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Proper scoring rules in Python

Project description

Proper scoring rules for evaluating probabilistic forecasts in Python. Evaluation methods that are “strictly proper” cannot be artificially improved through hedging, which makes them fair methods for accessing the accuracy of probabilistic forecasts. In particular, these rules are often used for evaluating weather forecasts.

properscoring runs on both Python 2 and 3. It requires NumPy (1.8 or later) and SciPy (any recent version should be fine). Numba is optional, but highly encouraged: it enables significant speedups (e.g., 20x faster) for crps_ensemble and threshold_brier_score.

To install, use pip: pip install properscoring.

Example: five ways to calculate CRPS

This library focuses on the closely related Continuous Ranked Probability Score (CRPS) and Brier Score. We like these scores because they are both interpretable (e.g., CRPS is a generalization of mean absolute error) and easily calculated from a finite number of samples of a probability distribution.

We will illustrate how to calculate CRPS against a forecast given by a Gaussian random variable. To begin, import properscoring:

import numpy as np
import properscoring as ps
from scipy.stats import norm

Exact calculation using crps_gaussian (this is the fastest method):

>>>> ps.crps_gaussian(0, mu=0, sig=1)

Numerical integration with crps_quadrature:

>>> ps.crps_quadrature(0, norm)

From a finite sample with crps_ensemble:

>>> ensemble = np.random.RandomState(0).randn(1000)
>>> ps.crps_ensemble(0, ensemble)

Weighted by PDF values with crps_ensemble:

>>> x = np.linspace(-5, 5, num=1000)
>>> ps.crps_ensemble(0, x, weights=norm.pdf(x))

Based on the threshold decomposition of CRPS with threshold_brier_score:

>>> threshold_scores = ps.threshold_brier_score(0, ensemble, threshold=x)
>>> (x[1] - x[0]) * threshold_scores.sum(axis=-1)

In this example, we only scored a single observation/forecast pair. But to reliably evaluate a forecast model, you need to average these scores across many observations. Fortunately, all scoring rules in properscoring happily accept and return observations as multi-dimensional arrays:

>>> ps.crps_gaussian([-2, -1, 0, 1, 2], mu=0, sig=1)
array([ 1.45279182,  0.60244136,  0.23369498,  0.60244136,  1.45279182])

Once you calculate an average score, is often useful to normalize them relative to a baseline forecast to calculate a so-called “skill score”, defined such that 0 indicates no improvement over the baseline and 1 indicates a perfect forecast. For example, suppose that our baseline forecast is to always predict 0:

>>> obs = [-2, -1, 0, 1, 2]
>>> baseline_score = ps.crps_ensemble(obs, [0, 0, 0, 0, 0]).mean()
>>> forecast_score = ps.crps_gaussian(obs, mu=0, sig=1).mean()
>>> skill = (baseline_score - forecast_score) / baseline_score
>>> skill

A standard normal distribution was 28% better at predicting these five observations.


properscoring contains optimized and extensively tested routines for scoring probability forecasts. These functions currently fall into two categories:

  • Continuous Ranked Probability Score (CRPS):

    • for an ensemble forecast: crps_ensemble

    • for a Gaussian distribution: crps_gaussian

    • for an arbitrary cumulative distribution function: crps_quadrature

  • Brier score:

    • for binary probability forecasts: brier_score

    • for threshold exceedances with an ensemble forecast: threshold_brier_score

All functions are robust to missing values represented by the floating point value NaN.


This library was written by researchers at The Climate Corporation. The original authors include Leon Barrett, Stephan Hoyer, Alex Kleeman and Drew O’Kane.


Copyright 2015 The Climate Corporation

Licensed under the Apache License, Version 2.0 (the “License”); you may not use this file except in compliance with the License. You may obtain a copy of the License at

Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an “AS IS” BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License.


Outside contributions (bug fixes or new features related to proper scoring rules) would be very welcome! Please open a GitHub issue to discuss your plans.

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