Forecast Error Reduction Research Articles

Scale‐ and Variable‐Dependent Localization for 3DEnVar Data Assimilation in the Rapid Refresh Forecast System

AbstractThis study demonstrates the advantages of scale‐ and variable‐dependent localization (SDL and VDL) on three‐dimensional ensemble variational data assimilation of the hourly‐updated high‐resolution regional forecast system, the Rapid Refresh Forecast System (RRFS). SDL and VDL apply different localization radii for each spatial scale and variable, respectively, by extended control vectors. Single‐observation assimilation tests and cycling experiments with RRFS indicated that SDL can enlarge the localization radius without increasing the sampling error caused by the small ensemble size and decreased associated imbalance of the analysis field, which was effective at decreasing the bias of temperature and humidity forecasts. Moreover, simultaneous assimilation of conventional and radar reflectivity data with VDL, where a smaller localization radius was applied only for hydrometeors and vertical wind, improved precipitation forecasts without introducing noisy analysis increments. Statistical verification showed that these impacts contributed to forecast error reduction, especially for low‐level temperature and heavy precipitation.

Adjoint‐based observation impact on meteorological forecast errors in the Arctic

AbstractAccurate weather forecasting using numerical weather prediction in the Arctic remains a challenge owing to the complex atmospheric dynamics and relatively sparse observations of the region. Nevertheless, the impact of observations used for data assimilation (DA) on forecast errors (observation impacts, OIs) in the Arctic has not been fully explored. This study evaluated the OIs on 24 hr forecast errors in the Arctic by the adjoint‐based forecast sensitivity OI (FSOI) method using the Polar Weather Research and Forecasting (WRF) model, three‐dimensional variational DA, and adjoint model of WRFPLUS. The time‐averaged FSOI during August 2018 was analyzed in terms of the type, variable, satellite channel, altitude, and horizontal location of the observations. The results showed that the total FSOI was greatest for aircraft observations (AIRCRAFT), followed by sonde observations, scatterometer sea‐surface winds, advanced microwave sounding unit‐A radiance data from each satellite, and surface synoptic observations. The impact of AIRCRAFT in reducing forecast errors was greater in the Arctic than in other regions of the globe. The total FSOI of the advanced microwave sounding unit‐A radiance data from all four satellites was the greatest among the total FSOI of the individual observation types. For both AIRCRAFT and sonde observations, the FSOI per observation increased at higher latitudes, indicating that forecast errors in the Arctic could be reduced when upper atmospheric in‐situ observations at higher latitudes are used for DA. The beneficial ratio of the Global Positioning System precipitable water in the Arctic was greater than that in the midlatitudes with relatively moist atmospheres that cause greater uncertainties in converting zenith wet delay. The results of this study can contribute to configuring an optimal observing system in the Arctic by presenting the relative importance of observations for reducing forecast errors.

Functional Background Diversity of Top Management Team and Managerial Forecasting Capability

ABSTRACT This study examines whether top management team (TMT) functional background diversity affects managerial forecasting capability. We find that TMTs with more diverse functional backgrounds issue more accurate earnings forecasts. Additional tests reveal that the negative association between TMT functional background diversity and management forecast errors becomes weaker when TMTs work together for a long time, when subgroups are more likely to develop within the team, when CEOs are more powerful, and when team size is larger. By contrast, the reduction in management forecast errors is more significant for functionally diverse TMTs when forecasting difficulty is higher. These findings suggest that TMT functional background diversity plays an important role in managerial forecasting capability. Data Availability: Data are available from the sources cited in the text. JEL Classifications: M14; M41.

On the use of kolmogorov–arnold networks for adapting wind numerical weather forecasts with explainability and interpretability: application to madeira international airport

Abstract This study examines the application of machine learning to enhance wind nowcasting by using a Kolmogorov-Arnold Network model to improve predictions from the Global Forecast System at Madeira International Airport, a site affected by complex terrain. The research addresses the limitations of traditional numerical weather prediction models, which often fail to accurately forecast localized wind patterns. Using the Kolmogorov-Arnold Network model led to a substantial reduction in wind speed and direction forecast errors, with a performance that reached a 48.5% improvement to the Global Forecast System 3 h nowcast, considering the mean squared error. A key outcome of this study comes from the model’s ability to generate mathematical formulas that provide insights into the physical and mathematical dynamics influencing local wind patterns and improve the transparency, explainability, and interpretability of the employed machine learning models for atmosphere modeling.

