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1import numpy as np

2from scipy.optimize import minimize

3from scipy.linalg import solve_triangular, cho_factor, cho_solve

4from ase.optimize.gpmin.kernel import SquaredExponential

5from ase.optimize.gpmin.prior import ZeroPrior

8class GaussianProcess():

9 """Gaussian Process Regression

10 It is recommended to be used with other Priors and Kernels from

11 ase.optimize.gpmin

13 Parameters:

15 prior: Prior class, as in ase.optimize.gpmin.prior

16 Defaults to ZeroPrior

18 kernel: Kernel function for the regression, as in

19 ase.optimize.gpmin.kernel

20 Defaults to the Squared Exponential kernel with derivatives

21 """

22 def __init__(self, prior=None, kernel=None):

23 if kernel is None:

24 self.kernel = SquaredExponential()

25 else:

26 self.kernel = kernel

28 if prior is None:

29 self.prior = ZeroPrior()

30 else:

31 self.prior = prior

33 def set_hyperparams(self, params):

34 """Set hyperparameters of the regression.

35 This is a list containing the parameters of the

36 kernel and the regularization (noise)

37 of the method as the last entry.

38 """

39 self.hyperparams = params

40 self.kernel.set_params(params[:-1])

41 self.noise = params[-1]

43 def train(self, X, Y, noise=None):

44 """Produces a PES model from data.

46 Given a set of observations, X, Y, compute the K matrix

47 of the Kernel given the data (and its cholesky factorization)

48 This method should be executed whenever more data is added.

50 Parameters:

52 X: observations (i.e. positions). numpy array with shape: nsamples x D

53 Y: targets (i.e. energy and forces). numpy array with

54 shape (nsamples, D+1)

55 noise: Noise parameter in the case it needs to be restated.

56 """

57 if noise is not None:

58 self.noise = noise # Set noise attribute to a different value

60 self.X = X.copy() # Store the data in an attribute

61 n = self.X.shape

62 D = self.X.shape

63 regularization = np.array(n * ([self.noise * self.kernel.l] +

64 D * [self.noise]))

66 K = self.kernel.kernel_matrix(X) # Compute the kernel matrix

67 K[range(K.shape), range(K.shape)] += regularization**2

69 self.m = self.prior.prior(X)

70 self.a = Y.flatten() - self.m

71 self.L, self.lower = cho_factor(K, lower=True, check_finite=True)

72 cho_solve((self.L, self.lower), self.a, overwrite_b=True,

73 check_finite=True)

75 def predict(self, x, get_variance=False):

76 """Given a trained Gaussian Process, it predicts the value and the

77 uncertainty at point x. It returns f and V:

78 f : prediction: [y, grady]

79 V : Covariance matrix. Its diagonal is the variance of each component

80 of f.

82 Parameters:

84 x (1D np.array): The position at which the prediction is computed

85 get_variance (bool): if False, only the prediction f is returned

86 if True, the prediction f and the variance V are

87 returned: Note V is O(D*nsample2)

88 """

89 n = self.X.shape

90 k = self.kernel.kernel_vector(x, self.X, n)

91 f = self.prior.prior(x) + np.dot(k, self.a)

92 if get_variance:

93 v = solve_triangular(self.L, k.T.copy(), lower=True,

94 check_finite=False)

95 variance = self.kernel.kernel(x, x)

96 # covariance = np.matmul(v.T, v)

97 covariance = np.tensordot(v, v, axes=(0, 0))

98 V = variance - covariance

99 return f, V

100 return f

102 def neg_log_likelihood(self, params, *args):

103 """Negative logarithm of the marginal likelihood and its derivative.

104 It has been built in the form that suits the best its optimization,

105 with the scipy minimize module, to find the optimal hyperparameters.

107 Parameters:

109 l: The scale for which we compute the marginal likelihood

110 *args: Should be a tuple containing the inputs and targets

111 in the training set-

112 """

113 X, Y = args

114 # Come back to this

115 self.kernel.set_params(np.array([params, params, self.noise]))

116 self.train(X, Y)

117 y = Y.flatten()

119 # Compute log likelihood

120 logP = (-0.5 * np.dot(y - self.m, self.a) -

121 np.sum(np.log(np.diag(self.L))) -

122 X.shape * 0.5 * np.log(2 * np.pi))

124 # Gradient of the loglikelihood

127 # vectorizing the derivative of the log likelihood

128 D_P_input = np.array([np.dot(np.outer(self.a, self.a), g)

130 D_complexity = np.array([cho_solve((self.L, self.lower), g)

133 DlogP = 0.5 * np.trace(D_P_input - D_complexity, axis1=1, axis2=2)

134 return -logP, -DlogP

136 def fit_hyperparameters(self, X, Y, tol=1e-2, eps=None):

137 """Given a set of observations, X, Y; optimize the scale

138 of the Gaussian Process maximizing the marginal log-likelihood.

139 This method calls TRAIN there is no need to call the TRAIN method

140 again. The method also sets the parameters of the Kernel to their

141 optimal value at the end of execution

143 Parameters:

145 X: observations(i.e. positions). numpy array with shape: nsamples x D

146 Y: targets (i.e. energy and forces).

147 numpy array with shape (nsamples, D+1)

148 tol: tolerance on the maximum component of the gradient of the

149 log-likelihood.

150 (See scipy's L-BFGS-B documentation:

151 https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.minimize.html)

152 eps: include bounds to the hyperparameters as a +- a percentage

153 if eps is None there are no bounds in the optimization

155 Returns:

157 result (dict) :

158 result = {'hyperparameters': (numpy.array) New hyperparameters,

159 'converged': (bool) True if it converged,

160 False otherwise

161 }

162 """

163 params = np.copy(self.hyperparams)[:2]

164 arguments = (X, Y)

165 if eps is not None:

166 bounds = [((1 - eps) * p, (1 + eps) * p) for p in params]

167 else:

168 bounds = None

170 result = minimize(self.neg_log_likelihood, params, args=arguments,

171 method='L-BFGS-B', jac=True, bounds=bounds,

172 options={'gtol': tol, 'ftol': 0.01 * tol})

174 if not result.success:

175 converged = False

176 else:

177 converged = True

178 self.hyperparams = np.array([result.x.copy(),

179 result.x.copy(), self.noise])

180 self.set_hyperparams(self.hyperparams)

181 return {'hyperparameters': self.hyperparams, 'converged': converged}