• Akaike, H., 1974: A new look at the statistical model identification. IEEE Trans. Autom. Control, 19, 716723, doi:10.1109/TAC.1974.1100705.

    • Search Google Scholar
    • Export Citation
  • Applequist, S., , Gahrs G. E. , , Pfeffer R. L. , , and Niu X. , 2002: Comparison of methodologies for probabilistic quantitative precipitation forecasting. Wea. Forecasting, 17, 783799, doi:10.1175/1520-0434(2002)017<0783:COMFPQ>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Ashley, S. T., , and Ashley W. S. , 2008: Flood fatalities in the United States. J. Appl. Meteor. Climatol., 47, 805818, doi:10.1175/2007JAMC1611.1.

    • Search Google Scholar
    • Export Citation
  • Breiman, L., 1984: Classification and Regression Trees.Wadsworth International Group, 358 pp.

  • Breiman, L., 2001: Random forests. Mach. Learn., 45, 532, doi:10.1023/A:1010933404324.

  • Bremnes, J. B., 2004: Probabilistic forecasts of precipitation in terms of quantiles using NWP model output. Mon. Wea. Rev., 132, 338347, doi:10.1175/1520-0493(2004)132<0338:PFOPIT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Brier, G. W., 1950: Verification of forecasts expressed in terms of probability. Mon. Wea. Rev., 78, 13, doi:10.1175/1520-0493(1950)078<0001:VOFEIT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Clark, A. J., , Gallus W. A. Jr., , Xue M. , , and Kong F. , 2009: A comparison of precipitation forecast skill between small convection-allowing and large convection-parameterizing ensembles. Wea. Forecasting, 24, 11211140, doi:10.1175/2009WAF2222222.1.

    • Search Google Scholar
    • Export Citation
  • Clark, A. J., and et al. , 2011: Probabilistic precipitation forecast skill as a function of ensemble size and spatial scale in a convection-allowing ensemble. Mon. Wea. Rev., 139, 14101418, doi:10.1175/2010MWR3624.1.

    • Search Google Scholar
    • Export Citation
  • Clark, A. J., and et al. , 2012: An overview of the 2010 Hazardous Weather Testbed Experimental Forecast Program Spring Experiment. Bull. Amer. Meteor. Soc., 93, 5574, doi:10.1175/BAMS-D-11-00040.1.

    • Search Google Scholar
    • Export Citation
  • Doswell, C. A., III, , Davies-Jones R. , , and Keller D. L. , 1990: On summary measures of skill in rare event forecasting based on contingency tables. Wea. Forecasting, 5, 576585, doi:10.1175/1520-0434(1990)005<0576:OSMOSI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Doswell, C. A., III, , Brooks H. E. , , and Maddox R. A. , 1996: Flash flood forecasting: An ingredients-based methodology. Wea. Forecasting, 11, 560581, doi:10.1175/1520-0434(1996)011<0560:FFFAIB>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Du, J., , McQueen J. , , DiMego G. , , Toth Z. , , Jovic D. , , Zhou B. , , and Chuang H. , 2006: New dimension of NCEP Short-Range Ensemble Forecasting (SREF) system: Inclusion of WRF members. Preprints, WMO Expert Team Meeting on Ensemble Prediction System, Exeter, United Kingdom, WMO. [Available online at http://www.emc.ncep.noaa.gov/mmb/SREF/reference.html.]

  • Ebert, E. E., 2001: Ability of a poor man’s ensemble to predict the probability and distribution of precipitation. Mon. Wea. Rev., 129, 24612480, doi:10.1175/1520-0493(2001)129<2461:AOAPMS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Ebert, E. E., 2009: Neighborhood verification: A strategy for rewarding close forecasts. Wea. Forecasting, 24, 14981510, doi:10.1175/2009WAF2222251.1.

    • Search Google Scholar
    • Export Citation
  • Eckel, F. A., , and Walters M. K. , 1998: Calibrated probabilistic quantitative precipitation forecasts based on the MRF ensemble. Wea. Forecasting, 13, 11321147, doi:10.1175/1520-0434(1998)013<1132:CPQPFB>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Gagne, D. J., II, , McGovern A. , , and Brotzge J. , 2009: Classification of convective areas using decision trees. J. Atmos. Oceanic Technol., 26, 13411353, doi:10.1175/2008JTECHA1205.1.

    • Search Google Scholar
    • Export Citation
  • Gagne, D. J., II, , McGovern A. , , and Xue M. , 2012: Machine learning enhancement of storm scale ensemble precipitation forecasts. Proc. Conf. on Intelligent Data Understanding, Boulder, CO, IEEE-CIS, 3946.

  • Gilleland, E., and et al. , 2010: Spatial forecast verification: Image warping. NCAR Tech. Rep. NCAR/TN-482+STR, 23 pp. [Available online at http://nldr.library.ucar.edu/repository/assets/technotes/TECH-NOTE-000-000-000-850.pdf.]

  • Glahn, H. R., , and Lowry D. A. , 1972: The use of model output statistics (MOS) in objective weather forecasts. J. Appl. Meteor., 11, 12031211, doi:10.1175/1520-0450(1972)011<1203:TUOMOS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Hall, T., , Brooks H. E. , , and Doswell C. A. III, 1999: Precipitation forecasting using a neural network. Wea. Forecasting, 14, 338345, doi:10.1175/1520-0434(1999)014<0338:PFUANN>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Hamill, T. M., , and Colucci S. J. , 1997: Verification of Eta–RSM short-range ensemble forecasts. Mon. Wea. Rev., 125, 13121327, doi:10.1175/1520-0493(1997)125<1312:VOERSR>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Hamill, T. M., , and Colucci S. J. , 1998: Evaluation of Eta–RSM ensemble probabilistic precipitation forecasts. Mon. Wea. Rev., 126, 711724, doi:10.1175/1520-0493(1998)126<0711:EOEREP>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Hamill, T. M., , Whitaker J. S. , , and Wei X. , 2004: Ensemble reforecasting: Improving medium-range forecast skill using retrospective forecasts. Mon. Wea. Rev., 132, 14341447, doi:10.1175/1520-0493(2004)132<1434:ERIMFS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Hamill, T. M., , Hagedorn R. , , and Whitaker J. S. , 2008: Probabilistic forecast calibration using ECMWF and GFS ensemble reforecasts. Part II: Precipitation. Mon. Wea. Rev., 136, 26202632, doi:10.1175/2007MWR2411.1.

    • Search Google Scholar
    • Export Citation
  • Hansen, A. W., , and Kuipers W. J. A. , 1965: On the relationship between the frequency of rain and various meteorological parameters. Meded. Verh.,81, 215.

    • Search Google Scholar
    • Export Citation
  • James, G., , Witten D. , , Hastie T. , , and Tibshirani R. , 2013: An Introduction to Statistical Learning with Applications in R.Springer, 430 pp.

  • Johnson, A., , and Wang X. , 2012: Verification and calibration of neighborhood and object-based probabilistic precipitation forecasts from a multimodel convection-allowing ensemble. Mon. Wea. Rev., 140, 30543077, doi:10.1175/MWR-D-11-00356.1.

    • Search Google Scholar
    • Export Citation
  • Koizumi, K., 1999: An objective method to modify numerical model forecasts with newly given weather data using an artificial neural network. Wea. Forecasting, 14, 109118, doi:10.1175/1520-0434(1999)014<0109:AOMTMN>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Kong, F., and et al. , 2011: Evaluation of CAPS multi-model storm-scale ensemble forecast for the NOAA HWT 2010 spring experiment. 25th Conf. on Severe Local Storms, Seattle, WA, Amer. Meteor. Soc., P4.18. [Available online at https://ams.confex.com/ams/25SLS/techprogram/paper_175822.htm.]

  • Krishnamurti, T. N., , Kishtawal C. M. , , LaRow T. E. , , Bachiochi D. R. , , Zhang Z. , , Williford C. E. , , Gadgil S. , , and Surendran S. , 1999: Improved weather and seasonal climate forecasts from multimodel superensemble. Science, 285, 15481550, doi:10.1126/science.285.5433.1548.

    • Search Google Scholar
    • Export Citation
  • Kusiak, A., , and Verma A. , 2011: Prediction of status patterns of wind turbines: A data-mining approach. J. Sol. Energy Eng., 133, 011008, doi:10.1115/1.4003188.

    • Search Google Scholar
    • Export Citation
  • Manzato, A., 2007: A note on the maximum Peirce skill score. Wea. Forecasting, 22, 11481154, doi:10.1175/WAF1041.1.

