Quantity of data available to train TANN2 models.
Artificial neural network (ANN) model classifiers were developed to generate ≤15h predictions of thunderstorms within three 400-km2 domains. The feed-forward, multi-layer perceptron and single hidden layer network topology, scaled conjugate gradient learning algorithm, and the sigmoid (linear) transfer function in the hidden (output) layer were used. The optimal number of neurons in the hidden layer was determined iteratively based on training set performance. Three sets of nine ANN models were developed: two sets based on predictors chosen from feature selection (FS) techniques and one set with all 36 predictors. The predictors were based on output from a numerical weather prediction (NWP) model. This study amends an earlier study and involves the increase in available training data by two orders of magnitude. ANN model performance was compared to corresponding performances of operational forecasters and multi-linear regression (MLR) models. Results revealed improvement relative to ANN models from the previous study. Comparative results between the three sets of classifiers, NDFD, and MLR models for this study were mixed—the best performers were a function of prediction hour, domain, and FS technique. Boosting the fraction of total positive target data (lightning strikes) in the training set did not improve generalization.
- thunderstorm prediction
- artificial neural networks
- correlation-based feature selection
- minimum redundancy maximum relevance
- multi-linear regression
However, the complexity of thunderstorm generation (hereafter
2. Thunderstorm development
To properly understand the process of thunderstorm development, it is essential to define the terms
Consider an environment of depth
The parcel will remain positively buoyant until it reaches the
Now, consider a separate case whereby the environment is
Air parcels extending above the LFC accelerate upward owing to positive buoyancy and draw energy for acceleration from CAPE. The relationship between maximum updraft velocity (
The moist updraft is integral to the development of a supersaturated condition that results in excess water vapor that (with the aid of hygroscopic aerosols that serve as cloud condensation nuclei or CNN) condenses to form water in liquid and solid form (condensate) manifested as the development of a
As saturated parcels rise to the region with environmental temperatures colder than −4°C, the likelihood that ice crystals will develop within the cloud increases. Further, a fraction of water remains in liquid form (supercooled water) until around −35°C . Thus, a region characterized by water in all three phases (vapor, liquid, and solid) develops. The development of the solid hydrometeors known as
Straight-line thunderstorm surface winds develop as negative buoyancy (owing to cooling associated with evaporation of water/melting of ice), condensate loading (weight of precipitation dragging air initially downward before negative buoyancy effects commence), and/or downward-directed pressure gradient force (associated with convection developing within strong environmental vertical wind shear) contribute to the generation of the
Given the foregoing thunderstorm development process, the simultaneous occurrence of three conditions are necessary for CI: sufficient atmospheric moisture, CAPE, and a lifting/triggering mechanism. Moisture is necessary for the development of the cumulonimbus cloud condensate which serves as a source material for the development of hydrometeors rain, ice crystals, graupel, and hail. The environmental moisture profile contributes to the development conditional and convective instability. CAPE provides the energy for updraft to heights necessary for the development of the cumulonimbus cloud and the associated mixed phase region that contributes to lightning via charge separation. A mechanism is necessary to lift surface-based parcels through the energy barrier to their LFC, and to lift convectively unstable layers necessary for the development of conditional instability. A myriad of phenomena can provide lift, including fronts, dry lines, sea breezes, gravity waves, PBL horizontal convective rolls, orography, and circulations associated with local soil moisture/vegetation gradients [11, 21].
A myriad of synoptic scale  patterns/processes can alter the thermodynamic structure of the environment at a particular location to one favorable for the development of CAPE or convective instability . One scenario in the USA involves the advection of lower-level moist air toward the north across the Southern Plains from the Gulf of Mexico in advance of an upper-level disturbance approaching from the west and advecting midtropospheric dry air originating from the desserts of northern Mexico. The thermodynamic profile over a location influenced by those air masses (e.g., Oklahoma City, Oklahoma) would become one characterized by both conditional and convective instabilities owing to the dry air mass moving over the moist air mass .
The foregoing discussion is not exhaustive with respect to thunderstorms. The transition to
3. Thunderstorm prediction methods
We classify thunderstorm prediction based on the following methods: (1) numerical weather prediction (NWP) models, (2) post-processing of NWP model ensemble output, (3) the post-processing of single deterministic NWP model output via statistical and artificial intelligence (AI)/machine learning (ML), and (4) classical statistical, AI/ML techniques.