The Ensemble Tangent Linear Model Applied to a Cloud Physics Parameterization

Abstract An Ensemble Tangent Linear Model (ETLM) is applied to a cloud physics scheme used in the Navy Global Environmental Model (NAVGEM). The ensemble is created using 3-hour forecasts from the Ensemble Transform method used in the NAVGEM data assimilation system. The model states are saved before and after applying the cloud physics parameterization (which includes condensation/evaporation of cloud ice and cloud liquid water and stratiform precipitation), and these states are used to construct linearized model tendencies for temperature, specific humidity, cloud liquid water, and cloud ice water. We examine separately the application of the ETLM to cloud physics components that are explicitly local versus non-local. For the local components, an ETLM is built using a single grid point. ETLMs from 50 to 1000 members are tested, and skillful forecasts can be obtained for both local and non-local physics even with a moderate sized ensemble (e.g., 100 members). At 1000 members, the globally-averaged forecast error reductions (relative to persistence errors) are ∼40% for temperature, water vapor, and cloud liquid water and ∼30% for cloud ice. When initial perturbations are reduced by a factor of 0.1, the error reductions increase to ∼65% for all variables. For physics with non-local components (stratiform precipitation) the covariances that comprise the ETLM are localized with a Schur product matrix using a Gaussian localization shape with tunable length. The optimal lengths increased with ensemble size from ∼2-3 km for 50 members to ∼10 km for 1000 members. ETLMs for “all cloud physics” are also constructed and evaluated.

Performance evaluation of multivariate deep-time convolution neural architectures for short-term electricity forecasting: Findings and failures

Deep learning (DL) models hold great promise in enhancing the decision-making abilities of electricity market participants and system operators in the short term, as they excel at capturing the intricate interdependencies and uncertainties in electricity consumption. Nevertheless, currently, the adoption of DL models for electricity prediction is not keeping up with the advancements in DL research. Within this framework, the aims of this study are to examine the probabilistic, multivariate, and multihorizon time series prediction of deep time convolution neural (DTCN) models. The effectiveness of DTCN in forecasting electricity consumption over daily, weekly and monthly time horizons is emperically assessed. We analyze both probabilistic and deterministic predictions. The evaluation data comprises real residential electricity consumption, average temperature and humidity with hourly granularity, collected over one year from a residential building. For evaluation, we considered accuracy and uncertainties, sensitivity to hyperparameters, the impact of exogeneous variables, as well as computational efficiency. Results show that the probabilistic models offer greater advantages when it comes to forecasting on multihorizons, especially for diverse datasets with non-uniform time series models. Unlike traditional models, the DTCN models demonstrated superior performance in capturing complex patterns and reducing forecast errors in short-term scenarios. However, their performance can be influenced by the sensitivity of hyperparameters. These findings can act as a benchmark for quantitatively assessing the performance of novel multivariate DL models in predicting electricity demand.

Improving short-term sea ice concentration forecasts using deep learning

Abstract. Reliable short-term sea ice forecasts are needed to support maritime operations in polar regions. While sea ice forecasts produced by physically based models still have limited accuracy, statistical post-processing techniques can be applied to reduce forecast errors. In this study, post-processing methods based on supervised machine learning have been developed for improving the skill of sea ice concentration forecasts from the TOPAZ4 prediction system for lead times from 1 to 10 d. The deep learning models use predictors from TOPAZ4 sea ice forecasts, weather forecasts, and sea ice concentration observations. Predicting the sea ice concentration for the next 10 d takes about 4 min (including data preparation), which is reasonable in an operational context. On average, the forecasts from the deep learning models have a root mean square error 41 % lower than TOPAZ4 forecasts and 29 % lower than forecasts based on persistence of sea ice concentration observations. They also significantly improve the forecasts for the location of the ice edges, with similar improvements as for the root mean square error. Furthermore, the impact of different types of predictors (observations, sea ice, and weather forecasts) on the predictions has been evaluated. Sea ice observations are the most important type of predictors, and the weather forecasts have a much stronger impact on the predictions than sea ice forecasts.

