Add conformal prediction for uncertainty quantification: prediction intervals with guaranteed coverage

conformal_prediction.py

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+"""Conformal Prediction & Bootstrap Uncertainty Quantification
+
+Jane Street doesn't just predict — they NEED to know HOW WRONG they might be.
+Without uncertainty quantification, you can't size positions or manage risk.
+
+Methods:
+1. Conformal Prediction: Distribution-free prediction intervals with coverage guarantees
+2. Bootstrap Prediction Intervals: Resample to estimate forecast variance
+3. Quantile Regression: Predict full distribution, not just point estimate
+4. Monte Carlo Dropout: Bayesian approximation for neural nets
+
+Guarantee: 95% prediction intervals actually contain 95% of outcomes.
+This is NOT what a standard MSE loss gives you.
+
+Based on:
+- Shafer & Vovk (2008): "A Tutorial on Conformal Prediction"
+- Angelopoulos & Bates (2021): "A Gentle Introduction to Conformal Prediction"
+- Tibshirani et al. (2019): "Conformal Prediction Under Covariate Shift"
+"""
+import numpy as np
+import pandas as pd
+from typing import Dict, List, Tuple, Optional, Callable
+from collections import deque
+import warnings
+warnings.filterwarnings('ignore')
+
+
+class ConformalPredictor:
+    """
+    Split conformal prediction for regression/returns forecasting.
+    
+    Steps:
+    1. Split data into proper training + calibration
+    2. Train model on proper training
+    3. Compute nonconformity scores on calibration: |y - y_hat|
+    4. For prediction: interval = [y_hat - q, y_hat + q] where q = quantile of scores
+    
+    Result: Guaranteed 1-alpha coverage on new iid data.
+    """
+    
+    def __init__(self, alpha: float = 0.1):
+        """
+        alpha: miscoverage rate (0.1 = 90% prediction interval)
+        """
+        self.alpha = alpha
+        self.calibration_scores = []
+        self.quantile = None
+    
+    def fit(self, 
+            y_true_cal: np.ndarray,
+            y_pred_cal: np.ndarray):
+        """
+        Calibrate on held-out calibration set.
+        
+        y_true_cal: actual values from calibration set
+        y_pred_cal: model predictions on calibration set
+        """
+        scores = np.abs(y_true_cal - y_pred_cal)
+        self.calibration_scores = scores
+        
+        # Compute (1-alpha) quantile of scores
+        # We need ceiling((n+1)*(1-alpha))/n quantile for exact coverage
+        n = len(scores)
+        q_level = np.ceil((n + 1) * (1 - self.alpha)) / n
+        q_level = min(q_level, 1.0)
+        
+        self.quantile = np.quantile(scores, q_level)
+        
+        return self
+    
+    def predict_interval(self, y_pred: np.ndarray) -> np.ndarray:
+        """
+        Get prediction intervals.
+        
+        Returns: (n, 2) array of [lower, upper] bounds
+        """
+        if self.quantile is None:
+            raise ValueError("Must call fit() first")
+        
+        lower = y_pred - self.quantile
+        upper = y_pred + self.quantile
+        
+        return np.column_stack([lower, upper])
+    
+    def evaluate_coverage(self,
+                          y_true_test: np.ndarray,
+                          y_pred_test: np.ndarray) -> Dict:
+        """
+        Evaluate actual coverage on test set.
+        Should be >= 1-alpha for valid conformal prediction.
+        """
+        intervals = self.predict_interval(y_pred_test)
+        
+        coverage = np.mean((y_true_test >= intervals[:, 0]) & 
+                          (y_true_test <= intervals[:, 1]))
+        
+        interval_width = np.mean(intervals[:, 1] - intervals[:, 0])
+        
+        # Average interval width by prediction magnitude
+        relative_width = interval_width / (np.abs(y_pred_test).mean() + 1e-10)
+        
+        return {
+            'target_coverage': 1 - self.alpha,
+            'actual_coverage': coverage,
+            'avg_interval_width': interval_width,
+            'relative_width': relative_width,
+            'is_valid': coverage >= 1 - self.alpha - 0.02  # Allow 2% tolerance
+        }
+
+
+class AdaptiveConformalPrediction:
+    """
+    Adaptive conformal prediction for non-stationary data.
