# This package fits RS and FMKL generalised lambda distributions using various methods. It also provides functions for fitting bimodal distributions using mixtures of generalised lambda distributions. The fitting algorithms considered in this package have two major objectives. One is to provide a smoothing device to fit distributions to data using the weight and unweighted discretised approach based on the bin width of the histogram. The other is to provide a definitive fit to the data set using the maximum likelihood estimation. Copyright Information: To ensure the stability of this package, this package ports other functions from other open sourced packages directly so that any changes in other packages will not cause this package to malfunction. All functions obtained from other sources have been acknowledged by the author in the authorship or the description sections of the help files and they are freely available online for all to use. Please contact the author for any copyright issues. Specifically the following functions have been modified from R: hist.su, ks.gof, pretty.su The following functions are taken from other open source packages in R: runif.pseudo, rnorm.pseudo, runif.halton, rnorm.halton, runif.sobol, rnorm.sobol by Diethelm Wuertz distributed under GPL. digitsBase, QUnif and sHalton written by Martin Maechler distributed under GPL. dgl, pgl, rgl, qgl, starship.adpativegrid, starship.obj and starship written by Robert King and some functions modified by Steve Su distributed under GPL. Lmoments and t1lmoments written by Juha Karvanen distributed under GPL. package ## Details This package allows a direct fitting method onto the data set using `fun.RMFMKL.ml`, `fun.RMFMKL.ml.m`, `fun.RMFMKL.hs`, `fun.RMFMKL.hs.nw`, `fun.RPRS.ml`, `fun.RPRS.ml.m`, `fun.RPRS.hs`, `fun.RPRS.hs.nw`, `fun.RMFMKL.qs`, `fun.RPRS.qs`, `fun.RMFMKL.mm`, `fun.RPRS.mm`, `fun.RMFMKL.lm`, `fun.RPRS.lm` and in the case of bimodal data set: `fun.auto.bimodal.qs`, `fun.auto.bimodal.ml`, `fun.auto.bimodal.pml` functions. The resulting fits can be graphically gauged by `fun.plot.fit` or `fun.plot.fit.bm` (for bimodal data), or examined by numbers using the Kolmogorov-Smirnoff resample tests (`fun.diag.ks.g`) and `fun.diag.ks.g.bimodal`). For unimodal data fits, it is also possible to compare the mean, variance, skewness and kurtosis of the fitted distribution with the data set using `fun.comp.moments.ml` and `fun.comp.moments.ml.2` functions. Similarly, for bimodal data fits, this is done via `fun.theo.bi.mv.gld` and `fun.moments.r`. Additionally, L moments for single generalised lambda distribution can be obtained using `fun.lm.theo.gld`. For graphical display of goodness of fit, quantile plots can be used, these can be done using `qqplot.gld` and `qqplot.gld.bi` for univariate and bimodal generalised lambda distribution fits respectively. ## Author(s) Steve Su ## References Asquith, W. (2007), L-moments and TL-moments of the generalized lambda distribution, Computational Statistics and Data Analysis 51(9): 4484-4496. Freimer, M., Mudholkar, G. S., Kollia, G. & Lin, C. T. (1988), A study of the generalized tukey lambda family, Communications in Statistics - Theory and Methods *17*, 3547-3567. Gilchrist, Warren G. (2000), Statistical Modelling with Quantile Functions, Chapman & Hall Karian, Z.A., Dudewicz, E.J., and McDonald, P. (1996), The extended generalized lambda distribution system for fitting distributions to data: history, completion of theory, tables, applications, the ``Final Word'' on Moment fits, Communications in Statistics - Simulation and Computation *25*, 611-642. Karian, Zaven A. and