# Multivariate Bias Correction of Climate Model Outputs Calibrate and apply multivariate bias correction algorithms for climate model simulations of multiple climate variables. Three iterative methods are supported: (i) MBC Pearson correlation (MBCp), (ii) MBC rank correlation (MBCr), and (iii) MBC N-dimensional probability density function transform (MBCn). The first two, MBCp and MBCr (Cannon, 2016), match marginal distributions and inter-variable dependence structure. Dependence structure can be measured either by the Pearson correlation (`MBCp`) or by the Spearman rank correlation (`MBCr`). The energy distance score (`escore`) is recommended for model selection. The third, `MBCn` (Cannon, 2018), which operates on the full multivariate distribution, is more flexible and can be considered to be a multivariate analogue of univariate quantile mapping. All aspects of the observed distribution are transferred to the climate model simulations. In each of the three methods, marginal distributions are corrected by the change-preserving quantile delta mapping (`QDM`) algorithm (Cannon et al., 2015). Finally, an implementation of the Rank Resampling for Distributions and Dependences (R2D2) method introduced by Vrac (2018) is also included. An example application of the three MBC methods using the `cccma` dataset can be run via: `example(MBC, run.dontrun=TRUE)` Note: these functions apply bias correction to the supplied data without reference to other conditioning variables (e.g., time of year or season). The user must partition their data in a way that makes sense for a particular application. Furthermore, if `MBCn` is being applied to multiple spatial locations or grid cells, it is recommended that the same sequence of random rotations be used for each location. This can be achived by generating the random rotations first and then passing the sequence using the `rot.seq` argument: `rot.seq <- replicate(niterations, list(rot.random(nvars)))` `bias_correction <- MBCn(..., rot.seq=rot.seq)` package ## Details ||| |:--|:--| |Package:|MBC| |Type:|Package| |License:|GPL-2| |LazyLoad:|yes| ## References Cannon, A.J., 2018. Multivariate quantile mapping bias correction: An N-dimensional probability density function transform for climate model simulations of multiple variables. Climate Dynamics, 50(1-2):31-49. doi:10.1007/s00382-017-3580-6 Cannon, A.J., 2016. Multivariate bias correction of climate model output: Matching marginal distributions and inter-variable dependence structure. Journal of Climate, 29:7045-7064. doi:10.1175/JCLI-D-15-0679.1 Cannon, A.J., S.R. Sobie, and T.Q. Murdock, 2015. Bias correction of simulated precipitation by quantile mapping: How well do methods preserve relative changes in quantiles and extremes? Journal of Climate, 28:6938-6959. doi:10.1175/JCLI-D-14-00754.1 Francois, B., M. Vrac, A.J. Cannon, Y. Robin, and D. Allard, 2020. Multivariate bias corrections of climate simulations: Which benefits for which losses? Earth System Dynamics, 11:537-562. doi:10.5194/esd-11-537-2020 Vrac, M., 2018. Multivariate bias adjustment of high-dimensional climate simulations: the Rank Resampling for Distributions and Dependences (R2D2) bias correction. Hydrology and Earth System Sciences, 22:3175-3196. doi:10.5194/hess-22-3175-2018 ## See Also `QDM, MBCp, MBCr, MBCn, R2D2, escore, rot.random, cccma` ## Examples ```r ## Not run: data(cccma) set.seed(1) # Univariate quantile mapping qdm.c <- cccma$gcm.c*0 qdm.p <- cccma$gcm.p*0 for(i in seq(ncol(cccma$gcm.c))){ fit.qdm <- QDM(o.c=cccma$rcm.c[,i], m.c=cccma$gcm.c[,i], m.p=cccma$gcm.p[,i], ratio=cccma$ratio.seq[i], trace=cccma$trace[i]) qdm.c[,i] <- fit.qdm$mhat.c qdm.p[,i] <- fit.qdm$mhat.p } # Multivariate MBCp bias correction fit.mbcp <- MBCp(o.c=cccma$rcm.c, m.c=cccma$gcm.c, m.p=cccma$gcm.p, ratio.seq=cccma$ratio.seq, trace=cccma$trace) mbcp.c <- fit.mbcp$mhat.c mbcp.p <- fit.mbcp$mhat.p # Multivariate MBCr bias correction fit.mbcr <- MBCr(o.c=cccma$rcm.c, m.c=cccma$gcm.c, m.p=cccma$gcm.p, ratio.seq=cccma$ratio.seq, trace=cccma$trace) mbcr.c <- fit.mbcr$mhat.c mbcr.p <- fit.mbcr$mhat.p # Multivariate MBCn bias correction fit.mbcn <- MBCn(o.c=cccma$rcm.c, m.c=cccma$gcm.c, m.p=cccma$gcm.p, ratio.seq=cccma$ratio.seq, trace=cccma$trace) mbcn.c <- fit.mbcn$mhat.c mbcn.p <- fit.mbcn$mhat.p