# Quantile mapping using distribution derived transformations `fitQmapDIST` fits a theoretical distribution to observed and to modelled time series and returns these parameters as well as a transfer function derived from the distribution. `doQmapDIST` uses the transfer function to transform the distribution of the modelled data to match the distribution of the observations. ```r fitQmapDIST(obs, mod, ...) ## Default S3 method: fitQmapDIST(obs,mod,distr="berngamma",start.fun, qstep=NULL,mlepar,...) ## S3 method for class 'matrix' fitQmapDIST(obs, mod, ...) ## S3 method for class 'data.frame' fitQmapDIST(obs, mod, ...) doQmapDIST(x,fobj,...) ## Default S3 method: doQmapDIST(x,fobj,...) ## S3 method for class 'matrix' doQmapDIST(x,fobj,...) ## S3 method for class 'data.frame' doQmapDIST(x,fobj,...) ``` ## Arguments - `obs`: `numeric` vector, column `matrix` or `data.frame` with observed time series. - `mod`: `numeric` vector, column `matrix` or `data.frame` with modelled time series, corresponding to `obs`. - `distr`: A character string `"name"` naming a distribution for which the corresponding density function (`dname`), the corresponding distribution function (`pname`) and the quantile function (`qname`) must be defined (see for example `GammaDist`, `berngamma` or `bernweibull`. - `start.fun`: function estimating starting values for parameter optimisation. Default starting values are provided for `berngamma`, `bernweibull`, `bernlnorm`, `bernexp` and the distributions mentioned in the documentation of `mledist`. - `qstep`: `NULL` or a numeric value between 0 and 1. If `!is.null(qstep)` than `mod` and `obs` are aggregated to quantiles before model identification as: `quantile(x,probs=seq(0,1,by=qstep)`. This effectively reduces the sample-size and can be used to speedup computations - but may render estimates less reliable. - `mlepar`: a named list. Names correspond to parameters passed to `mledist` note that `start` may be overwritten by `start.fun` See examples. - `x`: `numeric` vector or a column `matrix` of modelled time series - `fobj`: output from `fitQmapDIST` - ``...``: Further arguments passed to methods ## Details Quantile mapping using distribution derived transformations to adjust the distribution of a modelled variable ($P_m$) such that it matches the distribution of an observed variable ($P_o$). The distribution derived transfer function is defined as $$ P_o=F^{-1}_o(F_m(P_m)) $$ where $F$ is a CDF and $F^{-1}$ is the corresponding quantile function (inverse CDF). The subscripts $o$ and $m$ indicate parameters of the distribution that correspond to observed and modelled data respectively. ## Returns `fitQmapDIST` returns an object of class `fitQmapDIST` containing following elements: - **tfun**: The function used to transform the distribution of modelled values such that the distribution of observations. The function is build internally based on the distribution function ("pname") and quantile function ("qname") corresponding to `distr`. - **par**: A matrix. The (named) columns correspond to the parameters of the distribution specified in `distr` estimated for the observed (suffix `.o`) and the modelled (suffix `.m`) data. The rows correspond to each pair of time series in `obs` and `mod`. `doQmapDIST` returns a `numeric` vector, `matrix` or `data.frame` depending on the format of `x`. ## References Piani, C.; Haerter, J. & Coppola, E. Statistical bias correction for daily precipitation in regional climate models over Europe. Theoretical and