def reduced_likelihood_function(self, theta=None):
"""
This function determines the BLUP parameters and evaluates the reduced
likelihood function for the given autocorrelation parameters theta.
Maximizing this function wrt the autocorrelation parameters theta is
equivalent to maximizing the likelihood of the assumed joint Gaussian
distribution of the observations y evaluated onto the design of
experiments X.
Parameters
----------
theta : array_like, optional
An array containing the autocorrelation parameters at which the
Gaussian Process model parameters should be determined.
Default uses the built-in autocorrelation parameters
(ie ``theta = self.theta_``).
Returns
-------
reduced_likelihood_function_value : double
The value of the reduced likelihood function associated to the
given autocorrelation parameters theta.
par : dict
A dictionary containing the requested Gaussian Process model
parameters:
- ``sigma2`` is the Gaussian Process variance.
- ``beta`` is the generalized least-squares regression weights for
Universal Kriging or given beta0 for Ordinary Kriging.
- ``gamma`` is the Gaussian Process weights.
- ``C`` is the Cholesky decomposition of the correlation
matrix [R].
- ``Ft`` is the solution of the linear equation system
[R] x Ft = F
- ``G`` is the QR decomposition of the matrix Ft.
"""
check_is_fitted(self, "X")
if theta is None:
# Use built-in autocorrelation parameters
theta = self.theta_
# Initialize output
reduced_likelihood_function_value = -np.inf
par = {}
# Retrieve data
n_samples = self.X.shape[0]
D = self.D
ij = self.ij
F = self.F
if D is None:
# Light storage mode (need to recompute D, ij and F)
D, ij = l1_cross_distances(self.X)
if (np.min(np.sum(D, axis=1)) == 0.
and self.corr != correlation_models.pure_nugget):
raise Exception("Multiple X are not allowed")
F = self.regr(self.X)
# Set up R
r = self.corr(theta, D)
R = np.eye(n_samples) * (1. + self.nugget)
R[ij[:, 0], ij[:, 1]] = r
R[ij[:, 1], ij[:, 0]] = r
# Cholesky decomposition of R
try:
C = linalg.cholesky(R, lower=True)
except linalg.LinAlgError:
return reduced_likelihood_function_value, par
# Get generalized least squares solution
Ft = linalg.solve_triangular(C, F, lower=True)
Q, G = linalg.qr(Ft, mode='economic')
sv = linalg.svd(G, compute_uv=False)
rcondG = sv[-1] / sv[0]
if rcondG < 1e-10:
# Check F
sv = linalg.svd(F, compute_uv=False)
condF = sv[0] / sv[-1]
if condF > 1e15:
raise Exception("F is too ill conditioned. Poor combination "
"of regression model and observations.")
else:
# Ft is too ill conditioned, get out (try different theta)
return reduced_likelihood_function_value, par
Yt = linalg.solve_triangular(C, self.y, lower=True)
if self.beta0 is None:
# Universal Kriging
beta = linalg.solve_triangular(G, np.dot(Q.T, Yt))
else:
# Ordinary Kriging
beta = np.array(self.beta0)
rho = Yt - np.dot(Ft, beta)
sigma2 = (rho**2.).sum(axis=0) / n_samples
# The determinant of R is equal to the squared product of the diagonal
# elements of its Cholesky decomposition C
detR = (np.diag(C)**(2. / n_samples)).prod()
# Compute/Organize output
reduced_likelihood_function_value = -sigma2.sum() * detR
par['sigma2'] = sigma2 * self.y_std**2.
par['beta'] = beta
par['gamma'] = linalg.solve_triangular(C.T, rho)
par['C'] = C
par['Ft'] = Ft
par['G'] = G
return reduced_likelihood_function_value, par
def maxdiv_gp(X, intervals, mode='I_OMEGA', theta=30, train_step=5, **kwargs):
""" Scores given intervals by fitting a Gaussian Process to them.
`X` is a d-by-n matrix with `n` data points, each with `d` attributes.
`intervals` has to be an iterable of `(a, b, score)` tuples, which define an
interval `[a,b)` which is suspected to be an anomaly.
Returns: a list of `(a, b, score)` tuples. `a` and `b` are the same as in the given
`intervals` iterable, but the scores will indicate whether a given interval
is an anomaly or not.