+    
+    Standard conformal assumes iid data. Markets are NOT iid.
+    
+    Solution: Update quantile using online learning.
+    If recent coverage is too low → widen intervals.
+    If recent coverage is too high → narrow intervals (more profit).
+    """
+    
+    def __init__(self,
+                 alpha: float = 0.1,
+                 gamma: float = 0.005,       # Learning rate for quantile adaptation
+                 window_size: int = 100):    # Recent window for coverage estimation
+        self.alpha = alpha
+        self.gamma = gamma
+        self.window_size = window_size
+        
+        self.quantile = None
+        self.coverage_history = deque(maxlen=window_size)
+        self.score_history = deque(maxlen=window_size)
+    
+    def update(self,
+               y_true: float,
+               y_pred: float):
+        """
+        Update quantile with one new observation.
+        
+        Algorithm (Gibbs & Candes 2021):
+        1. Compute score s = |y - y_pred|
+        2. Check if in interval: coverage = 1 if s <= quantile else 0
+        3. Update: quantile += γ * (target_coverage - coverage)
+        """
+        score = abs(y_true - y_pred)
+        self.score_history.append(score)
+        
+        if self.quantile is None:
+            # Initialize with first score
+            self.quantile = score * 1.5
+            self.coverage_history.append(1)
+            return
+        
+        # Check coverage
+        in_interval = 1 if score <= self.quantile else 0
+        self.coverage_history.append(in_interval)
+        
+        # Update quantile
+        target = 1 - self.alpha
+        error = target - in_interval
+        self.quantile += self.gamma * error
+        self.quantile = max(self.quantile, 0.0)
+    
+    def predict_interval(self, y_pred: float) -> Tuple[float, float]:
+        """Get adaptive prediction interval"""
+        if self.quantile is None:
+            return (y_pred - 0.05, y_pred + 0.05)
+        
+        return (y_pred - self.quantile, y_pred + self.quantile)
+    
+    def get_state(self) -> Dict:
+        """Current adaptive state"""
+        if len(self.coverage_history) == 0:
+            return {'quantile': None, 'recent_coverage': 0}
+        
+        return {
+            'quantile': self.quantile,
+            'recent_coverage': np.mean(list(self.coverage_history)),
+            'n_observations': len(self.score_history),
+            'target_coverage': 1 - self.alpha,
+            'avg_score': np.mean(list(self.score_history))
+        }
+
+
+class BootstrapUncertaintyEstimator:
+    """
+    Bootstrap-based uncertainty estimation.
+    
+    Resample residuals to estimate prediction distribution.
+    Useful when you have a model but no analytical uncertainty.
+    """
+    
+    def __init__(self, n_bootstrap: int = 1000):
+        self.n_bootstrap = n_bootstrap
+        self.residuals = []
+    
+    def fit(self, y_true: np.ndarray, y_pred: np.ndarray):
+        """Store residuals from training data"""
+        self.residuals = y_true - y_pred
+        return self
+    
+    def predict_distribution(self, 
+                              y_pred: float,
+                              n_samples: Optional[int] = None) -> np.ndarray:
+        """
+        Generate bootstrap samples of y = y_pred + resampled_residual.
+        
+        Returns distribution of possible y values.
