def f_dlogl_pw(v_coef, v_basis, d_basis, sigma_curr, q1, q2):
vec = uf.f_basistofunction(v_basis["x"], 0, v_coef, v_basis)
psi = uf.f_exp1(vec)
N = q1.shape[0]
obs_domain = np.linspace(0, 1, N)
binsize = np.diff(obs_domain)
binsize = binsize.mean()
gamma = uf.f_phiinv(psi)
q2_warp = uf.warp_q_gamma(obs_domain, q2, gamma)
q2_warp_grad = np.gradient(q2_warp, binsize)
basismat = d_basis["matrix"]
g = np.zeros(N)
for i in range(0, basismat.shape[1]):
ubar = cumtrapz(basismat[:, i], obs_domain, initial=0)
integrand = (q1 - q2_warp) * (-2 * q2_warp_grad * ubar -
q2_warp * basismat[:, i])
tmp = 1 / sigma_curr * trapz(integrand, obs_domain)
g += tmp * basismat[:, i]
out, SSEv = f_vpostlogl_pw(vec, q1, q2, sigma_curr, 0)
nll = -1 * out
g_coef = v_basis["matrix"].T @ g
return nll, g_coef, SSEv
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 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_updateg_pw(g_coef_curr, g_basis, var1_curr, q1, q2, SSE_curr,
propose_g_coef):
g_coef_prop = propose_g_coef(g_coef_curr)
tst = uf.f_exp1(
uf.f_basistofunction(g_basis["x"], 0, g_coef_prop["prop"], g_basis))
while tst.min() < 0:
g_coef_prop = propose_g_coef(g_coef_curr)
tst = uf.f_exp1(
uf.f_basistofunction(g_basis["x"], 0, g_coef_prop["prop"],
g_basis))
if SSE_curr == 0:
SSE_curr = f_SSEg_pw(
uf.f_basistofunction(g_basis["x"], 0, g_coef_curr, g_basis), q1,
q2)
SSE_prop = f_SSEg_pw(
uf.f_basistofunction(g_basis["x"], 0, g_coef_prop["prop"], g_basis),
q1, q2)
logl_curr = f_logl_pw(
uf.f_basistofunction(g_basis["x"], 0, g_coef_curr, g_basis), q1, q2,
var1_curr, SSE_curr)
logl_prop = f_logl_pw(
uf.f_basistofunction(g_basis["x"], 0, g_coef_prop["prop"], g_basis),
q1, q2, var1_curr, SSE_prop)
ratio = np.minimum(1, np.exp(logl_prop - logl_curr))
u = np.random.rand()
if u <= ratio:
g_coef = g_coef_prop["prop"]
logl = logl_prop
SSE = SSE_prop
accept = True
zpcnInd = g_coef_prop["ind"]
if u > ratio:
g_coef = g_coef_curr
logl = logl_curr
SSE = SSE_curr
accept = False
zpcnInd = g_coef_prop["ind"]
return g_coef, logl, SSE, accept, zpcnInd
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)