Dudewicz, Edward J. (2000), Fitting statistical distributions: the Generalized Lambda Distribution and Generalized Bootstrap methods, Chapman & Hall Karvanen, J. and A. Nuutinen (2008), Characterizing the generalized lambda distribution by L-moments, Computational Statistics and Data Analysis 52(4): 1971-1983. doi:10.1016/j.csda.2007.06.021 King, R.A.R. & MacGillivray, H. L. (1999), A starship method for fitting the generalised lambda distributions, Australian and New Zealand Journal of Statistics, 41, 353-374 doi:10.1111/1467-842X.00089 Ramberg, J. S. & Schmeiser, B. W. (1974), An approximate method for generating asymmetric random variables, Communications of the ACM *17*, 78-82. Su, S. (2005). A Discretized Approach to Flexibly Fit Generalized Lambda Distributions to Data. Journal of Modern Applied Statistical Methods (November): 408-424. doi:10.22237/jmasm/1130803560 Su, S. (2007). Nmerical Maximum Log Likelihood Estimation for Generalized Lambda Distributions. Computational Statistics and Data Analysis: *51*, 8, 3983-3998. doi:10.1016/j.csda.2006.06.008 Su, S. (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. doi:10.18637/jss.v021.i09 Su, S. (2009). Confidence Intervals for Quantiles Using Generalized Lambda Distributions. Computational Statistics and Data Analysis *53*, 9, 3324-3333. doi:10.1016/j.csda.2009.02.014 Su, S. (2010). Chapter 14: Fitting GLDs and Mixture of GLDs to Data using Quantile Matching Method. Handbook of Distribution Fitting Methods with R. (Karian and Dudewicz) 557-583. doi:10.1201/b10159-3 Su, S. (2010). Chapter 15: Fitting GLD to data using GLDEX 1.0.4 in R. Handbook of Distribution Fitting Methods with R. (Karian and Dudewicz) 585-608. doi:10.1201/b10159-3 Su, S. (2015) "Flexible Parametric Quantile Regression Model" Statistics & Computing 25 (3). 635-650. doi:10.1007/s11222-014-9457-1 Su S. (2021) "Flexible parametric accelerated failure time model" J Biopharm Stat. 2021 Sep 31(5):650-667. doi:10.1080/10543406.2021.1934854 ## See Also `GLDreg` package in R for GLD regression models. ## Examples ```r ###Univariate distributions example: set.seed(1000) junk<-rweibull(300,3,2) ##A faster ML estimtion junk.fit1<-fun.RMFMKL.ml.m(junk) junk.fit2<-fun.RPRS.ml.m(junk) qqplot.gld(junk.fit1,data=junk,param="fmkl") qqplot.gld(junk.fit2,data=junk,param="rs") ##Using discretised approach, with 50 classes #Using discretised method with weights obj.fit1.hs<-fun.data.fit.hs(junk) #Plot the resulting fit. The fun.plot.fit function also works for singular fits #such as those by fun.plot.fit(obj.fit1.ml,junk,nclass=50, #param=c("rs","fmkl","fmkl"),xlab="x") fun.plot.fit(obj.fit1.hs,junk,nclass=50,param=c("rs","fmkl"),xlab="x") #Compare the mean, variance, skewness and kurtosis of the fitted distribution #with actual data fun.theo.mv.gld(obj.fit1.hs[1,1],obj.fit1.hs[2,1],obj.fit1.hs[3,1], obj.fit1.hs[4,1],param="rs") fun.theo.mv.gld(obj.fit1.hs[1,2],obj.fit1.hs[2,2],obj.fit1.hs[3,2], obj.fit1.hs[4,2],param="fmkl") fun.moments.r(junk) #Conduct resample KS tests fun.diag.ks.g(obj.fit1.hs[,1],junk,param="rs") fun.diag.ks.g(obj.fit1.hs[,2],junk,param="fmkl") ##Try another fit, say 15 classes obj.fit2.hs<-fun.data.fit.hs(junk,rs.default="N",fmkl.default="N",no.c.rs = 15, no.c.fmkl = 15) fun.plot.fit(obj.fit2.hs,junk,nclass=50,param=c("rs","fmkl"),xlab="x") fun.theo.mv.gld(obj.fit2.hs[1,1],obj.fit2.hs[2,1],obj.fit2.hs[3,1], obj.fit2.hs[4,1],param="rs") fun.theo.mv.gld(obj.fit2.hs[1,2],obj.fit2.hs[2,2],obj.fit2.hs[3,2], obj.fit2.hs[4,2],param="fmkl") fun.moments.r(junk) fun.diag.ks.g(obj.fit2.hs[,1],junk,param="rs") fun.diag.ks.g(obj.fit2.hs[,2],junk,param="fmkl") ##Uses the maximum