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GPSuz.py
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GPSuz.py
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import scipy, pylab, numpy
import scipy.spatial as ssp
import scipy.optimize as sop
# from planetc import transit
'''GP regression routines with a few different kernels, and
hyper-parameter tuning using max likelihood or a simple MCMC. Incudes
a transit mean function which requires Tom Evans's planetc module
(just comment out the relevant code if you don't have that module).'''
def GP_covmat(X1, X2, par, typ = 'SE', sigma = None):
'''
Compute covariance matrix with or without white noise for a range of
GP kernels. Currently implemented:
- SE (squared exponential 1D, default)
- SE_ARD (squared exponential with separate length scales for each input dimension)
- M32 (Matern 32, 1D)
- QP (quasi-periodic SE, 1D)
'''
if typ == 'QP':
DD = ssp.distance.cdist(X1, X2, 'euclidean')
K = par[0]**2 * \
scipy.exp(- (scipy.sin(scipy.pi * DD / par[1]))**2 / 2. / par[2]**2 \
- DD**2 / 2. / par[3]**2)
elif typ == 'Per':
DD = ssp.distance.cdist(X1, X2, 'euclidean')
K = par[0]**2 * \
scipy.exp(- (scipy.sin(scipy.pi * DD / par[1]))**2 / 2. / par[2]**2)
elif typ == 'M32':
DD = ssp.distance.cdist(X1, X2, 'euclidean')
arg = scipy.sqrt(3) * abs(DD) / par[1]
K = par[0]**2 * (1 + arg) * scipy.exp(- arg)
elif typ == 'SE_ARD':
V = numpy.abs(numpy.matrix( numpy.diag( 1. / numpy.sqrt(2) / par[1:]) ))
D2 = ssp.distance.cdist(X1 * V, X2 * V, 'sqeuclidean')
K = par[0]**2 * numpy.exp( -D2 )
elif typ == 'SE_SUM':
D = X1.shape[1]
for i in numpy.arange(D):
D2 = ssp.distance.cdist(X1[:,i], X2[:,i], 'sqeuclidean')
pp = par[i*2:i*2+2]
if i == 0:
K = pp[0]**2 * scipy.exp(- D2 / 2. / pp[1]**2)
else:
K += pp[0]**2 * scipy.exp(- D2 / 2. / pp[1]**2)
else: # 'SE (radial)'
D2 = ssp.distance.cdist(X1, X2, 'sqeuclidean')
K = par[0]**2 * scipy.exp(- D2 / 2. / par[1]**2)
if sigma != None:
N = X1.shape[0]
K += sigma**2 * scipy.identity(N)
return scipy.matrix(K)
# def transit_MF(p, x):
# '''Compute transit light curve for parameters p and times x'''
# pars = {}
# pars['P'] = p[0]
# pars['T0'] = p[1]
# pars['RpRs'] = p[2]
# pars['aRs'] = p[3]
# pars['incl'] = p[4]
# pars['ecc'] = p[5]
# pars['omega'] = p[6]
# pars['foot'] = p[7]
# pars['grad'] = p[8]
# if len(p) == 11:
# pars['ld'] = 'quad'
# pars['gam1'] = p[9]
# pars['gam2'] = p[10]
# else:
# pars['ld'] = 'nonlin'
# pars['c1'] = p[9]
# pars['c2'] = p[10]
# pars['c3'] = p[11]
# pars['c4'] = p[12]
# return transit.ma02_aRs(x, **pars)
def step_MF(p, x):
y = numpy.zeros(len(x)) + p[0]
l = x > p[1]
y[l] = p[2]
return y
def GP_negloglik(p, x, y, cov_func = None, cov_typ = 'SE', MF = None, n_MF_par = 0, \
MF_args = None, fixed = None, fixed_par = None, \
prior = None):
'''
Compute negative log likelihood for GP, optionally with:
- fixed parameters (indices in fixed, values in fixed_par)
- simple prior specs (mean and standard dev of Gaussian prior)
- mean function MF
The parameters of both covariance and mean function should be in
the p array, MF pars last. The total number of MF pars
(variable+fixed) should be given by n_MF_par. The separation
between cov and MF pars is done after reinserting the fixed pars.
