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 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 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 joint_gauss_model(self, n=1, no=3):
"""
This function models the functional data using a joint Gaussian model
extracted from the principal components of the srsfs
:param n: number of random samples
:param no: number of principal components (default = 3)
:type n: integer
:type no: integer
"""
# Parameters
fn = self.fn
time = self.time
qn = self.qn
gam = self.gam
M = time.size
# Perform PCA
jfpca = fpca.fdajpca(self)
jfpca.calc_fpca(no=no)
s = jfpca.latent
U = jfpca.U
C = jfpca.C
mu_psi = jfpca.mu_psi
# compute mean and covariance
mq_new = qn.mean(axis=1)
mididx = jfpca.id
m_new = np.sign(fn[mididx, :]) * np.sqrt(np.abs(fn[mididx, :]))
mqn = np.append(mq_new, m_new.mean())
# generate random samples
vals = np.random.multivariate_normal(np.zeros(s.shape), np.diag(s), n)
tmp = np.matmul(U, np.transpose(vals))
qhat = np.tile(mqn.T, (n, 1)).T + tmp[0:M + 1, :]
tmp = np.matmul(U, np.transpose(vals) / C)
vechat = tmp[(M + 1):, :]
psihat = np.zeros((M, n))
gamhat = np.zeros((M, n))
for ii in range(n):
psihat[:, ii] = geo.exp_map(mu_psi, vechat[:, ii])
gam_tmp = cumtrapz(psihat[:, ii]**2,
np.linspace(0, 1, M),
initial=0.0)
gamhat[:, ii] = (gam_tmp - gam_tmp.min()) / (gam_tmp.max() -
gam_tmp.min())
ft = np.zeros((M, n))
fhat = np.zeros((M, n))
for ii in range(n):
fhat[:, ii] = uf.cumtrapzmid(
time, qhat[0:M, ii] * np.fabs(qhat[0:M, ii]),
np.sign(qhat[M, ii]) * (qhat[M, ii] * qhat[M, ii]), mididx)
ft[:, ii] = uf.warp_f_gamma(np.linspace(0, 1, M), fhat[:, ii],
gamhat[:, ii])
self.rsamps = True
self.fs = fhat
self.gams = gamhat
self.ft = ft
self.qs = qhat[0:M, :]
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.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 jointfPCA(fn, time, qn, q0, gam, no=2, showplot=True):
"""
This function calculates joint functional principal component analysis
on aligned data
:param fn: numpy ndarray of shape (M,N) of N aligned functions with M
samples
:param time: vector of size N describing the sample points
:param qn: numpy ndarray of shape (M,N) of N aligned SRSF with M samples
:param no: number of components to extract (default = 2)
:param showplot: Shows plots of results using matplotlib (default = T)
:type showplot: bool
:type no: int
:rtype: tuple of numpy ndarray
:return q_pca: srsf principal directions
:return f_pca: functional principal directions
:return latent: latent values
:return coef: coefficients
:return U: eigenvectors
"""
coef = np.arange(-1., 2.)
Nstd = coef.shape[0]
# set up for fPCA in q-space
mq_new = qn.mean(axis=1)
M = time.shape[0]
mididx = int(np.round(M / 2))
m_new = np.sign(fn[mididx, :]) * np.sqrt(np.abs(fn[mididx, :]))
mqn = np.append(mq_new, m_new.mean())
qn2 = np.vstack((qn, m_new))
# calculate vector space of warping functions
mu_psi, gam_mu, psi, vec = uf.SqrtMean(gam)
# joint fPCA
C = fminbound(find_C,0,1e4,(qn2,vec,q0,no,mu_psi))
qhat, gamhat, a, U, s, mu_g = jointfPCAd(qn2, vec, C, no, mu_psi)
# geodesic paths
q_pca = np.ndarray(shape=(M, Nstd, no), dtype=float)
f_pca = np.ndarray(shape=(M, Nstd, no), dtype=float)
for k in range(0, no):
for l in range(0, Nstd):
qhat = mqn + dot(U[0:(M+1),k],coef[l]*np.sqrt(s[k]))
vechat = dot(U[(M+1):,k],(coef[l]*np.sqrt(s[k]))/C)
psihat = geo.exp_map(mu_psi,vechat)
gamhat = cumtrapz(psihat*psihat,np.linspace(0,1,M),initial=0)
gamhat = (gamhat - gamhat.min()) / (gamhat.max() - gamhat.min())
if (sum(vechat)==0):
gamhat = np.linspace(0,1,M)
fhat = uf.cumtrapzmid(time, qhat[0:M]*np.fabs(qhat[0:M]), np.sign(qhat[M])*(qhat[M]*qhat[M]), mididx)
