def regression_warp(nu, q, y, alpha):
"""
calculates optimal warping for function linear regression
:param nu: numpy ndarray of srvf (M,N) of M functions with N samples
:param q: numpy ndarray of srvf (M,N) of M functions with N samples
:param y: numpy ndarray of shape (1,N) of M functions with N samples
responses
:param alpha: numpy scalar
:rtype: numpy array
:return gamma_new: warping function
"""
T = q.shape[1]
qM, O_M, gam_M = cf.find_rotation_and_seed_q(nu, q, rotation=False)
y_M = cf.innerprod_q2(qM, nu)
qm, O_m, gam_m = cf.find_rotation_and_seed_q(-1 * nu, q, rotation=False)
y_m = cf.innerprod_q2(qm, nu)
if y > alpha + y_M:
O_hat = O_M
gamma_new = gam_M
elif y < alpha + y_m:
O_hat = O_m
gamma_new = gam_m
else:
gamma_new, O_hat = cf.curve_zero_crossing(y - alpha, q, nu, y_M, y_m,
gam_M, gam_m)
return (gamma_new, O_hat)
def karcher_calc(mu, q, basis, closed, rotation, method):
# Compute shooting vector from mu to q_i
qn_t, R, gamI = cf.find_rotation_and_seed_unique(mu, q, closed, rotation,
method)
qn_t = qn_t / sqrt(cf.innerprod_q2(qn_t, qn_t))
q1dotq2 = cf.innerprod_q2(mu, qn_t)
if (q1dotq2 > 1):
q1dotq2 = 1
d = arccos(q1dotq2)
u = qn_t - q1dotq2 * mu
normu = sqrt(cf.innerprod_q2(u, u))
if (normu > 1e-4):
w = u * arccos(q1dotq2) / normu
else:
w = zeros(qn_t.shape)
# Project to tangent space of manifold to obtain v_i
if closed == 0:
v = w
else:
v = cf.project_tangent(w, q, basis)
return (v, gamI, d)
def calculate_gradE(u, utilde, T=100, k=5):
"""
calculates gradient of energy along path
:param u: numpy ndarray of shape (2,M) of M samples
:param utilde: numpy ndarray of shape (2,M) of M samples
:param T: Number of samples of curve (Default = 100)
:param k: number of samples along path (Default = 5)
:rtype: numpy scalar
:return gradE: gradient of energy
:return normgradE: norm of gradient of energy
"""
gradE = zeros((2, T, k))
normgradE = zeros(k)
for tau in range(2, k + 1):
gradE[:, :,
tau - 1] = u[:, :, tau - 1] - ((tau - 1.) /
(k - 1.)) * utilde[:, :, tau - 1]
normgradE[tau - 1] = sqrt(
cf.innerprod_q2(gradE[:, :, tau - 1], gradE[:, :, tau - 1]))
return (gradE, normgradE)
def sample_shapes(mu, K, mode='O', no=3, numSamp=10):
"""
Computes sample shapes from mean and covariance
:param betamean: numpy ndarray of shape (n, M) describing the mean curve
:param mu: numpy ndarray of shape (n, M) describing the mean srvf
:param K: numpy ndarray of shape (M, M) describing the covariance
:param mode: Open ('O') or closed curve ('C') (default 'O')
:param no: number of direction (default 3)
:param numSamp: number of samples (default 10)
:rtype: tuple of numpy array
:return samples: sample shapes
"""
n, T = mu.shape
modes = ['O', 'C']
mode = [i for i, x in enumerate(modes) if x == mode]
if len(mode) == 0:
mode = 0
else:
mode = mode[0]
U, s, V = svd(K)
if mode == 0:
N = 2
else:
N = 10
epsilon = 1./(N-1)
samples = empty(numSamp, dtype=object)
for i in range(0, numSamp):
v = zeros((2, T))
for m in range(0, no):
v = v + randn()*sqrt(s[m])*vstack((U[0:T, m], U[T:2*T, m]))
q1 = mu
for j in range(0, N-1):
normv = sqrt(cf.innerprod_q2(v, v))
if normv < 1e-4:
q2 = mu
else:
q2 = cos(epsilon*normv)*q1+sin(epsilon*normv)*v/normv
if mode == 1:
q2 = cf.project_curve(q2)
# Parallel translate tangent vector
basis2 = cf.find_basis_normal(q2)
v = cf.parallel_translate(v, q1, q2, basis2, mode)
q1 = q2
samples[i] = cf.q_to_curve(q2)
return(samples)
def sample_shapes(mu, K, mode='O', no=3, numSamp=10):
"""
Computes sample shapes from mean and covariance
:param betamean: numpy ndarray of shape (n, M) describing the mean curve
:param mu: numpy ndarray of shape (n, M) describing the mean srvf
:param K: numpy ndarray of shape (M, M) describing the covariance
:param mode: Open ('O') or closed curve ('C') (default 'O')
:param no: number of direction (default 3)
:param numSamp: number of samples (default 10)
:rtype: tuple of numpy array
:return samples: sample shapes
"""
n, T = mu.shape
modes = ['O', 'C']
mode = [i for i, x in enumerate(modes) if x == mode]
if len(mode) == 0:
mode = 0
else:
mode = mode[0]
U, s, V = svd(K)
if mode == 0:
N = 2
else:
N = 10
epsilon = 1./(N-1)
samples = empty(numSamp, dtype=object)
for i in range(0, numSamp):
v = zeros((2, T))
for m in range(0, no):
v = v + randn()*sqrt(s[m])*vstack((U[0:T, m], U[T:2*T, m]))
q1 = mu
for j in range(0, N-1):
normv = sqrt(cf.innerprod_q2(v, v))
if normv < 1e-4:
q2 = mu
else:
q2 = cos(epsilon*normv)*q1+sin(epsilon*normv)*v/normv
if mode == 1:
q2 = cf.project_curve(q2)
# Parallel translate tangent vector
basis2 = cf.find_basis_normal(q2)
v = cf.parallel_translate(v, q1, q2, basis2, mode)
q1 = q2
samples[i] = cf.q_to_curve(q2)
return(samples)
def regression_warp(nu, beta, y, alpha):
"""
calculates optimal warping for function linear regression
:param nu: numpy ndarray of shape (M,N) of M functions with N samples
:param beta: numpy ndarray of shape (M,N) of M functions with N samples
:param y: numpy ndarray of shape (1,N) of M functions with N samples
responses
:param alpha: numpy scalar
:rtype: numpy array
:return gamma_new: warping function
"""
T = beta.shape[1]
betanu = cf.q_to_curve(nu)
betaM, O_M, tauM = cf.find_rotation_and_seed_coord(betanu, beta)
q = cf.curve_to_q(betaM)
gam_M = cf.optimum_reparam_curve(nu, q)
betaM = cf.group_action_by_gamma_coord(betaM, gam_M)
qM = cf.curve_to_q(betaM)
y_M = cf.innerprod_q2(qM, nu)
betam, O_m, taum = cf.find_rotation_and_seed_coord(-1 * betanu, beta)
q = cf.curve_to_q(betam)
