Non-homogeneous Gaussian Regression Research Articles

Probabilistic post-processing of short to mediumrange temperature forecasts: Implications for heatwave prediction in India.

Accurate and reliable air temperature forecasts are necessary for predicting and responding to thermal disasters such as heat strokes. Forecasts from Numerical Weather Prediction(NWP) models contain biases which require post-processing. Studies assessing the skill of probabilistic post-processing techniques (PPTs) on temperature forecasts in India are lacking. This study aims to evaluate probabilistic post-processing approaches such as Nonhomogeneous Gaussian Regression (NGR) and Bayesian Model Averaging (BMA) for improving daily temperature forecasts from two NWP models, namely, the European Centre for Medium Range Weather Forecasts (ECMWF) and the Global Ensemble Forecast System (GEFS), across the Indian subcontinent. Apart from that, the effect of probabilistic PPT on heatwave prediction skills across India is also evaluated. Results show that probabilistic PPT comprehensively outperform traditional approaches in forecasting temperatures across India at all lead times. In the Himalayan regions where the forecast skill of raw forecasts is low, the probabilistic techniques are not able to produce skillful forecasts even though they perform much better than traditional techniques. The NGR method is found to be the best performing PPT across the Indian region. Post-processing Tmax forecasts using the NGR approach was found to considerably improve the heatwave prediction skill across highly heatwave prone regions in India. The outcomes of this study will be helpful in setting up improved heatwave prediction and early warning systems in India.

Comparing calibrated analog and dynamical ensemble solar forecasts

Ensemble modeling is a chief strategy for probabilistic forecasting. In weather forecasting, analog ensemble, which operates under the principle that weather patterns often repeat, and dynamical ensemble, which generates equally likely trajectories of future weather by perturbing the initial and boundary conditions, constitute the two most common approaches to making ensembles. That said, in the field of solar forecasting, nor is there any head-to-head comparison made thus far in regard to understanding the relative performance of these two competing approaches; this work seeks to fill the gap. Four years (2017–2020) of operational forecasts, at seven locations, from the European Centre for Medium-Range Weather Forecasts (ECMWF) are used, and both the raw and post-processed versions of the ensemble irradiance forecasts are verified in a fair and thorough fashion. Three classical post-processing methods, namely, Bayesian model averaging, nonhomogeneous Gaussian regression, and quantile regression, are applied to both the analog ensemble forecasts derived from ECMWF’s high-resolution control forecasts and dynamical ensemble forecasts from ECMWF’s Ensemble Prediction System. It is found that analog ensemble before post-processing possesses some advantage in terms of calibration over dynamical ensemble; their average reliability values are 0.6 W/m2 and 8.2 W/m2, respectively. However, dynamical ensemble after post-processing becomes generally more attractive, obtaining an average continuous ranked probability score of 49.0 W/m2, against 51.7 W/m2 for AnEn.

Gaussian mixture models for clustering and calibration of ensemble weather forecasts

<p style='text-indent:20px;'>Nowadays, most weather forecasting centers produce ensemble forecasts. Ensemble forecasts provide information about probability distribution of the weather variables. They give a more complete description of the atmosphere than a unique run of the meteorological model. However, they may suffer from bias and under/over dispersion errors that need to be corrected. These distribution errors may depend on weather regimes. In this paper, we propose various extensions of the Gaussian mixture model and its associated inference tools for ensemble data sets. The proposed models are then used to identify clusters which correspond to different types of distribution errors. Finally, a standard calibration method known as Non homogeneous Gaussian Regression (NGR) is applied cluster by cluster in order to correct ensemble forecast distributions. It is shown that the proposed methodology is effective, interpretable and easy to use. The clustering algorithms are illustrated on simulated and real data. The calibration method is applied to real data of temperature and wind medium range forecast for 3 stations in France.</p>

Probabilistic extreme SST and marine heatwave forecasts in Chesapeake Bay: A forecast model, skill assessment, and potential value

