GLM - Poisson and Logistic
This notebook demonstrates Generalized Linear Models with Poisson and Logistic regression examples.
Notebook Contents
This notebook covers Generalized Linear Models with practical examples:
- Poisson Regression: For modeling count data
- Logistic Regression: For binary classification
- Link Functions: Understanding different GLM link functions
- Model Evaluation: Comparing model performance
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In [20]:
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import statsmodels.api as sm
from statsmodels.genmod.families import Poisson, Binomial
from sklearn.model_selection import train_test_split
from sklearn.metrics import confusion_matrix, classification_report
import seaborn as sns
from sklearn.metrics import mean_absolute_percentage_error
# Set random seed for reproducibility
np.random.seed(42)
# ============================================================================
# EXAMPLE 1: POISSON REGRESSION
# ============================================================================
# Use case: Predicting count data (e.g., number of customer visits per day)
print("="*70)
print("POISSON REGRESSION EXAMPLE")
print("="*70)
# Generate synthetic data
n_samples = 500
X_pois = np.random.randn(n_samples, 2)
# True coefficients
beta_pois = np.array([0.5, -0.3])
# Linear predictor
eta = 2 + X_pois @ beta_pois
# Expected count (lambda parameter)
lambda_true = np.exp(eta)
# Generate Poisson-distributed counts
y_pois = np.random.poisson(lambda_true)
# Create DataFrame
df_pois = pd.DataFrame({
'advertising_spend': X_pois[:, 0],
'competitor_activity': X_pois[:, 1],
'customer_visits': y_pois
})
print("\nFirst few rows of Poisson data:")
print(df_pois.head())
print(f"\nBasic statistics of customer visits:")
print(df_pois['customer_vbisits'].describe())
In [4]:
df_pois.customer_visits.hist()
Out[4]:
In [19]:
# Fit Poisson GLM
X_pois_model = sm.add_constant(X_pois)
poisson_model = sm.GLM(y_pois, X_pois_model, family=Gaussian())
poisson_results = poisson_model.fit()
print("\n" + "="*70)
print("POISSON MODEL SUMMARY")
print("="*70)
print(poisson_results.summary())
# Predictions
y_pred_pois = poisson_results.predict(X_pois_model)
# Visualization
fig, axes = plt.subplots(1, 2, figsize=(14, 5))
# Plot 1: Actual vs Predicted
axes[0].scatter(y_pois, y_pred_pois, alpha=0.5)
axes[0].plot([y_pois.min(), y_pois.max()],
[y_pois.min(), y_pois.max()],
'r--', lw=2)
axes[0].set_xlabel('Actual Count')
axes[0].set_ylabel('Predicted Count')
axes[0].set_title('Poisson GLM: Actual vs Predicted')
axes[0].grid(True, alpha=0.3)
# Plot 2: Residuals
residuals = y_pois - y_pred_pois
axes[1].scatter(y_pred_pois, residuals, alpha=0.5)
axes[1].axhline(y=0, color='r', linestyle='--', lw=2)
axes[1].set_xlabel('Predicted Count')
axes[1].set_ylabel('Residuals')
axes[1].set_title('Poisson GLM: Residual Plot')
axes[1].grid(True, alpha=0.3)
plt.tight_layout()
plt.savefig('poisson_glm.png', dpi=100, bbox_inches='tight')
print("\nPoisson GLM plots saved as 'poisson_glm.png'")
In [14]:
# ============================================================================
# EXAMPLE 2: LOGISTIC REGRESSION
# ============================================================================
# Use case: Binary classification (e.g., customer churn prediction)
print("\n\n" + "="*70)
print("LOGISTIC REGRESSION EXAMPLE")
print("="*70)
# Generate synthetic data
n_samples = 1000
X_log = np.random.randn(n_samples, 3)
# True coefficients
beta_log = np.array([1.5, -0.8, 0.6])
# Linear predictor
eta_log = -1 + X_log @ beta_log
# Probability of success
prob_true = 1 / (1 + np.exp(-eta_log))
# Generate binary outcomes
y_log = np.random.binomial(1, prob_true)
# Create DataFrame
df_log = pd.DataFrame({
'account_age': X_log[:, 0],
'support_tickets': X_log[:, 1],
'usage_frequency': X_log[:, 2],
'churned': y_log
})
print("\nFirst few rows of Logistic data:")
print(df_log.head())
print(f"\nClass distribution:")
print(df_log['churned'].value_counts())
print(f"Churn rate: {y_log.mean():.2%}")
In [15]:
df_log.churned.hist()
Out[15]:
In [22]:
mean_absolute_percentage_error(y_pois, y_pred_pois)
Out[22]:
In [18]:
# Split data
X_train, X_test, y_train, y_test = train_test_split(
X_log, y_log, test_size=0.3, random_state=42
)
# Fit Logistic GLM
X_train_const = sm.add_constant(X_train)
X_test_const = sm.add_constant(X_test)
logistic_model = sm.GLM(y_train, X_train_const, family=Binomial())
logistic_results = logistic_model.fit()
print("\n" + "="*70)
print("LOGISTIC MODEL SUMMARY")
print("="*70)
print(logistic_results.summary())
# Predictions
y_pred_prob = logistic_results.predict(X_test_const)
y_pred_class = (y_pred_prob > 0.5).astype(int)
# Model evaluation
print("\n" + "="*70)
print("LOGISTIC MODEL EVALUATION")
print("="*70)
print("\nConfusion Matrix:")
cm = confusion_matrix(y_test, y_pred_class)
print(cm)
print("\nClassification Report:")
print(classification_report(y_test, y_pred_class))
# Calculate odds ratios
odds_ratios = np.exp(logistic_results.params)
print("\nOdds Ratios:")
for name, or_val in zip(['Intercept', 'Feature 1', 'Feature 2', 'Feature 3'],
odds_ratios):
print(f"{name}: {or_val:.3f}")
# Visualization
fig, axes = plt.subplots(1, 2, figsize=(14, 5))
# Plot 1: ROC-like visualization
sorted_idx = np.argsort(y_pred_prob)
axes[0].plot(y_pred_prob[sorted_idx], marker='o', markersize=2, alpha=0.6)
axes[0].axhline(y=0.5, color='r', linestyle='--', lw=2, label='Threshold=0.5')
axes[0].set_xlabel('Sample (sorted by predicted probability)')
axes[0].set_ylabel('Predicted Probability')
axes[0].set_title('Logistic GLM: Predicted Probabilities')
axes[0].legend()
axes[0].grid(True, alpha=0.3)
# Plot 2: Probability distribution by actual class
axes[1].hist(y_pred_prob[y_test == 0], bins=30, alpha=0.6,
label='Actual: No Churn', color='blue')
axes[1].hist(y_pred_prob[y_test == 1], bins=30, alpha=0.6,
label='Actual: Churn', color='red')
axes[1].axvline(x=0.5, color='black', linestyle='--', lw=2,
label='Threshold=0.5')
axes[1].set_xlabel('Predicted Probability')
axes[1].set_ylabel('Frequency')
axes[1].set_title('Logistic GLM: Probability Distribution by Class')
axes[1].legend()
axes[1].grid(True, alpha=0.3)
plt.tight_layout()
plt.savefig('logistic_glm.png', dpi=100, bbox_inches='tight')
print("\nLogistic GLM plots saved as 'logistic_glm.png'")
print("\n" + "="*70)
print("ANALYSIS COMPLETE")
print("="*70)