gam_m = cf.optimum_reparam_curve(-1 * nu, q)
betam = cf.group_action_by_gamma_coord(betam, gam_m)
qm = cf.curve_to_q(betam)
y_m = cf.innerprod_q2(qm, nu)
if y > alpha + y_M:
O_hat = O_M
gamma_new = gam_M
tau = tauM
elif y < alpha + y_m:
O_hat = O_m
gamma_new = gam_m
tau = taum
else:
gamma_new, O_hat, tau = cf.curve_zero_crossing(y - alpha, beta, nu, y_M, y_m, gam_M,
gam_m)
return(gamma_new, O_hat, tau)
def sample_shapes(self, no=3, numSamp=10):
"""
Computes sample shapes from mean and covariance
:param no: number of direction (default 3)
:param numSamp: number of samples (default 10)
"""
n, T = self.q_mean.shape
modes = ['O', 'C']
mode = [i for i, x in enumerate(modes) if x == self.mode]
if len(mode) == 0:
mode = 0
else:
mode = mode[0]
U, s, V = svd(self.C)
if mode == 0:
N = 2
else:
N = 10
epsilon = 1. / (N - 1)
samples = empty(numSamp, dtype=object)
for i in range(0, numSamp):
v = zeros((2, T))
for m in range(0, no):
v = v + randn() * sqrt(s[m]) * vstack(
(U[0:T, m], U[T:2 * T, m]))
q1 = self.q_mean
for j in range(0, N - 1):
normv = sqrt(cf.innerprod_q2(v, v))
if normv < 1e-4:
q2 = self.q_mean
else:
q2 = cos(epsilon * normv) * q1 + sin(
epsilon * normv) * v / normv
if mode == 1:
q2 = cf.project_curve(q2)
# Parallel translate tangent vector
basis2 = cf.find_basis_normal(q2)
v = cf.parallel_translate(v, q1, q2, basis2, mode)
q1 = q2
samples[i] = cf.q_to_curve(q2)
self.samples = samples
return
def geod_dist_path_strt(beta, k=5):
"""
calculate geodisc distance for path straightening
:param beta: numpy ndarray of shape (2,M) of M samples
:param k: number of samples along path (Default = 5)
:rtype: numpy scalar
:return dist: geodesic distance
"""
dist = 0
for i in range(1, k):
beta1 = beta[:, :, i - 1]
beta2 = beta[:, :, i]
q1 = cf.curve_to_q(beta1)
q2 = cf.curve_to_q(beta2)
d = arccos(cf.innerprod_q2(q1, q2))
dist += d
return dist
def calculate_gradE(u, utilde, T=100, k=5):
"""
calculates gradient of energy along path
:param u: numpy ndarray of shape (2,M) of M samples
:param utilde: numpy ndarray of shape (2,M) of M samples
:param T: Number of samples of curve (Default = 100)
:param k: number of samples along path (Default = 5)
:rtype: numpy scalar
:return gradE: gradient of energy
:return normgradE: norm of gradient of energy
"""
gradE = zeros((2, T, k))
normgradE = zeros(k)
for tau in range(2, k + 1):
gradE[:, :, tau - 1] = u[:, :, tau - 1] - ((tau - 1.0) / (k - 1.0)) * utilde[:, :, tau - 1]
normgradE[tau - 1] = sqrt(cf.innerprod_q2(gradE[:, :, tau - 1], gradE[:, :, tau - 1]))
return (gradE, normgradE)
def geod_dist_path_strt(beta, k=5):
"""
calculate geodisc distance for path straightening
:param beta: numpy ndarray of shape (2,M) of M samples
:param k: number of samples along path (Default = 5)
:rtype: numpy scalar
:return dist: geodesic distance
"""
dist = 0
for i in range(1, k):
beta1 = beta[:, :, i-1]
beta2 = beta[:, :, i]
q1 = cf.curve_to_q(beta1)
q2 = cf.curve_to_q(beta2)
d = arccos(cf.innerprod_q2(q1, q2))
dist += d
return(dist)
def oc_elastic_prediction(beta, model, y=None):
"""
This function identifies a regression model with phase-variablity
using elastic methods
:param beta: numpy ndarray of shape (M,N) of M functions with N samples
:param model: identified 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 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
"""
T = model.q.shape[1]
n = beta.shape[2]
N = model.q.shape[2]
q, beta = preproc_open_curve(beta, T)
if model.type == 'oclinear' or model.type == 'oclogistic':
y_pred = np.zeros(n)
elif model.type == 'ocmlogistic':
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.linalg.norm(np.abs(diff), axis=(0, 1)) ** 2
dist = np.zeros(N)
for jj in range(0, N):
dist[jj] = np.linalg.norm(np.abs(diff[:, :, jj])) ** 2
if model.type == 'oclinear' or model.type == 'oclogistic':
# beta1 = cf.shift_f(beta[:, :, ii], int(model.tau[dist.argmin()]))
beta1 = beta[:, :, ii]
else:
beta1 = beta[:, :, ii]
beta1 = model.O[:, :, dist.argmin()].dot(beta1)
beta1 = cf.group_action_by_gamma_coord(beta1,
model.gamma[:, dist.argmin()])
q_tmp = cf.curve_to_q(beta1)
if model.type == 'oclinear':
y_pred[ii] = model.alpha + cf.innerprod_q2(q_tmp, model.nu)
elif model.type == 'oclogistic':
y_pred[ii] = model.alpha + cf.innerprod_q2(q_tmp, model.nu)
elif model.type == 'ocmlogistic':
for jj in range(0, m):
y_pred[ii, jj] = model.alpha[jj] + cf.innerprod_q2(q_tmp, model.nu[:, :, jj])
if y is None:
if model.type == 'oclinear':
SSE = None
elif model.type == 'oclogistic':
y_pred = phi(y_pred)
y_labels = np.ones(n)
y_labels[y_pred < 0.5] = -1
PC = None
elif model.type == 'ocmlogistic':
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 == 'oclinear':
SSE = sum((y - y_pred) ** 2)
elif model.type == 'oclogistic':
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 == 'ocmlogistic':
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 == 'oclinear':
prediction = collections.namedtuple('prediction', ['y_pred', 'SSE'])
out = prediction(y_pred, SSE)
elif model.type == 'oclogistic':
prediction = collections.namedtuple('prediction', ['y_prob',
'y_labels', 'PC'])
out = prediction(y_pred, y_labels, PC)
elif model.type == 'ocmlogistic':
prediction = collections.namedtuple('prediction', ['y_prob',
'y_labels', 'PC'])
out = prediction(y_pred, y_labels, PC)
return out
def karcher_mean(self, parallel=False, cores=-1):
"""
This calculates the mean of a set of curves
:param parallel: run in parallel (default = F)