  • Marsh, P. T., , Kain J. S. , , Lakshmanan V. , , Clark A. J. , , Hitchens N. , , and Hardy J. , 2012: A method for calibrating deterministic forecasts of rare events. Wea. Forecasting, 27, 531538, doi:10.1175/WAF-D-11-00074.1.

    • Search Google Scholar
    • Export Citation
  • Mason, I., 1982: A model for assessment of weather forecasts. Aust. Meteor. Mag., 30, 291303.

  • Molteni, F., , Buizza R. , , Palmer T. N. , , and Petroliagis T. , 1996: The ECMWF Ensemble Prediction System: Methodology and validation. Quart. J. Roy. Meteor. Soc., 122, 73119, doi:10.1002/qj.49712252905.

    • Search Google Scholar
    • Export Citation
  • Murphy, A. H., 1973: A new vector partition of the probability score. J. Appl. Meteor., 12, 595600, doi:10.1175/1520-0450(1973)012<0595:ANVPOT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Murphy, A. H., 1977: The value of climatological, categorical, and probabilistic forecasts in the cost–loss ratio situation. Mon. Wea. Rev., 105, 803816, doi:10.1175/1520-0493(1977)105<0803:TVOCCA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Peirce, C. S., 1884: The numerical measure of the success of predictions. Science, 4, 453454, doi:10.1126/science.ns-4.93.453-a.

  • Strobl, C., , Boulesteix A.-L. , , Kneib T. , , Augustin T. , , and Zeileis A. , 2008: Conditional variable importance for random forests. BMC Bioinf., 9, 307, doi:10.1186/1471-2105-9-307.

    • Search Google Scholar
    • Export Citation
  • Toth, Z., , and Kalnay E. , 1993: Ensemble forecasting at NMC: The generation of perturbations. Bull. Amer. Meteor. Soc., 74, 23172330, doi:10.1175/1520-0477(1993)074<2317:EFANTG>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Tracton, M. S., , and Kalnay E. , 1993: Operational ensemble prediction at the National Meteorological Center: Practical aspects. Wea. Forecasting, 8, 379398, doi:10.1175/1520-0434(1993)008<0379:OEPATN>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Vasiloff, S., and et al. , 2007: Improving QPE and very short term QPF: An initiative for a community-wide integrated approach. Bull. Amer. Meteor. Soc., 88, 18991911, doi:10.1175/BAMS-88-12-1899.

    • Search Google Scholar
    • Export Citation
  • Wilks, D. S., 2011: Statistical Methods in the Atmospheric Sciences.3rd ed. Academic Press, 676 pp.

  • Williams, J. K., 2013: Using random forests to diagnose aviation turbulence. Mach. Learn.,95, 51–70, doi:10.1007/s10994-013-5346-7.

  • Xue, M., , Droegemeier K. K. , , and Wong V. , 2000: The Advanced Regional Prediction System (ARPS)—A multiscale nonhydrostatic atmospheric simulation and prediction model. Part I: Model dynamics and verification. Meteor. Atmos. Phys., 75, 161193, doi:10.1007/s007030070003.

    • Search Google Scholar
    • Export Citation
  • Xue, M., and et al. , 2001: The Advanced Regional Prediction System (ARPS)—A multiscale nonhydrostatic atmospheric simulation and prediction model. Part II: Model physics and applications. Meteor. Atmos. Phys., 76, 143165, doi:10.1007/s007030170027.

    • Search Google Scholar
    • Export Citation
  • Xue, M., , Wang D. , , Gao J. , , Brewster K. , , and Droegemeier K. K. , 2003: The Advanced Regional Prediction System (ARPS), storm-scale numerical weather prediction and data assimilation. Meteor. Atmos. Phys., 82, 139170, doi:10.1007/s00703-001-0595-6.

    • Search Google Scholar
    • Export Citation
  • Xue, M., and et al. , 2011: CAPS realtime storm-scale ensemble and convection-resolving high-resolution forecasts for the NOAA Hazardous Weather Testbed 2010 Spring Experiment. 25th Conf. on Severe Local Storms, Seattle, WA, Amer. Meteor. Soc., 7B.3. [Available online at https://ams.confex.com/ams/pdfpapers/176056.pdf.]

  • Yuan, H., , Gao X. , , Mullen S. L. , , Sorooshian S. , , Du J. , , and Juang H. H. , 2007: Calibration of probabilistic quantitative precipitation forecasts with an artificial neural network. Wea. Forecasting, 22, 12871303, doi:10.1175/2007WAF2006114.1.

    • Search Google Scholar
    • Export Citation
  • Zhang, J., , Qi Y. , , Howard K. , , Langston C. , , and Kaney B. , 2011: Radar quality index (RQI)—A combined measure of beam blockage and VPR effects in a national network. Proc. Eighth Int. Symp. on Weather Radar and Hydrology, Exeter, United Kingdom, IAHS Publ. 351, 388–393. [Available online at http://iahs.info/uploads/dms/15970.351%20Abstracts%2074.pdf.]

  • View in gallery

    Map of the number of grid points sampled within a 400-km2 box spatially averaged with a 300-km-radius median filter. The domain subgrids are also shown and labeled.

  • View in gallery

    Histogram comparing the relative frequencies of the full precipitation distribution for each subgrid to the sampled rainfall distributions.

  • View in gallery

    For a precipitation threshold of 0.25 mm h−1, the OTs and binary verification statistics are calculated at the OT for the SSEF and each machine learning model in grid 1.

  • View in gallery

    As in Fig. 3, but for grid 2.

  • View in gallery

    Attributes diagrams for the raw ensemble and each machine learning model. The solid line with circles is the reliability curve, which indicates the observed relative frequency of rain events above the 6.35 mm h−1 threshold for each probability bin. The dotted–dashed line with triangles shows the relative frequency of all forecasts that fall in each probability bin. If a point on the reliability curve falls in the gray area, it contributes positively to the BSS. The horizontal and vertical dashed lines are located at the climatological frequency for a particular subgrid. The diagonal dashed line indicates perfect reliability.

  • View in gallery

    ROC curves for the raw ensemble and each machine learning model. The diagonal lines indicate PSS.

  • View in gallery

    BSS and AUC comparisons by hour.

  • View in gallery

    Evaluation using BSS of random forests trained with three precipitation thresholds at each forecast hour.

  • View in gallery

    Observed 1-h precipitation (mm h−1) at 0200 UTC 13 May 2010.

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    Filled contours of 2-m dewpoint and temperature for the 2- and 26-h SSEF forecasts valid at 0200 UTC 13 May 2010.

  • View in gallery

    The (left) 2- and (right) 26-h forecasts of the (top) SSEF ensemble probability and (bottom) simple logistic regression methods. The green contours indicate the observed areas of 1-h precipitation greater than 6.35 mm h−1.

  • View in gallery

    As in Fig. 11, but for (top) multiple logistic regression and (bottom) random forest methods.

  • View in gallery

    Maps of the SSEF 26-h forecasts of predictors considered important by the random forest model.

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Machine Learning Enhancement of Storm-Scale Ensemble Probabilistic Quantitative Precipitation Forecasts

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  • 1 School of Meteorology, University of Oklahoma, Norman, Oklahoma
  • | 2 School of Computer Science, University of Oklahoma, Norman, Oklahoma
  • | 3 Center for Analysis and Prediction of Storms, and School of Meteorology, University of Oklahoma, Norman, Oklahoma
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Abstract

Probabilistic quantitative precipitation forecasts challenge meteorologists due to the wide variability of precipitation amounts over small areas and their dependence on conditions at multiple spatial and temporal scales. Ensembles of convection-allowing numerical weather prediction models offer a way to produce improved precipitation forecasts and estimates of the forecast uncertainty. These models allow for the prediction of individual convective storms on the model grid, but they often displace the storms in space, time, and intensity, which results in added uncertainty. Machine learning methods can produce calibrated probabilistic forecasts from the raw ensemble data that correct for systemic biases in the ensemble precipitation forecast and incorporate additional uncertainty information from aggregations of the ensemble members and additional model variables. This study utilizes the 2010 Center for Analysis and Prediction of Storms Storm-Scale Ensemble Forecast system and the National Severe Storms Laboratory National Mosaic & Multi-Sensor Quantitative Precipitation Estimate as input data for training logistic regressions and random forests to produce a calibrated probabilistic quantitative precipitation forecast. The reliability and discrimination of the forecasts are compared through verification statistics and a case study.

