crps

crps, function

def pure crps(r: ranvar, n: number): number
def pure crps(r1: ranvar, r2: ranvar): number

Computes the Continuous Ranked Probability Score (CRPS). With crps(r, n), the score is between the predictive distribution r and the observed value n. With crps(r1, r2), it is the extended CRPS between two predictive distributions.

• r: the predictive distribution to compare to observation n.
• n: the observed value.
• r1: the first predictive distribution.
• r2: the second predictive distribution.

Examples

r = poisson(3)
r1 = poisson(3)
r2 = poisson(7)

show table "CRPS" with
  crps(r, 4) as "crps(r, 4)"
  crps(r1, r2) as "crps(r1, r2)"

Output:

Remarks

The scalar overload crps(r, n) scores the forecast against the exact observation n. The two-ranvar overload compares their cumulative distribution functions; it is not the average scalar CRPS obtained by drawing observations from r2.

Let $X_1$ and $X_2$ be two random variables with cumulative distribution functions $F_1$ and $F_2$. The two-ranvar overload computes:

$$CRPS(X_1, X_2) = \int_{-\infty}^{+\infty} (F_1(x) - F_2(x))^2 dx$$

Do not replace an exact observation with dirac(n) merely to use the two-ranvar overload. Away from zero, the ranvar’s wider buckets can approximate the point mass, whereas the scalar observation is not bucketed.