:param cores: number of cores for parallel (default = -1 (all))
"""
n, T, N = self.beta.shape
modes = ['O', 'C']
mode = [i for i, x in enumerate(modes) if x == self.mode]
if len(mode) == 0:
mode = 0
else:
mode = mode[0]
# Initialize mu as one of the shapes
mu = self.q[:, :, 0]
betamean = self.beta[:, :, 0]
itr = 0
gamma = zeros((T, N))
maxit = 20
sumd = zeros(maxit + 1)
v = zeros((n, T, N))
normvbar = zeros(maxit + 1)
delta = 0.5
tolv = 1e-4
told = 5 * 1e-3
print("Computing Karcher Mean of %d curves in SRVF space.." % N)
while itr < maxit:
print("updating step: %d" % (itr + 1))
if iter == maxit:
print("maximal number of iterations reached")
mu = mu / sqrt(cf.innerprod_q2(mu, mu))
if mode == 1:
self.basis = cf.find_basis_normal(mu)
else:
self.basis = []
sumv = zeros((n, T))
sumd[0] = inf
sumd[itr + 1] = 0
out = Parallel(n_jobs=cores)(delayed(karcher_calc)(
self.beta[:, :, n], self.q[:, :,
n], betamean, mu, self.basis, mode)
for n in range(N))
v = zeros((n, T, N))
for i in range(0, N):
v[:, :, i] = out[i][0]
sumd[itr + 1] = sumd[itr + 1] + out[i][1]**2
sumv = v.sum(axis=2)
# Compute average direction of tangent vectors v_i
vbar = sumv / float(N)
normvbar[itr] = sqrt(cf.innerprod_q2(vbar, vbar))
normv = normvbar[itr]
if normv > tolv and fabs(sumd[itr + 1] - sumd[itr]) > told:
# Update mu in direction of vbar
mu = cos(delta * normvbar[itr]) * mu + sin(
delta * normvbar[itr]) * vbar / normvbar[itr]
if mode == 1:
mu = cf.project_curve(mu)
x = cf.q_to_curve(mu)
a = -1 * cf.calculatecentroid(x)
betamean = x + tile(a, [T, 1]).T
else:
break
itr += 1
self.q_mean = mu
self.beta_mean = betamean
self.v = v
self.qun = sumd[0:(itr + 1)]
self.E = normvbar[0:(itr + 1)]
return
def oc_elastic_regression(beta, y, B=None, df=40, T=200, max_itr=20, cores=-1):
"""
This function identifies a regression model for open curves
using elastic methods
:param beta: numpy ndarray of shape (n, M, N) describing N curves
in R^M
:param y: numpy array of N responses
:param B: optional matrix describing Basis elements
:param df: number of degrees of freedom B-spline (default 20)
:param T: number of desired samples along curve (default 100)
:param max_itr: maximum number of iterations (default 20)
:param cores: number of cores for parallel processing (default all)
:type beta: 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
"""
n = beta.shape[0]
N = beta.shape[2]
time = np.linspace(0, 1, T)
if n > 500:
parallel = True
elif T > 100:
parallel = True
else:
parallel = False
# 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, beta = preproc_open_curve(beta, T)
beta0 = beta.copy()
qn = q.copy()
gamma = np.tile(np.linspace(0, 1, T), (N, 1))
gamma = gamma.transpose()
O_hat = np.tile(np.eye(n), (N, 1, 1)).T
itr = 1
SSE = np.zeros(max_itr)
while itr <= max_itr:
print("Iteration: %d" % itr)
# align data
# OLS using basis
Phi = np.ones((N, n * Nb + 1))
for ii in range(0, N):
for jj in range(0, n):
for kk in range(1, Nb + 1):
Phi[ii, jj * Nb + kk] = trapz(qn[jj, :, ii] * B[:, kk - 1], time)
xx = dot(Phi.T, Phi)
inv_xx = inv(xx)
xy = dot(Phi.T, y)
b = dot(inv_xx, xy)
alpha = b[0]
nu = np.zeros((n, T))
for ii in range(0, n):
nu[ii, :] = B.dot(b[(ii * Nb + 1):((ii + 1) * Nb + 1)])
# compute the SSE
int_X = np.zeros(N)
for ii in range(0, N):
int_X[ii] = cf.innerprod_q2(qn[:, :, ii], nu)
SSE[itr - 1] = sum((y.reshape(N) - alpha - int_X) ** 2)
# find gamma
gamma_new = np.zeros((T, N))
if parallel:
out = Parallel(n_jobs=cores)(delayed(regression_warp)(nu, beta0[:, :, n], y[n], alpha) for n in range(N))
for ii in range(0, N):
gamma_new[:, ii] = out[ii][0]
beta1n = cf.group_action_by_gamma_coord(out[ii][1].dot(beta0[:, :, ii]), out[ii][0])
beta[:, :, ii] = beta1n
O_hat[:, :, ii] = out[ii][1]
qn[:, :, ii] = cf.curve_to_q(beta[:, :, ii])
else:
for ii in range(0, N):
beta1 = beta0[:, :, ii]
gammatmp, Otmp, tau = regression_warp(nu, beta1, y[ii], alpha)
gamma_new[:, ii] = gammatmp
beta1n = cf.group_action_by_gamma_coord(Otmp.dot(beta0[:, :, ii]), gammatmp)
beta[:, :, ii] = beta1n
O_hat[:, :, ii] = Otmp
qn[:, :, ii] = cf.curve_to_q(beta[:, :, ii])
if np.abs(SSE[itr - 1] - SSE[itr - 2]) < 1e-15:
break
else:
gamma = gamma_new
itr += 1
tau = np.zeros(N)
model = collections.namedtuple('model', ['alpha', 'nu', 'betan' 'q', 'gamma',
'O', 'tau', 'B', 'b', 'SSE', 'type'])
out = model(alpha, nu, beta, q, gamma, O_hat, tau, B, b[1:-1], SSE[0:itr], 'oclinear')
return out
def curve_principal_directions(betamean, mu, K, mode='O', no=3, N=5):
"""
Computes principal direction of variation specified by no. N is
Number of shapes away from mean. Creates 2*N+1 shape sequence
:param betamean: numpy ndarray of shape (n, M) describing the mean curve
:param mu: numpy ndarray of shape (n, M) describing the mean srvf
:param K: numpy ndarray of shape (M, M) describing the covariance
:param mode: Open ('O') or closed curve ('C') (default 'O')
:param no: number of direction (default 3)
:param N: number of shapes (2*N+1) (default 5)
:rtype: tuple of numpy array
:return pd: principal directions
"""
n, T = betamean.shape
modes = ['O', 'C']
mode = [i for i, x in enumerate(modes) if x == mode]
if len(mode) == 0:
mode = 0
else:
mode = mode[0]
U, s, V = svd(K)
qarray = empty((no, 2*N+1), dtype=object)
qarray1 = empty(N, dtype=object)
qarray2 = empty(N, dtype=object)
pd = empty((no, 2*N+1), dtype=object)
pd1 = empty(N, dtype=object)
pd2 = empty(N, dtype=object)
for m in range(0, no):
princDir = vstack((U[0:T, m], U[T:2*T, m]))
