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.. only:: html
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.. rst-class:: sphx-glr-example-title
.. _sphx_glr_auto_examples_gaussian_process_plot_compare_gpr_krr.py:
==========================================================
Comparison of kernel ridge and Gaussian process regression
==========================================================
This example illustrates differences between a kernel ridge regression and a
Gaussian process regression.
Both kernel ridge regression and Gaussian process regression are using a
so-called "kernel trick" to make their models expressive enough to fit
the training data. However, the machine learning problems solved by the two
methods are drastically different.
Kernel ridge regression will find the target function that minimizes a loss
function (the mean squared error).
Instead of finding a single target function, the Gaussian process regression
employs a probabilistic approach : a Gaussian posterior distribution over
target functions is defined based on the Bayes' theorem, Thus prior
probabilities on target functions are being combined with a likelihood function
defined by the observed training data to provide estimates of the posterior
distributions.
We will illustrate these differences with an example and we will also focus on
tuning the kernel hyperparameters.
.. GENERATED FROM PYTHON SOURCE LINES 27-32
.. code-block:: default
# Authors: Jan Hendrik Metzen <jhm@informatik.uni-bremen.de>
# Guillaume Lemaitre <g.lemaitre58@gmail.com>
# License: BSD 3 clause
.. GENERATED FROM PYTHON SOURCE LINES 33-39
Generating a dataset
--------------------
We create a synthetic dataset. The true generative process will take a 1-D
vector and compute its sine. Note that the period of this sine is thus
:math:`2 \pi`. We will reuse this information later in this example.
.. GENERATED FROM PYTHON SOURCE LINES 39-45
.. code-block:: default
import numpy as np
rng = np.random.RandomState(0)
data = np.linspace(0, 30, num=1_000).reshape(-1, 1)
target = np.sin(data).ravel()
.. GENERATED FROM PYTHON SOURCE LINES 46-51
Now, we can imagine a scenario where we get observations from this true
process. However, we will add some challenges:
- the measurements will be noisy;
- only samples from the beginning of the signal will be available.
.. GENERATED FROM PYTHON SOURCE LINES 51-57
.. code-block:: default
training_sample_indices = rng.choice(np.arange(0, 400), size=40, replace=False)
training_data = data[training_sample_indices]
training_noisy_target = target[training_sample_indices] + 0.5 * rng.randn(
len(training_sample_indices)
)
.. GENERATED FROM PYTHON SOURCE LINES 58-59
Let's plot the true signal and the noisy measurements available for training.
.. GENERATED FROM PYTHON SOURCE LINES 59-76
.. code-block:: default
import matplotlib.pyplot as plt
plt.plot(data, target, label="True signal", linewidth=2)
plt.scatter(
training_data,
training_noisy_target,
color="black",
label="Noisy measurements",
)
plt.legend()
plt.xlabel("data")
plt.ylabel("target")
_ = plt.title(
"Illustration of the true generative process and \n"
"noisy measurements available during training"
)
.. image-sg:: /auto_examples/gaussian_process/images/sphx_glr_plot_compare_gpr_krr_001.png
:alt: Illustration of the true generative process and noisy measurements available during training
:srcset: /auto_examples/gaussian_process/images/sphx_glr_plot_compare_gpr_krr_001.png
:class: sphx-glr-single-img
.. GENERATED FROM PYTHON SOURCE LINES 77-83
Limitations of a simple linear model
------------------------------------
First, we would like to highlight the limitations of a linear model given
our dataset. We fit a :class:`~sklearn.linear_model.Ridge` and check the
predictions of this model on our dataset.
.. GENERATED FROM PYTHON SOURCE LINES 83-100
.. code-block:: default
from sklearn.linear_model import Ridge
ridge = Ridge().fit(training_data, training_noisy_target)
plt.plot(data, target, label="True signal", linewidth=2)
plt.scatter(
training_data,
training_noisy_target,
color="black",
label="Noisy measurements",
)
plt.plot(data, ridge.predict(data), label="Ridge regression")
plt.legend()
plt.xlabel("data")
plt.ylabel("target")
_ = plt.title("Limitation of a linear model such as ridge")
.. image-sg:: /auto_examples/gaussian_process/images/sphx_glr_plot_compare_gpr_krr_002.png
:alt: Limitation of a linear model such as ridge
:srcset: /auto_examples/gaussian_process/images/sphx_glr_plot_compare_gpr_krr_002.png
:class: sphx-glr-single-img
.. GENERATED FROM PYTHON SOURCE LINES 101-127
Such a ridge regressor underfits data since it is not expressive enough.
