crnGP

Methods

`dloglik`(X0, S0, Z, theta, g, rho, stype[, ...])	Gradient of log-likelihood under correlated noise
`get`(key)	General get item (retrives key from self.__dict__)
`loglik`(X0, S0, Z, theta, g, rho, stype[, ...])	Log Likelihood under correlated noise
`mlecrnGP`(X, Z[, T0, rhotype, stype, lower, ...])	Gaussian process modeling with correlated noise.
`predict`(x[, xprime, t0, interval, ...])	Prediction under correlated noise

dloglikT
loglikT
pairwise_rho

dloglik(X0, S0, Z, theta, g, rho, stype, beta0=None, covtype='Gaussian', eps=np.float64(1.4901161193847656e-08), components=('theta', 'g', 'rho'))[source]

Gradient of log-likelihood under correlated noise

Parameters:

X0 (ndarray_like) – matrix of designs (one point per row)
S0 (ndarray_like) – integer vector containing seed information
Z (ndarray_like) – vector of outputs
theta (ndarray_like) – scalar (isotropic) or vector (anisotropic) lengthscale values
g (float) – noise variance
rho (float) – seed correlation
stype (str) – type of kronecker structure of the data
beta0 (float) – trend
covtype (str) – covariance kernel to use
eps (float) – amount of jitter on diagonal of covariance matrix
components (tuple) – directions for partial derivatives, one or several of ‘theta’ (lengthscales) or ‘g’ (for noise) or ‘rho’ (seed correlation strength)

Return type:

gradient with respect to hyperparameters (specified via components)

get(key)[source]: General get item (retrives key from self.__dict__)

loglik(X0, S0, Z, theta, g, rho, stype, beta0=None, covtype='Gaussian', eps=np.float64(1.4901161193847656e-08))[source]

Log Likelihood under correlated noise

Parameters:

X0 (ndarray_like) – unique design matrix (of size nxd)
S0 (ndarray_like) – vector of seed information (size nx1)
Z (ndarray_like) – observation (output) vector (size N)
theta (ndarray_like) – lengthscale
g (float) – noise variance
rho (float) – seed correlation strength
stype (str) – type of kronecker structure of the data
beta0 (float) – trend
covtype (str) – covariance kernel to use
eps (float) – amount of jitter on diagonal of covariance matrix

Returns:

loglik – log-likelihood

Return type:

float

mlecrnGP(X, Z, T0=None, rhotype='simple', stype='none', lower=None, upper=None, known={}, noiseControl={'g_bounds': array([1.49011612e-07, 1.00000000e+02]), 'rho_bounds': None}, init={}, covtype='Gaussian', maxit=100, eps=np.float64(1.4901161193847656e-08), settings={'factr': 10000000.0, 'return_Ki': True})[source]

Gaussian process modeling with correlated noise.

You may also call this function as crnGP.mle

Gaussian process regression under correlated noise based on maximum likelihood estimation of the hyperparameters.

Parameters:

X (ndarray_like) – matrix of all designs, one per row, or list with elements. The last column is assumed to contain the integer seed value.
Z (ndarray_like) – Z vector of all observations.
T0 (ndarray_like:) – T0 optional vector of times (same for all X's). Not currently supported.
lower (ndarray_like) – optional bounds for the theta parameter (see :func: covariance_functions.cov_gen for the exact parameterization). In the multivariate case, it is possible to give vectors for bounds (resp. scalars) for anisotropy (resp. isotropy)
upper (ndarray_like) – optional bounds for the theta parameter (see :func: covariance_functions.cov_gen for the exact parameterization). In the multivariate case, it is possible to give vectors for bounds (resp. scalars) for anisotropy (resp. isotropy)
noiseControl (dict) –
dict with element:
- g_bounds vector providing minimal and maximal noise to signal ratio (default to (sqrt(MACHINE_DOUBLE_EPS), 100)).
settings (dict) –
dict for options about the general modeling procedure, with elements:
- return_Ki boolean to include the inverse covariance matrix in the object for further use (e.g., prediction).
- factr (default to 1e7) and pgtol are available to be passed to options for L-BFGS-B in :func: scipy.optimize.minimize.
eps (float) – jitter used in the inversion of the covariance matrix for numerical stability
known (dict) – optional dict of known parameters (e.g. beta0, theta, g)
init (dict) –
optional lists of starting values for mle optimization:
- theta_init initial value of the theta parameters to be optimized over (default to 10% of the range determined with lower and upper)
- g_init vector of nugget parameter to be optimized over
covtype (str) – covariance kernel type, either 'Gaussian', 'Matern5_2' or 'Matern3_2', see :func: ~covariance_functions.cov_gen
maxit (int) – maximum number of iterations for L-BFGS-B of :func: scipy.optimize.minimize dedicated to maximum likelihood optimization

