import matplotlib.pyplot as plt
import numpy as np
import torch
import torch.nn as nn
import pandas as pd
# Retina mode
%matplotlib inline
%config InlineBackend.figure_format = 'retina'PMF and their applications
ML
# Set random seed
torch.manual_seed(42)<torch._C.Generator at 0x11e27cab0>
# Generate 2D input features
n_samples = 1000
X1 = torch.distributions.Uniform(0, 5).sample((n_samples, 1)) # Feature 1
X2 = torch.distributions.Uniform(0, 5).sample((n_samples, 1)) # Feature 2# True weights and bias
w1, w2, b = 1.2, -0.8, -2.5 # Compute logits and apply sigmoid
logits = w1 * X1 + w2 * X2 + b
prob_Y = torch.sigmoid(logits) # Probabilitiesprob_Ytensor([[0.4690],
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[0.0870],
[0.4037],
[0.0158]])# Sample class labels
Y = torch.distributions.Bernoulli(prob_Y).sample()# Convert to NumPy for visualization
X1_np, X2_np, Y_np = X1.numpy(), X2.numpy(), Y.numpy()
# Plot data points
plt.scatter(X1_np[Y_np == 0], X2_np[Y_np == 0], color="blue", label="Class 0", alpha=0.5)
plt.scatter(X1_np[Y_np == 1], X2_np[Y_np == 1], color="red", label="Class 1", alpha=0.5)
plt.xlabel("Feature 1 (X1)")
plt.ylabel("Feature 2 (X2)")
plt.legend()
plt.title("Generated 2D Logistic Regression Data")Text(0.5, 1.0, 'Generated 2D Logistic Regression Data')
import torch.nn as nn
import torch.optim as optim
# Stack X1 and X2 into a single tensor
X_train = torch.cat((X1, X2), dim=1)
# Define logistic regression model
class LogisticRegression2D(nn.Module):
def __init__(self):
super(LogisticRegression2D, self).__init__()
self.linear = nn.Linear(2, 1) # Two inputs, one output
def forward(self, x):
return torch.sigmoid(self.linear(x)) # Sigmoid activation
# Initialize model
model = LogisticRegression2D()
loss_fn = nn.BCELoss() # Binary cross-entropy loss
optimizer = optim.SGD(model.parameters(), lr=0.01)
# Training loop
epochs = 1000
for epoch in range(epochs):
optimizer.zero_grad()
Y_pred = model(X_train) # Forward pass
loss = loss_fn(Y_pred, Y.view(-1, 1)) # Compute loss
loss.backward() # Backpropagation
optimizer.step() # Update weights
if epoch % 200 == 0:
print(f"Epoch {epoch}, Loss: {loss.item():.4f}")
# Extract learned parameters
w1_learned, w2_learned = model.linear.weight[0].detach().numpy()
b_learned = model.linear.bias[0].detach().numpy()
print(f"Learned Parameters: w1 = {w1_learned:.4f}, w2 = {w2_learned:.4f}, b = {b_learned:.4f}")Epoch 0, Loss: 2.0864
Epoch 200, Loss: 0.4884
Epoch 400, Loss: 0.4536
Epoch 600, Loss: 0.4395
Epoch 800, Loss: 0.4318
Learned Parameters: w1 = 0.6339, w2 = -0.8716, b = -0.4170def plot_decision_boundary(model, X1_np, X2_np, Y_np, w1_true, w2_true, b_true):
"""Plots the true and learned decision boundaries."""
# Generate mesh grid
x1_vals = np.linspace(0, 5, 100)
x2_vals = np.linspace(0, 5, 100)
X1_grid, X2_grid = np.meshgrid(x1_vals, x2_vals)
# Compute model's learned decision boundary
with torch.no_grad():
Z = model(torch.tensor(np.c_[X1_grid.ravel(), X2_grid.ravel()], dtype=torch.float32))
Z = Z.view(X1_grid.shape).numpy()
# Compute true decision boundary
x1_boundary = np.linspace(0,5, 100)
x2_boundary_true = - (w1_true / w2_true) * x1_boundary - (b_true / w2_true)
# Plot data points
plt.scatter(X1_np[Y_np == 0], X2_np[Y_np == 0], color="blue", label="Class 0", alpha=0.5)
plt.scatter(X1_np[Y_np == 1], X2_np[Y_np == 1], color="red", label="Class 1", alpha=0.5)
# Plot learned decision boundary
plt.contour(X1_grid, X2_grid, Z, levels=[0.5], colors="black", linestyles="dashed", label="Learned Boundary")
# Plot true decision boundary
plt.plot(x1_boundary, x2_boundary_true, color="green", linestyle="solid", label="True Boundary")
plt.xlabel("Feature 1 (X1)")
plt.ylabel("Feature 2 (X2)")
plt.legend()
plt.title("Logistic Regression Decision Boundary")
plt.ylim([0, 5])
# Call the function with true and learned parameters
plot_decision_boundary(model, X1_np, X2_np, Y_np, w1_true=1.2, w2_true=-0.8, b_true=-2.5)/var/folders/z8/gpvqr8mn3w9_f38byxhnsk780000gn/T/ipykernel_8863/319328462.py:23: UserWarning: The following kwargs were not used by contour: 'label'
plt.contour(X1_grid, X2_grid, Z, levels=[0.5], colors="black", linestyles="dashed", label="Learned Boundary")