K-Fold Cross-Validation

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import numpy as np
import matplotlib.pyplot as plt
from sklearn.model_selection import cross_val_score, KFold
from sklearn.linear_model import LogisticRegression
from sklearn.datasets import make_classification
 
# Generate data
np.random.seed(42)
X, y = make_classification(n_samples=300, n_features=20, 
                           n_informative=10, random_state=42)
 
# Run 10-fold CV with many different random partitions
n_runs = 100
cv_means = []
 
for seed in range(n_runs):
    kfold = KFold(n_splits=10, shuffle=True, random_state=seed)
    scores = cross_val_score(
        LogisticRegression(max_iter=1000, random_state=42),  # Fixed model seed
        X, y, cv=kfold, scoring='accuracy'
    )
    cv_means.append(scores.mean())
 
cv_means = np.array(cv_means)
 
# Analyze the distribution
print("Partition Dependence Analysis (10-fold CV, 100 runs)")
print("=" * 60)
print(f"Mean of CV estimates: {cv_means.mean():.4f}")
print(f"Std of CV estimates:  {cv_means.std():.4f}")
print(f"Range: [{cv_means.min():.4f}, {cv_means.max():.4f}]")
print(f"Span: {(cv_means.max() - cv_means.min()):.4f}")
print()
print("If we reported just one run, we might get anything from")
print(f"{cv_means.min():.4f} to {cv_means.max():.4f} - a {(cv_means.max()-cv_means.min())*100:.1f}% point range!")
 
# Visualize
plt.figure(figsize=(12, 4))
 
plt.subplot(1, 2, 1)
plt.hist(cv_means, bins=20, edgecolor='black', alpha=0.7)
plt.axvline(cv_means.mean(), color='red', linestyle='--', 
            label=f'Mean = {cv_means.mean():.4f}')
plt.xlabel('CV Accuracy')
plt.ylabel('Frequency')
plt.title('Distribution of 10-Fold CV Estimates\n(100 different partitions)')
plt.legend()
 
plt.subplot(1, 2, 2)
plt.plot(range(n_runs), cv_means, 'o-', markersize=3, alpha=0.7)
plt.axhline(cv_means.mean(), color='red', linestyle='--')
plt.fill_between(range(n_runs), 
                 cv_means.mean() - cv_means.std(),
                 cv_means.mean() + cv_means.std(),
                 alpha=0.2, color='red')
plt.xlabel('Run Number (different random seed)')
plt.ylabel('CV Accuracy')
plt.title('CV Estimate Varies with Partition')
plt.tight_layout()
 
# Show that model randomness is not the issue
print("\n" + "=" * 60)
print("Confirming: Model randomness is NOT the source of variance")
print("=" * 60)
 
# Same partition, different model seeds
fixed_kfold = KFold(n_splits=10, shuffle=True, random_state=42)
model_var_scores = []
for model_seed in range(50):
    scores = cross_val_score(
        LogisticRegression(max_iter=1000, random_state=model_seed),
        X, y, cv=fixed_kfold
    )
    model_var_scores.append(scores.mean())
 
print(f"Fixed partition, varying model seed:")
print(f"  Std of CV estimates: {np.std(model_var_scores):.6f}")
print(f"\nVarying partition, fixed model seed:")
print(f"  Std of CV estimates: {cv_means.std():.6f}")
print(f"\nPartition variance is {cv_means.std()/np.std(model_var_scores):.1f}x larger!")

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import numpy as np
from sklearn.model_selection import (
    cross_val_score, RepeatedKFold, RepeatedStratifiedKFold
)
from sklearn.ensemble import RandomForestClassifier
from sklearn.datasets import make_classification
from typing import List, Tuple
import time
 
class RepeatedCVAnalysis:
    """
    Comprehensive repeated cross-validation analysis.
    """
    
    def __init__(self, n_splits: int = 10, n_repeats: int = 5,
                 random_state: int = 42):
        self.n_splits = n_splits
        self.n_repeats = n_repeats
        self.random_state = random_state
        
    def run_repeated_cv(self, X, y, model_factory, scoring='accuracy',
                        stratified=True, verbose=True):
        """
        Run repeated cross-validation with detailed analysis.
        """
        # Choose CV strategy
        if stratified:
            cv = RepeatedStratifiedKFold(
                n_splits=self.n_splits,
                n_repeats=self.n_repeats,
                random_state=self.random_state
            )
            cv_name = f"Repeated Stratified {self.n_splits}-Fold"
        else:
            cv = RepeatedKFold(
                n_splits=self.n_splits,
                n_repeats=self.n_repeats,
                random_state=self.random_state
            )
            cv_name = f"Repeated {self.n_splits}-Fold"
        