"""
dimension, n = X.shape
scores = []
# Fit gaussian process parameters to time-series
if theta is None:
gp = GaussianProcess(thetaL=0.1,
thetaU=1000,
nugget=1e-8,
normalize=False)
else:
gp = GaussianProcess(theta0=theta, nugget=1e-8, normalize=False)
gp.fit(np.linspace(0, 1, n, endpoint=True).reshape(n, 1), X.T)
# Compute characteristic length scale
ls = int(np.sqrt(0.5 / gp.theta_).flat[0] * n + 0.5)
# Compute regression function
f = gp.regr(gp.X)
# Compute correlation matrix
D, ij = l1_cross_distances(gp.X)
r = gp.corr(gp.theta_, D)
corr = np.eye(n) * (1. + gp.nugget)
corr[ij[:, 0], ij[:, 1]] = r
corr[ij[:, 1], ij[:, 0]] = r
# Search for maximally divergent intervals
eps = 1e-7
for a, b, base_score in intervals:
score = 0.0
extreme_interval_length = b - a
non_extreme_points = n - extreme_interval_length
timesteps_extreme = np.arange(a, b)
#timesteps_non_extreme = np.setdiff1d(np.arange(n), timesteps_extreme)
timesteps_non_extreme = np.concatenate(
(np.arange(max(0, a - ls), b,
train_step), np.arange(b, min(n, b + ls), train_step)))
mu, sigma = condition_gp(gp, f, corr, timesteps_non_extreme,
timesteps_extreme)
ll = multivariate_normal.logpdf(gp.y[timesteps_extreme, :].T - mu.T,
None, sigma +
np.eye(sigma.shape[0]) * eps).sum()
if mode != 'TS':
score -= ll / extreme_interval_length
else:
score -= ll
scores.append((a, b, score))
return scores
def fit(self, X, y):
"""
The Gaussian Process model fitting method.
Parameters
----------
X : double array_like
An array with shape (n_samples, n_features) with the input at which
observations were made.
y : double array_like
An array with shape (n_samples, ) or shape (n_samples, n_targets)
with the observations of the output to be predicted.
Returns
-------
gp : self
A fitted Gaussian Process model object awaiting data to perform
predictions.
"""
# Run input checks
self._check_params()
self.random_state = check_random_state(self.random_state)
# Force data to 2D numpy.array
X, y = check_X_y(X, y, multi_output=True, y_numeric=True)
self.y_ndim_ = y.ndim
if y.ndim == 1:
y = y[:, np.newaxis]
# Check shapes of DOE & observations
n_samples, n_features = X.shape
_, n_targets = y.shape
# Run input checks
self._check_params(n_samples)
# Normalize data or don't
if self.normalize:
X_mean = np.mean(X, axis=0)
X_std = np.std(X, axis=0)
y_mean = np.mean(y, axis=0)
y_std = np.std(y, axis=0)
X_std[X_std == 0.] = 1.
y_std[y_std == 0.] = 1.
# center and scale X if necessary
X = (X - X_mean) / X_std
y = (y - y_mean) / y_std
else:
X_mean = np.zeros(1)
X_std = np.ones(1)
y_mean = np.zeros(1)
y_std = np.ones(1)
# Calculate matrix of distances D between samples
D, ij = l1_cross_distances(X)
if (np.min(np.sum(D, axis=1)) == 0.
and self.corr != correlation_models.pure_nugget):
raise Exception("Multiple input features cannot have the same"
" target value.")
# Regression matrix and parameters
F = self.regr(X)
n_samples_F = F.shape[0]
if F.ndim > 1:
p = F.shape[1]
else:
p = 1
if n_samples_F != n_samples:
raise Exception("Number of rows in F and X do not match. Most "
"likely something is going wrong with the "
"regression model.")
if p > n_samples_F:
raise Exception(("Ordinary least squares problem is undetermined "
"n_samples=%d must be greater than the "
"regression model size p=%d.") % (n_samples, p))
if self.beta0 is not None:
if self.beta0.shape[0] != p:
raise Exception("Shapes of beta0 and F do not match.")
# Set attributes
self.X = X
self.y = y
self.D = D
self.ij = ij
self.F = F
self.X_mean, self.X_std = X_mean, X_std
self.y_mean, self.y_std = y_mean, y_std
# Determine Gaussian Process model parameters
if self.thetaL is not None and self.thetaU is not None:
# Maximum Likelihood Estimation of the parameters
if self.verbose:
print("Performing Maximum Likelihood Estimation of the "
"autocorrelation parameters...")
self.theta_, self.reduced_likelihood_function_value_, par = \
self._arg_max_reduced_likelihood_function()
if np.isinf(self.reduced_likelihood_function_value_):
raise Exception("Bad parameter region. "
"Try increasing upper bound")
else:
# Given parameters
if self.verbose:
print("Given autocorrelation parameters. "
"Computing Gaussian Process model parameters...")
self.theta_ = self.theta0
self.reduced_likelihood_function_value_, par = \
self.reduced_likelihood_function()
if np.isinf(self.reduced_likelihood_function_value_):
raise Exception("Bad point. Try increasing theta0.")
self.beta = par['beta']
self.gamma = par['gamma']
self.sigma2 = par['sigma2']
self.C = par['C']
self.Ft = par['Ft']
self.G = par['G']
if self.storage_mode == 'light':
# Delete heavy data (it will be computed again if required)
# (it is required only when MSE is wanted in self.predict)
if self.verbose:
print("Light storage mode specified. "
"Flushing autocorrelation matrix...")
self.D = None
self.ij = None
self.F = None
self.C = None
self.Ft = None
self.G = None
return self