Observational meteorological data have been growing in size and complexity. This wealth of data can be used to improve prediction and/or understanding of events, but the amount of data also provides many challenges to processing and learning from it. The challenge of analyzing large data volumes is not unique to meteorology. Computer scientists—and specifically machine learning and data mining researchers—are developing frameworks for analyzing big data for a range of applications. The AMS Committee on Artificial Intelligence and Its Applications to Environmental Science aims to bring AI researchers and environmental scientists together to increase the synergy between the two fields. The AI Committee has sponsored four previous contests on a variety of meteorological problems including wind energy, air pollution, winter hydrometeor classification, and storm classification (Lakshmanan et al. 2010), with the goal of bringing together the two fields of research to discuss a common challenge from multiple perspectives. The winners of the past contests presented in a special session at the AMS Annual Meeting that featured both the results and discussions of the various techniques used, as well as how they could be applied to similar problems. While the discussions had been fruitful and attracted people from different backgrounds, participation in the contests declined from year to year. For the 2013–14 contest, we made significant changes to the contest format in order to increase participation and reach a much wider audience.
Our goal for the 2013–14 contest was to determine which approach produces the best total daily solar energy forecast. We changed three key features of the contest organization. First, we used the year prior to the contest to gather and format a larger and more complex dataset for predictions. Second, we hosted the contest website on Kaggle, a popular platform for AI contests with a worldwide audience. Third, we extended the time window of the contest from just the fall to July through November, and allowed contestants to submit and evaluate entries every day throughout the period. These changes resulted in an order-of-magnitude increase in the number of participants and a broadening of the participant pool from those in the existing meteorological community to scientists and engineers around the world.
The forecast data used in this study came from the second-generation NCEP Global Ensemble Forecast System (GEFS) reforecast dataset described in Hamill et al (2013). These data consist of an 11-member global ensemble initialized at 0000 UTC every day from 1985 to the present. Forecasts extend to +16 days lead time. The modeling system closely replicates the GEFS as it was implemented in 2012. The initial conditions for most of the dataset used the Climate Forecast System Reanalysis (CFSR; Saha et al. 2010) for the control initial condition and the ensemble transform with rescaling (Wei et al. 2008) for generating perturbed initial conditions. Forecast data were archived every 3 h to +72 h lead time, and every 6 h thereafter. More details are available in Hamill et al (2013).
The Oklahoma Mesonet is a permanent mesoscale surface observing network of 120 remote meteorological stations across Oklahoma (Brock et al. 1995; McPherson et al. 2007). The Mesonet represents a partnership of Oklahoma State University and the University of Oklahoma, and is managed by the Oklahoma Climatological Survey (OCS). Each station measures more than 20 environmental variables, including wind at 2 m and 10 m, air temperature at 1.5 m and 9 m, relative humidity, rainfall, pressure, solar radiation, and soil temperature and moisture at various depths. All sensors are mounted on or near a 10-m tower supported by three guy wires and powered via solar energy.
Downwelling, global solar radiation is measured by the LI-COR LI-200 pyranometer mounted on a boom that extends southward from the tower. Even so, measurements of solar radiation during early morning and late afternoon into the evening may be sensitive to obstructions to the east and west of the station. All solar radiation data are collected and transmitted to a central point every 5 min where (1) sensor-specific calibration coefficients are applied and (2) the data are quality-controlled via automated algorithms and human inspection prior to distribution and archiving (Shafer et al. 2000; McPherson et al. 2007).
The locations of the GEFS and Mesonet stations are shown in Fig. 1. Due to the coarseness of the GEFS grid relative to the Mesonet station spacing, contestants were provided with additional grid points well outside the Oklahoma state boundaries so that any interpolation techniques would not experience any interference from edge conditions.
The contest was hosted by Kaggle, a company that developed a platform for hosting data mining competitions in addition to providing modeling support for a variety of Fortune 500 companies. For each competition hosted on the site, Kaggle provides pages for describing the competition and the rules, downloading the data, displaying real-time rankings of the participants, and discussions about the contests. The site also automatically manages submission of contestant entries and evaluation of the predictions. The continuous stream of contests on Kaggle has led to the development of a large community of contest participants who come from a wide range of backgrounds and from around the world. For these services and for access to its large user community, Kaggle charges a fee to companies who wish to host their contest through the site, but Kaggle also hosts research competitions for smaller contests organized by academic groups for a small fee. EarthRisk Technologies sponsored the contest and provided the prize money.
