample to same number of points as the input f1 and f2
interp = interp1d(np.linspace(0, 1,
result_posterior_psi_simDomain.shape[0]),
result_posterior_psi_simDomain,
fill_value="extrapolate")
result_posterior_psi = interp(np.linspace(0, 1, f1i.shape[0]))
# transform posterior mean of psi to gamma
result_posterior_gamma = uf.f_phiinv(result_posterior_psi)
result_posterior_gamma = uf.norm_gam(result_posterior_gamma)
# warped f2
f2_warped = uf.warp_f_gamma(time, f2i, result_posterior_gamma)
if mcmcopts["extrainfo"]:
M, N = pw_sim_est_psi_matrix.shape
gamma_mat = np.zeros((time.shape[0], N))
one_v = np.ones(M)
Dx = np.zeros(N)
Dy = Dx
for ii in range(0, N):
interp = interp1d(np.linspace(
0, 1, result_posterior_psi_simDomain.shape[0]),
pw_sim_est_psi_matrix[:, ii],
fill_value="extrapolate")
result_i = interp(time)
tmp = uf.f_phiinv(result_i)
gamma_mat[:, ii] = uf.norm_gam(tmp)
v, theta = geo.inv_exp_map(one_v, pw_sim_est_psi_matrix[:, ii])
Dx[ii] = np.sqrt(trapz(v**2, pw_sim_global_domain_par))
q2warp = uf.warp_q_gamma(pw_sim_global_domain_par, q2,
gamma_mat[:, ii])
Dy[ii] = np.sqrt(trapz((q1i - q2warp)**2, time))
gamma_stats = uf.statsFun(gamma_mat)
results_o = collections.namedtuple('align_bayes', [
'f2_warped', 'gamma', 'g_coef', 'psi', 'sigma1', 'accept', 'betas_ind',
'logl', 'gamma_mat', 'gamma_stats', 'xdist', 'ydist'
])
out = results_o(f2_warped, result_posterior_gamma, g_coef,
result_posterior_psi, sigma1, accept[1:], accept_betas[1:],
logl, gamma_mat, gamma_stats, Dx, Dy)
return (out)
def predict(self, newdata=None):
"""
This function performs prediction on regression model on new data if available or current stored data in object
Usage: obj.predict()
obj.predict(newdata)
:param newdata: dict containing new data for prediction (needs the keys below, if None predicts on training data)
:type newdata: dict
:param f: (M,N) matrix of functions
:param time: vector of time points
:param y: truth if available
:param smooth: smooth data if needed
:param sparam: number of times to run filter
"""
if newdata != None:
f = newdata['f']
time = newdata['time']
y = newdata['y']
q = uf.f_to_srsf(f, time, newdata['smooth'])
n = f.shape[1]
m = self.n_classes
yhat = np.zeros((n, m))
for ii in range(0, n):
diff = self.q - q[:, ii][:, np.newaxis]
dist = np.sum(np.abs(diff)**2, axis=0)**(1. / 2)
q_tmp = uf.warp_q_gamma(time, q[:, ii],
self.gamma[:, dist.argmin()])
for jj in range(0, m):
yhat[ii, jj] = self.alpha[jj] + trapz(
q_tmp * self.beta[:, jj], time)
if y is None:
yhat = phi(yhat.ravel())
yhat = yhat.reshape(n, m)
y_labels = yhat.argmax(axis=1) + 1
self.PC = None
else:
yhat = phi(yhat.ravel())
yhat = yhat.reshape(n, m)
y_labels = yhat.argmax(axis=1) + 1
PC = np.zeros(m)
cls_set = np.arange(1, m + 1)
for ii in range(0, m):
cls_sub = np.delete(cls_set, ii)
TP = sum(y[y_labels == (ii + 1)] == (ii + 1))
FP = sum(y[np.in1d(y_labels, cls_sub)] == (ii + 1))
TN = sum(y[np.in1d(y_labels, cls_sub)] == y_labels[np.in1d(
y_labels, cls_sub)])
FN = sum(np.in1d(y[y_labels == (ii + 1)], cls_sub))
PC[ii] = (TP + TN) / float(TP + FP + FN + TN)
self.PC = sum(y == y_labels) / float(y_labels.size)
self.y_pred = yhat
self.y_labels = y_labels
else:
n = self.f.shape[1]
m = self.n_classes
yhat = np.zeros((n, m))
for ii in range(0, n):
diff = self.q - self.q[:, ii][:, np.newaxis]
dist = np.sum(np.abs(diff)**2, axis=0)**(1. / 2)
q_tmp = uf.warp_q_gamma(self.time, self.q[:, ii],
self.gamma[:, dist.argmin()])
for jj in range(0, m):
yhat[ii, jj] = self.alpha[jj] + trapz(
q_tmp * self.beta[:, jj], self.time)
yhat = phi(yhat.ravel())
yhat = yhat.reshape(n, m)
y_labels = yhat.argmax(axis=1) + 1
PC = np.zeros(m)
cls_set = np.arange(1, m + 1)
for ii in range(0, m):
cls_sub = np.delete(cls_set, ii)
TP = sum(self.y[y_labels == (ii + 1)] == (ii + 1))
FP = sum(self.y[np.in1d(y_labels, cls_sub)] == (ii + 1))
TN = sum(self.y[np.in1d(y_labels, cls_sub)] == y_labels[
np.in1d(y_labels, cls_sub)])
FN = sum(np.in1d(self.y[y_labels == (ii + 1)], cls_sub))
PC[ii] = (TP + TN) / float(TP + FP + FN + TN)
self.PC = sum(self.y == y_labels) / float(y_labels.size)
self.y_pred = yhat
self.y_labels = y_labels
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 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