3.1. Secondary output variables/parameters from Numerical Weather Prediction (NWP) models
NWP models are based on the concept of determinism which posits that future states of a system evolve from earlier states in accordance with physical laws . Meteorologists describe atmospheric motion by a set of nonlinear partial differential conservation equations—derived from Newton’s second law of motion for a fluid, the continuity equation, the equation of state, and the thermodynamic energy equation—that describe atmospheric heat, momentum, water, and mass referred to as
The state-of-the-art high-resolution NWP models have the ability to explicitly predict/simulate individual thunderstorm cells rather than parameterize the effects of sub-grid-scale convection . NWP output identified as thunderstorm activity involves assessment of NWP output parameter/secondary variable known as
Despite the utility of NWP models, there exist fundamental limitations. In particular, the atmosphere is chaotic—a property of the class of deterministic systems characterized by sensitive dependence on the system’s initial condition [13, 14, 23]. Thus, minor errors between the initial atmospheric state and the NWP model representation of the initial state can result in a future NWP solution divergent from the future true atmospheric state. Unfortunately, a true, exact, and complete representation of the initial state of the atmosphere using the state-of-the-art NWP models is not possible. Even if the NWP model could perfectly represent the initial atmospheric state, errors associated with imperfections inherent in model formulation and time integration would grow. Model discretization and physics parameterizations introduce errors. Further, the gradient terms in the finite difference equations are approximated using a Taylor series expansion of only a few orders [26, 34], thus introducing
Accurately predicting the exact
3.2. Post processing of NWP model ensembles
Methods exist to generate more optimal or skillful thunderstorm predictions/forecasts when using NWP models. One such method is known as ensemble forecasting , which is essentially a Monte Carlo approximation to
One limitation of NWP ensembles to support operational forecasters is the tremendous computational cost necessary to run since each ensemble member is a separate NWP model run. Another limitation is the realization that the true PD of the initial condition uncertainty is unknown and changes daily .
3.3. Post-processing of single deterministic NWP model output using other statistical and artificial intelligence/machine learning techniques
MOS involves the development of data-driven models to predict the future state of a target based on a data set of past NWP output (features/predictors) and the corresponding target/predictand. Following , a regression function
One limitation of the MOS technique involves NWP model changes made by the developers. If NWP model adjustments alter
The NWS uses multiple linear regression (MLR) (with forward selection) applied to operational NWP model predictors and corresponding weather observations (target) to derive MOS equations to support forecast operations . The NWS provides high-resolution gridded MOS products which include a 3-h probability of thunderstorms  and thunderstorm probability forecasts as part of the NWS
Logistic regression is a method to relate the predicted probability
The regression parameters are determined via the method of
Logistic regression models were used by  to develop MOS equations to generate probability of thunderstorms and the conditional probability of severe thunderstorms in twelve 7200 km2 regions at 6-h projections out to 48-h in the Netherlands. The NWP output was provided by both the High-Resolution Limited-Area Model (HIRLAM) and the European Centre for Medium-Range Weather Forecasts (ECMWF) NWP model. The Surveillance et d’Alerte Foudre par Interférométrie Radioélectrique (SAFIR) lightning network provided the target data. Verification results suggest that the prediction system possessed good skills.
A random forest  is a classifier resulting from an ensemble/forest of tree-structured classifiers, whereby each tree is developed from independent random subsamples of the data set (including a random selection of features from which the optimum individual tree predictors are selected) drawn from the original training set. The generalization error (which depends on individual tree strength and tree-tree correlations) convergences to a minimum as the number of trees becomes large, yet overfitting does not occur owing to the law of large numbers. After training, the classifier prediction is the result of a synthesis of the votes of each tree.
In one study, selected thermodynamic and kinematic output from an Australian NWP model served as input for an expert system using the decision tree method to assess the likelihood of thunderstorms and severe thunderstorms . Further, an artificial neural network model to predict significant thunderstorms that require the issuance of the Convective SIGMET product (called WST), issued by the National Weather Service’s Aviation Weather Center, for the 3–7 h period after 1800 UTC; the model demonstrated skill, including the ability to narrow the WST outlook region while still capturing the subsequent WST issuance region . Logistic regression and random forests were used to develop models to predict convective initiation (CI) ≤1 h in advance. The features/predictors included NWP output and selected Geostationary Operational Environmental Satellite (GOES)-R data. The performance of these models was an improvement of an earlier model developed based on GOES-R data alone .