v = sqrt(s[m]) * princDir
q1 = mu
epsilon = 2./N
# Forward direction from mean
for i in range(0, N):
normv = sqrt(cf.innerprod_q2(v, v))
if normv < 1e-4:
q2 = mu
else:
q2 = cos(epsilon*normv)*q1 + sin(epsilon*normv)*v/normv
if mode == 1:
q2 = cf.project_curve(q2)
qarray1[i] = q2
p = cf.q_to_curve(q2)
centroid1 = -1*cf.calculatecentroid(p)
beta_scaled, scale = cf.scale_curve(p + tile(centroid1, [T, 1]).T)
pd1[i] = beta_scaled
# Parallel translate tangent vector
basis2 = cf.find_basis_normal(q2)
v = cf.parallel_translate(v, q1, q2, basis2, mode)
q1 = q2
# Backward direction from mean
v = -sqrt(s[m])*princDir
q1 = mu
for i in range(0, N):
normv = sqrt(cf.innerprod_q2(v, v))
if normv < 1e-4:
q2 = mu
else:
q2 = cos(epsilon*normv)*q1+sin(epsilon*normv)*v/normv
if mode == 1:
q2 = cf.project_curve(q2)
qarray2[i] = q2
p = cf.q_to_curve(q2)
centroid1 = -1*cf.calculatecentroid(p)
beta_scaled, scale = cf.scale_curve(p + tile(centroid1, [T, 1]).T)
pd2[i] = beta_scaled
# Parallel translate tangent vector
basis2 = cf.find_basis_normal(q2)
v = cf.parallel_translate(v, q1, q2, basis2, mode)
q1 = q2
for i in range(0, N):
qarray[m, i] = qarray2[(N-1)-i]
pd[m, i] = pd2[(N-1)-i]
qarray[m, N] = mu
centroid1 = -1*cf.calculatecentroid(betamean)
beta_scaled, scale = cf.scale_curve(betamean +
tile(centroid1, [T, 1]).T)
pd[m, N] = beta_scaled
for i in range(N+1, 2*N+1):
qarray[m, i] = qarray1[i-(N+1)]
pd[m, i] = pd1[i-(N+1)]
return(pd)
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 beta: (n, M,N) matrix of curves
:param y: truth if available
"""
T = self.warp_data.beta_mean.shape[1]
if newdata != None:
beta = newdata['beta']
y = newdata['y']
n = beta.shape[2]
beta1 = np.zeros(beta.shape)
q = np.zeros(beta.shape)
for ii in range(0,n):
if (beta.shape[1] != T):
beta1[:,:,ii] = cf.resamplecurve(beta[:,:,ii],T)
else:
beta1[:,:,ii] = beta[:,:,ii]
a = -cf.calculatecentroid(beta1[:,:,ii])
beta1[:,:,ii] += np.tile(a, (T,1)).T
q[:,:,ii] = cf.curve_to_q(beta1[:,:,ii])[0]
mu = self.warp_data.q_mean
v = np.zeros(q.shape)
for ii in range(0,n):
qn_t, R, gamI = cf.find_rotation_and_seed_unique(mu, q[:,:,ii], 0, self.rotation)
qn_t = qn_t / np.sqrt(cf.innerprod_q2(qn_t,qn_t))
q1dotq2 = cf.innerprod_q2(mu,qn_t)
if (q1dotq2 > 1):
q1dotq2 = 1
d = np.arccos(q1dotq2)
u = qn_t - q1dotq2*mu
normu = np.sqrt(cf.innerprod_q2(u,u))
if (normu>1e-4):
v[:,:,ii] = u*np.arccos(q1dotq2)/normu
else:
v[:,:,ii] = np.zeros(qn_t.shape)
Utmp = self.warp_data.U.T
no = self.warp_data.U.shape[1]
VM = np.mean(self.warp_data.v,2)
VM = VM.flatten()
x = np.zeros((no,n))
for i in range(0,n):
tmp = v[:,:,i]
tmpv1 = tmp.flatten()
x[:,i] = Utmp.dot((tmpv1- VM))
self.y_pred = np.zeros(n)
for ii in range(0,n):
self.y_pred[ii] = self.alpha + np.dot(x[:,ii],self.b)
if y is None:
self.SSE = np.nan
else:
self.SSE = np.sum((y-self.y_pred)**2)
else:
n = self.warp_data.coef.shape[1]
self.y_pred = np.zeros(n)
for ii in range(0,n):
self.y_pred[ii] = self.alpha + np.dot(self.warp_data.coef[:,ii],self.b)
self.SSE = np.sum((self.y-self.y_pred)**2)
return
def geod_sphere(beta1, beta2, k=5):
"""
This function caluclates the geodecis between open curves beta1 and
beta2 with k steps along path
:param beta1: numpy ndarray of shape (2,M) of M samples
:param beta2: numpy ndarray of shape (2,M) of M samples
:param k: number of samples along path (Default = 5)
:rtype: numpy ndarray
:return dist: geodesic distance
:return path: geodesic path
:return O: rotation matrix
"""
lam = 0.0
elastic = 1
rotation = 1
returnpath = 1
n, T = beta1.shape
beta1 = cf.resamplecurve(beta1, T)
beta2 = cf.resamplecurve(beta2, T)
centroid1 = cf.calculatecentroid(beta1)
beta1 = beta1 - tile(centroid1, [T, 1]).T
centroid2 = cf.calculatecentroid(beta2)
beta2 = beta2 - tile(centroid2, [T, 1]).T
q1 = cf.curve_to_q(beta1)
if rotation:
beta2, O1, tau = cf.find_rotation_and_seed_coord(beta1, beta2)
q2 = cf.curve_to_q(beta2)
else:
O1 = eye(2)
q2 = cf.curve_to_q(beta2)
if elastic:
# Find the optimal coorespondence
gam = cf.optimum_reparam_curve(q2, q1, lam)
gamI = uf.invertGamma(gam)
# Applying optimal re-parameterization to the second curve
beta2n = cf.group_action_by_gamma_coord(beta2, gamI)
q2n = cf.curve_to_q(beta2n)
if rotation:
beta2n, O2, tau = cf.find_rotation_and_seed_coord(beta1, beta2n)
centroid2 = cf.calculatecentroid(beta2n)
beta2n = beta2n - tile(centroid2, [T, 1]).T
q2n = cf.curve_to_q(beta2n)
O = O1.dot(O2)
else:
q2n = q2
O = O1
# Forming geodesic between the registered curves
dist = arccos(cf.innerprod_q2(q1, q2n))
if returnpath:
PsiQ = zeros((n, T, k))
PsiX = zeros((n, T, k))
for tau in range(0, k):
s = dist * tau / (k - 1.0)
PsiQ[:, :, tau] = (sin(dist - s) * q1 + sin(s) * q2n) / sin(dist)
PsiX[:, :, tau] = cf.q_to_curve(PsiQ[:, :, tau])
path = PsiQ
else:
path = 0
return (dist, path, O)
def curve_karcher_mean(beta, mode='O'):
"""
This claculates the mean of a set of curves
:param beta: numpy ndarray of shape (n, M, N) describing N curves
in R^M
:param mode: Open ('O') or closed curve ('C') (default 'O')
:rtype: tuple of numpy array
:return mu: mean srvf
:return betamean: mean curve
:return v: shooting vectors
:return q: srvfs
"""
n, T, N = beta.shape
q = zeros((n, T, N))
for ii in range(0, N):
q[:, :, ii] = cf.curve_to_q(beta[:, :, ii])
modes = ['O', 'C']
mode = [i for i, x in enumerate(modes) if x == mode]
if len(mode) == 0:
mode = 0
else:
mode = mode[0]
# Initialize mu as one of the shapes
mu = q[:, :, 0]
betamean = beta[:, :, 0]