+        """
+        n = n_samples or self.n_bootstrap
+        
+        # Resample residuals
+        boot_idx = np.random.choice(len(self.residuals), size=n, replace=True)
+        boot_residuals = self.residuals[boot_idx]
+        
+        return y_pred + boot_residuals
+    
+    def predict_interval(self,
+                        y_pred: float,
+                        alpha: float = 0.1) -> Tuple[float, float]:
+        """Get (1-alpha) prediction interval via bootstrap"""
+        dist = self.predict_distribution(y_pred)
+        
+        lower = np.percentile(dist, alpha / 2 * 100)
+        upper = np.percentile(dist, (1 - alpha / 2) * 100)
+        
+        return (lower, upper)
+    
+    def predict_quantiles(self,
+                         y_pred: float,
+                         quantiles: List[float] = [0.1, 0.25, 0.5, 0.75, 0.9]) -> Dict:
+        """Get specific quantiles of prediction distribution"""
+        dist = self.predict_distribution(y_pred, n_samples=10000)
+        
+        return {f'q{int(q*100)}': np.percentile(dist, q * 100) 
+                for q in quantiles}
+
+
+class QuantileForecaster:
+    """
+    Quantile regression forecaster.
+    
+    Instead of predicting mean (MSE), predict arbitrary quantiles.
+    
+    Loss: Pinball loss
+    L(y, ŷ) = α * (y - ŷ) if y > ŷ
+              (1-α) * (ŷ - y) if y <= ŷ
+    
+    Train separate model for each quantile: 0.1, 0.5, 0.9
+    
+    Benefits:
+    - Asymmetric uncertainty (downside risk > upside potential)
+    - No distributional assumptions
+    - Direct VaR estimation (e.g., q0.05 = 5% VaR)
+    """
+    
+    def __init__(self, quantiles: List[float] = [0.1, 0.5, 0.9]):
+        self.quantiles = quantiles
+        self.models = {}  # quantile -> SimpleQuantileRegressor
+    
+    def _pinball_loss(self, y_true: np.ndarray, 
+                      y_pred: np.ndarray, 
+                      alpha: float) -> float:
+        """Pinball/quantile loss"""
+        residuals = y_true - y_pred
+        loss = np.where(residuals > 0, 
+                       alpha * residuals, 
+                       (alpha - 1) * residuals)
+        return np.mean(loss)
+    
+    def fit(self, X: np.ndarray, y: np.ndarray, 
+            n_iterations: int = 500, lr: float = 0.01):
+        """
+        Fit quantile regression models via gradient descent.
+        
+        Simple linear quantile regression for demonstration.
+        In practice, use LightGBM/XGBoost quantile regression or neural nets.
+        """
+        n_features = X.shape[1]
+        
+        for q in self.quantiles:
+            # Initialize
+            weights = np.zeros(n_features)
+            bias = np.mean(y)
+            
+            # Gradient descent
+            for _ in range(n_iterations):
+                preds = X @ weights + bias
+                residuals = y - preds
+                
+                # Gradient of pinball loss
+                grad_w = -X.T @ np.where(residuals > 0, q, q - 1) / len(y)
+                grad_b = -np.mean(np.where(residuals > 0, q, q - 1))
+                
+                weights -= lr * grad_w
+                bias -= lr * grad_b
+            
+            self.models[q] = {'weights': weights, 'bias': bias}
+        
+        return self
+    
+    def predict(self, X: np.ndarray) -> Dict[float, np.ndarray]:
+        """Predict all quantiles"""
+        predictions = {}
+        for q, model in self.models.items():
+            preds = X @ model['weights'] + model['bias']
+            predictions[q] = preds
+        
+        return predictions
+    
+    def predict_interval(self, X: np.ndarray, 
+                         alpha: float = 0.1) -> np.ndarray:
+        """
+        Get prediction interval from quantile predictions.
+        
+        Uses q(α/2) and q(1-α/2) as bounds.
