Note

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Advanced Plotting With Partial Dependence

The PartialDependenceDisplay object can be used for plotting without needing to recalculate the partial dependence. In this example, we show how to plot partial dependence plots and how to quickly customize the plot with the visualization API.

Note

# Authors: The scikit-learn developers
# SPDX-License-Identifier: BSD-3-Clause

import matplotlib.pyplot as plt
import pandas as pd

from sklearn.datasets import load_diabetes
from sklearn.inspection import PartialDependenceDisplay
from sklearn.neural_network import MLPRegressor
from sklearn.pipeline import make_pipeline
from sklearn.preprocessing import StandardScaler
from sklearn.tree import DecisionTreeRegressor

Train models on the diabetes dataset

First, we train a decision tree and a multi-layer perceptron on the diabetes dataset.

diabetes = load_diabetes()
X = pd.DataFrame(diabetes.data, columns=diabetes.feature_names)
y = diabetes.target

tree = DecisionTreeRegressor()
mlp = make_pipeline(
    StandardScaler(),
    MLPRegressor(hidden_layer_sizes=(100, 100), tol=1e-2, max_iter=500, random_state=0),
)
tree.fit(X, y)
mlp.fit(X, y)

Pipeline(steps=[('standardscaler', StandardScaler()),
                ('mlpregressor',
                 MLPRegressor(hidden_layer_sizes=(100, 100), max_iter=500,
                              random_state=0, tol=0.01))])

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Pipeline

?Documentation for PipelineiFitted

Parameters

	steps steps: list of tuples List of (name of step, estimator) tuples that are to be chained in sequential order. To be compatible with the scikit-learn API, all steps must define `fit`. All non-last steps must also define `transform`. See :ref:`Combining Estimators <combining_estimators>` for more details.	[('standardscaler', ...), ('mlpregressor', ...)]
	transform_input transform_input: list of str, default=None The names of the :term:`metadata` parameters that should be transformed by the pipeline before passing it to the step consuming it. This enables transforming some input arguments to ``fit`` (other than ``X``) to be transformed by the steps of the pipeline up to the step which requires them. Requirement is defined via :ref:`metadata routing <metadata_routing>`. For instance, this can be used to pass a validation set through the pipeline. You can only set this if metadata routing is enabled, which you can enable using ``sklearn.set_config(enable_metadata_routing=True)``. .. versionadded:: 1.6	None
	memory memory: str or object with the joblib.Memory interface, default=None Used to cache the fitted transformers of the pipeline. The last step will never be cached, even if it is a transformer. By default, no caching is performed. If a string is given, it is the path to the caching directory. Enabling caching triggers a clone of the transformers before fitting. Therefore, the transformer instance given to the pipeline cannot be inspected directly. Use the attribute ``named_steps`` or ``steps`` to inspect estimators within the pipeline. Caching the transformers is advantageous when fitting is time consuming. See :ref:`sphx_glr_auto_examples_neighbors_plot_caching_nearest_neighbors.py` for an example on how to enable caching.	None
	verbose verbose: bool, default=False If True, the time elapsed while fitting each step will be printed as it is completed.	False

Fitted attributes

Name	Type	Value
feature_names_in_ feature_names_in_: ndarray of shape (`n_features_in_`,) Names of features seen during :term:`fit`. Only defined if the underlying estimator exposes such an attribute when fit. .. versionadded:: 1.0	ndarray[object](10,)	['age','sex','bmi',...,'s4','s5','s6']
n_features_in_ n_features_in_: int Number of features seen during :term:`fit`. Only defined if the underlying first estimator in `steps` exposes such an attribute when fit. .. versionadded:: 0.24	int	10

StandardScaler

?Documentation for StandardScaler

Parameters

	copy copy: bool, default=True If False, try to avoid a copy and do inplace scaling instead. This is not guaranteed to always work inplace; e.g. if the data is not a NumPy array or scipy.sparse CSR matrix, a copy may still be returned.	True
	with_mean with_mean: bool, default=True If True, center the data before scaling. This does not work (and will raise an exception) when attempted on sparse matrices, because centering them entails building a dense matrix which in common use cases is likely to be too large to fit in memory.	True
	with_std with_std: bool, default=True If True, scale the data to unit variance (or equivalently, unit standard deviation).	True

