Continuous Ranked Probability Score Research Articles

Predictive Inference Based on Markov Chain Monte Carlo Output

SummaryIn Bayesian inference, predictive distributions are typically in the form of samples generated via Markov chain Monte Carlo or related algorithms. In this paper, we conduct a systematic analysis of how to make and evaluate probabilistic forecasts from such simulation output. Based on proper scoring rules, we develop a notion of consistency that allows to assess the adequacy of methods for estimating the stationary distribution underlying the simulation output. We then provide asymptotic results that account for the salient features of Bayesian posterior simulators and derive conditions under which choices from the literature satisfy our notion of consistency. Importantly, these conditions depend on the scoring rule being used, such that the choices of approximation method and scoring rule are intertwined. While the logarithmic rule requires fairly stringent conditions, the continuous ranked probability score yields consistent approximations under minimal assumptions. These results are illustrated in a simulation study and an economic data example. Overall, mixture‐of‐parameters approximations that exploit the parametric structure of Bayesian models perform particularly well. Under the continuous ranked probability score, the empirical distribution function is a simple and appealing alternative option.

Forecasting stock market in high and low volatility periods: a modified multifractal volatility approach

We propose a new multifractal volatility measure (MVA) and accordingly construct new types of multifractal volatility models. We compare these models to some alternatives, and consider long memory features, the benefits of leverage effects, and the presence of jumps. The analysis is applied to the multifractal volatility and realized volatility series of three representative indices, adopting six loss functions as guidance for the model selection. The forecast performance is evaluated and tested using model confident set (MCS) in both high- and low-volatility periods, as well as based on robustness checks on the continuous rank probability score statistics. The results show satisfactory performances for the new multifractal volatility models. Specifically, we find that our new multifractal volatility model significantly improves the one-day-ahead volatility forecasts in the high-volatility period. While in the low-volatility periods, the out-of-sample test results highlight the superiority of the traditional multifractal volatility models in the accuracy of volatility forecasting. These findings will be helpful for market volatility prediction and investment strategies.

Added-value of ensemble prediction system on the quality of solar irradiance probabilistic forecasts

Abstract Accurate solar forecasts is one of the most effective solution to enhance grid operations. As the solar resource is intrinsically uncertain, a growing interest for solar probabilistic forecasts is observed in the solar research community. In this work, we compare two approaches for the generation of day-ahead solar irradiance probabilistic forecasts. The first class of models termed as deterministic-based models generates probabilistic forecasts from a deterministic value of the irradiance predicted by a Numerical Weather Prediction (NWP) model. The second type of models denoted by ensemble-based models issues probabilistic forecasts through the calibration of an Ensemble Prediction System (EPS) or from information (such as mean and variance) derived from the ensemble. The verification of the probabilistic forecasts is made using a sound framework. A numerical score, the Continuous Ranked Probability Score (CRPS), is used to assess the overall performance of the different models. The decomposition of the CRPS into reliability and resolution provides a further detailed insight into the quality of the probabilistic forecasts. In addition, a new diagnostic tool which evaluates the contribution of the statistical moments of the forecast distributions to the CRPS is proposed. This tool denoted by MC-CRPS allows identifying the characteristics of an ensemble that have an impact on the quality of the probabilistic forecasts. The assessment of the different models is done on several sites experiencing very different climatic conditions. Results show a general superior performance of ensemble-based models as the gain in forecast quality measured by the CRPS ranges from 4% to 16% depending on the site.

Ensemble model output statistics for the separation of direct and diffuse components from 1-min global irradiance

Separation models split diffuse and direct components of solar radiation from the global horizontal radiation. At the moment, all separation models only issue predictions that are deterministic (as opposed to probabilistic). Since the best prediction is necessarily probabilistic, a parametric post-processing framework called the ensemble model output statistics (EMOS) is introduced in this paper, to make probabilistic predictions. EMOS takes the diffuse fractions predicted by an ensemble of existing 1-min separation models, and outputs a predictive distribution, with parameters optimized by maximum likelihood estimation. Clearly, the EMOS-based separation modeling goes beyond the current literature, in terms of uncertainty quantification.Eight popular separation models from the literature, with different architectures, are used to demonstrate the predictive power of EMOS. Using 1-min high-quality radiometric data from seven stations in the USA and four stations in Europe, it is found that Yang2 is the best stand-alone model with an average RMSE of 21.8%, in terms of direct normal irradiance prediction, contrasting the 26.3% of the previously reported best model, namely, Engerer2. On the other hand, the EMOS post-processed predictions have an average RMSE of 20.8%, which is lower than that of the best stand-alone model. Moreover, EMOS is shown superior to simple model averaging, in terms of continuous ranked probability score and ignorance score.

