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import numpy as np
from scipy.optimize import minimize
from scipy.linalg import solve_triangular, cho_factor, cho_solve
from ase.optimize.gpmin.kernel import SquaredExponential
from ase.optimize.gpmin.prior import ZeroPrior
class GaussianProcess():
"""Gaussian Process Regression
It is recommended to be used with other Priors and Kernels from
ase.optimize.gpmin
Parameters:
prior: Prior class, as in ase.optimize.gpmin.prior
Defaults to ZeroPrior
kernel: Kernel function for the regression, as in
ase.optimize.gpmin.kernel
Defaults to the Squared Exponential kernel with derivatives
"""
def __init__(self, prior=None, kernel=None):
if kernel is None:
self.kernel = SquaredExponential()
else:
self.kernel = kernel
if prior is None:
self.prior = ZeroPrior()
else:
self.prior = prior
def set_hyperparams(self, params):
"""Set hyperparameters of the regression.
This is a list containing the parameters of the
kernel and the regularization (noise)
of the method as the last entry.
"""
self.hyperparams = params
self.kernel.set_params(params[:-1])
self.noise = params[-1]
def train(self, X, Y, noise=None):
"""Produces a PES model from data.
Given a set of observations, X, Y, compute the K matrix
of the Kernel given the data (and its cholesky factorization)
This method should be executed whenever more data is added.
Parameters:
X: observations (i.e. positions). numpy array with shape: nsamples x D
Y: targets (i.e. energy and forces). numpy array with
shape (nsamples, D+1)
noise: Noise parameter in the case it needs to be restated.
"""
if noise is not None:
self.noise = noise # Set noise attribute to a different value
self.X = X.copy() # Store the data in an attribute
n = self.X.shape[0]
D = self.X.shape[1]
regularization = np.array(n * ([self.noise * self.kernel.l] +
D * [self.noise]))
K = self.kernel.kernel_matrix(X) # Compute the kernel matrix
K[range(K.shape[0]), range(K.shape[0])] += regularization**2
self.m = self.prior.prior(X)
self.a = Y.flatten() - self.m
self.L, self.lower = cho_factor(K, lower=True, check_finite=True)
cho_solve((self.L, self.lower), self.a, overwrite_b=True,
check_finite=True)
def predict(self, x, get_variance=False):
"""Given a trained Gaussian Process, it predicts the value and the
uncertainty at point x. It returns f and V:
f : prediction: [y, grady]
V : Covariance matrix. Its diagonal is the variance of each component
of f.
Parameters:
x (1D np.array): The position at which the prediction is computed
get_variance (bool): if False, only the prediction f is returned
if True, the prediction f and the variance V are
returned: Note V is O(D*nsample2)
"""
n = self.X.shape[0]
k = self.kernel.kernel_vector(x, self.X, n)
f = self.prior.prior(x) + np.dot(k, self.a)
if get_variance:
v = solve_triangular(self.L, k.T.copy(), lower=True,
check_finite=False)
variance = self.kernel.kernel(x, x)
# covariance = np.matmul(v.T, v)
covariance = np.tensordot(v, v, axes=(0, 0))
V = variance - covariance
return f, V
return f
def neg_log_likelihood(self, params, *args):
"""Negative logarithm of the marginal likelihood and its derivative.
It has been built in the form that suits the best its optimization,
with the scipy minimize module, to find the optimal hyperparameters.
Parameters:
l: The scale for which we compute the marginal likelihood
*args: Should be a tuple containing the inputs and targets
in the training set-
"""
X, Y = args
# Come back to this
self.kernel.set_params(np.array([params[0], params[1], self.noise]))
self.train(X, Y)
y = Y.flatten()
# Compute log likelihood
logP = (-0.5 * np.dot(y - self.m, self.a) -
np.sum(np.log(np.diag(self.L))) -
X.shape[0] * 0.5 * np.log(2 * np.pi))
# Gradient of the loglikelihood
grad = self.kernel.gradient(X)
# vectorizing the derivative of the log likelihood
D_P_input = np.array([np.dot(np.outer(self.a, self.a), g)
for g in grad])
D_complexity = np.array([cho_solve((self.L, self.lower), g)
for g in grad])
DlogP = 0.5 * np.trace(D_P_input - D_complexity, axis1=1, axis2=2)
return -logP, -DlogP
def fit_hyperparameters(self, X, Y, tol=1e-2, eps=None):
"""Given a set of observations, X, Y; optimize the scale
of the Gaussian Process maximizing the marginal log-likelihood.
This method calls TRAIN there is no need to call the TRAIN method
again. The method also sets the parameters of the Kernel to their
optimal value at the end of execution
Parameters:
X: observations(i.e. positions). numpy array with shape: nsamples x D
Y: targets (i.e. energy and forces).
numpy array with shape (nsamples, D+1)
tol: tolerance on the maximum component of the gradient of the
log-likelihood.
(See scipy's L-BFGS-B documentation:
https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.minimize.html)
eps: include bounds to the hyperparameters as a +- a percentage
if eps is None there are no bounds in the optimization
Returns:
result (dict) :
result = {'hyperparameters': (numpy.array) New hyperparameters,
'converged': (bool) True if it converged,
False otherwise
}
"""
params = np.copy(self.hyperparams)[:2]
arguments = (X, Y)
if eps is not None:
bounds = [((1 - eps) * p, (1 + eps) * p) for p in params]
else:
bounds = None
result = minimize(self.neg_log_likelihood, params, args=arguments,
method='L-BFGS-B', jac=True, bounds=bounds,
options={'gtol': tol, 'ftol': 0.01 * tol})
if not result.success:
converged = False
else:
converged = True
self.hyperparams = np.array([result.x.copy()[0],
result.x.copy()[1], self.noise])
self.set_hyperparams(self.hyperparams)
return {'hyperparameters': self.hyperparams, 'converged': converged}
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