Quality measure for surrogate model comparison

In this example we use the quality measure class to compare the performances of a mixture of experts (MoE) and a random forest algorithm under different circumstances. We will consider two different datasets: A 1D function, and the Rosenbrock dataset (two inputs and one output).

Import

from __future__ import annotations

import matplotlib.pyplot as plt
from numpy import hstack
from numpy import linspace
from numpy import meshgrid
from numpy import sin

from gemseo import configure_logger
from gemseo import create_benchmark_dataset
from gemseo.datasets.io_dataset import IODataset
from gemseo.mlearning import create_regression_model
from gemseo.mlearning.quality_measures.mse_measure import MSEMeasure
from gemseo.mlearning.transformers.scaler.min_max_scaler import MinMaxScaler

configure_logger()

<RootLogger root (INFO)>

Test on 1D dataset

In this section we create a dataset from an analytical expression of a 1D function, and compare the errors of the two regression models.

Create 1D dataset from expression

def data_gen(x):
    return 3 + 0.5 * sin(14 * x) * (x <= 0.7) + (x > 0.7) * (0.8 + 6 * (x - 1) ** 2)


x = linspace(0, 1, 25)
y = data_gen(x)


data = hstack((x[:, None], y[:, None]))
variables = ["x", "y"]
sizes = {"x": 1, "y": 1}
groups = {"x": IODataset.INPUT_GROUP, "y": IODataset.OUTPUT_GROUP}

dataset = IODataset.from_array(data, variables, sizes, groups)

Plot 1D data

x_refined = linspace(0, 1, 500)
y_refined = data_gen(x_refined)
plt.plot(x_refined, y_refined)
plt.scatter(x, y)
plt.show()

Create regression algorithms

moe = create_regression_model(
    "MOERegressor", dataset, transformer={"outputs": MinMaxScaler()}
)

moe.set_clusterer("GaussianMixture", n_components=4)
moe.set_classifier("KNNClassifier", n_neighbors=3)
moe.set_regressor(
    "PolynomialRegressor", degree=5, l2_penalty_ratio=1, penalty_level=0.00005
)


randfor = create_regression_model(
    "RandomForestRegressor",
    dataset,
    transformer={"outputs": MinMaxScaler()},
    n_estimators=50,
)

Compute measures (Mean Squared Error)

measure_moe = MSEMeasure(moe)
measure_randfor = MSEMeasure(randfor)

Evaluate on training set directly (keyword: ‘learn’)

print("Learn:")
print("Error MoE:", measure_moe.compute_learning_measure())
print("Error Random Forest:", measure_randfor.compute_learning_measure())

plt.figure()
plt.plot(x_refined, moe.predict(x_refined[:, None]).flatten(), label="MoE")
plt.plot(x_refined, randfor.predict(x_refined[:, None]).flatten(), label="RndFr")
plt.scatter(x, y)
plt.legend()
plt.ylim(2, 5)
plt.show()

plt.figure()
plt.plot(
    x_refined, moe.predict_local_model(x_refined[:, None], 0).flatten(), label="MoE 0"
)
plt.plot(
    x_refined, moe.predict_local_model(x_refined[:, None], 1).flatten(), label="MoE 1"
)
plt.plot(
    x_refined, moe.predict_local_model(x_refined[:, None], 2).flatten(), label="MoE 2"
)
plt.plot(
    x_refined, moe.predict_local_model(x_refined[:, None], 3).flatten(), label="MoE 3"
)
plt.plot(x_refined, moe.predict(x_refined[:, None]).flatten(), label="MoE")
plt.plot(x_refined, randfor.predict(x_refined[:, None]).flatten(), label="RndFr")
plt.scatter(x, y)
plt.legend()
plt.ylim(2, 5)
plt.show()

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• 

Learn:
Error MoE: [0.00130025]
Error Random Forest: [0.0135322]

Evaluate using cross validation (keyword: ‘kfolds’)

In order to better consider the generalization error, perform a k-folds cross validation algorithm. We also plot the predictions from the last iteration of the algorithm.