'''
if fixed == None:
par = scipy.copy(p)
else:
par = scipy.zeros(len(fixed))
par[fixed == True] = fixed_par
par[fixed == False] = p
if (MF != None):
r = y - scipy.matrix([MF(par[-n_MF_par:], MF_args)]).T
else:
r = y[:]
if cov_func == None:
K = GP_covmat(x, x, par[:-n_MF_par-1], typ = cov_typ, sigma = par[-n_MF_par-1])
else:
if n_MF_par == 0:
covpar_end = len(par)
else:
covpar_end = -n_MF_par
K = cov_func(x, x, par[:covpar_end], white_noise = True)
try:
L = scipy.linalg.cho_factor(K)
except scipy.linalg.LinAlgError:
pylab.clf()
pylab.imshow(K, interpolation = 'nearest')
print 'Warning: covariance matrix was not positive definite'
raw_input('continue?')
return 1e20
a = numpy.log(numpy.diag(L[0])).sum()
b = scipy.linalg.cho_solve(L, r)
a += 0.5 * r.T * scipy.matrix(b)
if prior != None:
for i in scipy.arange(len(p)):
if prior[i,1] > 0:
a -= (par[i] - prior[i,0])**2 / 2.0 / (prior[i,1])**2
return a
def GP_train(x, y, cov_par, cov_func = None, cov_typ ='SE', \
cov_fixed = None, prior = None, \
MF = None, MF_par = None, MF_args = None, \
MF_fixed = None):
'''
Max likelihood optimization of GP hyper-parameters. Calls
GP_negloglik. Takes care of merging / splitting the fixed /
variable and cov / MF parameters
'''
if MF != None:
merged_par = scipy.append(cov_par, MF_par)
n_MF_par = len(MF_par)
fixed = scipy.append(scipy.zeros(len(cov_par), 'bool'), \
scipy.zeros(n_MF_par, 'bool'))
if (cov_fixed != None): fixed[0:-n_MF_par] = cov_fixed
if (MF_fixed != None): fixed[-n_MF_par:] = MF_fixed
if MF_args == None: MF_args = x[:]
else:
merged_par = cov_par[:]
n_MF_par = 0
fixed = scipy.zeros(len(cov_par), 'bool')
if cov_fixed != None: fixed[:] = cov_fixed
var_par_in = merged_par[fixed == False]
fixed_par = merged_par[fixed == True]
args = (x, y, cov_func, cov_typ, MF, n_MF_par, MF_args, fixed, \
fixed_par, prior)
var_par_out = \
sop.fmin(GP_negloglik, var_par_in, args, disp = 0)
par_out = scipy.copy(merged_par)
par_out[fixed == False] = var_par_out
par_out[fixed == True] = fixed_par
if MF != None:
return par_out[:-n_MF_par], par_out[-n_MF_par:]
else:
return par_out
def GP_train_MCMC(Nstep, x, y, cov_par, cov_scales, cov_func = None, \
cov_typ = 'SE', cov_prior = None, \
MF = None, MF_par = None, MF_scales = None, MF_args = None, \
MF_prior = None):
'''
MCMC over GP hyper-parameters. Calls GP_negloglik. Takes care of
merging / splitting the fixed / variable and cov / MF parameters
Returns a Nstep x (M+1) array where M is the number of *variable*
(scale > 0) hyper-parameters. The first column of the return array
contains the neg log likelihood values, then the other columns the
parameters that were varied along the chain.