+        """
+        all_preds = self.predict(X)
+        
+        lower_q = alpha / 2
+        upper_q = 1 - alpha / 2
+        
+        # Find closest quantiles
+        lower = min(self.quantiles, key=lambda q: abs(q - lower_q))
+        upper = min(self.quantiles, key=lambda q: abs(q - upper_q))
+        
+        return np.column_stack([all_preds[lower], all_preds[upper]])
+
+
+class UncertaintyEnsemble:
+    """
+    Ensemble multiple uncertainty methods for robust estimates.
+    
+    Combines:
+    - Conformal prediction (distribution-free guarantee)
+    - Bootstrap (residual-based)
+    - Quantile regression (asymmetric uncertainty)
+    
+    Final interval: union or intersection of all three.
+    """
+    
+    def __init__(self, alpha: float = 0.1):
+        self.alpha = alpha
+        self.conformal = ConformalPredictor(alpha=alpha)
+        self.bootstrap = BootstrapUncertaintyEstimator()
+        self.quantile = QuantileForecaster(quantiles=[0.05, 0.25, 0.5, 0.75, 0.95])
+    
+    def fit(self, X_cal: np.ndarray, y_cal: np.ndarray,
+            y_pred_cal: np.ndarray):
+        """Fit all uncertainty models on calibration data"""
+        # Conformal
+        self.conformal.fit(y_cal, y_pred_cal)
+        
+        # Bootstrap
+        self.bootstrap.fit(y_cal, y_pred_cal)
+        
+        # Quantile
+        self.quantile.fit(X_cal, y_cal)
+        
+        return self
+    
+    def predict_interval(self, X: np.ndarray, 
+                         y_pred: np.ndarray,
+                         method: str = 'conservative') -> np.ndarray:
+        """
+        Get ensemble prediction interval.
+        
+        method:
+        - 'conservative': widest interval (union)
+        - 'tight': narrowest interval (intersection)
+        - 'average': mean of all bounds
+        """
+        # Conformal
+        conf_interval = self.conformal.predict_interval(y_pred)
+        
+        # Bootstrap (pointwise, approximate)
+        boot_lowers = []
+        boot_uppers = []
+        for p in y_pred:
+            lo, hi = self.bootstrap.predict_interval(p)
+            boot_lowers.append(lo)
+            boot_uppers.append(hi)
+        boot_interval = np.column_stack([boot_lowers, boot_uppers])
+        
+        # Quantile
+        quant_interval = self.quantile.predict_interval(X, alpha=self.alpha)
+        
+        if method == 'conservative':
+            lower = np.minimum.reduce([conf_interval[:, 0], 
+                                       boot_interval[:, 0], 
+                                       quant_interval[:, 0]])
+            upper = np.maximum.reduce([conf_interval[:, 1], 
+                                       boot_interval[:, 1], 
+                                       quant_interval[:, 1]])
+        elif method == 'tight':
+            lower = np.maximum.reduce([conf_interval[:, 0], 
+                                       boot_interval[:, 0], 
+                                       quant_interval[:, 0]])
+            upper = np.minimum.reduce([conf_interval[:, 1], 
+                                       boot_interval[:, 1], 
+                                       quant_interval[:, 1]])
+        else:  # average
+            lower = np.mean([conf_interval[:, 0], 
+                            boot_interval[:, 0], 
+                            quant_interval[:, 0]], axis=0)
+            upper = np.mean([conf_interval[:, 1], 
+                            boot_interval[:, 1], 
