Bayesian Model Averaging Method Research Articles

Blending gauge, multi-satellite and atmospheric reanalysis precipitation products to facilitate drought monitoring

Long-term, continuous, and highly accurate precipitation estimation is crucial for reliable drought monitoring. Precipitation estimation based on gauge measurements, satellite retrieval, and reanalysis datasets exhibits heterogeneous uncertainties across different regions of mainland China. This study introduces two modifiable weighting schemes (Cheng-Kling-Gupta Efficiency (CKGE) weighted-ensemble model (CWEM) and Bayesian Model Averaging (BMA)), utilizing posterior probabilities generated by BMA and the performance of CKGE, to merge 7 monthly precipitation datasets into new weighted precipitation (BMA Ensemble Precipitation (BMAEP) and CKGE Weighted-Ensemble Precipitation (CWEP)) using various quantity combination schemes. Subsequently, the precision and drought monitoring utility of the weighted precipitation are evaluated and compared with the representative fused precipitation product Multi-Source Weighted-Ensemble Precipitation (MSWEP) in mainland China using gauged data. The amalgamated results demonstrate that, compared to individual datasets, certain weighted precipitation schemes exhibit superiority over MSWEP. In particular, BMAEP-2P demonstrates superior performance in the composite index CKGE, achieving a value of 0.828. CWEP-4P exhibits a higher correlation coefficient (CC), reaching 0.905. CWEP-2P excels in terms of relative bias (BIAS) and root mean square error (RMSE), with values of 0.579 % and 20.755 mm, respectively. Furthermore, BMAEP or CWEP performs optimally in drought monitoring applications across all sub-regions of mainland China at different time scales (1, 3, 6, 12 and 24 months), with the average of the highest values of CC and probability of detection (POD) reached 0.919 and 0.844, respectively. Further contribution analysis reveals CPC as the dominant factor contributing to the greatest improvement in the performance (excluding CKGE, CC_SPEI1, CC_SPEI24 and POD_SPEI24) of the fusion models, among which it boosted MERRA2′s performance on POD_SPEI6 by 9.41 %. In conclusion, the merging schemes based on CWEM and BMA methods effectively generate a new precipitation dataset, integrating information from multiple products into drought monitoring applications.

Battery state-of-health estimation incorporating model uncertainty based on Bayesian model averaging

Accurately estimating the state-of-health (SOH) of lithium-ion batteries is crucial for efficient, reliable, and safe use. However, the degradation of these batteries involves complex and intricate failure mechanisms that cannot be fully captured by a single model. To tackle this challenge, we propose an SOH estimation method based on Bayesian Model Averaging (BMA), which effectively accounts for both parameter and model uncertainties by combining estimations from different model implementations. It provides both point-value estimates and the probability distribution estimates, delivering results in a fraction of a second. Evaluated on three open-source datasets, the maximum absolute error of SOH estimation is within 0.03, and the Continuous Ranked Probability Score is within 0.015. Compared to optimal individual models, the proposed BMA method reduces the prediction error of point estimates by half to two-thirds. Additionally, the prediction error for probability estimation decreases by an order of magnitude. Moreover, the comparison studies with respect to the Gaussian Process Regression model and the Quantile Regression Forests model demonstrate the applicability and superiority of proposed method. These results highlight the potential of the BMA method to advance battery SOH estimation and facilitate reliable battery management.

Quantifying and reducing flood forecast uncertainty by the CHUP-BMA method

Abstract. The Bayesian model averaging (BMA), hydrological uncertainty processor (HUP), and HUP-BMA methods have been widely used to quantify flood forecast uncertainty. This study proposes the copula-based hydrological uncertainty processor BMA (CHUP-BMA) method by introducing a copula-based HUP in the framework of BMA to bypass the need for a normal quantile transformation of the HUP-BMA method. The proposed ensemble forecast scheme consists of eight members (two forecast precipitation inputs; two advanced long short-term memory, LSTM, models; and two objective functions used to calibrate parameters) and is applied to the interval basin between the Xiangjiaba and Three Gorges Reservoir (TGR) dam sites. The ensemble forecast performance of the HUP-BMA and CHUP-BMA methods is explored in the 6–168 h forecast horizons. The TGR inflow forecasting results show that the two methods can improve the forecast accuracy over the selected member with the best forecast accuracy and that the CHUP-BMA performs much better than the HUP-BMA. Compared with the HUP-BMA method, the forecast interval width and continuous ranked probability score metrics of the CHUP-BMA method are reduced by a maximum of 28.42 % and 17.86 % within all forecast horizons, respectively. The probability forecast of the CHUP-BMA method has better reliability and sharpness and is more suitable for flood ensemble forecasts, providing reliable risk information for flood control decision-making.