3.4. Classical statistical and artificial intelligence/machine learning techniques
Statistical methods not involving the use of NWP model output (classical statistical) that have been used to predict thunderstorm occurrence include
MDA is essentially a form of multiple linear regression used to predict an event. In particular, a discriminant function relates a nonnumerical predictand to a set of predictors; the value of the function that distinguishes between event groups [1, 55]. An MDA was used to obtain 12 h prediction functions to distinguish between the following two or three member groups within selected domains in portions of the Central and Eastern USA: thunderstorm/no-thunderstorm, thunderstorm/severe thunderstorm, and thunderstorm/severe thunderstorm/no-thunderstorm. The verification domains were within 1° latitude radius relative to the position of the data source that provided the predictor variables. The MDA prediction system provided skill in the 0-12 h period .
In one study, the utility of both a graphical method and multiple regression was tested to predict thunderstorms . The graphical method involved scatter diagrams that were used to analyze multiple pairs of atmospheric stability index parameters in order to discover any diagram(s) whereby the majority of thunderstorm occurrence cases were clustered within a zone while the majority of the non-thunderstorm occurrence cases were outside of the zone. Two such diagrams were found—scatter diagrams of Showalter index versus Total Totals index, and Jefferson’s modified index versus the George index. The multiple regression technique involved the stepwise screening of 274 potential predictors to 9 remaining. Both prediction models provided thunderstorm predictions in probabilistic terms. Objective techniques were used to convert probabilistic predictions to binary predictions for the purpose of verification. Results indicate that the multiple regression model performed better.
An expert system is a form of artificial intelligence that attempts to mimic the performance of a human expert when making decisions. The expert system includes a knowledge base and an inference engine. The knowledge base contains the combination of human knowledge and experience. Once the system is developed, questions are given to the system and the inference engine uses the knowledge base and renders a decision . An expert system was developed using the decision tree method to forecast the development of thunderstorms and severe thunderstorms ; the tree was based on physical reasoning using the observation of meteorological parameters considered essential for convective development. An expert system named
The artificial neural network (ANN) has been used to predict thunderstorms. Thermodynamic data from Udine Italy rawinsondes, surface observations, and lightning data were used to train/optimize an ANN model to predict thunderstorms 6 h in advance over a 5000-km2 domain in the Friuli Venezia Giulia region . An ANN model was developed (also using thermodynamic data) to predict severe thunderstorms over Kolkata India during the pre-monsoon season (April–May) .
Logistic regression was used to develop a binary classifier to predict thunderstorms 6–12 h in advance within a 6825–km2 region in Spain. A total of 15 predictors (combination of stability indices and other parameters) were chosen to adequately describe the pre-convective environment. The classifier generated satisfactory results on the novel data set .
A limitation of classical statistics to weather prediction is that utility is bimodal in time; confined to very short time periods (less than a few hours) or long time periods (10 days) . The utility of AI/ML techniques without NWP may not be as restrictive. Convective storm prediction accuracies associated with expert systems may be similar to that of NWP models .
4. The utility of post-processing NWP model output with artificial neural networks to predict thunderstorm occurrence, timing, and location
The rapid drop in prediction/forecast accuracy per unit time for classical statistical techniques renders it less than optimal for thunderstorm predictions over time scales greater than nowcasting (≤ 3 h).
Predicting thunderstorms with the use of deterministic NWP models allows for the prediction of atmospheric variables/parameters in the ambient environment that directly influence thunderstorm development. As discussed previously, limitations of NWP models include predictability limitations owing to a chaotic atmosphere and inherent model error growth [13, 14, 23]. NWP model ensembles attempt to account for the uncertainty in the NWP model’s initial condition which contributes to predictability limitations in NWP models . The post-processing of NWP model ensembles can generate useful probabilistic thunderstorm output . However, there is a much greater computational cost to generate an ensemble of deterministic runs relative to a single run. The post-processing of output from a single deterministic NWP model can improve skill by minimizing certain systematic model biases, yet predictability limitations associated with single deterministic NWP model output remain.
