Package 'qmap'

Statistical Transformations for Post-Processing Climate Model Output

Description

Empirical adjustment (bias correction) of variables originating from (regional) climate model simulations using quantile mapping. The workhorse functions of this package are fitQmap and doQmap which offer an easy to use interface to different statistical transformations, also referred to as quantile mapping methods.

Details

Package:	qmap
Type:	Package
Version:	1.0-4
Date:	2016-05-03
License:	GPL >= 2
LazyLoad:	yes

Author(s)

Lukas Gudmundsson

References

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, doi:10.5194/hess-16-3383-2012.

---

The Bernoulli-Exponential distribution

Description

Density, distribution function, quantile function and random generation for the Bernoulli-Exponential distribution with parameters prob, and rate.

Usage

dbernexp(x, prob, rate)
pbernexp(q, prob, rate)
qbernexp(p, prob, rate)
rbernexp(n, prob, rate)

Arguments

x, q	vector of quantiles.
p	vector of probabilities.
prob	probability of non-zero event.
n	number of random samples.
rate	rate parameter of the Exponential distribution.

Details

Mixture of the Bernoulli and the Exponential distribution. The mixture is analogue to the one described for the berngamma distribution.

Value

dbernexp gives the density (pdf), pbernexp gives the distribution function (cdf), qbernexp gives the quantile function (inverse cdf), and rbernexp generates random numbers.

Author(s)

Lukas Gudmundsson

See Also

Exponential, berngamma

Examples

data(obsprecip)

(ts <- startbernexp(obsprecip[,1]))
hist(obsprecip[,1],freq=FALSE) 
lines(seq(0,max(obsprecip[,1])),
      dbernexp(seq(0,max(obsprecip[,1])),
               prob=ts$prob,
               rate=ts$rate),
      col="red")

pp <- seq(0.01,0.99,by=0.01)
qq <-quantile(obsprecip[,1],probs=pp) 

plot(qq,pp)
lines(qbernexp(pp,
               prob=ts$prob,
               rate=ts$rate),
      pp,col="red")

plot(qq,pp)
lines(qq,
      pbernexp(qq,
               prob=ts$prob,
               rate=ts$rate),
      col="red")

hist(rbernexp(1000,prob=ts$prob,
              rate=ts$rate),freq=FALSE)

---

The Bernoulli-Gamma distribution

Description

Density, distribution function, quantile function and random generation for the Bernoulli-Gamma distribution with parameters prob, shape, and scale.

Usage

dberngamma(x, prob, scale, shape)
pberngamma(q, prob, scale, shape)
qberngamma(p, prob, scale, shape)
rberngamma(n, prob, scale, shape)

Arguments

x, q	vector of quantiles.
p	vector of probabilities.
prob	probability of non-zero event.
n	number of random samples.
scale, shape	shape and scale parameters of the gamma distribution.

Details

Mixture of the Bernoulli and the Gamma distribution. The Bernoulli distribution is used to model the occurrence of zero values with the probability of 1-prob. Non-zero values follow the Gamma distribution with shape and scale parameters.

The probability density function (PDF) is defined as:

$g(x) = \left\{ \begin{array}{l l} \pi*\gamma(x) & \quad \mbox{if } x > 0 \\ 1- \pi & \quad \mbox{if } x \leq 0 \end{array} \right.$

where $\gamma(x)$ is the probability density function of the gamma distribution and $\pi$ is probability of a non-zero event.

The cumulative distribution function (CDF) is defined as:

$G(x) = \left\{ \begin{array}{l l} 1 - \pi + \pi*\Gamma(x) & \quad \mbox{if } x > 0 \\ 1 - \pi & \quad \mbox{if } x \leq 0 \end{array} \right.$

where $\Gamma(x)$ is the cumulative distribution function of the gamma distribution.

The quantile function (inverse of the CDF) is defined as

$G^{-1}(p) = \left\{ \begin{array}{l l} \Gamma^{-1}\left(\frac{p-1+\pi}{\pi}\right) & \quad \mbox{if } \pi > 1-p \\ 0 & \quad \mbox{if } p \leq 1-p \end{array} \right.$

where $\Gamma^{-1}(p)$ is the inverse CDF of the gamma distribution and $p$ is a probability.

Value

dberngamma gives the density (pdf), pberngamma gives the distribution function (cdf), qberngamma gives the quantile function (inverse cdf), and rberngamma generates random deviates.

Note

The implementation is largely based on the bgamma family in the CaDENCE-package (Cannon, 2012) that was only available as test version at time of implementation (Mar. 2012). The CaDENCE-package is available at http://www.eos.ubc.ca/~acannon/CaDENCE/.

For further details and meteorological application of Bernoulli-Gamma distributions (also referred to as 'Mixed Gamma' distribution) see Burger et al. 2012, Cannon 2008, Li et al. 2010, Mooley 1973, Piani et al. 2010, Thom 1968, Sloughter et al. 2007.