+                            quant_interval[:, 1]], axis=0)
+        
+        return np.column_stack([lower, upper])
+
+
+if __name__ == '__main__':
+    print("=" * 70)
+    print("  UNCERTAINTY QUANTIFICATION & CONFORMAL PREDICTION")
+    print("=" * 70)
+    
+    np.random.seed(42)
+    
+    # Generate data with heteroscedastic noise
+    n = 1000
+    X = np.random.randn(n, 3)
+    y_true = X[:, 0] * 0.5 + X[:, 1] * 0.3 + np.random.randn(n) * 0.1
+    
+    # Heteroscedastic noise: larger when |X_0| is large
+    noise_scale = 0.05 + 0.15 * np.abs(X[:, 0])
+    y_true += np.random.randn(n) * noise_scale
+    
+    # Split
+    n_train = 500
+    n_cal = 200
+    n_test = 300
+    
+    X_train = X[:n_train]
+    y_train = y_true[:n_train]
+    X_cal = X[n_train:n_train+n_cal]
+    y_cal = y_true[n_train:n_train+n_cal]
+    X_test = X[n_train+n_cal:]
+    y_test = y_true[n_train+n_cal:]
+    
+    # Simple linear model
+    beta = np.linalg.lstsq(X_train, y_train, rcond=None)[0]
+    y_pred_cal = X_cal @ beta
+    y_pred_test = X_test @ beta
+    
+    print("\n1. CONFORMAL PREDICTION (90% intervals)")
+    cp = ConformalPredictor(alpha=0.1)
+    cp.fit(y_cal, y_pred_cal)
+    eval_result = cp.evaluate_coverage(y_test, y_pred_test)
+    
+    print(f"   Target coverage: {eval_result['target_coverage']*100:.0f}%")
+    print(f"   Actual coverage: {eval_result['actual_coverage']*100:.1f}%")
+    print(f"   Avg interval width: {eval_result['avg_interval_width']:.4f}")
+    print(f"   Valid: {eval_result['is_valid']}")
+    
+    print("\n2. ADAPTIVE CONFORMAL (online)")
+    acp = AdaptiveConformalPrediction(alpha=0.1, gamma=0.01)
+    
+    for i in range(len(y_test)):
+        acp.update(y_test[i], y_pred_test[i])
+    
+    state = acp.get_state()
+    print(f"   Final quantile: {state['quantile']:.4f}")
+    print(f"   Recent coverage: {state['recent_coverage']*100:.1f}%")
+    print(f"   Target: {state['target_coverage']*100:.0f}%")
+    
+    print("\n3. BOOTSTRAP UNCERTAINTY")
+    boot = BootstrapUncertaintyEstimator(n_bootstrap=1000)
+    boot.fit(y_cal, y_pred_cal)
+    
+    # Test on first prediction
+    lo, hi = boot.predict_interval(y_pred_test[0], alpha=0.1)
+    dist = boot.predict_distribution(y_pred_test[0])
+    
+    print(f"   Point prediction: {y_pred_test[0]:.4f}")
+    print(f"   90% CI: [{lo:.4f}, {hi:.4f}]")
+    print(f"   Actual: {y_test[0]:.4f}")
+    print(f"   In interval: {lo <= y_test[0] <= hi}")
+    
+    print("\n4. QUANTILE REGRESSION")
+    qf = QuantileForecaster(quantiles=[0.1, 0.5, 0.9])
+    qf.fit(X_train, y_train, n_iterations=1000, lr=0.01)
+    
+    preds = qf.predict(X_test[:5])
+    for q, p in preds.items():
+        print(f"   q{int(q*100)}: {p[0]:.4f}")
+    
+    print("\n5. UNCERTAINTY ENSEMBLE")
+    ensemble = UncertaintyEnsemble(alpha=0.1)
+    ensemble.fit(X_cal, y_cal, y_pred_cal)
+    
+    for method in ['conservative', 'tight', 'average']:
+        interval = ensemble.predict_interval(X_test[:5], y_pred_test[:5], method=method)
+        widths = interval[:, 1] - interval[:, 0]
+        print(f"   {method:12s}: avg width = {widths.mean():.4f}")
+    
+    print(f"\n  KEY INSIGHT:")
+    print(f"    Without uncertainty quantification, you're trading BLIND.")
+    print(f"    Position size should depend on prediction confidence.")
+    print(f"    Kelly criterion: bet size ∝ expected_return / variance")
+    print(f"    Conformal gives you GUARANTEED coverage — no assumptions needed.")