        # Run CV
        start = time.time()
        all_scores = cross_val_score(
            model_factory(), X, y, cv=cv, scoring=scoring
        )
        elapsed = time.time() - start
        
        # Reshape to (n_repeats, n_splits)
        scores_matrix = all_scores.reshape(self.n_repeats, self.n_splits)
        
        # Compute statistics
        results = self._compute_statistics(scores_matrix, all_scores)
        results['cv_name'] = cv_name
        results['time'] = elapsed
        results['total_folds'] = len(all_scores)
        
        if verbose:
            self._print_report(results)
        
        return results
    
    def _compute_statistics(self, scores_matrix, all_scores):
        """Compute comprehensive statistics."""
        # Overall statistics
        mean_score = all_scores.mean()
        std_score = all_scores.std()
        se_score = std_score / np.sqrt(len(all_scores))
        
        # Per-repetition means
        rep_means = scores_matrix.mean(axis=1)
        
        # Variance decomposition
        within_rep_var = scores_matrix.var(axis=1).mean()  # Avg variance within reps
        between_rep_var = rep_means.var()  # Variance between rep means
        
        # Confidence intervals
        ci_95 = (mean_score - 1.96 * se_score, mean_score + 1.96 * se_score)
        
        return {
            'mean': mean_score,
            'std': std_score,
            'se': se_score,
            'ci_95': ci_95,
            'rep_means': rep_means,
            'within_rep_var': within_rep_var,
            'between_rep_var': between_rep_var,
            'scores_matrix': scores_matrix,
            'all_scores': all_scores
        }
    
    def _print_report(self, results):
        """Print detailed report."""
        print(f"\n{'='*60}")
        print(f"Repeated CV Results: {results['cv_name']}")
        print(f"{'='*60}")
        print(f"Configuration: {self.n_repeats} repeats × {self.n_splits} folds "
              f"= {results['total_folds']} evaluations")
        print(f"Time: {results['time']:.2f}s")
        print()
        print("Overall Statistics:")
        print(f"  Mean: {results['mean']:.4f}")
        print(f"  Std:  {results['std']:.4f}")
        print(f"  SE:   {results['se']:.4f}")
        print(f"  95% CI: [{results['ci_95'][0]:.4f}, {results['ci_95'][1]:.4f}]")
        print()
        print("Variance Decomposition:")
        print(f"  Within-repetition variance:  {results['within_rep_var']:.6f}")
        print(f"  Between-repetition variance: {results['between_rep_var']:.6f}")
        print()
        print("Per-Repetition Means:")
        for i, mean in enumerate(results['rep_means']):
            print(f"  Rep {i+1}: {mean:.4f}")
 
# Demo
np.random.seed(42)
X, y = make_classification(n_samples=500, n_features=20,
                           n_informative=10, random_state=42)
 
analyzer = RepeatedCVAnalysis(n_splits=10, n_repeats=5, random_state=42)
results = analyzer.run_repeated_cv(
    X, y,
    lambda: RandomForestClassifier(n_estimators=100, random_state=42)
)