We test whether skillful 35-day probabilistic forecasts of estuarine sea surface temperature (SST) are possible and whether these forecasts could potentially be used to reduce the economic damages associated with extreme SST events. Using an ensemble of 35-day retrospective forecasts of atmospheric temperature and a simple model that predicts daily mean SST from past SST and forecast atmospheric temperature, we create an equivalent ensemble of retrospective SST forecasts. We compare these SST forecasts with reference forecasts of climatology and damped persistence and find that the SST forecasts are skillful for up to two weeks in the summer. Then, we post-process the forecasts using nonhomogeneous Gaussian regression and assess whether the resulting calibrated probabilistic forecasts are more accurate than the probability implied by the raw model ensemble. Finally, we use an idealized framework to assess whether these probabilistic forecasts can valuably inform decisions to take protective action to mitigate the effects of extreme temperatures and heatwaves. We find that the probabilistic forecasts provide value relative to a naive climatological forecast for 1-2 weeks of lead time, and the value is particularly high in cases where the cost of protection is small relative to the preventable losses suffered when a heatwave occurs. In most cases, the calibrated probabilistic forecasts are also more valuable than deterministic forecasts based on the ensemble mean and naive probabilistic forecasts based on damped persistence. Probabilistic SST forecasts could provide substantial value if applied to adaptively manage the rapid impacts of extreme SSTs, including managing the risks of catch-and-release mortality in fish and Vibrio bacteria in oysters.

Winter Subseasonal Wind Speed Forecasts for Finland from ECMWF

Abstract. The subseasonal forecasts from the ECMWF (European Centre for Medium-Range Weather Forecasts) were used to construct weekly mean wind speed forecasts for the spatially aggregated area in Finland. Reforecasts for the winters (November, December and January) of 2016–2017 and 2017–2018 were analysed. The ERA-Interim reanalysis was used as observations and climatological forecasts. We evaluated two types of forecasts, the deterministic forecasts and the probabilistic forecasts. Non-homogeneous Gaussian regression was used to bias-adjust both types of forecasts. The forecasts proved to be skilful until the third week, but the longest skilful lead time depends on the reference data sets and the verification scores used.

A comparison of statistical and dynamical downscaling methods for short‐term weather forecasts in the US Northeast

AbstractThe Weather Research and Forecasting model (WRF) was used to produce both 9 and 3 km resolution ensemble forecasts from the deterministic Global Forecast System (GFS) model for microclimatic, agricultural regions in New York State. The forecasts were then statistically post‐processed to generate probabilistic forecasts for surface temperature (T), specific humidity (q), incoming solar radiation (SR) and precipitation (P). T was post‐processed with non‐homogeneous Gaussian regression (NGR), q and SR with truncated NGR, and P with extended logistic regression. A comparison of forecast skill was conducted between these post‐processed forecasts, the raw WRF output, the GFS forecasts and forecasts from the National Weather Service's deterministic National Digital Forecast Database (NDFD). Overall, significant improvement was observed in post‐processed WRF forecasts over all other methods for all locations and variables. Furthermore, raw WRF ensembles were found to outperform deterministic NDFD, so that if observational data are unavailable for post‐processing, dynamically downscaled WRF should be selected over deterministic, human‐altered NDFD forecasts. Finally, the 9 km post‐processed WRF had the same forecast skill as the 3 km post‐processed WRF, except for precipitation, rendering the 3 km WRF unnecessary if observational data are available, saving computational cost.

Comparison of probabilistic post-processing approaches for improving numerical weather prediction-based daily and weekly reference evapotranspiration forecasts

Abstract. Reference evapotranspiration (ET0) forecasts play an important role in agricultural, environmental, and water management. This study evaluated probabilistic post-processing approaches, including the nonhomogeneous Gaussian regression (NGR), affine kernel dressing (AKD), and Bayesian model averaging (BMA) techniques, for improving daily and weekly ET0 forecasting based on single or multiple numerical weather predictions (NWPs) from the THORPEX Interactive Grand Global Ensemble (TIGGE), which includes the European Centre for Medium-Range Weather Forecasts (ECMWF), the National Centers for Environmental Prediction (NCEP) Global Forecast System (GFS), and the United Kingdom Meteorological Office (UKMO) forecasts. The approaches were examined for the forecasting of summer ET0 at 101 US Regional Climate Reference Network stations distributed all over the contiguous United States (CONUS). We found that the NGR, AKD, and BMA methods greatly improved the skill and reliability of the ET0 forecasts compared with a linear regression bias correction method, due to the considerable adjustments in the spread of ensemble forecasts. The methods were especially effective when applied over the raw NCEP forecasts, followed by the raw UKMO forecasts, because of their low skill compared with that of the raw ECMWF forecasts. The post-processed weekly forecasts had much lower rRMSE values (between 8 % and 11 %) than the persistence-based weekly forecasts (22 %) and the post-processed daily forecasts (between 13 % and 20 %). Compared with the single-model ensemble, ET0 forecasts based on ECMWF multi-model ensemble ET0 forecasts showed higher skill at shorter lead times (1 or 2 d) and over the southern and western regions of the US. The improvement was higher at a daily timescale than at a weekly timescale. The NGR and AKD methods showed the best performance; however, unlike the AKD method, the NGR method can post-process multi-model forecasts and is easier to interpret than the other methods. In summary, this study demonstrated that the three probabilistic approaches generally outperform conventional procedures based on the simple bias correction of single-model forecasts, with the NGR post-processing of the ECMWF and ECMWF–UKMO forecasts providing the most cost-effective ET0 forecasting.

