import numpy as np
import matplotlib.pyplot as plt
import japanize_matplotlib
from sklearn.ensemble import GradientBoostingRegressor
Intuition (formulas) #
Minimize a loss $L(y,F(x))$ by stagewise additive modeling:
- Initialize $F_0(x)$.
- For $m=1,\dots,M$: compute negative gradients (pseudo-residuals) $r_{im} = -\left[\partial L(y_i, F(x_i))/\partial F\right]{F=F{m-1}}$, fit a tree $h_m$ to $r_{im}$, and update $F_m(x) = F_{m-1}(x) + \nu, \rho_m, h_m(x)$ ($\nu$: learning rate).
Fit on synthetic data #
X = np.linspace(-10, 10, 500)[:, np.newaxis]
noise = np.random.rand(X.shape[0]) * 10
y = (
(np.sin(X).ravel() + np.cos(4 * X).ravel()) * 10
+ 10
+ np.linspace(-10, 10, 500)
+ noise
)
reg = GradientBoostingRegressor(n_estimators=50, learning_rate=0.5)
reg.fit(X, y)
y_pred = reg.predict(X)
plt.figure(figsize=(10, 5))
plt.scatter(X, y, c="k", marker="x", label="train")
plt.plot(X, y_pred, c="r", label="prediction", linewidth=1)
plt.xlabel("x")
plt.ylabel("y")
plt.title("Fitting on training data")
plt.legend()
plt.show()
Effect of the loss function (outliers) #
Compare loss in { "squared_error", "absolute_error", "huber", "quantile" }.
X = np.linspace(-10, 10, 500)[:, np.newaxis]
noise = np.random.rand(X.shape[0]) * 10
for i in range(0, X.shape[0], 80):
noise[i] = 70 + np.random.randint(-10, 10)
y = (
(np.sin(X).ravel() + np.cos(4 * X).ravel()) * 10
+ 10
+ np.linspace(-10, 10, 500)
+ noise
)
for loss in ["squared_error", "absolute_error", "huber", "quantile"]:
reg = GradientBoostingRegressor(n_estimators=50, learning_rate=0.5, loss=loss)
reg.fit(X, y)
y_pred = reg.predict(X)
plt.figure(figsize=(10, 5))
plt.scatter(X, y, c="k", marker="x", label="train")
plt.plot(X, y_pred, c="r", label="prediction", linewidth=1)
plt.xlabel("x")
plt.ylabel("y")
plt.title(f"loss={loss}")
plt.legend()
plt.show()