colnames(mbcn.c) <- colnames(mbcn.p) <- colnames(cccma$rcm.c) # Correlation matrices (Pearson and Spearman) # MBCp dev.new() par(mfrow=c(2, 2)) plot(c(cor(cccma$rcm.c)), c(cor(qdm.c)), col='black', pch=19, xlim=c(-1, 1), ylim=c(-1, 1), xlab='CanRCM4', ylab='CanESM2 MBCp', main='Pearson correlation\nMBCp calibration') abline(0, 1) grid() points(c(cor(cccma$rcm.c)), c(cor(mbcp.c)), col='red') plot(c(cor(cccma$rcm.p)), c(cor(qdm.p)), col='black', pch=19, xlim=c(-1, 1), ylim=c(-1, 1), xlab='CanRCM4', ylab='CanESM2 MBCp', main='Pearson correlation\nMBCp evaluation') abline(0, 1) grid() points(c(cor(cccma$rcm.p)), c(cor(mbcp.p)), col='red') plot(c(cor(cccma$rcm.c, m='s')), c(cor(qdm.c, m='s')), col='black', pch=19, xlim=c(-1, 1), ylim=c(-1, 1), xlab='CanRCM4', ylab='CanESM2 MBCp', main='Spearman correlation\nMBCp calibration') abline(0, 1) grid() points(c(cor(cccma$rcm.c, m='s')), c(cor(mbcp.c, m='s')), col='red') plot(c(cor(cccma$rcm.p, m='s')), c(cor(qdm.p, m='s')), col='black', pch=19, xlim=c(-1, 1), ylim=c(-1, 1), xlab='CanRCM4', ylab='CanESM2 MBCp', main='Spearman correlation\nMBCp evaluation') abline(0, 1) grid() points(c(cor(cccma$rcm.p, m='s')), c(cor(mbcp.p, m='s')), col='red') # MBCr dev.new() par(mfrow=c(2, 2)) plot(c(cor(cccma$rcm.c)), c(cor(qdm.c)), col='black', pch=19, xlim=c(-1, 1), ylim=c(-1, 1), xlab='CanRCM4', ylab='CanESM2 MBCr', main='Pearson correlation\nMBCr calibration') abline(0, 1) grid() points(c(cor(cccma$rcm.c)), c(cor(mbcr.c)), col='blue') plot(c(cor(cccma$rcm.p)), c(cor(qdm.p)), col='black', pch=19, xlim=c(-1, 1), ylim=c(-1, 1), xlab='CanRCM4', ylab='CanESM2 MBCr', main='Pearson correlation\nMBCr evaluation') abline(0, 1) grid() points(c(cor(cccma$rcm.p)), c(cor(mbcr.p)), col='blue') plot(c(cor(cccma$rcm.c, m='s')), c(cor(qdm.c, m='s')), col='black', pch=19, xlim=c(-1, 1), ylim=c(-1, 1), xlab='CanRCM4', ylab='CanESM2 MBCr', main='Spearman correlation\nMBCr calibration') abline(0, 1) grid() points(c(cor(cccma$rcm.c, m='s')), c(cor(mbcr.c, m='s')), col='blue') plot(c(cor(cccma$rcm.p, m='s')), c(cor(qdm.p, m='s')), col='black', pch=19, xlim=c(-1, 1), ylim=c(-1, 1), xlab='CanRCM4', ylab='CanESM2 MBCr', main='Spearman correlation\nMBCr evaluation') abline(0, 1) grid() points(c(cor(cccma$rcm.p, m='s')), c(cor(mbcr.p, m='s')), col='blue') # MBCn dev.new() par(mfrow=c(2, 2)) plot(c(cor(cccma$rcm.c)), c(cor(qdm.c)), col='black', pch=19, xlim=c(-1, 1), ylim=c(-1, 1), xlab='CanRCM4', ylab='CanESM2 MBCn', main='Pearson correlation\nMBCn calibration') abline(0, 1) grid() points(c(cor(cccma$rcm.c)), c(cor(mbcn.c)), col='orange') plot(c(cor(cccma$rcm.p)), c(cor(qdm.p)), col='black', pch=19, xlim=c(-1, 1), ylim=c(-1, 1), xlab='CanRCM4', ylab='CanESM2 MBCn', main='Pearson correlation\nMBCn evaluation') abline(0, 1) grid() points(c(cor(cccma$rcm.p)), c(cor(mbcn.p)), col='orange') plot(c(cor(cccma$rcm.c, m='s')), c(cor(qdm.c, m='s')), col='black', pch=19, xlim=c(-1, 1), ylim=c(-1, 1), xlab='CanRCM4', ylab='CanESM2 MBCn', main='Spearman correlation\nMBCn calibration') abline(0, 1) grid() points(c(cor(cccma$rcm.c, m='s')), c(cor(mbcn.c, m='s')), col='orange') plot(c(cor(cccma$rcm.p, m='s')), c(cor(qdm.p, m='s')), col='black', pch=19, xlim=c(-1, 1), ylim=c(-1, 1), xlab='CanRCM4', ylab='CanESM2 MBCn', main='Spearman correlation\nMBCn evaluation') abline(0, 1) grid() points(c(cor(cccma$rcm.p, m='s')), c(cor(mbcn.p, m='s')), col='orange') # Pairwise scatterplots dev.new() pairs(cccma$gcm.c, main='CanESM2 calibration', col='#0000001A') dev.new() pairs(cccma$rcm.c, main='CanRCM4 calibration', col='#0000001A') dev.new() pairs(qdm.c, main='QDM calibration', col='#0000001A') dev.new() pairs(mbcp.c, main='MBCp calibration', col='#FF00001A') dev.new() pairs(mbcr.c, main='MBCr calibration', col='#0000FF1A') dev.new() pairs(mbcn.c, main='MBCn calibration', col='#FFA5001A') # Energy distance skill score relative to univariate QDM escore.qdm <- escore(cccma$rcm.p, qdm.p, scale.x=TRUE) escore.mbcp <- escore(cccma$rcm.p, mbcp.p, scale.x=TRUE) escore.mbcr <- escore(cccma$rcm.p, mbcr.p, scale.x=TRUE) escore.mbcn <- escore(cccma$rcm.p, mbcn.p, scale.x=TRUE) cat('ESS (MBCp):', 1-escore.mbcp/escore.qdm, '\n') cat('ESS (MBCr):', 1-escore.mbcr/escore.qdm, '\n') cat('ESS (MBCn):', 1-escore.mbcn/escore.qdm, '\n') ## End(Not run) ```

MBC-package