Corresponding author address: David John Gagne II, 120 David L. Boren Blvd., Ste. 5900, Norman, OK 73072. E-mail: djgagne@ou.edu

Abstract

Probabilistic quantitative precipitation forecasts challenge meteorologists due to the wide variability of precipitation amounts over small areas and their dependence on conditions at multiple spatial and temporal scales. Ensembles of convection-allowing numerical weather prediction models offer a way to produce improved precipitation forecasts and estimates of the forecast uncertainty. These models allow for the prediction of individual convective storms on the model grid, but they often displace the storms in space, time, and intensity, which results in added uncertainty. Machine learning methods can produce calibrated probabilistic forecasts from the raw ensemble data that correct for systemic biases in the ensemble precipitation forecast and incorporate additional uncertainty information from aggregations of the ensemble members and additional model variables. This study utilizes the 2010 Center for Analysis and Prediction of Storms Storm-Scale Ensemble Forecast system and the National Severe Storms Laboratory National Mosaic & Multi-Sensor Quantitative Precipitation Estimate as input data for training logistic regressions and random forests to produce a calibrated probabilistic quantitative precipitation forecast. The reliability and discrimination of the forecasts are compared through verification statistics and a case study.

Corresponding author address: David John Gagne II, 120 David L. Boren Blvd., Ste. 5900, Norman, OK 73072. E-mail: djgagne@ou.edu

1. Introduction

Most flooding fatalities occur due to flash floods, in which waters rise and fall rapidly due to concentrated rainfall over a small area (Ashley and Ashley 2008). The first step to anticipating flash floods is quantitative forecasting of the amount, location, and timing of precipitation. These quantitative precipitation forecasts are challenging due to the wide variability of precipitation amounts over small areas, the dependence of precipitation amounts on processes at a wide range of scales, and the dependence of extreme precipitation on any precipitation actually occurring (Bremnes 2004; Ebert 2001; Doswell et al. 1996). Recent advances in numerical modeling and machine learning are working to address these challenges.

Numerical weather prediction (NWP) models are now being run experimentally at 4-km horizontal grid spacing, or storm scale, allowing for the formation of individual convective cells without a convective parameterization scheme. These models better represent storm processes and output hourly predictions, but they have the challenge of correctly placing and timing the precipitation compared to models with coarser grid spacing and temporal resolution. An ensemble of storm-scale NWP models can provide improved estimates of uncertainty compared to coarser ensembles due to better sampling of the spatiotemporal errors associated with individual storms (Clark et al. 2009). As each ensemble member produces predictions of precipitation and precipitation ingredients, the question then becomes how best to combine those predictions into the most accurate and useful consensus guidance.

The final product should highlight areas most likely to be impacted by heavy rain and also provide an uncertainty estimate for that impact. Probabilistic quantitative precipitation forecasts (PQPFs) incorporate both of these qualities. A good PQPF should have reliable probabilities, such that a 40% chance of rain verifies 40% of the time over a large sample (Murphy 1977). PQPFs should also discriminate between extreme and trace precipitation events consistently, so most extreme events occur with higher probabilities, and most trace precipitation events are associated with low probabilities. Since individual ensemble members may be biased in different situations, a simple count of the ensemble members that exceed a threshold will often result in unreliable forecasts that discriminate poorly. Incorporating trends from past forecasts and additional information from other model variables can offset these biases and produce an enhanced PQPF.

Ensemble weather prediction (Toth and Kalnay 1993; Tracton and Kalnay 1993; Molteni et al. 1996) has required various forms of statistical postprocessing to produce accurate precipitation forecasts and uncertainty estimates from the ensembles, but most previous studies used coarser ensembles and longer forecast windows for calibration. The rank histogram method (Hamill and Colucci 1997) showed that ensemble precipitation forecasts tended to be underdispersive. Linear regression calibration methods have shown some skill improvements in Hamill and Colucci (1998), Eckel and Walters (1998), Krishnamurti et al. (1999), and Ebert (2001). Hall et al. (1999), Koizumi (1999), and Yuan et al. (2007) applied neural networks to precipitation forecasts and found increases in performance over linear regression. Logistic regression, a transform of a linear regression to fit an S-shaped curve ranging from 0 to 1, has shown more promise, as in Applequist et al. (2002), which tested linear regression, logistic regression, neural networks, and genetic algorithms on 24-h PQPFs and found that logistic regression consistently outperformed the other methods. Hamill et al. (2004, 2008) also utilized the logistic regression with an extended training period for added skill.

Storm-scale ensemble precipitation forecasts have been postprocessed with smoothing algorithms that produce reliable probabilities within longer forecast windows. Clark et al. (2011) applied smoothing algorithms at different spatial scales to the Storm-Scale Ensemble Forecast (SSEF) of precipitation and compared verification scores. Johnson and Wang (2012) compared the skill of multiple calibration methods on neighborhood and object-based probabilistic forecasts from the 2009 SSEF. Marsh et al. (2012) applied a Gaussian kernel density estimation function to the National Severe Storms Laboratory (NSSL) 4-km Weather Research and Forecasting Model (WRF) to derive probabilities from deterministic forecasts. These methods are helpful for predicting larger-scale events but smooth out the threats from extreme precipitation in individual convective cells. Gagne et al. (2012) took the first step in examining how multiple machine learning approaches performed in producing probabilistic, deterministic, and quantile precipitation forecasts over the central United States at individual grid points.

The purpose of this paper is to analyze PQPF predictions produced by multiple machine learning techniques incorporating data from the Center for Analysis and Prediction of Storms (CAPS) 2010 SSEF system (Xue et al. 2011; Kong et al. 2011). In addition to the choice of algorithm, some variations in the algorithm setups are also examined. The strengths and weaknesses of the machine learning algorithms are shown through the analysis of verification statistics, variables chosen by the machine learning models, and a case study. This paper expands upon the work presented in Gagne et al. (2012) by including statistics for the eastern United States, a larger training set more representative of the precipitation probabilities, a different case study day with comparisons of multiple runs, multiple precipitation thresholds, and more physical justification for the performance of the machine learning algorithms.

2. Data

a. Ensemble data

The CAPS SSEF system (Xue et al. 2011; Kong et al. 2011) provides the input data for the machine learning algorithms. The 2010 SSEF consists of 19 individual model members from the Advanced Research WRF (ARW), 5 members from the WRF Nonhydrostatic Mesoscale Model (NMM), and 2 members from the CAPS Advanced Research Prediction System (ARPS; Xue et al. 2000, 2001, 2003). Each member has a varied combination of microphysics schemes, land surface models, and planetary boundary layer schemes. The SSEF ran every weekday at 0000 UTC in support of the 2010 National Oceanic and Atmospheric Administration/Hazardous Weather Testbed Spring Experiment (Clark et al. 2012), which ran from 3 May to 18 June, for a total of 34 runs. The SSEF provides hourly model output over the contiguous United States at 4-km horizontal grid spacing out to 30 h. Of the 26 members, the 14 members included initial condition perturbations derived from the National Centers for Environmental Prediction Short Range Ensemble Forecast (SREF; Du et al. 2006) members, and only they are included in our postprocessing procedure. The 12 other members used the same control initial conditions although with different physics options and models; designed to examine forecast sensitivities to physics parameterizations, they do not contain the full set of initial conditions and model uncertainties and are therefore excluded from our study.

b. Verification data

A radar-based verification dataset was used as the verifying observations for the SSEF. The National Mosaic & Multi-Sensor Quantitative Precipitation Estimation (NMQ; Vasiloff et al. 2007) derives precipitation estimates from the reflectivity data of the Next Generation Weather Radar (NEXRAD) network. The estimates are made on a grid with 1-km horizontal spacing over the conterminous United States (CONUS). The original grid has been bilinearly interpolated onto the same grid as the SSEF.

c. Data selection and aggregation

The relative performance of any machine learning algorithm is conditioned on the distribution of its training data. The sampling scheme for the SSEF is conditioned on the constraints of 34 ensemble runs over a short, homogenous time period with 840 849 grid points from each of the 30 time steps. The short training period and large number of grid points preclude training a single model at each grid point, so a regional approach was used.

The SSEF domain was split into thirds (280 283 points per time step), and points were selected with a uniform random sample from each subdomain in areas with quality radar coverage. The gridded Radar Quality Index (RQI; Zhang et al. 2011) was evaluated at each grid point to determine the trustworthiness of the verification data. Points with an RQI > 0 were located within the useful range of a NEXRAD radar and included in the sampling. For points with precipitation values less than 0.25 mm, 0.04% were sampled, and for points with more precipitation, 0.4% were sampled. Grid 0 corresponds to the western third of the CONUS, grid 1 corresponds to the central third, and grid 2 corresponds to the eastern third. Figure 1 shows that the north-central section of the United States and the states south of the Great Lakes were most heavily sampled. Grid 0 was excluded from postprocessing due to the low frequency of heavy precipitation for most of the region.

Fig. 1.
Fig. 1.

