Distributions that are sometimes used in hydrology

Description

This package contains several distributions that are sometimes useful in hydrology

Author(s)

Francois Aucoin

Maintainer: Thomas Petzoldt <thomas.petzoldt@tu-dresden.de> in agreement with the original author.

Internal functions of FAdist

Description

Internal functions of the package.

Details

These are not to be called by the user.

Three-Parameter Gamma Distribution (also known as Pearson type III distribution)

Description

Density, distribution function, quantile function and random generation for the 3-parameter gamma distribution with shape, scale, and threshold (or shift) parameters equal to shape, scale, and thres, respectively.

Usage

dgamma3(x,shape=1,scale=1,thres=0,log=FALSE)
pgamma3(q,shape=1,scale=1,thres=0,lower.tail=TRUE,log.p=FALSE)
qgamma3(p,shape=1,scale=1,thres=0,lower.tail=TRUE,log.p=FALSE)
rgamma3(n,shape=1,scale=1,thres=0)

Arguments

Details

If Y is a random variable distributed according to a gamma distribution (with shape and scale parameters), then X = Y+m has a 3-parameter gamma distribution with the same shape and scale parameters, and with threshold (or shift) parameter m.

Value

dgamma3 gives the density, pgamma3 gives the distribution function, qgamma3 gives the quantile function, and rgamma3 generates random deviates.

References

Bobee, B. and F. Ashkar (1991). The Gamma Family and Derived Distributions Applied in Hydrology. Water Resources Publications, Littleton, Colo., 217 p.

See Also

dgamma, pgamma, qgamma, rgamma

Examples

thres <- 10
x <- rgamma3(n=10,shape=2,scale=11,thres=thres)
dgamma3(x,2,11,thres)
dgamma(x-thres,2,1/11)

Generalized Extreme Value Distribution (for maxima)

Description

Density, distribution function, quantile function and random generation for the generalized extreme value distribution (for maxima) with shape, scale, and location parameters equal to shape, scale, and location, respectively.

Usage

dgev(x,shape=1,scale=1,location=0,log=FALSE)
pgev(q,shape=1,scale=1,location=0,lower.tail=TRUE,log.p=FALSE)
qgev(p,shape=1,scale=1,location=0,lower.tail=TRUE,log.p=FALSE)
rgev(n,shape=1,scale=1,location=0)

Arguments

Details

If X is a random variable distributed according to a generalized extreme value distribution, it has density
f(x) = 1/scale*(1+shape*((x-location)/scale))^(-1/shape-1)*exp(-(1+shape*((x-location)/scale))^(-1/shape))

Value

dgev gives the density, pgev gives the distribution function, qgev gives the quantile function, and rgev generates random deviates.

References

Coles, S. (2001) An introduction to statistical modeling of extreme values. Springer

Examples

x <- rgev(1000,-.1,3,100)
hist(x,freq=FALSE,col='gray',border='white')
curve(dgev(x,-.1,3,100),add=TRUE,col='red4',lwd=2)

Gumbel Distribution (for maxima)

Description

Density, distribution function, quantile function and random generation for the Gumbel distribution (for maxima) with scale and location parameters equal to scale and location, respectively.

Usage

dgumbel(x,scale=1,location=0,log=FALSE)
pgumbel(q,scale=1,location=0,lower.tail=TRUE,log.p=FALSE)
qgumbel(p,scale=1,location=0,lower.tail=TRUE,log.p=FALSE)
rgumbel(n,scale=1,location=0)

Arguments

Details

If X is a random variable distributed according to a Gumbel distribution, it has density
f(x) = 1/scale*exp(-(x-location)/scale-exp(-(x-location)/scale))

Value

dgumbel gives the density, pgumbel gives the distribution function, qgumbel gives the quantile function, and rgumbel generates random deviates.

References

Coles, S. (2001) An introduction to statistical modeling of extreme values. Springer

Examples

x <- rgumbel(1000,3,100)
hist(x,freq=FALSE,col='gray',border='white')
curve(dgumbel(x,3,100),add=TRUE,col='red4',lwd=2)

Generalized Pareto Distribution

Description

Density, distribution function, quantile function and random generation for the generalized Pareto distribution with shape and scale parameters equal to shape and scale, respectively.