Identifying and Interpreting Hydrological Model Structural Nonstationarity Using the Bayesian Model Averaging Method

Understanding hydrological nonstationarity under climate change is important for runoff prediction and it enables more robust decisions. Regarding the multiple structural hypotheses, this study aims to identify and interpret hydrological structural nonstationarity using the Bayesian Model Averaging (BMA) method by (i) constructing a nonstationary model through the Bayesian weighted averaging of two lumped conceptual rainfall–runoff (RR) models (the Xinanjiang and GR4J model) with time-varying weights; and (ii) detecting the temporal variation in the optimized Bayesian weights under climate change conditions. By combining the BMA method with period partition and time sliding windows, the efficacy of adopting time-varying model structures is investigated over three basins located in the U.S. and Australia. The results show that (i) the nonstationary ensemble-averaged model with time-varying weights surpasses both individual models and the ensemble-averaged model with time-invariant weights, improving NSE[Q] from 0.04 to 0.15; (ii) the optimized weights of Xinanjiang model increase and that of GR4J declines with larger precipitation, and vice versa; (iii) the change in the optimized weights is proportional to that of precipitation under monotonic climate change, as otherwise the mechanism changes significantly. Overall, it is recommended to adopt nonstationary structures in hydrological modeling.

A generalisation of the method of regression calibration and comparison with Bayesian and frequentist model averaging methods

For many cancer sites low-dose risks are not known and must be extrapolated from those observed in groups exposed at much higher levels of dose. Measurement error can substantially alter the dose–response shape and hence the extrapolated risk. Even in studies with direct measurement of low-dose exposures measurement error could be substantial in relation to the size of the dose estimates and thereby distort population risk estimates. Recently, there has been considerable attention paid to methods of dealing with shared errors, which are common in many datasets, and particularly important in occupational and environmental settings. In this paper we test Bayesian model averaging (BMA) and frequentist model averaging (FMA) methods, the first of these similar to the so-called Bayesian two-dimensional Monte Carlo (2DMC) method, and both fairly recently proposed, against a very newly proposed modification of the regression calibration method, the extended regression calibration (ERC) method, which is particularly suited to studies in which there is a substantial amount of shared error, and in which there may also be curvature in the true dose response. The quasi-2DMC with BMA method performs well when a linear model is assumed, but very poorly when a linear-quadratic model is assumed, with coverage probabilities both for the linear and quadratic dose coefficients that are under 5% when the magnitude of shared Berkson error is large (50%). For the linear model the bias is generally under 10%. However, using a linear-quadratic model it produces substantially biased (by a factor of 10) estimates of both the linear and quadratic coefficients, with the linear coefficient overestimated and the quadratic coefficient underestimated. FMA performs as well as quasi-2DMC with BMA when a linear model is assumed, and generally much better with a linear-quadratic model, although the coverage probability for the quadratic coefficient is uniformly too high. However both linear and quadratic coefficients have pronounced upward bias, particularly when Berkson error is large. By comparison ERC yields coverage probabilities that are too low when shared and unshared Berkson errors are both large (50%), although otherwise it performs well, and coverage is generally better than the quasi-2DMC with BMA or FMA methods, particularly for the linear-quadratic model. The bias of the predicted relative risk at a variety of doses is generally smallest for ERC, and largest for the quasi-2DMC with BMA and FMA methods (apart from unadjusted regression), with standard regression calibration and Monte Carlo maximum likelihood exhibiting bias in predicted relative risk generally somewhat intermediate between ERC and the other two methods. In general ERC performs best in the scenarios presented, and should be the method of choice in situations where there may be substantial shared error, or suspected curvature in the dose response.