Map of the number of grid points sampled within a 400-km2 box spatially averaged with a 300-km-radius median filter. The domain subgrids are also shown and labeled.

Citation: Weather and Forecasting 29, 4; 10.1175/WAF-D-13-00108.1

Comparisons of the sampled rainfall distributions and the full rainfall distributions for each subgrid are shown in Fig. 2 and Table 1. Undersampling of the 0 and trace (below 0.25 mm) precipitation points was necessary because of the large number of no-precipitation events, which overwhelmed the signal from the actual precipitation events. The random sampling of grid points helps reduce the chance of sampling multiple grid points from the same storm without explicitly filtering subsets of the domain, which was performed by Hamill et al. (2008). Filtering grid points surrounding each sampled point is extremely computationally expensive because filtering requires multiple passes over the grid while random sampling only requires one grid reordering.

Fig. 2.
Fig. 2.

Histogram comparing the relative frequencies of the full precipitation distribution for each subgrid to the sampled rainfall distributions.

Citation: Weather and Forecasting 29, 4; 10.1175/WAF-D-13-00108.1

Table 1.

Frequencies of total and sampled grid points for the different precipitation thresholds.

Table 1.

Relevant model output variables, called predictors (Table 2), were also extracted from each ensemble member at each sampled point. These predictors captured additional information about the mesoscale and synoptic conditions in each model. The predictors from each ensemble member were then aggregated into four descriptive statistics for each predictor. The mean provided the most likely forecast value, the standard deviation estimated the spread, and the minimum and maximum showed the extent of the forecasted values.

Table 2.

The names and descriptions of model predictors sampled from the SSEF runs. CAPE is convective available potential energy, and CIN is convective inhibition. Variables U and V wind refer to the west–east and south–north components of the wind at a particular constant-pressure level, respectively.

Table 2.

3. Methods

a. Machine learning methods

1) Logistic regression

One of the goals of this study is to compare the skill of more advanced machine learning methods with more traditional statistical methods. Logistic regression was used as the baseline statistical method. Logistic regressions are linear regression models in which a logit transformation is applied to the data so that the predicted values will range between 0 and 1. Two formulations of logistic regression were used: simple and multiple. The first formulation uses the ensemble-mean precipitation forecast expressed as
e1
where R is the precipitation amount, t is the threshold, x1 is the ensemble-mean precipitation forecast, β0 is the intercept term, and β1 is the weight given to the ensemble-mean precipitation forecast. Observed precipitation with an amount greater than or equal to the chosen threshold is assigned a probability of 1, and precipitation amounts below the threshold are assigned a probability of 0. The β coefficients are estimated by iteratively maximizing the log likelihood of the transformed training data. This formulation will adjust the probability of the ensemble forecast based on systematic biases in the mean but will not change the areal coverage of the precipitation forecasts. The second formulation incorporates multiple variables with stepwise variable selection similar to standard model output statistics (Glahn and Lowry 1972) approaches:
e2
where from x1 through xn are the predictors chosen from Table 2 and from β1 through βn are the weights for those predictors. The terms are determined through a forward selection process that finds the set of up to 15 terms that minimize the Akaike information criterion (Akaike 1974), which rewards goodness of fit but penalizes for large numbers of terms. This approach does produce the best-fit regression model given the available parameters, but the searching process can take extensive time given the large dimensionality of the training set. The generalized linear model (glm) and stepwise variable selection (step) functions in the R statistical package are used to generate the logistic regressions. A more detailed description of logistic regression and the fitting process can be found in James et al. (2013).

2) Random forest

A nonparametric, nonlinear alternative to linear regression is the classification and regression decision tree (Breiman 1984). Decision trees recursively partition a multidimensional dataset into successively smaller subdomains by selecting variables and decision thresholds that maximize a dissimilarity metric. At each node in the tree, every predictor is evaluated with the dissimilarity metric, and the predictor and threshold with the highest metric value are selected as the splitting criteria for that node. After enough partitions, each subdomain is similar enough that the prediction can be approximated with a single value. The primary advantages of decision trees are that they can be human readable and perform variable selection as part of the model growing process. The disadvantages lie in the brittleness of the trees. Trees can undergo significant structural shifts due to small variations in the training dataset, which results in large error variance.

Random forests (Breiman 2001) consist of an ensemble of classification and regression trees (Breiman 1984) with two key modifications. First, the training data cases are bootstrap resampled with replacement for each tree in the ensemble. Second, a random subset of the predictors is selected for evaluation at each node. The final prediction from the forest is the mean of the predicted probabilities from each tree. Random forests can produce both probabilistic and regression predictions through this method. The random forest method contains a few advantages that often lead to performance increases over traditional regression methods. The averaging of the results from multiple trees produces a smoother range of values than individual decision trees while also reducing the sensitivity of the model predictions to minor differences in the training set (Strobl et al. 2008). The random selection of predictors within the tree-building process allows for less optimal predictors to be included in the model and increases the likelihood of the discovery of interaction effects among predictors that would be missed by the stepwise selection method used in logistic regression (Strobl et al. 2008). Random forests have been shown to improve predictive performance on multiple problem domains in meteorology, including storm classification (Gagne et al. 2009), aviation turbulence (Williams 2013), and wind energy forecasting (Kusiak and Verma 2011). For this project, we used the R randomForest library, which implements the original approach (Breiman 2001). For the parameter settings, we chose to use 100 trees, a minimum node size of 20, and the default values for all other parameters. A more detailed description of the random forest method can be found in James et al. (2013).

In addition to gains in performance, the random forest methodology can also be used to rank the importance of each input variable (Breiman 2001). Variable importance is computed by first calculating the accuracy of each tree in the forest on classifying the cases that were not selected for training, known as the out-of-bag cases. Within the out-of-bag cases, the values of each variable are randomly rearranged, or permuted, and those cases are then reevaluated by each tree. The mean variable importance score is then the difference in prediction accuracy on the out-of-bag cases averaged over all trees. Variable importance scores can vary randomly among forests trained on the same dataset, so the variable importance scores from each of the 34 forests trained for cross validation were averaged together for a more robust ranking.

b. Evaluation methods

Two scores were used to assess the probabilistic forecasts. The Brier skill score (BSS; Brier 1950) is one method used to evaluate probabilistic forecasts. The Brier skill score can be decomposed into three terms (Murphy 1973):
e3
where N is the number of forecasts, K is the number of probability bins, nk is the number of forecasts in each probability bin, is the observed relative frequency for each bin, is the climatological frequency, and pk is the forecast probability for a particular bin k. The first term in the numerator describes the resolution of the forecast probability, which should be maximized and increases as the observed relative frequency differs more from climatology. The second term in the numerator describes the reliability of the forecast probability, which should be minimized and decreases with smaller differences between the forecast probability and observed relative frequency. The denominator term is the uncertainty, which is based on the climatological probability of precipitation and cannot be reduced through calibration. The BSS increases with skill from 0. The components of the BSS can be displayed graphically with an attributes diagram (Wilks 2011), in which the observed relative frequency of binned probability forecasts are plotted against lines showing perfect reliability, no skill where the reliability and resolution are equal, and no resolution where the observed relative frequency and probability equal climatology.

The area under the relative operating characteristic (ROC) curve (AUC; Mason 1982), evaluates how well a probabilistic forecast correctly identifies heavy precipitation events compared to identifying the events based on random chance. To calculate AUC, first, the decision probability threshold is varied from 0 to 1 in equal steps. At each step, a contingency table is constructed by splitting the probabilities into two categories with the decision threshold (Table 3). Using Table 3, the following scores can be computed (Table 4). Probability of detection (POD) is the ratio of hits to the total number of observed events. The false alarm ratio (FAR) accounts for the number of false alarms compared to total yes forecasts. The probability of false detection (POFD) is the ratio of false alarms to total no forecasts. The equitable threat score (ETS) is the skill score officially used to assess the skill of 24-h precipitation forecasts, and is sensitive to the climatological base rate of the validation data. The Peirce skill score (PSS) is another measure of skill that is insensitive to the climatological base rate but is sensitive to hedging by overforecasting rare events (Doswell et al. 1990). The bias ratio determines if false alarms or misses are more prevalent. From the contingency table, the POD and POFD are calculated and plotted against each other, forming an ROC curve. The AUC is the area between the right and bottom sides of the plot and the curve itself. AUCs with values above 0.5 have positive skill. The AUC only determines how well the forecast discriminates between two categories, so it does not take the reliability of the forecast into account.

Table 3.

An example binary contingency table for whether or not rain is forecast.

Table 3.
Table 4.

The scores calculated from the binary contingency table for use with the optimal threshold predictions. The arandom score is the number of hits due to random chance.

