fitdistrplus (version 0.1-2)

fitdist: Fitting of univariate distributions to non-censored data and goodness-of-fit statistics

Description

Fits a univariate distribution to non-censored data by maximum likelihood or matching moments, and computes goodness-of-fit statistics.

Usage

fitdist(data, distr, method=c("mle", "mme"), start, chisqbreaks, meancount,...)
## S3 method for class 'fitdist':
print(x,...)
## S3 method for class 'fitdist':
plot(x,breaks="default",...)
## S3 method for class 'fitdist':
summary(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

method

A character string coding for the fitting method: "mle" for 'maximum likelihood estimation and "mme" for 'matching moment estimation'.

start

A named list giving the initial values of parameters of the named distribution. This argument will not be taken into account if method="mme", and may be omitted for some distributions for which reasonable starting values are

chisqbreaks

A numeric vector defining the breaks of the cells used to compute the chi-squared statistic. If omitted, these breaks are automatically computed from the data in order to reach roughly the same number of observations per cell, roughly equal to

meancount

The mean number of observations per cell expected for the definition of the breaks of the cells used to compute the chi-squared statistic. This argument will not be taken into account if the breaks are directly defined in the argument ch

x

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 discre

...

further arguments to be passed to generic functions, or to the function "mledist" if 'maximum likelihood' is the chosen method, in order to control the optimization method.

Value

• fitdist returns an object of class 'fitdist', a list with following components,
• estimate
  the parameter estimates
• method
  the character string coding for the fitting method : "mle" for 'maximum likelihood estimation' and "mme" for 'matching moment estimation'
• sd
  the estimated standard errors or NULL if method="mme"
• cor
  the estimated correlation matrix or NULL if method="mme"
• loglik
  the log-likelihood or NULL if method="mme"
• aic
  the Akaike information criterion or NULL if method="mme"
• bic
  the the so-called BIC or SBC (Schwarz Bayesian criterion) or NULL if method="mme"
• n
  the length of the data set
• data
  the dataset
• distname
  the name of the distribution
• chisq
  the Chi-squared statistic or NULL if not computed
• chisqbreaks
  breaks used to define cells in the Chi-squared statistic
• chisqpvalue
  p-value of the Chi-squared statistic or NULL if not computed
• chisqdf
  degree of freedom of the Chi-squared distribution or NULL if not computed
• chisqtable
  a table with observed and theoretical counts used for the Chi-squared calculations
• ad
  the Anderson-Darling statistic or NULL if not computed
• adtest
  the decision of the Anderson-Darling test or NULL if not computed
• ks
  the Kolmogorov-Smirnov statistic or NULL if not computed
• kstest
  the decision of the Kolmogorov-Smirnov test or NULL if not computed

Details

When method="mle", maximum likelihood estimations of the distribution parameters are computed using the function mledist. By default direct optimization of the log-likelihood 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 method used in optim may be chosen or another optimization method may be chosen using ... argument (see mledist for details). For the following named distributions, reasonable starting values will be computed if start is omitted : "norm", "lnorm", "exp" and "pois", "cauchy", "gamma", "logis", "nbinom" (parametrized by mu and size), "geom", "beta" and "weibull". Note that these starting values may not be good enough if the fit is poor. The function is not able to fit a uniform distribution. With the parameter estimates, the function returns the log-likelihood and 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. When method="mme", the estimated values of the distribution parameters are provided only for the following distributions : "norm", "lnorm", "pois", "exp", "gamma", "nbinom", "geom", "beta", "unif" and "logis". For distributions characterized by one parameter ("geom", "pois" and "exp"), this parameter is simply estimated by matching theoretical and observed means, and for distributions characterized by two parameters, these parameters are estimated by matching theoretical and observed means and variances (Vose, 2000). Goodness-of-fit statistics are computed. The Chi-squared statistic is computed using cells defined by the argument chisqbreaks or cells automatically defined from the data in order to reach roughly the same number of observations per cell, roughly equal to the argument meancount, or sligthly more if there are some ties. If chisqbreaks and meancount are both omitted, meancount is fixed in order to obtain roughly $(4n)^{2/5}$ cells, with $n$ the length of the dataset (Vose, 2000). The Chi-squared statistic is not computed if the program fails to define enough cells due to a too small dataset. When the Chi-squared statistic is computed, and if the degree of freedom (nb of cells - nb of parameters - 1) of the corresponding distribution is strictly positive, the p-value of the Chi-squared test is returned. For the distributions assumed continuous (all but "binom", "nbinom", "geom", "hyper" and "pois" for R base distributions), Kolmogorov-Smirnov and Anderson-Darling statistics are also computed, as defined by Cullen and Frey (1999). An approximate Kolmogorov-Smirnov test is performed by assuming the distribution parameters known. The critical value defined by Stephens (1986) for a completely specified distribution is used to reject or not the distribution at the significance level 0.05. Because of this approximation, the result of the test (decision of rejection of the distribution or not) is returned only for datasets with more than 30 observations. Note that this approximate test may be too conservative. For datasets with more than 5 observations and for distributions for which the test is described by Stephens (1986) ("norm", "lnorm", "exp", "cauchy", "gamma", "logis" and "weibull"), the Anderson-darling test is performed as described by Stephens (1986). This test takes into account the fact that the parameters are not known but estimated from the data. The result is the decision to reject or not the distribution at the significance level 0.05. The plot of an object of class "fitdist" returned by fitdist uses the function plotdist.