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import numpy as np
from sklearn.model_selection import cross_val_score, StratifiedKFold
from sklearn.ensemble import RandomForestClassifier
from sklearn.datasets import make_classification
 
def analyze_variance_components(X, y, model_factory, k=10, 
                                n_repetitions=30, random_state=42):
    """
    Decompose variance into within-fold and between-repetition components.
    """
    np.random.seed(random_state)
    
    all_fold_scores = []
    rep_means = []
    
    for rep in range(n_repetitions):
        # Different partition each repetition
        cv = StratifiedKFold(n_splits=k, shuffle=True, 
                            random_state=random_state + rep)
        
        fold_scores = cross_val_score(
            model_factory(), X, y, cv=cv, scoring='accuracy'
        )
        
        all_fold_scores.append(fold_scores)
        rep_means.append(fold_scores.mean())
    
    all_fold_scores = np.array(all_fold_scores)  # Shape: (n_reps, k)
    rep_means = np.array(rep_means)
    
    # Variance decomposition
    # Within-repetition: average variance across folds within each rep
    within_variances = all_fold_scores.var(axis=1)
    avg_within_var = within_variances.mean()
    
    # Between-repetition: variance of repetition means
    between_var = rep_means.var()
    
    # Total variance of individual fold scores
    total_var = all_fold_scores.var()
    
    print("Variance Decomposition Analysis")
    print("=" * 60)
    print(f"Configuration: {n_repetitions} repetitions × {k} folds")
    print(f"Total fold evaluations: {n_repetitions * k}")
    print()
    print(f"Within-repetition variance (avg): {avg_within_var:.6f}")
    print(f"Between-repetition variance:      {between_var:.6f}")
    print(f"Total variance of fold scores:    {total_var:.6f}")
    print()
    print(f"Between/Total ratio: {between_var/total_var:.2%}")
    print(f"Within/Total ratio:  {avg_within_var/total_var:.2%}")
    print()
    
    # Effective variance reduction from averaging
    # Variance of mean over r repetitions
    var_of_mean_1_rep = all_fold_scores[0].var() / k
    var_of_mean_r_reps = total_var / (n_repetitions * k)
    
    print("Variance of the mean estimate:")
    print(f"  Single repetition (k={k}): {var_of_mean_1_rep:.8f}")
    print(f"  {n_repetitions} repetitions:          {var_of_mean_r_reps:.8f}")
    print(f"  Variance reduction: {var_of_mean_1_rep/var_of_mean_r_reps:.1f}x")
    
    return {
        'within_var': avg_within_var,
        'between_var': between_var,
        'total_var': total_var,
        'all_scores': all_fold_scores,
        'rep_means': rep_means
    }
 
# Generate data
np.random.seed(42)
X, y = make_classification(n_samples=500, n_features=20, 
                           n_informative=10, random_state=42)
 
# Analyze variance for different models
print("\n" + "=" * 70)
print("Random Forest (moderate stability)")
print("=" * 70)
rf_results = analyze_variance_components(
    X, y,
    lambda: RandomForestClassifier(n_estimators=100, random_state=42)
)
 
print("\n" + "=" * 70)
print("How variance decreases with more repetitions")
print("=" * 70)
 
for r in [1, 2, 5, 10, 20]:
    rep_means = rf_results['rep_means'][:r]
    var_of_mean = np.var(rep_means) / r + rf_results['within_var'] / (r * 10)
    se = np.sqrt(var_of_mean)
    print(f"r={r:2d}: SE ≈ {se:.4f}, 95% CI width ≈ {3.92*se:.4f}")

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import numpy as np
import matplotlib.pyplot as plt
from sklearn.model_selection import RepeatedStratifiedKFold, cross_val_score
from sklearn.ensemble import RandomForestClassifier
from sklearn.datasets import make_classification
 
def analyze_repetition_effect(X, y, model_factory, k=10, 
                               max_reps=30, random_state=42):
    """
    Analyze how estimation quality improves with more repetitions.
    """
    # Run many repetitions
    all_scores = []
    for rep in range(max_reps):
        cv = RepeatedStratifiedKFold(
            n_splits=k, n_repeats=1, 
            random_state=random_state + rep
        )
        scores = cross_val_score(model_factory(), X, y, cv=cv)
        all_scores.extend(scores)
    
    all_scores = np.array(all_scores).reshape(max_reps, k)
    
