Implementation of the Gaussian Process Autoregressive Regression Model. See the paper, and see the docs.
See the instructions here. Then simply
pip install gpar
A simple instance of GPAR can be created as follows:
from gpar import GPARRegressor
gpar = GPARRegressor(
replace=True,
impute=True,
scale=1.0,
linear=True,
linear_scale=100.0,
nonlinear=True,
nonlinear_scale=1.0,
noise=0.1,
normalise_y=True
)
Here the keyword arguments have the following meaning:
replace=True
: Replace data points with the posterior mean of the previous
layer before feeding them into the next layer. This helps the model deal
with noisy data, but may discard important structure in the data if the
fit is bad.
impute=True
: GPAR requires that data is closed downwards. If this is
not the case, the model will be unable to use part of the data. Setting
impute
to True
lets GPAR impute data points to ensure that the data is
closed downwards.
scale=1.0
: Initialisation of the length scale with respect to the inputs.
linear=True
: Use linear dependencies between outputs.
linear_scale=100.0
: Initialisation of the length scale of the linear
dependencies between outputs.
nonlinear=True
: Also use nonlinear dependencies between outputs.
nonlinear_scale=1.0
: Initialisation of the length scale of the nonlinear
dependencies between outputs. Important: this length scale applies
after possible normalisation of the outputs (see below), in which case
nonlinear_scale=1.0
corresponds to a simple, but nonlinear relationship.
noise=0.1
: Initialisation of the variance of the observation noise.
normalise_y=True
: Internally, work with a normalised version of the
outputs by subtracting the mean and dividing by the standard deviation.
Predictions will be transformed back appropriately.
In the above, any scalar hyperparameter may be replaced by a list of values
to initialise each layer separately, e.g. scale=[1.0, 2.0]
. See the
documentation for a full overview of the keywords that may be passed to
GPARRegressor
.
By default, GPAR models dependencies between outputs as follows:
1.
the first output y1
is modelled as a function of only the inputs x
:
y1 = y1(x)
;
2.
the second output y2
is modelled as a function of the previous output y1
and
the inputs x
: y2 = y2(y1, x)
;
3.
the third output y3
is modelled as a function of the previous outputs y2
and
y1
and the inputs x
: y3 = y3(y2, y1, x)
;
To fit GPAR, call gpar.fit(x_train, y_train)
where x_train
are the training
inputs and y_train
the training outputs.
The inputs x_train
must have shape (n,)
or (n, m)
, where n
is the
number of data points and m
the number of input features, and the outputs
y_train
must have shape (n,)
or (n, p)
, where p
is the number of
outputs.
Missing data can simply be set to np.nan
s.
To condition GPAR on data without optimising its hyperparameters, use
gpar.condition(x_train, y_train)
instead.
Finally, to make predictions, call
means = gpar.predict(x_test, num_samples=100)
to get the predictive means, or
means, lowers, uppers = gpar.predict(
x_test,
num_samples=100,
credible_bounds=True
)
to also get lower and upper 95% central marginal credible bounds. If you wish
to predict the underlying latent function instead of the observed values, set
latent=True
in the call to GPARRegressor.predict
.
Using keywords arguments, GPARRegressor
can be configured to specify the
dependencies with respect to the inputs and between the outputs. The following
dependencies can be specified:
Linear input dependencies: Set linear_input=True
and specify the
length scale with linear_input_scale
.
Nonlinear input dependencies: This is enabled by default. The length
scale can be specified using scale
. To tie these length scales across all
layers, set scale_tie=True
.
Locally periodic input dependencies: Set per=True
and specify the period
with per_period
, the length scale with per_scale
, and the length
scale on which the periodicity changes with per_decay
.
Linear output dependencies: Set linear=True
and specify the length
scale with linear_scale
.
Nonlinear output dependencies: Set nonlinear=True
and specify the
length scale with nonlinear_scale
.
All nonlinear kernels are exponentiated quadratic kernels. If you wish to
instead use rational quadratic kernels, set rq=True
.