Tracking shallow and deep groundwater storage changes in North China Plain with improved fusion method and hybrid spectral analysis approach

The North China Plain (NCP) is a critical agricultural hub in China, confronting a significant issue of groundwater depletion. However, different terrestrial water storage (TWS) change products from the Gravity Recovery and Climate Experiment (GRACE) satellites differ considerably in estimating groundwater depletion rates in the NCP. It is urgent to optimize these different products to obtain an elevated understanding of regional groundwater storage (GWS) changes. The generalized three-cornered hat (GTCH) method is used to quantify uncertainties of different GRACE products, and this prior information is integrated with the Bayesian model averaging (BMA) method to fuse GRACE products. Results demonstrate that the GTCH + BMA method can generate uniform GWS products with heightened precision, enhanced coherence, and diminished uncertainty in the NCP. After that, the shallow and deep GWS are meticulously differentiated based on the high-coverage in-situ data, and the shallow GWS (2005.1–2016.8) (-13.7 ± 4.6 mm/yr) declined faster than deep GWS (-6.05 ± 3.9 mm/yr) in the NCP. To further explore the primary factors driving these changes, the latest Hankel Spectrum Analysis (HSA) method combined with ridge regression simulation is employed to decouple the change components of human and natural factors in the shallow and deep GWS. The first attempt to give the contribution of human and natural factors to shallow GWS is 77.18 %/22.82 %, and to deep GWS is 87.59 %/12.41 %.

Nuclear mass predictions with the naive Bayesian model averaging method

A naive Bayesian model averaging (NBMA) method is developed to predict nuclear masses. In the NBMA method, the weights of different models may be different for each nucleus, which are sensitive to the model accuracies to describe the nuclear masses of the isotopes and isotones with the same proton and neutron numbers of that nucleus. Therefore, there are remarkable local structures for the weights of different models on the nuclear chart, which well eliminates the local deviations between the model predictions and the experimental masses and thus achieves better accuracy of mass predictions than the traditional arithmetic mean method (AMM) and weighted average method (WAM). Based on the latest atomic mass evaluation of AME2020, the root-mean-square (rms) mass deviation of the NBMA method is 0.293 MeV, while the rms deviations of AMM and WAM are 0.634 and 0.361 MeV, respectively. This accuracy of the NBMA method is even 28% better than the best accuracy of the mass models used in the NBMA method. The extrapolation ability of the NBMA method is also verified with the experimental nuclear masses which are not used in the training of the NBMA method.

Frequentist model averaging for analysis of dose-response in epidemiologic studies with complex exposure uncertainty.

In epidemiologic studies, association estimates of an exposure with disease outcomes are often biased when the uncertainties of exposure are ignored. Consequently, corresponding confidence intervals (CIs) will not have correct coverage. This issue is particularly problematic when exposures must be reconstructed from physical measurements, for example, for environmental or occupational radiation doses that were received by a study population for which radiation doses cannot be measured directly. To incorporate complex uncertainties in reconstructed exposures, the two-dimensional Monte Carlo (2DMC) dose estimation method has been proposed and used in various dose reconstruction efforts. The 2DMC method generates multiple exposure realizations from dosimetry models that incorporate various sources of errors to reflect the uncertainty of the dose distribution as well as the uncertainties in individual doses in the exposed population. Traditional measurement-error model approaches, typically based on using mean doses in the dose-exposure analysis, do not fully account exposure uncertainties. A recently developed statistical approach that overcomes many of these limitations by analyzing multiple exposure realizations in relation to disease risk is Bayesian model averaging (BMA). The analytic advantage of the BMA is its ability to better accommodate complex exposure uncertainty in the risk estimation, but a practical. Drawback is its significant computational complexity. In this present paper, we propose a novel frequentist model averaging (FMA) approach which has all the analytical advantages of the BMA method but is much simpler to implement and computationally faster. We show in simulations that, like BMA, FMA yields 95% confidence intervals for association parameters that close to 95% coverage rate. In simulations, the FMA has shorter length of CIs than those of another frequentist approach, the corrected information matrix (CIM) method. We illustrate the similarities in performance of BMA and FMA from a study of exposures from radioactive fallout in Kazakhstan.