The authors explored the use of the artificial neural network (ANN) to post-process output from a single deterministic NWP model in an effort to improve thunderstorm predictive skill, based on an adjustment to the author’s previous research . As mentioned earlier, this approach involves a much lower computational cost relative to model ensembles. In particular, the single NWP model used in the development of the thunderstorm ANN (TANN) models discussed in this chapter is the 12-km NAM (North American Mesoscale), which refers to either the Eta, NEMS-NMMB, or WRF-NMM model (only one model used at a time; see Appendix C). The model integration cost to run the 21-member SREF ensemble would be of the order of 20 times the cost required to run the 12-km NAM (Israel Jirak 2016, personal communication.)
Given that ANN was developed to capture the parallel distributed processing thought to occur in the human brain and has tremendous pattern recognition capabilities [63–65], we posit that the ANN will learn the limitations/errors associated with a single NWP deterministic model solution to generate skillful forecasts, notwithstanding NWP predictability limitations. Thus, AI/ML rather than NWP model ensembles would be utilized to deal with atmospheric chaos. The remainder of this chapter focuses on the development of ANN models to predict thunderstorms (TANN) that are part of the ongoing research.
5. Artificial neural network models to predict thunderstorms within three South Texas 400-km2 domains based solely on a data set within the same domain of each
In an earlier study , a thunderstorm ANN (TANN) was developed to predict thunderstorm occurrence in three separate 400-km2 square (box) domains in South Texas (USA), 9, 12, and 15 h (+/−2 h) in advance, by post-processing 12-km NWP model output from a single deterministic NWP model (Appendix C) and 4-km sub-grid-scale output from soil moisture magnitude and heterogeneity estimates. A framework was established to predict thunderstorms in 286 box domains (Figure 1), yet predictions were only performed for three. The three box regions were strategically located to access model performance within the
Eighteen (18) TANN classifiers were developed with half based on a reduced set of predictors/features due to feature selection (FS) and the other half using the full set of 43 potential predictors. FS involves a determination of the subset of potential predictors that describes much of the variability of the target/predictand. This is very important for very large dimensional models which may suffer from the
The optimized TANN binary classifiers were evaluated on a novel data set (2007–2008), and performance was compared to MLR-based binary classifiers, and to operational forecasts from the NWS (National Digital Forecast Database, NDFD, Appendix D).
The MLR models were developed via stepwise forward linear regression (SFLR). For each MLR model, the SFLR process began with an empty set of features (constant value
The results were mixed—the TANN, MLR, and human forecasters performed better than the other two depending on the domain, prediction hour, and performance metric used. Results revealed the utility of an automated TANN model to support forecast operations with the limitation that a larger number of individual ANNs must be calibrated in order to generate operational predictions over a large area. Further, the utility of sub-grid-scale soil moisture data appeared limited given the fact that only 1/9 of the TANN models with reduced features retained any of the sub-grid parameters as a feature. The NWP model convective precipitation (CP) feature was retained in all the nine feature selection TANN models, suggesting that CP adequately accounted for the initiation of sub-grid-scale convection. This result is consistent with another study  which found that model CP was the most relevant predictor of thunderstorm activity.
6. Artificial neural network models to predict thunderstorms within three South Texas 400-km2 domains based on data set from two-hundred and eighty-six 400-km2 domains
With respect to TANN skill, the endeavor to predict thunderstorms in a small domain relative to domains used in other studies restricted the amount of CG lightning cases. A large number of thunderstorm cases would be beneficial to the model calibration and verification process; the amount of target data in  may have been insufficient to train this data-driven model with sufficient skill owing to the curse of dimensionality . Further, the method of training/optimizing TANN models for each 400-km2 domain limits operational applicability since it would require the development of literally thousands of TANN models at a country scale. In order to retain thunderstorm predictions in 400-km2 domains while increasing predictive skill, a new approach was developed (TANN2), whereby for each prediction hour (9, 12, 15), a single TANN model is trained over all two-hundred and eighty six 400-km2 continuous domains. This approach dramatically increased the amount of positive target data (thunderstorm cases) and total cases/instances. Table 1 depicts the quantity of data utilized in this project. The total number of cases over the study period was 1,148,576 with 939,510 cases used for model calibration and 209,066 cases contained in the 2007–2008 testing set.
