fitdist: Fit of univariate distributions to non-censored data

Description

Fit of univariate distributions to non-censored data by maximum likelihood (mle), moment matching (mme), quantile matching (qme) or maximizing goodness-of-fit estimation (mge). The latter is also known as minimizing distance estimation. Generic methods are print, plot, summary, quantile, logLik, vcov and coef.

Usage

fitdist(data, distr, method = c("mle", "mme", "qme", "mge"),  start=NULL, fix.arg=NULL, discrete, keepdata = TRUE, keepdata.nb=100, ...) 
"print"(x, ...)
"plot"(x, breaks="default", ...)
"summary"(object, ...)
"logLik"(object, ...)
"vcov"(object, ...)
"coef"(object, ...)

Arguments

data

A numeric vector.

distr

A character string "name" naming a distribution for which the corresponding density function dname, the corresponding distribution function pname and the corresponding quantile function qname must be defined, or directly the density function.

method

A character string coding for the fitting method: "mle" for 'maximum likelihood estimation', "mme" for 'moment matching estimation', "qme" for 'quantile matching estimation' and "mge" for 'maximum goodness-of-fit estimation'.

start

A named list giving the initial values of parameters of the named distribution or a function of data computing initial values and returning a named list. This argument may be omitted (default) for some distributions for which reasonable starting values are computed (see the 'details' section of mledist). It may not be into account for closed-form formulas.

fix.arg

An optional named list giving the values of fixed parameters of the named distribution or a function of data computing (fixed) parameter values and returning a named list. Parameters with fixed value are thus NOT estimated by this maximum likelihood procedure. The use of this argument is not possible if method="mme" and a closed-form formula is used.

An object of class "fitdist".

object

An object of class "fitdist".

breaks

If "default" the histogram is plotted with the function hist with its default breaks definition. Else breaks is passed to the function hist. This argument is not taken into account with discrete distributions: "binom", "nbinom", "geom", "hyper" and "pois".

keepdata

a logical. If TRUE, dataset is returned, otherwise only a sample subset is returned.

keepdata.nb

When keepdata=FALSE, the length (>1) of the subset returned.

discrete

If TRUE, the distribution is considered as discrete. If discrete is missing, discrete is automaticaly set to TRUE when distr belongs to "binom", "nbinom", "geom", "hyper" or "pois" and to FALSE in the other cases. It is thus recommended to enter this argument when using another discrete distribution. This argument will not directly affect the results of the fit but will be passed to functions gofstat, plotdist and cdfcomp.

...

Further arguments to be passed to generic functions, or to one of the functions "mledist", "mmedist", "qmedist" or "mgedist" depending of the chosen method. See mledist, mmedist, qmedist, mgedist for details on parameter estimation.

Value

fitdist returns an object of class "fitdist", a list with the following components:Generic functions:

Details

It is assumed that the distr argument specifies the distribution by the probability density function, the cumulative distribution function and the quantile function (d, p, q).

The four possible fitting methods are described below:

By default, direct optimization of the log-likelihood (or other criteria depending of the chosen method) is performed using optim, with the "Nelder-Mead" method for distributions characterized by more than one parameter and the "BFGS" method for distributions characterized by only one parameter. The optimization algorithm used in optim can be chosen or another optimization function can be specified using ... argument (see mledist for details). start may be omitted (i.e. NULL) for some classic distributions (see the 'details' section of mledist). Note that when errors are raised by optim, it's a good idea to start by adding traces during the optimization process by adding control=list(trace=1, REPORT=1) in ... argument. Once the parameter(s) is(are) estimated, fitdist computes the log-likelihood for every estimation method and for maximum likelihood estimation the standard errors of the estimates calculated from the Hessian at the solution found by optim or by the user-supplied function passed to mledist.

By default (keepdata = TRUE), the object returned by fitdist contains the data vector given in input. When dealing with large datasets, we can remove the original dataset from the output by setting keepdata = FALSE. In such a case, only keepdata.nb points (at most) are kept by random subsampling keepdata.nb-2 points from the dataset and adding the minimum and the maximum. If combined with bootdist, and use with non-parametric bootstrap be aware that bootstrap is performed on the subset randomly selected in fitdist. Currently, the graphical comparisons of multiple fits is not available in this framework. Weighted version of the estimation process is available for method = "mle", "mme", "qme" by using weights=.... See the corresponding man page for details. Weighted maximum GOF estimation (when method = "mge") is not allowed. It is not yet possible to take into account weighths in functions plotdist, plot.fitdist, cdfcomp, denscomp, ppcomp, qqcomp, gofstat and descdist (developments planned in the future).

