CRPS CDF Loss

Overview

When the target is a probability distribution (e.g., "what is the CDF of yards gained?"), encode the label as a step function over discrete bins, use a softmax output layer, then train with Continuous Ranked Probability Score (CRPS) loss. CRPS penalizes the squared difference between predicted and true CDFs, rewarding both calibration and sharpness.

Quick Start

import numpy as np
import tensorflow.keras.backend as K

def crps_loss(y_true, y_pred):
    """CRPS loss on cumulative softmax output."""
    return K.mean(K.square(y_true - K.cumsum(y_pred, axis=1)), axis=1)

def encode_cdf_target(values, n_bins=199, offset=99):
    """Encode scalar targets as step-function CDFs.
    
    Args:
        values: array of integer targets
        n_bins: number of discrete bins
        offset: bin index corresponding to target=0
    """
    y = np.zeros((len(values), n_bins))
    for i, v in enumerate(values):
        y[i, v + offset:] = 1.0
    return y

# Model
output = Dense(199, activation='softmax')(hidden)
model.compile(optimizer='adam', loss=crps_loss)

# CRPS evaluation callback
y_pred_cdf = np.clip(np.cumsum(model.predict(X_val), axis=1), 0, 1)
y_true_cdf = np.clip(np.cumsum(y_val, axis=1), 0, 1)
crps = np.mean(np.sum((y_true_cdf - y_pred_cdf) ** 2, axis=1)) / n_bins

Key Decisions

• Softmax output: guarantees predicted PDF sums to 1; cumsum gives valid CDF
• Bin count: 199 for yards (−99 to +99); adjust to match your target range
• Clip cumsum: numerical safety — ensures CDF stays in [0, 1]
• CRPS vs MSE: CRPS is a proper scoring rule for distributions; MSE on CDF works but lacks calibration incentive

References

• Source: neural-networks-feature-engineering-for-the-win
• Competition: NFL Big Data Bowl