Combining physical-based model and machine learning to forecast chlorophyll-a concentration in freshwater lakes

Increasing algal blooms in freshwater lakes have become a serious challenge facing the world. Short-term forecast of chlorophyll-a concentration (Chla) is essential for providing early warnings and taking action to mitigate the risks of algal blooms in freshwater lakes. At present, a variety of data-driven models and physical-based models have been developed for Chla forecast, yet how to effectively combine multiple models for improving the forecast accuracy remains largely unknown. Here we developed an effective model by combining a physical-based model and machine learning algorithms (long short-term memory, LSTM; random forest, RF; support vector machine, SVM) to forecast the Chla in a freshwater lake, and a Bayesian model averaging (BMA) ensemble forecasting method was further proposed to improve the accuracy and reliability of the forecast results. We found that, with the increase of time steps of advance forecast from 1-day to 7-day, the forecast accuracy as measured by R2 of the machine learning algorithms is decreased from 0.95 to 0.68. The combination of physical-based modeling with LSTM had great capability in short-term forecast of Chla, owing to the fact that the physical-based model can provide high-frequency Chla data and LSTM is skilled at forecasting in the sequence. This is also evidenced by the weights in the BMA method. The proposed BMA short-term ensemble forecasting results had the robust performance when compared to each individual machine learning forecast model for the 7-day advance forecast, with the largest R2 (0.834) and the smallest RMSE (0.267 μg/L). In particular, the uncertainty of a single machine learning model can be effectively reduced by the BMA method.

Determinants of Foreign Direct Investment: A Bayesian Model Averaging Approach From 55 Countries

Purpose: This study aims to investigate the determinants of Foreign Direct Investment (FDI) for the 55 countries from 4 different continents and 3 different development levels. 
 Method: A panel dataset is constructed over the period of 1995-2019 with 17 candidate independent variables and Bayesian Model Averaging (BMA) is employed for each continent and for each development level to reveal the determinants of FDI. 
 Results: Results showed that globalization is the primary factor to attract FDI regardless of the development level and continent. But there are also other factors that affect FDI which differ based on the development level and geographic location of the country. FDI in transition economies benefits from qualified labor force which negatively affects developed countries and has no effect on developing countries. Corruption is only effective in developing countries. Secondary education is effective at European and Asian countries, tertiary education is effective on American countries. This study contributes the literature by using a BMA method in a panel data context. BMA uses weighted average of all the possible models which gives more balanced results than a single model. Applying BMA to the countries in different continents and development levels separately allows to make inferences based on region and wealth.
 Originality: This study used a fairly large dataset on terms of selected time period, number of countries and selected variables which most of the studies lack of. Also, the advantage of applying BMA to a panel dataset, is obvious as both controls the cross section variation and also considers a weighted average of all possible linear regression results rather than giving only a single linear regression model result. Thus applying BMA on a panel dataset give a better insight as to which variables should be considered as determinants of FDI.

Multi-parameter approaches for improved ensemble prediction accuracy in hydrology and water quality modeling

Ensemble approaches can be a quick way to improve hydrological prediction accuracy by considering multiple plausible modeling representations of a system. Previous studies showed that the use of multiple types of models could increase prediction accuracy, but model selection subjectivity and parameter equifinality issues remain unsolved. This study proposes to consider multiple parameter sets identified in the model calibration process and information on their performance for improved accuracy of ensemble hydrological prediction. In this study, we investigate whether multi-parameter ensemble (MP) compared to multi-model ensemble (MM) approaches can improve the accuracy of hydrology and water quality prediction. A variety of ensemble methods, including simple averaging, median filtering, weighted averaging, Bayesian model averaging (BMA), and their variants, were tested in watershed modeling implemented to predict daily water discharge and total phosphorus (TP) loads of a study watershed. The parameter spaces of two watershed loading models, Soil and Water Assessment Tool (SWAT) and Hydrological Simulation Program-FORTRAN (HSPF), were sampled using a multi-objective heuristic optimization algorithm, namely A Multi-Algorithm Genetically Adaptive Multiobjective (AMALGAM). The modeling experiment showed that the MP approach more effectively improved modeling accuracy statistics compared to the MM of both models as well as single-model and single-parameter approaches. The ensemble approaches did not always improve modeling accuracy, and their efficiency depended on ways to determine weights for ensemble members. The BMA method outperformed the other weighting schemes tested in this study, and its performance was responsive to the number and diversity (or variance) of ensemble member candidates and the computational cost to create the candidates. The findings demonstrate how the information of parameter sets sampled in model calibration and their performance statistics could help improve hydrological prediction accuracy.