NB: if your data values are particularly small or large, a scaling may be needed before the optimization process. See example (14) in this man page and examples (14,15) in the test file of the package. Please also take a look at the Rmpfr package available on CRAN for numerical accuracy issues.

References

Cullen AC and Frey HC (1999), Probabilistic techniques in exposure assessment. Plenum Press, USA, pp. 81-155.

Venables WN and Ripley BD (2002), Modern applied statistics with S. Springer, New York, pp. 435-446.

Vose D (2000), Risk analysis, a quantitative guide. John Wiley & Sons Ltd, Chischester, England, pp. 99-143.

Delignette-Muller ML and Dutang C (2015), fitdistrplus: An R Package for Fitting Distributions. Journal of Statistical Software, 64(4), 1-34.

Examples

Run this code


# (1) fit of a gamma distribution by maximum likelihood estimation
#

data(groundbeef)
serving <- groundbeef$serving
fitg <- fitdist(serving, "gamma")
summary(fitg)
plot(fitg)
plot(fitg, demp = TRUE)
plot(fitg, histo = FALSE, demp = TRUE)
cdfcomp(fitg, addlegend=FALSE)
denscomp(fitg, addlegend=FALSE)
ppcomp(fitg, addlegend=FALSE)
qqcomp(fitg, addlegend=FALSE)


# (2) use the moment matching estimation (using a closed formula)
#

fitgmme <- fitdist(serving, "gamma", method="mme")
summary(fitgmme)

# (3) Comparison of various fits 
#

fitW <- fitdist(serving, "weibull")
fitg <- fitdist(serving, "gamma")
fitln <- fitdist(serving, "lnorm")
summary(fitW)
summary(fitg)
summary(fitln)
cdfcomp(list(fitW, fitg, fitln), legendtext=c("Weibull", "gamma", "lognormal"))
denscomp(list(fitW, fitg, fitln), legendtext=c("Weibull", "gamma", "lognormal"))
qqcomp(list(fitW, fitg, fitln), legendtext=c("Weibull", "gamma", "lognormal"))
ppcomp(list(fitW, fitg, fitln), legendtext=c("Weibull", "gamma", "lognormal"))
gofstat(list(fitW, fitg, fitln), fitnames=c("Weibull", "gamma", "lognormal"))

# (4) defining your own distribution functions, here for the Gumbel distribution
# for other distributions, see the CRAN task view 
# dedicated to probability distributions
#

dgumbel <- function(x, a, b) 1/b*exp((a-x)/b)*exp(-exp((a-x)/b))
pgumbel <- function(q, a, b) exp(-exp((a-q)/b))
qgumbel <- function(p, a, b) a-b*log(-log(p))

fitgumbel <- fitdist(serving, "gumbel", start=list(a=10, b=10))
summary(fitgumbel)
plot(fitgumbel)

# (5) fit discrete distributions (Poisson and negative binomial)
#

data(toxocara)
number <- toxocara$number
fitp <- fitdist(number,"pois")
summary(fitp)
plot(fitp)
fitnb <- fitdist(number,"nbinom")
summary(fitnb)
plot(fitnb)

cdfcomp(list(fitp,fitnb))
gofstat(list(fitp,fitnb))

# (6) how to change the optimisation method?
#

data(groundbeef)
serving <- groundbeef$serving
fitdist(serving, "gamma", optim.method="Nelder-Mead")
fitdist(serving, "gamma", optim.method="BFGS") 
fitdist(serving, "gamma", optim.method="SANN")

# (7) custom optimization function
#

#create the sample
set.seed(1234)
mysample <- rexp(100, 5)
mystart <- list(rate=8)

res1 <- fitdist(mysample, dexp, start= mystart, optim.method="Nelder-Mead")

#show the result
summary(res1)

#the warning tell us to use optimise, because the Nelder-Mead is not adequate.