Author(s)

Lukas Gudmundsson

References

Burger, G.; Murdock, T. Q.; Werner, A. T.; Sobie, S. R. & Cannon, A. J. Downscaling extremes - an intercomparison of multiple statistical methods for present climate. Journal of Climate, American Meteorological Society, early online release, 2012, doi:10.1175/JCLI-D-11-00408.1.

Cannon, A. J. Probabilistic Multisite Precipitation Downscaling by an Expanded Bernoulli-Gamma Density Network. Journal of Hydrometeorology, American Meteorological Society, 2008, 9, 1284-1300, doi:10.1175/2008JHM960.1.

Cannon, A. J. Neural networks for probabilistic environmental prediction: Conditional Density Estimation Network Creation and Evaluation (CaDENCE) in R. Computers & Geosciences, 2012, 41, 126 - 135, doi:10.1016/j.cageo.2011.08.023.

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., AGU, 2010, 115, D10101, doi:10.1029/2009JD012882.

Mooley, D. A. Gamma Distribution Probability Model for Asian Summer Monsoon Monthly Rainfall. Monthly Weather Review, 1973, 101, 160-176, doi:10.1175/1520-0493(1973)101<0160:GDPMFA>2.3.CO;2.

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, doi:10.1007/s00704-009-0134-9.

Thom, H. C. S. Approximate convolution of the gamma and mixed gamma distributions. Monthly Weather Review, 1968, 96, 883-886, doi:10.1175/1520-0493(1968)096<0883:ACOTGA>2.0.CO;2.

Sloughter, J. M. L.; Raftery, A. E.; Gneiting, T. & Fraley, C. Probabilistic Quantitative Precipitation Forecasting Using Bayesian Model Averaging. Monthly Weather Review, 2007, 135, 3209-3220, doi:10.1175/MWR3441.1.

See Also

GammaDist

Examples

data(obsprecip)

(ts <- startberngamma(obsprecip[,1]))
hist(obsprecip[,1],freq=FALSE) 
lines(seq(0,20),dberngamma(0:20,
                           prob=ts$prob,
                           scale=ts$scale,
                           shape=ts$shape),
      col="red")

pp <- seq(0.01,0.99,by=0.01)
qq <-quantile(obsprecip[,1],probs=pp) 

plot(qq,pp)
lines(qberngamma(pp,
                 prob=ts$prob,
                 scale=ts$scale,
                 shape=ts$shape),
      pp,col="red")

plot(qq,pp)
lines(qq,
      pberngamma(qq,
                 prob=ts$prob,
                 scale=ts$scale,
                 shape=ts$shape),
      col="red")

hist(rberngamma(1000,
                prob=ts$prob,
                scale=ts$scale,
                shape=ts$shape),freq=FALSE)

---

The Bernoulli-Log-Normal distribution

Description

Density, distribution function, quantile function and random generation for the Bernoulli-Log-Normal distribution with parameters prob, meanlog, and sdlog.

Usage

dbernlnorm(x, prob, meanlog, sdlog)
pbernlnorm(q, prob, meanlog, sdlog)
qbernlnorm(p, prob, meanlog, sdlog)
rbernlnorm(n, prob, meanlog, sdlog)

Arguments

x, q	vector of quantiles.
p	vector of probabilities.
prob	probability of non-zero event.
n	number of random samples.
meanlog, sdlog	meanlog and sdlog parameters of the Log-Normal distribution.

Details

Mixture of Bernoulli and Log-Normal distribution. The mixture is analogue to the one described for the berngamma distribution.

Value

dbernlnorm gives the density (pdf), pbernlnorm gives the distribution function (cdf), qbernlnorm gives the quantile function (inverse cdf), and rbernlnorm generates random deviates.

Note

The implementation is largely based on the blnorm family in the CaDENCE-package (Cannon, 2012) that was only available as test version at time of implementation (Mar. 2012). The CaDENCE-package is available at http://www.eos.ubc.ca/~acannon/CaDENCE/.

Author(s)

Lukas Gudmundsson

References

Cannon, A. J. Neural networks for probabilistic environmental prediction: Conditional Density Estimation Network Creation and Evaluation (CaDENCE) in R. Computers & Geosciences, 2012, 41, 126 - 135, doi:10.1016/j.cageo.2011.08.023.