Kernel methods: kernel ridge and Gaussian process
-------------------------------------------------
Kernel ridge
............
We can make the previous linear model more expressive by using a so-called
kernel. A kernel is an embedding from the original feature space to another
one. Simply put, it is used to map our original data into a newer and more
complex feature space. This new space is explicitly defined by the choice of
kernel.
In our case, we know that the true generative process is a periodic function.
We can use a :class:`~sklearn.gaussian_process.kernels.ExpSineSquared` kernel
which allows recovering the periodicity. The class
:class:`~sklearn.kernel_ridge.KernelRidge` will accept such a kernel.
Using this model together with a kernel is equivalent to embed the data
using the mapping function of the kernel and then apply a ridge regression.
In practice, the data are not mapped explicitly; instead the dot product
between samples in the higher dimensional feature space is computed using the
"kernel trick".
Thus, let's use such a :class:`~sklearn.kernel_ridge.KernelRidge`.
.. GENERATED FROM PYTHON SOURCE LINES 127-140
.. code-block:: default
import time
from sklearn.gaussian_process.kernels import ExpSineSquared
from sklearn.kernel_ridge import KernelRidge
kernel_ridge = KernelRidge(kernel=ExpSineSquared())
start_time = time.time()
kernel_ridge.fit(training_data, training_noisy_target)
print(
f"Fitting KernelRidge with default kernel: {time.time() - start_time:.3f} seconds"
)
.. rst-class:: sphx-glr-script-out
.. code-block:: none
Fitting KernelRidge with default kernel: 0.001 seconds
.. GENERATED FROM PYTHON SOURCE LINES 141-163
.. code-block:: default
plt.plot(data, target, label="True signal", linewidth=2, linestyle="dashed")
plt.scatter(
training_data,
training_noisy_target,
color="black",
label="Noisy measurements",
)
plt.plot(
data,
kernel_ridge.predict(data),
label="Kernel ridge",
linewidth=2,
linestyle="dashdot",
)
plt.legend(loc="lower right")
plt.xlabel("data")
plt.ylabel("target")
_ = plt.title(
"Kernel ridge regression with an exponential sine squared\n "
"kernel using default hyperparameters"
)
.. image-sg:: /auto_examples/gaussian_process/images/sphx_glr_plot_compare_gpr_krr_003.png
:alt: Kernel ridge regression with an exponential sine squared kernel using default hyperparameters
:srcset: /auto_examples/gaussian_process/images/sphx_glr_plot_compare_gpr_krr_003.png
:class: sphx-glr-single-img
.. GENERATED FROM PYTHON SOURCE LINES 164-166
This fitted model is not accurate. Indeed, we did not set the parameters of
the kernel and instead used the default ones. We can inspect them.
.. GENERATED FROM PYTHON SOURCE LINES 166-168
.. code-block:: default
kernel_ridge.kernel
.. rst-class:: sphx-glr-script-out
.. code-block:: none
ExpSineSquared(length_scale=1, periodicity=1)
.. GENERATED FROM PYTHON SOURCE LINES 169-178
Our kernel has two parameters: the length-scale and the periodicity. For our
dataset, we use `sin` as the generative process, implying a
:math:`2 \pi`-periodicity for the signal. The default value of the parameter
being :math:`1`, it explains the high frequency observed in the predictions of
our model.
Similar conclusions could be drawn with the length-scale parameter. Thus, it
tell us that the kernel parameters need to be tuned. We will use a randomized
search to tune the different parameters the kernel ridge model: the `alpha`
parameter and the kernel parameters.
.. GENERATED FROM PYTHON SOURCE LINES 180-199
.. code-block:: default
from scipy.stats import loguniform
from sklearn.model_selection import RandomizedSearchCV
param_distributions = {
"alpha": loguniform(1e0, 1e3),
"kernel__length_scale": loguniform(1e-2, 1e2),
"kernel__periodicity": loguniform(1e0, 1e1),
}
kernel_ridge_tuned = RandomizedSearchCV(
kernel_ridge,
param_distributions=param_distributions,
n_iter=500,
random_state=0,
)
start_time = time.time()
kernel_ridge_tuned.fit(training_data, training_noisy_target)
print(f"Time for KernelRidge fitting: {time.time() - start_time:.3f} seconds")
.. rst-class:: sphx-glr-script-out
.. code-block:: none
Time for KernelRidge fitting: 3.733 seconds
.. GENERATED FROM PYTHON SOURCE LINES 200-203
Fitting the model is now more computationally expensive since we have to try
several combinations of hyperparameters. We can have a look at the
hyperparameters found to get some intuitions.
