# Quantile mapping using parametric transformations `fitQmapPTF` fits a parametric transformations to the quantile-quantile relation of observed and modelled values. `doQmapPTF` uses the transformation to adjust the distribution of the modelled data to match the distribution of the observations. ```r fitQmapPTF(obs, mod, ...) ## Default S3 method: fitQmapPTF(obs, mod, transfun=c("power","linear","expasympt", "scale","power.x0","expasympt.x0"), wet.day=TRUE, cost=c("RSS","MAE"), qstep=0.001,opar,...) ## S3 method for class 'matrix' fitQmapPTF(obs, mod, ...) ## S3 method for class 'data.frame' fitQmapPTF(obs, mod, ...) doQmapPTF(x,fobj,...) ## Default S3 method: doQmapPTF(x,fobj,...) ## S3 method for class 'matrix' doQmapPTF(x,fobj,...) ## S3 method for class 'data.frame' doQmapPTF(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`. - `transfun`: either a character string specifying a predefined function used for the transformation (see Details) or a function with `x` as first argument e.g. `function(x,a,b){a*x^b}` - `wet.day`: `logical` indicating whether to perform wet day correction or not. OR a `numeric` threshold below which all values are set to zero. See Details. - `cost`: Criterion for optimisation. "RSS" minimises the residual sum of squares and produces a least square fit. "MAE" minimises the mean absolute error, which is less sensitive to outliers. - `qstep`: `NULL` or a numeric value between 0 and 1. See Details. - `opar`: a named list with arguments passed to `optim`. Note that `method` is chosen automatically. If `transfun` is a character string default values for `par` are available (but can be overwritten). See examples. - `x`: `numeric` vector or a column `matrix` of modelled time series - `fobj`: output from `fitQmapDIST` - ``...``: Further arguments passed to methods ## Details Before further computations the empirical cumulative distribution functions (CDF) of the observed (`obs`) and modelled (`mod`) are estimated. If `!is.null(qstep)` than `mod` and `obs` are aggregated to quantiles before model identification as: `quantile(x,probs=seq(0,1,by=qstep)`. If `!is.null(qstep)` than `mod` and `obs` are sorted to produce an estimate of the empirical CDF. In case of different length of `mod` and `obs` than `quantile(x,probs=seq(0,1,len=n)]` is used, where `n <- min(length(obs),length(mod))`. NOTE that large values of `qstep` effectively reduce the sample-size and can be used to speedup computations - but may render estimates less reliable. `wet.day` is intended for the use for precipitation data. Wet day correction attempts to equalise the fraction of days with precipitation between the observed and the modelled data. If `wet.day=TRUE` the empirical probability of nonzero observations is found (`obs>=0`) and the corresponding modelled value is selected as a threshold. All modelled values below this threshold are set to zero. If `wet.day` is `numeric` the same procedure is performed after setting all `obs. Dosio, A. & Paruolo, P. Bias correction of the ENSEMBLES high-resolution climate change projections for use by impact models: Evaluation on the present climate. J. Geophys. Res., AGU, 2011, 116, D16106, . 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 `fitQmap`, `optim` ## Examples ```r data(obsprecip) data(modprecip) ## data.frame example qm.fit <- fitQmapPTF(obsprecip,modprecip, transfun="power.x0", cost="RSS",wet.day=TRUE, qstep=0.001) qm <- doQmapPTF(modprecip,qm.fit) ## application to "single time series" qm.b.fit <- fitQmapPTF(obsprecip[,1],modprecip[,1], transfun="expasympt.x0", cost="RSS",wet.day=0.1, qstep=0.001) qm.b <- doQmapPTF(modprecip[,1],qm.b.fit) qm.c.fit <- fitQmapPTF(obsprecip[,1],modprecip[,1], transfun="expasympt", cost="RSS",wet.day=TRUE, qstep=0.001) qm.c <- doQmapPTF(modprecip[,1],qm.c.fit) ## user defined transfer function ## and usage of the 'opar' argument ## (same as transfun="power") myff <- function(x,a,b) a*x^b qm3.fit <- fitQmapPTF(obsprecip[,1],modprecip[,1], transfun=myff, opar=list(par=c(a=1,b=1)), cost="RSS",wet.day=TRUE, qstep=0.001) qm3 <- doQmapPTF(modprecip[,1],qm3.fit) 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[,1]),col="red") lines(sqrtquant(modprecip[,1]), sqrtquant(qm.b),col="blue") lines(sqrtquant(modprecip[,1]), sqrtquant(qm.c),col="green") lines(sqrtquant(modprecip[,1]), sqrtquant(qm3),col="orange") legend("topleft", legend=c("power.x0","expasympt.x0", "expasympt","myff"), col=c("red","blue","green","orange"),lty=1) ```

fitqmapptf function