A Bayesian model averaging method for software reliability modeling and assessment

AbstractThe software reliability modeling is of great significance in improving software quality and managing the software development process. However, the existing methods are not able to accurately model software reliability improvement behavior because existing single model methods rely on restrictive assumptions and combination models cannot well deal with model uncertainties. In this article, we propose a Bayesian model averaging (BMA) method to model software reliability. First, the existing reliability modeling methods are selected as the candidate models, and the Bayesian theory is used to obtain the posterior probabilities of each reliability model. Then, the posterior probabilities are used as weights to average the candidate models. Both Markov Chain Monte Carlo (MCMC) algorithm and the Expectation–Maximization (EM) algorithm are used to evaluate a candidate model's posterior probability and for comparison purpose. The results show that the BMA method has superior performance in software reliability modeling, and the MCMC algorithm performs better than EM algorithm when they are used to estimate the parameters of BMA method.

Improving rice phenology simulations based on the Bayesian model averaging method

Crop simulation models play an increasingly important role in agricultural management. Therefore, improving crop models’ accuracy and reliability has become a key issue. Previous studies have shown that multi-model ensembles (MMEs) perform better than individual models and are capable of assessing model uncertainty. Here, we study the use of the Bayesian model averaging (BMA) method to improve the simulation performance of rice phenology models. The main objective of this paper is to apply BMA to five individual rice phenology models (CERES-Rice, the beta model, ORYZA2000, the rice clock model, and the simulation model for the rice-weather relationship) and to compare its simulation performance with an MME based on the mean value (Mean-MME). Furthermore, we quantify and evaluate the uncertainty of the individual and ensemble models. Finally, we analyze the prediction results of two ensemble models and five individual models under future climate scenarios. The result indicates that, compared with the accuracies of individual models, the accuracies of BMA and Mean-MME are 12.75% and 11.48% higher, respectively. BMA performs the best when the individual models had the best parameters, effectively optimizing the phenology simulation results of the individual models. For individual and ensemble models, the squared bias accounts for 48–88%, which is the largest contributor to overall prediction uncertainty. Under the future climate scenarios, the five models predict a range of 0.40–9.78 days in the phenological period, whereas the two ensemble models predict a smaller range of 0.54–1.63 days, which shows that the MMEs reduce the uncertainty of the individual models. We conclude that BMA is suitable for improving the accuracy and reliability of crop modeling.

Multi-source rainfall merging and reservoir inflow forecasting by ensemble technique and artificial intelligence

Study regionThe Shihmen reservoir, the second largest reservoir in northern Taiwan, is flooded frequently, and featured by short period of flood peak due to uneven distribution of rainfall and mountainous topography. Study focusWe proposed a novel streamflow-oriented ensemble recurrent neural networks (ERNN) method to merge three precipitation products, namely gauge measurements, radar and satellite rainfall products. We analyzed (1) the effect of artificial intelligence method for bias correction of precipitation products with reference to streamflow, (2) the comparison of two widely used arithmetic average (AA) and Bayesian model averaging (BMA) methods with ERNN, (3) the performance of merged rainfalls and the original three precipitation products in inflow forecasting during 13 typhoon events. New hydrological insightsWe found that all precipitation products are biased and can be appropriately adjusted with an improvement in inflow forecasting of about 2.5 %, 21.8 % and 60.7 % for gauge, radar and satellite rainfall products in terms of RMSE. After rainfall merging, RMSE for radar and satellite are further improved by 14% and 36% at least. ERNN merging method indicates that the optimal merging weights for gauge, radar and satellite are within the range of 0.52–0.56, 0.35–0.39 and 0.08–0.09, respectively according to 95 % confidence interval. The merged rainfall product skillfully predicts the inflow with a lead time of five hours, and ERNN merging method is superior to traditional AA and BMA methods, with NS up to 0.85 and CRPS reduced by near 50 % compared to BMA. Additionally, ERNN is found to capture the peak times and peak flows with the smallest error. Overall, this study provides a potential approach to merge multiple rainfall products and to obtain the effective rainfall by fitting the hydrological responses over mountainous watersheds, where observational biases may frequently occur in gauge, radar and satellite measurements.

