Package logitnorm. R topics documented: February 20, 2015

Transcription

1 Title Functions for the al distribution. Version Date Author Package February 20, 2015 Maintainer Density, distribution, quantile and random generation function for the al distribution. Estimation of the mode and the first two moments. Estimation of distribution parameters. Depends Suggests RUnit License GPL-2 LazyLoad yes Repository CRAN Repository/R-Forge/Project Repository/R-Forge/Revision 31 Date/Publication :45:36 NeedsCompilation no R topics documented: -package d invlogit logit modelogitnorm momentslogitnorm p q r twcoeflogitnorm

2 2 d twcoeflogitnormci twcoeflogitnorme twcoeflogitnormmle twcoeflogitnormn twsigmalogitnorm Index 15 -package Functions for the al distribution. Details Density, distribution, quantile and random generation function for the al distribution. Estimation of the mode and the first two moments. Estimation of distribution parameters. Package: Title: Functions for the al distribution. Version: Date: Author: Maintainer: <twutz@bgc-jena.mpg.de> Depends: Suggests: RUnit License: GPL-2 LazyLoad: yes d d Density function of al distribution d(q, mu = 0, sigma = 1,...)

3 invlogit 3 q mu Details sigma quantiles distribution parameters... further arguments passed to dnorm: mean, and sd for mu and sigma respectively. Logitnorm distribution density function: d distribution function: p quantile function: q random generation function: r Transformation functions (0,1) -> (-Inf,Inf): logit (-Inf,Inf) -> (0,1): invlogit Moments and mode Expected value and variance: momentslogitnorm Mode: modelogitnorm Estimating parameters from mode and upper quantile: twcoeflogitnormmle from median and upper quantile: twcoeflogitnorm from expected value, i.e. mean and upper quantile: twcoeflogitnorme from a confidence interval which is symmetric at normal scale: twcoeflogitnormci from prescribed quantiles: twcoeflogitnormn invlogit invlogit Transforming (-Inf,Inf) to original scale (0,1) invlogit(q,...) q...

4 4 logit Details function f(z) = ez e z +1 = 1 1+e z logit logit logit Transforming (0,1) to normal scale (-Inf Inf) logit(p,...) p... Details function logit(p) = log ( ) p 1 p = log(p) log(1 p) invlogit

5 modelogitnorm 5 modelogitnorm modelogitnorm Mode of the al distribution by numerical optimization modelogitnorm(mu, sigma, tol = invlogit(mu)/1000) mu sigma tol parameter mu parameter sigma precisions of the estimate momentslogitnorm momentslogitnorm First two moments of the al distribution by numerical integration momentslogitnorm(mu, sigma, abs.tol = 0,...) mu parameter mu sigma parameter sigma abs.tol chaning default to integrate... further parameters to the integrate function

6 6 p Value named numeric vector with components mean: expected value, i.e. first moment var: variance, i.e. second moment Examples (res <- momentslogitnorm(4,1)) (res <- momentslogitnorm(5,0.1)) p p Distribution function for al distribution p(q, mu = 0, sigma = 1,...) q mu sigma... distribution parameters

7 q 7 q q Quantiles of al distribution. q(p, mu = 0, sigma = 1,...) p mu sigma... distribution parameters r r Random number generation for al distribution r(mu = 0, sigma = 1,...) mu distribution parameters sigma... arguments to rnorm

8 8 twcoeflogitnorm twcoeflogitnorm twcoeflogitnorm Estimating coefficients of al distribution from median and upper quantile twcoeflogitnorm(median, quant, perc = 0.975, method = "BFGS", theta0 = c(mu = 0, sigma = 1), returndetails = FALSE,...) median quant perc method theta0 returndetails... numeric vector: the median of the density function numeric vector: the upper quantile value numeric vector: the probability for which the quantile was specified method of optimization (see optim) starting parameters if TRUE, the full output of optim is attached as attributes resoptim Value numeric matrix with columns c("mu","sigma") rows correspond to rows in median, quant, and perc

9 twcoeflogitnormci 9 Examples # estimate the parameters, with median at 0.7 and upper quantile at 0.9 (theta <- twcoeflogitnorm(0.7,0.9)) x <- seq(0,1,length.out=41)[-c(1,41)] # plotting grid px <- p(x,mu=theta[1],sigma=theta[2]) #percentiles function plot(px~x); abline(v=c(0.7,0.9),col="gray"); abline(h=c(0.5,0.975),col="gray") dx <- d(x,mu=theta[1],sigma=theta[2]) #density function plot(dx~x); abline(v=c(0.7,0.9),col="gray") # vectorized (theta <- twcoeflogitnorm(seq(0.4,0.8,by=0.1),0.9)) twcoeflogitnormci twcoeflogitnormci Calculates mu and sigma of the al distribution from lower and upper quantile, i.e. confidence interval. twcoeflogitnormci(lower, upper, perc = 0.975, sigmafac = qnorm(perc), istransscale = FALSE) lower upper perc sigmafac istransscale value at the lower quantile, i.e. practical minimum value at the upper quantile, i.e. practical maximum numeric vector: the probability for which the quantile was specified sigmafac=2 is 95% sigmafac=2.6 is 99% interval if true lower and upper are already on logit scale Value named numeric vector: mu and sigma parameter of the al distribution.