A multi-energy load forecasting method based on complementary ensemble empirical model decomposition and composite evaluation factor reconstruction

Ensuring precise multi-energy load forecasting is crucial for the effective planning, management, and operation of Integrated Energy Systems (IES). This study proposes a novel multivariate load forecasting model based on time-series decomposition and reconstruction to handle the complex, high-dimensional multi-energy load data in IES and enhance forecasting accuracy. Initially, the model conducts a thorough correlation analysis and variable screening to minimize irrelevant data interference. It then applies denoising by decomposing the load sequence into modal components with distinct characteristics, using the complementary ensemble empirical mode decomposition (CEEMD). To overcome the unstable prediction accuracy inherent in time-domain decomposition methods, this study introduces an innovative composite evaluation factor (CEF) that reconstructs the modal components after considering their complexity, coupling, and frequency. The final predictions are generated using the proposed MTL-CNN-BiLSTM model, optimized with the attention mechanism. The results show that the proposed model significantly reduces error accumulation compared to traditional time-domain analysis methods, achieving a 37.40% reduction in average forecasting error and a 30.73% increase in forecasting efficiency.

Signalling through managerial tone and analysts’ response

ABSTRACT This study investigates the interaction between a tone signaller and receiver through managerial tone on a conference call. Based on a signalling framework, we examine the following research questions: Does the signal (tone) match the latent information it intends to convey? Can a receiver (analyst) interpret the signal correctly and send feedback to confirm (or disconfirm) it? Does the signalling environment impact signalling effectiveness? Using a sample of 3680 transcripts of earnings conference calls from 241 UK firms in the FTSE 350 Index, we find that managerial tone can produce effective signals that match private information from managers. Analysts understand tone signals accurately and send confirmation feedback to managers. They revise their forecasts upward (downward) following more (less) optimistic managerial tones. They consistently confirm and respond to managers with a similar tone. Analysts with superior expertise, experience and ability better understand the tone signal, leading to a larger reduction in forecast error. The information environment can moderate signalling effectiveness. This shows that managerial tone represents a reliable signalling device for transmitting latent information on a firm’s quality and future performance and that receivers can understand this signal correctly. This study highlights the signalling role of managerial tone in managers and analysts’ communication. HIGHLIGHTS Managerial tone effectively communicates private information to analysts. Analysts accurately interpret tone signals and adjust forecasts accordingly. Analysts with superior expertise better understand tone, reducing forecast errors. Managerial tone's signalling effectiveness varies with information environment.

Life cycle dynamics of Greenland blocking from a potential vorticity perspective

Abstract. Blocking over Greenland stands out in comparison to blocking in other regions, as it favors accelerated Greenland Ice Sheet melting and has substantial impacts on surface weather in adjacent regions, particularly in Europe and North America. Climate models notoriously underestimate the frequency of blocking over Greenland in historical periods, but the reasons for this are not entirely clear, as we are still lacking a full dynamical understanding of Greenland blocking from formation through maintenance to decay. This study investigates the dynamics of blocking life cycles over Greenland based on ERA5 reanalysis data from 1979–2021. A year-round weather regime definition allows us to identify Greenland blocking as consistent life cycles with an objective onset, maximum, and decay stage. By applying a new quasi-Lagrangian potential vorticity (PV) perspective, following the negative, upper-tropospheric PV anomalies (PVAs−) associated with the block, we examine and quantify the contribution from different physical processes, including dry and moist dynamics, to the evolution of the PVA− amplitude. We find that PVAs− linked to blocking do not form locally over Greenland but propagate into the region along two distinct pathways (termed “upstream” and “retrogression”) during the days before the onset. The development of PVAs− differs more between the pathways than between seasons. Moist processes play a key role in the amplification of PVAs− before the onset and are linked to midlatitude warm conveyor belts. Interestingly, we find moist processes supporting the westward propagation of retrograding PVAs− from Europe, too, previously thought to be a process dominated by dry-barotropic Rossby wave propagation. After onset, moist processes remain the main contribution to PVA− amplification and maintenance. However, moist processes weaken markedly after the maximum stage, and dry processes, i.e., barotropic, nonlinear wave dynamics, dominate the decay of the PVAs− accompanied by a general decrease in blocking area. Our results corroborate the importance of moist processes in the formation and maintenance of Greenland blocking and suggest that a correct representation of moist processes might help reduce forecast errors linked to blocking in numerical weather prediction models and blocking biases in climate models.

