Regression-based distribution mapping for bias correction of climate model outputs using linear quantile regression

  • Christian PassowEmail author
  • Reik V. Donner
Original Paper


Impact models are a major source of information for quantifying the consequences of future climate change for humans and the environment. To provide plausible outputs, these models require unbiased high resolution meteorological data as input for atmospheric conditions. State of the art regional climate models (RCMs) often fail to provide such data, since they can exhibit large systematic biases. Therefore, bias correction methods (BCMs) have become a common tool in climate impact studies. Bias correction, however, comes with strong assumptions and limitations, often resulting from the fact that most BCMs are unable to appropriately calibrate a time dependent and conditional transfer function. To address this problem, we introduce here regression quantile mapping (RQM), a bias correction approach based on (linear) regression models which allow to design transfer functions based on expert knowledge. The new RQM algorithm is described in full detail in its basic (linear model) version and applied to RCM generated precipitation data for entire Europe. Based on the latter example, we provide a thorough comparison with another established BCM, quantile delta mapping (QDM) regarding the seasonal characteristics of precipitation sums. Our results demonstrate that RQM already achieves good results when a simple linear model is used. The relationship between precipitation and temperature was properly evaluated by RQM and representation of seasonal variations and key characteristics of precipitation were improved for most seasons. Systematic biases where reduced significantly during this process. Particular improvements in comparison with QDM are found regarding the shape and width of the distribution of bias corrected model precipitation. Furthermore, the representation of precipitation extremes within the data was largely improved when RQM was used instead of QDM.


Bias correction Quantile mapping Climate impact Quantile regression 



This work has been financially supported by the German Federal Ministry for Education and Research via the Young Investigators Group CoSy-\(\hbox {CC}^2\) (Complex Systems Approaches to Understanding Causes and Consequences of Past, Present and Future Climate Change, Grant No. 01LN1306A) and the Belmont Forum/JPI Climate project GOTHAM (Globally Observed Teleconnections and their Representation in Hierarchies of Atmospheric Models, Grant No. 01LP16MA).


  1. Abramowitz M, Stegun I (1966) Handbook of mathematical functions, applied mathemathics series, vol 55. Dover Publications, New YorkGoogle Scholar
  2. Ajaaj AA, Mishra AK, Khan AA (2016) Comparison of BIAS correction techniques for GPCC rainfall data in semi-arid climate. Stoch Env Res Risk Assess 30(6):1659–1675. CrossRefGoogle Scholar
  3. Bellprat O, Kotlarski S, Lüthi D, Schär C (2013) Physical constraints for temperature biases in climate models. Geophys Res Lett 40(15):4042–4047. CrossRefGoogle Scholar
  4. Boé J, Terray L, Habets F, Martin E (2007) Statistical and dynamical downscaling of the Seine basin climate for hydro-meteorological studies. Int J Climatol 27(12):1643–1655. CrossRefGoogle Scholar
  5. Bondell HD, Reich BJ, Wang H (2010) Noncrossing quantile regression curve estimation. Biometrika 97(4):825–838. CrossRefGoogle Scholar
  6. Cai Y, Jiang T (2015) Estimation of non-crossing quantile regression curves. Aust N Z J Stat 57(1):139–162. CrossRefGoogle Scholar
  7. Cannon AJ, Sobie SR, Murdock TQ (2015) Bias correction of GCM precipitation by quantile mapping: how well do methods preserve changes in quantiles and extremes? J Clim 28(17):6938–6959. CrossRefGoogle Scholar
