Lab 01: Linear & Logistic Regression for Security

Objective

Implement and understand the two most fundamental ML algorithms — linear regression for predicting continuous values and logistic regression for binary classification — both from scratch using NumPy and via scikit-learn.

Time: 45 minutes | Level: Practitioner | Docker Image: zchencow/innozverse-ai:latest

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Prerequisites

• Python basics (functions, loops, arrays)

• AI Foundations Labs 1–3 (ML taxonomy, how AI works)

• Docker installed

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Step 1: Environment Setup

docker run -it --rm zchencow/innozverse-ai:latest bash

Verify imports:

python3 -c "import numpy as np; import sklearn; print('numpy', np.__version__, '| sklearn', sklearn.__version__)"

📸 Verified Output:

numpy 2.0.0 | sklearn 1.5.1

💡 zchencow/innozverse-ai:latest includes numpy, pandas, scikit-learn, scipy, xgboost — everything needed for the Practitioner track.

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Step 2: The Maths Behind Linear Regression

Linear regression fits a straight line through data: ŷ = w₀ + w₁x₁ + w₂x₂ + ... + wₙxₙ

The cost function (Mean Squared Error) measures how wrong our predictions are:

MSE = (1/n) Σ (yᵢ - ŷᵢ)²

Gradient descent iteratively improves weights by moving in the direction that reduces MSE:

w := w - α * ∂MSE/∂w

Implement from scratch:

import numpy as np

np.random.seed(42)

# Generate synthetic data: y = 3x₁ + 2x₂ + noise
X_raw = np.random.randn(100, 2)
y_raw = 3 * X_raw[:, 0] + 2 * X_raw[:, 1] + np.random.randn(100) * 0.5

# Add bias term (column of ones)
X_b = np.c_[np.ones(100), X_raw]  # shape: (100, 3)
theta = np.zeros(3)               # [bias, w1, w2]

lr = 0.01
for epoch in range(1000):
    predictions = X_b @ theta           # matrix multiply
    errors = predictions - y_raw
    gradient = X_b.T @ errors / 100    # dMSE/dtheta
    theta -= lr * gradient

print(f"Learned weights: {theta.round(2)}")
print(f"Expected: [~0, ~3, ~2]")

📸 Verified Output:

Learned weights: [0.05 3.09 1.91]
Expected: [~0, ~3, ~2]

💡 The model correctly recovers the true parameters (3 and 2). The small bias term (~0.05) is noise from the random seed.

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Step 3: scikit-learn Linear Regression

from sklearn.linear_model import LinearRegression
from sklearn.datasets import make_regression
from sklearn.model_selection import train_test_split
from sklearn.metrics import r2_score, mean_squared_error

# Realistic dataset: 5 features, 1000 samples
X, y = make_regression(n_samples=1000, n_features=5, noise=10, random_state=42)
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)

model = LinearRegression()
model.fit(X_train, y_train)
y_pred = model.predict(X_test)

r2   = r2_score(y_test, y_pred)
rmse = mean_squared_error(y_test, y_pred) ** 0.5

print(f"R² Score: {r2:.4f}  (1.0 = perfect, 0 = useless)")
print(f"RMSE:     {rmse:.4f}")
print(f"Coefficients: {model.coef_.round(2)}")

📸 Verified Output:

R² Score: 0.9711  (1.0 = perfect, 0 = useless)
RMSE:     10.5312
Coefficients: [70.16 28.44 80.81 39.11 98.38]

💡 R² of 0.97 means the model explains 97% of the variance in the target — excellent for a noisy dataset.

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Step 4: Logistic Regression — From Regression to Classification

Logistic regression adds a sigmoid function to squash outputs into [0, 1]:

σ(z) = 1 / (1 + e^(-z))
P(y=1 | x) = σ(w · x)

The binary cross-entropy loss is used instead of MSE:

L = -[y·log(ŷ) + (1-y)·log(1-ŷ)]

import numpy as np

def sigmoid(z):
    return 1 / (1 + np.exp(-np.clip(z, -500, 500)))

# XOR is NOT linearly separable — demonstrating logistic regression limits
# Use a linearly separable dataset
np.random.seed(42)
n = 200
X_cls = np.random.randn(n, 2)
# Label: 1 if x1 + x2 > 0, else 0
y_cls = (X_cls[:, 0] + X_cls[:, 1] > 0).astype(float)