|Prediction hour||Total instances (training sample)||Positive target data||Percent positive target|
Relative to the previous study , only the NWP model and Julian date predictor variables (features) were retained (resulting in 36 potential features used in this study; see Tables 2 and 3.) Given that in , only one sub-grid parameter was chosen for only 1/9 of the box/prediction hour combinations, and that the model including this sub-grid-scale parameter did not result in classifier performance improvement, the utility of the sub-grid-scale data appeared very limited. As mentioned in , the NWP model CP parameter was a ubiquitous output of the FS technique and thus considered a skillful predictor of convection. Physically, it was surmised that CP adequately accounted for the effects of sub-grid-scale convection. The use of FS was retained in order to eliminate irrelevant and redundant features to improve model skill and to reduce model size. The reduction in model size/dimension (owing to FS) and the increase in both the training data set and the amount of target CG lightning cases are expected to result in a more accurate/skillful model when considering the curse of dimensionality.
|Abbreviation||Description (Units)||Justification as thunderstorm predictor|
|PWAT||Total precipitable water|
|MR850||Mixing ratio at 850 hPa|
|Lower level moisture necessary for convective cell to reach horizontal scale ≥4 km in order to overcome dissipative effects |
|RH850||Relative humidity at 850 hPa|
|When combined with CAPE, predictor of subsequent thunderstorm location independent of synoptic pattern |
available potential energy
|Instability proxy; the quantity is the theoretical limit of thunderstorm updraft velocity |
|CIN||Convective inhibition (J kg−1)||Surface-based convective updraft magnitude must exceed for parcels to reach level of free convection |
|LI||Lifted index (K)||Atmospheric instability proxy; utility in thunderstorm prediction |
surface, 850 hPa (LEVEL =
surface, 850 hPa) (ms−1)
|Strong wind can modulate or preclude surface heterogeneity induced mesoscale circulations [87, 88]|
|VVLEVEL||Vertical velocity at 925, 700,|
and 500 hPa (LEVEL = 925,
700, 500 hPa) (Pa s−1)
|Account for mesoscale and synoptic scale thunderstorm triggering mechanisms (sea breezes, fronts, upper level disturbances) that are resolved by the NAM|
|DROPOFFPROXY||Potential temperature dropoff|
|Atmospheric instability proxy; highly sensitive to CI |
|LCL||Lifted condensation level (m)||Proxy for cloud base height; positive correlation between cloud base height and CAPE to convective updraft conversion efficiency |
|T_LCL||Temperature at the LCL (K)||T_LCL |
|By-product of the |
|VSHEARS8||Vertical wind shear: 10 m to|
800 hPa layer (×10−3 s−1)
|The combination of horizontal vorticity (associated with ambient 0–2 km vertical shear), and density current (e.g., gust front) generated horizontal vorticity (associated with 0–2 km vertical shear of opposite sign than that of ambient shear can trigger new convection |
|VSHEAR86||Vertical wind shear: 800–600 hPa layer|
|Convective updraft must exceed vertical shear immediately above the boundary layer for successful thunderstorm development [58, 89]|
|Abbreviation||Description (units)||Justification as thunderstorm predictor|
900, 800, 700, 600, 500
hPa levels (LEVEL=
surface, 900, 800, 700,
600, 500) (ms−1)
|Thermodynamic profile modification owing to veering|
of wind (warming) or backing of wind (cooling); backing
(veering) of wind in the lowest 300 hPa can
suppress (enhance) convective development 
|HILOW||Humidity index (°C)||Both a constraint on afternoon convection and an|
atmospheric control on the interaction between soil
moisture and convection 
|CTP Proxy||Proxy for convective|
|Both a constraint on afternoon convection and an|
atmospheric control on the interaction between soil
moisture and convection 
|VSHEARS7||Vertical wind shear:|
surface to 700 hPa
layer (×10−3 s−1)
|Strong vertical shear in the lowest 300 hPa|
can suppress convective development 
|VSHEAR75||Vertical wind shear: 700|
to 500 hPa layer (×10−
|Convective updraft must exceed vertical shear immediately|
above the boundary layer for successful
thunderstorm development [58, 89]
With respect to model training, validating, optimizing, and testing, the same strategy was utilized as in , with two differences. First, when determining the optimal number of hidden layer neurons, the range of neurons was extended to
Figure 2 depicts an example of how the optimal Y is chosen. The light red highlight identifies the number of hidden neurons leading to the largest mean PSS while the green highlight indicates the number of hidden neurons selected as the two cases standard errors overlap. Table 4 depicts the optimal topologies for the TANN2 X-Y-1 and 36-Y-1 models.
