Partial Dependence Plots

Partial dependence plots show the dependence between the target function 2 and a set of ‘target’ features, marginalizing over the values of all other features (the complement features). Due to the limits of human perception the size of the target feature set must be small (usually, one or two) thus the target features are usually chosen among the most important features.

This example shows how to obtain partial dependence plots from a MLPRegressor and a HistGradientBoostingRegressor trained on the California housing dataset. The example is taken from 1.

The plots show four 1-way and two 1-way partial dependence plots (ommitted for MLPRegressor due to computation time). The target variables for the one-way PDP are: median income (MedInc), average occupants per household (AvgOccup), median house age (HouseAge), and average rooms per household (AveRooms).

1

T. Hastie, R. Tibshirani and J. Friedman, “Elements of Statistical Learning Ed. 2”, Springer, 2009.

2

For classification you can think of it as the regression score before the link function.

print(__doc__)

from time import time
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

from sklearn.model_selection import train_test_split
from sklearn.preprocessing import QuantileTransformer
from sklearn.pipeline import make_pipeline

from sklearn.inspection import partial_dependence
from sklearn.inspection import plot_partial_dependence
from sklearn.experimental import enable_hist_gradient_boosting  # noqa
from sklearn.ensemble import HistGradientBoostingRegressor
from sklearn.neural_network import MLPRegressor
from sklearn.datasets.california_housing import fetch_california_housing

California Housing data preprocessing

Center target to avoid gradient boosting init bias: gradient boosting with the ‘recursion’ method does not account for the initial estimator (here the average target, by default)

cal_housing = fetch_california_housing()
names = cal_housing.feature_names
X, y = cal_housing.data, cal_housing.target

y -= y.mean()

X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.1,
                                                    random_state=0)

Partial Dependence computation for multi-layer perceptron

Let’s fit a MLPRegressor and compute single-variable partial dependence plots

print("Training MLPRegressor...")
tic = time()
est = make_pipeline(QuantileTransformer(),
                    MLPRegressor(hidden_layer_sizes=(50, 50),
                                 learning_rate_init=0.01,
                                 max_iter=200,
                                 early_stopping=True,
                                 n_iter_no_change=10,
                                 validation_fraction=0.1))
est.fit(X_train, y_train)
print("done in {:.3f}s".format(time() - tic))
print("Test R2 score: {:.2f}".format(est.score(X_test, y_test)))

print('Computing partial dependence plots...')
tic = time()
# We don't compute the 2-way PDP (5, 1) here, because it is a lot slower
# with the brute method.
features = [0, 5, 1, 2]
plot_partial_dependence(est, X_train, features, feature_names=names,
                        n_jobs=3, grid_resolution=20)
print("done in {:.3f}s".format(time() - tic))
fig = plt.gcf()
fig.suptitle('Partial dependence of house value on non-location features\n'
             'for the California housing dataset, with MLPRegressor')
fig.tight_layout(rect=[0, 0.03, 1, 0.95])
../../_images/sphx_glr_plot_partial_dependence_001.png

Out:

Training MLPRegressor...
done in 5.208s
Test R2 score: 0.82
Computing partial dependence plots...
done in 1.494s

Partial Dependence computation for Gradient Boosting

Let’s now fit a GradientBoostingRegressor and compute the partial dependence plots either or one or two variables at a time.

print("Training GradientBoostingRegressor...")
tic = time()
est = HistGradientBoostingRegressor(max_iter=100, max_leaf_nodes=64,
                                    learning_rate=0.1, random_state=1)
est.fit(X_train, y_train)
print("done in {:.3f}s".format(time() - tic))
print("Test R2 score: {:.2f}".format(est.score(X_test, y_test)))

print('Computing partial dependence plots...')
tic = time()
features = [0, 5, 1, 2, (5, 1)]
plot_partial_dependence(est, X_train, features, feature_names=names,
                        n_jobs=3, grid_resolution=20)
print("done in {:.3f}s".format(time() - tic))
fig = plt.gcf()
fig.suptitle('Partial dependence of house value on non-location features\n'
             'for the California housing dataset, with Gradient Boosting')
fig.tight_layout(rect=[0, 0.03, 1, 0.95])
../../_images/sphx_glr_plot_partial_dependence_002.png

Out:

Training GradientBoostingRegressor...
done in 0.671s
Test R2 score: 0.86
Computing partial dependence plots...
done in 0.209s

Analysis of the plots

We can clearly see that the median house price shows a linear relationship with the median income (top left) and that the house price drops when the average occupants per household increases (top middle). The top right plot shows that the house age in a district does not have a strong influence on the (median) house price; so does the average rooms per household. The tick marks on the x-axis represent the deciles of the feature values in the training data.

We also observe that MLPRegressor has much smoother predictions than HistGradientBoostingRegressor. For the plots to be comparable, it is necessary to subtract the average value of the target y: The ‘recursion’ method, used by default for HistGradientBoostingRegressor, does not account for the initial predictor (in our case the average target). Setting the target average to 0 avoids this bias.

Partial dependence plots with two target features enable us to visualize interactions among them. The two-way partial dependence plot shows the dependence of median house price on joint values of house age and average occupants per household. We can clearly see an interaction between the two features: for an average occupancy greater than two, the house price is nearly independent of the house age, whereas for values less than two there is a strong dependence on age.

3D interaction plots

Let’s make the same partial dependence plot for the 2 features interaction, this time in 3 dimensions.

fig = plt.figure()

target_feature = (1, 5)
pdp, axes = partial_dependence(est, X_train, target_feature,
                               grid_resolution=20)
XX, YY = np.meshgrid(axes[0], axes[1])
Z = pdp[0].T
ax = Axes3D(fig)
surf = ax.plot_surface(XX, YY, Z, rstride=1, cstride=1,
                       cmap=plt.cm.BuPu, edgecolor='k')
ax.set_xlabel(names[target_feature[0]])
ax.set_ylabel(names[target_feature[1]])
ax.set_zlabel('Partial dependence')
#  pretty init view
ax.view_init(elev=22, azim=122)
plt.colorbar(surf)
plt.suptitle('Partial dependence of house value on median\n'
             'age and average occupancy, with Gradient Boosting')
plt.subplots_adjust(top=0.9)

plt.show()
../../_images/sphx_glr_plot_partial_dependence_003.png

Total running time of the script: ( 0 minutes 8.599 seconds)

Estimated memory usage: 11 MB

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