Logistic Regression — Classification with Probabilities

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from sklearn.linear_model import LogisticRegression
from sklearn.datasets import load_breast_cancer
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import StandardScaler
from sklearn.metrics import classification_report, roc_auc_score

# THE SIGMOID FUNCTION -- core of logistic regression
z = np.linspace(-8, 8, 200)
sigmoid = 1 / (1 + np.exp(-z))

plt.figure(figsize=(8, 4))
plt.plot(z, sigmoid, "steelblue", linewidth=2.5)
plt.axhline(0.5, color="red", linestyle="--", alpha=0.7, label="Decision boundary (p=0.5)")
plt.fill_between(z, sigmoid, 0.5, where=(sigmoid > 0.5), alpha=0.1, color="green", label="Predict Class 1")
plt.fill_between(z, sigmoid, 0.5, where=(sigmoid < 0.5), alpha=0.1, color="red", label="Predict Class 0")
plt.xlabel("Linear combination z = w0 + w1*x1 + w2*x2 + ...")
plt.ylabel("Probability P(y=1)")
plt.title("Sigmoid Function -- converts linear score to probability")
plt.legend()
plt.tight_layout()
plt.savefig("sigmoid.png", dpi=100, bbox_inches="tight")
plt.show()

# BINARY CLASSIFICATION -- breast cancer
cancer = load_breast_cancer()
X, y = cancer.data, cancer.target
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42, stratify=y)

scaler = StandardScaler()
X_train_sc = scaler.fit_transform(X_train)
X_test_sc  = scaler.transform(X_test)

model = LogisticRegression(C=1.0, max_iter=1000, random_state=42)
model.fit(X_train_sc, y_train)

# PROBABILITIES (not just class labels!)
y_prob = model.predict_proba(X_test_sc)[:, 1]  # probability of class 1 (benign)
y_pred = model.predict(X_test_sc)

print("Binary Classification -- Breast Cancer:")
print(classification_report(y_test, y_pred, target_names=cancer.target_names))
print(f"AUC-ROC: {roc_auc_score(y_test, y_prob):.4f}")

# CUSTOM THRESHOLD -- adjust when recall vs precision matters
threshold = 0.35  # lower threshold = more positive predictions = higher recall
y_pred_thresh = (y_prob > threshold).astype(int)
print(f"\nWith threshold={threshold} (more sensitive to malignant):")
print(classification_report(y_test, y_pred_thresh, target_names=cancer.target_names))

# COEFFICIENT INTERPRETATION
coeff_df = pd.DataFrame({
    "Feature": cancer.feature_names[:10],
    "Coefficient": model.coef_[0][:10],
    "Odds_Ratio": np.exp(model.coef_[0][:10]),
}).sort_values("Coefficient", key=abs, ascending=False)
print("\nCoefficients and Odds Ratios (first 10 features):")
print(coeff_df.round(4).to_string(index=False))
print("  Odds ratio > 1: feature increases P(benign)")
print("  Odds ratio < 1: feature decreases P(benign)")

# MULTICLASS: one-vs-rest or multinomial
from sklearn.datasets import load_iris
iris = load_iris()
lr_multi = LogisticRegression(multi_class="multinomial", max_iter=1000, C=1.0, random_state=42)
lr_multi.fit(StandardScaler().fit_transform(iris.data), iris.target)
print(f"\nMulticlass accuracy on Iris: {lr_multi.score(StandardScaler().fit_transform(iris.data), iris.target):.4f}")