10 10 twcoeflogitnorme Examples mu=2 sd=c(1,0.8) p=0.99 lower <- l <- q(1-p, mu, sd ) # p-confidence interval upper <- u <- q(p, mu, sd ) # p-confidence interval cf <- twcoeflogitnormci(lower,upper) all.equal( cf[,"mu"], c(mu,mu) ) all.equal( cf[,"sigma"], sd ) twcoeflogitnorme twcoeflogitnorme Estimating coefficients of al distribution from expected value, i.e. mean, and upper quantile. twcoeflogitnorme(mean, quant, perc = c(0.975), method = "BFGS", theta0 = c(mu = 0, sigma = 1), returndetails = FALSE,...) mean quant perc method theta0 returndetails... the expected value of the density function the quantile values the probabilites for which the quantiles were specified method of optimization (see optim) starting parameters if TRUE, the full output of optim is returned with attribut resoptim Value named numeric matrix with estimated parameters of the al distrubtion. colnames: c("mu","sigma")

11 twcoeflogitnormmle 11 Examples # estimate the parameters (thetae <- twcoeflogitnorme(0.7,0.9)) x <- seq(0,1,length.out=41)[-c(1,41)] # plotting grid px <- p(x,mu=thetae[1],sigma=thetae[2]) #percentiles function plot(px~x); abline(v=c(0.7,0.9),col="gray"); abline(h=c(0.5,0.975),col="gray") dx <- d(x,mu=thetae[1],sigma=thetae[2]) #density function plot(dx~x); abline(v=c(0.7,0.9),col="gray") z <- r(1e5, mu=thetae[1],sigma=thetae[2]) mean(z) # about 0.7 # vectorized (theta <- twcoeflogitnorme(mean=seq(0.4,0.8,by=0.1),quant=0.9)) twcoeflogitnormmle twcoeflogitnormmle Estimating coefficients of al distribution from mode and upper quantile twcoeflogitnormmle(mle, quant, perc = 0.999) mle quant perc numeric vector: the mode of the density function numeric vector: the upper quantile value numeric vector: the probability for which the quantile was specified Value numeric matrix with columns c("mu","sigma") rows correspond to rows in mle, quant, and perc

12 12 twcoeflogitnormn Examples # estimate the parameters, with mode 0.7 and upper quantile 0.9 (theta <- twcoeflogitnormmle(0.7,0.9)) x <- seq(0,1,length.out=41)[-c(1,41)] # plotting grid px <- p(x,mu=theta[1],sigma=theta[2]) #percentiles function plot(px~x); abline(v=c(0.7,0.9),col="gray"); abline(h=c(0.999),col="gray") dx <- d(x,mu=theta[1],sigma=theta[2]) #density function plot(dx~x); abline(v=c(0.7,0.9),col="gray") # vectorized (theta <- twcoeflogitnormmle(mle=seq(0.4,0.8,by=0.1),quant=0.9)) twcoeflogitnormn twcoeflogitnormn Estimating coefficients of al distribution from a vector of quantiles and perentiles (nonvectorized). twcoeflogitnormn(quant, perc = c(0.5, 0.975), method = "BFGS", theta0 = c(mu = 0, sigma = 1), returndetails = FALSE,...) Value quant perc method theta0 returndetails the quantile values the probabilites for which the quantiles were specified method of optimization (see optim) starting parameters if TRUE, the full output of optim is returned instead of only entry par... further parameters passed to optim, e.g. control=list(maxit=1000) named numeric vector with estimated parameters of the al distrubtion. names: c("mu","sigma")

13 twsigmalogitnorm 13 Examples # estimate the parameters quant=c(0.7,0.8,0.9) perc=c(0.5,0.75,0.975) (theta <- twcoeflogitnormn( quant=quant, perc=perc )) x <- seq(0,1,length.out=41)[-c(1,41)] # plotting grid px <- p(x,mu=theta[1],sigma=theta[2]) #percentiles function plot(px~x); abline(v=quant,col="gray"); abline(h=perc,col="gray") twsigmalogitnorm twsigmalogitnorm Estimating coefficients of al distribution from mode and given mu twsigmalogitnorm(mle, mu = 0) mle mu numeric vector: the mode of the density function for mu=0 the distribution will be the flattest case (maybe bimodal) Value numeric matrix with columns c("mu","sigma") rows correspond to rows in mle and mu Examples # estimate the parameters, with mode 0.7 and upper quantile 0.9 (theta <- twcoeflogitnormmle(0.7,0.9)) x <- seq(0,1,length.out=41)[-c(1,41)] # plotting grid px <- p(x,mu=theta[1],sigma=theta[2]) #percentiles function plot(px~x); abline(v=c(0.7,0.9),col="gray"); abline(h=c(0.999),col="gray") dx <- d(x,mu=theta[1],sigma=theta[2]) #density function plot(dx~x); abline(v=c(0.7,0.9),col="gray")

14 14 twsigmalogitnorm # vectorized (theta <- twcoeflogitnormmle(mle=seq(0.4,0.8,by=0.1),quant=0.9))

15 Index Topic package -package, 2 d, 2 dnorm, 3 integrate, 5 invlogit, 3, 3, 4 logit, 3, 4, 4, 4 13 (d), 2 -package, 2 modelogitnorm, 3, 5 momentslogitnorm, 3, 5 optim, 8, 10, 12 p, 3, 6 q, 3, 7 r, 3, 7 rnorm, 7 twcoeflogitnorm, 3, 8 twcoeflogitnormci, 3, 9 twcoeflogitnorme, 3, 10 twcoeflogitnormmle, 3, 11 twcoeflogitnormn, 3, 12 twsigmalogitnorm, 13 15