See Also

Lognormal, berngamma

Examples

data(obsprecip)

(ts <- startbernlnorm(obsprecip[,1]))
hist(obsprecip[,1],freq=FALSE)
lines(seq(0,20),dbernlnorm(0:20,
                           prob=ts$prob,
                           meanlog=ts$meanlog,
                           sdlog=ts$sdlog),
      col="red")

pp <- seq(0.01,0.99,by=0.01)
qq <-quantile(obsprecip[,1],probs=pp) 

plot(qq,pp)
lines(qbernlnorm(pp,
                 prob=ts$prob,
                 meanlog=ts$meanlog,
                 sdlog=ts$sdlog),
      pp,col="red")

plot(qq,pp)
lines(qq,
      pbernlnorm(qq,
                 prob=ts$prob,
                 meanlog=ts$meanlog,
                 sdlog=ts$sdlog),
      col="red")

hist(rbernlnorm(1000,prob=ts$prob,
                meanlog=ts$meanlog,
                sdlog=ts$sdlog),freq=FALSE)

---

The Bernoulli-Weibull distribution

Description

Density, distribution function, quantile function and random generation for the Bernoulli-Weibull distribution with parameters prob, shape, and scale.

Usage

dbernweibull(x, prob, scale, shape)
pbernweibull(q, prob, scale, shape)
qbernweibull(p, prob, scale, shape)
rbernweibull(n, prob, scale, shape)

Arguments

x, q	vector of quantiles.
p	vector of probabilities.
prob	probability of non-zero event.
n	number of random samples.
scale, shape	shape and scale parameters of the weibull distribution.

Details

Mixture of Bernoulli and Weibull distribution. The mixture is analogue to the one described for the berngamma distribution.

Value

dbernweibull gives the density (pdf), pbernweibull gives the distribution function (cdf), qbernweibull gives the quantile function (inverse cdf), and rbernweibull generates random deviates.

Note

The implementation is largely based on the bweibull family in the CaDENCE-package (Cannon, 2012) that was only available as test version at time of implementation (Mar. 2012). The CaDENCE-package is available at http://www.eos.ubc.ca/~acannon/CaDENCE/.

Author(s)

Lukas Gudmundsson

References

Cannon, A. J. Neural networks for probabilistic environmental prediction: Conditional Density Estimation Network Creation and Evaluation (CaDENCE) in R. Computers & Geosciences, 2012, 41, 126 - 135, doi:10.1016/j.cageo.2011.08.023.

See Also

Weibull, berngamma

Examples

data(obsprecip)

(ts <- startbernweibull(obsprecip[,1]))
hist(obsprecip[,1],freq=FALSE) 
lines(seq(0,max(obsprecip[,1])),
      dbernweibull(seq(0,max(obsprecip[,1])),
                   prob=ts$prob,
                   shape=ts$shape,
                   scale=ts$scale),
      col="red")

pp <- seq(0.01,0.99,by=0.01)
qq <-quantile(obsprecip[,1],probs=pp) 

plot(qq,pp)
lines(qbernweibull(pp,
                   prob=ts$prob,
                   scale=ts$scale,
                   shape=ts$shape),
      pp,col="red")

plot(qq,pp)
lines(qq,
      pbernweibull(qq,
                   prob=ts$prob,
                   scale=ts$scale,
                   shape=ts$shape),
      col="red")

hist(rbernweibull(1000,prob=ts$prob,
                      shape=ts$shape,
                      scale=ts$scale),freq=TRUE)

---

Quantile mapping

Description

fitQmap identifyes the parameters of different quantile mapping methods. doQmap performs quantile mapping using previously identified parameters.

Usage

fitQmap(obs,mod,method=c("PTF","DIST","RQUANT","QUANT","SSPLIN"),...)
doQmap(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.
method	A character string indicating the method to be used. See Details.
x	numeric vector or a column matrix of modelled time series. Should have the same number of columns as obs.
fobj	output from fitQmap (or of method defined via method).
...	arguments passed to the method specified by method.

Details

The method argument decides upon which method for quantile mapping is used:

"PTF" selects fitQmapPTF.

"DIST" selects fitQmapDIST

"RQUANT" selects fitQmapRQUANT

"QUANT" selects fitQmapQUANT

"SSPLIN" selects fitQmapSSPLIN

doQmap investigates the class of fobj and chooses the appropriate method for quantile mapping.

Value

fitQmap returns an object which class and structure depends on the selected method (see Details).

doQmap returns a numeric vector, matrix or data.frame depending on the format of x.

Author(s)

Lukas Gudmundsson

References

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, doi:10.5194/hess-16-3383-2012.