Distributional regression forests for probabilistic precipitation forecasting in complex terrain

To obtain a probabilistic model for a dependent variable based on some set of explanatory variables, a distributional approach is often adopted where the parameters of the distribution are linked to regressors. In many classical models this only captures the location of the distribution but over the last decade there has been increasing interest in distributional regression approaches modeling all parameters including location, scale and shape. Notably, so-called nonhomogeneous Gaussian regression (NGR) models both mean and variance of a Gaussian response and is particularly popular in weather forecasting. Moreover, generalized additive models for location, scale and shape (GAMLSS) provide a framework where each distribution parameter is modeled separately capturing smooth linear or nonlinear effects. However, when variable selection is required and/or there are nonsmooth dependencies or interactions (especially unknown or of high-order), it is challenging to establish a good GAMLSS. A natural alternative in these situations would be the application of regression trees or random forests but, so far, no general distributional framework is available for these. Therefore, a framework for distributional regression trees and forests is proposed that blends regression trees and random forests with classical distributions from the GAMLSS framework as well as their censored or truncated counterparts. To illustrate these novel approaches in practice, they are employed to obtain probabilistic precipitation forecasts at numerous sites in a mountainous region (Tyrol, Austria) based on a large number of numerical weather prediction quantities. It is shown that the novel distributional regression forests automatically select variables and interactions, performing on par or often even better than GAMLSS specified either through prior meteorological knowledge or a computationally more demanding boosting approach.

Skewed logistic distribution for statistical temperature post-processing in mountainous areas

Abstract. Nonhomogeneous post-processing is often used to improve the predictive performance of probabilistic ensemble forecasts. A common quantity used to develop, test, and demonstrate new methods is the near-surface air temperature, which is frequently assumed to follow a Gaussian response distribution. However, Gaussian regression models with only a few covariates are often not able to account for site-specific local features leading to uncalibrated forecasts and skewed residuals. This residual skewness remains even if many covariates are incorporated. Therefore, a simple refinement of the classical nonhomogeneous Gaussian regression model is proposed to overcome this problem by assuming a skewed response distribution to account for possible skewness. This study shows a comprehensive analysis of the performance of nonhomogeneous post-processing for the 2 m temperature for three different site types, comparing Gaussian, logistic, and skewed logistic response distributions. The logistic and skewed logistic distributions show satisfying results, in particular for sharpness, but also in terms of the calibration of the probabilistic predictions.

Machine Learning Techniques for Predicting the Energy Consumption/Production and Its Uncertainties Driven by Meteorological Observations and Forecasts

Reliable predictions of the energy consumption and production is important information for the management and integration of renewable energy sources. Several different Machine Learning (ML) methodologies have been tested for predicting the energy consumption/production based on the information of hydro-meteorological data. The methods analysed include Multivariate Adaptive Regression Splines (MARS) and various Quantile Regression (QR) models like Quantile Random Forest (QRF) and Gradient Boosting Machines (GBM). Additionally, a Nonhomogeneous Gaussian Regression (NGR) approach has been tested for combining and calibrating monthly ML based forecasts driven by ensemble weather forecasts. The novelty and main focus of this study is the comparison of the capability of ML methods for producing reliable predictive uncertainties and the application of monthly weather forecasts. Different skill scores have been used to verify the predictions and their uncertainties and first results for combining the ML methods applying the NGR approach and coupling the predictions with monthly ensemble weather forecasts are shown for the southern Switzerland (Canton of Ticino). These results highlight the possibilities of improvements using ML methods and the importance of optimally combining different ML methods for achieving more accurate estimates of future energy consumptions and productions with sharper prediction uncertainty estimates (i.e., narrower prediction intervals).