Fitted attributes

Name	Type	Value
feature_names_in_ feature_names_in_: ndarray of shape (`n_features_in_`,) Names of features seen during :term:`fit`. Defined only when `X` has feature names that are all strings. .. versionadded:: 1.0	ndarray[object](10,)	['age','sex','bmi',...,'s4','s5','s6']
mean_ mean_: ndarray of shape (n_features,) or None The mean value for each feature in the training set. Equal to ``None`` when ``with_mean=False`` and ``with_std=False``.	ndarray[float64](10,)	[-0., 0.,-0.,...,-0., 0., 0.]
n_features_in_ n_features_in_: int Number of features seen during :term:`fit`. .. versionadded:: 0.24	int	10
n_samples_seen_ n_samples_seen_: int or ndarray of shape (n_features,) The number of samples processed by the estimator for each feature. If there are no missing samples, the ``n_samples_seen`` will be an integer, otherwise it will be an array of dtype int. If `sample_weights` are used it will be a float (if no missing data) or an array of dtype float that sums the weights seen so far. Will be reset on new calls to fit, but increments across ``partial_fit`` calls.	float64	442
scale_ scale_: ndarray of shape (n_features,) or None Per feature relative scaling of the data to achieve zero mean and unit variance. Generally this is calculated using `np.sqrt(var_)`. If a variance is zero, we can't achieve unit variance, and the data is left as-is, giving a scaling factor of 1. `scale_` is equal to `None` when `with_std=False`. .. versionadded:: 0.17 scale_	ndarray[float64](10,)	[0.05,0.05,0.05,...,0.05,0.05,0.05]
var_ var_: ndarray of shape (n_features,) or None The variance for each feature in the training set. Used to compute `scale_`. Equal to ``None`` when ``with_mean=False`` and ``with_std=False``.	ndarray[float64](10,)	[0.,0.,0.,...,0.,0.,0.]

10 features

age

sex

bmi

MLPRegressor

?Documentation for MLPRegressor

Parameters

	hidden_layer_sizes hidden_layer_sizes: array-like of shape(n_layers - 2,), default=(100,) The ith element represents the number of neurons in the ith hidden layer.	(100, ...)
	max_iter max_iter: int, default=200 Maximum number of iterations. The solver iterates until convergence (determined by 'tol') or this number of iterations. For stochastic solvers ('sgd', 'adam'), note that this determines the number of epochs (how many times each data point will be used), not the number of gradient steps.	500
	random_state random_state: int, RandomState instance, default=None Determines random number generation for weights and bias initialization, train-test split if early stopping is used, and batch sampling when solver='sgd' or 'adam'. Pass an int for reproducible results across multiple function calls. See :term:`Glossary <random_state>`.	0
	tol tol: float, default=1e-4 Tolerance for the optimization. When the loss or score is not improving by at least ``tol`` for ``n_iter_no_change`` consecutive iterations, unless ``learning_rate`` is set to 'adaptive', convergence is considered to be reached and training stops.	0.01
	loss loss: {'squared_error', 'poisson'}, default='squared_error' The loss function to use when training the weights. Note that the "squared error" and "poisson" losses actually implement "half squares error" and "half poisson deviance" to simplify the computation of the gradient. Furthermore, the "poisson" loss internally uses a log-link (exponential as the output activation function) and requires ``y >= 0``. .. versionchanged:: 1.7 Added parameter `loss` and option 'poisson'.	'squared_error'
	activation activation: {'identity', 'logistic', 'tanh', 'relu'}, default='relu' Activation function for the hidden layer. - 'identity', no-op activation, useful to implement linear bottleneck, returns f(x) = x - 'logistic', the logistic sigmoid function, returns f(x) = 1 / (1 + exp(-x)). - 'tanh', the hyperbolic tan function, returns f(x) = tanh(x). - 'relu', the rectified linear unit function, returns f(x) = max(0, x)	'relu'
	solver solver: {'lbfgs', 'sgd', 'adam'}, default='adam' The solver for weight optimization. - 'lbfgs' is an optimizer in the family of quasi-Newton methods. - 'sgd' refers to stochastic gradient descent. - 'adam' refers to a stochastic gradient-based optimizer proposed by Kingma, Diederik, and Jimmy Ba For a comparison between Adam optimizer and SGD, see :ref:`sphx_glr_auto_examples_neural_networks_plot_mlp_training_curves.py`. Note: The default solver 'adam' works pretty well on relatively large datasets (with thousands of training samples or more) in terms of both training time and validation score. For small datasets, however, 'lbfgs' can converge faster and perform better.	'adam'
	alpha alpha: float, default=0.0001 Strength of the L2 regularization term. The L2 regularization term is divided by the sample size when added to the loss.	0.0001
	batch_size batch_size: int, default='auto' Size of minibatches for stochastic optimizers. If the solver is 'lbfgs', the regressor will not use minibatch. When set to "auto", `batch_size=min(200, n_samples)`.	'auto'
	learning_rate learning_rate: {'constant', 'invscaling', 'adaptive'}, default='constant' Learning rate schedule for weight updates. - 'constant' is a constant learning rate given by 'learning_rate_init'. - 'invscaling' gradually decreases the learning rate ``learning_rate_`` at each time step 't' using an inverse scaling exponent of 'power_t'. effective_learning_rate = learning_rate_init / pow(t, power_t) - 'adaptive' keeps the learning rate constant to 'learning_rate_init' as long as training loss keeps decreasing. Each time two consecutive epochs fail to decrease training loss by at least tol, or fail to increase validation score by at least tol if 'early_stopping' is on, the current learning rate is divided by 5. Only used when solver='sgd'.	'constant'
	learning_rate_init learning_rate_init: float, default=0.001 The initial learning rate used. It controls the step-size in updating the weights. Only used when solver='sgd' or 'adam'.	0.001
	power_t power_t: float, default=0.5 The exponent for inverse scaling learning rate. It is used in updating effective learning rate when the learning_rate is set to 'invscaling'. Only used when solver='sgd'.	0.5
	shuffle shuffle: bool, default=True Whether to shuffle samples in each iteration. Only used when solver='sgd' or 'adam'.	True
	verbose verbose: bool, default=False Whether to print progress messages to stdout.	False
	warm_start warm_start: bool, default=False When set to True, reuse the solution of the previous call to fit as initialization, otherwise, just erase the previous solution. See :term:`the Glossary <warm_start>`.	False
	momentum momentum: float, default=0.9 Momentum for gradient descent update. Should be between 0 and 1. Only used when solver='sgd'.	0.9
	nesterovs_momentum nesterovs_momentum: bool, default=True Whether to use Nesterov's momentum. Only used when solver='sgd' and momentum > 0.	True
	early_stopping early_stopping: bool, default=False Whether to use early stopping to terminate training when validation score is not improving. If set to True, it will automatically set aside ``validation_fraction`` of training data as validation and terminate training when validation score is not improving by at least ``tol`` for ``n_iter_no_change`` consecutive epochs. Only effective when solver='sgd' or 'adam'.	False
	validation_fraction validation_fraction: float, default=0.1 The proportion of training data to set aside as validation set for early stopping. Must be between 0 and 1. Only used if early_stopping is True.	0.1
	beta_1 beta_1: float, default=0.9 Exponential decay rate for estimates of first moment vector in adam, should be in [0, 1). Only used when solver='adam'.	0.9
	beta_2 beta_2: float, default=0.999 Exponential decay rate for estimates of second moment vector in adam, should be in [0, 1). Only used when solver='adam'.	0.999
	epsilon epsilon: float, default=1e-8 Value for numerical stability in adam. Only used when solver='adam'.	1e-08
	n_iter_no_change n_iter_no_change: int, default=10 Maximum number of epochs to not meet ``tol`` improvement. Only effective when solver='sgd' or 'adam'. .. versionadded:: 0.20	10
	max_fun max_fun: int, default=15000 Only used when solver='lbfgs'. Maximum number of function calls. The solver iterates until convergence (determined by ``tol``), number of iterations reaches max_iter, or this number of function calls. Note that number of function calls will be greater than or equal to the number of iterations for the MLPRegressor. .. versionadded:: 0.22	15000