See Also

fitQmapDIST, fitQmapPTF, fitQmapRQUANT, fitQmapQUANT, fitQmapSSPLIN

Examples

data(obsprecip)
data(modprecip)

## call to fitQmapPTF and doQmapPTF
qm1.fit <- fitQmap(obsprecip,modprecip,
              method="PTF",
              transfun="expasympt",
              cost="RSS",wett.day=TRUE)
qm1 <- doQmap(modprecip,qm1.fit)

## call to fitQmapDIST and doQmapDIST
qm2.fit <- fitQmap(sqrt(obsprecip),sqrt(modprecip),
              method="DIST",qstep=0.001,
              transfun="berngamma")
qm2 <- doQmap(sqrt(modprecip),qm2.fit)^2

## call to fitQmapRQUANT and doQmapRQUANT
qm3.fit <- fitQmap(obsprecip,modprecip,
              method="RQUANT",qstep=0.01)
qm3 <- doQmap(modprecip,qm3.fit,type="linear")


## call to fitQmapRQUANT and doQmapRQUANT
qm4.fit <- fitQmap(obsprecip,modprecip,
              method="QUANT",qstep=0.01)
qm4 <- doQmap(modprecip,qm4.fit,type="tricub")

## call to fitQmapSSPLIN and doQmapSSPLIN
qm5.fit <- fitQmap(obsprecip,modprecip,qstep=0.01,
                   method="SSPLIN")
qm5 <- doQmap(modprecip,qm5.fit)

sqrtquant <- function(x,qstep=0.001){
  qq <- quantile(x,prob=seq(0,1,by=qstep))
  sqrt(qq)
}

op <- par(mfrow=c(1,3))
for(i in 1:3){
  plot(sqrtquant(modprecip[,i]),
       sqrtquant(obsprecip[,i]),pch=19,col="gray",
       main=names(obsprecip)[i])
  lines(sqrtquant(modprecip[,i]),
        sqrtquant(qm1[,i]),col=1)
  lines(sqrtquant(modprecip[,i]),
        sqrtquant(qm2[,i]),col=2)
  lines(sqrtquant(modprecip[,i]),
        sqrtquant(qm3[,i]),col=3)
  lines(sqrtquant(modprecip[,i]),
        sqrtquant(qm4[,i]),col=4)
  lines(sqrtquant(modprecip[,i]),
        sqrtquant(qm5[,i]),col=5)
}
legend("topleft",
       legend=c("PTF","DIST","RQUANT","QUANT","SSPLIN"),
       lty=1, col=1:5)
par(op)

---

Quantile mapping using distribution derived transformations

Description

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.

Usage

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.

Value

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.

Author(s)

Lukas Gudmundsson

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, doi:10.1007/s00704-009-0134-9.

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, doi:10.1029/2009JD012882.

Ines, A. V. & Hansen, J. W. Bias correction of daily GCM rainfall for crop simulation studies. Agricultural and Forest Meteorology, 2006, 138, 44 - 53, doi: 10.1016/j.agrformet.2006.03.009.

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, doi:10.5194/hess-16-3383-2012.

See Also

doQmap, startberngamma, berngamma, startbernweibull, bernweibull, startbernlnorm, bernlnorm, startbernexp, bernexp, mledist, fitdist

Examples

data(obsprecip)
data(modprecip)

qm.fit <- fitQmapDIST(obsprecip[,1],modprecip[,1],
                      distr="berngamma",
                      qstep=0.001)
qm <- doQmapDIST(modprecip[,1],qm.fit)


qm.lnorm.fit <- fitQmapDIST(obsprecip[,1],modprecip[,1],
                      distr="bernlnorm",
                      qstep=0.001)
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)
}

---

Quantile mapping using parametric transformations

Description

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.

Usage

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<wet.day to zero. The transformations are then only fitted to the portion of the distributions corresponding to observed wet days. See Piani et. al (2010) for further explanations.

Transformations (transfun):

NOTE: If wet day correction is performed (see wet.day), the transformations are only fitted to the portion of the empirical CDF with nonzero observations.

A series of predefined transformations are available and can be accessed by setting transfun to one of the following options ($P_o$ refers to observed and $P_m$ to modelled CDFs):

"power":

$P_o=b*P_m^c$

"linear":

$P_o=a+b*P_m$

"expasympt" (exponential tendency to an asymptote):

$P_o=(a+b*P_m)*(1-exp(-P_m/\tau))$

"scale":

$P_o=b*P_m$

"power.x0":

$P_o=b*(P_m-x0)^c$

"expasympt.x0" (exponential tendency to an asymptote):

$P_o=(a+b*P_m)*(1-exp(-(P_m-x0)/\tau))$

Value

fitQmapPTF returns an object of class fitQmapPTF containing following elements:

tfun	The function used to transform the distribution of the modelled values to match the distribution of the observations.
par	A matrix. The (named) columns correspond to the parameters of the transfer function. The rows correspond to pairs of time series in obs and mod.
wet.day	logical, indicating whether to perform wet day correction or not. OR a numeric threshold below which all values are set to zero.

doQmapPTF returns a numeric vector, matrix or data.frame depending on the format of x.

Author(s)

Lukas Gudmundsson

References

The implementation is closely related to the methods published in:

Piani, C.; Weedon, G.; Best, M.; Gomes, S.; Viterbo, P.; Hagemann, S. & Haerter, J. Statistical bias correction of global simulated daily precipitation and temperature for the application of hydrological models. Journal of Hydrology, 2010, 395, 199 - 215, doi:10.1016/j.jhydrol.2010.10.024.