likelihood estimation method #Fit the function using fun.data.fit.ml obj.fit1.ml<-fun.data.fit.ml(junk) #Plot the resulting fit fun.plot.fit(obj.fit1.ml,junk,nclass=50,param=c("rs","fmkl","fmkl"),xlab="x", name=".ML") #Compare the mean, variance, skewness and kurtosis of the fitted distribution #with actual data fun.comp.moments.ml(obj.fit1.ml,junk) #Do a quantile plot qqplot.gld(junk,obj.fit1.ml[,1],param="rs",name="RS ML fit") #Run a KS resample test on the resulting fit fun.diag2(obj.fit1.ml,junk,1000) #It is possible to use say fun.data.fit.ml(junk,rs.leap=409,fmkl.leap=409, #FUN="QUnif") to find solution under a different set of low discrepancy number #generators. ###Bimodal distributions example: #Fitting mixture of generalised lambda distributions on the data set using both #the maximum likelihood and partition maximum likelihood and plot the resulting #fits opar <- par() par(mfrow=c(2,1)) junk<-fun.auto.bimodal.ml(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) fun.plot.fit.bm(nclass=50,fit.obj=junk,data=faithful[,1], name="Maximum likelihood using",xlab="faithful1",param.vec=c("rs","fmkl")) #Do a quantile plot qqplot.gld.bi(faithful[,1],junk$par,param1="rs",param2="fmkl", name="RS FMKL ML fit",range=c(0.001,0.999)) par(opar) junk<-fun.auto.bimodal.pml(faithful[,1],clustering.m=clara,init1.sel="rprs", init2.sel="rmfmkl",init1=c(-1.5,1.5),init2=c(-0.25,1.5),leap1=3,leap2=3) fun.plot.fit.bm(nclass=50,fit.obj=junk,data=faithful[,1], name="Partition maximum likelihood using",xlab="faithful1", param.vec=c("rs","fmkl")) #Fit the faithful[,1] data from the dataset library using sobol sequence #generator for the first distribution fit and halton sequence for the second #distribution fit. fit1<-fun.auto.bimodal.ml(faithful[,1],init1.sel="rmfmkl",init2.sel="rmfmkl", init1=c(-0.25,1.5),init2=c(-0.25,1.5),leap1=3,leap2=3,fun1="runif.sobol", fun2="runif.halton") #Run diagnostic KS tests fun.diag.ks.g.bimodal(fit1$par[1:4],fit1$par[5:8],prop1=fit1$par[9], data=faithful[,1],param1="fmkl",param2="fmkl") #Find the theoretical moments of the fit fun.theo.bi.mv.gld(fit1$par[1],fit1$par[2],fit1$par[3],fit1$par[4],"fmkl", fit1$par[5],fit1$par[6],fit1$par[7],fit1$par[8],"fmkl",fit1$par[9]) #Compare this with the empirical moments from the data set. fun.moments.r(faithful[,1]) #Do a quantile plot qqplot.gld.bi(faithful[,1],fit1$par,param1="fmkl",param2="fmkl", name="FMKL FMKL ML fit") #Quantile matching method #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%. #Fitting normal(3,2) distriution using the default setting junk<-rnorm(50,3,2) fun.data.fit.qs(junk) 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) #L Moment matching #Fitting normal(3,2) distriution using the default setting junk<-rnorm(50,3,2) fun.data.fit.lm(junk) # Moment matching method #Fitting normal(3,2) distriution using the default setting junk<-rnorm(50,3,2) fun.data.fit.mm(junk) # Example on fitting mixture of normal distributions data1<-c(rnorm(1500,-1,2/3),rnorm(1500,1,2/3)) junk<-fun.auto.bimodal.ml(data1,per.of.mix=0.01,clustering.m=data1>0, init1.sel="rprs",init2.sel="rmfmkl",init1=c(-1.5,1.5),init2=c(-0.25,1.5), leap1=3,leap2=3) fun.plot.fit.bm(nclass=50,fit.obj=junk,data=data1, name="Maximum likelihood using",xlab="faithful1",param.vec=c("rs","fmkl")) qqplot.gld.bi(data1,junk$par,param1="rs",param2="fmkl", name="RS FMKL ML fit",range=c(0.001,0.999)) # Generate random observations from FMKL generalised lambda distributions with # parameters (1,2,3,4) and (4,3,2,1) with 50% of data from each distribution. fun.simu.bimodal(c(1,2,3,4),c(4,3,2,1),prop1=0.5,param1="fmkl",param2="fmkl") ```

GLDEX.package function