    # Analyze cumulative effect of adding repetitions
    results = []
    for r in range(1, max_reps + 1):
        scores_so_far = all_scores[:r].flatten()
        
        mean = scores_so_far.mean()
        se = scores_so_far.std() / np.sqrt(len(scores_so_far))
        ci_width = 3.92 * se  # 95% CI
        
        results.append({
            'r': r,
            'mean': mean,
            'se': se,
            'ci_width': ci_width
        })
    
    # Print key points
    print("Effect of Repetitions on Estimate Quality")
    print("=" * 60)
    print(f"{'Reps':>5} {'Mean':>8} {'SE':>8} {'CI Width':>10} {'vs r=1':>10}")
    print("-" * 60)
    
    baseline_ci = results[0]['ci_width']
    for r in [1, 2, 3, 5, 10, 15, 20, 30]:
        if r <= max_reps:
            res = results[r-1]
            reduction = res['ci_width'] / baseline_ci
            print(f"{r:>5} {res['mean']:>8.4f} {res['se']:>8.4f} "
                  f"{res['ci_width']:>10.4f} {reduction:>10.1%}")
    
    return results
 
# Generate data
np.random.seed(42)
X, y = make_classification(n_samples=500, n_features=20, 
                           n_informative=10, random_state=42)
 
print()
results = analyze_repetition_effect(
    X, y,
    lambda: RandomForestClassifier(n_estimators=100, random_state=42)
)
 
# Visualize convergence
plt.figure(figsize=(12, 4))
 
plt.subplot(1, 2, 1)
reps = [r['r'] for r in results]
means = [r['mean'] for r in results]
ses = [r['se'] for r in results]
 
plt.plot(reps, means, 'b-', linewidth=2)
plt.fill_between(reps, 
                 np.array(means) - 1.96*np.array(ses),
                 np.array(means) + 1.96*np.array(ses), 
                 alpha=0.3)
plt.xlabel('Number of Repetitions')
plt.ylabel('Mean Accuracy')
plt.title('CV Estimate Stabilizes with More Repetitions')
plt.axhline(means[-1], color='red', linestyle='--', alpha=0.5)
 
plt.subplot(1, 2, 2)
plt.plot(reps, [r['ci_width'] for r in results], 'g-', linewidth=2)
plt.xlabel('Number of Repetitions')
plt.ylabel('95% CI Width')
plt.title('Confidence Interval Shrinks with Repetitions')
 
plt.tight_layout()
 
# Recommendation function
def recommend_repetitions(compute_budget, importance, model_stability='medium'):
    """
    Recommend number of repetitions based on constraints.
    """
    if importance == 'critical':
        base_r = 10
    elif importance == 'publication':
        base_r = 5
    elif importance == 'development':
        base_r = 3
    else:  # exploration
        base_r = 1
    
    # Adjust for model stability
    if model_stability == 'low':  # Unstable models need more
        base_r = int(base_r * 1.5)
    elif model_stability == 'high':  # Stable models need fewer
        base_r = max(1, int(base_r * 0.7))
    
    # Adjust for compute budget
    if compute_budget == 'low':
        base_r = min(base_r, 3)
    elif compute_budget == 'very_low':
        base_r = 1
    
    return base_r
 
print("\nRecommendation Examples:")
print("-" * 40)
for imp in ['exploration', 'development', 'publication', 'critical']:
    r = recommend_repetitions('medium', imp)
    print(f"Importance='{imp}': r={r}")

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import numpy as np
from sklearn.model_selection import RepeatedStratifiedKFold, cross_validate
from sklearn.base import clone
from dataclasses import dataclass
from typing import Dict, List, Any, Callable, Optional
import time
from joblib import Parallel, delayed
 
