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, …)
    
# S3 method for fitdist
print(x, …)
# S3 method for fitdist
plot(x, breaks="default", …)
# S3 method for fitdist
summary(object, …)
# S3 method for fitdist
logLik(object, …)
# S3 method for fitdist
vcov(object, …)
# S3 method for fitdist
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.

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.

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".

…

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:

estimate

the parameter estimates.

method

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

the estimated standard errors, NA if numerically not computable or NULL if not available.

cor

the estimated correlation matrix, NA if numerically not computable or NULL if not available.

vcov

the estimated variance-covariance matrix, NULL if not available.

loglik

the log-likelihood.

aic

the Akaike information criterion.

bic

the the so-called BIC or SBC (Schwarz Bayesian criterion).

the length of the data set.

data

the data set.

distname

the name of the distribution.

fix.arg

the named list giving the values of parameters of the named distribution that must be kept fixed rather than estimated by maximum likelihood or NULL if there are no such parameters.

fix.arg.fun

the function used to set the value of fix.arg or NULL.

dots

the list of further arguments passed in … to be used in bootdist in iterative calls to mledist, mmedist, qmedist, mgedist or NULL if no such arguments.

convergence

an integer code for the convergence of optim/constrOptim defined as below or defined by the user in the user-supplied optimization function. 0 indicates successful convergence. 1 indicates that the iteration limit of optim has been reached. 10 indicates degeneracy of the Nealder-Mead simplex. 100 indicates that optim encountered an internal error.

discrete

the input argument or the automatic definition by the function to be passed to functions gofstat, plotdist and cdfcomp.

weights

the vector of weigths used in the estimation process or NULL.

Generic functions:

print

The print of a "fitdist" object shows few traces about the fitting method and the fitted distribution.

summary

The summary provides the parameter estimates of the fitted distribution, the log-likelihood, AIC and BIC statistics and when the maximum likelihood is used, the standard errors of the parameter estimates and the correlation matrix between parameter estimates.

plot

The plot of an object of class "fitdist" returned by fitdist uses the function plotdist. An object of class "fitdist" or a list of objects of class "fitdist" corresponding to various fits using the same data set may also be plotted using a cdf plot (function cdfcomp), a density plot(function denscomp), a density Q-Q plot (function qqcomp), or a P-P plot (function ppcomp).

logLik

Extracts the estimated log-likelihood from the "fitdist" object.

vcov

Extracts the estimated var-covariance matrix from the "fitdist" object (only available When method = "mle").

coef

Extracts the fitted coefficients from the "fitdist" object.

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:

When method="mle": Maximum likelihood estimation consists in maximizing the log-likelihood. A numerical optimization is carried out in mledist via optim to find the best values (see mledist for details).
When method="mme": Moment matching estimation consists in equalizing theoretical and empirical moments. Estimated values of the distribution parameters are computed by a closed-form formula for the following distributions : "norm", "lnorm", "pois", "exp", "gamma", "nbinom", "geom", "beta", "unif" and "logis". Otherwise the theoretical and the empirical moments are matched numerically, by minimization of the sum of squared differences between observed and theoretical moments. In this last case, further arguments are needed in the call to fitdist: order and memp (see mmedist for details).
When method = "qme": Quantile matching estimation consists in equalizing theoretical and empirical quantile. A numerical optimization is carried out in qmedist via optim to minimize of the sum of squared differences between observed and theoretical quantiles. The use of this method requires an additional argument probs, defined as the numeric vector of the probabilities for which the quantile(s) is(are) to be matched (see qmedist for details).
When method = "mge": Maximum goodness-of-fit estimation consists in maximizing a goodness-of-fit statistics. A numerical optimization is carried out in mgedist via optim to minimize the goodness-of-fit distance. The use of this method requires an additional argument gof coding for the goodness-of-fit distance chosen. One can use the classical Cramer-von Mises distance ("CvM"), the classical Kolmogorov-Smirnov distance ("KS"), the classical Anderson-Darling distance ("AD") which gives more weight to the tails of the distribution, or one of the variants of this last distance proposed by Luceno (2006) (see mgedist for more details). This method is not suitable for discrete distributions.

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

# NOT RUN {
# (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
#
# }
# NOT RUN {
#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)
# }
# NOT RUN {

# (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)
# }
# 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)

# }
# NOT RUN {
# (12) Fit of a Weibull distribution to serving size data by maximum 
# goodness-of-fit estimation using all the distances available
# 
# }
# NOT RUN {
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))
# }
# NOT RUN {
# (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))
# }
# 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))
# }
# 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')
# }
# NOT RUN {
# }

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