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, doi:10.1029/2011JD015934.

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, doi:10.5194/hess-16-3383-2012.

See Also

fitQmap, optim

Examples

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)

---

Non-parametric quantile mapping using empirical quantiles.

Description

fitQmapQUANT estimates values of the empirical cumulative distribution function of observed and modelled time series for regularly spaced quantiles. doQmapQUANT uses these estimates to perform quantile mapping.

Usage

fitQmapQUANT(obs, mod, ...)
## Default S3 method:
fitQmapQUANT(obs,mod,wet.day=TRUE,qstep=0.01,
nboot = 1,...)
## S3 method for class 'matrix'
fitQmapQUANT(obs, mod, ...)
## S3 method for class 'data.frame'
fitQmapQUANT(obs, mod, ...)
doQmapQUANT(x,fobj,...)
## Default S3 method:
doQmapQUANT(x,fobj, type=c("linear","tricub"),...)
## S3 method for class 'matrix'
doQmapQUANT(x,fobj,...)
## S3 method for class 'data.frame'
doQmapQUANT(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.
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.
qstep	a numeric value between 0 and 1. The quantile mapping is fitted only for the quantiles defined by quantile(0,1,probs=seq(0,1,by=qstep).
nboot	number of bootstrap samples used for estimation of the observed quantiles. If nboot==1 the estimation is based on all (and not resampled) data. See details.
x	numeric vector or a column matrix of modelled time series
fobj	output from fitQmapQUANT
type	type of interpolation between the fitted transformed values. See details.
...	Further arguments passed to methods

Details

fitQmapQUANT estimates the empirical cumulative distribution function of mod and obs for the quantiles defined by seq(0,1,by=qstep). The quantiles of mod are estimated using the empirical quantiles. If nboot>1 the quantiles of obs are estimated as the mean of nboot bootstrap samples (if nboot>1).

doQmapQUANT transforms the variable x based on the transformation identified using fitQmapQUANT. For all values that are not in quantile(mod,probs=seq(0,1,by=qstep)) the transformation is estimated using interpolation of the fitted values. Available interpolation options are:

type="linear": linear interpolation using approx, but using the extrapolation suggested by Boe et al. (2007) for values of x larger than max(mod) (constant correction).

type="tricube": monotonic tricubic spline interpolation using splinefun. Spline interpolation is performed using a _monotone_ Hermite spline (method="monoH.FC" in splinefun).

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<wet.day to zero.

Value

fitQmapQUANT returns an object of class fitQmapQUANT containing following elements:

par	A list containing:
par$modq	a matrix. Each column i corresponds to the output of quantile(mod[,i],probs=seq(0,1,by=qstep)).
par$fitq	observed empirical quantiles corresponding to par$modq.
wet.day	logical, indicating whether to perform wet day correction or not. OR a numeric threshold below which all values are set to zero.

doQmapQUANT returns a numeric vector or matrix depending on the format of x.

Author(s)

Lukas Gudmundsson

References

Boe, J.; Terray, L.; Habets, F. & Martin, E. Statistical and dynamical downscaling of the Seine basin climate for hydro-meteorological studies. International Journal of Climatology, 2007, 27, 1643-1655, doi: 10.1002/joc.1602.

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, doi:10.5194/hess-16-3383-2012.

See Also

fitQmap

Examples

data(obsprecip)
data(modprecip)

qm.fit <- fitQmapQUANT(obsprecip[,2],modprecip[,2],
                      qstep=0.1,nboot=1,wet.day=TRUE)
qm.a <- doQmapQUANT(modprecip[,2],qm.fit,type="linear")
qm.s <- doQmapQUANT(modprecip[,2],qm.fit,type="tricub")


sqrtquant <- function(x,qstep=0.01){
  qq <- quantile(x,prob=seq(0,1,by=qstep))
  sqrt(qq)
}

plot(sqrtquant(modprecip[,2]),
     sqrtquant(obsprecip[,2]))
lines(sqrtquant(modprecip[,2]),
      sqrtquant(qm.a),col="red")
lines(sqrtquant(modprecip[,2]),
      sqrtquant(qm.s),col="blue")
points(sqrt(qm.fit$par$modq),sqrt(qm.fit$par$fitq),
       pch=19,cex=0.5,col="green")
legend("topleft",
       legend=c("linear","tricub","support"),
       lty=c(1,1,NA),pch=c(NA,NA,19),
       col=c("red","blue","green"))
         

qm2.fit <- fitQmapQUANT(obsprecip,modprecip,
                      qstep=0.01,nboot=1,wet.day=TRUE)
qm2 <- doQmapQUANT(modprecip,qm2.fit,type="tricub")

op <- par(mfrow=c(1,3))
for(i in 1:3){
  plot(sqrtquant(modprecip[,i]),
       sqrtquant(obsprecip[,i]),
       main=names(qm2)[i])
  lines(sqrtquant(modprecip[,i]),
        sqrtquant(qm2[,i]),col="red")
  points(sqrt(qm2.fit$par$modq[,i]),
         sqrt(qm2.fit$par$fitq[,i]),
       pch=19,cex=0.5,col="green")

}
par(op)

---

Non-parametric quantile mapping using robust empirical quantiles.

