# MSE example - test-train split¶

In this example we consider a polynomial linear regression, splitting the data into two sets. We measure the quality of the regression by comparing the predictions with the output on the test set.

from __future__ import annotations

import matplotlib.pyplot as plt
from gemseo.api import configure_logger
from gemseo.api import create_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 numpy import arange
from numpy import argmin
from numpy import hstack
from numpy import linspace
from numpy import sort
from numpy.random import choice
from numpy.random import normal
from numpy.random import seed

configure_logger()

<RootLogger root (INFO)>


## Define parameters¶

seed(12345)
n_samples = 10
noise = 0.3**2
max_pow = 5
amount_train = 0.8


## Construct data¶

We construct a parabola with added noise, on the interval [0, 1].

def f(x):
return -4 * (x - 0.5) ** 2 + 3

x = linspace(0, 1, n_samples)
y = f(x) + normal(0, noise, n_samples)


## Indices for test-train split¶

samples = arange(n_samples)
n_train = int(amount_train * n_samples)
n_test = n_samples - n_train
train = sort(choice(samples, n_train, False))
test = sort([sample for sample in samples if sample not in train])
print("Train:", train)
print("Test:", test)

Train: [1 3 4 5 6 7 8 9]
Test: [0 2]


## Build datasets¶

data = hstack([x[:, None], y[:, None]])
variables = ["x", "y"]
groups = {"x": Dataset.INPUT_GROUP, "y": Dataset.OUTPUT_GROUP}
dataset = create_dataset("synthetic_data", data[train], variables, groups=groups)
dataset_test = create_dataset("synthetic_data", data[test], variables, groups=groups)


## Build regression model¶

model = create_regression_model("PolynomialRegressor", dataset, degree=max_pow)
print(model)

PolynomialRegressor(degree=5, fit_intercept=True, l2_penalty_ratio=1.0, penalty_level=0.0)
based on the scikit-learn library


## Predictions errors¶

measure = MSEMeasure(model)

mse_train = measure.evaluate("learn")
mse_test = measure.evaluate("test", test_data=dataset_test)

print("Training error:", mse_train)
print("Test error:", mse_test)

Training error: [0.0003947]
Test error: [2.29565983]


## Compute predictions¶

measure = MSEMeasure(model)
model.learn()

n_refined = 1000
x_refined = linspace(0, 1, n_refined)
y_refined = model.predict({"x": x_refined[:, None]})["y"].flatten()


## Plot data points¶

plt.plot(x_refined, f(x_refined), label="Exact function")
plt.scatter(x, y, label="Data points")
plt.legend()
plt.show()


## Plot predictions¶

plt.plot(x_refined, y_refined, label=f"Prediction (x^{max_pow})")
plt.scatter(x[train], y[train], label="Train")
plt.scatter(x[test], y[test], color="r", label="Test")
plt.legend()
plt.show()


## Compare different parameters¶

powers = [1, 2, 3, 4, 5, 7]
test_errors = []
for power in powers:
model = create_regression_model("PolynomialRegressor", dataset, degree=power)
measure = MSEMeasure(model)

test_mse = measure.evaluate("test", test_data=dataset_test)
test_errors += [test_mse]

y_refined = model.predict({"x": x_refined[:, None]})["y"].flatten()

plt.plot(x_refined, y_refined, label=f"x^{power}")

plt.scatter(x[train], y[train], label="Train")
plt.scatter(x[test], y[test], color="r", label="Test")
plt.legend()
plt.show()


Grid search

print(test_errors)
print("Power for minimal test error:", argmin(test_errors))

[array([0.54513687]), array([0.00518409]), array([0.00584647]), array([0.06387849]), array([2.29565983]), array([1.16961302])]
Power for minimal test error: 1


Total running time of the script: ( 0 minutes 0.451 seconds)

Gallery generated by Sphinx-Gallery