For this contest, a small spatial subset of the 11-member ensemble data were extracted over Oklahoma and surrounding regions, consisting of forecasts at the +12-, +15-, +18-, +21-, and +24-h lead times. To be coincident with the observational data, the reforecast data were extracted only back to 1994. These pervaded the forecast training data for the contest’s 1-day solar energy predictions. The forecast variables saved were mean sea level pressure, skin and 2-m temperature, 2-m specific humidity, daily maximum and minimum 2-m temperature, total precipitation in the last 3 h, total column precipitable water, total column integrated condensate, total cloud cover, downward and upward short- and long-wave radiation flux at the surface, and upward long-wave radiation flux at the top of the atmosphere.
The data were split into training, public testing, and private testing sets. The training set time frame extended from 1 January 1994 to 31 December 2007; the public testing set ranged from 1 January 2008 to 31 December 2009, and the private testing set ranged from 1 January 2010 to 30 November 2012. Teams could evaluate their predictions on the public testing set up to 5 times per day and optimize their algorithm based on the evaluation score. The final ranking of the teams was determined from the private testing set results, and the scores were not revealed until the contest concluded. Mean absolute error (MAE) over all stations and days was chosen as the evaluation metric because it does not penalize extreme forecasts as greatly as root mean squared error.
In addition to the contest data, participants also received the results and source code for three benchmark methods that indicated how random selection and interpolation methods would perform on the dataset. The random normal benchmark input random numbers sampled from a normal distribution with a mean of 16 MJ m–2 and a standard deviation of 8 MJ m–2. The other two benchmarks interpolated the GEFS mean total daily incoming solar radiation to the Mesonet sites using nonlinear approaches. One method fit a set of scaled Gaussian mixture models to the GEFS data with an expectation-maximization iterative approach similar to the method of Lakshmanan and Kain (2010). It produced a smoothed field that could be evaluated at any point in the domain and had an MAE of 4.02 MJ m–2. The second approach was to use Catmull-Rom cubic splines to interpolate the nearest four grid points to each Mesonet site. The splines performed significantly better than the Gaussian mixture model approach, with an MAE of 2.61 MJ m–2, although they did have a tendency to have larger extremes than the observed data. Once the spline code was provided to the contestants, 118 of the 160 teams were able to either equal or improve on their performance.
GRADIENT BOOSTED REGRESSION TREES.
One of the surprising outcomes of the contest was that all of the winning methods made use of the same regression technique, Gradient Boosted Regression Trees (GBRT) (Friedman 2001). GBRT robustly models the (volatile) daily solar energy output from spatiotemporal input variables. For this data, GBRT proved to be an accurate and effective off-the-shelf regression technique because (1) it natively handles data of mixed type, (2) it is robust to outliers (through robust loss functions), and (3) it is nonparametric and has high predictive power.
Mathematically, GBRT is a generalization of boosting (Freund and Schapire 1995) to arbitrary differentiable loss functions L. The method considers additive models of the form
where hm(x) are basis functions called weak learners. In GBRT, weak learners are regression trees (Breiman et al. 1984) that are learned sequentially using a forward stagewise procedure. More specifically, at each stage, hm(x) is chosen to minimize the loss function L via steepest descent (using the negative gradient of L at the current model Fm–1), while the step length γm is chosen using line search.
The top contestant methods exhibited similar monthly error characteristics. The monthly MAE for all stations (Fig. 2) follows the average magnitude of solar energy by month, with the smallest error in December and January, then increasing to the highest error in May and June. All of the contestants have very similar monthly errors, with Eustaquio and Titericz (first-place winners, see sidebar) consistently having the lowest error. The monthly mean error shows a very small amount of bias relative to the magnitude of the mean absolute error. Each contestant follows a similar monthly trend in the mean error. Eustaquio and Titericz have a consistently higher mean error than the other models, which is due to the multiplicative factor applied to their results.
The winning approach creatively combined the predictions from models that focused on different aspects of the input data as well as information about their spatial and temporal variability. At each Mesonet site, 13 GBRT models were trained. The first 11 models used input data from each GEFS ensemble member, and the other 2 used the medians and maximums of the GEFS variable values over all ensemble members. The models trained on each member incorporated data from the 4 GEFS grid points that surrounded each Mesonet site. The 5 intraday values for all 15 input weather variables were used from the 4 nearest grid points, resulting in 300 input values per day. Additional descriptive variables (latitude and longitude from the GEFS and Mesonet, the station ID, and the distances between the Mesonet site and GEFS points) were also included. The aggregated models were trained on either the median or the maximum value of the ensemble variables and on the sum of the intraday values. All of the models were trained and optimized with threefold continuous cross-validation over consecutive 4-year periods. The Python implementation of the GBRT was used.
