fun.auto.bimodal.qs {GLDEX}

Fitting mixtures of generalied lambda distribtions to data using quantile matching method

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 quantile matching method. It is a two step optimization procedure, each unimodal part of the bimodal distribution is modelled using quantile matching method. The initial values obtained are then used to maximise the theoretical and empirical quantile match 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.qs(data, per.of.mix = 0.01, 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", trial.n = 100, 
len = 1000, type = 7, no = 10000, maxit = 5000)

Arguments

data

A numerical vector representing the dataset.

per.of.mix

Level of mix between two parts of the distribution, usually 1-2% of cross mix is sufficient.

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" or "rmfmkl", representing the choice (RS or FMKL) of the first distribution

init2.sel

This can be "rprs" or "rmfmkl", representing the choice (RS or FMKL) of 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".

trial.n

Number of evenly spaced quantile ranging from 0 to 1 to be used in the checking phase, to find the best set of initial values for optimisation, this is intended to be lower than len to speed up the fitting algorithm. Default is 100.

len

Number of evenly spaced quantile ranging from 0 to 1 to be used, default is 1000

type

Type of quantile to be used, default is 7, see quantile

no

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

maxit

Maximum number of iterations for numerical optimisation. Default is 5000.

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.

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 -ML 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 span a valid generalised lambda distribution, try another set of initial values.

Author(s)

Steve Su

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 (2008). Fitting GLD to data via quantile matching method. (Book chapter to appear)

See Also

fun.auto.bimodal.pml, 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. 
# The percentage of data mix is 1%.

fun.auto.bimodal.qs(faithful[,1],per.of.mix=0.01,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)