Latent component Gaussian process
Project description
Latent component Gaussian process (LCGP)
Implementation of latent component Gaussian process (LCGP). LCGP handles the emulation of multivariate stochastic simulation outputs.
List of Contents:
Installation
The implementation of LCGP can be installed through
pip install lcgp
Note on LBFGS optimizer:
It is strongly recommended that PyTorch-LBFGS is installed to fully utilize this implementation. Installation guide on PyTorch-LBFGS can be found on its repository. Note that PyTorch-LBFGS has an additional requirement
matplotlib
. The source code of a version of PyTorch-LBFGS that does not requirematplotlib
is included in reference_code.
Basic usage
What most of us need:
import numpy as np
from lcgp import LCGP
from lcgp import evaluation # optional evaluation module
# Generate fifty 2-dimensional input and 4-dimensional output
x = np.random.randn(50, 2)
y = np.random.randn(4, 50)
# Define LCGP model
model = LCGP(y=y, x=x)
# Estimate error covariance and hyperparameters
model.fit()
# Prediction
p = model.predict(x0=x) # mean and variance
rmse = evaluation.rmse(y, p[0].numpy())
dss = evaluation.dss(y, p[0].numpy(), p[1].numpy(), use_diag=True)
print('Root mean squared error: {:.3E}'.format(rmse))
print('Dawid-Sebastiani score: {:.3f}'.format(dss))
# Access parameters
print(model)
Specifying number of latent components
There are two ways to specify the number of latent components by passing one of the following arguments in initializing an LCGP instance:
q = 5
: Five latent components will be used.q
must be less than or equal to the output dimension.var_threshold = 0.99
: Include $q$ latent components such that 99% of the output variance are explained, using a singular value decomposition.
Note: Only one of the options should be provided at a time.
model_q = LCGP(y=y, x=x, q=5)
model_var = LCGP(y=y, x=x, var_threshold=0.99)
Specifying diagonal error groupings
If errors of multiple output dimensions are expected to be similar, the error variances can be grouped in estimation.
For example, the 6-dimensional output is split into two groups: the first two have low errors and the remaining four have high errors.
import numpy as np
x = np.linspace(0, 1, 100)
y = np.row_stack((
np.sin(x), np.cos(x), np.tan(x),
np.sin(x/2), np.cos(x/2), np.tan(x/2)
))
y[:2] += np.random.normal(2, 1e-3, size=(2, 100))
y[2:] += np.random.normal(-2, 1e-1, size=(4, 100))
Then, LCGP can be defined with the argument diag_error_structure
as a list
of output dimensions to group. The following code groups the first 2 and the remaining
4 output dimensions.
model_diag = LCGP(y=y, x=x, diag_error_structure=[2, 4])
By default, LCGP assigns a separate error variance to each dimension, equivalent to
model_diag = LCGP(y=y, x=x, diag_error_structure=[1]*6)
Define LCGP using different submethod
Three submethods are implemented under LCGP:
- Full posterior (
full
) - ELBO (
elbo
) - Profile likelihood (
proflik
)
Under circumstances where the simulation outputs are stochastic, the full posterior approach should perform the best. If the simulation outputs are deterministic, the profile likelihood method should suffice.
LCGP_models = []
submethods = ['full', 'elbo', 'proflik']
for submethod in submethods:
model = LCGP(y=y, x=x, submethod=submethod)
LCGP_models.append(model)
Standardization choices
LCGP standardizes the simulation output by each dimension to facilitate hyperparameter
training. The two choices are implemented through robust_mean = True
or
robust_mean = False
.
robust_mean = False
: The empirical mean and standard deviation are used.robust_mean = True
: The empirical median and median absolute error are used.
model = LCGP(y=y, x=x, robust_mean=False)
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