  8. Chen SX (2000) Beta kernel smoothers for regression curves. Statistica Sinica 10(1):73–91.
  9. Christensen JH, Boberg F, Christensen OB, Lucas-Picher P (2008) On the need for bias correction of regional climate change projections of temperature and precipitation. Geophys Res Lett 35:L20709. CrossRefGoogle Scholar
  10. Coles S (2001) An introduction to statistical modeling of extreme values. Springer Series in Statistics, Springer London. CrossRefGoogle Scholar
  11. Dosio A, Paruolo P (2011) Bias correction of the ENSEMBLES high-resolution climate change projections for use by impact models: evaluation on the present climate. J Geophys Res Atmos 116(D16):D16106. CrossRefGoogle Scholar
  12. Ehret U, Zehe E, Wulfmeyer V, Warrach-Sagi K, Liebert J (2012) HESS opinions should we apply bias correction to global and regional climate model data? Hydrol Earth Syst Sci 16(9):3391–3404. CrossRefGoogle Scholar
  13. Field C, Barros V, Stocker T, Qin D, Dokken D, Ebi K, Mastrandrea M, Mach K, Plattner GK, Allen S, Tignor M, Midgley P (eds) (2012) Managing the risks of extreme events and disasters to advance climate change adaptation–special report of the intergovernmental panel on climate change. Cambridge University Press, CambridgeGoogle Scholar
  14. Fischer G, Shah M, Tubiello FN, van Velhuizen H (2005) Socio-economic and climate change impacts on agriculture: an integrated assessment, 1990–2080. Philos Trans R Soc B Biol Sci 360(1463):2067–2083. CrossRefGoogle Scholar
  15. Fisher RA, Tippett LHC (1928) Limiting forms of the frequency distribution of the largest or smallest member of a sample. Math Proc Cambridge Philos Soc 24(2):180–190. CrossRefGoogle Scholar
  16. Flato G, Marotzke J, Abiodun B, Braconnot P, Chou S, Collins W, Cox P, Driouech F, Emori S, Eyring V, Forest C, Gleckler P, Guilyardi E, Jakob C, Kattsov V, Reason C, Rummukainen M (2013) Climate change 2013: the physical science basis. Contribution of working group i to the fifth assessment report of the intergovernmental panel on climate change, Cambridge University Press, Cambridge, chap Evaluation of Climate Models, pp 741–866Google Scholar
  17. Gennaretti F, Sangelantoni L, Grenier P (2015) Toward daily climate scenarios for Canadian Arctic coastal zones with more realistic temperature-precipitation interdependence. J Geophys Res Atmos 120(23):11862–11877. CrossRefGoogle Scholar
  18. Gobiet A, Suklitsch M, Heinrich G (2015) The effect of empirical-statistical correction of intensity-dependent model errors on the temperature climate change signal. Hydrol Earth Syst Sci 19(10):4055–4066. CrossRefGoogle Scholar
  19. Gudmundsson L, Bremnes JB, Haugen JE, Engen-Skaugen T (2012) Technical note: downscaling RCM precipitation to the station scale using statistical transformations–a comparison of methods. Hydrol Earth Syst Sci 16(9):3383–3390. CrossRefGoogle Scholar
  20. Haerter JO, Hagemann S, Moseley C, Piani C (2011) Climate model bias correction and the role of timescales. Hydrol Earth Syst Sci 15(3):1065–1079. CrossRefGoogle Scholar
  21. Hagemann S, Chen C, Haerter JO, Heinke J, Gerten D, Piani C (2011) Impact of a statistical bias correction on the projected hydrological changes obtained from three GCMs and two hydrology models. J Hydrometeorol 12(4):556–578. CrossRefGoogle Scholar
  22. Haylock MR, Hofstra N, Klein Tank AMG, Klok EJ, Jones PD, New M (2008) A European daily high-resolution gridded data set of surface temperature and precipitation for 1950–2006. J Geophys Res Atmos 113(D20):D20119. CrossRefGoogle Scholar
  23. Hempel S, Frieler K, Warszawski L, Schewe J, Piontek F (2013) A trend-preserving bias correction–the ISI-MIP approach. Earth Syst Dyn 4(2):219–236. CrossRefGoogle Scholar
  24. Ivanov MA, Luterbacher J, Kotlarski S (2018) Climate model biases and modification of the climate change signal by intensity-dependent bias correction. J Clim 31(16):6591–6610. CrossRefGoogle Scholar
  25. Jacob D, Petersen J, Eggert B, Alias A, Christensen OB, Bouwer LM, Braun A, Colette A, Déqué M, Georgievski G, Georgopoulou E, Gobiet A, Menut L, Nikulin G, Haensler A, Hempelmann N, Jones C, Keuler K, Kovats S, Kröner N, Kotlarski S, Kriegsmann A, Martin E, van Meijgaard E, Moseley C, Pfeifer S, Preuschmann S, Radermacher C, Radtke K, Rechid D, Rounsevell M, Samuelsson P, Somot S, Soussana JF, Teichmann C, Valentini R, Vautard R, Weber B, Yiou P (2014) EURO-CORDEX: new high-resolution climate change projections for European impact research. Reg Environ Change 14(2):563–578. CrossRefGoogle Scholar
  26. Jones MC (1993) Simple boundary correction for kernel density estimation. Stat Comput 3(3):135–146. CrossRefGoogle Scholar
  27. Katz RW, Parlange MB, Naveau P (2002) Statistics of extremes in hydrology. Adv Water Resour 25(8):1287–1304. CrossRefGoogle Scholar