Ensemble solar forecasting using data-driven models with probabilistic post-processing through GAMLSS

Forecast performance of data-driven models depends on the local weather and climate regime, which makes model selection a tedious task for forecast practitioners. Ensemble forecasting, or forecast combination, is beneficial in such cases, in that, forecasts from multiple models are combined to form a final forecast. In ensemble forecasting, additional to the final deterministic-style forecasts, predictive distributions are also available, which can be used by grid operators for better decision-making. Such empirical predictive distributions are useful to represent the uncertainty associated with the forecasts. However, raw ensemble forecasts are often not calibrated, e.g., due to the lack of diversity in the ensemble members. The lack of ensemble spread is known as underdispersion, and it can be ameliorated through post-processing.This study aims to calibrate hourly ensemble clear-sky index forecasts, generated by 20 data-driven models, using both parametric and nonparametric post-processing techniques. Four years of data collected at 7 research-grade sites are used in the empirical part of the paper. Quantitative and qualitative methods are used to evaluate the performance of post-processing techniques in terms of calibration and sharpness. Post-processed ensemble forecasts outperform raw ensemble forecasts under all verification metrics. The proposed parametric post-processing technique, namely, generalized additive models for location, scale and shape, substantially reduces the continuous ranked probability score (CRPS) of the raw ensemble forecasts from 32–59 W/m2 to 25–45 W/m2 and quantile score from 16–30 W/m2 to 13–23 W/m2. In terms of CRPS skill score, the proposed method achieved 38–58% improvements over a climatology reference.

Prediction Skill of Extended Range 2-m Maximum Air Temperature Probabilistic Forecasts Using Machine Learning Post-Processing Methods

The extended range temperature prediction is of great importance for public health, energy and agriculture. The two machine learning methods, namely, the neural networks and natural gradient boosting (NGBoost), are applied to improve the prediction skills of the 2-m maximum air temperature with lead times of 1–35 days over East Asia based on the Environmental Modeling Center, Global Ensemble Forecast System (EMC-GEFS), under the Subseasonal Experiment (SubX) of the National Centers for Environmental Prediction (NCEP). The ensemble model output statistics (EMOS) method is conducted as the benchmark for comparison. The results show that all the post-processing methods can efficiently reduce the prediction biases and uncertainties, especially in the lead week 1–2. The two machine learning methods outperform EMOS by approximately 0.2 in terms of the continuous ranked probability score (CRPS) overall. The neural networks and NGBoost behave as the best models in more than 90% of the study area over the validation period. In our study, CRPS, which is not a common loss function in machine learning, is introduced to make probabilistic forecasting possible for traditional neural networks. Moreover, we extend the NGBoost model to atmospheric sciences of probabilistic temperature forecasting which obtains satisfying performances.

Load probability density forecasting by transforming and combining quantile forecasts

Compared with traditional deterministic load forecasting, probabilistic load forecasting (PLF) help us understand the potential risks in the power system operation by providing more information about future uncertainties of the loads. Quantile forecasting, as a kind of non-parametric probabilistic forecasting method, has been well developed and widely used in PLF. However, the results of quantile forecasts are discrete, which contain fewer details than density forecasts which provide the most comprehensive information. This paper proposes a novel day-ahead load probability density forecasting method by transforming and combining multiple quantile forecasts. The proposed method includes two main steps: transformation and combination. In the first step, the kernel density estimation method is used to transform the individual quantile forecast into the probability density curve; in the second step, an optimization problem is established to obtain the weighted combination of different probability density forecasts. The perturbation search method is applied to determine the optimal weight of each individual forecast. We demonstrate the effectiveness and superiority of our proposed method using comprehensive case studies on the real-world load data from Guangdong province in China, ISO New England (ISO-NE) in the US and Irish smart meter data. Case studies show that the combined model is robust to kernel function selection in the transformation step and has better forecasting performance. Compared with the best individual model, the purposed combined model has an accuracy improvement of 1.54% in the Guangdong dataset and 2.9% in the ISO-NE dataset in terms of the continuous ranked probability score. The proposed combination forecasting method can be robust in high volatility scenarios.