Table 4.

The ROC curve also can be used to determine an “optimal” decision threshold to convert a probabilistic forecast to a deterministic forecast. As the decision probability increases, the POD decreases from 1 to 0 while 1 − POFD increases from 0 to 1. At some decision threshold, POD and 1 − POFD should be equal. At this optimal threshold (OT), the user has an equal chance of correctly detecting precipitation events and no-precipitation events. This point occurs where the ROC curve is farthest from the positive diagonal “no skill” line. In addition, the vertical distance between the ROC curve and the positive diagonal is the PSS (Peirce 1884; Hansen and Kuipers 1965) since PSS = POD − POFD (Manzato 2007). Because the PSS can be calculated from the contingency table at each decision threshold, finding the maximum PSS also reveals the OT. The performance of the threshold choice can be validated by treating the predictions as a binary classification problem and using the binary verification statistics derived from the contingency table (Table 4).

c. Experimental procedure

Each model was trained and evaluated using a leave-one-day-out cross-validation procedure. For the 34 daily model runs in the training set, each machine learning model was trained on 33 days and tested on 1. Separate models were trained for each 6-h forecast period and each subgrid to better capture the trends in the weather and ensemble dispersiveness. Models were primarily trained at two thresholds: 0.25 and 6.35 mm h−1. The 0.25 mm h−1 threshold models were used to determine if rain was to occur at a point or not, and the 6.35 mm h−1 models were trained and evaluated only on the conditions that either rain occurred or the ensemble-mean precipitation forecast predicted rain. Deterministic rain forecasts were derived from the 0.25 mm h−1 models with the ROC curve optimal threshold technique to evaluate any biases in the areal forecasts. Additional models were trained at the 2.54 and 12.70 mm h−1 thresholds to evaluate the skill in predicting lighter and heavier precipitation, but only the 6.35 mm h−1 models were physically evaluated with variable importance and the case study. The 0.25, 2.54, 6.35, and 12.70 mm h−1 values were chosen because they correspond to the 0.01, 0.1, 0.25, and 0.5 in. h−1 thresholds, respectively, that are used for determining trace and heavy precipitation amounts. The probabilities shown in the case study are the joint probabilities of precipitation greater than or equal to 6.35 and 0.25 mm h−1. They were calculated by multiplying the conditional probability of precipitation greater than or equal to 6.35 mm h−1 with the probability of precipitation greater than or equal to 0.25 mm h−1.

4. Results

a. Deterministic rain model evaluation

Evaluation of the deterministic forecasts of precipitation location shows that the multiple logistic regression and random forest approaches do add skill compared to the uncalibrated ensemble probability and the simple logistic regression. Figure 3 shows only slight variations in the optimal threshold with forecast hour and that the thresholds for each machine learning algorithm are similar. The low threshold for the raw ensemble indicates that the best precipitation coverage is found when any ensemble member forecasts rain. PSS and ETS show similar hourly trends, but PSS is higher than ETS since ETS tends to maximize at a higher probability threshold. The multipredictor machine learning algorithms (random forest and multiple logistic regression) provide the most improvement for the first 15 h of the forecast, with only a slight improvement for the afternoon and evening. The improvement comes from an increased POD without also increasing the FAR. Similar results are seen in grid 2 in terms of thresholds and score values, and the multipredictor algorithms are able to provide consistent improvement over all forecast hours (Fig. 4). Similar results to these are found in the SSEF verification from Kong et al. (2011) although the ETS is lower in that paper as a result of that study verifying against all grid points, including points with no radar coverage.

Fig. 3.
Fig. 3.

For a precipitation threshold of 0.25 mm h−1, the OTs and binary verification statistics are calculated at the OT for the SSEF and each machine learning model in grid 1.

Citation: Weather and Forecasting 29, 4; 10.1175/WAF-D-13-00108.1

Fig. 4.
Fig. 4.

As in Fig. 3, but for grid 2.

Citation: Weather and Forecasting 29, 4; 10.1175/WAF-D-13-00108.1

b. Probabilistic threshold exceedance model evaluation

1) Evaluation over all forecast hours

The attributes diagrams for the raw ensemble and machine learning algorithms show how the forecasts were altered to improve their calibration and skill. Figure 5 contains the attribute diagrams for the ensemble and algorithms applied to grids 1 and 2 at the 6.35 mm h−1 threshold for all forecast hours. The raw ensemble in both domains tends to be overconfident with probabilistic forecasts above climatology and underconfident with forecasts below climatology, resulting in a negative BSS. In spatial terms, the ensemble probabilities are too high for areas where it forecasts heavy rain, and it is missing some areas where rain actually occurred. The simple logistic regression improves the ensemble forecast by rescaling the probabilities based on the ensemble-mean rain amount. In grids 1 and 2 this generally results in all of the probabilities being scaled closer to the climatological probability to address the general overconfidence of the ensemble. This scaling brings the observed relative frequencies above climatology into the positive skill zone, but they are still overconfident.

Fig. 5.
Fig. 5.

Attributes diagrams for the raw ensemble and each machine learning model. The solid line with circles is the reliability curve, which indicates the observed relative frequency of rain events above the 6.35 mm h−1 threshold for each probability bin. The dotted–dashed line with triangles shows the relative frequency of all forecasts that fall in each probability bin. If a point on the reliability curve falls in the gray area, it contributes positively to the BSS. The horizontal and vertical dashed lines are located at the climatological frequency for a particular subgrid. The diagonal dashed line indicates perfect reliability.

Citation: Weather and Forecasting 29, 4; 10.1175/WAF-D-13-00108.1

The multiple logistic regression and random forest methods incorporate additional ensemble variables to add more information to the probability estimates. This does result in a significant improvement in BSS over the simple logistic regression. The multiple logistic regression method performs the best in grid 1 because it produces nearly perfect reliability between 0% and 50% while maintaining positive skill from 60% to 90%. The random forest also places its forecasts close to the perfect reliability line but slightly farther away than the stepwise logistic regression, and the forecast probabilities do not exceed 70%, so the random forest has a lower maximum resolution than the multiple logistic regression.

In grid 2, the BSSs for all three models decrease. The random forest overforecasts probabilities below 50% and underforecasts probabilities above 50%. The magnitude of the underforecasting may be overestimated because of the small number of high probability forecasts. The multiple logistic regression method performs better with the lower-probability forecasts but has negative skill with the higher probabilities. This poorer performance may be due to having fewer heavy rain events in grid 2 during the study time period.

The ROC curves show that the multiple-predictor machine learning algorithms enhance the discrimination abilities of the ensemble. The raw ensembles in both subgrids have slightly positive skill in terms of AUC (Fig. 6). At the optimal PSS threshold, the raw ensemble detects only 67% of heavy rain events (POD) in grid 1, and 86% of its positive forecasts are false alarms (FAR). In grid 2, the detection ability is worse, with a 66% POD and a 91% FAR. Bias scores greater than 1 indicate a larger proportion of false alarms than misses. Since the simple logistic regression rescales the predictions over a smaller probability range, it has a slightly lower AUC than does the raw ensemble. Its optimal threshold is higher than the raw ensemble, so it has a correspondingly lower POD, POFD, and FAR. It also has a smaller bias score at the optimal threshold. The multiple logistic regression and random forest have a much larger AUC and POD and similar FAR compared to the raw ensemble. The multiple logistic regression and random forest approaches have very similar scores with only slight differences in their POFD, FAR, and bias. The same relationships hold in grid 2 except that there is a slightly larger difference in the scores of the multiple logistic regression and random forest methods. The overall scores are also slightly lower for grid 2, which is likely due to the lower frequency of convective precipitation events in the training dataset.

Fig. 6.
Fig. 6.

ROC curves for the raw ensemble and each machine learning model. The diagonal lines indicate PSS.

Citation: Weather and Forecasting 29, 4; 10.1175/WAF-D-13-00108.1

2) Evaluation by forecast hour

Comparisons of BSS and AUC by hour and by subgrid show additional trends in the probability of precipitation forecasts. The raw ensemble consistently has the worst BSS (Fig. 7). For all models, the best performance occurs at forecast hour 1 then decreases sharply at hour 2 before stabilizing. This initial decrease is likely due to the radar data assimilation placing the storms in the same place initially and then having the individual storms diverge from the predicted storm motions. There is a slight increase in performance between hours 6 and 12 for grid 2. This increase may be due to the diurnal cycle of convection resulting in larger storm clusters during this time period, or it may be due to the model fully spinning up. There is another major decrease in performance in grid 1 between hours 12 and 24. This time period is when the greatest uncertainty exists due to convective initiation and the tendency for initial convection to be isolated. The multiple logistic regression and the random forest methods do not have any statistically significant (α < 0.05) differences in their AUC and BSS results. The biggest departure from the ensemble mean in terms of AUC occurred in the 18–24-h range for both grids, which corresponds with peak convective activity. The simple logistic regression method did still improve on the ensemble forecast in terms of reliability but not to the same extent as the multiple-predictor models, and it made the AUC worse. The temporal trends in the BSS match those found in the 2009 SSEF by Johnson and Wang (2012). The multiple-predictor methods appeared to produce similar increases in BSS when compared to the single-predictor neighborhood and object-based methods.