Spring Onset Predictability in the North American Multimodel Ensemble

AbstractThe predictability of spring onset is assessed using an index of its interannual variability (the “extended spring index” or SI‐x) and output from the North American Multimodel Ensemble reforecast experiment. The input data to compute SI‐x were treated with a daily joint bias correction approach, and the SI‐x outputs computed from the North American Multimodel Ensemble were postprocessed using an ensemble model output statistic approach—nonhomogeneous Gaussian regression. This ensemble model output statistic approach was used to quantify the effects of training period length and ensemble size on forecast skill. The lead time for predicting the timing of spring onset is found to be from 10 to 60 days, with the higher end of this range located along a narrow band between 35°N to 45°N in the eastern United States. Using continuous rank probability scores and skill score (SS) thresholds, this study demonstrates that ranges of positive predictability of SI‐x fall into two categories: 10–40 and 40–60 days. Using higher skill thresholds (SS equal to 0.1 and 0.2), predictability is confined to a lower range with values around 10–30 days. The postprocessing work using joint bias correction improves the predictive skill for SI‐x relative to the untreated input data set. Using nonhomogeneous Gaussian regression, a positive change in the SS is noted in regions where the skill with joint bias correction shows evidence of improvement. These findings suggest that the start of spring might be predictable on intraseasonal time horizons, which in turn could be useful for farmers, growers, and stakeholders making decisions on these time scales.

Technical note: Combining quantile forecasts and predictive distributions of streamflows

Abstract. The enhanced availability of many different hydro-meteorological modelling and forecasting systems raises the issue of how to optimally combine this great deal of information. Especially the usage of deterministic and probabilistic forecasts with sometimes widely divergent predicted future streamflow values makes it even more complicated for decision makers to sift out the relevant information. In this study multiple streamflow forecast information will be aggregated based on several different predictive distributions, and quantile forecasts. For this combination the Bayesian model averaging (BMA) approach, the non-homogeneous Gaussian regression (NGR), also known as the ensemble model output statistic (EMOS) techniques, and a novel method called Beta-transformed linear pooling (BLP) will be applied. By the help of the quantile score (QS) and the continuous ranked probability score (CRPS), the combination results for the Sihl River in Switzerland with about 5 years of forecast data will be compared and the differences between the raw and optimally combined forecasts will be highlighted. The results demonstrate the importance of applying proper forecast combination methods for decision makers in the field of flood and water resource management.

Multimodel probabilistic prediction of 2 m-temperature anomalies on the monthly timescale

Abstract. The 2 m-temperature anomalies from the reforecasts of the CNR-ISAC and ECMWF monthly prediction systems have been combined in a multimodel super-ensemble. Tercile probability predictions obtained from the multimodel have been constructed using direct model outputs (DMO) and model output statistics (MOS), like logistic and nonhomogeneous Gaussian regression, for the 1990–2010 winter seasons. Verification with ERA-Interim reanalyses indicates that logistic regression gives the best results in terms of ranked probability skill scores (RPSS) and reliability diagrams for low–medium forecast probabilities. Also, it is argued that the logistic regression would not yield further improvements if a larger dataset was used.

Meta-heuristic CRPS minimization for the calibration of short-range probabilistic forecasts

This paper deals with the probabilistic short-range temperature forecasts over synoptic meteorological stations across Iran using non-homogeneous Gaussian regression (NGR). NGR creates a Gaussian forecast probability density function (PDF) from the ensemble output. The mean of the normal predictive PDF is a bias-corrected weighted average of the ensemble members and its variance is a linear function of the raw ensemble variance. The coefficients for the mean and variance are estimated by minimizing the continuous ranked probability score (CRPS) during a training period. CRPS is a scoring rule for distributional forecasts. In the paper of Gneiting et al. (Mon Weather Rev 133:1098–1118, 2005), Broyden–Fletcher–Goldfarb–Shanno (BFGS) method is used to minimize the CRPS. Since BFGS is a conventional optimization method with its own limitations, we suggest using the particle swarm optimization (PSO), a robust meta-heuristic method, to minimize the CRPS. The ensemble prediction system used in this study consists of nine different configurations of the weather research and forecasting model for 48-h forecasts of temperature during autumn and winter 2011 and 2012. The probabilistic forecasts were evaluated using several common verification scores including Brier score, attribute diagram and rank histogram. Results show that both BFGS and PSO find the optimal solution and show the same evaluation scores, but PSO can do this with a feasible random first guess and much less computational complexity.