References

Cullen AC and Frey HC (1999) Probabilistic techniques in exposure assessment. Plenum Press, USA, pp. 81-155. Stephens MA (1986) Tests based on edf statistics. In Goodness-of-fit techniques (D'Agostino RB and Stephens MA, eds), Marcel dekker, New York, pp. 97-194. 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.

See Also

plotdist, optim, mledist, mmedist and fitdistcens.

Examples

# (1) basic fit of a normal distribution with maximum likelihood estimation
#

x1 <- c(6.4,13.3,4.1,1.3,14.1,10.6,9.9,9.6,15.3,22.1,13.4,
13.2,8.4,6.3,8.9,5.2,10.9,14.4)
f1 <- fitdist(x1,"norm")
print(f1)
plot(f1)
summary(f1)
f1$chisqtable

# (2) use the moment matching estimation
#

f1b <- fitdist(x1,"norm",method="mme",meancount=6)
summary(f1b)
f1b$chisqtable

# (3) MME for log normal distribution
#

f1c <- fitdist(x1,"lnorm",method="mme",meancount=6)
summary(f1c)
f1c$chisqtable

# (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))

f1c <- fitdist(x1,"gumbel",start=list(a=10,b=5))
print(f1c)
plot(f1c)

# (5) fit a discrete distribution (Poisson)
#

x2<-c(rep(4,1),rep(2,3),rep(1,7),rep(0,12))
f2<-fitdist(x2,"pois",chisqbreaks=c(0,1))
plot(f2)
summary(f2)
f2$chisqtable



# (5) comparison of fits of various distributions
#
xw<-rweibull(n=100,shape=2,scale=1)
fa<-fitdist(xw,"weibull")
summary(fa)
fa$chisqtable
fb<-fitdist(xw,"gamma")
summary(fb)
fc<-fitdist(xw,"exp")
summary(fc)


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

fitdist(x1,"gamma",optim.method="Nelder-Mead")
fitdist(x1,"gamma",optim.method="BFGS") 
fitdist(x1,"gamma",optim.method="L-BFGS-B",lower=c(0,0))
fitdist(x1,"gamma",optim.method="SANN")

# (7) custom optimisation function
#

#create the sample
mysample <- rexp(100, 5)
mystart <- 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, dexp, start=mystart, custom.optim=myoptimize, interval=c(0, 100))

#show the result
summary(res2)


# (8) custom optimisation function - another example with the genetic algorithm
#
#set a sample
    x1 <- c(6.4, 13.3, 4.1, 1.3, 14.1, 10.6, 9.9, 9.6, 15.3, 22.1, 13.4, 13.2, 8.4, 6.3, 8.9, 5.2, 10.9, 14.4) 
    fit1 <- fitdist(x1, "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(x1, "gamma", custom.optim=mygenoud, nvars=2,    
        Domains=cbind(c(0,0), c(10, 10)), boundary.enforcement=1, 
        print.level=1, hessian=TRUE)

    summary(fit2)