delta = 0.5
tolv = 1e-4
told = 5*1e-3
maxit = 20
itr = 0
sumd = zeros(maxit+1)
v = zeros((n, T, N))
normvbar = zeros(maxit+1)
while itr < maxit:
print("Iteration: %d" % itr)
mu = mu / sqrt(cf.innerprod_q2(mu, mu))
sumv = zeros((2, T))
sumd[itr+1] = 0
out = Parallel(n_jobs=-1)(delayed(karcher_calc)(beta[:, :, n],
q[:, :, n], betamean, mu, mode) for n in range(N))
v = zeros((n, T, N))
for i in range(0, N):
v[:, :, i] = out[i][0]
sumd[itr+1] = sumd[itr+1] + out[i][1]**2
sumv = v.sum(axis=2)
# Compute average direction of tangent vectors v_i
vbar = sumv/float(N)
normvbar[itr] = sqrt(cf.innerprod_q2(vbar, vbar))
normv = normvbar[itr]
if normv > tolv and fabs(sumd[itr+1]-sumd[itr]) > told:
# Update mu in direction of vbar
mu = cos(delta*normvbar[itr])*mu + sin(delta*normvbar[itr]) * vbar/normvbar[itr]
if mode == 1:
mu = cf.project_curve(mu)
x = cf.q_to_curve(mu)
a = -1*cf.calculatecentroid(x)
betamean = x + tile(a, [T, 1]).T
else:
break
itr += 1
return(mu, betamean, v, q)
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 beta: numpy ndarray of shape (M,N) of M functions with N samples
:param y: truth if available
"""
if newdata != None:
beta = newdata['beta']
y = newdata['y']
T = self.q.shape[1]
n = beta.shape[2]
N = self.q.shape[2]
q, beta = preproc_open_curve(beta, T)
m = self.n_classes
y_pred = np.zeros((n, m))
for ii in range(0, n):
diff = self.q - q[:, :, ii][:, :, np.newaxis]
dist = np.zeros(N)
for jj in range(0, N):
dist[jj] = np.linalg.norm(np.abs(diff[:, :, jj]))**2
beta1 = beta[:, :, ii]
beta1 = self.O[:, :, dist.argmin()].dot(beta1)
beta1 = cf.group_action_by_gamma_coord(
beta1, self.gamma[:, dist.argmin()])
q_tmp = cf.curve_to_q(beta1)[0]
for jj in range(0, m):
y_pred[ii, jj] = self.alpha[jj] + cf.innerprod_q2(
q_tmp, self.nu[:, :, jj])
if y is None:
y_pred = phi(y_pred.ravel())
y_pred = y_pred.reshape(n, m)
y_labels = y_pred.argmax(axis=1) + 1
self.PC = None
else:
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)
self.PC = sum(y == y_labels) / float(y_labels.size)
self.y_pred = y_pred
self.y_labels = y_labels
else:
n = self.q.shape[2]
N = self.q.shape[2]
m = self.n_classes
y_pred = np.zeros((n, m))
for ii in range(0, n):
diff = self.q - self.q[:, :, ii][:, :, np.newaxis]
dist = np.zeros(N)
for jj in range(0, N):
dist[jj] = np.linalg.norm(np.abs(diff[:, :, jj]))**2
beta1 = self.beta0[:, :, ii]
beta1 = self.O[:, :, dist.argmin()].dot(beta1)
beta1 = cf.group_action_by_gamma_coord(
beta1, self.gamma[:, dist.argmin()])
q_tmp = cf.curve_to_q(beta1)[0]
for jj in range(0, m):
y_pred[ii, jj] = self.alpha[jj] + cf.innerprod_q2(
q_tmp, self.nu[:, :, jj])
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)
self.PC = sum(self.y == y_labels) / float(y_labels.size)
self.y_pred = y_pred
self.y_labels = y_labels
return
def init_path_rand(beta1, beta_mid, beta2, T=100, k=5):
"""
Initializes a path in \cal{C}. beta1, beta_mid beta2 are already
standardized curves. Creates a path from beta1 to beta_mid to beta2 in
shape space, then projects to the closed shape manifold.
:param beta1: numpy ndarray of shape (2,M) of M samples (first curve)
:param betamid: numpy ndarray of shape (2,M) of M samples (mid curve)
:param beta2: numpy ndarray of shape (2,M) of M samples (end curve)
:param T: Number of samples of curve (Default = 100)
:param k: number of samples along path (Default = 5)
:rtype: numpy ndarray
:return alpha: a path between two q-functions
:return beta: a path between two curves
:return O: rotation matrix
"""
alpha = zeros((2, T, k))
beta = zeros((2, T, k))
q1 = cf.curve_to_q(beta1)
q_mid = cf.curve_to_q(beta_mid)
# find optimal rotation of q2
beta2, O1, tau1 = cf.find_rotation_and_seed_coord(beta1, beta2)
q2 = cf.curve_to_q(beta2)
# find the optimal coorespondence
gam = cf.optimum_reparam_curve(q2, q1)
gamI = uf.invertGamma(gam)
# apply optimal reparametrization
beta2n = cf.group_action_by_gamma_coord(beta2, gamI)
# find optimal rotation of q2
beta2n, O2, tau1 = cf.find_rotation_and_seed_coord(beta1, beta2n)
centroid2 = cf.calculatecentroid(beta2n)
beta2n = beta2n - tile(centroid2, [T, 1]).T
q2n = cf.curve_to_q(beta2n)
O = O1.dot(O2)
# Initialize a path as a geodesic through q1 --- q_mid --- q2
theta1 = arccos(cf.innerprod_q2(q1, q_mid))
theta2 = arccos(cf.innerprod_q2(q_mid, q2n))
tmp = arange(2, int((k - 1) / 2) + 1)
t = zeros(tmp.size)
alpha[:, :, 0] = q1
beta[:, :, 0] = beta1
i = 0
for tau in range(2, int((k - 1) / 2) + 1):
t[i] = (tau - 1.0) / ((k - 1) / 2.0)
qnew = (1 / sin(theta1)) * (sin((1 - t[i]) * theta1) * q1 + sin(t[i] * theta1) * q_mid)
alpha[:, :, tau - 1] = cf.project_curve(qnew)
x = cf.q_to_curve(alpha[:, :, tau - 1])
a = -1 * cf.calculatecentroid(x)
beta[:, :, tau - 1] = x + tile(a, [T, 1]).T
i += 1
alpha[:, :, int((k - 1) / 2)] = q_mid
beta[:, :, int((k - 1) / 2)] = beta_mid
i = 0
for tau in range(int((k - 1) / 2) + 1, k - 1):
qnew = (1 / sin(theta2)) * (sin((1 - t[i]) * theta2) * q_mid + sin(t[i] * theta2) * q2n)
alpha[:, :, tau] = cf.project_curve(qnew)
x = cf.q_to_curve(alpha[:, :, tau])
a = -1 * cf.calculatecentroid(x)
beta[:, :, tau] = x + tile(a, [T, 1]).T
i += 1
alpha[:, :, k - 1] = q2n
beta[:, :, k - 1] = beta2n
return (alpha, beta, O)
def init_path_rand(beta1, beta_mid, beta2, T=100, k=5):
r"""
Initializes a path in :math:`\cal{C}`. beta1, beta_mid beta2 are already
standardized curves. Creates a path from beta1 to beta_mid to beta2 in
shape space, then projects to the closed shape manifold.