@dataclass
class RepeatedCVResult:
    """Container for repeated CV results."""
    mean: float
    std: float
    se: float
    ci_95: tuple
    all_scores: np.ndarray
    per_rep_means: np.ndarray
    config: Dict[str, Any]
    time_seconds: float
    
    def summary(self) -> str:
        return (f"{self.mean:.4f} ± {self.std:.4f} "
                f"(95% CI: [{self.ci_95[0]:.4f}, {self.ci_95[1]:.4f}])")
    
    def detailed_report(self) -> str:
        lines = [
            "=" * 60,
            "Repeated Cross-Validation Report",
            "=" * 60,
            f"Configuration: {self.config['n_repeats']}×{self.config['n_splits']} CV",
            f"Total evaluations: {len(self.all_scores)}",
            f"Time: {self.time_seconds:.2f}s",
            "",
            "Results:",
            f"  Mean:     {self.mean:.4f}",
            f"  Std:      {self.std:.4f}",
            f"  SE:       {self.se:.4f}",
            f"  95% CI:   [{self.ci_95[0]:.4f}, {self.ci_95[1]:.4f}]",
            "",
            "Per-Repetition Means:",
        ]
        for i, m in enumerate(self.per_rep_means):
            lines.append(f"  Rep {i+1}: {m:.4f}")
        
        return "\n".join(lines)
 
def run_repeated_cv(
    X: np.ndarray,
    y: np.ndarray,
    model_factory: Callable,
    n_splits: int = 10,
    n_repeats: int = 5,
    scoring: str = 'accuracy',
    random_state: int = 42,
    n_jobs: int = -1,
    return_estimators: bool = False
) -> RepeatedCVResult:
    """
    Production-quality repeated cross-validation.
    
    Parameters:
    -----------
    X : array-like of shape (n_samples, n_features)
    y : array-like of shape (n_samples,)
    model_factory : callable returning unfitted estimator
    n_splits : number of folds
    n_repeats : number of repetitions
    scoring : scoring metric
    random_state : random seed for reproducibility
    n_jobs : parallel jobs (-1 = all cores)
    return_estimators : whether to keep fitted models
    
    Returns:
    --------
    RepeatedCVResult with comprehensive results
    """
    cv = RepeatedStratifiedKFold(
        n_splits=n_splits,
        n_repeats=n_repeats,
        random_state=random_state
    )
    
    start_time = time.time()
    
    cv_results = cross_validate(
        model_factory(),
        X, y,
        cv=cv,
        scoring=scoring,
        return_train_score=True,
        return_estimator=return_estimators,
        n_jobs=n_jobs
    )
    
    elapsed = time.time() - start_time
    
    # Extract test scores
    all_scores = cv_results['test_score']
    
    # Reshape to (n_repeats, n_splits)
    scores_matrix = all_scores.reshape(n_repeats, n_splits)
    per_rep_means = scores_matrix.mean(axis=1)
    
    # Statistics
    mean = all_scores.mean()
    std = all_scores.std()
    se = std / np.sqrt(len(all_scores))
    ci_95 = (mean - 1.96 * se, mean + 1.96 * se)
    
    return RepeatedCVResult(
        mean=mean,
        std=std,
        se=se,
        ci_95=ci_95,
        all_scores=all_scores,
        per_rep_means=per_rep_means,
        config={
            'n_splits': n_splits,
            'n_repeats': n_repeats,
            'scoring': scoring,
            'random_state': random_state
        },
        time_seconds=elapsed
    )
 
# Example usage
from sklearn.ensemble import RandomForestClassifier
from sklearn.datasets import make_classification
 
# Generate data
np.random.seed(42)
X, y = make_classification(n_samples=1000, n_features=20,
                           n_informative=10, random_state=42)
 
# Run repeated CV
result = run_repeated_cv(
    X, y,
    lambda: RandomForestClassifier(n_estimators=100, random_state=42),
    n_splits=10,
    n_repeats=5,
    random_state=42
)
 
print(result.detailed_report())
 
# Compare models properly
print("\n" + "=" * 60)
print("Comparing Models with Repeated CV")
print("=" * 60)
 
from sklearn.linear_model import LogisticRegression
from sklearn.ensemble import GradientBoostingClassifier
 
models = [
    (lambda: LogisticRegression(max_iter=1000), "Logistic Regression"),
    (lambda: RandomForestClassifier(n_estimators=100), "Random Forest"),
    (lambda: GradientBoostingClassifier(n_estimators=100), "Gradient Boosting")
]
 
for model_factory, name in models:
    result = run_repeated_cv(X, y, model_factory, n_splits=10, n_repeats=5)
    print(f"{name:25s}: {result.summary()}")

