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gp.py
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from __future__ import division
import numpy as np
from numpy.linalg import solve
from scipy import optimize
#from scipy.optimize import approx_fprime
#from checkgrad import checkgrad
MEAN_SHIFT = True
def cov_mat(x1, x2, a, b):
return a * np.exp(-b * (x1[:, np.newaxis] - x2)**2)
def reg_cov_mat(x, a, b, c):
return cov_mat(x, x, a, b) + c * np.eye(x.shape[0])
def posterior_mean_cov(t_train, y_train, t_test, parms, mean_shift=MEAN_SHIFT):
"""Prediction for Gaussian process regression
Returns the predictive mean and covariance matrix
Parameters
----------
t_train : array_like, shape (n_training_samples, )
time points of training samples
y_train : array_like, shape (n_training_samples, )
values of corresponding time points of training samples
t_test : array_like, shape (n_test_samples, )
time points of test samples
parms : tuple, length 3
hyperparameters (a, b, c) for covariance function of Gaussian
process.
[K(t)]_ij = a * exp(-b * (t_i - t_j)^2) + c * I(i == j)
Returns
-------
mean_test : array, shape (n_test_samples, )
predictive mean
cov_test : array, shape (n_test_samples, n_test_samples)
predictive covariance matrix
"""
#assert t_train.shape[0] == y_train.shape[0]
a, b, c = parms
if len(y_train) == 0:
return np.zeros_like(t_test), cov_mat(t_test, t_test, a, b)
K_train = reg_cov_mat(t_train, a, b, c)
K_train_test = cov_mat(t_train, t_test, a, b)
Ktr_inv_Ktt = solve(K_train, K_train_test)
if mean_shift:
mu = np.mean(y_train)
mean_test = mu + Ktr_inv_Ktt.T.dot(y_train - mu)
else:
mean_test = Ktr_inv_Ktt.T.dot(y_train)
cov_test = cov_mat(t_test, t_test, a, b) - K_train_test.T.dot(Ktr_inv_Ktt)
return mean_test, cov_test
def pointwise_posterior_mean_var(t_train, y_train, t_test, parms,
mean_shift=MEAN_SHIFT):
"""Prediction for Gaussian process regression
Returns the pointwise predictive mean and variance
Parameters
----------
t_train : array_like, shape (n_training_samples, )
time points of training samples
y_train : array_like, shape (n_training_samples, )
values of corresponding time points of training samples
t_test : array_like, shape (n_test_samples, )
time points of test samples
parms : tuple, length 3
hyperparameters (a, b, c) for covariance function of Gaussian
process.
[K(t)]_ij = a * exp(-b * (t_i - t_j)^2) + c * I(i == j)
Returns
-------
mean_test : array, shape (n_test_samples, )
predictive mean
var_test : array, shape (n_test_samples, )
predictive variance
"""
#assert t_train.shape[0] == y_train.shape[0]
a, b, c = parms
if len(y_train) == 0:
return np.zeros_like(t_test), np.ones_like(t_test) * a
K_train = reg_cov_mat(t_train, a, b, c)
K_train_test = cov_mat(t_train, t_test, a, b)
Ktr_inv_Ktt = solve(K_train, K_train_test)
if mean_shift:
mu = np.mean(y_train)
mean_test = mu + Ktr_inv_Ktt.T.dot(y_train - mu)
else:
mean_test = Ktr_inv_Ktt.T.dot(y_train)
var_test = a * np.ones_like(t_test) - (K_train_test * Ktr_inv_Ktt).sum(axis=0)
return mean_test, var_test
def posterior_mean(t_train, y_train, t_test, parms, mean_shift=MEAN_SHIFT):
a, b, c = parms
K_train = reg_cov_mat(t_train, a, b, c)
K_test_train = cov_mat(t_test, t_train, a, b)
if mean_shift:
mu = np.mean(y_train)
post_mean = mu + K_test_train.dot(solve(K_train, y_train - mu))
else:
post_mean = K_test_train.dot(solve(K_train, y_train))
return post_mean
def _dfunc(dk, cov, Kinv_y):
return (Kinv_y.dot(dk).dot(Kinv_y) - np.trace(solve(cov, dk))) * 0.5
def learn_hyperparms(ts,
a_shape=None, a_mean=None,
b_shape=None, b_mean=None,
c_shape=None, c_mean=None,
mean_shift=MEAN_SHIFT):
if a_shape is not None:
a_scale = a_mean / a_shape
if b_shape is not None:
b_scale = b_mean / b_shape
if c_shape is not None:
c_scale = c_mean / c_shape
#@checkgrad
def neg_mloglik(w):
#print 'parm', w
a, b, c = w
f = 0
df = np.zeros_like(w)
for t, y_orig in ts:
if mean_shift:
y = y_orig - np.mean(y_orig)
else:
y = y_orig
n = t.shape[0]
tsq = (t[:, np.newaxis] - t)**2
da = np.exp(-b * tsq)
cov = a * da
db = -tsq * cov
cov += c * np.eye(n)
dc = np.eye(n)
log_det = np.linalg.slogdet(cov)[1]
Kinv_y = solve(cov, y)
f += -0.5 * (y.dot(Kinv_y) + log_det)
df[0] += _dfunc(da, cov, Kinv_y)
df[1] += _dfunc(db, cov, Kinv_y)
df[2] += _dfunc(dc, cov, Kinv_y)
if a_shape is not None:
f += (a_shape - 1) * np.log(a) - a / a_scale
df[0] += (a_shape - 1) / a - 1 / a_scale
if b_shape is not None:
f += (b_shape - 1) * np.log(b) - b / b_scale
df[1] += (b_shape - 1) / b - 1 / b_scale
if c_shape is not None:
f += (c_shape - 1) * np.log(c) - c / c_scale
df[2] += (c_shape - 1) / c - 1 / c_scale
#print -f, -df
return -f, -df
w0 = np.array([1, 200, 1e-1])
opt_parms = optimize.fmin_l_bfgs_b(neg_mloglik, w0,
#bounds=[(1e-3, None), (1e-3, None), (1e-3, None)],
bounds=[(1e-3, None), (1e-5, None), (1e-5, None)],
factr=1e3, pgtol=1e-07, #disp=1,
)[0]
#print neg_mloglik(opt_parms)[1]
#def _func(x, *args): return neg_mloglik(x, *args)[0]
#print approx_fprime(opt_parms, _func, 1e-8)
print opt_parms
return opt_parms