:param beta1: numpy ndarray of shape (2,M) of M samples (first curve)
:param betamid: numpy ndarray of shape (2,M) of M samples (mid curve)
:param beta2: numpy ndarray of shape (2,M) of M samples (end curve)
:param T: Number of samples of curve (Default = 100)
:param k: number of samples along path (Default = 5)
:rtype: numpy ndarray
:return alpha: a path between two q-functions
:return beta: a path between two curves
:return O: rotation matrix
"""
alpha = zeros((2, T, k))
beta = zeros((2, T, k))
q1 = cf.curve_to_q(beta1)
q_mid = cf.curve_to_q(beta_mid)
# find optimal rotation of q2
beta2, O1, tau1 = cf.find_rotation_and_seed_coord(beta1, beta2)
q2 = cf.curve_to_q(beta2)
# find the optimal coorespondence
gam = cf.optimum_reparam_curve(q2, q1)
gamI = uf.invertGamma(gam)
# apply optimal reparametrization
beta2n = cf.group_action_by_gamma_coord(beta2, gamI)
# find optimal rotation of q2
beta2n, O2, tau1 = cf.find_rotation_and_seed_coord(beta1, beta2n)
centroid2 = cf.calculatecentroid(beta2n)
beta2n = beta2n - tile(centroid2, [T, 1]).T
q2n = cf.curve_to_q(beta2n)
O = O1.dot(O2)
# Initialize a path as a geodesic through q1 --- q_mid --- q2
theta1 = arccos(cf.innerprod_q2(q1, q_mid))
theta2 = arccos(cf.innerprod_q2(q_mid, q2n))
tmp = arange(2, int((k-1)/2)+1)
t = zeros(tmp.size)
alpha[:, :, 0] = q1
beta[:, :, 0] = beta1
i = 0
for tau in range(2, int((k-1)/2)+1):
t[i] = (tau-1.)/((k-1)/2.)
qnew = (1/sin(theta1))*(sin((1-t[i])*theta1)*q1+sin(t[i]*theta1)*q_mid)
alpha[:, :, tau-1] = cf.project_curve(qnew)
x = cf.q_to_curve(alpha[:, :, tau-1])
a = -1*cf.calculatecentroid(x)
beta[:, :, tau-1] = x + tile(a, [T, 1]).T
i += 1
alpha[:, :, int((k-1)/2)] = q_mid
beta[:, :, int((k-1)/2)] = beta_mid
i = 0
for tau in range(int((k-1)/2)+1, k-1):
qnew = (1/sin(theta2))*(sin((1-t[i])*theta2)*q_mid
+ sin(t[i]*theta2)*q2n)
alpha[:, :, tau] = cf.project_curve(qnew)
x = cf.q_to_curve(alpha[:, :, tau])
a = -1*cf.calculatecentroid(x)
beta[:, :, tau] = x + tile(a, [T, 1]).T
i += 1
alpha[:, :, k-1] = q2n
beta[:, :, k-1] = beta2n
return(alpha, beta, O)
def curve_karcher_mean(beta, mode='O'):
"""
This claculates the mean of a set of curves
:param beta: numpy ndarray of shape (n, M, N) describing N curves
in R^M
:param mode: Open ('O') or closed curve ('C') (default 'O')
:rtype: tuple of numpy array
:return mu: mean srvf
:return betamean: mean curve
:return v: shooting vectors
:return q: srvfs
"""
n, T, N = beta.shape
q = zeros((n, T, N))
for ii in range(0, N):
q[:, :, ii] = cf.curve_to_q(beta[:, :, ii])
modes = ['O', 'C']
mode = [i for i, x in enumerate(modes) if x == mode]
if len(mode) == 0:
mode = 0
else:
mode = mode[0]
# Initialize mu as one of the shapes
mu = q[:, :, 0]
betamean = beta[:, :, 0]
delta = 0.5
tolv = 1e-4
told = 5*1e-3
maxit = 20
itr = 0
sumd = zeros(maxit+1)
v = zeros((n, T, N))
normvbar = zeros(maxit+1)
while itr < maxit:
print("Iteration: %d" % itr)
mu = mu / sqrt(cf.innerprod_q2(mu, mu))
sumv = zeros((2, T))
sumd[itr+1] = 0
out = Parallel(n_jobs=-1)(delayed(karcher_calc)(beta[:, :, n],
q[:, :, n], betamean, mu, mode) for n in range(N))
v = zeros((n, T, N))
for i in range(0, N):
v[:, :, i] = out[i][0]
sumd[itr+1] = sumd[itr+1] + out[i][1]**2
sumv = v.sum(axis=2)
# Compute average direction of tangent vectors v_i
vbar = sumv/float(N)
normvbar[itr] = sqrt(cf.innerprod_q2(vbar, vbar))
normv = normvbar[itr]
if normv > tolv and fabs(sumd[itr+1]-sumd[itr]) > told:
# Update mu in direction of vbar
mu = cos(delta*normvbar[itr])*mu + sin(delta*normvbar[itr]) * vbar/normvbar[itr]
if mode == 1:
mu = cf.project_curve(mu)
x = cf.q_to_curve(mu)
a = -1*cf.calculatecentroid(x)
betamean = x + tile(a, [T, 1]).T
else:
break
itr += 1
return(mu, betamean, v, q)
def curve_principal_directions(betamean, mu, K, mode='O', no=3, N=5):
"""
Computes principal direction of variation specified by no. N is
Number of shapes away from mean. Creates 2*N+1 shape sequence
:param betamean: numpy ndarray of shape (n, M) describing the mean curve
:param mu: numpy ndarray of shape (n, M) describing the mean srvf
:param K: numpy ndarray of shape (M, M) describing the covariance
:param mode: Open ('O') or closed curve ('C') (default 'O')
:param no: number of direction (default 3)
:param N: number of shapes (2*N+1) (default 5)
:rtype: tuple of numpy array
:return pd: principal directions
"""
n, T = betamean.shape
modes = ['O', 'C']
mode = [i for i, x in enumerate(modes) if x == mode]
if len(mode) == 0:
mode = 0
else:
mode = mode[0]
U, s, V = svd(K)
qarray = empty((no, 2*N+1), dtype=object)
qarray1 = empty(N, dtype=object)
qarray2 = empty(N, dtype=object)
pd = empty((no, 2*N+1), dtype=object)
pd1 = empty(N, dtype=object)
pd2 = empty(N, dtype=object)
for m in range(0, no):
princDir = vstack((U[0:T, m], U[T:2*T, m]))
v = sqrt(s[m]) * princDir
q1 = mu
epsilon = 2./N
# Forward direction from mean
for i in range(0, N):
normv = sqrt(cf.innerprod_q2(v, v))
if normv < 1e-4:
q2 = mu