Fitted attributes

Name	Type	Value
best_loss_ best_loss_: float The minimum loss reached by the solver throughout fitting. If `early_stopping=True`, this attribute is set to `None`. Refer to the `best_validation_score_` fitted attribute instead. Only accessible when solver='sgd' or 'adam'.	float64	1110
best_validation_score_ best_validation_score_: float or None The best validation score (i.e. R2 score) that triggered the early stopping. Only available if `early_stopping=True`, otherwise the attribute is set to `None`. Only accessible when solver='sgd' or 'adam'.	NoneType	None
coefs_ coefs_: list of shape (n_layers - 1,) The ith element in the list represents the weight matrix corresponding to layer i.	list	[array([[ 2.26...1628603e-01]]), array([[ 0.16...pe=(100, 100)), array([[ 0.22... 0.43165108]])]
intercepts_ intercepts_: list of shape (n_layers - 1,) The ith element in the list represents the bias vector corresponding to layer i + 1.	list	[array([ 0.369... 0.27554867]), array([ 0.313... 0.03395789]), array([0.3372181])]
loss_ loss_: float The current loss computed with the loss function.	float64	1110
loss_curve_ loss_curve_: list of shape (`n_iter_`,) Loss value evaluated at the end of each training step. The ith element in the list represents the loss at the ith iteration. Only accessible when solver='sgd' or 'adam'.	list	[np.float64(14432.825030436637), np.float64(14372.921005126946), np.float64(14311.387710317184), np.float64(14245.964671913125), ...]
n_features_in_ n_features_in_: int Number of features seen during :term:`fit`. .. versionadded:: 0.24	int	10
n_iter_ n_iter_: int The number of iterations the solver has run.	int	500
n_layers_ n_layers_: int Number of layers.	int	4
n_outputs_ n_outputs_: int Number of outputs.	int	1
out_activation_ out_activation_: str Name of the output activation function.	str	'id...ty'
t_ t_: int The number of training samples seen by the solver during fitting. Mathematically equals `n_iters * X.shape[0]`, it means `time_step` and it is used by optimizer's learning rate scheduler.	int	221000
validation_scores_ validation_scores_: list of shape (`n_iter_`,) or None The score at each iteration on a held-out validation set. The score reported is the R2 score. Only available if `early_stopping=True`, otherwise the attribute is set to `None`. Only accessible when solver='sgd' or 'adam'.	NoneType	None