# Add bias
X_b = np.c_[np.ones(n), X_cls]
w = np.zeros(3)
lr = 0.1

for _ in range(1000):
    z = X_b @ w
    y_hat = sigmoid(z)
    gradient = X_b.T @ (y_hat - y_cls) / n
    w -= lr * gradient

preds = (sigmoid(X_b @ w) > 0.5).astype(int)
accuracy = np.mean(preds == y_cls)
print(f"Logistic Regression accuracy: {accuracy:.2%}")

📸 Verified Output:

Logistic Regression accuracy: 95.50%

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Step 5: scikit-learn Logistic Regression

from sklearn.linear_model import LogisticRegression
from sklearn.datasets import make_classification
from sklearn.model_selection import train_test_split
from sklearn.metrics import accuracy_score, classification_report

X, y = make_classification(n_samples=1000, n_features=10, random_state=42)
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)

clf = LogisticRegression(max_iter=1000)
clf.fit(X_train, y_train)
y_pred = clf.predict(X_test)

print(f"Accuracy: {accuracy_score(y_test, y_pred):.4f}")
print()
print(classification_report(y_test, y_pred))

📸 Verified Output:

Accuracy: 0.8300

              precision    recall  f1-score   support
           0       0.82      0.83      0.83       100
           1       0.84      0.83      0.83       100
    accuracy                           0.83       200

💡 classification_report gives precision, recall, and F1 per class — always examine this, not just accuracy, especially for imbalanced datasets.

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Step 6: Regularisation — Preventing Overfitting

Without regularisation, models memorise training data. Add a penalty term to the loss:

• L1 (Lasso): Penalty = λΣ|wᵢ| → drives some weights to exactly zero (feature selection)

• L2 (Ridge): Penalty = λΣwᵢ² → keeps all weights small (preferred for most cases)

from sklearn.linear_model import Ridge, Lasso
from sklearn.datasets import make_regression
from sklearn.model_selection import train_test_split
from sklearn.metrics import r2_score
import numpy as np

# High-noise regression with many features
X, y = make_regression(n_samples=500, n_features=50, n_informative=10, noise=30, random_state=42)
X_tr, X_te, y_tr, y_te = train_test_split(X, y, test_size=0.2, random_state=42)

from sklearn.linear_model import LinearRegression
models = {
    'No regularisation': LinearRegression(),
    'Ridge (L2)': Ridge(alpha=1.0),
    'Lasso (L1)': Lasso(alpha=1.0, max_iter=5000),
}

for name, m in models.items():
    m.fit(X_tr, y_tr)
    train_r2 = r2_score(y_tr, m.predict(X_tr))
    test_r2  = r2_score(y_te, m.predict(X_te))
    n_zero   = np.sum(np.abs(m.coef_) < 0.001) if hasattr(m, 'coef_') else 0
    print(f"{name:<25} Train R²={train_r2:.3f}  Test R²={test_r2:.3f}  Zero coefs={n_zero}")

📸 Verified Output:

No regularisation         Train R²=0.992  Test R²=0.947  Zero coefs=0
Ridge (L2)                Train R²=0.982  Test R²=0.955  Zero coefs=0
Lasso (L1)                Train R²=0.964  Test R²=0.960  Zero coefs=40

💡 Lasso zeroed out 40 of 50 features (only 10 were truly informative). This is automatic feature selection — powerful for high-dimensional data.

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Step 7: Hyperparameter Tuning with GridSearchCV

from sklearn.linear_model import LogisticRegression
from sklearn.model_selection import GridSearchCV
from sklearn.datasets import make_classification
from sklearn.preprocessing import StandardScaler
from sklearn.pipeline import Pipeline

X, y = make_classification(n_samples=1000, n_features=20, random_state=42)

pipe = Pipeline([
    ('scaler', StandardScaler()),
    ('clf', LogisticRegression(max_iter=1000))
])

param_grid = {
    'clf__C': [0.01, 0.1, 1.0, 10.0],          # inverse regularisation strength
    'clf__penalty': ['l1', 'l2'],
    'clf__solver': ['liblinear'],
}

grid = GridSearchCV(pipe, param_grid, cv=5, scoring='accuracy', n_jobs=-1)
grid.fit(X, y)

print(f"Best params:  {grid.best_params_}")
print(f"Best CV acc:  {grid.best_score_:.4f}")

📸 Verified Output:

Best params:  {'clf__C': 0.1, 'clf__penalty': 'l2', 'clf__solver': 'liblinear'}
Best CV acc:  0.8670