Description

fitQmapRQUANT estimates the values of the quantile-quantile relation of observed and modelled time series for regularly spaced quantiles using local linear least square regression. doQmapRQUANT performs quantile mapping by interpolating the empirical quantiles.

Usage

fitQmapRQUANT(obs, mod, ...)
## Default S3 method:
fitQmapRQUANT(obs,mod,wet.day=TRUE,qstep=0.01,
nlls = 10,nboot = 10,...)
## S3 method for class 'matrix'
fitQmapRQUANT(obs, mod, ...)
## S3 method for class 'data.frame'
fitQmapRQUANT(obs, mod, ...)
doQmapRQUANT(x,fobj,...)
## Default S3 method:
doQmapRQUANT(x,fobj,slope.bound=c(lower=0,upper=Inf), 
type=c("linear","linear2","tricub"),...)
## S3 method for class 'matrix'
doQmapRQUANT(x,fobj,...)
## S3 method for class 'data.frame'
doQmapRQUANT(x,fobj,...)

Arguments

obs	numeric vector, column matrix or data.frame with observed time series.
mod	numeric vector or column matrix/data.frame with modelled time series, corresponding to obs
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.
qstep	A numeric value between 0 and 1. The values quantile-quantile plot are estimated at the position of the values defined by: quantile(mod,probs=seq(0,1,by=qstep).
nlls	number of nearest data points to apply in the local regression
nboot	number of bootstrap samples in the estimation of the transformation. If nboot==1 the estimation is based on all (and not resampled) data.
x	numeric vector or a column matrix of modelled time series
fobj	output from fitQmapRQUANT
slope.bound	bounds for the slopes in case of extrapolation. Applies only if type="linear2"
type	type of interpolation between the fitted transformed values. See details
...	Further arguments passed to methods

Details

fitQmapRQUANT produces a robust estimate of the empirical quantile-quantile plot (QQ-plot) of mod vs obs for the seq(0,1,by=qstep) quantiles mod. The corresponding value of the quantiles of obs is estimated using local linear least squares regression. For each quantile of mod the nlls nearest data points in the QQ-plot are identified and used to fit a local regression line. This regression line is then used to estimate value of the quantile of obs. The estimation is replicated for nboot bootstrap samples and the mean of the bootstrap replicates is returned.

This procedure results in a table with empirical quantiles of mod and a corresponding table with robust estimates of the empirical quantiles of obs.

doQmapRQUANT uses the tables of robust empirical quantiles identified using fitQmapRQUANT to transform the variable x. For values that are not in

quantile(mod,probs=seq(0,1,by=qstep)) the transformation is estimated using interpolation of the fitted values. Available interpolation options are:

type="linear": linear interpolation using approx, but using the extrapolation suggested by Boe et al. (2007) for values of x larger than max(mod) (constant correction).

type="linear2": linear interpolation using approx. For any value of x outside range(mod) the transformation is extrapolated using the slope of the local linear least squares regression at the outer most points.

type="tricube": monotonic tricubic spline interpolation using splinefun. Spline interpolation is performed using a _monotone_ Hermite spline (method="monoH.FC" in splinefun).

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<wet.day to zero.

Value

fitQmapRQUANT returns an object of class fitQmapRQUANT containing following elements:

par	A list containing:
par$modq	a matrix. Each column i corresponds to the output of quantile(mod[,i],probs=seq(0,1,by=qstep)).
par$fitq	the fitted values of the local linear least square regression corresponding to par$modq
par$slope	a matrix. the columns correspond to the columns of mod. The rows contain the slope of the "lower" and the "upper" extreme points of the local linear fit and is used for extrapolation if type="linear2".
wet.day	logical, indicating whether to perform wet day correction or not. OR a numeric threshold below which all values are set to zero.

doQmapRQUANT returns a numeric vector or matrix depending on the format of x.

Author(s)

John Bjornar Bremnes and Lukas Gudmundsson

References

Boe, J.; Terray, L.; Habets, F. & Martin, E. Statistical and dynamical downscaling of the Seine basin climate for hydro-meteorological studies. International Journal of Climatology, 2007, 27, 1643-1655, doi: 10.1002/joc.1602.