Fig. 7.
Fig. 7.

BSS and AUC comparisons by hour.

Citation: Weather and Forecasting 29, 4; 10.1175/WAF-D-13-00108.1

c. Evaluation of multiple precipitation thresholds

Machine learning models were trained at three precipitation thresholds (2.54, 6.35, and 12.70 mm h−1) in order to evaluate how skill varies with precipitation intensity. Figure 8 shows the random forest BSS by hour for the three precipitation thresholds. All three follow the same diurnal patterns, but the 2.54 mm h−1 forecasts consistently have much higher skill than do the other thresholds. While the 2.54 and 6.35 mm h−1 forecasts show skill for all hours, the 12.70 mm h−1 forecasts do not show any skill from 14 to 26 h. The decreasing skill with threshold size is likely due to the smaller heavy precipitation areas and spatial and timing errors in placement of convection in the ensembles. Similar results were found for the other machine learning methods (not shown).

Fig. 8.
Fig. 8.

Evaluation using BSS of random forests trained with three precipitation thresholds at each forecast hour.

Citation: Weather and Forecasting 29, 4; 10.1175/WAF-D-13-00108.1

d. Variable importance

Variable importance scores were calculated and averaged over each random forest and subgrid to determine if the random forests were choosing relevant variables and how the choice of variables was affected by region. Variable importance is indicative of how randomizing the value of each variable affects the random forest performance. This process accounts for how often a variable is used in the model, the depth of the variable in the tree, and the number of cases that transit through the branch containing that variable, but the importance score cannot be decomposed into those factors. The top eight variable importance scores for the random forests trained on each 6-h period in grid 1 are shown in Table 5. In the first 6 h, all of the top five variables are aggregations of the hour precipitation or precipitable water, with the rest being hour maximum upward wind. In this time period, the ensemble members are very similar, so there is great overlap among the precipitation regions. By hours 7–12, the maximum upward wind becomes more dominant in the rankings, although the precipitation maximum and standard deviation are still found among the top five variables. Vertical velocities become the most common feature in hours 13–18 with only precipitable water and specific humidity contributing moisture information. For hours 19–30, the standard deviation of surface-based CAPE appears, which is likely associated with the presence of nearby boundaries.

Table 5.

The top eight variable importance scores for the random forests trained over 6-hourly periods in grid 1.

Table 5.
Table 6.

The top eight variable importance scores for the random forests trained over 6-hourly periods in grid 2.

Table 6.

The grid 2 variable importance scores (Table 6) highlight the greater importance of moisture and lower importance of vertical velocities in the eastern United States. The predicted precipitation is again only important in hours 1–12, but the precipitable water and specific humidity show high importance through the entire forecast period. Upward and downward winds are in the rankings for each time period, but they tend to be toward the bottom of the top eight. Surface-based CAPE is also important in the later hours of the forecasts, but the mean and maximum are selected instead of the standard deviation.

e. Case study: 13 May 2010

The case of 13 May 2010 illustrates the spatial characteristics, strengths, and weaknesses of the precipitation forecasts from the SSEF and the machine learning methods. Since SSEF is run through forecast hour 30, the last six forecast hours of one run overlap with the first six hours of the next run. This overlap allows for the comparison of two runs on the same observations and illustrates the effects of lead time on the forecast probabilities. Figure 9 shows the distribution of the observed 1-h precipitation at 0200 UTC 13 May. The bulk of the precipitation originates from a broken line of discrete supercell thunderstorms positioned along a stationary front stretching from northern Oklahoma into northern Missouri with additional storms in Iowa and ahead of the dryline in Oklahoma and Texas. Additional precipitation is falling in Virginia from a mesoscale convective system. A comparison of the 2- and 26-h dewpoint and temperature SSEF forecasts (Fig. 10) shows major differences in the placement and strength of the fronts and dryline. The cold front is farther north and more diffuse in the 26-h forecast while the dryline is farther east, moving from the central Texas Panhandle to western Oklahoma. The differences in the placement of the surface boundaries also affect the placement of the precipitation areas in both forecasts.

Fig. 9.
Fig. 9.

Observed 1-h precipitation (mm h−1) at 0200 UTC 13 May 2010.

Citation: Weather and Forecasting 29, 4; 10.1175/WAF-D-13-00108.1

Fig. 10.
Fig. 10.

Filled contours of 2-m dewpoint and temperature for the 2- and 26-h SSEF forecasts valid at 0200 UTC 13 May 2010.

Citation: Weather and Forecasting 29, 4; 10.1175/WAF-D-13-00108.1

Comparisons of the raw ensemble probabilities and the simple logistic regression methods (Fig. 11) show the effects of calibrating the probabilities only on precipitation. All of the precipitation probabilities shown are generated by multiplying the conditional PQPF of greater than 6.35 mm h−1 with the probability of precipitation, which removed spurious low conditional probabilities of precipitation. In the 2-h forecast, there is little dispersion among the ensemble members, so there is very high confidence in the raw ensemble. In this case though, the high-confidence areas are displaced westward from the actual heavy precipitation regions, resulting in some missed precipitation areas. The simple logistic regression corrects for this overconfidence by lowering the SSEF probabilities. The area with a greater than 10% probability is noticeably smaller and approximates the area occupied by the observed rain areas. Since the simple logistic regression only rescales the probabilities, the regression does not translate the predicted rain areas, resulting in more misses. At 26 h, the raw ensemble probabilities are lower and spread over a wider area, indicating greater ensemble dispersion. The probabilities in Oklahoma and Kansas are shifted farther east due to the positioning of the surface boundaries. In this case, it results in the observed precipitation being captured better than they were in the 2-h forecast with fewer misses but more false alarms. The simple logistic regression again reduces the probabilities and the area covered by them so that it does not have significantly more false alarms than at 2 h, but it misses some precipitation that was captured by the raw ensemble. Of the machine learning methods, the simple logistic approach is best for accurately depicting areal coverage, but it overcorrected in terms of downscaling the raw probabilities.

Fig. 11.
Fig. 11.

The (left) 2- and (right) 26-h forecasts of the (top) SSEF ensemble probability and (bottom) simple logistic regression methods. The green contours indicate the observed areas of 1-h precipitation greater than 6.35 mm h−1.

Citation: Weather and Forecasting 29, 4; 10.1175/WAF-D-13-00108.1

The results from the multiple-predictor methods show the advantages of incorporating additional predictors. In Fig. 12, the multiple logistic regression method is compared with the random forest method at 2 and 26 h. In the 2-h forecasts, both the multiple logistic regression and random forest methods expand their probabilities over a wider area and shift them slightly east compared to the raw ensemble, enabling them to capture the heavy precipitation fully. The random forest method also captures the precipitation areas north of the cold front better than the logistic regression method, which may weight Surface-based (SB) CAPE too highly and not handle cold sector precipitation as well. At 26 h, the areas increase and the probabilities are generally lower. Both algorithms capture nearly all of the precipitation areas, but they also produce more false alarms. Precipitation did extend through central Texas into Mexico ahead of the dryline, but none of that precipitation was heavy. The multiple logistic regression method highlighted that precipitation as well as ahead of the dryline. Unlike the 2-h forecast, it did cover the precipitation north of the front in Iowa. The random forest method covered all of the precipitation areas in the plains and seemed to handle the East Coast precipitation coverage better.

Fig. 12.
Fig. 12.

As in Fig. 11, but for (top) multiple logistic regression and (bottom) random forest methods.

Citation: Weather and Forecasting 29, 4; 10.1175/WAF-D-13-00108.1

The extended probability areas of the random forest and multiple logistic regression methods can be explained by examining some of the variables deemed important by the random forest scheme. In Fig. 13, the hour maximum upward wind, SBCAPE, precipitable water, and 850-mb specific humidity are shown. The hour maximum upward wind was the most important variable for the 25–30-h random forest simulation (Table 5), and its spatial distribution closely matches that of the random forest result, including the positive areas in southwest Texas. It also supports the precipitation north of the cold front in Iowa. SBCAPE was highest in central Texas in the area where both machine learning models placed probabilities but no heavy rain occurred. The machine learning algorithms were able to account for situations in which high CAPE and no precipitation did not occur together by not assigning probabilities in the Gulf of Mexico. Precipitable water and 850-mb specific humidity were also high over a wide area and generally correlated well with precipitation but occur over a much larger area than the actual precipitation. By weighting these variables together, the multipredictor algorithms were able to grow the forecasted area to include points at which the mix of ingredients was favorable for precipitation.