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import numpy as np
from scipy import stats
from sklearn.model_selection import StratifiedKFold
from sklearn.ensemble import RandomForestClassifier, GradientBoostingClassifier
from sklearn.datasets import make_classification
 
def paired_cv_comparison(X, y, model_a_factory, model_b_factory,
                         n_splits=10, n_repeats=5, random_state=42):
    """
    Properly compare two models using paired repeated CV.
    """
    np.random.seed(random_state)
    
    all_diffs = []
    
    for rep in range(n_repeats):
        cv = StratifiedKFold(n_splits=n_splits, shuffle=True, 
                            random_state=random_state + rep)
        
        for train_idx, val_idx in cv.split(X, y):
            X_train, X_val = X[train_idx], X[val_idx]
            y_train, y_val = y[train_idx], y[val_idx]
            
            # Same train/val split for both models
            model_a = model_a_factory()
            model_b = model_b_factory()
            
            model_a.fit(X_train, y_train)
            model_b.fit(X_train, y_train)
            
            score_a = model_a.score(X_val, y_val)
            score_b = model_b.score(X_val, y_val)
            
            all_diffs.append(score_a - score_b)
    
    all_diffs = np.array(all_diffs)
    
    # Statistics
    mean_diff = all_diffs.mean()
    std_diff = all_diffs.std()
    se_diff = std_diff / np.sqrt(len(all_diffs))
    
    # One-sample t-test: is mean_diff significantly different from 0?
    t_stat, p_value = stats.ttest_1samp(all_diffs, 0)
    
    # Corrected t-test (Nadeau & Bengio, 2003)
    # Accounts for CV fold correlation
    n = len(X)
    n_test = n // n_splits
    n_train = n - n_test
    
    corrected_variance = (
        all_diffs.var() * (1/len(all_diffs) + n_test/n_train)
    )
    corrected_t = mean_diff / np.sqrt(corrected_variance)
    df = len(all_diffs) - 1
    corrected_p = 2 * (1 - stats.t.cdf(abs(corrected_t), df))
    
    print("Paired CV Comparison Results")
    print("=" * 60)
    print(f"Configuration: {n_repeats}×{n_splits} CV = {len(all_diffs)} pairs")
    print()
    print(f"Mean difference (A - B): {mean_diff:+.4f}")
    print(f"Std of differences: {std_diff:.4f}")
    print(f"95% CI for difference: "
          f"[{mean_diff - 1.96*se_diff:+.4f}, {mean_diff + 1.96*se_diff:+.4f}]")
    print()
    print(f"Naive t-test: t = {t_stat:.3f}, p = {p_value:.4f}")
    print(f"Corrected t-test: t = {corrected_t:.3f}, p = {corrected_p:.4f}")
    print()
    
    if corrected_p < 0.05:
        winner = "Model A" if mean_diff > 0 else "Model B"
        print(f"Conclusion: {winner} is significantly better (α=0.05)")
    else:
        print("Conclusion: No significant difference between models")
    
    return {
        'mean_diff': mean_diff,
        'std_diff': std_diff,
        'all_diffs': all_diffs,
        'naive_p': p_value,
        'corrected_p': corrected_p
    }
 
# Demo
np.random.seed(42)
X, y = make_classification(n_samples=500, n_features=20, 
                           n_informative=10, random_state=42)
 
result = paired_cv_comparison(
    X, y,
    lambda: RandomForestClassifier(n_estimators=100, random_state=42),
    lambda: GradientBoostingClassifier(n_estimators=100, random_state=42),
    n_splits=10,
    n_repeats=5
)

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import numpy as np
from sklearn.model_selection import (
    RepeatedStratifiedKFold, ShuffleSplit,
    cross_val_score
)
from sklearn.ensemble import RandomForestClassifier
from sklearn.datasets import make_classification
 
def compare_cv_strategies(X, y, model_factory, n_iterations=50):
    """
    Compare different repeated CV strategies.
    """
    strategies = [
        # Standard repeated 10-fold
        ("10×5 Repeated KFold", 
         RepeatedStratifiedKFold(n_splits=10, n_repeats=5, random_state=42)),
        
