CDF for Discrete Random Variables

Understanding cumulative distribution functions for discrete random variables with categorical distributions and practical examples of logistic regression

Keywords

CDF, cumulative distribution function, discrete random variables, categorical distribution, logistic regression, PyTorch

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'

# Set random seed
torch.manual_seed(42)

<torch._C.Generator at 0x135180ab0>

probs = [0.1, 0.4, 0.2, 0.3]
dist = torch.distributions.Categorical(probs=torch.tensor(probs))

dist

Categorical(probs: torch.Size([4]))

pd.Series(dist.sample(torch.Size([1000]))).value_counts().sort_index()

0     90
1    409
2    187
3    314
Name: count, dtype: int64

# 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)  # Probabilities

prob_Y

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        [0.2377],
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        [0.5074],
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        [0.8787],
        [0.0406],
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        [0.6637],
        [0.8729],
        [0.2455],
        [0.4179],
        [0.0462],
        [0.9018],
        [0.1551],
        [0.0173],
        [0.1364],
        [0.2652],
        [0.2519],
        [0.2691],
        [0.8527],
        [0.0068],
        [0.0701],
        [0.0393],
        [0.1062],
        [0.1793],
        [0.0372],
        [0.5522],
        [0.0402],
        [0.7888],
        [0.6758],
        [0.0772],
        [0.1758],
        [0.1189],
        [0.8798],
        [0.2317],
        [0.0974],
        [0.7143],
        [0.0202],
        [0.4330],
        [0.6808],
        [0.7392],
        [0.1329],
        [0.4755],
        [0.1541],
        [0.3953],
        [0.0859],
        [0.0016],
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        [0.0707],
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        [0.3144],
        [0.0426],
        [0.0029],
        [0.0657],
        [0.4495],
        [0.1725],
        [0.1724],
        [0.4098],
        [0.0512],
        [0.8470],
        [0.1267],
        [0.5247],
        [0.1115],
        [0.7646],
        [0.7213],
        [0.0759],
        [0.3569],
        [0.3375],
        [0.8672],
        [0.7596],
        [0.1475],
        [0.0044],
        [0.4780],
        [0.0581],
        [0.0251],
        [0.4568],
        [0.2024],
        [0.7446],
        [0.6320],
        [0.0314],
        [0.0765],
        [0.0181],
        [0.1966],
        [0.5667],
        [0.8577],
        [0.9013],
        [0.1643],
        [0.0776],
        [0.1805],
        [0.0631],
        [0.1894],
        [0.3725],
        [0.0367],
        [0.1032],
        [0.8510],
        [0.3935],
        [0.8488],
        [0.9634],
        [0.0439],
        [0.0918],
        [0.5004],
        [0.0232],
        [0.2922],
        [0.0592],
        [0.6902],
        [0.0502],
        [0.2931],
        [0.5467],
        [0.0902],
        [0.0438],
        [0.7579],
        [0.0144],
        [0.0310],
        [0.9566],
        [0.0983],
        [0.1246],
        [0.0175],
        [0.3579],
        [0.1215],
        [0.0063],
        [0.8350],
        [0.0783],
        [0.0673],
        [0.0044],
        [0.1932],
        [0.0166],
        [0.0619],
        [0.0126],
        [0.0981],
        [0.1415],
        [0.0975],
        [0.0028],
        [0.7217],
        [0.1342],
        [0.0666],
        [0.8163],
        [0.5906],
        [0.8264],
        [0.0062],
        [0.1267],
        [0.0576],
        [0.4529],
        [0.1712],
        [0.0178],
        [0.1451],
        [0.0375],
        [0.1448],
        [0.1852],
        [0.9204],
        [0.6197],
        [0.0180],
        [0.3411],
        [0.0445],
        [0.1539],
        [0.4608],
        [0.5943],
        [0.2118],
        [0.8577],
        [0.0662],
        [0.2722],
        [0.6509],
        [0.5460],
        [0.4635],
        [0.2292],
        [0.0513],
        [0.2066],
        [0.5374],
        [0.0059],
        [0.2224],
        [0.1254],
        [0.3754],
        [0.5258],
        [0.0939],
        [0.1778],
        [0.2772],
        [0.0344],
        [0.0388],
        [0.9426],
        [0.1209],
        [0.2947],
        [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 = -0.4170

def 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)