Once the individual models had been trained, and once each produced solar energy predictions over the training time period, two optimized weighted ensembles were produced to create a consensus solar energy prediction for each site. The forecasts for each station were combined using the Nelder and Mead (1965) nonlinear optimization algorithm to minimize the MAE of the consensus prediction. A second optimized ensemble was created by optimally weighting the predictions at nearby Mesonet sites to match the predictions at a particular site. The two weighted ensemble predictions were then simply averaged and multiplied by 1.01 as a final bias correction. All of the models took 12 h to run and resulted in an error of 2.11 MJ m–2. For comparison, the mean daily production of all Mesonet sites was 16.7 MJ m–2, resulting in a mean global error of 13%. It should be noted that no manual feature engineering was performed; the GBRT and the optimization routines did all of the feature selection and distance weighting on their own.
As is often the case in predictive analytics, data preparation was the most important step in this project. Since the localization of the Mesonet stations did not coincide exactly with the position of the GEFS nodes (Fig. SB1), some transformations were necessary in the training and testing datasets. For each of the 98 Mesonet stations, a linear interpolation of the four nearest GEFS points (weighted by distance) was carried out using the following formula:
in which wi = max (0,1 – di) and di is the Euclidian distance from the Mesonet station to the nearest GEFS node (assuming that the smallest distance between 2 GEFS nodes is equal to 1).
Fifteen meteorological variables forecast each day at 0000 UTC for five different hours (at 1200, 1500, 1800, 2100, and 0000 UTC the following day) were provided. The 75 weather features were used without any prior selection. Additional features were created by spatially or temporally averaging the original 75 weather variables. The elevation, latitude, and longitude of the Mesonet stations, and the month of the observation, were also included. In total, 128 explanatory variables were defined.
All the data from the 98 Mesonet stations were gathered to obtain a single training set, a single testing set, and finally a single model for all stations. Some trials have been performed with separate datasets for each station, but they never gave more accurate predictions. Consequently, the training dataset had 501,074 rows and the testing dataset had 176,068 rows.
The best accuracy was achieved with GBRT, using the implementation directly available in R (gbm package) with the mean absolute error (MAE). Random Forests (Breiman 2001) were also evaluated, but were not retained because they were less accurate.
For each of the boosted trees, the following training settings were used: Mean Absolute Error (distribution = “laplace”), number of expansions between 2,000 and 3,000 (n.trees = 2,000 or 3,000), depth of the trees between 6 or 8 (interaction.depth = 6, 7 or 8), a learning rate of 0.05 (shrinkage = 0.05), an out of the bag proportion: 30% (bag.fraction = 0.7). An ensemble of 12 distinct gradient boosted regression tree models improved the accuracy by reducing the risks of overfitting.
The mean absolute error of the second-place model was 2,128,116 J m–2, as evaluated on the private test set. Knowing that the average daily incoming solar energy of the stations in the Mesonet is around 16,500,000 J m–2, it therefore corresponds to a mean absolute error of about 12.8%.
Some variables clearly appeared to be particularly important: the downward shortwave radiative flux average at the surface (dswrf) and the precipitable water over the entire depth of the atmosphere (pwat). Even if the other variables are less influential, they contribute to improve the global accuracy of the model. Table S1 gives the top 10 most important variables.
Figure SB2 is a 3D graphical representation of the model. The shape of the curve is typical of models obtained by combining several regression trees, and shows the dependence of the model on incoming solar radiation and precipitable water and how the two terms interact. As physically expected, increased precipitable water results in lower observed solar energy for a given amount of incoming solar radiation.
The third-place approach also used GBRTs, with the differences coming in the data preprocessing for training. Before training, the 11 forecast members were averaged to minimize the training data for efficiency purposes. Two GBRTs were trained, each on slightly different data. The first was trained on the data from the GEFS model point closest to the prediction point and the second was trained on a weighted average of the nearest 4 GEFS points. The data at each model point p was distance-weighted by the longitude (φ) and latitude (λ) distance to each Mesonet site (s) according to Eq. SB2.
Both models were trained on all 75 of the available features. Additional features included the day of the year, the longitude and latitude, and a derived feature called “daily differences in downward shortwave solar radiation (ΔSd).” This feature was defined in Eq. SB3 as a weighted sum of the downward shortwave solar radiation for each available hour (Sh):
The final prediction was a weighted vote of the two GBRTs. The weights were determined using cross-validation. Denoting the GBRT trained on the nearest data points as GBRTn and the one trained on the weighted average as GBRTwa, the final prediction for a Mesonet site s was
This approach was similar in principle to the first-place winner (Eustaquio and Titericz) but made use of robust regression techniques to take uncertainty into account. It comprises two steps: First, a nonlinear interpolation technique, Gaussian Process regression (also known as kriging in geostatistics), is used to interpolate the coarse GEFS grid to the location of the solar energy production facilities. Second, GBRT is used to predict the daily solar energy output based on the interpolated model and additional spatiotemporal features.
