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import numpy as np
import matplotlib
import matplotlib.pyplot as plt
from sklearn.ensemble import BaggingRegressor
from sklearn.tree import DecisionTreeRegressor
import gradio as gr
matplotlib.use('agg')
# Generate data
def f(x):
x = x.ravel()
return np.exp(-(x**2)) + 1.5 * np.exp(-((x - 2) ** 2))
def generate(n_samples, noise, n_repeat=1):
X = np.random.rand(n_samples) * 10 - 5
X = np.sort(X)
if n_repeat == 1:
y = f(X) + np.random.normal(0.0, noise, n_samples)
else:
y = np.zeros((n_samples, n_repeat))
for i in range(n_repeat):
y[:, i] = f(X) + np.random.normal(0.0, noise, n_samples)
X = X.reshape((n_samples, 1))
return X, y
def train_model(n_train, noise):
# Settings
n_repeat = 50 # Number of iterations for computing expectations
# n_train = 50 # Size of the training set
n_test = 1000 # Size of the test set
# noise = noise # Standard deviation of the noise
np.random.seed(0)
# Change this for exploring the bias-variance decomposition of other
# estimators. This should work well for estimators with high variance (e.g.,
# decision trees or KNN), but poorly for estimators with low variance (e.g.,
# linear models).
estimators = [
("Tree", DecisionTreeRegressor()),
("Bagging(Tree)", BaggingRegressor(DecisionTreeRegressor())),
]
n_estimators = len(estimators)
X_train = []
y_train = []
for i in range(n_repeat):
X, y = generate(n_samples=n_train, noise=noise)
X_train.append(X)
y_train.append(y)
X_test, y_test = generate(n_samples=n_test, noise=noise, n_repeat=n_repeat)
fig = plt.figure(figsize=(10, 8))
out_str = ""
# Loop over estimators to compare
for n, (name, estimator) in enumerate(estimators):
# Compute predictions
y_predict = np.zeros((n_test, n_repeat))
for i in range(n_repeat):
estimator.fit(X_train[i], y_train[i])
y_predict[:, i] = estimator.predict(X_test)
# Bias^2 + Variance + Noise decomposition of the mean squared error
y_error = np.zeros(n_test)
for i in range(n_repeat):
for j in range(n_repeat):
y_error += (y_test[:, j] - y_predict[:, i]) ** 2
y_error /= n_repeat * n_repeat
y_noise = np.var(y_test, axis=1)
y_bias = (f(X_test) - np.mean(y_predict, axis=1)) ** 2
y_var = np.var(y_predict, axis=1)
out_str += f"{name}: {np.mean(y_error):.4f} (error) = {np.mean(y_bias):.4f} (bias^2) + {np.mean(y_var):.4f} (var) + {np.mean(y_noise):.4f} (noise)\n"
# Plot figures
plt.subplot(2, n_estimators, n + 1)
plt.plot(X_test, f(X_test), "b", label="$f(x)$")
plt.plot(X_train[0], y_train[0], ".b", label="LS ~ $y = f(x)+noise$")
for i in range(n_repeat):
if i == 0:
plt.plot(X_test, y_predict[:, i], "r", label=r"$\^y(x)$")
else:
plt.plot(X_test, y_predict[:, i], "r", alpha=0.05)
plt.plot(X_test, np.mean(y_predict, axis=1), "c", label=r"$\mathbb{E}_{LS} \^y(x)$")
plt.xlim([-5, 5])
plt.title(name)
if n == n_estimators - 1:
plt.legend(loc=(1.1, 0.5))
plt.subplot(2, n_estimators, n_estimators + n + 1)
plt.plot(X_test, y_error, "r", label="$error(x)$")
plt.plot(X_test, y_bias, "b", label="$bias^2(x)$"),
plt.plot(X_test, y_var, "g", label="$variance(x)$"),
plt.plot(X_test, y_noise, "c", label="$noise(x)$")
plt.xlim([-5, 5])
plt.ylim([0, noise])
if n == n_estimators - 1:
plt.legend(loc=(1.1, 0.5))
plt.subplots_adjust(right=0.75)
return fig, out_str
title = "Single estimator versus bagging: bias-variance decomposition ⚖️"
description = """This example illustrates and compares the bias-variance decomposition of the \
expected mean squared error of a single estimator (Decision Tree Regressor) \
against a bagging ensemble of Tree Regressors. \
The dataset used for this demo is a one-dimensional synthetic dataset generated \
for a regression problem. In the top two figures, the blue line represents the true \
function and the blue dots represent the training data that are obtained by adding some \
random noise (user selected). The prediction of the models is represented by the red line. \
The average prediction of each estimator is presented in cyan.
In the two lower figures, we can see the decomposition of the expected mean squared error \
(red) into the bias (blue) and variance (green), as well as the noise part of the error (cyan).
"""
with gr.Blocks() as demo:
gr.Markdown(f"## {title}")
gr.Markdown(description)
num_samples = gr.Slider(minimum=50, maximum=200, step=50, value=50, label="Number of samples")
noise = gr.Slider(minimum=0.05, maximum=0.2, step=0.05, value=0.1, label="Noise")
with gr.Row():
with gr.Row():
with gr.Column(scale=2):
plot = gr.Plot()
with gr.Column(scale=1):
results = gr.Textbox(label="Results")
num_samples.change(fn=train_model, inputs=[num_samples, noise], outputs=[plot, results])
noise.change(fn=train_model, inputs=[num_samples, noise], outputs=[plot, results])
demo.launch(enable_queue=True)
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