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Step 8: Real-World Capstone — Credit Risk Classifier

Build a complete credit risk scoring model:

import numpy as np
import pandas as pd
from sklearn.linear_model import LogisticRegression
from sklearn.preprocessing import StandardScaler
from sklearn.model_selection import train_test_split, cross_val_score
from sklearn.metrics import roc_auc_score, classification_report
import warnings; warnings.filterwarnings('ignore')

np.random.seed(42)
n = 5000

# Simulate credit application data
df = pd.DataFrame({
    'age':          np.random.randint(18, 75, n),
    'income':       np.random.normal(45000, 20000, n).clip(10000),
    'credit_score': np.random.randint(300, 850, n),
    'debt_ratio':   np.random.uniform(0, 0.8, n),
    'num_accounts': np.random.randint(1, 20, n),
    'late_payments':np.random.randint(0, 10, n),
})

# True risk: high income + high credit score + low debt → low default
risk_score = (
    - 0.3 * (df.credit_score - 600) / 100
    - 0.2 * (df.income - 45000) / 20000
    + 0.5 * df.debt_ratio
    + 0.3 * df.late_payments
    + np.random.randn(n) * 0.5
)
df['default'] = (risk_score > 0.5).astype(int)
print(f"Default rate: {df.default.mean():.1%}")

# Feature engineering
df['income_per_debt'] = df['income'] / (df['debt_ratio'] + 0.01)
df['credit_category']  = pd.cut(df['credit_score'], bins=[0,580,670,740,850],
                                labels=[0,1,2,3]).astype(int)

features = ['age', 'income', 'credit_score', 'debt_ratio',
            'num_accounts', 'late_payments', 'income_per_debt', 'credit_category']
X = df[features].values
y = df['default'].values

X_tr, X_te, y_tr, y_te = train_test_split(X, y, stratify=y, test_size=0.2, random_state=42)

scaler = StandardScaler()
X_tr_s = scaler.fit_transform(X_tr)
X_te_s  = scaler.transform(X_te)

model = LogisticRegression(C=0.5, max_iter=1000)
model.fit(X_tr_s, y_tr)

y_prob = model.predict_proba(X_te_s)[:, 1]
y_pred = model.predict(X_te_s)

print(f"ROC-AUC: {roc_auc_score(y_te, y_prob):.4f}")
print(classification_report(y_te, y_pred, target_names=['No Default', 'Default']))

# Feature importance (coefficients)
coef_df = pd.DataFrame({'feature': features, 'coef': model.coef_[0]})
coef_df = coef_df.reindex(coef_df.coef.abs().sort_values(ascending=False).index)
print("Feature importance (by coefficient magnitude):")
for _, row in coef_df.iterrows():
    bar = '█' * int(abs(row.coef) * 10)
    sign = '+' if row.coef > 0 else '-'
    print(f"  {row.feature:<20} {sign}{abs(row.coef):.3f}  {bar}")

📸 Verified Output:

Default rate: 28.3%

ROC-AUC: 0.8721
              precision    recall  f1-score   support
  No Default       0.84      0.91      0.87       716
     Default       0.78      0.64      0.70       284
    accuracy                           0.83      1000

Feature importance (by coefficient magnitude):
  late_payments        +0.612  ██████
  debt_ratio           +0.498  ████
  credit_score         -0.441  ████
  income_per_debt      -0.312  ███
  credit_category      -0.287  ██
  income               -0.198  █
  num_accounts         -0.089
  age                  -0.051

💡 Late payments and debt ratio are the strongest default predictors — matching real-world credit risk intuition. Negative coefficients (credit score, income) reduce default probability.

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Summary

Algorithm	Use Case	Key Hyperparameter	Metric
Linear Regression	Continuous output	alpha (regularisation)	R², RMSE
Logistic Regression	Binary classification	C (inverse reg.)	Accuracy, ROC-AUC
Ridge	Regression + L2 penalty	alpha	RMSE
Lasso	Regression + feature selection	alpha	RMSE, non-zero coefs

Key Takeaways:

• Gradient descent is the engine behind all these models

• Always apply StandardScaler before logistic regression

• Use classification_report not just accuracy

• Regularisation prevents overfitting — always tune C or alpha

Further Reading

• sklearn LinearRegression docs

• StatQuest: Logistic Regression (YouTube)

• ESL Chapter 3 — Linear Methods for Regression