Probabilistic 2-meter surface temperature forecasting over Xinjiang based on Bayesian model averaging

Based on Bayesian model averaging (BMA), the suitability and characteristics of the BMA model for forecasting 2-m temperature in Xinjiang of China were analyzed by using the forecast results of the Desert Oasis Gobi Regional Analysis Forecast System (DOGRAFS) and Rapid-refresh Multiscale Analysis and Prediction System (RMAPS) developed by the Urumqi Institute of Desert Meteorology of the China Meteorological Administration, China Meteorological Administration–Global Forecast System (CMA-GFS) developed by the China Meteorological Administration, and the European Center for Medium-Range Weather Forecasts (ECMWF) developed by the European Center. The results showed that (1) the weight of ECMWF to the 2-m temperature forecast is maintained at about 0.6–0.7 under different lengths of training periods, and the weight of other model products is below 0.15. (2) The forecasts of each model at the four representative stations are quite different, and the maximum forecast error reaches 6.9°C. However, the maximum error of the BMA forecast is only about 2°C. In addition, the forecast uncertainty in southern Xinjiang is greater than that in northern Xinjiang. (3) Compared with multi-model ensembles, the overall prediction performance of the BMA method is more consistent in spatial distribution. Additionally, the standard deviation and correlation coefficient between the BMA forecast and observation were greater than 0.98, and the RMSE decreased significantly. It is feasible to use the BMA method to correct the accuracy of the 2-m temperature forecast in Xinjiang.

Bayesian Model Averaging for Improving the Accuracy of Cuffless Blood Pressure Estimation.

In recent decades, many researches have proposed various models for continuous, cuffless blood pressure (BP) estimation. However, due to aleatoric uncertainty and epistemic uncertainty existing in the problem, it is very challenging to evaluate cuffless BP with acceptable accuracy. This paper innovatively proposes a cuffless BP ensemble estimation model based on Bayesian Model Average (BMA) method to reduce the epistemic uncertainty. We combine four most frequently cited physiological models and four regression models based on Photoplethysmogram (PPG) and Electrocardiogram (ECG) signals, and use the BMA method to assign weights to each model to achieve accurate cuffless BP prediction. The proposed method was validated on 17 healthy and 13 hypertensive subjects with continuous Finometer BP as a reference. The results showed that the error mean ± SD (standard deviations) of both SBP and DBP predicted by the proposed method were 2.13 ± 5.68 mmHg and 1.42 ± 5.11 mmHg, respectively, which were both lower than each of the model. And the MAE was 6% and 8% lower than the best member of the model ensemble. We also analyzed the relationship between the number of training epochs and model prediction performance. When 15 cardiac cycles were choosed for training, it could get a good balance between model prediction accuracy and algorithm complexity. Therefore, the proposed BMA method can solve the model uncertainty problem, providing robust and deterministic BP prediction. Clinical relevance- This paper proposes a new method for wearable BP estimation which enables BP monitoring in both clinical settings and home settings. It offers a stable way to monitor BP to help patients detect disease early.

Flood risk analysis of reservoirs based on full-series ARIMA model under climate change

• A novel full-series ARIMA stochastic model has been built. • This model can reflect the statistical characteristics of past and future floods. • The BMA method was utilized to analyze the uncertainty of flood frequency curves. • More comprehensive and reliable flood risks considering the weight of each GCM have been obtained. To assess the flood risks of reservoirs in a changing environment, this study considers the uncertainty of future floods and argues that the future flood projected by Global Climate Models (GCMs) is only one of the possibilities. This possibility contains some main statistical characteristics of future floods. The autoregressive integrated moving average (ARIMA) model was used to extract such statistical characteristics of the future flood series. Moreover, the initial values of the model were assumed to obey the probability distribution derived from historical floods. Hence, a novel full-series ARIMA stochastic model which can simultaneously reflect the statistical characteristics of past and future floods has been built to simulate the possible future extreme floods. Synthetic flood series under different future climate scenarios were randomly generated by the proposed model. Then the flood risks exceeding the design flood and check flood can be obtained. The Bayesian model averaging (BMA) method was used to weight the flood risks projected by each GCM to better incorporate the uncertainty of GCMs. The Yangfanggou Reservoir in Yalong River basin of China was selected as a case study. The future flood risks under three Representative Concentration Pathways (RCPs) were assessed. The results highlight that future flood risks have an increasing trend. Under future RCP2.6, RCP4.5 and RCP8.5 scenarios, the risk rates of exceeding design flood are in the range of 0.63%–0.74%, 1.31%–4.58% and 2.63%–5.23%, respectively. The risk rates of exceeding check flood are in the range of 0.20%–0.25%, 0.41%–2.14% and 0.97%–2.48%, respectively. This study provides a comprehensive evaluation of flood risk and could be useful when planning to fortify existing structures and update design standards.