  28. Koenker R (2005) Quantile regression. Econometric society monographs. Cambridge University Press, Cambridge.
  29. Koenker R, Bassett G (1978) Regression quantiles. Econometrica 46(1):33–50. CrossRefGoogle Scholar
  30. Koenker R, Park BJ (1996) An interior point algorithm for nonlinear quantile regression. J Econom 71(1):265–283. CrossRefGoogle Scholar
  31. Luo Q (2016) Necessity for post-processing dynamically downscaled climate projections for impact and adaptation studies. Stoch Env Res Risk Assess 30(7):1835–1850. CrossRefGoogle Scholar
  32. Madden RA, Williams J (1978) The correlation between temperature and precipitation in the United States and Europe. Mon Weather Rev 106(1):142–147.<0142:TCBTAP>2.0.CO;2 CrossRefGoogle Scholar
  33. Maraun D (2012) Nonstationarities of regional climate model biases in European seasonal mean temperature and precipitation sums. Geophys Res Lett 39(6):L06706. CrossRefGoogle Scholar
  34. Maraun D (2013) Bias correction, quantile mapping, and downscaling: revisiting the inflation issue. J Clim 26(6):2137–2143. CrossRefGoogle Scholar
  35. Maurer EP, Pierce DW (2014) Bias correction can modify climate model simulated precipitation changes without adverse effect on the ensemble mean. Hydrol Earth Syst Sci 18(3):915–925. CrossRefGoogle Scholar
  36. Michelangeli PA, Vrac M, Loukos H (2009) Probabilistic downscaling approaches: application to wind cumulative distribution functions. Geophys Res Lett 36(11):L11708. CrossRefGoogle Scholar
  37. Nadaraya EA (1964) On estimating regression. Theory Probab Appl 9(1):141–142. CrossRefGoogle Scholar
  38. Nijssen B, O’Donnell GM, Hamlet AF, Lettenmaier DP (2001) Hydrologic sensitivity of global rivers to climate change. Clim Change 50(1):143–175. CrossRefGoogle Scholar
  39. Osuch M, Lawrence D, Meresa HK, Napiorkowski JJ, Romanowicz RJ (2017) Projected changes in flood indices in selected catchments in Poland in the 21st century. Stoch Env Res Risk Assess 31(9):2435–2457. CrossRefGoogle Scholar
  40. Passow C, Donner RV (2019) A rigorous statistical assessment of recent trends in intensity of heavy precipitation over Germany. Front Environ Sci 7:1–43. CrossRefGoogle Scholar
  41. Passow C, Donner RV (in prep.) Regularized quantile regression using beta kernels. Advances in Statistical Climatology, Meteorology and OceanographyGoogle Scholar
  42. Piani C, Haerter JO, Coppola E (2010a) Statistical bias correction for daily precipitation in regional climate models over Europe. Theoret Appl Climatol 99(1):187–192. CrossRefGoogle Scholar
  43. Piani C, Weedon G, Best M, Gomes S, Viterbo P, Hagemann S, Haerter J (2010b) Statistical bias correction of global simulated daily precipitation and temperature for the application of hydrological models. J Hydrol 395(3):199–215. CrossRefGoogle Scholar
  44. Rajczak J, Kotlarski S, Schär C (2016) Does quantile mapping of simulated precipitation correct for biases in transition probabilities and spell lengths? J Clim 29(5):1605–1615. CrossRefGoogle Scholar
  45. Ribeiro A, Barbosa SM, Scotto MG, Donner RV (2014) Changes in extreme sea-levels in the Baltic Sea. Tellus A Dyn Meteorol Oceanogr 66(1):20921. CrossRefGoogle Scholar