A Weighting Scheme in A Multi-Model Ensemble for Bias-Corrected Climate Simulation

A model weighting scheme is important in multi-model climate ensembles for projecting future changes. The climate model output typically needs to be bias corrected before it can be used. When a bias-correction (BC) is applied, equal model weights are usually derived because some BC methods cause the observations and historical simulation to match perfectly. This equal weighting is sometimes criticized because it does not take into account the model performance. Unequal weights reflecting model performance may be obtained from raw data before BC is applied. However, we have observed that certain models produce excessively high weights, while the weights generated in all other models are extremely low. This phenomenon may be partly due to the fact that some models are more fit or calibrated to the observations for a given applications. To address these problems, we consider, in this study, a hybrid weighting scheme including both equal and unequal weights. The proposed approach applies an “imperfect” correction to the historical data in computing their weights, while it applies ordinary BC to the future data in computing the ensemble prediction. We employ a quantile mapping method for the BC and a Bayesian model averaging for performance-based weighting. Furthermore, techniques for selecting the optimal correction rate based on the chi-square test statistic and the continuous ranked probability score are examined. Comparisons with ordinary ensembles are provided using a perfect model test. The usefulness of the proposed method is illustrated using the annual maximum daily precipitation as observed in the Korean peninsula and simulated by 21 models from the Coupled Model Intercomparison Project Phase 6.

Predictability of China’s Stock Market Returns Based on Combination of Distribution Forecasting Models

No consensus exists in the literature on whether stock prices can be predicted, with most existing studies employing point forecasting to predict returns. By contrast, this study adopts the new perspective of distribution forecasting to investigate the predictability of the stock market using the model combination strategy. Specifically, the Shanghai Composite Index and the Shenzhen Component Index are selected as research objects. Seven models – GARCH-norm, GARCH-sstd, EGARCH-sstd, EGARCH-sstd-M, one-component Beta-t-EGARCH, two-component Beta-t-EGARCH, and the EWMA-based nonparametric model – are employed to perform distribution forecasting of the returns. The results of out-of-sample forecasting evaluation show that none of the individual models is “qualified” in terms of predictive power. Therefore, three combinations of individual models were constructed: equal weight combination, log-likelihood score combination, and continuous ranked probability score combination. The latter two combinations were found to always have significant directional predictability and excess profitability, which indicates that the two combined models may be closer to the real data generation process; from the perspective of economic evaluation, they may have a predictive effect on the conditional return distribution in China’s stock market.

An approach to the verification of high-resolution ocean models using spatial methods

Abstract. The Met Office currently runs two operational ocean forecasting configurations for the North West European Shelf: an eddy-permitting model with a resolution of 7 km (AMM7) and an eddy-resolving model at 1.5 km (AMM15). Whilst qualitative assessments have demonstrated the benefits brought by the increased resolution of AMM15, particularly in the ability to resolve finer-scale features, it has been difficult to show this quantitatively, especially in forecast mode. Applications of typical assessment metrics such as the root mean square error have been inconclusive, as the high-resolution model tends to be penalised more severely, referred to as the double-penalty effect. This effect occurs in point-to-point comparisons whereby features correctly forecast but misplaced with respect to the observations are penalised twice: once for not occurring at the observed location, and secondly for occurring at the forecast location, where they have not been observed. An exploratory assessment of sea surface temperature (SST) has been made at in situ observation locations using a single-observation neighbourhood-forecast (SO-NF) spatial verification method known as the High-Resolution Assessment (HiRA) framework. The primary focus of the assessment was to capture important aspects of methodology to consider when applying the HiRA framework. Forecast grid points within neighbourhoods centred on the observing location are considered as pseudo ensemble members, so that typical ensemble and probabilistic forecast verification metrics such as the continuous ranked probability score (CRPS) can be utilised. It is found that through the application of HiRA it is possible to identify improvements in the higher-resolution model which were not apparent using typical grid-scale assessments. This work suggests that future comparative assessments of ocean models with different resolutions would benefit from using HiRA as part of the evaluation process, as it gives a more equitable and appropriate reflection of model performance at higher resolutions.