Fig. 13.
Fig. 13.

Maps of the SSEF 26-h forecasts of predictors considered important by the random forest model.

Citation: Weather and Forecasting 29, 4; 10.1175/WAF-D-13-00108.1

5. Discussion

The results of the machine learning postprocessing of storm-scale ensemble precipitation forecasts displayed not only improvements to the forecasts but also some of the limitations of the ensemble, the algorithms, and the gridpoint-based framework. First, the postprocessing model performance is constrained by the information available from the ensemble. If most of the ensemble members are predicting precipitation in the wrong place or not at all, and the environmental conditions are also displaced, then the machine learning algorithm will not be able to provide much additional skill. Second, the machine learning model will only make predictions based on the range of cases it has previously seen. For higher rain thresholds, the algorithms will need more independent samples in order to make skilled predictions.Third, the gridpoint framework does not fully account for the spatial information and error in the ensemble. The spatial error could be incorporated further by smoothing the verification grid with a neighborhood filter (Ebert 2009) or warping the ensemble forecast to more closely match the observed precipitation (Gilleland et al. 2010) and then training. Machine learning algorithms could also be used to improve the calibration from object-based frameworks (Johnson and Wang 2012) and could incorporate information from the additional predictors within the bounds of the objects.

6. Conclusions

Multiple machine learning algorithms were applied to the 2010 CAPS Storm Scale Ensemble Forecast (SSEF) system in order to improve the calibration and skill of its probabilistic heavy precipitation forecasts. Two types of machine learning methods were compared over a period from 3 May through 18 June with both verification statistics and a case study. Verification statistics showed that all of the machine learning methods improved the calibration of the SSEF precipitation forecasts but only the multiple-predictor methods were able to calibrate the models better and discriminate more skillfully between light and heavy precipitation cases. Hourly performance varied with diurnal storm cycles and the increasing dispersiveness of the ensemble members. Comparisons of the rankings of predictors indicated that the ensemble predicted precipitation was only important for the first 12 h of the model runs. After that period, the upward wind and atmospheric moisture variables became better indicators of the placement of precipitation. The case study showed that the multiple-predictor machine learning methods could shift the probability maxima to better match the actual precipitation areas, but they would also produce more false alarm areas in the process. For shorter-term forecasts, the false corrections were made without a significant increase in the false alarm area. Calibrating the probabilities with only ensemble rainfall predictions results in predicted areas that are too small and still displaced from the observed precipitation. The multiple-predictor machine learning algorithms did prove especially beneficial in that situation. Ultimately, machine learning techniques can provide an enhancement to precipitation forecasts by consistently maximizing the potential of the available information.

Acknowledgments

Special thanks go to DJGII’s master’s committee members: Fanyou Kong and Michael Richman. Zac Flamig provided assistance with the NMQ data. CAPS SSEF forecasts were supported by a grant (NWSPO-2010-201696) from the NOAA Collaborative Science, Technology, and Applied Research (CSTAR) Program and the forecasts were produced at the National Institute for Computational Science (http://www.nics.tennessee.edu/). Scientists at CAPS, including Fanyou Kong, Kevin Thomas, Yunheng Wang, and Keith Brewster contributed to the design and production of the CAPS ensemble forecasts. This study was funded by the NSF Graduate Research Fellowship under Grant 2011099434 and by NSF Grant AGS-0802888. The editor, Michael Baldwin, and the two anonymous reviewers provided very helpful feedback that strengthened the quality of the paper.

REFERENCES

  • Akaike, H., 1974: A new look at the statistical model identification. IEEE Trans. Autom. Control, 19, 716723, doi:10.1109/TAC.1974.1100705.

    • Search Google Scholar
    • Export Citation
  • Applequist, S., , Gahrs G. E. , , Pfeffer R. L. , , and Niu X. , 2002: Comparison of methodologies for probabilistic quantitative precipitation forecasting. Wea. Forecasting, 17, 783799, doi:10.1175/1520-0434(2002)017<0783:COMFPQ>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Ashley, S. T., , and Ashley W. S. , 2008: Flood fatalities in the United States. J. Appl. Meteor. Climatol., 47, 805818, doi:10.1175/2007JAMC1611.1.

    • Search Google Scholar
    • Export Citation
  • Breiman, L., 1984: Classification and Regression Trees.Wadsworth International Group, 358 pp.

  • Breiman, L., 2001: Random forests. Mach. Learn., 45, 532, doi:10.1023/A:1010933404324.

  • Bremnes, J. B., 2004: Probabilistic forecasts of precipitation in terms of quantiles using NWP model output. Mon. Wea. Rev., 132, 338347, doi:10.1175/1520-0493(2004)132<0338:PFOPIT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Brier, G. W., 1950: Verification of forecasts expressed in terms of probability. Mon. Wea. Rev., 78, 13, doi:10.1175/1520-0493(1950)078<0001:VOFEIT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Clark, A. J., , Gallus W. A. Jr., , Xue M. , , and Kong F. , 2009: A comparison of precipitation forecast skill between small convection-allowing and large convection-parameterizing ensembles. Wea. Forecasting, 24, 11211140, doi:10.1175/2009WAF2222222.1.

    • Search Google Scholar
    • Export Citation
  • Clark, A. J., and et al. , 2011: Probabilistic precipitation forecast skill as a function of ensemble size and spatial scale in a convection-allowing ensemble. Mon. Wea. Rev., 139, 14101418, doi:10.1175/2010MWR3624.1.

    • Search Google Scholar
    • Export Citation
  • Clark, A. J., and et al. , 2012: An overview of the 2010 Hazardous Weather Testbed Experimental Forecast Program Spring Experiment. Bull. Amer. Meteor. Soc., 93, 5574, doi:10.1175/BAMS-D-11-00040.1.

    • Search Google Scholar
    • Export Citation
  • Doswell, C. A., III, , Davies-Jones R. , , and Keller D. L. , 1990: On summary measures of skill in rare event forecasting based on contingency tables. Wea. Forecasting, 5, 576585, doi:10.1175/1520-0434(1990)005<0576:OSMOSI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Doswell, C. A., III, , Brooks H. E. , , and Maddox R. A. , 1996: Flash flood forecasting: An ingredients-based methodology. Wea. Forecasting, 11, 560581, doi:10.1175/1520-0434(1996)011<0560:FFFAIB>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Du, J., , McQueen J. , , DiMego G. , , Toth Z. , , Jovic D. , , Zhou B. , , and Chuang H. , 2006: New dimension of NCEP Short-Range Ensemble Forecasting (SREF) system: Inclusion of WRF members. Preprints, WMO Expert Team Meeting on Ensemble Prediction System, Exeter, United Kingdom, WMO. [Available online at http://www.emc.ncep.noaa.gov/mmb/SREF/reference.html.]

  • Ebert, E. E., 2001: Ability of a poor man’s ensemble to predict the probability and distribution of precipitation. Mon. Wea. Rev., 129, 24612480, doi:10.1175/1520-0493(2001)129<2461:AOAPMS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Ebert, E. E., 2009: Neighborhood verification: A strategy for rewarding close forecasts. Wea. Forecasting, 24, 14981510, doi:10.1175/2009WAF2222251.1.

    • Search Google Scholar
    • Export Citation
  • Eckel, F. A., , and Walters M. K. , 1998: Calibrated probabilistic quantitative precipitation forecasts based on the MRF ensemble. Wea. Forecasting, 13, 11321147, doi:10.1175/1520-0434(1998)013<1132:CPQPFB>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Gagne, D. J., II, , McGovern A. , , and Brotzge J. , 2009: Classification of convective areas using decision trees. J. Atmos. Oceanic Technol., 26, 13411353, doi:10.1175/2008JTECHA1205.1.

    • Search Google Scholar
    • Export Citation
  • Gagne, D. J., II, , McGovern A. , , and Xue M. , 2012: Machine learning enhancement of storm scale ensemble precipitation forecasts. Proc. Conf. on Intelligent Data Understanding, Boulder, CO, IEEE-CIS, 3946.

  • Gilleland, E., and et al. , 2010: Spatial forecast verification: Image warping. NCAR Tech. Rep. NCAR/TN-482+STR, 23 pp. [Available online at http://nldr.library.ucar.edu/repository/assets/technotes/TECH-NOTE-000-000-000-850.pdf.]

