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Hailiang Du

Abstract

The evaluation of probabilistic forecasts plays a central role both in the interpretation and in the use of forecast systems and their development. Probabilistic scores (scoring rules) provide statistical measures to assess the quality of probabilistic forecasts. Often, many probabilistic forecast systems are available while evaluations of their performance are not standardized, with different scoring rules being used to measure different aspects of forecast performance. Even when the discussion is restricted to strictly proper scoring rules, there remains considerable variability between them; indeed strictly proper scoring rules need not rank competing forecast systems in the same order when none of these systems are perfect. The locality property is explored to further distinguish scoring rules. The nonlocal strictly proper scoring rules considered are shown to have a property that can produce “unfortunate” evaluations. Particularly the fact that Continuous Rank Probability Score prefers the outcome close to the median of the forecast distribution regardless the probability mass assigned to the value at/near the median raises concern to its use. The only local strictly proper scoring rules, the logarithmic score, has direct interpretations in terms of probabilities and bits of information. The nonlocal strictly proper scoring rules, on the other hand, lack meaningful direct interpretation for decision support. The logarithmic score is also shown to be invariant under smooth transformation of the forecast variable, while the nonlocal strictly proper scoring rules considered may, however, change their preferences due to the transformation. It is therefore suggested that the logarithmic score always be included in the evaluation of probabilistic forecasts.

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Hailiang Du and Leonard A. Smith

Abstract

Data assimilation and state estimation for nonlinear models is a challenging task mathematically. Performing this task in real time, as in operational weather forecasting, is even more challenging as the models are imperfect: the mathematical system that generated the observations (if such a thing exists) is not a member of the available model class (i.e., the set of mathematical structures admitted as potential models). To the extent that traditional approaches address structural model error at all, most fail to produce consistent treatments. This results in questionable estimates both of the model state and of its uncertainty. A promising alternative approach is proposed to produce more consistent estimates of the model state and to estimate the (state dependent) model error simultaneously. This alternative consists of pseudo-orbit data assimilation with a stopping criterion. It is argued to be more efficient and more coherent than one alternative variational approach [a version of weak-constraint four-dimensional variational data assimilation (4DVAR)]. Results that demonstrate the pseudo-orbit data assimilation approach can also outperform an ensemble Kalman filter approach are presented. Both comparisons are made in the context of the 18-dimensional Lorenz96 flow and the two-dimensional Ikeda map. Many challenges remain outside the perfect model scenario, both in defining the goals of data assimilation and in achieving high-quality state estimation. The pseudo-orbit data assimilation approach provides a new tool for approaching this open problem.

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Hailiang Du and Leonard A. Smith

Abstract

State estimation lies at the heart of many meteorological tasks. Pseudo-orbit-based data assimilation provides an attractive alternative approach to data assimilation in nonlinear systems such as weather forecasting models. In the perfect model scenario, noisy observations prevent a precise estimate of the current state. In this setting, ensemble Kalman filter approaches are hampered by their foundational assumptions of dynamical linearity, while variational approaches may fail in practice owing to local minima in their cost function. The pseudo-orbit data assimilation approach improves state estimation by enhancing the balance between the information derived from the dynamic equations and that derived from the observations. The potential use of this approach for numerical weather prediction is explored in the perfect model scenario within two deterministic chaotic systems: the two-dimensional Ikeda map and 18-dimensional Lorenz96 flow. Empirical results demonstrate improved performance over that of the two most common traditional approaches of data assimilation (ensemble Kalman filter and four-dimensional variational assimilation).

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Leonard A. Smith, Hailiang Du, and Sarah Higgins

Abstract

Probabilistic forecasting is common in a wide variety of fields including geoscience, social science, and finance. It is sometimes the case that one has multiple probability forecasts for the same target. How is the information in these multiple nonlinear forecast systems best “combined”? Assuming stationarity, in the limit of a very large forecast–outcome archive, each model-based probability density function can be weighted to form a “multimodel forecast” that will, in expectation, provide at least as much information as the most informative single model forecast system. If one of the forecast systems yields a probability distribution that reflects the distribution from which the outcome will be drawn, Bayesian model averaging will identify this forecast system as the preferred system in the limit as the number of forecast–outcome pairs goes to infinity. In many applications, like those of seasonal weather forecasting, data are precious; the archive is often limited to fewer than 26 entries. In addition, no perfect model is in hand. It is shown that in this case forming a single “multimodel probabilistic forecast” can be expected to prove misleading. These issues are investigated in the surrogate model (here a forecast system) regime, where using probabilistic forecasts of a simple mathematical system allows many limiting behaviors of forecast systems to be quantified and compared with those under more realistic conditions.

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