Assessing the Impact of Stochastic Perturbations in Cloud Microphysics using GOES-16 Infrared Brightness Temperatures

AbstractIn this study, infrared brightness temperatures (BTs) are used to examine how applying stochastic perturbed parameter (SPP) methodology to the widely used Thompson–Eidhammer cloud microphysics scheme impacts the cloud field in high-resolution forecasts. Modifications are made to add stochastic perturbations to three parameters controlling cloud generation and dissipation processes. Two five-member ensembles are generated, one using the microphysics parameter perturbations (SPP-MP) and another where white noise perturbations were added to potential temperature fields at initialization time (Control). The impact of the SPP method was assessed using simulated and observed GOES-16 BTs. This analysis uses pixel-based and object-based methods to assess the impact on the cloud field. Pixel-based methods revealed that the SPP-MP BTs are slightly more accurate than the Control BTs. However, too few pixels with a BT lower than 270 K result in a positive bias compared to the observations. A negative bias compared to the observations is observed when only analyzing lower BTs. The spread of the ensemble BTs was analyzed using the continuous ranked probability score differences, with the SPP-MP ensemble BTs having less (more) spread during May (January) compared to the Control. Object-based analysis using the Method for Object-Based Diagnostic Evaluation revealed the upper-level cloud objects are smaller in the SPP-MP ensemble than the Control but a lower bias exists in the SPP-MP BTs compared to the Control BTs when overlapping matching objects. However, there is no clear distinction between the SPP-MP and Control ensemble members during the evolution of objects, Overall, the SPP-MP perturbations result in lower BTs compared to the Control ensemble and more cloudy pixels.

A spatio-temporal model for Red Sea surface temperature anomalies

This paper details the approach of team Lancaster to the 2019 EVA data challenge, dealing with spatio-temporal modelling of Red Sea surface temperature anomalies. We model the marginal distributions and dependence features separately; for the former, we use a combination of Gaussian and generalised Pareto distributions, while the dependence is captured using a localised Gaussian process approach. We also propose a space-time moving estimate of the cumulative distribution function that takes into account spatial variation and temporal trend in the anomalies, to be used in those regions with limited available data. The team’s predictions are compared to results obtained via an empirical benchmark. Our approach performs well in terms of the threshold-weighted continuous ranked probability score criterion, chosen by the challenge organiser.

Leveraging Turbine-Level Data for Improved Probabilistic Wind Power Forecasting

This paper describes two methods for creating improved probabilistic wind power forecasts through the use of turbine-level data. The first is a feature engineering approach whereby deterministic power forecasts from the turbine level are used as explanatory variables in a wind farm level forecasting model. The second is a novel bottom-up hierarchical approach where the wind farm forecast is inferred from the joint predictive distribution of the power output from individual turbines. Notably, the latter produces probabilistic forecasts that are coherent across both turbine and farm levels, which the former does not. The methods are tested at two utility scale wind farms and are shown to provide consistent improvements of up to 5%, in terms of continuous ranked probability score compared to the best performing state-of-the-art benchmark model. The bottom-up hierarchical approach provides greater improvement at the site characterized by a complex layout and terrain, while both approaches perform similarly at the second location. We show that there is a clear benefit in leveraging readily available turbine-level information for wind power forecasting.

How to create an operational multi-model of seasonal forecasts?

Seasonal forecasts of variables like near-surface temperature or precipitation are becoming increasingly important for a wide range of stakeholders. Due to the many possibilities of recalibrating, combining, and verifying ensemble forecasts, there are ambiguities of which methods are most suitable. To address this we compare approaches how to process and verify multi-model seasonal forecasts based on a scientific assessment performed within the framework of the EU Copernicus Climate Change Service (C3S) Quality Assurance for Multi-model Seasonal Forecast Products (QA4Seas) contract C3S 51 lot 3. Our results underpin the importance of processing raw ensemble forecasts differently depending on the final forecast product needed. While ensemble forecasts benefit a lot from bias correction using climate conserving recalibration, this is not the case for the intrinsically bias adjusted multi-category probability forecasts. The same applies for multi-model combination. In this paper, we apply simple, but effective, approaches for multi-model combination of both forecast formats. Further, based on existing literature we recommend to use proper scoring rules like a sample version of the continuous ranked probability score and the ranked probability score for the verification of ensemble forecasts and multi-category probability forecasts, respectively. For a detailed global visualization of calibration as well as bias and dispersion errors, using the Chi-square decomposition of rank histograms proved to be appropriate for the analysis performed within QA4Seas.