All parameters can be set to a list of values to initialise the value for each layer separately.
To let every layer depend only the k
th previous layers, set markov=k
.
One may want to apply a transformation to the data before fitting the model,
e.g. $y\mapsto\log(y)$ in the case of positive data. Such a transformation can
be specified by setting the transform_y
keyword argument for GPARRegressor
.
The following transformations are available:
log_transform
: $y \mapsto \log(y)$.
squishing_transform
: $y \mapsto \operatorname{sign}(y) \log(1 + |y|)$.
Sampling from the model can be done with GPARRegressor.sample
. The keyword
argument num_samples
specifies the number of samples, and latent
specifies whether to sample from the underlying latent function or the
observed values. Sampling from the prior and posterior (model must be fit
first) can be done as follows:
sample = gpar.sample(x, p=2) # Sample two outputs from the prior.
sample = gpar.sample(x, posterior=True) # Sample from the posterior.
The logpdf of data can be computed with GPARRegressor.logpdf
. To compute the
logpdf under the posterior, set posterior=True
. To sample missing data to
compute an unbiased estimate of the pdf, not logpdf, set
sample_missing=True
.
Inducing points can be used to scale GPAR to large data sets. Simply set x_ind
to the locations of the inducing points in GPARRegressor
.
examples/paper/synthetic.py
)import matplotlib.pyplot as plt
import numpy as np
from gpar.regression import GPARRegressor
from wbml.experiment import WorkingDirectory
import wbml.plot
if __name__ == "__main__":
wd = WorkingDirectory("_experiments", "synthetic", seed=1)
# Create toy data set.
n = 200
x = np.linspace(0, 1, n)
noise = 0.1
# Draw functions depending on each other in complicated ways.
f1 = -np.sin(10 * np.pi * (x + 1)) / (2 * x + 1) - x ** 4
f2 = np.cos(f1) ** 2 + np.sin(3 * x)
f3 = f2 * f1 ** 2 + 3 * x
f = np.stack((f1, f2, f3), axis=0).T
# Add noise and subsample.
y = f + noise * np.random.randn(n, 3)
x_obs, y_obs = x[::8], y[::8]
# Fit and predict GPAR.
model = GPARRegressor(
scale=0.1,
linear=True,
linear_scale=10.0,
nonlinear=True,
nonlinear_scale=0.1,
noise=0.1,
impute=True,
replace=False,
normalise_y=False,
)
model.fit(x_obs, y_obs)
means, lowers, uppers = model.predict(
x, num_samples=100, credible_bounds=True, latent=True
)
# Fit and predict independent GPs: set `markov=0` in GPAR.
igp = GPARRegressor(
scale=0.1,
linear=True,
linear_scale=10.0,
nonlinear=True,
nonlinear_scale=0.1,
noise=0.1,
markov=0,
normalise_y=False,
)
igp.fit(x_obs, y_obs)
igp_means, igp_lowers, igp_uppers = igp.predict(
x, num_samples=100, credible_bounds=True, latent=True
)
# Plot the result.
plt.figure(figsize=(15, 3))
for i in range(3):
plt.subplot(1, 3, i + 1)
# Plot observations.
plt.scatter(x_obs, y_obs[:, i], label="Observations", style="train")
plt.plot(x, f[:, i], label="Truth", style="test")
# Plot GPAR.
plt.plot(x, means[:, i], label="GPAR", style="pred")
plt.fill_between(x, lowers[:, i], uppers[:, i], style="pred")
# Plot independent GPs.
plt.plot(x, igp_means[:, i], label="IGP", style="pred2")
plt.fill_between(x, igp_lowers[:, i], igp_uppers[:, i], style="pred2")
plt.xlabel("$t$")
plt.ylabel(f"$y_{i + 1}$")
wbml.plot.tweak(legend=i == 2)
plt.tight_layout()
plt.savefig(wd.file("synthetic.pdf"))