.. GENERATED FROM PYTHON SOURCE LINES 203-205
.. code-block:: default
kernel_ridge_tuned.best_params_
.. rst-class:: sphx-glr-script-out
.. code-block:: none
{'alpha': 1.9915849773450223, 'kernel__length_scale': 0.7986499491396727, 'kernel__periodicity': 6.607275806426107}
.. GENERATED FROM PYTHON SOURCE LINES 206-209
Looking at the best parameters, we see that they are different from the
defaults. We also see that the periodicity is closer to the expected value:
:math:`2 \pi`. We can now inspect the predictions of our tuned kernel ridge.
.. GENERATED FROM PYTHON SOURCE LINES 209-213
.. code-block:: default
start_time = time.time()
predictions_kr = kernel_ridge_tuned.predict(data)
print(f"Time for KernelRidge predict: {time.time() - start_time:.3f} seconds")
.. rst-class:: sphx-glr-script-out
.. code-block:: none
Time for KernelRidge predict: 0.001 seconds
.. GENERATED FROM PYTHON SOURCE LINES 214-236
.. code-block:: default
plt.plot(data, target, label="True signal", linewidth=2, linestyle="dashed")
plt.scatter(
training_data,
training_noisy_target,
color="black",
label="Noisy measurements",
)
plt.plot(
data,
predictions_kr,
label="Kernel ridge",
linewidth=2,
linestyle="dashdot",
)
plt.legend(loc="lower right")
plt.xlabel("data")
plt.ylabel("target")
_ = plt.title(
"Kernel ridge regression with an exponential sine squared\n "
"kernel using tuned hyperparameters"
)
.. image-sg:: /auto_examples/gaussian_process/images/sphx_glr_plot_compare_gpr_krr_004.png
:alt: Kernel ridge regression with an exponential sine squared kernel using tuned hyperparameters
:srcset: /auto_examples/gaussian_process/images/sphx_glr_plot_compare_gpr_krr_004.png
:class: sphx-glr-single-img
.. GENERATED FROM PYTHON SOURCE LINES 237-251
We get a much more accurate model. We still observe some errors mainly due to
the noise added to the dataset.
Gaussian process regression
...........................
Now, we will use a
:class:`~sklearn.gaussian_process.GaussianProcessRegressor` to fit the same
dataset. When training a Gaussian process, the hyperparameters of the kernel
are optimized during the fitting process. There is no need for an external
hyperparameter search. Here, we create a slightly more complex kernel than
for the kernel ridge regressor: we add a
:class:`~sklearn.gaussian_process.kernels.WhiteKernel` that is used to
estimate the noise in the dataset.
.. GENERATED FROM PYTHON SOURCE LINES 251-264
.. code-block:: default
from sklearn.gaussian_process import GaussianProcessRegressor
from sklearn.gaussian_process.kernels import WhiteKernel
kernel = 1.0 * ExpSineSquared(1.0, 5.0, periodicity_bounds=(1e-2, 1e1)) + WhiteKernel(
1e-1
)
gaussian_process = GaussianProcessRegressor(kernel=kernel)
start_time = time.time()
gaussian_process.fit(training_data, training_noisy_target)
print(
f"Time for GaussianProcessRegressor fitting: {time.time() - start_time:.3f} seconds"
)
.. rst-class:: sphx-glr-script-out
.. code-block:: none
Time for GaussianProcessRegressor fitting: 0.051 seconds
.. GENERATED FROM PYTHON SOURCE LINES 265-268
The computation cost of training a Gaussian process is much less than the
kernel ridge that uses a randomized search. We can check the parameters of
the kernels that we computed.
.. GENERATED FROM PYTHON SOURCE LINES 268-270
.. code-block:: default
gaussian_process.kernel_
.. rst-class:: sphx-glr-script-out
.. code-block:: none
0.675**2 * ExpSineSquared(length_scale=1.34, periodicity=6.57) + WhiteKernel(noise_level=0.182)
.. GENERATED FROM PYTHON SOURCE LINES 271-275
Indeed, we see that the parameters have been optimized. Looking at the
`periodicity` parameter, we see that we found a period close to the
theoretical value :math:`2 \pi`. We can have a look now at the predictions of
our model.