Usage

dgp(x,shape=1,scale=1,log=FALSE)
pgp(q,shape=1,scale=1,lower.tail=TRUE,log.p=FALSE)
qgp(p,shape=1,scale=1,lower.tail=TRUE,log.p=FALSE)
rgp(n,shape=1,scale=1)

Arguments

Details

If X is a random variable distributed according to a generalized Pareto distribution, it has density
f(x) = 1/scale*(1-shape*x/scale)^((1-shape)/shape)

Value

dgp gives the density, pgp gives the distribution function, qgp gives the quantile function, and rgp generates random deviates.

References

Coles, S. (2001) An introduction to statistical modeling of extreme values. Springer

Examples

x <- rgp(1000,-.2,10)
hist(x,freq=FALSE,col='gray',border='white')
curve(dgp(x,-.2,10),add=TRUE,col='red4',lwd=2)

Kappa Distribution

Description

Density, distribution function, quantile function and random generation for the kappa distribution with shape and scale parameters equal to shape and scale, respectively.

Usage

dkappa(x,shape=1,scale=1,log=FALSE)
pkappa(q,shape=1,scale=1,lower.tail=TRUE,log.p=FALSE)
qkappa(p,shape=1,scale=1,lower.tail=TRUE,log.p=FALSE)
rkappa(n,shape=1,scale=1)

Arguments

Details

If X is a random variable distributed according to a kappa distribution, it has density
f(x) = shape/scale*(shape+(x/scale)^shape)^(-(shape+1)/shape)

Value

dkappa gives the density, pkappa gives the distribution function, qkappa gives the quantile function, and rkappa generates random deviates.

Examples

x <- rkappa(1000,12,10)
hist(x,freq=FALSE,col='gray',border='white')
curve(dkappa(x,12,10),add=TRUE,col='red4',lwd=2)

Four-Parameter Kappa Distribution

Description

Density, distribution function, quantile function and random generation for the four-parameter kappa distribution with shape1, shape2, scale, and location parameters equal to shape1, shape2, scale, and location, respectively.

Usage

dkappa4(x,shape1,shape2,scale=1,location=0,log=FALSE)
pkappa4(q,shape1,shape2,scale=1,location=0,lower.tail=TRUE,log.p=FALSE)
qkappa4(p,shape1,shape2,scale=1,location=0,lower.tail=TRUE,log.p=FALSE)
rkappa4(n,shape1,shape2,scale=1,location=0)

Arguments

Details

See References

Value

dkappa4 gives the density, pkappa4 gives the distribution function, qkappa4 gives the quantile function, and rkappa4 generates random deviates.

References

Hosking, J.R.M. (1994). The four-parameter kappa distribution. IBM Journal of Research and Development, 38(3), 251-258.

Examples

x <- rkappa4(1000,.1,.2,12,110)
hist(x,freq=FALSE,col='gray',border='white')
curve(dkappa4(x,.1,.2,12,110),add=TRUE,col='red4',lwd=2)

Log-Pearson Type III Distribution

Description

Density, distribution function, quantile function and random generation for the log-Pearson type III distribution with shape1, shape2, and scale parameters equal to shape, scale, and thres, respectively.

Usage

dlgamma3(x,shape=1,scale=1,thres=1,log=FALSE)
plgamma3(q,shape=1,scale=1,thres=1,lower.tail=TRUE,log.p=FALSE)
qlgamma3(p,shape=1,scale=1,thres=1,lower.tail=TRUE,log.p=FALSE)
rlgamma3(n,shape=1,scale=1,thres=1)

Arguments

Details

If Y is a random variable distributed according to a gamma distribution (with shape and scale parameters), then X = exp(Y+m) has a log-Pearson type III distribution with shape1 and shape2 parameters corresponding to the shape and 1/scale parameteres of Y, and with scale parameter m.

Value

dlgamma3 gives the density, plgamma3 gives the distribution function, qlgamma3 gives the quantile function, and rlgamma3 generates random deviates.

References

BOBEE, B. and F. ASHKAR (1991). The Gamma Family and Derived Distributions Applied in Hydrology. Water Resources Publications, Littleton, Colo., 217 p.

See Also

dgamma, pgamma, qgamma, rgamma, dgamma3, pgamma3, qgamma3, rgamma3

Examples

thres <- 10
x <- rlgamma3(n=10,shape=2,scale=11,thres=thres)
dlgamma3(x,2,11,thres)
dgamma3(log(x),2,1/11,thres)/x
dgamma(log(x)-thres,2,11)/x

Log-Logistic Distribution

Description

Density, distribution function, quantile function and random generation for the log-logistic distribution with shape and scale parameters equal to shape and scale, respectively.