With respect to FS, an exhaustive search involving the 36 potential features, although ideal, would have been computationally unrealistic. The FS methods used for this work were filter based, information theoretic, and designed to choose feature subsets relevant to the corresponding target while non-redundant to each feature in the subset. The methods used were multi-variate in the sense that feature-feature relationships were also considered, rather than the univariate strategy of assessing only feature-target relationships sequentially. The methods used are CFS (described earlier) and
Thus, we initialize the set of selected features
The mRMR classic function requires the user to select the size of
|U800(0), HILOW, CP, VV925, VV500 RH850, CIN, LCL, T_LCL, CAPE,|
VSHEAR86, V600(0), PWAT, CTP_PROXY, VSHEAR75(0)
|12 h||CP||CP, VV500 ,VV925, PWAT, RH850, CTP_PROXY, CAPE, VSHEAR86,|
HILOW, DROPOFFPROXY, U800(0), VV700
|15 h||CP||CP, LI, CAPE,VV925, PWAT,|
RH850, CTP_PROXY, VV700 V500(0), USFC(0), VV500, HILOW
|Prediction hour||Optimal 36-Y-1 topology||Optimal X-Y-1 topology (CFS)||Optimal X-Y-1 topology (mRMR)|
|Performance metric (value range)||Symbol||Equation|
|Probability of detection [0,1]||POD|
|False alarm rate [0,1]||F|
|False alarm ratio [0,1]||FAR|
|Critical success index [0,1]||CSI|
|Peirce skill score [−1,1]||PSS|
|Heidke skill score [−1,1]||HSS|
|Yule’s Q (odds ratio skill score) [−1,1]||ORSS|
|Clayton skill score [−1,1]||CSS|
|Gilbert skill score [−1/3,1]||GSS|
Tables 8–11 depict the performance results of the TANN2 models, trained over all 286 boxes, and applied to boxes 73, 103, 238, and 1–286 (all boxes.) For each skill-based performance metric (PSS, CSI, HSS, ORSS, CSS, GSS), the
There is a significant improvement in the value of selected performance metrics for TANN2 over TANN in absolute terms. For example, with respect to the TANN models developed without FS, the PSS metric for the TANN2 models increased over the corresponding TANN models, by approximately 10–70%, 55–74%, and 10–120%, respectively, for boxes 238, 103, and 73.
When comparing TANN2 model performance relative to the operational forecasters (NDFD), and defining superior performance as statistically significant superior performance with respect to at least one skill-based performance metric (PSS, CSI, HSS, ORSS, CSS, and GSS), the results are as follows: For
Conducting the same analysis with respect to TANN2 model compared to MLR, namely assessing statistically significant superior performance with respect to at least one skill-based performance metric, the results are as follows: For
|TANN 2-60-1||TANN 36-125-1||NDFD|
|TANN 36-150-1||NDFD||TANN 36-70-1|
|TANN 2-60-1||TANN 36-125-1||NDFD|
An alternative analysis was performed to determine the single best-performing classifiers for each box and prediction hour based only on performance metrics PSS, HSS, and ORSS. HSS and PSS are truly equitable and ORSS is asymptotically equitable (approaches equitability as size of the data set approaches infinity) and truly equitable for the condition (see contingency matrix.) . Equitability includes the desirable condition whereby the performance metric is zero for random or constant forecasts. For each of the three performance metrics and for each box and prediction hour combination, comparisons were made between the TANN2 models and NDFD. Such a comparison was performed separately between the TANN2 models and MLR. The best performers were determined in the following manner: First, the output from the
|TANN 2-60-1||MLR||TANN 1-1-1|
|TANN 36-150-1||TAN 1-1-1||TAN 36-70-1|
The increase in data as compared to the previous study  likely contributed to performance enhancements. Figure 3 depicts the change in performance of the 36-Y-1 12 h model as a function of training data used. Note that performance improvement was positively correlated with the quantity of training data. This adds credence to the argument that the amount of data in  may have been insufficient. Further, note that for ≤1% of available data (~6000 instances), performance decreased after the number of neurons in the hidden layer (Y) exceeded 100, possibly due to the curse of dimensionality. Figure 4 depicts the relationship between an overfitting metric, defined as , and the number of hidden layer neurons. Note that overfitting generally increases with the number of hidden layer neurons when the training set is reduced to ≤ 10 of the total available training data, and particularly apparent when only 1 of available, or about 6000 cases. In , only a few thousand cases were available. This partially explains why the original TANN models are relatively smaller (