.. GENERATED FROM PYTHON SOURCE LINES 275-284
.. code-block:: default
start_time = time.time()
mean_predictions_gpr, std_predictions_gpr = gaussian_process.predict(
data,
return_std=True,
)
print(
f"Time for GaussianProcessRegressor predict: {time.time() - start_time:.3f} seconds"
)
.. rst-class:: sphx-glr-script-out
.. code-block:: none
Time for GaussianProcessRegressor predict: 0.002 seconds
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.. code-block:: default
plt.plot(data, target, label="True signal", linewidth=2, linestyle="dashed")
plt.scatter(
training_data,
training_noisy_target,
color="black",
label="Noisy measurements",
)
# Plot the predictions of the kernel ridge
plt.plot(
data,
predictions_kr,
label="Kernel ridge",
linewidth=2,
linestyle="dashdot",
)
# Plot the predictions of the gaussian process regressor
plt.plot(
data,
mean_predictions_gpr,
label="Gaussian process regressor",
linewidth=2,
linestyle="dotted",
)
plt.fill_between(
data.ravel(),
mean_predictions_gpr - std_predictions_gpr,
mean_predictions_gpr + std_predictions_gpr,
color="tab:green",
alpha=0.2,
)
plt.legend(loc="lower right")
plt.xlabel("data")
plt.ylabel("target")
_ = plt.title("Comparison between kernel ridge and gaussian process regressor")
.. image-sg:: /auto_examples/gaussian_process/images/sphx_glr_plot_compare_gpr_krr_005.png
:alt: Comparison between kernel ridge and gaussian process regressor
:srcset: /auto_examples/gaussian_process/images/sphx_glr_plot_compare_gpr_krr_005.png
:class: sphx-glr-single-img
.. GENERATED FROM PYTHON SOURCE LINES 321-344
We observe that the results of the kernel ridge and the Gaussian process
regressor are close. However, the Gaussian process regressor also provide
an uncertainty information that is not available with a kernel ridge.
Due to the probabilistic formulation of the target functions, the
Gaussian process can output the standard deviation (or the covariance)
together with the mean predictions of the target functions.
However, it comes at a cost: the time to compute the predictions is higher
with a Gaussian process.
Final conclusion
----------------
We can give a final word regarding the possibility of the two models to
extrapolate. Indeed, we only provided the beginning of the signal as a
training set. Using a periodic kernel forces our model to repeat the pattern
found on the training set. Using this kernel information together with the
capacity of the both models to extrapolate, we observe that the models will
continue to predict the sine pattern.
Gaussian process allows to combine kernels together. Thus, we could associate
the exponential sine squared kernel together with a radial basis function
kernel.
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.. code-block:: default
from sklearn.gaussian_process.kernels import RBF
kernel = 1.0 * ExpSineSquared(1.0, 5.0, periodicity_bounds=(1e-2, 1e1)) * RBF(
length_scale=15, length_scale_bounds="fixed"
) + WhiteKernel(1e-1)
gaussian_process = GaussianProcessRegressor(kernel=kernel)
gaussian_process.fit(training_data, training_noisy_target)
mean_predictions_gpr, std_predictions_gpr = gaussian_process.predict(
data,
return_std=True,
)
.. GENERATED FROM PYTHON SOURCE LINES 357-392
.. code-block:: default
plt.plot(data, target, label="True signal", linewidth=2, linestyle="dashed")
plt.scatter(
training_data,
training_noisy_target,
color="black",
label="Noisy measurements",
)
# Plot the predictions of the kernel ridge
plt.plot(
data,
predictions_kr,
label="Kernel ridge",
linewidth=2,
linestyle="dashdot",
)
# Plot the predictions of the gaussian process regressor
plt.plot(
data,
mean_predictions_gpr,
label="Gaussian process regressor",
linewidth=2,
linestyle="dotted",
)
plt.fill_between(
data.ravel(),
mean_predictions_gpr - std_predictions_gpr,
mean_predictions_gpr + std_predictions_gpr,
color="tab:green",
alpha=0.2,
)
plt.legend(loc="lower right")
plt.xlabel("data")
plt.ylabel("target")
_ = plt.title("Effect of using a radial basis function kernel")
.. image-sg:: /auto_examples/gaussian_process/images/sphx_glr_plot_compare_gpr_krr_006.png
:alt: Effect of using a radial basis function kernel
:srcset: /auto_examples/gaussian_process/images/sphx_glr_plot_compare_gpr_krr_006.png
:class: sphx-glr-single-img
.. GENERATED FROM PYTHON SOURCE LINES 393-398
The effect of using a radial basis function kernel will attenuate the
periodicity effect once that no sample are available in the training.
As testing samples get further away from the training ones, predictions
are converging towards their mean and their standard deviation
also increases.
.. rst-class:: sphx-glr-timing
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