Plotting partial dependence for two features

We plot partial dependence curves for features “age” and “bmi” (body mass index) for the decision tree. With two features, from_estimator expects to plot two curves. Here the plot function place a grid of two plots using the space defined by ax .

fig, ax = plt.subplots(figsize=(12, 6))
ax.set_title("Decision Tree")
tree_disp = PartialDependenceDisplay.from_estimator(tree, X, ["age", "bmi"], ax=ax)

The partial dependence curves can be plotted for the multi-layer perceptron. In this case, line_kw is passed to from_estimator to change the color of the curve.

fig, ax = plt.subplots(figsize=(12, 6))
ax.set_title("Multi-layer Perceptron")
mlp_disp = PartialDependenceDisplay.from_estimator(
    mlp, X, ["age", "bmi"], ax=ax, line_kw={"color": "red"}
)

Plotting partial dependence of the two models together

The tree_disp and mlp_disp PartialDependenceDisplay objects contain all the computed information needed to recreate the partial dependence curves. This means we can easily create additional plots without needing to recompute the curves.

One way to plot the curves is to place them in the same figure, with the curves of each model on each row. First, we create a figure with two axes within two rows and one column. The two axes are passed to the plot functions of tree_disp and mlp_disp. The given axes will be used by the plotting function to draw the partial dependence. The resulting plot places the decision tree partial dependence curves in the first row of the multi-layer perceptron in the second row.

fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(10, 10))
tree_disp.plot(ax=ax1)
ax1.set_title("Decision Tree")
mlp_disp.plot(ax=ax2, line_kw={"color": "red"})
ax2.set_title("Multi-layer Perceptron")

Text(0.5, 1.0, 'Multi-layer Perceptron')

Another way to compare the curves is to plot them on top of each other. Here, we create a figure with one row and two columns. The axes are passed into the plot function as a list, which will plot the partial dependence curves of each model on the same axes. The length of the axes list must be equal to the number of plots drawn.

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 6))
tree_disp.plot(ax=[ax1, ax2], line_kw={"label": "Decision Tree"})
mlp_disp.plot(
    ax=[ax1, ax2], line_kw={"label": "Multi-layer Perceptron", "color": "red"}
)
ax1.legend()
ax2.legend()

plot partial dependence visualization api

<matplotlib.legend.Legend object at 0x784116e5f230>

tree_disp.axes_ is a numpy array container the axes used to draw the partial dependence plots. This can be passed to mlp_disp to have the same affect of drawing the plots on top of each other. Furthermore, the mlp_disp.figure_ stores the figure, which allows for resizing the figure after calling plot. In this case tree_disp.axes_ has two dimensions, thus plot will only show the y label and y ticks on the left most plot.

tree_disp.plot(line_kw={"label": "Decision Tree"})
mlp_disp.plot(
    line_kw={"label": "Multi-layer Perceptron", "color": "red"}, ax=tree_disp.axes_
)
tree_disp.figure_.set_size_inches(10, 6)
tree_disp.axes_[0, 0].legend()
tree_disp.axes_[0, 1].legend()
plt.show()

Plotting partial dependence for one feature

Here, we plot the partial dependence curves for a single feature, “age”, on the same axes. In this case, tree_disp.axes_ is passed into the second plot function.

tree_disp = PartialDependenceDisplay.from_estimator(tree, X, ["age"])
mlp_disp = PartialDependenceDisplay.from_estimator(
    mlp, X, ["age"], ax=tree_disp.axes_, line_kw={"color": "red"}
)

Total running time of the script: (0 minutes 2.614 seconds)

Related examples

Partial Dependence and Individual Conditional Expectation Plots

Monotonic Constraints

Using KBinsDiscretizer to discretize continuous features

Release Highlights for scikit-learn 0.24

© 2007–2026 The scikit-learn developers
Licensed under the 3-clause BSD License.
https://scikit-learn.org/1.9/auto_examples/inspection/plot_partial_dependence_visualization_api.html