DDOS-attacks detection using an efficient measurement-based statistical mechanism

A monitoring mechanism is vital for detecting malicious attacks against cyber systems. Detecting denial of service (DOS) and distributed DOS (DDOS) is one of the most important security challenges facing network technologies. This paper introduces a reliable detection mechanism based on the continuous ranked probability score (CRPS) statistical metric and exponentially smoothing (ES) scheme for enabling efficient detection of DOS and DDOS attacks. In this regard, the CRPS is used to quantify the dissimilarity between a new observation and the distribution of normal traffic. The ES scheme, which is sensitive in detecting small changes, is applied to CRPS measurements for anomaly detection. Moreover, in CRPS-ES approach, a nonparametric decision threshold computed via kernel density estimation is used to suitably detect anomalies. Tests on three publically available datasets proclaim the efficiency of the proposed mechanism in detecting cyber-attacks.

Assessment of NCMRWF Global Ensemble System with differing ensemble populations for Tropical cyclone prediction

Ensemble Prediction System (EPS) estimates the probability distribution of atmospheric states through multiple integrations of the Numerical Weather Prediction (NWP) model/s with number of perturbed initial conditions. The choice of an ensemble size can affect the skill of the ensemble forecast. Since more ensemble members require more computational requirements, it is important to determine whether the larger ensemble size significantly improves the forecast uncertainty estimates. The EPS at National Centre for Medium Range Weather Forecasting (NEPS) is assessed for the prediction of track and intensity of Tropical Cyclones (TC) over Bay of Bengal (BoB). Medium range forecast of four tropical cyclones over BoB with three different ensemble sizes namely E11 (11 perturbed members), E22 (22 perturbed members) and E44 (44 perturbed members) are presented here. The results of the study indicate that doubling of the ensemble population reduced the mean initial position errors by 25% and increase of ensemble size by four times (E11 to E44) reduced the errors by about 38%. The ensemble mean forecast track has lower errors compared to the deterministic forecast (CNTL) at shorter lead times for all ensemble sizes. The impact of ensemble size is significant on the improvement of mean track prediction skill at longer lead times. Median of central pressure error is very close to zero for all the sizes and the forecasts are slightly underestimated compared to the observed values. The initial error in Maximum Sustained surface Wind speed (MSW) is smaller for E44 as compared to E22 and E11. Interestingly, in all lead time forecasts and all the ensemble sizes, ensemble mean MSW errors are nearly within ±5 ms−1. There is only slight improvement in the ensemble mean precipitation forecasts with the increase of ensemble size. NEPS fails to resolve the heavier rainfall distribution and has the tendency to over-predict light precipitation. Ensembles with larger size predict the TC strike area of intense storms with higher probabilities due to better consensus among member tracks fueled by improved steering flow. But few cases have shown E11 or E22 perform equally well in forecasting TC propagation and in some cases even better compared to E44. Threshold oriented verification was carried out for winds at 850 hPa considering all the cases collectively. Brier Score (BS) and its components are very close to one another for higher thresholds of winds. In general, E44 has lower BS for all different thresholds with small lead time forecasts. Smaller ensembles have slightly better reliability while large ensembles improved the resolution for lower thresholds of wind speed. Continuous ranked probability score indicate that predicted probability distribution agrees well with the analyzed cumulative distribution. Area under relative operating characteristic curve exhibits better discriminating ability of probabilistic events with larger ensemble sizes. This study clearly showed that an increase in the ensemble size has significant beneficial impact on the reduction of the percentage of outliers thereby improving the potential predictive skill of the unexpected events.

Online Aggregation of Probabilistic Forecasts Based on the Continuous Ranked Probability Score

Abstract—Methods for generating predictions online and in the form of probability distributions of future outcomes are considered. The difference between the probabilistic forecast (probability distribution) and the numerical outcome is measured using the loss function (scoring rule). In practical statistics, the continuous ranked probability score (CRPS) is often used to estimate the discrepancy between probabilistic forecasts and (quantitative) outcomes. The paper considers the case when several competing methods (experts) give their online predictions as distribution functions. An algorithm is proposed for online aggregation of these distribution functions. The performance bounds of the proposed algorithm are obtained in the form of a comparison of the cumulative loss of the algorithm and the loss of expert hypotheses. Unlike existing estimates, the proposed estimates do not depend on time. The results of numerical experiments illustrating the proposed methods are presented.