else:
q2 = cos(epsilon*normv)*q1 + sin(epsilon*normv)*v/normv
if mode == 1:
q2 = cf.project_curve(q2)
qarray1[i] = q2
p = cf.q_to_curve(q2)
centroid1 = -1*cf.calculatecentroid(p)
beta_scaled, scale = cf.scale_curve(p + tile(centroid1, [T, 1]).T)
pd1[i] = beta_scaled
# Parallel translate tangent vector
basis2 = cf.find_basis_normal(q2)
v = cf.parallel_translate(v, q1, q2, basis2, mode)
q1 = q2
# Backward direction from mean
v = -sqrt(s[m])*princDir
q1 = mu
for i in range(0, N):
normv = sqrt(cf.innerprod_q2(v, v))
if normv < 1e-4:
q2 = mu
else:
q2 = cos(epsilon*normv)*q1+sin(epsilon*normv)*v/normv
if mode == 1:
q2 = cf.project_curve(q2)
qarray2[i] = q2
p = cf.q_to_curve(q2)
centroid1 = -1*cf.calculatecentroid(p)
beta_scaled, scale = cf.scale_curve(p + tile(centroid1, [T, 1]).T)
pd2[i] = beta_scaled
# Parallel translate tangent vector
basis2 = cf.find_basis_normal(q2)
v = cf.parallel_translate(v, q1, q2, basis2, mode)
q1 = q2
for i in range(0, N):
qarray[m, i] = qarray2[(N-1)-i]
pd[m, i] = pd2[(N-1)-i]
qarray[m, N] = mu
centroid1 = -1*cf.calculatecentroid(betamean)
beta_scaled, scale = cf.scale_curve(betamean +
tile(centroid1, [T, 1]).T)
pd[m, N] = beta_scaled
for i in range(N+1, 2*N+1):
qarray[m, i] = qarray1[i-(N+1)]
pd[m, i] = pd1[i-(N+1)]
return(pd)
def geod_sphere(beta1, beta2, k=5):
"""
This function calculates the geodesics between open curves beta1 and
beta2 with k steps along path
:param beta1: numpy ndarray of shape (2,M) of M samples
:param beta2: numpy ndarray of shape (2,M) of M samples
:param k: number of samples along path (Default = 5)
:rtype: numpy ndarray
:return dist: geodesic distance
:return path: geodesic path
:return O: rotation matrix
"""
lam = 0.0
elastic = 1
rotation = 1
returnpath = 1
n, T = beta1.shape
beta1 = cf.resamplecurve(beta1, T)
beta2 = cf.resamplecurve(beta2, T)
centroid1 = cf.calculatecentroid(beta1)
beta1 = beta1 - tile(centroid1, [T, 1]).T
centroid2 = cf.calculatecentroid(beta2)
beta2 = beta2 - tile(centroid2, [T, 1]).T
q1 = cf.curve_to_q(beta1)
if rotation:
beta2, O1, tau = cf.find_rotation_and_seed_coord(beta1, beta2)
q2 = cf.curve_to_q(beta2)
else:
O1 = eye(2)
q2 = cf.curve_to_q(beta2)
if elastic:
# Find the optimal coorespondence
gam = cf.optimum_reparam_curve(q2, q1, lam)
gamI = uf.invertGamma(gam)
# Applying optimal re-parameterization to the second curve
beta2n = cf.group_action_by_gamma_coord(beta2, gamI)
q2n = cf.curve_to_q(beta2n)
if rotation:
beta2n, O2, tau = cf.find_rotation_and_seed_coord(beta1, beta2n)
centroid2 = cf.calculatecentroid(beta2n)
beta2n = beta2n - tile(centroid2, [T, 1]).T
q2n = cf.curve_to_q(beta2n)
O = O1.dot(O2)
else:
q2n = q2
O = O1
# Forming geodesic between the registered curves
dist = arccos(cf.innerprod_q2(q1, q2n))
if returnpath:
PsiQ = zeros((n, T, k))
PsiX = zeros((n, T, k))
for tau in range(0, k):
s = dist * tau / (k - 1.)
PsiQ[:, :, tau] = (sin(dist-s)*q1+sin(s)*q2n)/sin(dist)
PsiX[:, :, tau] = cf.q_to_curve(PsiQ[:, :, tau])
path = PsiQ
else:
path = 0
return(dist, path, O)
def 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 beta: numpy ndarray of shape (M,N) of M functions with N samples
:param y: truth if available
"""
if newdata != None:
beta = newdata['beta']
y = newdata['y']
T = self.q.shape[1]
n = beta.shape[2]
N = self.q.shape[2]
q, beta = preproc_open_curve(beta, T)
y_pred = np.zeros(n)
for ii in range(0, n):
diff = self.q - q[:, :, ii][:, :, np.newaxis]
dist = np.zeros(N)
for jj in range(0, N):
dist[jj] = np.linalg.norm(np.abs(diff[:, :, jj]))**2
beta1 = beta[:, :, ii]
beta1 = self.O[:, :, dist.argmin()].dot(beta1)
beta1 = cf.group_action_by_gamma_coord(
beta1, self.gamma[:, dist.argmin()])
q_tmp = cf.curve_to_q(beta1)[0]
y_pred[ii] = self.alpha + cf.innerprod_q2(q_tmp, self.nu)
if y is None:
y_pred = phi(y_pred)
y_labels = np.ones(n)
y_labels[y_pred < 0.5] = -1
self.PC = None
else:
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)
self.PC = (TP + TN) / float(TP + FP + FN + TN)
self.y_pred = y_pred
self.y_labels = y_labels
else:
n = self.q.shape[2]
N = self.q.shape[2]
y_pred = np.zeros(n)
for ii in range(0, n):
diff = self.q - self.q[:, :, ii][:, :, np.newaxis]
dist = np.zeros(N)
for jj in range(0, N):
dist[jj] = np.linalg.norm(np.abs(diff[:, :, jj]))**2
beta1 = self.beta0[:, :, ii]
beta1 = self.O[:, :, dist.argmin()].dot(beta1)
beta1 = cf.group_action_by_gamma_coord(
beta1, self.gamma[:, dist.argmin()])
q_tmp = cf.curve_to_q(beta1)[0]
y_pred[ii] = self.alpha + cf.innerprod_q2(q_tmp, self.nu)
y_pred = phi(y_pred)
y_labels = np.ones(n)
y_labels[y_pred < 0.5] = -1
TP = sum(self.y[y_labels == 1] == 1)
FP = sum(self.y[y_labels == -1] == 1)
TN = sum(self.y[y_labels == -1] == -1)
FN = sum(self.y[y_labels == 1] == -1)
self.PC = (TP + TN) / float(TP + FP + FN + TN)
self.y_pred = y_pred
self.y_labels = y_labels
return
def calc_model(self, B=None, lam=0, df=40, T=200, max_itr=20, cores=-1):
"""
This function identifies a regression model for open curves