Applied Climatology, 2010, 99, 187-192, . Li, H.; Sheffield, J. & Wood, E. F. Bias correction of monthly precipitation and temperature fields from Intergovernmental Panel on Climate Change AR4 models using equidistant quantile matching. J. Geophys. Res., 2010, 115, D10101, . Ines, A. V. & Hansen, J. W. Bias correction of daily GCM rainfall for crop simulation studies. Agricultural and Forest Meteorology, 2006, 138, 44 - 53, . For a general assessment of the methods see: Gudmundsson, L.; Bremnes, J. B.; Haugen, J. E. & Engen-Skaugen, T. Technical Note: Downscaling RCM precipitation to the station scale using statistical transformations - a comparison of methods. Hydrology and Earth System Sciences, 2012, 16, 3383-3390, . ## Author(s) Lukas Gudmundsson ## See Also `doQmap`, `startberngamma`, `berngamma`, `startbernweibull`, `bernweibull`, `startbernlnorm`, `bernlnorm`, `startbernexp`, `bernexp`, `mledist`, `fitdist` ## Examples ```r data(obsprecip) data(modprecip) qm.fit <- fitQmapDIST(obsprecip[,1],modprecip[,1], distr="berngamma", qstep=0.001) qm <- doQmapDIST(modprecip[,1],qm.fit) # adjusting settings of the maximum likelyhood estimator # here changing the convergence criterion in the optimizer 'optim' # Note: the selected value might be to large for true applications. # Please check for your context. qm.lnorm.fit <- fitQmapDIST(obsprecip[,1],modprecip[,1], distr="bernlnorm", qstep=0.001, mlepar = list(control = list( reltol = 1e-2 ))) qm.lnorm <- doQmapDIST(modprecip[,1],qm.lnorm.fit) qm.weibu.fit <- fitQmapDIST(obsprecip[,1],modprecip[,1], distr="bernweibull", qstep=0.001) qm.weibu <- doQmapDIST(modprecip[,1],qm.weibu.fit) qm.exp.fit <- fitQmapDIST(sqrt(obsprecip[,1]),sqrt(modprecip[,1]), distr="bernexp", qstep=0.001) qm.exp <- doQmapDIST(sqrt(modprecip[,1]),qm.exp.fit)^2 ## utility function. ## plots are easier to investigate if ## precipitation data are sqrt transformed sqrtquant <- function(x,qstep=0.01){ qq <- quantile(x,prob=seq(0,1,by=qstep)) sqrt(qq) } plot(sqrtquant(modprecip[,1]), sqrtquant(obsprecip[,1])) lines(sqrtquant(modprecip[,1]), sqrtquant(qm),col="red") lines(sqrtquant(modprecip[,1]), sqrtquant(qm.lnorm),col="blue") lines(sqrtquant(modprecip[,1]), sqrtquant(qm.weibu),col="green") lines(sqrtquant(modprecip[,1]), sqrtquant(qm.exp),col="orange") legend("topleft", legend=c("berngamma","bernlnorm","bernweibull","bernexp"), lty=1, col=c("red","blue","green","orange")) ## effect of qstep on speed of fitting process: system.time( qm.a.fit <- fitQmapDIST(obsprecip[,2],modprecip[,2], distr="berngamma", start.fun=startberngamma, qstep=0.001) ) system.time( qm.b.fit <- fitQmapDIST(obsprecip[,2],modprecip[,2], distr="berngamma", start.fun=startberngamma, qstep=0.01) ) qm.a <- doQmapDIST(modprecip[,2],qm.a.fit) qm.b <- doQmapDIST(modprecip[,2],qm.b.fit) plot(sqrtquant(modprecip[,2]), sqrtquant(obsprecip[,2])) lines(sqrtquant(modprecip[,2]), sqrtquant(qm.a),col="red") lines(sqrtquant(modprecip[,2]), sqrtquant(qm.b),col="blue") legend("topleft", legend=c("qstep=0.001","qstep=0.01"), col=c("red","blue"), lty=1) ## method for matrix ## the sqrt() transformation renders the ## fitting procedure more stable qm2.fit <- fitQmapDIST(sqrt(obsprecip),sqrt(modprecip), distr="berngamma", qstep=0.001) qm2 <- doQmapDIST(sqrt(modprecip),qm2.fit)^2 if(!any(is.na(qm2.fit$par))){ op <- par(mfrow=c(1,3)) for(i in 1:3){ plot(sqrtquant(modprecip[,i]), sqrtquant(obsprecip[,i])) lines(sqrtquant(modprecip[,i]), sqrtquant(qm2[,i]),col="red") } par(op) } ```

fitqmapdist function