Predictably Unpredictable? How Judgmental and Machine Learning Forecasts Complement Each Other

Demand forecasting for seasonal products becomes especially challenging in the case of fast innovations, where the product portfolio is upgraded every season. In addition to the problem of forecasting demand without any historical data, companies also have to deal with frequent stockouts, which bias past sales and provide an unreliable anchor for making new forecasts. We show how one can use machine learning models to leverage information on comparable products from the past together with experts’ forecasts to improve forecasting accuracy. A machine learning forecast using only statistical features results in a forecast error reduction of 24%, measured by weighted mean absolute percentage error, compared to a purely judgmental prediction on data from Canyon Bicycles. Better yet, an integrated human-machine forecast leads to a further 14% reduction in forecast error, indicating that experts’ predictions remain essential for forecasting demand for rapidly innovating seasonal products. The combination of the experts’ knowledge of the future and the machine learning algorithms’ ability to leverage historical information works best in this setting.

Volatility and jump with intraday periodicity and truncated power variation in Chinese yuan-US dollar exchange rates

ABSTRACT This study analyzes the discrete jump volatility of the Chinese yuan/U.S. dollar exchange rate returns using high-frequency five-minute returns from June 2012 to April 2021. Using periodicity filters of volatility, we conduct volatility jump tests for finite and infinite exchange rate jumps, and the truncated power variation does not exhibit upward or downward jumps. We use combined intraday jump statistics with local robust variance. All empirical results show that if we do not include periodicity filters of volatility, such as MAD and ShortH, the five-minute returns of the Chinese yuan/US dollar exchange rates have much lower intraday jump probabilities. Thus, the volatility and jumps of the Chinese yuan/US dollar ratio will be underestimated if we do not consider the periodicity filters of the volatility in the 2010s. We need to include periodicity filters of volatility and jumps to reduce forecast errors in future Chinese Yuan/US dollar exchange rates so that we obtain a better measure of current volatility and generate better hedging strategies for future volatility.

Introducing large‐scale analysis constraints in regional hybrid EnVar data assimilation for the prediction of triple typhoons

AbstractLarge‐scale environment fields play an important role in the accurate prediction of typhoons. However, regional predictions for typhoons often suffer from inadequate representation of large‐scale flow pattern such as those from global models due to limited domain size and observations employed in regional models, especially when multiple typhoons that interact concurrently occur in a regional domain. This study merges the large‐scale information from global model forecasts with the mesoscale information from regional model forecasts in the hybrid ensemble‐variational (EnVar) data assimilation by adding an analysis constraint in the EnVar cost function, which is defined by the departure of the regional model EnVar analysis from the global model fields and takes advantage of flow‐dependent ensemble background error covariance for the introduction of large scales using data assimilation. The EnVar assimilation impacts of the large‐scale fields on predictions of triple typhoons are assessed by conducting cycling assimilation and forecast experiments for a 13‐day‐long period in July 2015 when three typhoons concurrently occurred. Results show that the large‐scale constraint for EnVar can clearly improve the triple‐typhoons' track and intensity forecasts of the regional model. The large‐scale information introduced by the proposed method is also shown to reduce forecast errors of wind, temperature and humidity, respectively. Predictions of the rainfall caused by typhoons are also ameliorated. Besides, the analysis‐constrained regional predictions provide better model dynamic fields in terms of sea surface pressure, geopotential height, and water vapor transport, as well as developed typhoon structures. In addition, the adaptive bias correction for radiance assimilation presents a stable performance under the influence of introducing extra background large‐scale fields. The results indicate that the large‐scale analysis constraint introduced in the hybrid EnVar takes advantages of the multiscale information from the global model and the regional model respectively, thus improving the final results of the predictions of multiple typhoons.

Combined MIMO Deep Learning Method for ACOPF with High Wind Power Integration

The higher penetration of renewable energy sources in current and future power grids requires effective optimization models to solve economic dispatch (ED) and optimal power flow (OPF) problems. Data-driven optimization models have shown promising results compared to classical algorithms because they can address complex and computationally demanding problems and obtain the most cost-effective solution for dispatching generators. This study compares the forecast performance of selected data-driven models using the modified IEEE 39 benchmark system with high penetration of wind power generation. The active and reactive power load data of each bus are generated using Monte Carlo simulations, and synthetic wind power data are generated by utilizing a physical wind turbine model and wind speed samples withdrawn from a Weibull distribution. The objective is to design and evaluate an enhanced deep learning approach for the nonlinear, nonconvex alternating current optimal power flow (ACOPF) problem. The study attempts to establish relationships between loads, generators, and bus outcomes, utilizing a multiple-input, multiple-output (MIMO) workflow. Specifically, the study compares the forecast error reduction of convolutional neural networks (CNNs), deep feed-forward neural networks (DFFNNs), combined/hybrid CNN-DFFNN models, and the transfer learning (TL) approach. The results indicate that the proposed combined model outperforms the CNN, hybrid CNN-DFFNN, and TL models by a small margin and the DFFNN by a large margin.