Notes

The global covariance matrix of the model is parameterized as nu_hat * (Cx + g Id) * Cs = nu_hat * K, with Cx the spatial correlation matrix between unique designs, depending on the family of kernel used (see ~covariance_functions.cov_gen for available choices) and values of lengthscale parameters. Cs is the correlation matrix between seed values, equal to 1 if the seeds are equal, rho otherwise. nu_hat is the plugin estimator of the variance of the process.

It is generally recommended to rescale the inputs to the unit cube and to normalize the outputs.

Returns:

theta: unless given, maximum likelihood estimate (mle) of the lengthscale parameter(s),
nu_hat: plugin estimator of the variance,
g: unless given, mle of the nugget of the noise/log-noise process,
trendtype: either "SK" if beta0 is provided, else "OK",
beta0: constant trend of the mean process, plugin-estimator unless given,
ll: log-likelihood value,
nit_opt, msg: counts and message returned by :func:scipy.optimize.minimize
used_args: list with arguments provided in the call to the function,
- Ki, Kgi: inverse of the covariance matrices of the mean and noise processes (not scaled by nu_hat and nu_hat_var),
- X0, Z0, Z, eps, logN, covtype: values given in input,
time: time to train the model, in seconds.

Return type:

self, with the following attributes

References

A Fadikar, M Binois, N Collier, A Stevens, KB Toh, J Ozik. Trajectory-oriented optimization of stochastic epidemiological models. 2023 Winter Simulation Conference (WSC), San Antonio, TX, USA, 2023, pp. 1244-1255, doi: 10.1109/WSC60868.2023.10408258

Examples

>>> from hetgpy import crnGP
>>> import numpy as np
>>> rng = np.random.default_rng(1)
>>> pps = 10 # points per seed
>>> x = np.linspace(0,2*np.pi,pps).reshape(-1,1)
>>> X = np.vstack([x,x])
>>> seeds = ([1] * pps) + ([5] * pps)
>>> X = np.hstack([X,np.array(seeds).reshape(-1,1)])
>>> # amplitude and phase shift
>>> Z = np.sin(X[:,0] + (np.pi/2)*(X[:,-1]==5)) + X[:,-1]
>>> GP = crnGP()
>>> GP.mle(X,Z,covtype="Matern5_2")

predict(x, xprime=None, t0=None, interval=None, interval_lower=None, interval_upper=None, **kw)[source]

Prediction under correlated noise

Parameters:

x (ndarray_like) – matrix of designs locations to predict at (one point per row)
xprime (ndarray_like) – optional second matrix of predictive locations to obtain the predictive covariance matrix between x and xprime
t0 (ndarray_like) – vector of times (not currently supported)
interval (str) – one of ‘confidence’ or ‘predictive’ which is a convenience method to return confidence/predictive intervals corresponding to interval_lower and interval_upper
interval_lower (float) – lower of confidence/predictive interval
interval_upper (float) – upper of confidence/predictive interval
nugs_only (bool (default False)) – if True, only return noise variance prediction
kwargs (dict) – optional additional elements (only used for nugs_only)

Returns:

mean: kriging mean;
sd2: kriging variance (filtered, e.g. without the nugget values)
nugs: noise variance prediction
cov: (returned if xprime is given) predictive covariance matrix between x and xprime
confidence_interval: prediction with kriging variance only
predictive_interval: prediction with kriging and noise variance

Return type:

dict with elements