Forecast variables measured at the GEFS locations are interpolated nonlinearly onto the Mesonet stations using Gaussian Processes (Rasmussen and Williams 2005). More specifically, for a given day and a given time period, a Gaussian Process models the value of a given forecast variable (e.g., temperature, humidity, etc.) with respect to the location of a station. Uncertainty in the forecast variables is taken into account by modeling the average value over the 11 members of the ensemble, where uncertainty in the ensemble measurements is specified as confidence intervals through the nugget parameter of the Gaussian Process. Using this technique, 75 forecast variables were interpolated per day in the dataset.
To enhance the final model, spatiotemporal variables were engineered and added to the 75 variables, including:
Solar features (delta between sunrise and sunset)
Temporal features (day of year, month of year)
Spatial features (latitude, longitude, elevation)
Nonlinear combinations of measurement estimates
Daily mean estimates
Variance of the measurement estimates, as produced by the Gaussian Processes
The best accuracy was achieved with GBRT. The least absolute deviation loss function was used for robustness, and all hyperparameters were optimized on an internal learning set. To further decrease variance of the model, several GBRT instances were built (using different random seeds) and their predictions averaged to form the final predictions. The ability of GBRT to handle outliers in the outputs by using robust loss functions is crucial in this context, due to the volatile nature of solar energy output. This pipeline was built on top of the Scikit-Learn machine learning library (Pedregosa et al. 2011), offering efficient implementations for both Gaussian Processes and GBRT.
The approach was evaluated on a dataset of daily solar energy measurements from 98 stations in Oklahoma. The results show a relative improvement of 17.17% and 46.19% over the baselines, Spline Interpolation, and Gaussian Mixture Models, respectively.
Analysis of the station error shows the effects of geography on the predictions. For all contestants, eastern Oklahoma featured generally higher mean absolute errors compared to western Oklahoma, with the Oklahoma Panhandle featuring some of the lowest errors (Fig. 3). This solar error distribution mirrors the annual precipitation distribution in Oklahoma. Since the presence of clouds and rain has a large impact on solar energy amounts, and since precipitation location and duration are challenging to predict, this factor is likely a large component of the increased error in eastern Oklahoma. A subset of the stations buck the geographical trend, and analysis of the contest observations shows that some of these stations recorded extended periods of missing data that were filled with the mean solar radiation value for that site. Only a few stations had these discrepancies, so it did not have a significant impact on the overall contest results.
A bootstrap statistical analysis of the forecast errors was performed on the top four contestants to determine if there were statistically significant differences in their forecasts. The confidence intervals (Table 1) indicated large amounts of overlap and no statistically significant differences in the top four contestants. The scores of the top seven contestants fall within the confidence interval of the first place winner, and the top sixteen contestants fall within the confidence interval of the fourth-place winner.
DISCUSSION AND LESSONS LEARNED.
By hosting the forecasting challenge on Kaggle, we dramatically increased the participation and the diversity of the participants from prior years. This diversity includes a significant increase in international participation, as well as participation from people outside of meteorology. This broader participation was valuable in highlighting meteorological applications for machine learning and data mining. However, it also provided some challenges from the perspective of running a contest with a final session at an AMS Annual Meeting. Because the winners were largely international participants, they were not able to travel to the meeting. Although most of the winners were able to send a prerecorded video of their talks and there was an informative discussion in the AMS session, future contests could benefit from better use of video technology to engage the winners in discussions in real time.
The data, evaluation system, and results from the contest have broader applicability for meteorologists in the renewable energy forecasting sector. The contest results showcased GBRT, which has not been used extensively in the atmospheric science community to this point. Optimized GBRTs have been shown to provide superior performance on this dataset compared to random forests, linear regressions, and neural networks, which were all used by other contestants. In addition to desirable performance characteristics, GBRTs use different optimization functions depending on the problem, and can be tuned for both computational and accuracy constraints. Due to its decision tree roots, GBRT can also be used to extract information about its input variables through variable influence rankings and partial dependence plots. We hope that the results of this contest and the availability of GBRT in both Python and R open-source machine learning libraries encourage the atmospheric science community to apply the algorithm to their existing datasets.
In the spirit of open data and reproducibility, the contest website (www.kaggle.com/c/ams-2014-solar-energy-prediction-contest), data, and evaluation system will continue to be available to anyone wishing to compare their approaches against the contest winners. While new submissions will not appear on the leaderboard, people are still invited to compare their algorithm and discuss new findings on the contest forum.
The contest was sponsored by the AMS Committees on Artificial Intelligence Applications to Environmental Science, Probability and Statistics, and Earth and Energy, and by EarthRisk Technologies. Will Cukierski from Kaggle helped set up the contest website and provided extensive technical support.