With regard to data set size and composition, it was hypothesized that performance gains may be obtained when artificially increasing the proportion of positive targets (CTG lightning strikes) in the training set. All possible inputs were included and the total number of training cases was maintained constant with positive and negative cases randomly selected (with replacement) to create target vectors with 5, 10, 25, and 50% of CTG lightning strikes. Substantial increases in performance were obtained for the training sets for all prediction lead times; Figure 5 depicts the 12 h prediction example. Maximum PSS increased progressively while increasing proportion of lightning strikes in the data set. However, as the percent of positive targets was raised, the performance over the 2007–2008 independent testing decreased. Efforts are continuing to further modify the training of the TANN to improve performance.
We presented here the results of an ANN approach to the post processing of single deterministic NWP model output for the prediction of thunderstorms at high spatial resolution, 9, 12, and 15 h in advance (TANN2.) ANNs were selected to take advantage of a large data set of over 1 million cases with multiple predictors and attempt to capture the complex relationships between the predictors and the generation of thunderstorms. This study represents an adjustment to a previous ANN model framework, resulting in the generation of a significantly larger data set. The larger data set allowed for more complex ANN models (by increasing the number of neurons in the hidden layer). Three groups of TANN2 model variants were generated based on two filtering-based feature selection methods (designed to retain only relevant and non-redundant features) and one group based on models calibrated with all predictors.
The skills of these TANN2 models within each of the three 400-km2 boxes were substantially improved over previous work with the improvements attributed to the increase of the size of the data set. TANN2 model performance was compared to that of NWS operational forecasters and to MLR models. Results regarding the best-performing classifiers per prediction hour and box were mixed. Several attempts were made to further improve model performance or decrease training time. Training the models using a small fraction of the data set reduced model calibration time yet resulted in lower performance skill. Altering the target by artificially boosting the proportion of positive outcomes (lightning strikes) resulted in substantial performance improvements over the training sets but did not lead to substantial improvements of performance on the independent 2007–2008 cases.
Given that the atmosphere is chaotic, or deterministic with highly sensitive dependence on the initial condition, one future research plan includes the prediction of thunderstorms by the post-processing of single deterministic NWP model output using ANN models that account for chaotic systems . Such a strategy would be an alternative to the state of the art practice of using NWP model ensembles to account for the sensitive dependence on the initial condition. In addition, another plan involves the development of ensemble ANN models . Specifically, an optimal TANN prediction can be developed by integrating output from 50 unique TANN models.
The total mean mass of the atmosphere: 
Total mol of dry air:
Total molecules of dry air:
(Mass of dry air: ; Avogadro’s number: )
Version 1.04 of the HRRR uses the Advanced Research WRF (ARW) dynamic core within the WRF modeling framework (version 3.4.1 WRF-ARW). The HRRR uses GSI 3D-VAR data assimilation. With respect to parameterizations, the RRTM longwave radiation, Goddard shortwave radiation, Thompson microphysics (version 3.4.1), no cumulus/convective parameterization, MYNN planetary boundary layer, and the rapid update cycle (RUC) land surface model .
At any given time, only one NWP model was utilized in the TANN. Yet, three different modeling systems were used, each during a unique time period—the hydrostatic Eta model  (1 March 2004 to 19 June 2006), the Weather Research and Forecasting Non-hydrostatic Mesoscale Model (WRF-NMM)  (20 June 2006 to 30 September 2011), and the NOAA Environmental Modeling System Non-hydrostatic Multiscale Model (NEMS-NMMB) (October 2011 to December 2013.)
NWS operational forecasts were obtained from the NWS
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