fun.auto.bimodal.pml {GLDEX}

Fitting mixture of generalied lambda distribtions to data using parition maximum likelihood estimation

Description

This function will fit mixture of generalised lambda distributions to dataset. It is restricted to two generalised lambda distributions. The method of fitting is parition maximum likelihood. It is a two step optimization procedure, each unimodal part of the bimodal distribution is modelled using the maximum likelihood method or the starship method (FMKL GLD only). These initial values the used to "maximise" the complete log likelihood for the entire bimodal distribution. It fits mixture of the form p*(f1)+(1-p)*(f2) where f1 and f2 are pdfs of the generalised lambda distributions.

Usage

fun.auto.bimodal.pml(data, clustering.m = clara, init1.sel = "rprs", 
init2.sel = "rprs", init1=c(-1.5, 1.5), init2=c(-1.5, 1.5), leap1=3, leap2=3,
fun1="runif.sobol", fun2="runif.sobol",no=10000,max.it=5000, optim.further="Y")

Arguments

data

A numerical vector representing the dataset.

clustering.m

Clustering method used in classifying the dataset into two parts. Valid arguments include clara, fanny and pam from the cluster library. Default is clara. Or a logical vector specifying how data should be split.

init1.sel

This can be "rprs", "rmfmkl" or "star", the initial method used to fit the first distribution.

init2.sel

This can be "rprs", "rmfmkl" or "star", the initial method used to fit the second distribution.

init1

Inititial values lambda3 and lambda4 for the first generalised lambda distribution.

init2

Inititial values lambda3 and lambda4 for the second generalised lambda distribution.

leap1

See scrambling argument in fun.gen.qrn.

fun1

A character string of either "runif.sobol" (default), "runif.sobol.owen", "runif.halton" or "QUnif".

leap2

See scrambling argument in fun.gen.qrn.

fun2

A character string of either "runif.sobol" (default), "runif.sobol.owen", "runif.halton" or "QUnif".

no

Number of initial random values to find the best initial values for optimisation.

max.it

Maximum number of iterations for numerical optimisation.

optim.further

Whether to optimise the function further using full maximum likelihood method, recommended setting is "Y"

Details

The initial values that work well for RPRS are c(-1.5,1.5) and for RMFMKL are c(-0.25,1.5). For scrambling, if 1, 2 or 3 the sequence is scrambled otherwise not. If 1, Owen type type of scrambling is applied, if 2, Faure-Tezuka type of scrambling, is applied, and if 3, both Owen+Faure-Tezuka type of scrambling is applied. The star method uses the same initial values as rmfmkl since it uses the FMKL generalised lambda distribution. Nelder-Simplex algorithm is used in the numerical optimization. rprs stands for revised percentile method for RS generalised lambda distribution and "rmfmkl" stands for revised method of moment for FMKL generalised lambda distribution. These acronyms represents the initial optimization algorithm used to get a reasonable set of initial values for the subsequent optimization procedues. This function is an improvement from Su (2007) in Journal of Statistical Software.

Value

par

The best set of parameters found, the first four corresponds to the first distribution fit, the second four corresponds to the second distribution fit, the last value correspond to p for the first distribution fit.

value

The value of -PML for the paramters obtained.

counts

A two-element integer vector giving the number of calls to "fn" and "gr" respectively. This excludes those calls needed to compute the Hessian, if requested, and any calls to 'fn' to compute a finite-difference approximation to the gradient.

convergence

0 indicates successful convergence, 1 indicates the iteration limit maxit had been reached, 10 indicates degeneracy of the Nelder-Mead simplex.

message

A character string giving any additional information returned by the optimizer, or NULL.

Note

If the number of observations is small, rprs can sometimes fail as the percentiles may not exist for this data. Also, if the initial values do not result in a valid generalised lambda distribution, try another set of initial values.

References

Bratley P. and Fox B.L. (1988) Algorithm 659: Implementing Sobol's quasi random sequence generator, ACM Transactions on Mathematical Software 14, 88-100.

Joe S. and Kuo F.Y. (1998) Remark on Algorithm 659: Implementing Sobol's quasi random Sequence Generator.

Nelder, J. A. and Mead, R. (1965) A simplex algorithm for function minimization. Computer Journal *7*, 308-313.

Su (2007). Fitting Single and Mixture of Generalized Lambda Distributions to Data via Discretized and Maximum Likelihood Methods: GLDEX in R. Journal of Statistical Software: *21* 9.

See Also

fun.auto.bimodal.ml,fun.plot.fit.bm, fun.diag.ks.g.bimodal

Examples

# Fitting faithful data from the dataset library, with the clara clustering 
# regime. The first distribution is RS and the second distribution is fmkl. 

fun.auto.bimodal.pml(faithful[,1],clustering.m=clara,init1.sel="rprs",
init2.sel="rmfmkl",init1=c(-1.5,1,5),init2=c(-0.25,1.5),leap1=3,leap2=3)