f_pca[:,l,k] = uf.warp_f_gamma(np.linspace(0,1,M), fhat, gamhat)
q_pca[:,l,k] = uf.warp_q_gamma(np.linspace(0,1,M), qhat[0:M], gamhat)
jfpca_results = collections.namedtuple('jfpca', ['q_pca', 'f_pca', 'latent', 'coef', 'U'])
jfpca = jfpca_results(q_pca, f_pca, s, a, U)
if showplot:
CBcdict = {
'Bl': (0, 0, 0),
'Or': (.9, .6, 0),
'SB': (.35, .7, .9),
'bG': (0, .6, .5),
'Ye': (.95, .9, .25),
'Bu': (0, .45, .7),
'Ve': (.8, .4, 0),
'rP': (.8, .6, .7),
}
cl = sorted(CBcdict.keys())
fig, ax = plt.subplots(2, no)
for k in range(0, no):
axt = ax[0, k]
for l in range(0, Nstd):
axt.plot(time, q_pca[0:M, l, k], color=CBcdict[cl[l]])
axt.set_title('q domain: PD %d' % (k + 1))
axt = ax[1, k]
for l in range(0, Nstd):
axt.plot(time, f_pca[:, l, k], color=CBcdict[cl[l]])
axt.set_title('f domain: PD %d' % (k + 1))
fig.set_tight_layout(True)
cumm_coef = 100 * np.cumsum(s) / sum(s)
idx = np.arange(0, s.shape[0]) + 1
plot.f_plot(idx, cumm_coef, "Coefficient Cumulative Percentage")
plt.xlabel("Percentage")
plt.ylabel("Index")
plt.show()
return jfpca
def 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 calc_fpca(self,
no=3,
stds=np.arange(-1., 2.),
id=None,
parallel=False,
cores=-1):
"""
This function calculates joint functional principal component analysis
on aligned data
:param no: number of components to extract (default = 3)
:param id: point to use for f(0) (default = midpoint)
:param stds: number of standard deviations along gedoesic to compute (default = -1,0,1)
:param parallel: run in parallel (default = F)
:param cores: number of cores for parallel (default = -1 (all))
:type no: int
:type id: int
:type parallel: bool
:type cores: int
:rtype: fdajpca object of numpy ndarray
:return q_pca: srsf principal directions
:return f_pca: functional principal directions
:return latent: latent values
:return coef: coefficients
:return U: eigenvectors
"""
fn = self.warp_data.fn
time = self.warp_data.time
qn = self.warp_data.qn
q0 = self.warp_data.q0
gam = self.warp_data.gam
M = time.shape[0]
if id is None:
mididx = int(np.round(M / 2))
else:
mididx = id
Nstd = stds.shape[0]
# set up for fPCA in q-space
mq_new = qn.mean(axis=1)
m_new = np.sign(fn[mididx, :]) * np.sqrt(np.abs(fn[mididx, :]))
mqn = np.append(mq_new, m_new.mean())
qn2 = np.vstack((qn, m_new))
# calculate vector space of warping functions
mu_psi, gam_mu, psi, vec = uf.SqrtMean(gam, parallel, cores)
# joint fPCA
C = fminbound(find_C, 0, 1e4,
(qn2, vec, q0, no, mu_psi, parallel, cores))
qhat, gamhat, a, U, s, mu_g, g, cov = jointfPCAd(
qn2, vec, C, no, mu_psi, parallel, cores)
# geodesic paths
q_pca = np.ndarray(shape=(M, Nstd, no), dtype=float)
f_pca = np.ndarray(shape=(M, Nstd, no), dtype=float)
for k in range(0, no):
for l in range(0, Nstd):
qhat = mqn + np.dot(U[0:(M + 1), k], stds[l] * np.sqrt(s[k]))
vechat = np.dot(U[(M + 1):, k], (stds[l] * np.sqrt(s[k])) / C)
psihat = geo.exp_map(mu_psi, vechat)
gamhat = cumtrapz(psihat * psihat,
np.linspace(0, 1, M),
initial=0)
gamhat = (gamhat - gamhat.min()) / (gamhat.max() -
gamhat.min())
if (sum(vechat) == 0):
gamhat = np.linspace(0, 1, M)
fhat = uf.cumtrapzmid(time, qhat[0:M] * np.fabs(qhat[0:M]),
np.sign(qhat[M]) * (qhat[M] * qhat[M]),
mididx)
f_pca[:, l, k] = uf.warp_f_gamma(np.linspace(0, 1, M), fhat,
gamhat)
q_pca[:, l, k] = uf.warp_q_gamma(np.linspace(0, 1, M),
qhat[0:M], gamhat)
self.q_pca = q_pca
self.f_pca = f_pca
self.latent = s[0:no]
self.coef = a
self.U = U[:, 0:no]
self.mu_psi = mu_psi
self.mu_g = mu_g
self.id = mididx
self.C = C
self.time = time
self.g = g
self.cov = cov
self.no = no
self.stds = stds
return