#to meet the standard 'fn' argument and specific name arguments, we wrap optimize, 
myoptimize <- function(fn, par, ...) 
{
    res <- optimize(f=fn, ..., maximum=FALSE)  
    #assume the optimization function minimize
    
    standardres <- c(res, convergence=0, value=res$objective, 
        par=res$minimum, hessian=NA)
    
    return(standardres)
}

#call fitdist with a 'custom' optimization function
res2 <- fitdist(mysample, "exp", start=mystart, custom.optim=myoptimize, 
    interval=c(0, 100))

#show the result
summary(res2)


# (8) custom optimization function - another example with the genetic algorithm
#
## Not run: 
#     #set a sample
#     fit1 <- fitdist(serving, "gamma")
#     summary(fit1)
# 
#     #wrap genoud function rgenoud package
#     mygenoud <- function(fn, par, ...) 
#     {
#         require(rgenoud)
#         res <- genoud(fn, starting.values=par, ...)        
#         standardres <- c(res, convergence=0)
#             
#         return(standardres)
#     }
# 
#     #call fitdist with a 'custom' optimization function
#     fit2 <- fitdist(serving, "gamma", custom.optim=mygenoud, nvars=2,    
#         Domains=cbind(c(0, 0), c(10, 10)), boundary.enforcement=1, 
#         print.level=1, hessian=TRUE)
# 
#     summary(fit2)
# ## End(Not run)

# (9) estimation of the standard deviation of a gamma distribution 
# by maximum likelihood with the shape fixed at 4 using the argument fix.arg
#

data(groundbeef)
serving <- groundbeef$serving
f1c  <- fitdist(serving,"gamma",start=list(rate=0.1),fix.arg=list(shape=4))
summary(f1c)
plot(f1c)

# (10) fit of a Weibull distribution to serving size data 
# by maximum likelihood estimation
# or by quantile matching estimation (in this example 
# matching first and third quartiles)
#

data(groundbeef)
serving <- groundbeef$serving
fWmle <- fitdist(serving, "weibull")
summary(fWmle)
plot(fWmle)
gofstat(fWmle)

fWqme <- fitdist(serving, "weibull", method="qme", probs=c(0.25, 0.75))
summary(fWqme)
plot(fWqme)
gofstat(fWqme)


# (11) Fit of a Pareto distribution by numerical moment matching estimation
#

## Not run: 
#     require(actuar)
#     #simulate a sample
#     x4 <- rpareto(1000, 6, 2)
# 
#     #empirical raw moment
#     memp <- function(x, order) mean(x^order)
# 
#     #fit
#     fP <- fitdist(x4, "pareto", method="mme", order=c(1, 2), memp="memp", 
#       start=list(shape=10, scale=10), lower=1, upper=Inf)
#     summary(fP)
#     plot(fP)
# 
# ## End(Not run)

# (12) Fit of a Weibull distribution to serving size data by maximum 
# goodness-of-fit estimation using all the distances available
# 

data(groundbeef)
serving <- groundbeef$serving
(f1 <- fitdist(serving, "weibull", method="mge", gof="CvM"))
(f2 <- fitdist(serving, "weibull", method="mge", gof="KS"))
(f3 <- fitdist(serving, "weibull", method="mge", gof="AD"))
(f4 <- fitdist(serving, "weibull", method="mge", gof="ADR"))
(f5 <- fitdist(serving, "weibull", method="mge", gof="ADL"))
(f6 <- fitdist(serving, "weibull", method="mge", gof="AD2R"))
(f7 <- fitdist(serving, "weibull", method="mge", gof="AD2L"))
(f8 <- fitdist(serving, "weibull", method="mge", gof="AD2"))
cdfcomp(list(f1, f2, f3, f4, f5, f6, f7, f8))
cdfcomp(list(f1, f2, f3, f4, f5, f6, f7, f8), 
  xlogscale=TRUE, xlim=c(8, 250), verticals=TRUE)
denscomp(list(f1, f2, f3, f4, f5, f6, f7, f8))

