Scoring rules assess the quality of probabilistic forecasts, by assigning a numerical score based on the predictive distribution and on the event or value that materializes. A scoring rule is proper if it encourages truthful reporting. It is local of order λ if the score depends on the predictive density only through its value and its derivatives of order up to λ at the observation. Previously, only a single local proper scoring rule had been known, namely the logarithmic score, which is local of order λ = 0. Here we introduce the Fisher score, which is a local proper scoring rule of order λ = 2. It relates to the Fisher information in the same way that the logarithmic score relates to the Kullback-Leibler information. The convex cone generated by the logarithmic score and the Fisher score exhausts the class of the local proper scoring rules of order λ ≤ 2, up to equivalence and regularity conditions. In a data example, we use local and non-local proper scoring rules to assess statistically postprocessed ensemble weather forecasts. Finally, we develop a multivariate version of the Fisher score. 1

Optimal probabilistic forecasts of integer-valued random variables are derived. The optimality is achieved by estimating the forecast distribution nonparametrically over a given broad model class and proving asymptotic efficiency in that setting. The ideas are demonstrated within the context of the integer autoregressive class of models, which is a suitable class for any count data that can be interpreted as a queue, stock, birth and death process or branching process. The theoretical proofs of asymptotic optimality are supplemented by simulation results which demonstrate the overall superiority of the nonparametric method relative to a misspecified parametric maximum likelihood estimator, in large but finite samples. The method is applied to counts of wage claim benefits, stock market iceberg orders and civilian deaths in Iraq, with bootstrap methods used to quantify sampling variation in the estimated forecast distributions.

was supported by NICHD grant R01 HD54511. Its contents are solely the responsibility of the authors and do not necessarily represent the official views of the National Institute of Child Health and Human Development. Also, the views expressed in this paper are those of the authors and do not necessarily reflect the views of the United Nations. Its contents have not been formally edited and cleared by the United Nations. The designations employed and the presentation of material in this paper do not imply the expression of any opinion whatsoever on the part of the United Nations concerning the legal status of any country, territory, city or area or of its authorities, or concerning the delimitation of its frontiers or boundaries. The authors are grateful to Leontine

Description Bayesian Model Averaging to create probabilistic forecasts from ensemble forecasts and

Abstract. In this article, we discuss a class of stochastic boosting algorithms, which corrects and develops the work of [23], showing how to perform statistical inference in a computationally efficient manner. Sequential Monte Carlo (SMC) methods are used to illustrate that the stochastic boosting methods can provide better predictions, for a higher computational cost, than the corresponding boosting algorithm. A theoretical result is also given, which expresses an upper-bound of the posterior-predictive test error, in terms of that of boosting. The result shows that the averaged predictions used, are relatively stable with respect to boosting, when the latter provides the single best prediction. We also investigate the method on a real case study from machine learning and in a regression context, showing that it can be a useful tool for data exploration.

Daily streamflow forecasting by machine learning methods with weather and climate inputs

The purpose of this paper is to outline a proposed seasonal water availability prediction service and, in particular, to describe the modelling components behind the service and their future development. In terms of its climate, Australia has experienced a remarkable decade. The continent has had its warmest period since records began and southern areas have been extremely dry. A seasonal water availability prediction service has been needed in Australia for many years and the Australian Government’s recent investment in water information will help address this need. A seasonal climate prediction service has been operating in the Bureau of Meteorology since 1989 but its primary focus has been on rainfall and temperature rather than water availability. Reliable seasonal predictions of streamflows are highly valuable and will have uses for providing water allocation outlooks, informing water markets, planning and managing water use and managing drought. The seasonal water availability prediction service will rely on the development and integration of a number of modelling systems. A statistical prediction system will be based on a Bayesian Joint Probability modelling approach and is expected to provide reliable predictions of ‘seasonal ’ streamflow at lead times of up to several months. Skill (or accuracy) in these predictions will generally exceed those for both temperature and rainfall. In parallel, a ‘dynamical ’ modelling approach will be developed whereby the outputs from climate

Given a nonlinear deterministic model, a density forecast is obtained by evolving forward an ensemble of starting values and doing density estimation with the final ensemble. The density forecasts will inevitably be downgraded by model misspecification. To mitigate model misspecification and enhance the quality of the predictive densities, one can mix them with the system’s climatology. This paper examines the effect of including the climatology on the sharpness and calibration of density forecasts at various time horizons. The density forecasts are estimated using a non-parametric approach. The findings have positive implications for issuing early warnings in different disciplines including economic applications and weather forecasting, but a non-linear electronic circuit is used as a test bed.

Short-range forecasts of precipitation fields are needed in a wealth of agricultural, hydrological, ecological and other applications. Forecasts from numerical weather prediction models are often biased and do not provide uncertainty information. Here we present a postprocessing technique for such numerical forecasts that produces correlated probabilistic forecasts of precipitation accumulation at multiple sites simultaneously. The statistical model is a spatial version of a two-stage model that represents the distribution of precipitation by a mixture of a point mass at zero and a Gamma density for the continuous distribution of precipitation accumulation. Spatial correlation is captured by assuming that two Gaussian processes drive precipitation occurrence and precipitation amount, respectively. The first process is latent and drives precipitation occurrence via a threshold. The second process explains the spatial correlation in precipitation accumulation. It is related to precipitation via a site-specific transformation function, so as to retain the marginal right-skewed distribution of precipitation while modeling spatial dependence. Both processes take into account the information contained in the numerical weather forecast and are modeled as stationary isotropic spatial processes with an exponential correlation function. The two-stage spatial model was applied to 48-hour-ahead forecasts of daily precipitation accumulation over the Pacific Northwest

Abstract. This note proves a weak type of the sharpness principle as conjectured by Gneiting, Balabdaoui, and Raftery [9] in connection with probabilistic forecasting subject to calibration constraints. A strong version of such a principle still awaits a proper formulation. 1.