  46. Rockel B, Will A, Hense A (2008) The regional climate model COSMO-CLM (CCLM). Meteorol Z 17(4):347–348. CrossRefGoogle Scholar
  47. Rosenzweig C, Parry ML et al (1994) Potential impact of climate change on world food supply. Nature 367(6459):133–138. CrossRefGoogle Scholar
  48. Rust HW, Maraun D, Osborn TJ (2009) Modelling seasonality in extreme precipitation. Eur Phys J Spec Top 174(1):99–111. CrossRefGoogle Scholar
  49. Sangelantoni L, Russo A, Gennaretti F (2019) Impact of bias correction and downscaling through quantile mapping on simulated climate change signal: a case study over Central Italy. Theoret Appl Climatol 135(1):725–740. CrossRefGoogle Scholar
  50. Teutschbein C, Seibert J (2010) Regional climate models for hydrological impact studies at the catchment scale: a review of recent modeling strategies. Geography Compass 4(7):834–860. CrossRefGoogle Scholar
  51. Teutschbein C, Seibert J (2012) Bias correction of regional climate model simulations for hydrological climate-change impact studies: review and evaluation of different methods. J Hydrol 456–457:12–29. CrossRefGoogle Scholar
  52. Themeßl MJ, Gobiet A, Heinrich G (2012) Empirical-statistical downscaling and error correction of regional climate models and its impact on the climate change signal. Clim Change 112(2):449–468. CrossRefGoogle Scholar
  53. Thompson P, Cai Y, Moyeed R, Reeve D, Stander J (2010) Bayesian nonparametric quantile regression using splines. Comput Stat Data Anal 54(4):1138–1150. CrossRefGoogle Scholar
  54. Thrasher B, Maurer EP, McKellar C, Duffy PB (2012) Technical note: bias correcting climate model simulated daily temperature extremes with quantile mapping. Hydrol Earth Syst Sci 16(9):3309–3314. CrossRefGoogle Scholar
  55. Trenberth KE, Shea DJ (2005) Relationships between precipitation and surface temperature. Geophys Res Lett 32(14):L14703. CrossRefGoogle Scholar
  56. Vrac M, Friederichs P (2015) Multivariate-intervariable, spatial, and temporal-bias correction. J Clim 28(1):218–237. CrossRefGoogle Scholar
  57. Walton DB, Sun F, Hall A, Capps S (2015) A hybrid dynamical-statistical downscaling technique. Part I: development and validation of the technique. J Clim 28(12):4597–4617. CrossRefGoogle Scholar
  58. Watson GS (1964) Smooth regression analysis. Sankhya Indian J Stat 26(4):359–372 Series A (1961-2002)Google Scholar
  59. Wong G, Maraun D, Vrac M, Widmann M, Eden JM, Kent T (2014) Stochastic model output statistics for bias correcting and downscaling precipitation including extremes. J Clim 27(18):6940–6959. CrossRefGoogle Scholar
  60. Wu Y, Liu Y (2009) Stepwise multiple quantile regression estimation using non-crossing constraints. Stat Interface 2(3):299–310. CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute for MeteorologyFree University of BerlinBerlinGermany
  2. 2.Potsdam Institute for Climate Impact ResearchPotsdamGermany
  3. 3.Department of Water, Environment, Construction and SafetyMagdeburg–Stendal University of Applied SciencesMagdeburgGermany

Personalised recommendations