.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/gaussian_process/plot_compare_gpr_krr.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note Click :ref:here  to download the full example code or to run this example in your browser via Binder .. 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 # Guillaume Lemaitre # 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 127-139 .. 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 140-162 .. 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 163-165 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 165-167 .. 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 168-177 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 179-197 .. code-block:: default from sklearn.model_selection import RandomizedSearchCV from sklearn.utils.fixes import loguniform 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: 2.530 seconds .. GENERATED FROM PYTHON SOURCE LINES 198-201 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 201-203 .. code-block:: default kernel_ridge_tuned.best_params_ .. rst-class:: sphx-glr-script-out .. code-block:: none {'alpha': 1.9915849773450223, 'kernel__length_scale': 0.7986499491396728, 'kernel__periodicity': 6.607275806426108} .. GENERATED FROM PYTHON SOURCE LINES 204-207 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 207-211 .. 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 212-234 .. 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 235-249 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 249-262 .. 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: 0.027 seconds .. GENERATED FROM PYTHON SOURCE LINES 263-266 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 266-268 .. 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 269-273 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 273-282 .. 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: 0.001 seconds .. GENERATED FROM PYTHON SOURCE LINES 283-318 .. 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 319-342 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. .. GENERATED FROM PYTHON SOURCE LINES 342-354 .. 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 355-390 .. 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 391-396 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 **Total running time of the script:** ( 0 minutes 3.078 seconds) .. _sphx_glr_download_auto_examples_gaussian_process_plot_compare_gpr_krr.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: binder-badge .. image:: images/binder_badge_logo.svg :target: https://mybinder.org/v2/gh/scikit-learn/scikit-learn/1.1.X?urlpath=lab/tree/notebooks/auto_examples/gaussian_process/plot_compare_gpr_krr.ipynb :alt: Launch binder :width: 150 px .. container:: sphx-glr-download sphx-glr-download-python :download:Download Python source code: plot_compare_gpr_krr.py  .. container:: sphx-glr-download sphx-glr-download-jupyter :download:Download Jupyter notebook: plot_compare_gpr_krr.ipynb  .. only:: html .. rst-class:: sphx-glr-signature Gallery generated by Sphinx-Gallery _