'''
# Sort out the fixed / variable and cov / MF parameters
if MF != None:
params = scipy.append(cov_par, MF_par)
scales = scipy.append(cov_scales, MF_scales)
n_MF_par = len(MF_par)
if MF_prior == None:
if cov_prior == None:
prior = None
else:
prior = numpy.copy(cov_prior)
else:
if cov_prior == None:
prior = numpy.copy(MF_prior)
else:
prior = scipy.append(cov_prior, MF_prior)
if MF_args == None: MF_args = x
else:
params = cov_par[:]
scales = cov_scales[:]
n_MF_par = 0
prior = numpy.copy(cov_prior)
# No do MCMC proper
fixed = scales == 0
var = scales > 0
nvar = var.sum()
var_par = params[scales > 0]
var_scales = scales[scales > 0]
fixed_par = scales[scales == 0]
chain = scipy.zeros((Nstep, nvar+1)) - 1
logL = - GP_negloglik(var_par, x, y, covfunc = cov_func, covtyp = cov_typ, \
MF = MF, n_MF_par = n_MF_par, \
MF_args = MF_args, fixed = fixed, fixed_par = params[fixed])
randnos = scipy.log(scipy.rand(Nstep))
for i in range(Nstep):
shift = pylab.normal(0., 1., nvar) * scales[var]
var_par_new = var_par + shift
print var_par_new
logL_new = \
- GP_negloglik(var_par_new, x, y, covfunc = cov_func, \
covtyp = cov_typ, MF = MF, n_MF_par = n_MF_par, \
MF_args = MF_args, fixed = fixed, fixed_par = params[fixed], \
prior = prior)
dlogL = logL_new - logL
if (randnos[i] <= dlogL):
print '%8.6f %11.6f %11.6f %8.6f %8.6f %1s' % \
(i/float(Nstep), logL, logL_new, dlogL, randnos[i], 'A')
var_par = var_par_new
logL = logL_new
else:
print '%8.6f %11.6f %11.6f %8.6f %8.6f %1s' % \
(i/float(Nstep), logL, logL_new, dlogL, randnos[i], 'R')
# Store the new values of the parameters and the new merit function
chain[i,0] = logL
chain[i,1:] = scipy.array(var_par)
return chain
def GP_predict(p, xpred, x, y, cov_func = None, cov_typ = 'SE', MF = None, n_MF_par = 0, \
MF_args = None, MF_args_pred = None, \
WhiteNoise = False, ReturnCov = False):
'''
Compute predictive distribution for GP with hyper-parameters p,
conditioned on (x,y) at test inputs xpred, with (optional) mean
function MF. The parameters of both covariance and mean function
should be in the p array, MF pars last. The number of MF pars should
be given by n_MF_par.
'''
if MF == None:
r = y[:]
else:
if MF_args == None:
MF_args = x
r = y - scipy.matrix([MF(p[-n_MF_par:], MF_args)]).T
if cov_func == None:
K = GP_covmat(x, x, p[:-n_MF_par-1], typ = cov_typ, sigma = p[-n_MF_par-1])
Ks = GP_covmat(xpred, x, p[:-n_MF_par-1], typ = cov_typ)
if WhiteNoise == True:
Kss = GP_covmat(xpred, xpred, p[:-n_MF_par-1], typ = cov_typ, \
sigma = p[-n_MF_par-1])
else:
Kss = GP_covmat(xpred, xpred, p[:-n_MF_par-1], typ = cov_typ)
else:
if n_MF_par == 0:
covpar_end = len(p)
else:
covpar_end = -n_MF_par
K = cov_func(x, x, p[:covpar_end], white_noise = True)
Ks = cov_func(xpred, x, p[:covpar_end], white_noise = False)
Kss = cov_func(xpred, xpred, p[:covpar_end], white_noise = WhiteNoise)
L = scipy.linalg.cho_factor(K)
b = scipy.linalg.cho_solve(L, r)
prec_mean = scipy.array(Ks * scipy.matrix(b)).flatten()
if MF != None:
if MF_args_pred == None:
MF_args_pred = xpred