Data Assimilation for Streamflow Forecasting Using Extreme Learning Machines and Multilayer Perceptrons

AbstractData assimilation allows for updating state variables in a model to represent the initial condition of a catchment more accurately than the initial OpenLoop simulation. In hydrology, data assimilation is often a prerequisite for forecasting. According to Hornik (1991,https://doi.org/10.1016/0893-6080(91)90009-T) artificial neural networks can learn any nonlinear relationship between inputs and outputs. Here, we hypothesize that neural networks could learn the relationship between the simulated streamflow (from a hydrological model) and the corresponding state variables. Once learned, this relationship can be used to obtain corrected state variables by applying it to observed rather than simulated streamflow. Based on this, we propose a novel, ensemble‐based, data assimilation approach. As a proof of concept and to verify the abovementioned hypothesis, we used an international testbed comprising four hydrologically dissimilar catchments. We applied the new data assimilation method to the lumped hydrological model GR4J, which has two state variables. Within this framework, we compared two types of neural networks, namely, Extreme Learning Machine and the Multilayer Perceptron. Using well‐known metrics such as the continuous ranked probability score, we compared the assimilated streamflow series with the OpenLoop streamflow series and with the observed streamflow. We show that neural networks can be successfully used for data assimilation, with a noticeable improvement over the OpenLoop simulation for all catchments.

Ensemble forecasting system for the management of the Senegal River discharge: application upstream the Manantali dam

Providing useful inflow forecasts of the Manantali dam is critical for zonal consumption and agricultural water supply, power production, flood and drought control and management (Shin et al., Meteorol Appl 27:e1827, 2019). Probabilistic approaches through ensemble forecasting systems are often used to provide more rational and useful hydrological information. This paper aims at implementing an ensemble forecasting system at the Senegal River upper the Manantali dam. Rainfall ensemble is obtained through harmonic analysis and an ARIMA stochastic process. Cyclical errors that are within rainfall cyclical behavior from the stochastic modeling are settled and processed using multivariate statistic tools to dress a rainfall ensemble forecast. The rainfall ensemble is used as input to run the HBV-light to product streamflow ensemble forecasts. A number of 61 forecasted rainfall time series are then used to run already calibrated hydrological model to produce hydrological ensemble forecasts called raw ensemble. In addition, the affine kernel dressing method is applied to the raw ensemble to obtain another ensemble. Both ensembles are evaluated using on the one hand deterministic verifications such the linear correlation, the mean error, the mean absolute error and the root-mean-squared error, and on the other hand, probabilistic scores (Brier score, rank probability score and continuous rank probability score) and diagrams (attribute diagram and relative operating characteristics curve). Results are satisfactory as at deterministic than probabilistic scale, particularly considering reliability, resolution and skill of the systems. For both ensembles, correlation between the averages of the members and corresponding observations is about 0.871. In addition, the dressing method globally improved the performances of ensemble forecasting system. Thus, both schemes system can help decision maker of the Manantali dam in water resources management.

Ensemble model output statistics as a probabilistic site-adaptation tool for solar irradiance: A revisit

Previously, in the study by Yang [J. Renewable Sustainable Energy 12, 016102 (2020)], probabilistic site adaptation was demonstrated for the first time. This technique leverages the ensemble model output statistics (EMOS), post-processes the empirical distribution formed by m gridded solar irradiance estimates from different satellite-derived and reanalysis databases, and thus obtains a final predictive distribution of the site-adapted irradiance, which has a normal density. That said, three questions were later thought of: (1) can post-processing the clear-sky index, instead of irradiance, lead to better site-adaptation performance; (2) will the parameter estimation strategy substantially affect model performance; and (3) how does the normality assumption hold in reality? In this paper, I revisit the probabilistic site-adaptation problem and aim to address these questions. In summary, it is found that (1) building EMOS models on irradiance and on the clear-sky index leads to similar model performance; (2) the choice of minimizing the continuous ranked probability score and the ignorance score needs to be tailored to the problem at hand; and (3) using a truncated normal predictive distribution in EMOS does not seem to possess an advantage over using a normal predictive distribution.