Research on Numerical Weather Prediction Wind Speed Correction Based on Stacking Ensemble Learning Algorithm

The transportation of wind fields plays a crucial role in the dispersion and distribution of air pollutants, and accurate wind speed prediction is essential for assessing pollutant concentrations. In this study, we constructed a Stacking ensemble learning model using three models, namely Random Forest, LightGBM, and XGBoost, as base learners, and the LASSO regression model as the meta-learner to optimize wind speed data forecasting for Zhengzhou City’s urban area using WRF. Firstly, based on Pearson correlation coefficients, we selected meteorological variables that have a significant impact on near-surface wind speed and derived historical lagged features. Bayesian TPE was utilized for hyperparameter tuning and model building. Finally, the performance of the trained model was evaluated by comparing it with ground observation data. The results showed that compared to the WRF model, Random Forest, LightGBM, and XGBoost effectively reduced forecast errors and significantly improved wind speed predictions. Both LightGBM and XGBoost demonstrated similar performance in the correction models across the 11 stations and outperformed Random Forest. The Stacking method integrated the advantages of the base learners and exhibited improvement capabilities over individual models, highlighting the potential of machine learning in enhancing localized and accurate weather forecasting.

Forecasting the price of oil: A cautionary note

We study the out-of-sample predictability of monthly crude oil prices using forecast combinations constructed from several individual predictor forecasts. Our empirical results indicate that combination forecasts of monthly average oil prices are more accurate than the no-change forecast with statistically significant reductions in mean square forecast errors (MSFE) and significant directional accuracy at every horizon up to 24 months, consistent with earlier evidence that forecast combinations greatly enhance the forecastability of oil prices. In contrast, we find no significant MSFE reductions or directional accuracy for forecasts of end-of-month oil prices at almost all horizons. Furthermore, we document that end-of-month forecasts when used to guide investment and hedging decisions of investors, statistically, do not deliver superior economic value to investors. Overall, the implication of our results is that the statistical and economic significance of forecasts of oil prices is heavily influenced by the construction of the underlying oil price series and provide a cautionary note on which oil price series to use in forecasting.

Evaluation of observation impact on the meteorological forecasts associated with heat wave in 2018 over East Asia using observing system experiments

Heat waves are meteorological disasters that inflict damage on public health and societal systems over extensive areas. The frequency and intensity of heat waves are increasing in many regions worldwide. However, insufficient research has been conducted to reduce forecast errors for heat waves in terms of short-range predictions. In this study, observing system experiments were performed using the Weather Research and Forecasting model and a three-dimensional variational data assimilation (DA), and the effects of observations used for DA on forecast errors for meteorological variables (i.e., upper atmospheric geopotential height, upper temperature, upper wind, and near-surface 2-m temperature) associated with heat wave were analyzed. As the forecast time increased, the 200 and 500 hPa geopotential heights in East Asia and the 2-m temperatures around Korea, Japan, and eastern China tended to be underestimated. All observations used for DA reduced the forecast errors for the meteorological variables associated with heat wave. Upper atmospheric observations (i.e., Advanced Microwave Sounding Unit-A (AMSU-A), aircraft, atmospheric motion vector, and radiosonde) played a more significant role in reducing forecast errors than near-surface observations. Among the upper atmospheric observations, AMSU-A had the greatest impact on reducing underestimation in forecasts for 200 and 500 hPa geopotential heights and the second greatest impact on 2-m temperature. The contraction of Tibetan high area at 200 hPa and Northern Pacific high area at 500 hPa in the experiments without using specific observations, compared to the analysis from the experiment using all observations, caused the deflection of the upper winds clockwise through the geostrophic relationship. Radiosonde observations had the greatest impact on reducing forecast errors of 2-m temperatures. Therefore, upper atmospheric observations are important for reducing errors in heat wave simulations in East Asia. The results of this study could help in designing an optimal observing system that reduces heat wave forecast errors in East Asia.

