Copula-based statistical post-processing for multi-site temperature forecasts

[Departement_IRSTEA]Eaux [TR1_IRSTEA]ARCEAU

By the end of 2022, the renewable energy share of the global electricity capacity reached 40.3% and the new installations were dominated by solar energy, showing a global increase of 21.7%. Due to the high volatility of photovoltaic energy sources, their successful integration into the electrical grid requires accurate short-term power forecasts. These forecasts are obtained from the predictions of solar irradiance, where the most advanced method is the probabilistic approach based on ensemble forecasts.  However, ensemble forecasts are often underdispersive and subject to systematic bias. Hence, they require some form of statistical post-processing, where parametric models provide full predictive distributions of the weather variables at hand.We propose a general two-step machine learning-based approach to calibrating ensemble weather forecasts, where, in the first step, improved point forecasts are generated, which then together with various ensemble statistics serve as input features of the neural network estimating the parameters of the predictive distribution [1]. In a case study based on global horizontal irradiance forecasts of the operational ensemble prediction system of the Hungarian Meteorological Service, the predictive performance of this novel method is compared with the forecast skill of the raw ensemble and the state-of-the-art ensemble model output statistics approaches [2]. We show that at least up to 48h, statistical post-processing substantially improves the predictive performance of the raw ensemble for all forecast horizons considered; the maximal gain e.g. in terms of the mean continuous ranked probability score is above 20%. Furthermore, the proposed two-step machine learning-based approach outperforms in skill its competitors.

In modeling and analyzing multivariate data, the conventionally used measure of dependence structure is the Pearson's correlation coeffi- cient. However use of the correlation as a dependence measure has several pitfalls. Copulas recently have emerged as an alternative measure of the de- pendence, overcoming most of the drawbacks of the correlation. We discuss Archimedean copulas and their relationships with tail dependence. An algo- rithm to construct empirical and Archimedean copulas is described. Monte Carlo simulations are carried out to replicate and analyze data sets by iden- tifying the appropriate copula. We apply the Archimedean copula based methodology to assess the accuracy of Doppler echocardiography in deter- mining aortic valve area from the Aortic Stenosis: Simultaneous Doppler - Catheter Correlative study carried out at the King Faisal Specialist Hospital and Research Centre, Riyadh, KSA.

In modeling spatial extremes, the dependence structure is classically inferred by assuming that block maxima derive from max-stable processes. Weather stations provide daily records rather than just block maxima. The point process approach for univariate extreme value analysis, which uses more historical data and is preferred by some practitioners, does not adapt easily to the spatial setting. We propose a two-step approach with a composite likelihood that utilizes site-wise daily records in addition to block maxima. The procedure separates the estimation of marginal parameters and dependence parameters into two steps. The first step estimates the marginal parameters with an independence likelihood from the point process approach using daily records. Given the marginal parameter estimates, the second step estimates the dependence parameters with a pairwise likelihood using block maxima. In a simulation study, the two-step approach was found to be more efficient than the pairwise likelihood approach using only block maxima. The method was applied to study the effect of El Niño-Southern Oscillation on extreme precipitation in California with maximum daily winter precipitation from 35 sites over 55 years. Using site-specific generalized extreme value models, the two-step approach led to more sites detected with the El Niño effect, narrower confidence intervals for return levels and tighter confidence regions for risk measures of jointly defined events.

In multivariate survival analysis, estimating the multivariate distribution functions and then measuring the association between survival times are of great interest. Copula functions, such as Archimedean Copulas, are commonly used to estimate the unknown bivariate distributions based on known marginal functions. In this paper the feasibility of using the idea of local dependence to identify the most efficient copula model, which is used to construct a bivariate Weibull distribution for bivariate Survival times, among some Archimedean copulas is explored. Furthermore, to evaluate the efficiency of the proposed procedure, a simulation study is implemented. It is shown that this approach is useful for practical situations and applicable for real datasets. Moreover, when the proposed procedure implemented on Diabetic Retinopathy Study (DRS) data, it is found that treated eyes have greater chance for non-blindness compared to untreated eyes.

Conditional Value-at-Risk (CoVaR) is defined as the Value-at-Risk of a certain risk given that the related risk equals a given threshold (CoVaR=) or is smaller/larger than a given threshold (CoVaR</CoVaR≥). We extend the notion of Conditional Value-at-Risk to quantile based co-risk measures that are weighted mixtures of CoVaR at different levels and hence involve the stochastic dependence that occurs among the risks and that is captured by copulas. We show that every quantile based co-risk measure is a quantile based risk measure and hence fulfills all related properties. We further discuss continuity results of quantile based co-risk measures from which consistent estimators for CoVaR< and CoVaR≥ based risk measures immediately follow when plugging in empirical copulas. Although estimating co-risk measures based on CoVaR= is a nontrivial endeavour since conditioning on events with zero probability is necessary we show that working with so-called empirical checkerboard copulas allows to construct strongly consistent estimators for CoVaR= and related co-risk measures under very mild regularity conditions. A small simulation study illustrates the performance of the obtained estimators for special classes of copulas.