Usage

dllog(x,shape=1,scale=1,log=FALSE)
pllog(q,shape=1,scale=1,lower.tail=TRUE,log.p=FALSE)
qllog(p,shape=1,scale=1,lower.tail=TRUE,log.p=FALSE)
rllog(n,shape=1,scale=1)

Arguments

Details

If Y is a random variable distributed according to a logistic distribution (with location and scale parameters), then X = exp(Y) has a log-logistic distribution with shape and scale parameters corresponding to the scale and location parameteres of Y, respectively.

Value

dllog gives the density, pllog gives the distribution function, qllog gives the quantile function, and rllog generates random deviates.

See Also

dlogis, plogis, qlogis, rlogis

Examples

x <- rllog(10,1,0)
dllog(x,1,0)
dlogis(log(x),0,1)/x

Three-Parameter Log-Logistic Distribution

Description

Density, distribution function, quantile function and random generation for the 3-parameter log-logistic distribution with shape, scale, and threshold (or shift) parameters equal to shape, scale, and thres, respectively.

Usage

dllog3(x,shape=1,scale=1,thres=0,log=FALSE)
pllog3(q,shape=1,scale=1,thres=0,lower.tail=TRUE,log.p=FALSE)
qllog3(p,shape=1,scale=1,thres=0,lower.tail=TRUE,log.p=FALSE)
rllog3(n,shape=1,scale=1,thres=0)

Arguments

Details

If Y is a random variable distributed according to a logistic distribution (with location and scale parameters), then X = exp(Y)+m has a 3-parameter log-logistic distribution with shape and scale parameters corresponding to the scale and location parameteres of Y, respectively; and threshold parameter m.

Value

dllog3 gives the density, pllog3 gives the distribution function, qllog3 gives the quantile function, and rllog3 generates random deviates.

See Also

dlogis, plogis, qlogis, rlogis, dllog, pllog, qllog, rllog

Examples

m <- 100
x <- rllog3(10,1,0,m)
dllog3(x,1,0,m)
dllog(x-m,1,0)
dlogis(log(x-m),0,1)/(x-m)

Three-Parameter Lognormal Distribution

Description

Density, distribution function, quantile function and random generation for the 3-parameter lognormal distribution with shape, scale, and threshold (or shift) parameters equal to shape, scale, and thres, respectively.

Usage

dlnorm3(x,shape=1,scale=1,thres=0,log=FALSE)
plnorm3(q,shape=1,scale=1,thres=0,lower.tail=TRUE,log.p=FALSE)
qlnorm3(p,shape=1,scale=1,thres=0,lower.tail=TRUE,log.p=FALSE)
rlnorm3(n,shape=1,scale=1,thres=0)

Arguments

Details

If Y is a random variable distributed according to a normal distribution (with location(mean) and scale(standard deviation) parameters), then X = exp(Y)+m has a 3-parameter lognormal distribution with shape and scale parameters corresponding to the scale and location parameteres of Y, respectively; and threshold parameter m.

Value

dlnorm3 gives the density, plnorm3 gives the distribution function, qlnorm3 gives the quantile function, and rlnorm3 generates random deviates.

See Also

dnorm, pnorm, qnorm, rnorm, dlnorm, plnorm, qlnorm, rlnorm

Examples

m <- 100
x <- rlnorm3(10,1,0,m)
dlnorm3(x,1,0,m)
dlnorm(x-m,0,1)
dnorm(log(x-m),0,1)/(x-m)

Three-Parameter Weibull Distribution

Description

Density, distribution function, quantile function and random generation for the 3-parameter Weibull distribution with shape, scale, and threshold (or shift) parameters equal to shape, scale, and thres, respectively.

Usage

dweibull3(x,shape,scale=1,thres=0,log=FALSE)
pweibull3(q,shape,scale=1,thres=0,lower.tail=TRUE,log.p=FALSE)
qweibull3(p,shape,scale=1,thres=0,lower.tail=TRUE,log.p=FALSE)
rweibull3(n,shape,scale=1,thres=0)

Arguments

Details

If Y is a random variable distributed according to a Weibull distribution (with shape and scale parameters), then X = Y+m has a 3-parameter Weibull distribution with shape and scale parameters corresponding to the shape and scale parameteres of Y, respectively; and threshold parameter m.

Value

dweibull3 gives the density, pweibull3 gives the distribution function, qweibull3 gives the quantile function, and rweibull3 generates random deviates.

See Also

dweibull, pweibull, qweibull, rweibull

Examples

m <- 100
x <- rweibull3(10,3,1,m)
dweibull3(x,3,1,m)
dweibull(x-m,3,1)