Educated decision making involves two major ingredients: probabilistic forecasts for future events or quantities and an assessment of predictive performance. This thesis focuses on the latter topic and illustrates its importance and implications from both theoretical and applied perspectives.
Receiver operating characteristic (ROC) curves are key tools for the assessment of predictions for binary events. Despite their popularity and ubiquitous use, the mathematical understanding of ROC curves is still incomplete. We establish the equivalence between ROC curves and cumulative distribution functions (CDFs) on the unit interval and elucidate the crucial role of concavity in interpreting and modeling ROC curves. Under this essential requirement, the classical binormal ROC model is strongly inhibited in its flexibility and we propose the novel beta ROC model as an alternative. For a class of models that includes the binormal and the beta model, we derive the large sample distribution of the minimum distance estimator. This allows for uncertainty quantification and statistical tests of goodness-of-fit or equal predictive ability. Turning to empirical examples, we analyze the suitability of both models and find empirical evidence for the increased flexibility of the beta model. A freely available software package called betaROC is currently prepared for release for the statistical programming language R.
Throughout the tropics, probabilistic forecasts for accumulated precipitation are of economic importance. However, it is largely unknown how skillful current numerical weather prediction (NWP) models are at timescales of one to a few days. For the first time, we systematically assess the quality of nine global operational NWP ensembles for three regions in northern tropical Africa, and verify against station and satellite-based observations and for the monsoon seasons 2007-2014. All examined NWP models are uncalibrated and unreliable, in particular for high probabilities of precipitation, and underperform in the prediction of amount and occurrence of precipitation when compared to a climatological reference forecast. Statistical postprocessing corrects systematic deficiencies and realizes the full potential of ensemble forecasts. Postprocessed forecasts are calibrated and reliable and outperform raw ensemble forecasts in all regions and monsoon seasons. Disappointingly however, they have predictive performance only equal to the climatological reference. This assessment is robust and holds for all examined NWP models, all monsoon seasons, accumulation periods of 1 to 5 days, and station and spatially aggregated satellite-based observations. Arguably, it implies that current NWP ensembles cannot translate information about the atmospheric state into useful information regarding occurrence or amount of precipitation. We suspect convective parameterization as likely cause of the poor performance of NWP ensemble forecasts as it has been shown to be a first-order error source for the realistic representation of organized convection in NWP models.
One may ask if the poor performance of NWP ensembles is exclusively confined to northern tropical Africa or if it applies to the tropics in general. In a comprehensive study, we assess the quality of two major NWP ensemble prediction systems (EPSs) for 1 to 5-day accumulated precipitation for ten climatic regions in the tropics and the period 2009-2017. In particular, we investigate their skill regarding the occurrence and amount of precipitation as well as the occurrence of extreme events. Both ensembles exhibit clear calibration problems and are unreliable and overconfident. Nevertheless, they are (slightly) skillful for most climates when compared to the climatological reference, except tropical and northern arid Africa and alpine climates. Statistical postprocessing corrects for the lack of calibration and reliability, and improves forecast quality. Postprocessed ensemble forecasts are skillful for most regions except the above mentioned ones.
The lack of NWP forecast skill in tropical and northern arid Africa and alpine climates calls for alternative approaches for the prediction of precipitation. In a pilot study for northern tropical Africa, we investigate whether it is possible to construct skillful statistical models that rely on information about recent rainfall events. We focus on the prediction of the probability of precipitation and find clear evidence for its modulation by recent precipitation events. The spatio-temporal correlation of rainfall coincides with meteorological assumptions, is reasonably pronounced and stable, and allows to construct meaningful statistical forecasts. We construct logistic regression based forecasts that are reliable, have a higher resolution than the climatological reference forecast, and yield an average improvement of 20% for northern tropical Africa and the period 1998-2014