print("K-folds:")
print("Error MoE:", measure_moe.compute_cross_validation_measure())
print("Error Random Forest:", measure_randfor.compute_cross_validation_measure())

print("Loo:")
print("Error MoE:", measure_moe.compute_leave_one_out_measure())
print("Error Random Forest:", measure_randfor.compute_leave_one_out_measure())

plt.plot(x_refined, moe.predict(x_refined[:, None]).flatten(), label="MoE")
plt.plot(
    x_refined, randfor.predict(x_refined[:, None]).flatten(), label="Random Forest"
)
plt.scatter(x, y)
plt.legend()
plt.show()

K-folds:
Error MoE: [0.21137583]
Error Random Forest: [0.1151925]
Loo:
Error MoE: [0.13231133]
Error Random Forest: [0.08229462]

Test on 2D dataset (Rosenbrock)

In this section, we load the Rosenbrock dataset, and compare the error measures for the two regression models.

Load dataset

dataset = create_benchmark_dataset("RosenbrockDataset", opt_naming=False)
x = dataset.input_dataset.to_numpy()
y = dataset.output_dataset.to_numpy()
Y = y.reshape((10, 10))

refinement = 100
x_refined = linspace(-2, 2, refinement)
X_1_refined, X_2_refined = meshgrid(x_refined, x_refined)
x_1_refined, x_2_refined = X_1_refined.flatten(), X_2_refined.flatten()
x_refined = hstack((x_1_refined[:, None], x_2_refined[:, None]))

dataset

GROUP	inputs		outputs
VARIABLE	x		rosen
COMPONENT	0	1	0
0	-2.000000	-2.0	3609.000000
1	-1.555556	-2.0	1959.952599
2	-1.111111	-2.0	1050.699741
3	-0.666667	-2.0	600.308642
4	-0.222222	-2.0	421.490779
...	...	...	...
95	0.222222	2.0	381.095717
96	0.666667	2.0	242.086420
97	1.111111	2.0	58.600975
98	1.555556	2.0	17.927907
99	2.000000	2.0	401.000000

100 rows × 3 columns

Create regression algorithms

moe = create_regression_model(
    "MOERegressor", dataset, transformer={"outputs": MinMaxScaler()}
)
moe.set_clusterer("KMeans", n_clusters=3)
moe.set_classifier("KNNClassifier", n_neighbors=5)
moe.set_regressor(
    "PolynomialRegressor", degree=5, l2_penalty_ratio=1, penalty_level=0.1
)


randfor = create_regression_model(
    "RandomForestRegressor",
    dataset,
    transformer={"outputs": MinMaxScaler()},
    n_estimators=200,
)

Compute measures (Mean Squared Error)

measure_moe = MSEMeasure(moe)
measure_randfor = MSEMeasure(randfor)

print("Learn:")
print("Error MoE:", measure_moe.compute_learning_measure())
print("Error Random Forest:", measure_randfor.compute_learning_measure())

print("K-folds:")
print("Error MoE:", measure_moe.compute_cross_validation_measure())
print("Error Random Forest:", measure_randfor.compute_cross_validation_measure())

Learn:
Error MoE: [9.87065269]
Error Random Forest: [4478.29264563]
K-folds:
Error MoE: [266.03322933]
Error Random Forest: [60783.64211188]

Plot data

plt.imshow(Y, interpolation="nearest")
plt.colorbar()
plt.show()

Plot predictions

moe.learn()
randfor.learn()
Y_pred_moe = moe.predict(x_refined).reshape((refinement, refinement))
Y_pred_moe_0 = moe.predict_local_model(x_refined, 0).reshape((refinement, refinement))
Y_pred_moe_1 = moe.predict_local_model(x_refined, 1).reshape((refinement, refinement))
Y_pred_moe_2 = moe.predict_local_model(x_refined, 2).reshape((refinement, refinement))
Y_pred_randfor = randfor.predict(x_refined).reshape((refinement, refinement))

Plot mixture of experts predictions

plt.imshow(Y_pred_moe)
plt.colorbar()
plt.show()

Plot local models

plt.figure()
plt.imshow(Y_pred_moe_0)
plt.colorbar()
plt.show()

plt.figure()
plt.imshow(Y_pred_moe_1)
plt.colorbar()
plt.show()

plt.figure()
plt.imshow(Y_pred_moe_2)
plt.colorbar()
plt.show()

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• 

Plot random forest predictions

plt.imshow(Y_pred_randfor)
plt.colorbar()
plt.show()