Forecast ensembles are typically employed to account for prediction uncertainties in numerical weather prediction models. However, ensembles often exhibit biases and dispersion errors, thus they require statistical post-processing to improve their predictive performance. Two popular univariate post-processing models are the Bayesian model averaging (BMA) and the ensemble model output statistics (EMOS). In the last few years increased interest has emerged in developing multivariate post-processing models, incorporating dependencies between weather quantities, such as for example a bivariate distribution for wind vectors or even a more general setting allowing to combine any types of weather variables. In line with a recently proposed approach to model temperature and wind speed jointly by a bivariate BMA model, this paper introduces a bivariate EMOS model for these weather quantities based on a truncated normal distribution. The bivariate EMOS model is applied to temperature and wind speed forecasts of the eight-member University of Washington mesoscale ensemble and of the eleven-member ALADIN-HUNEPS ensemble of the Hungarian Meteorological Service and its predictive performance is compared to the performance of the bivariate BMA model and a multivariate Gaussian copula approach, post-processing the margins with univariate EMOS. While the predictive skills of the compared methods are similar, the bivariate EMOS model requires considerably lower computation times than the bivariate BMA method.

Copulas represent a useful approach to understanding and modeling dependent components of random variables that allows us to focus explicitly on the dependence structure. This paper aims to seek out the most appropriate copula which can model the dependence structure and measure the risk characteristic for the tail dependent asset returns. Based on the empirical data from the financial market, we begin with the analysis of the marginal choice for the copulas by comparing three different Archimedean copulas with respect to nonparametric kernel density estimation, semiparametric estimation and the estimation based on full empirical assumption of the margins, on the basis of which, we conduct the statistical estimation of the copula parameters using inference functions for margins (IFM) and canonical maximum likelihood (CML) methods. A procedure is thereafter proposed for identifying the most suitable copula. We then calibrate copula functions to recover the joint tail distribution and to quantify the magnitude of tail dependence by comparing different Archimedean copulas with the nonparametric empirical one. We present in detail from different aspects that Gumbel among three Archimedean members is the most suitable copula that has the desired property which is in accordance with the empirical behavior of our market data.

Joint-life insurance pays a sum insured when the first death occurs. This insurance has a case based on the order of exit from the cohort, namely joint life and last survivor. The former means that one of the insured leaves the cohort, while the latter means the last member of the insured has left his or her cohort. For some reasons of simplicity, the insurance premium is usually calculated with the assumption that the husband and wife are mutually independent. However, this assumption is considered unrealistic. Couples are open to the same risks, hence explaining joint survival model should involve dependence structures between the distribution of spouse mortality. In line with this, to understand the dependence structure of multiple random variables, the approach used is Copula. In this context, Copula relates the marginal distribution function of these variables to the joint life distribution. One of the advantages from Copula is that the random variables do not have to come from the same distribution, hence Copula is considered good enough to explain the dependence of the mortality rate between husband and wife. This study aimed to develop a joint survival model for calculating joint life insurance premiums using the concept of Archimedean Copula to discover the minimum premium value by conducting the following steps: first, identifying the marginal distributions of mortality for genders using Indonesian Mortality Table IV (TMI/Tabel Mortalitas Indonesia IV); second, Archimedean copula function-based constructing survival models that captures the relationship between these variables; third, setting dependency parameter θ; fourth, calculating the joint life premium using Archimedean copula based survival modeled for each correlation dependency level; and carrying out optimization to find the minimum premium value. This can be achieved by formulating the problem as an optimization problem, considering an objective function that yields the lowest premium till satisfying the financial requirements of the insurance company.

The spatial variations of water and energy budgets are highly influenced by the heterogeneity of land-surface characteristics. We investigate the spatiotemporal variability and dependence structure patterns of water and energy fluxes along an elevation gradient. Our analysis is based on the application of the GEOtop model and empirical Copulas. It is performed for the Rott (∼55 km2) and Upper-Ammer (∼300 km2) catchments in the TERrestrial ENvironmental Observatories prealpine region over two recent summer episodes, as a test case. We found that GEOtop is capable of quantifying the spatiotemporal variability of the water and energy budgets with consideration for the elevation-gradient effect of this heterogeneous landscape, which is confirmed by the linear statistical metrics. Furthermore, the empirical Copula-based function reveals that the dependence structures between the measured and simulated hydrometeorological variables are similar either at upper or lower density maxima. This suggests a reasonable performance of the model, as the interaction of variables is described properly; however, the model shows poorer performance in the middle ranks of the data. It is concluded that the presented Copula-based model performance analysis is a valuable complement to traditional global performance model analyses.

