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 division, unicode_literals

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

from gemseo.api import configure_logger, load_dataset
from gemseo.core.dataset import Dataset
from gemseo.mlearning.api import create_regression_model
from gemseo.mlearning.qual_measure.mse_measure import MSEMeasure
from gemseo.mlearning.transform.scaler.min_max_scaler import MinMaxScaler

configure_logger()

Out:

<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": Dataset.INPUT_GROUP, "y": Dataset.OUTPUT_GROUP}

dataset = Dataset("dataset_name")
dataset.set_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(
    "MixtureOfExperts", dataset, transformer={"outputs": MinMaxScaler()}
)

moe.set_clusterer("GaussianMixture", n_components=4)
moe.set_classifier("KNNClassifier", n_neighbors=3)
moe.set_regressor(
    "PolynomialRegression", 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.evaluate(method="learn"))
print("Error Random Forest:", measure_randfor.evaluate(method="learn"))

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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Out:

Learn:
Error MoE: [0.00130025]
Error Random Forest: [0.01388602]

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.evaluate("kfolds"))
print("Error Random Forest:", measure_randfor.evaluate("kfolds"))

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

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()

Out:

K-folds:
Error MoE: [0.29601814]
Error Random Forest: [0.14403934]
Loo:
Error MoE: [0.13216698]
Error Random Forest: [0.07299357]

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 = load_dataset("RosenbrockDataset", opt_naming=False)
x = dataset.get_data_by_group(dataset.INPUT_GROUP)
y = dataset.get_data_by_group(dataset.OUTPUT_GROUP)
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]))

print(dataset)

Out:

Rosenbrock
   Number of samples: 100
   Number of variables: 2
   Variables names and sizes by group:
      inputs: x (2)
      outputs: rosen (1)
   Number of dimensions (total = 3) by group:
      inputs: 2
      outputs: 1

Create regression algorithms

moe = create_regression_model(
    "MixtureOfExperts", dataset, transformer={"outputs": MinMaxScaler()}
)
moe.set_clusterer("KMeans", n_clusters=3)
moe.set_classifier("KNNClassifier", n_neighbors=5)
moe.set_regressor(
    "PolynomialRegression", 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.evaluate(method="learn"))
print("Error Random Forest:", measure_randfor.evaluate(method="learn"))

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

Out:

Learn:
Error MoE: [8.19866424]
Error Random Forest: [4567.22027112]
K-folds:
Error MoE: [8983.49635387]
Error Random Forest: [104270.92710389]

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()