        # 5×2 CV for statistical testing
        ("5×2 CV (for t-test)",
         RepeatedStratifiedKFold(n_splits=2, n_repeats=5, random_state=42)),
        
        # Monte Carlo (Shuffle-Split)
        ("Monte Carlo (50 × 20%)",
         ShuffleSplit(n_splits=50, test_size=0.2, random_state=42)),
        
        # More folds, fewer repeats
        ("20×2 Repeated KFold",
         RepeatedStratifiedKFold(n_splits=20, n_repeats=2, random_state=42)),
    ]
    
    print("Comparison of CV Strategies")
    print("=" * 70)
    print(f"{'Strategy':<25} {'Mean':>8} {'Std':>8} {'SE':>8} {'Evals':>8}")
    print("-" * 70)
    
    for name, cv in strategies:
        scores = cross_val_score(model_factory(), X, y, cv=cv)
        print(f"{name:<25} {scores.mean():>8.4f} {scores.std():>8.4f} "
              f"{scores.std()/np.sqrt(len(scores)):>8.4f} {len(scores):>8}")
    
    return strategies
 
# Generate data
np.random.seed(42)
X, y = make_classification(n_samples=500, n_features=20, 
                           n_informative=10, random_state=42)
 
strategies = compare_cv_strategies(
    X, y,
    lambda: RandomForestClassifier(n_estimators=100, random_state=42)
)
 
# 5×2 CV t-test implementation
from scipy import stats
 
def five_times_two_cv_test(X, y, model_a_factory, model_b_factory, 
                            random_state=42):
    """
    Dietterich's 5×2 CV paired t-test.
    
    Designed to have approximately correct Type I error rate.
    """
    p_values = []  # Differences in first fold of each repetition
    differences_sq_sum = 0
    
    for rep in range(5):
        # Create 2-fold split
        np.random.seed(random_state + rep)
        indices = np.random.permutation(len(X))
        mid = len(X) // 2
        
        idx1, idx2 = indices[:mid], indices[mid:]
        
        X1, y1 = X[idx1], y[idx1]
        X2, y2 = X[idx2], y[idx2]
        
        # Fold 1: train on 1, test on 2
        model_a = model_a_factory().fit(X1, y1)
        model_b = model_b_factory().fit(X1, y1)
        p1_a = model_a.score(X2, y2)
        p1_b = model_b.score(X2, y2)
        
        # Fold 2: train on 2, test on 1
        model_a = model_a_factory().fit(X2, y2)
        model_b = model_b_factory().fit(X2, y2)
        p2_a = model_a.score(X1, y1)
        p2_b = model_b.score(X1, y1)
        
        d1 = p1_a - p1_b
        d2 = p2_a - p2_b
        
        p_values.append(d1)
        
        # Variance estimate for this repetition
        d_mean = (d1 + d2) / 2
        s_sq = (d1 - d_mean)**2 + (d2 - d_mean)**2
        differences_sq_sum += s_sq
    
    # 5×2 CV t-statistic
    p1 = p_values[0]
    t_stat = p1 / np.sqrt(differences_sq_sum / 5)
    
    # Approximation: t-distribution with 5 degrees of freedom
    p_value = 2 * (1 - stats.t.cdf(abs(t_stat), 5))
    
    print("5×2 CV Paired t-test Results")
    print("=" * 50)
    print(f"t-statistic: {t_stat:.4f}")
    print(f"p-value: {p_value:.4f}")
    print(f"Significant at α=0.05: {'Yes' if p_value < 0.05 else 'No'}")
    
    return {'t_stat': t_stat, 'p_value': p_value}
 
print("\n" + "=" * 70)
print("5×2 CV Paired t-test (Dietterich)")
print("=" * 70)
from sklearn.ensemble import GradientBoostingClassifier
 
result = five_times_two_cv_test(
    X, y,
    lambda: RandomForestClassifier(n_estimators=100, random_state=None),
    lambda: GradientBoostingClassifier(n_estimators=100, random_state=None)
)