<strong class="journal-contentHeaderColor">Abstract.</strong> To understand the impact of different assimilated observations on convection-allowing model forecast skill, a diverse range of observing system simulation experiment (OSSE) case studies are required (different storm modes and environments). Many previous convection-allowing OSSEs predicted the evolution of an isolated supercell generated via a warm air perturbation in a horizontally homogenous environment. This study introduces a new methodology where a quasi-linear convective system is generated in a highly-sheared and modestly unstable environment. Wind, temperature, and moisture perturbations superimposed on a horizontally homogeneous environment simulate a cold front that initiates an organized storm system that spawns multiple mesovortices. Mature boundary layer turbulence is also superimposed onto the initial environment to account for typical convective scale uncertainties. Creating an initial forecast ensemble remains a challenge for convection-allowing OSSEs because mesoscale uncertainties are difficult to quantify and represent. The generation of the forecast ensemble is described in detail. 24 hour full-physics simulations (e.g., radiative forcing, surface friction, microphysics) initialize the forecast ensemble. The simulations assume different surface conditions to alter surface moisture and heat fluxes and modify the effects of friction. The subsequent forecast ensemble contains robust non-gaussian errors that persist until corrected by the data assimilation system. An example OSSE suggests a combination of radar and conventional (surface and soundings) observations are required to produce a skilled quasi-linear convective system forecast, which is consistent with real case studies. The OSSE framework introduced in this study will be used to understand the impact of assimilated environmental observations on forecast skill.

This paper deals with the assessment of hydrological predictions driven by probabilistic hybrid rainfall forecasts obtained by merging numerical weather prediction and radar‐based rainfall forecasts. Deterministic and ensemble hybrid rainfall forecasts have been utilized to provide river flow predictions on a catchment study located in the UK. A novel assessment methodology has been developed and implemented for the purpose of comparing deterministic and ensemble river flow predictions. The assessment methodology provides a measure of the degree of underestimation/overestimation characterizing the forecasts in comparison with a series of normalized reference thresholds. The analysis of the results obtained by implementing this evaluation methodology shows that the overall performance of the deterministic forecasts is very similar to that of the ensemble forecasts, with a slightly larger degree of underestimation affecting the ensemble forecasts in comparison with the deterministic forecasts. Copyright © 2012 John Wiley &amp; Sons, Ltd.

Statistical postprocessing of global ensemble weather forecasts is revisited by leveraging recent developments in machine learning. Verification of past forecasts is exploited to learn systematic deficiencies of numerical weather predictions in order to boost postprocessed forecast performance. Here, we introduce postprocessing of ensembles with transformers (PoET), a postprocessing approach based on hierarchical transformers. PoET has two major characteristics: 1) the postprocessing is applied directly to the ensemble members rather than to a predictive distribution or a functional of it, and 2) the method is ensemble-size agnostic in the sense that the number of ensemble members in training and inference mode can differ. The PoET output is a set of calibrated members that has the same size as the original ensemble but with improved reliability. Performance assessments show that PoET can bring up to 20% improvement in skill globally for 2-m temperature and 2% for precipitation forecasts and outperforms the simpler statistical member-by-member method, used here as a competitive benchmark. PoET is also applied to the ENS-10 benchmark dataset for ensemble postprocessing and provides better results when compared to other deep learning solutions that are evaluated for most parameters. Furthermore, because each ensemble member is calibrated separately, downstream applications should directly benefit from the improvement made on the ensemble forecast with postprocessing.

China has highly emphasized the research and operational application of numerical weather prediction. This paper determines the objective function parameters, such as CAPE and SRH, to apply an ensemble numerical prediction model in weather forecasting. Preprocessing and evaluating rainfall data is necessary to construct the WRF-ARW numerical weather prediction model. The WRF-ARW model is applied to simulate the weather forecasts in Henan Province, and the difficulties and challenges faced in the efficient implementation of the parameterized scheme are outlined. The WRFARW model’s prediction errors for the maximum rainfall and total rainfall in Henan Province range from 1.78%-13.51% and 0.16%-3.78%, respectively, which are significantly less than 15%, and the model is more predictive than the others. The raw data test set’s credibility ranges from 0.957 to 0.997, which is close to 1, indicating that the raw data collected in this paper are highly credible. The WRF-ARW model’s qualification rates for forecasting maximum rainfall and total rainfall are 86.7% and 93.3%, respectively, and its overall accuracy is grade B and grade A, respectively. The pass rates for the peak occurrence time of maximum rainfall and total rainfall were 93.3% and 86.7%, respectively, and the overall prediction accuracy was Grade A and Grade B, respectively. The WRF-ARW model is effective in weather forecasting throughout Henan Province. In summary, the WRF-ARW model is very effective in improving the efficiency of ensemble numerical weather prediction and parameterization schemes in Henan Province.

Urban agglomeration polders (UAPs) are often used to control flooding in eastern China. The impacts of UAPs on individual flood events have been extensively examined, but how flood risks are influenced by UAPs is much less examined. This study aimed to explore a three-dimensional joint distribution of annual flood volume, peak flow and water level to examine UAPs’ impact on flood risks based on hydrological simulations. The dependence between pairwise hydrological characteristics are measured by rank correlation coefficients and graphs. An Archimedean Copula is applied to model the dependence structure. This approach is applied to the Qinhuai River Basin where UAPs are used proactively for flood control. The result shows that the Frank Copula can better represent the dependence structure in the Qinhuai River Basin. UAPs increase risks of individual flood characteristics and integrated risks. UAPs have a relatively greater impact on water level than the other two flood characteristics. It is noted that the impact on flood risk levels off for greater floods.