Deterministic and probabilistic projections and their credibility in analyzing future precipitation variations in the Yellow River Basin, China

AbstractIt remains a key challenge to obtain reliable future precipitation estimates and their reliability under different climate scenarios. In this study, the deterministic projection of future precipitation in the Yellow River Basin (YRB) was obtained within the Bayesian model averaging (BMA) framework. A probability estimation method based on the BMA weighting scheme was proposed to obtain the probabilistic projection of precipitation. We also analyzed the credibility of these two projections. The results showed that four indexes projected by the BMA method showed an increasing trend with a higher probability. The probabilities of increasing with varying degrees were more than those for decreasing for all the precipitation indexes. The credibility of the precipitation estimation under specific climate scenarios was testified by the lower ED (the mean of long-term annual relative simulation deviation) and VD (the variance of long-term annual relative simulation deviation). The estimation based on the BMA model is more trustworthy than any other model. For the four precipitation indicators, the accuracy between the calculated VR (Variation range, to describe the interval of variation of the indicators) with the greatest likelihood and the actual VR was 38.31–53.74%. In 81.93–94.70% of grids, the deviations were smaller than one level. Both the deterministic and probabilistic projections have high geographic distribution and variation trend consistency.

Multi-model integrated error correction for streamflow simulation based on Bayesian model averaging and dynamic system response curve

Error correction methods play an important role in improving the reliability and accuracy of hydrological modeling. The Dynamic system response curve (DSRC) is a novel and effective error correction method, but it may have the problem of over-correction. Therefore, two multi-model integrated error correction models based on DSRC and Bayesian model averaging (BMA) were proposed in this paper, namely DSRC-BMA and BMA-DSRC. The Sunshui River catchment is selected for a case study. First, three hydrological models including Xinanjiang model (XAJ), Hydrologiska Fyrans Vattenbalans modell (HBV) and vertically hybrid yield model (VHY) were employed. Then, a standard BMA model and three DSRC-based models were constructed separately. Finally, two multi-model integrated error correction models (DSRC-BMA and BMA-DSRC) were applied. The performance of these nine models was compared by Nash-Sutcliffe Efficiency coefficient (NSE), root mean squared error (RMSE) and percent bias (PBIAS). Results showed that the DSRC-based models presented better results than the standard BMA method and most DSRC-based models. Moreover, the uncertainty in BMA, DSRC-BMA and BMA-DSRC models were assessed. The 90% confidence interval of the BMA-DSRC model had high containing ratio values and low average relative bandwidth. Overall, the proposed multi-model integrated error correction methods are effective and can be applicable in improving streamflow modeling.

Probabilistic Forecast of the Extended Range Heatwave Over Eastern China

Several probabilistic forecast methods for heatwave (HW) in extended-range scales over China are constructed using four models (ECMWF, CMA, UKMO, and NCEP) from the Subseasonal-to-Seasonal (S2S) database. The methods include four single-model ensembles (SME; ECMWF, CMA, UKMO, and NCEP), multi-model ensemble (MME), and Bayesian model averaging (BMA). The construction and verification of reforecasts are implemented by a defined heat wave index (HWI) which is not only able to reflect the actual occurrence of heatwaves, but also to facilitate forecast and verification. The performance is measured by traditional verification method at each grid point of the 105°E to 132°E; 20°N to 45°N domain for the July, August, and September (JAS) of 1999–2010. For deterministic evaluations of HWI forecast, BMA shows a better pattern correlation coefficient than SME and MME and comparable equitable threat score (ETS) with ECMWF and MME. The good performance of ECMWF and MME take advantage of setting the percentile thresholds for forecasting HW. For the probabilistic forecast, the Brier score of BMA is comparable (superior) to that of MME and ECMWF at short (long) lead-time. BMA also demonstrates an improvement on the reliability of probabilistic forecast, indicating that BMA method is a useful tool for an extended-range forecast of HW. Meanwhile, in the real-time extended-range probabilistic forecast, the beginning date, end date, and probability of HW event can be predicted by the HWI probabilistic forecast of BMA.