See Also

fitQmap

Examples

data(obsprecip)
data(modprecip)

## single series example
qm.fit <- fitQmapRQUANT(obsprecip[,2],modprecip[,2],
                      qstep=0.1,nboot=10,wet.day=TRUE)
qm.a <- doQmapRQUANT(modprecip[,2],qm.fit,type="linear")
qm.b <- doQmapRQUANT(modprecip[,2],qm.fit,type="tricub")

sqrtquant <- function(x,qstep=0.01){
  qq <- quantile(x,prob=seq(0,1,by=qstep))
  sqrt(qq)
}

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")
points(sqrt(qm.fit$par$modq),sqrt(qm.fit$par$fitq),
       pch=19,cex=1,col="green")
legend("topleft",
       legend=c("linear","tricub","support","data"),
       lty=c(1,1,NA,NA),pch=c(NA,NA,19,21),
       col=c("red","blue","green","black"))

qm2.fit <- fitQmapRQUANT(obsprecip,modprecip,
                       qstep=0.02,nboot=1,
                       wet.day=TRUE)
qm2 <- doQmapRQUANT(modprecip,qm2.fit,type="tricub")

op <- par(mfrow=c(1,3))
for(i in 1:3){
  plot(sqrtquant(modprecip[,i]),
       sqrtquant(obsprecip[,i]),
       main=names(qm2)[i])
  lines(sqrtquant(modprecip[,i]),
        sqrtquant(qm2[,i]),col="red")
  points(sqrt(qm2.fit$par$modq[,i]),
         sqrt(qm2.fit$par$fitq[,i]),
       pch=19,cex=0.5,col="green")

}
par(op)

---

Quantile mapping using a smoothing spline

Description

fitQmapSSPLIN fits a smoothing spline to the quantile-quantile plot of observed and modelled time series. doQmapSSPLIN uses the spline function to adjust the distribution of the modelled data to match the distribution of the observations.

Usage

fitQmapSSPLIN(obs, mod, ...)
## Default S3 method:
fitQmapSSPLIN(obs,mod,wet.day=TRUE,qstep=0.01,
spline.par,...)
## S3 method for class 'matrix'
fitQmapSSPLIN(obs, mod, ...)
## S3 method for class 'data.frame'
fitQmapSSPLIN(obs, mod, ...)
doQmapSSPLIN(x,fobj,...)
## Default S3 method:
doQmapSSPLIN(x,fobj,...)
## S3 method for class 'matrix'
doQmapSSPLIN(x,fobj,...)
## S3 method for class 'data.frame'
doQmapSSPLIN(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.
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.
qstep	NULL or a numeric value between 0 and 1. See Details.
spline.par	a named list with parameters passed to smooth.spline.
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<wet.day to zero. The transformations are then only fitted to the portion of the distributions corresponding to observed wet days.

Value

fitQmapSSPLIN returns an object of class fitQmapSSPLIN containing following elements:

par	A list containing objects of class smooth.spline.fit, which are equivalent to the value of the element fit in the output of smooth.spline. The spline coefficients are checked for monotony and adjusted if necessary by replacement with the previous value. If mod is a matrix the names of par correspond to colnames(mod).
wet.day	logical, indicating whether to perform wet day correction or not. OR a numeric threshold below which all values are set to zero.

doQmapSSPLIN returns a numeric vector or matrix depending on the format of x.

Author(s)

Lukas Gudmundsson

References

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, doi:10.5194/hess-16-3383-2012.

See Also

fitQmap, smooth.spline

Examples

data(obsprecip)
data(modprecip)

qm.a.fit <- fitQmapSSPLIN(obsprecip[,2],modprecip[,2],
                      qstep=0.01,wet.day=TRUE)
qm.a <- doQmapSSPLIN(modprecip[,2],qm.a.fit)

## example on how to use spline.par
## (this example has little effect)
qm.b.fit <- fitQmapSSPLIN(obsprecip[,2],modprecip[,2],
                          qstep=0.01,wet.day=TRUE,
                          spline.par=list(cv=TRUE))
qm.b <- doQmapSSPLIN(modprecip[,2],qm.b.fit)

sqrtquant <- function(x,qstep=0.01){
  qq <- quantile(x,prob=seq(0,1,by=qstep))
  sqrt(qq)
}

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("cv=FALSE","cv=TRUE"),
       lty=1,col=c("red","blue"))

qm2.fit <- fitQmapSSPLIN(obsprecip,modprecip,
                      qstep=0.1,wet.day=TRUE)
qm2 <- doQmapSSPLIN(modprecip,qm2.fit)


op <- par(mfrow=c(1,3))
for(i in 1:3){
  plot(sqrtquant(modprecip[,i]),
       sqrtquant(obsprecip[,i]),
       main=names(qm2)[i])
  lines(sqrtquant(modprecip[,i]),
        sqrtquant(qm2[,i]),col="red")
}
par(op)

---

Daily precipitation data at three locations in Norway.

Description

Observed (obsprecip) and simulated (modprecip) daily precipitation data for three locations in Norway covering the 1961 - 1990 period.