Spatial Postprocessing of Ensemble Forecasts for Temperature Using Nonhomogeneous Gaussian Regression

Abstract Statistical postprocessing techniques are commonly used to improve the skill of ensembles from numerical weather forecasts. This paper considers spatial extensions of the well-established nonhomogeneous Gaussian regression (NGR) postprocessing technique for surface temperature and a recent modification thereof in which the local climatology is included in the regression model to permit locally adaptive postprocessing. In a comparative study employing 21-h forecasts from the Consortium for Small Scale Modelling ensemble predictive system over Germany (COSMO-DE), two approaches for modeling spatial forecast error correlations are considered: a parametric Gaussian random field model and the ensemble copula coupling (ECC) approach, which utilizes the spatial rank correlation structure of the raw ensemble. Additionally, the NGR methods are compared to both univariate and spatial versions of the ensemble Bayesian model averaging (BMA) postprocessing technique.

On the skill of various ensemble spread estimators for probabilistic short range wind forecasting

Abstract. A variety of applications ranging from civil protection associated with severe weather to economical interests are heavily dependent on meteorological information. For example, a precise planning of the energy supply with a high share of renewables requires detailed meteorological information on high temporal and spatial resolution. With respect to wind power, detailed analyses and forecasts of wind speed are of crucial interest for the energy management. Although the applicability and the current skill of state-of-the-art probabilistic short range forecasts has increased during the last years, ensemble systems still show systematic deficiencies which limit its practical use. This paper presents methods to improve the ensemble skill of 10-m wind speed forecasts by combining deterministic information from a nowcasting system on very high horizontal resolution with uncertainty estimates from a limited area ensemble system. It is shown for a one month validation period that a statistical post-processing procedure (a modified non-homogeneous Gaussian regression) adds further skill to the probabilistic forecasts, especially beyond the nowcasting range after +6 h.

Probabilistic Wind Gust Forecasting Using Nonhomogeneous Gaussian Regression

Abstract A joint probabilistic forecasting framework is proposed for maximum wind speed, the probability of gust, and, conditional on gust being observed, the maximum gust speed in a setting where only the maximum wind speed forecast is available. The framework employs the nonhomogeneous Gaussian regression (NGR) statistical postprocessing method with appropriately truncated Gaussian predictive distributions. For wind speed, the distribution is truncated at zero, the location parameter is a linear function of the wind speed ensemble forecast, and the scale parameter is a linear function of the ensemble variance. The gust forecasts are derived from the wind speed forecast using a gust factor, and the predictive distribution for gust speed is truncated according to its definition. The framework is applied to 48-h-ahead forecasts of wind speed over the North American Pacific Northwest obtained from the University of Washington mesoscale ensemble. The resulting density forecasts for wind speed and gust speed are calibrated and sharp, and offer substantial improvement in predictive performance over the raw ensemble or climatological reference forecasts.