Enhanced Polar Vortex Predictability Following Sudden Stratospheric Warming Events

AbstractSudden stratospheric warming (SSW) events can form a window of forecast opportunity for polar vortex predictions on subseasonal‐to‐seasonal time scales. Analyzing numerical ensemble simulations, we quantify the associated enhanced predictability due to reduced upward planetary wave fluxes during the mostly radiatively driven recovery phase following SSWs. Ensembles that predict an SSW show reduced ensemble spread in terms of polar vortex strength for several weeks to follow, as well as a corresponding reduction in forecast errors. This increased predictability is particularly pronounced for strong SSWs and even occurs if not all ensemble members predict a major SSW. Furthermore, we found a direct impact of the occurrence of SSWs on the date of the final warming (FW): the decrease in upward wave fluxes delays the FW significantly. The reduced spread after SSWs and the delay in FW date have potentially further implications for (subseasonal) predictions of the tropospheric and mesospheric circulations.

Impact of assimilating Aeolus observations in the global model ICON: A global statistical overview

AbstractGlobal wind profiles provided by the satellite mission Aeolus are an important recent supplement to the Global Observing System. This study investigates the impact of Aeolus horizontal line‐of‐sight wind observations in the operational global assimilation and forecasting system of Deutscher Wetterdienst that is based on the Icosahedral Nonhydrostatic (ICON) model. For this purpose, an observing system experiment was conducted and evaluated for a 3‐month period from July 2020 to October 2020. The Aeolus Rayleigh clear and Mie cloudy data quality and consistency were derived from observation minus background statistics. To correct for an altitude‐dependent bias, a model‐based bias correction scheme has been implemented. Comparisons of the systematic changes in the analysis and the respective forecasts provide an overview of the overall impact of the Aeolus horizontal line‐of‐sight wind assimilation in ICON. Increased influence of Aeolus wind profiles is found in jet regimes (e.g., amplification of the zonal wind component), around large‐scale circulation systems, and convectively active areas in the Tropics. The reduction in forecast error is largest in the tropical upper troposphere and stratosphere, as well as in the mid and upper troposphere of the Southern Hemisphere. The Northern Hemisphere shows a somewhat smaller but still beneficial impact of Aeolus observations. The verification with other conventional observations shows a mean relative reduction in short‐range forecast error between 0.1% and 0.6% in the Northern Hemisphere and up to 1.6% in the Tropics and the Southern Hemisphere. When verifying against the European Centre for Medium‐Range Weather Forecast Reanalysis v5, forecast errors of zonal wind, temperature, and geopotential up to 5 days lead time are reduced by 2–4% on global average and up to 5–8% around the tropical tropopause.

Improving Atmospheric Models by Accounting for Chaotic Physics

Abstract It is well known that randomly perturbing an atmospheric model’s diabatic tendencies can increase its probabilistic forecast skill, mainly by increasing the spread of ensemble forecasts and making it more consistent with the errors of ensemble-mean forecasts. Less obvious and less well established is that such perturbations can also reduce the errors of the ensemble-mean forecasts and improve the model’s mean climate, variability, and sensitivity to forcing. A clear reduction in ensemble-mean forecast errors is demonstrated here in large ensembles of 15-day forecasts made with NOAA’s Global Forecast System model. The nearly ubiquitous reduction around the globe, obtained throughout the forecast range, is interpreted as arising in effect from a modification of the model’s deterministic evolution operator by a stochastic noise-induced drift. The effect is general in systems with state-dependent noise, and occurs even if the noise is not white. In the atmospheric context considered here, the effect is suggested to arise largely from noise-induced reductions of mechanical and thermal damping by chaotic boundary layer and cloud-radiative processes, which also tend to increase model sensitivity to forcing. The results presented here are consistent with many previous studies performed with models ranging from simple stochastically forced models to comprehensive global weather and climate models. They suggest that the diabatic interactions in most current global atmospheric models may not be sufficiently chaotic and this deficiency could be partly remedied by specifying additional stochastic terms. Using some empirical guidance in such specifications may be unavoidable, given the generally intractable complexity of the diabatic interactions.