using elastic methods
: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 T: number of desired samples along curve (default 100)
:param max_itr: maximum number of iterations (default 20)
:param cores: number of cores for parallel processing (default all)
"""
n = self.beta.shape[0]
N = self.beta.shape[2]
time = np.linspace(0, 1, T)
if n > 500:
parallel = True
elif T > 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((T, Nb))
for ii in range(0, Nb):
Bdiff[:, ii] = np.gradient(np.gradient(B[:, ii], binsize), binsize)
q, beta = preproc_open_curve(self.beta, T)
self.q = q
beta0 = beta.copy()
qn = q.copy()
gamma = np.tile(np.linspace(0, 1, T), (N, 1))
gamma = gamma.transpose()
O_hat = np.tile(np.eye(n), (N, 1, 1)).T
itr = 1
self.SSE = np.zeros(max_itr)
while itr <= max_itr:
print("Iteration: %d" % itr)
# align data
# OLS using basis
Phi = np.ones((N, n * Nb + 1))
for ii in range(0, N):
for jj in range(0, n):
for kk in range(1, Nb + 1):
Phi[ii,
jj * Nb + kk] = trapz(qn[jj, :, ii] * B[:, kk - 1],
time)
R = np.zeros((n * Nb + 1, n * Nb + 1))
for kk in range(0, n):
for ii in range(1, Nb + 1):
for jj in range(1, Nb + 1):
R[kk * Nb + ii, kk * Nb + jj] = trapz(
Bdiff[:, ii - 1] * Bdiff[:, jj - 1], 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]
nu = np.zeros((n, T))
for ii in range(0, n):
nu[ii, :] = B.dot(b[(ii * Nb + 1):((ii + 1) * Nb + 1)])
# compute the SSE
int_X = np.zeros(N)
for ii in range(0, N):
int_X[ii] = cf.innerprod_q2(qn[:, :, ii], nu)
self.SSE[itr - 1] = sum((self.y.reshape(N) - alpha - int_X)**2)
# find gamma
gamma_new = np.zeros((T, N))
if parallel:
out = Parallel(n_jobs=cores)(
delayed(regression_warp)(nu, q[:, :, n], self.y[n], alpha)
for n in range(N))
for ii in range(0, N):
gamma_new[:, ii] = out[ii][0]
beta1n = cf.group_action_by_gamma_coord(
out[ii][1].dot(beta0[:, :, ii]), out[ii][0])
beta[:, :, ii] = beta1n
O_hat[:, :, ii] = out[ii][1]
qn[:, :, ii] = cf.curve_to_q(beta1n)[0]
else:
for ii in range(0, N):
q1 = q[:, :, ii]
gammatmp, Otmp = regression_warp(nu, q1, self.y[ii], alpha)
gamma_new[:, ii] = gammatmp
beta1n = cf.group_action_by_gamma_coord(
Otmp.dot(beta0[:, :, ii]), gammatmp)
beta[:, :, ii] = beta1n
O_hat[:, :, ii] = Otmp
qn[:, :, ii] = cf.curve_to_q(beta1n)[0]
if np.abs(self.SSE[itr - 1] - self.SSE[itr - 2]) < 1e-15:
break
else:
gamma = gamma_new
itr += 1
tau = np.zeros(N)
self.alpha = alpha
self.nu = nu
self.beta0 = beta0
self.betan = beta
self.gamma = gamma
self.qn = qn
self.B = B
self.O = O_hat
self.b = b[1:-1]
self.SSE = self.SSE[0:itr]
return
def geod_sphere(beta1, beta2, k=5, scale=False, rotation=True, center=True):
"""
This function calculates the geodesics between open curves beta1 and
beta2 with k steps along path
:param beta1: numpy ndarray of shape (2,M) of M samples
:param beta2: numpy ndarray of shape (2,M) of M samples
:param k: number of samples along path (Default = 5)
:param scale: include length (Default = False)
:param rotation: include rotation (Default = True)
:param center: center curves at origin (Default = True)
:rtype: numpy ndarray
:return dist: geodesic distance
:return path: geodesic path
:return O: rotation matrix
"""
lam = 0.0
returnpath = 1
n, T = beta1.shape
if center:
centroid1 = cf.calculatecentroid(beta1)
beta1 = beta1 - tile(centroid1, [T, 1]).T
centroid2 = cf.calculatecentroid(beta2)
beta2 = beta2 - tile(centroid2, [T, 1]).T
q1, len1, lenq1 = cf.curve_to_q(beta1)
if scale:
q2, len2, lenq2 = cf.curve_to_q(beta2)
beta2, q2n, O1, gamI = cf.find_rotation_and_seed_coord(beta1,
beta2,
rotation=rotation)
# Forming geodesic between the registered curves
val = cf.innerprod_q2(q1, q2n)
if val > 1:
if val < 1.0001: # assume numerical error
import warnings
warnings.warn(
f"Corrected a numerical error in geod_sphere: rounded {val} to 1"
)
val = 1
else:
raise Exception(
f"innerpod_q2 computed an inner product of {val} which is much greater than 1"
)
elif val < -1:
if val > -1.0001: # assume numerical error
import warnings
warnings.warn(
f"Corrected a numerical error in geod_sphere: rounded {val} to -1"
)
val = -1
else:
raise Exception(
f"innerpod_q2 computed an inner product of {val} which is much less than -1"
)
dist = arccos(val)
if isnan(dist):
raise Exception("geod_sphere computed a dist value which is NaN")
if returnpath:
PsiQ = zeros((n, T, k))
PsiX = zeros((n, T, k))
for tau in range(0, k):
if tau == 0:
tau1 = 0
else:
tau1 = tau / (k - 1.)
s = dist * tau1
if dist > 0:
PsiQ[:, :,
tau] = (sin(dist - s) * q1 + sin(s) * q2n) / sin(dist)
elif dist == 0:
PsiQ[:, :, tau] = (1 - tau1) * q1 + (tau1) * q2n
else:
raise Exception("geod_sphere computed a negative distance")
if scale:
scl = len1**(1 - tau1) * len2**(tau1)
else:
scl = 1
beta = scl * cf.q_to_curve(PsiQ[:, :, tau])
if center:
centroid = cf.calculatecentroid(beta)
beta = beta - tile(centroid, [T, 1]).T
PsiX[:, :, tau] = beta
path = PsiX
else:
path = 0
return (dist, path, PsiQ)
