CRPS for Normal Forecast

Problem

Compute the mean continuous ranked probability score for a normal predictive distribution: for each observation y, with z = (y - mu) / sigma, CRPS = sigma * (z * (2*Phi(z) - 1) + 2*phi(z) - 1/sqrt(pi)), where Phi and phi are the standard normal CDF and density.

Return 0.0 for empty observations or non-positive sigma.

Hint

CRPS generalizes absolute error and rewards sharp, calibrated forecasts.

Test cases

case 1

in: [[0],0,1]

expected: 0.23369497725510913

case 2

in: [[1.96],0,1]

expected: 1.414700556138123

case 3

in: [[0,1,-1],0,2]

expected: 0.5976680265098805

case 4

in: [[5],5,0]

expected: 0

Starter codePython

import math
def crps_normal_forecast(observations, mu, sigma):
    # Your code here
    pass

4 test cases · run your code against them in the editor