# (13) Fit of a uniform distribution using maximum likelihood 
# (a closed formula is used in this special case where the loglikelihood is not defined),
# or maximum goodness-of-fit with Cramer-von Mises or Kolmogorov-Smirnov distance
# 

set.seed(1234)
u <- runif(50, min=5, max=10)

fumle <- fitdist(u, "unif", method="mle")
summary(fumle)
plot(fumle)
gofstat(fumle)

fuCvM <- fitdist(u, "unif", method="mge", gof="CvM")
summary(fuCvM)
plot(fuCvM)
gofstat(fuCvM)

fuKS <- fitdist(u, "unif", method="mge", gof="KS")
summary(fuKS)
plot(fuKS)
gofstat(fuKS)

# (14) scaling problem
# the simulated dataset (below) has particularly small values, hence without scaling (10^0),
# the optimization raises an error. The for loop shows how scaling by 10^i
# for i=1,...,6 makes the fitting procedure work correctly.

set.seed(1234)
x2 <- rnorm(100, 1e-4, 2e-4)

for(i in 0:6)
        cat(i, try(fitdist(x2*10^i, "cauchy", method="mle")$estimate, silent=TRUE), "\n")

# (15) Fit of a normal distribution on acute toxicity values of endosulfan in log10 for
# nonarthropod invertebrates, using maximum likelihood estimation
# to estimate what is called a species sensitivity distribution 
# (SSD) in ecotoxicology, followed by estimation of the 5 percent quantile value of 
# the fitted distribution (which is called the 5 percent hazardous concentration, HC5,
# in ecotoxicology) and estimation of other quantiles.
#
data(endosulfan)
ATV <- subset(endosulfan, group == "NonArthroInvert")$ATV
log10ATV <- log10(subset(endosulfan, group == "NonArthroInvert")$ATV)
fln <- fitdist(log10ATV, "norm")

quantile(fln, probs = 0.05)
quantile(fln, probs = c(0.05, 0.1, 0.2))


# (16) Fit of a triangular distribution using Cramer-von Mises or
# Kolmogorov-Smirnov distance
# 

## Not run: 
# set.seed(1234)
# require(mc2d)
# t <- rtriang(100, min=5, mode=6, max=10)
# fCvM <- fitdist(t, "triang", method="mge", start = list(min=4, mode=6,max=9), gof="CvM")
# fKS <- fitdist(t, "triang", method="mge", start = list(min=4, mode=6,max=9), gof="KS")
# cdfcomp(list(fCvM,fKS))
# ## End(Not run)

# (17) fit a non classical discrete distribution (the zero inflated Poisson distribution)
#
## Not run: 
# require(gamlss.dist)
# set.seed(1234)
# x <- rZIP(n = 30, mu = 5, sigma = 0.2)
# plotdist(x, discrete = TRUE)
# fitzip <- fitdist(x, "ZIP", start =  list(mu = 4, sigma = 0.15), discrete = TRUE, 
#   optim.method = "L-BFGS-B", lower = c(0, 0), upper = c(Inf, 1))
# summary(fitzip)
# plot(fitzip)
# fitp <- fitdist(x, "pois")
# cdfcomp(list(fitzip, fitp))
# gofstat(list(fitzip, fitp))
# ## End(Not run)



# (18) examples with distributions in actuar (predefined starting values)
#
## Not run: 
# require(actuar)
# x <- c(2.3,0.1,2.7,2.2,0.4,2.6,0.2,1.,7.3,3.2,0.8,1.2,33.7,14.,
#        21.4,7.7,1.,1.9,0.7,12.6,3.2,7.3,4.9,4000.,2.5,6.7,3.,63.,
#        6.,1.6,10.1,1.2,1.5,1.2,30.,3.2,3.5,1.2,0.2,1.9,0.7,17.,
#        2.8,4.8,1.3,3.7,0.2,1.8,2.6,5.9,2.6,6.3,1.4,0.8)
# #log logistic
# ft_llogis <- fitdist(x,'llogis')
# 
# x <- c(0.3837053, 0.8576858, 0.3552237, 0.6226119, 0.4783756, 0.3139799, 0.4051403, 
#        0.4537631, 0.4711057, 0.5647414, 0.6479617, 0.7134207, 0.5259464, 0.5949068, 
#        0.3509200, 0.3783077, 0.5226465, 1.0241043, 0.4384580, 1.3341520)
# #inverse weibull
# ft_iw <- fitdist(x,'invweibull')
# ## End(Not run)

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