Selección de hiperparámetros teniendo en cuenta la generalización — 9:40

• Ultima modificación: 2023-02-27 | YouTube

Una pregunta de interés corresponde a ¿cómo seleccionar el grado del polinomio teniendo en cuenta que el error de generalización es una cantidad aleatoria?

Importación de librerías

import matplotlib.pyplot as plt
import numpy as np
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error
from sklearn.pipeline import make_pipeline
from sklearn.preprocessing import MinMaxScaler, PolynomialFeatures

#
# Función a aproximar
# (Proceso geneador de datos)
#
def f(x):
    return 2 * np.abs(np.sin(x * np.pi / 4 + 0.75)) / (1 + 0.1 * x)


#
# Datos reales.
# (No disponibles en la realidad)
#
x_real = np.linspace(0, 10, 100)
x_real = x_real[:, np.newaxis]
y_real = f(x_real)

#
# Muestra de datos.
# (Información disponible en la realidad)
#
rng = np.random.default_rng(12345)

x_sample = np.linspace(0, 10, 100)
rng.shuffle(x_sample)
x_sample = x_sample[:25]
x_sample = np.sort(x_sample)

y_sample = f(x_sample)
X_sample = x_sample[:, np.newaxis]

#
# Se computa el promedio del SSE sobre 16 muestras
#
plt.figure(figsize=(12, 4))

n_samples = len(x_sample)
n_test = 5
n_train = n_samples - n_test

degrees = list(range(1, 16))

mse_train = []
mse_test = []

for i_degree, degree in enumerate(degrees):

    np.random.seed(12345)

    mse_train_by_sample = []
    mse_test_by_sample = []

    for i_sample in range(17):

        #
        # Modelo
        #
        indexes = np.random.choice(
            n_samples,
            n_train,
            replace=False,
        )

        X_sample_train = X_sample[indexes]
        y_sample_train = y_sample[indexes]

        X_sample_test = np.delete(X_sample, indexes)
        X_sample_test = X_sample_test[:, np.newaxis]
        y_sample_test = np.delete(y_sample, indexes)

        model = make_pipeline(
            PolynomialFeatures(degree, include_bias=False),
            MinMaxScaler(),
            LinearRegression(),
        )
        model.fit(X_sample_train, y_sample_train)

        y_pred_train = model.predict(X_sample_train)
        y_pred_test = model.predict(X_sample_test)

        mse_train_by_sample.append(mean_squared_error(y_sample_train, y_pred_train))
        mse_test_by_sample.append(mean_squared_error(y_sample_test, y_pred_test))

    mse_train.append(np.mean(mse_train_by_sample))
    mse_test.append(np.mean(mse_test_by_sample))

plt.plot(degrees, mse_train, color="tab:blue", linewidth=2, label="fit")
plt.plot(degrees, mse_test, color="tab:orange", linewidth=2, label="test")

plt.yscale("log")


plt.xlabel("Degree")
plt.ylabel("MSE")
plt.grid()
plt.legend()
plt.gca().spines["left"].set_color("gray")
plt.gca().spines["bottom"].set_color("gray")
plt.gca().spines["top"].set_visible(False)
plt.gca().spines["right"].set_visible(False)


plt.show()

plt.figure(figsize=(12, 13))

optimal_degree = 8

n_samples = len(x_sample)
n_test = 5
n_train = n_samples - n_test

for i_plot in range(16):

    plt.subplot(4, 4, i_plot + 1)

    #
    # Datos
    #
    plt.plot(x_real, y_real, "--", color="black", alpha=1.0, zorder=10)
    plt.plot(x_sample, y_sample, "o", color="black", alpha=1.0, zorder=10)

    #
    # Modelo
    #
    indexes = np.random.choice(
        n_samples,
        n_train,
        replace=False,
    )

    X_sample_train = X_sample[indexes]
    y_sample_train = y_sample[indexes]

    X_sample_test = np.delete(X_sample, indexes)
    y_sample_test = np.delete(y_sample, indexes)

    model = make_pipeline(
        PolynomialFeatures(optimal_degree),
        MinMaxScaler(),
        LinearRegression(),
    )
    model.fit(X_sample_train, y_sample_train)
    y_predicted = model.predict(x_real)

    plt.plot(x_real, y_predicted, color="tab:blue", linewidth=3, zorder=2, alpha=0.8)

    plt.plot(
        X_sample_test,
        y_sample_test,
        "o",
        color="black",
        fillstyle="none",
        markersize=11,
    )

    plt.xticks([], [])
    plt.yticks([], [])

    plt.gca().spines["left"].set_color("gray")
    plt.gca().spines["bottom"].set_color("gray")
    plt.gca().spines["top"].set_visible(False)
    plt.gca().spines["right"].set_visible(False)


plt.show()