Combining 2-m temperature nowcasting and short range ensemble forecasting

Abstract. During recent years, numerical ensemble prediction systems have become an important tool for estimating the uncertainties of dynamical and physical processes as represented in numerical weather models. The latest generation of limited area ensemble prediction systems (LAM-EPSs) allows for probabilistic forecasts at high resolution in both space and time. However, these systems still suffer from systematic deficiencies. Especially for nowcasting (0–6 h) applications the ensemble spread is smaller than the actual forecast error. This paper tries to generate probabilistic short range 2-m temperature forecasts by combining a state-of-the-art nowcasting method and a limited area ensemble system, and compares the results with statistical methods. The Integrated Nowcasting Through Comprehensive Analysis (INCA) system, which has been in operation at the Central Institute for Meteorology and Geodynamics (ZAMG) since 2006 (Haiden et al., 2011), provides short range deterministic forecasts at high temporal (15 min–60 min) and spatial (1 km) resolution. An INCA Ensemble (INCA-EPS) of 2-m temperature forecasts is constructed by applying a dynamical approach, a statistical approach, and a combined dynamic-statistical method. The dynamical method takes uncertainty information (i.e. ensemble variance) from the operational limited area ensemble system ALADIN-LAEF (Aire Limitée Adaptation Dynamique Développement InterNational Limited Area Ensemble Forecasting) which is running operationally at ZAMG (Wang et al., 2011). The purely statistical method assumes a well-calibrated spread-skill relation and applies ensemble spread according to the skill of the INCA forecast of the most recent past. The combined dynamic-statistical approach adapts the ensemble variance gained from ALADIN-LAEF with non-homogeneous Gaussian regression (NGR) which yields a statistical \\mbox{correction} of the first and second moment (mean bias and dispersion) for Gaussian distributed continuous variables. Validation results indicate that all three methods produce sharp and reliable probabilistic 2-m temperature forecasts. However, the statistical and combined dynamic-statistical methods slightly outperform the pure dynamical approach, mainly due to the under-dispersive behavior of ALADIN-LAEF outside the nowcasting range. The training length does not have a pronounced impact on forecast skill, but a spread re-scaling improves the forecast skill substantially. Refinements of the statistical methods yield a slight further improvement.

A unified linear Model Output Statistics scheme for both deterministic and ensemble forecasts

An extension of the classical linear Model Output Statistics (MOS) technique is proposed allowing for the post-processing of ensemble forecasts. In this new approach, the cost function on which the least square parameter estimation is based takes into account the presence of errors in both observations and model observables (referred to as Error-in-Variables MOS, EVMOS), unlike the classical linear MOS cost function whose implicit assumption is the absence of errors in the model observables. It allows for the maintenance of an appropriate variability for the corrected forecasts, even for long lead times and for providing a framework in which both deterministic and probabilistic forecasts can be corrected. The scheme is successfully tested for ensemble correction in the context of an idealized low-order chaotic system, the Lorenz atmospheric model, in the presence of model errors, and compared with a classical technique, known as the non-homogeneous Gaussian regression (NGR) method. The potential use of this approach is also briefly discussed. Copyright © 2009 Royal Meteorological Society

Calibrating 2-m Temperature of Limited-Area Ensemble Forecasts Using High-Resolution Analysis

Abstract Although the quality of numerical ensemble prediction systems (EPS) has greatly improved during the last few years, these systems still show systematic deficiencies. Specifically, they are underdispersive and lack both reliability and sharpness. A variety of statistical postprocessing methods allows for improving direct model output. Since 2007, Aire Limitée Adaptation Dynamique Développement International Limited Area Ensemble Forecasting (ALADIN-LAEF) has been in operation at the Central Institute for Meteorology and Geodynamics (ZAMG), and its 2-m temperature model output subject to calibration. This work follows the approach of nonhomogeneous Gaussian regression (NGR) that addresses a statistical correction of the first and second moment (mean bias and dispersion) for Gaussian-distributed continuous variables. It is based on the multiple linear regression technique and provides a predictive probability density function (PDF) in terms of a normal distribution. Fitting the regression coefficients, a minimum continuous ranked probability score (CRPS) estimation has been chosen instead of the more traditional maximum likelihood technique. The use of high-resolution analysis data on a 1 km × 1 km grid as training data improves the forecast skill in terms of CRPS by about 35%, especially on the local scale. The percentage of outliers decreases significantly without loss of sharpness. Sensitivity studies confirm that about half of the total improvement can be attributed to the effect of a bias correction. The training length plays a minor role, at least for the chosen verification period. A rescaling of the predictive PDF is important in order to obtain sharp forecasts, especially in the short range. Applying the same method to the global ensemble from the European Centre for Medium-Range Weather Forecasts (ECMWF) gives improvements of similar magnitude. However, the calibrated 2-m temperature of ALADIN-LAEF still remains slightly better than the 2-m temperature from calibrated ECMWF-EPS, which leads to the conclusion that statistical downscaling of EPS cannot replace dynamical downscaling. Finally, an advanced version of NGR, the so-called NGR-TD, which uses time-weighted averaging within minimum CRPS estimation, is able to yield a further improvement of about 5% in terms of the CRPS.