prec_mean += MF(p[-n_MF_par:], MF_args_pred)
b = scipy.linalg.cho_solve(L, Ks.T)
prec_cov = Kss - Ks * scipy.matrix(b)
if ReturnCov == True:
return prec_mean, scipy.array(prec_cov)
else:
return prec_mean, \
scipy.array(scipy.sqrt(numpy.diag(prec_cov))).flatten()
def GP_plotpred(xpred, x, y, cov_par, cov_func = None, cov_typ = 'SE',
MF = None, MF_par = None, MF_args = None, MF_args_pred = None, \
WhiteNoise = False, plot_color = None):
'''
Wrapper for GP_predict that takes care of merging the
covariance and mean function parameters, and (optionally) plots
the predictive distribution (as well as returning it)
'''
if MF != None:
merged_par = scipy.append(cov_par, MF_par)
n_MF_par = len(MF_par)
else:
merged_par = cov_par[:]
n_MF_par = 0
fpred, fpred_err = GP_predict(merged_par, xpred, x, y, \
cov_func = cov_func, cov_typ = cov_typ, \
MF = MF, n_MF_par = n_MF_par, \
MF_args = MF_args, MF_args_pred = MF_args_pred, \
WhiteNoise = WhiteNoise)
xpl = scipy.array(xpred[:,0]).flatten()
if plot_color != None:
pylab.fill_between(xpl, fpred + 2 * fpred_err, fpred - 2 * fpred_err, \
color = plot_color, alpha = 0.1)
pylab.fill_between(xpl, fpred + fpred_err, fpred - fpred_err, \
color = plot_color, alpha = 0.1)
pylab.plot(xpl, fpred, '-', color = plot_color)
return fpred, fpred_err
# def transit_test(N = 50):
# # Generate a fake dataset
# xpred = scipy.r_[-0.12:0.1205:0.0005]
# MF = transit_MF
# MF_par = scipy.array([4.47, 0.0, 0.12, 9.85, \
# # P, T0, Rp/Rs, a/Rs
# 85.634, 0.0, 0.0, 1.0, 0.0, \
# # incl, ecc, omega, foot, grad
# 0.4397, 0.3754, 0.1005, -0.0622])
# # limb-darkening parameters
# cov_typ = 'SE_ARD'
# cov_par = scipy.array([0.0003, 0.01, 0.0001])
# Xpred = scipy.matrix([xpred]).T
# x = scipy.copy(xpred)
# numpy.random.shuffle(x)
# x = scipy.sort(x[:N])
# X = scipy.matrix([x]).T
# K = GP_covmat(X, X, cov_par[:-1], typ = cov_typ, sigma = cov_par[-1])
# y = MF(MF_par, x)
# y += numpy.random.multivariate_normal(scipy.zeros(N), K).flatten()
# Y = scipy.matrix([y]).T
# # Alter some parameters away from true values, and make prediction
# # using those
# print 'True values:', MF_par[2], cov_par[1]
# MF_par[2] = 0.1
# cov_par[1] = 0.1
# print 'Altered values:', MF_par[2], cov_par[1]
# pylab.clf()
# pylab.plot(x, y, 'k.')
# GP_plotpred(Xpred, X, Y, cov_par, cov_typ = cov_typ,
# MF = MF, MF_par = MF_par, MF_args = x, MF_args_pred = xpred, \
# WhiteNoise = True, plot_color = 'b')
# # Now try to fit for the altered parameters, holding the others fixed
# cov_fixed = scipy.ones(len(cov_par), 'bool')
# cov_fixed[1] = False
# MF_fixed = scipy.ones(len(MF_par), 'bool')
# MF_fixed[2] = False
# cov_par, MF_par = GP_train(X, Y, cov_par, None, cov_typ, cov_fixed, \
# MF = MF, MF_par = MF_par, MF_args = x, \
# MF_fixed = MF_fixed)
# print 'Fitted values:', MF_par[2], cov_par[1]
# # and make a prediction using the fitted values
# GP_plotpred(Xpred, X, Y, cov_par, cov_typ = cov_typ,
# MF = MF, MF_par = MF_par, MF_args = x, MF_args_pred = xpred, \
# WhiteNoise = True, plot_color = 'r')
# return