### Abstract

For probability forecasts, the Brier score and Brier skill score are commonly used verification measures of forecast accuracy and skill. Using sampling theory, analytical expressions are derived to estimate their sampling uncertainties. The Brier score is an unbiased estimator of the accuracy, and an exact expression defines its sampling variance. The Brier skill score (with climatology as a reference forecast) is a biased estimator, and approximations are needed to estimate its bias and sampling variance. The uncertainty estimators depend only on the moments of the forecasts and observations, so it is easy to routinely compute them at the same time as the Brier score and skill score. The resulting uncertainty estimates can be used to construct error bars or confidence intervals for the verification measures, or perform hypothesis testing. Monte Carlo experiments using synthetic forecasting examples illustrate the performance of the expressions. In general, the estimates provide very reliable information on uncertainty. However, the quality of an estimate depends on both the sample size and the occurrence frequency of the forecast event. The examples also illustrate that with infrequently occurring events, verification sample sizes of a few hundred forecast-observation pairs are needed to establish that a forecast is skillful because of the large uncertainties that exist.

Original language | English |
---|---|

Pages (from-to) | 992-1006 |

Number of pages | 15 |

Journal | Weather and Forecasting |

Volume | 23 |

Issue number | 5 |

DOIs | |

Publication status | Published - Dec 1 2008 |

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### All Science Journal Classification (ASJC) codes

- Atmospheric Science

### Cite this

*Weather and Forecasting*,

*23*(5), 992-1006. https://doi.org/10.1175/2007WAF2007049.1

**Sampling uncertainty and confidence intervals for the brier score and brier skill score.** / Bradley, A. Allen; Schwartz, Stuart S.; Hashino, Tempei.

Research output: Contribution to journal › Article

*Weather and Forecasting*, vol. 23, no. 5, pp. 992-1006. https://doi.org/10.1175/2007WAF2007049.1

}

TY - JOUR

T1 - Sampling uncertainty and confidence intervals for the brier score and brier skill score

AU - Bradley, A. Allen

AU - Schwartz, Stuart S.

AU - Hashino, Tempei

PY - 2008/12/1

Y1 - 2008/12/1

N2 - For probability forecasts, the Brier score and Brier skill score are commonly used verification measures of forecast accuracy and skill. Using sampling theory, analytical expressions are derived to estimate their sampling uncertainties. The Brier score is an unbiased estimator of the accuracy, and an exact expression defines its sampling variance. The Brier skill score (with climatology as a reference forecast) is a biased estimator, and approximations are needed to estimate its bias and sampling variance. The uncertainty estimators depend only on the moments of the forecasts and observations, so it is easy to routinely compute them at the same time as the Brier score and skill score. The resulting uncertainty estimates can be used to construct error bars or confidence intervals for the verification measures, or perform hypothesis testing. Monte Carlo experiments using synthetic forecasting examples illustrate the performance of the expressions. In general, the estimates provide very reliable information on uncertainty. However, the quality of an estimate depends on both the sample size and the occurrence frequency of the forecast event. The examples also illustrate that with infrequently occurring events, verification sample sizes of a few hundred forecast-observation pairs are needed to establish that a forecast is skillful because of the large uncertainties that exist.

AB - For probability forecasts, the Brier score and Brier skill score are commonly used verification measures of forecast accuracy and skill. Using sampling theory, analytical expressions are derived to estimate their sampling uncertainties. The Brier score is an unbiased estimator of the accuracy, and an exact expression defines its sampling variance. The Brier skill score (with climatology as a reference forecast) is a biased estimator, and approximations are needed to estimate its bias and sampling variance. The uncertainty estimators depend only on the moments of the forecasts and observations, so it is easy to routinely compute them at the same time as the Brier score and skill score. The resulting uncertainty estimates can be used to construct error bars or confidence intervals for the verification measures, or perform hypothesis testing. Monte Carlo experiments using synthetic forecasting examples illustrate the performance of the expressions. In general, the estimates provide very reliable information on uncertainty. However, the quality of an estimate depends on both the sample size and the occurrence frequency of the forecast event. The examples also illustrate that with infrequently occurring events, verification sample sizes of a few hundred forecast-observation pairs are needed to establish that a forecast is skillful because of the large uncertainties that exist.

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UR - http://www.scopus.com/inward/citedby.url?scp=65549136074&partnerID=8YFLogxK

U2 - 10.1175/2007WAF2007049.1

DO - 10.1175/2007WAF2007049.1

M3 - Article

AN - SCOPUS:65549136074

VL - 23

SP - 992

EP - 1006

JO - Weather and Forecasting

JF - Weather and Forecasting

SN - 0882-8156

IS - 5

ER -