  • Glahn, H. R., , and Lowry D. A. , 1972: The use of model output statistics (MOS) in objective weather forecasts. J. Appl. Meteor., 11, 12031211, doi:10.1175/1520-0450(1972)011<1203:TUOMOS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Hall, T., , Brooks H. E. , , and Doswell C. A. III, 1999: Precipitation forecasting using a neural network. Wea. Forecasting, 14, 338345, doi:10.1175/1520-0434(1999)014<0338:PFUANN>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Hamill, T. M., , and Colucci S. J. , 1997: Verification of Eta–RSM short-range ensemble forecasts. Mon. Wea. Rev., 125, 13121327, doi:10.1175/1520-0493(1997)125<1312:VOERSR>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Hamill, T. M., , and Colucci S. J. , 1998: Evaluation of Eta–RSM ensemble probabilistic precipitation forecasts. Mon. Wea. Rev., 126, 711724, doi:10.1175/1520-0493(1998)126<0711:EOEREP>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Hamill, T. M., , Whitaker J. S. , , and Wei X. , 2004: Ensemble reforecasting: Improving medium-range forecast skill using retrospective forecasts. Mon. Wea. Rev., 132, 14341447, doi:10.1175/1520-0493(2004)132<1434:ERIMFS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Hamill, T. M., , Hagedorn R. , , and Whitaker J. S. , 2008: Probabilistic forecast calibration using ECMWF and GFS ensemble reforecasts. Part II: Precipitation. Mon. Wea. Rev., 136, 26202632, doi:10.1175/2007MWR2411.1.

    • Search Google Scholar
    • Export Citation
  • Hansen, A. W., , and Kuipers W. J. A. , 1965: On the relationship between the frequency of rain and various meteorological parameters. Meded. Verh.,81, 215.

    • Search Google Scholar
    • Export Citation
  • James, G., , Witten D. , , Hastie T. , , and Tibshirani R. , 2013: An Introduction to Statistical Learning with Applications in R.Springer, 430 pp.

  • Johnson, A., , and Wang X. , 2012: Verification and calibration of neighborhood and object-based probabilistic precipitation forecasts from a multimodel convection-allowing ensemble. Mon. Wea. Rev., 140, 30543077, doi:10.1175/MWR-D-11-00356.1.

    • Search Google Scholar
    • Export Citation
  • Koizumi, K., 1999: An objective method to modify numerical model forecasts with newly given weather data using an artificial neural network. Wea. Forecasting, 14, 109118, doi:10.1175/1520-0434(1999)014<0109:AOMTMN>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Kong, F., and et al. , 2011: Evaluation of CAPS multi-model storm-scale ensemble forecast for the NOAA HWT 2010 spring experiment. 25th Conf. on Severe Local Storms, Seattle, WA, Amer. Meteor. Soc., P4.18. [Available online at https://ams.confex.com/ams/25SLS/techprogram/paper_175822.htm.]

  • Krishnamurti, T. N., , Kishtawal C. M. , , LaRow T. E. , , Bachiochi D. R. , , Zhang Z. , , Williford C. E. , , Gadgil S. , , and Surendran S. , 1999: Improved weather and seasonal climate forecasts from multimodel superensemble. Science, 285, 15481550, doi:10.1126/science.285.5433.1548.

    • Search Google Scholar
    • Export Citation
  • Kusiak, A., , and Verma A. , 2011: Prediction of status patterns of wind turbines: A data-mining approach. J. Sol. Energy Eng., 133, 011008, doi:10.1115/1.4003188.

    • Search Google Scholar
    • Export Citation
  • Manzato, A., 2007: A note on the maximum Peirce skill score. Wea. Forecasting, 22, 11481154, doi:10.1175/WAF1041.1.

  • Marsh, P. T., , Kain J. S. , , Lakshmanan V. , , Clark A. J. , , Hitchens N. , , and Hardy J. , 2012: A method for calibrating deterministic forecasts of rare events. Wea. Forecasting, 27, 531538, doi:10.1175/WAF-D-11-00074.1.

    • Search Google Scholar
    • Export Citation
  • Mason, I., 1982: A model for assessment of weather forecasts. Aust. Meteor. Mag., 30, 291303.

  • Molteni, F., , Buizza R. , , Palmer T. N. , , and Petroliagis T. , 1996: The ECMWF Ensemble Prediction System: Methodology and validation. Quart. J. Roy. Meteor. Soc., 122, 73119, doi:10.1002/qj.49712252905.

    • Search Google Scholar
    • Export Citation
  • Murphy, A. H., 1973: A new vector partition of the probability score. J. Appl. Meteor., 12, 595600, doi:10.1175/1520-0450(1973)012<0595:ANVPOT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Murphy, A. H., 1977: The value of climatological, categorical, and probabilistic forecasts in the cost–loss ratio situation. Mon. Wea. Rev., 105, 803816, doi:10.1175/1520-0493(1977)105<0803:TVOCCA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Peirce, C. S., 1884: The numerical measure of the success of predictions. Science, 4, 453454, doi:10.1126/science.ns-4.93.453-a.

  • Strobl, C., , Boulesteix A.-L. , , Kneib T. , , Augustin T. , , and Zeileis A. , 2008: Conditional variable importance for random forests. BMC Bioinf., 9, 307, doi:10.1186/1471-2105-9-307.

    • Search Google Scholar
    • Export Citation
  • Toth, Z., , and Kalnay E. , 1993: Ensemble forecasting at NMC: The generation of perturbations. Bull. Amer. Meteor. Soc., 74, 23172330, doi:10.1175/1520-0477(1993)074<2317:EFANTG>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Tracton, M. S., , and Kalnay E. , 1993: Operational ensemble prediction at the National Meteorological Center: Practical aspects. Wea. Forecasting, 8, 379398, doi:10.1175/1520-0434(1993)008<0379:OEPATN>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Vasiloff, S., and et al. , 2007: Improving QPE and very short term QPF: An initiative for a community-wide integrated approach. Bull. Amer. Meteor. Soc., 88, 18991911, doi:10.1175/BAMS-88-12-1899.

    • Search Google Scholar
    • Export Citation
  • Wilks, D. S., 2011: Statistical Methods in the Atmospheric Sciences.3rd ed. Academic Press, 676 pp.

  • Williams, J. K., 2013: Using random forests to diagnose aviation turbulence. Mach. Learn.,95, 51–70, doi:10.1007/s10994-013-5346-7.

  • Xue, M., , Droegemeier K. K. , , and Wong V. , 2000: The Advanced Regional Prediction System (ARPS)—A multiscale nonhydrostatic atmospheric simulation and prediction model. Part I: Model dynamics and verification. Meteor. Atmos. Phys., 75, 161193, doi:10.1007/s007030070003.

    • Search Google Scholar
    • Export Citation
  • Xue, M., and et al. , 2001: The Advanced Regional Prediction System (ARPS)—A multiscale nonhydrostatic atmospheric simulation and prediction model. Part II: Model physics and applications. Meteor. Atmos. Phys., 76, 143165, doi:10.1007/s007030170027.

    • Search Google Scholar
    • Export Citation
  • Xue, M., , Wang D. , , Gao J. , , Brewster K. , , and Droegemeier K. K. , 2003: The Advanced Regional Prediction System (ARPS), storm-scale numerical weather prediction and data assimilation. Meteor. Atmos. Phys., 82, 139170, doi:10.1007/s00703-001-0595-6.

    • Search Google Scholar
    • Export Citation
  • Xue, M., and et al. , 2011: CAPS realtime storm-scale ensemble and convection-resolving high-resolution forecasts for the NOAA Hazardous Weather Testbed 2010 Spring Experiment. 25th Conf. on Severe Local Storms, Seattle, WA, Amer. Meteor. Soc., 7B.3. [Available online at https://ams.confex.com/ams/pdfpapers/176056.pdf.]

  • Yuan, H., , Gao X. , , Mullen S. L. , , Sorooshian S. , , Du J. , , and Juang H. H. , 2007: Calibration of probabilistic quantitative precipitation forecasts with an artificial neural network. Wea. Forecasting, 22, 12871303, doi:10.1175/2007WAF2006114.1.

    • Search Google Scholar
    • Export Citation
  • Zhang, J., , Qi Y. , , Howard K. , , Langston C. , , and Kaney B. , 2011: Radar quality index (RQI)—A combined measure of beam blockage and VPR effects in a national network. Proc. Eighth Int. Symp. on Weather Radar and Hydrology, Exeter, United Kingdom, IAHS Publ. 351, 388–393. [Available online at http://iahs.info/uploads/dms/15970.351%20Abstracts%2074.pdf.]

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