Usage

data(obsprecip)
data(modprecip)

Format

Data frame(s) with rows representing days and with the following 3 variables.

MOSS

Daily Precipitation at Moss [mm/day]

GEIRANGER

Daily Precipitation at Geiranger [mm/day]

BARKESTAD

Daily Precipitation at Barkestad [mm/day]

Details

The time series in obsprecip stem from the observation-system of the Norwegian Meteorological Institute.

The time series in modprecip are based on simulations of HIRHAM2/NorACIA regional climate model forced with simulation the HadAM3H. The simulation setup is further described in Forland et al. 2011. The simulations are free-running and there is consequently no direct correspondence in the temporal evolution of modprecip and obsprecip.

NOTE that all months in the modelled data (modprecip) have 30 days (in contrast to the observations (obsprecip) which have true calender days.

Source

The observations are taken form the observation network of the Norwegian meteorological institute (www.met.no). The data are available for download at http://eklima.met.no.

References

Forland, E. J.; Benestad, R.; Hanssen-Bauer, I.; Haugen, J. E. & Skaugen, T. E. Temperature and Precipitation Development at Svalbard 1900-2100. Advances in Meteorology, 2011, Volume 2011, 893790, doi: 10.1155/2011/893790.

Examples

data(obsprecip)
data(modprecip)

---

Rough parameter estimate for the Bernoulli-Exponential distribution

Description

Estimates rough starting values for the Bernoulli-Exponential distribution using the method of moments for the rate parameter. The probability of non-zero events is estimated as the fraction of values that are larger than zero.

Usage

startbernexp(x)

Arguments

x	numeric vector.

Value

A list containing:

prob	probability of non-zero event.
rate	rate parameter of the Exponential distribution.

Note

In this package startbernexp is intended to be used in conjunction with fitQmapDIST (and mledist) with parameter distr="bernexp".

Author(s)

Lukas Gudmundsson

See Also

fitQmapDIST, bernexp, fitdist

Examples

gg <- rbernexp(n=300, prob=0.2, rate=1)
startbernexp(gg)
mledist(gg,"bernexp",startbernexp(gg))

---

Rough parameter estimate for the Bernoulli-Gamma distribution

Description

Estimates rough starting values for the Bernoulli-Gamma distribution using the method of moments for the shape and the scale parameters. The probability of non-zero events is estimated as the fraction of values that are larger than zero.

Usage

startberngamma(x)

Arguments

x	numeric vector.

Value

A list containing:

prob	probability of non-zero event.
scale	scale parameter of the gamma distribution.
shape	shape parameter of the gamma distribution.

Note

In this package startberngamma is intended to be used in conjunction with fitQmapDIST (and mledist) with parameter distr="berngamma".

Author(s)

Lukas Gudmundsson

See Also

fitQmapDIST, berngamma, fitdist

Examples

gg <- rberngamma(n=300, prob=0.2, scale=1, shape=1)
startberngamma(gg)
mledist(gg,"berngamma",startberngamma(gg))

---

Rough parameter estimate for the Bernoulli-Log-Normal distribution

Description

Estimates rough starting values for the Bernoulli-Log-Normal distribution using the method of moments for the meanlog and the sdlog parameters. The probability of non-zero events is estimated as the fraction of values that are larger than zero.

Usage

startbernlnorm(x)

Arguments

x	numeric vector.

Value

A list containing:

prob	probability of non-zero event.
meanlog	meanlog parameter of the Log-Normal distribution.
sdlog	sdlog parameter of the Log-Normal distribution.

Note

In this package startbernlnorm is intended to be used in conjunction with fitQmapDIST (and mledist) with parameter distr="bernlnorm".

Author(s)

Lukas Gudmundsson

See Also

fitQmapDIST, bernlnorm, fitdist

Examples

gg <- rbernlnorm(n=300, prob=0.2, meanlog=1, sdlog=1)
startbernlnorm(gg)
mledist(gg,"bernlnorm",startbernlnorm(gg))

---

Rough parameter estimate for the Bernoulli-Weibull distribution

Description

Estimates rough starting values for the Bernoulli-Weibull distribution using the method of moments for the shape and the scale parameters. The probability of non-zero events is estimated as the fraction of values that are larger than zero.

Usage

startbernweibull(x)

Arguments

x	numeric vector.

Value

A list containing:

prob	probability of non-zero event.
scale	scale parameter of the weibull distribution.
shape	shape parameter of the weibull distribution.

Note

In this package startbernweibull is intended to be used in conjunction with fitQmapDIST (and mledist) with parameter distr="bernweibull".

Author(s)

Lukas Gudmundsson

See Also

fitQmapDIST, bernweibull, fitdist

Examples

gg <- rbernweibull(n=300, prob=0.2, scale=1, shape=1)
startbernweibull(gg)
mledist(gg,"bernweibull",startbernweibull(gg))

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