A.2. Packages for mixture models
A.2.5. Flexmix
This package, uses a general framework for finite mixtures of regression models. They use the EM algorithm to do so. This package allows to compute information criteria, it has examples of applications, making it easier to understand allows to cluster Gaussian distributions this can use the maximum likelihood of a wide family of distributions as Gaussian, binomial, Poisson and Gamma.
B.1. Algorithm for number of populations known
EM. r e s u l t s= function ( d a t o s 1 , k , i t e r , expnumber ) { t r y C a t c h (
normalmixEM ( d a t o s 1 , lambda = NULL, mu = NULL, sigma = NULL, k = k , sd . c o n s t r = NULL,
e p s i l o n = 1 e −08 , maxit = 5 0 0 , m a x r e s t a r t s =20 , v e r b = FALSE, f a s t=FALSE, ECM = FALSE,
arbmean = TRUE, a r b v a r = TRUE) ,
warning = function (w) {
l i s t ( lambda=rep (NA, k ) ,mu=rep (NA, k ) , sigma=rep (NA, k ) ) } ,
e r r o r = function ( e ) { l i s t ( lambda=rep (NA, k ) ,mu=rep (NA, k ) , sigma=rep (NA, k ) ) } )
}
GAestimationk<−function ( cromosoma ) {
cromosoma=t ( round ( cromosoma , d i g i t s =0))
datosGA = d a t o s
datosGak = matrix (nrow=nrow( d a t o s ) , ncol=(ncol ( d a t o s ) + 1 ) )
f o r ( i i n 1 : 3 ) { datosGak [ , i ]= d a t o s [ , i ] }
datosGak [ , 4 ] = t ( cromosoma )
colnames ( datosGak)=c ( ” O b s e r v a t i o n ” , ” Value ” , ” R e a l P o p u l a t i o n ” , ”Chromosome” )
ParameterValue=D a t a S e t t i n g s [ , , expnumber ]
tablacromosoma=table ( cromosoma )
B.1 Algorithm for number of populations known 81
R e s u l t s=matrix (NA, nrow=max( cromosoma ) , ncol =6)
c u e n t a cromosoma=rep (NA,max( cromosoma ) )
f o r ( h i n 1 :max( cromosoma ) ) { c u e n t a cromosoma [ h]=sum( cromosoma==h ) }
f o r ( j i n 1 :max( cromosoma ) ) {
i f ( c u e n t a cromosoma [ j ] >3){
P o p u l a t i o n=datosGak [ cromosoma==j , 3 ]
Param= f i t d i s t r ( P o p u l a t i o n , d e n s f u n=” normal ” )
R e s u l t s [ j , 1 ] = ParameterValue [ j , 1 ]
R e s u l t s [ j , 2 ] = coef ( Param ) [ 1 ] #p a r a m e t e r 1 (mu o r a l p h a )
R e s u l t s [ j , 3 ] = coef ( Param ) [ 2 ] # p a r a m e t e r 2 ( sigma o r b e t a )
R e s u l t s [ j , 4 ] = abs ( ParameterValue [ j , 2 ] − ( c u e n t a cromosoma [ j ] / sum( c u e n t a cromosoma ) ) ) / ParameterValue [ j , 2 ]
R e s u l t s [ j , 5 ] = abs ( ParameterValue [ j , 3 ] − coef ( Param ) [ 1 ] ) / ParameterValue [ j , 3 ]
R e s u l t s [ j , 6 ] = abs ( ParameterValue [ j , 4 ] − coef ( Param ) [ 2 ] ) / ParameterValue [ j , 4 ]
}
}
m e a n e r r o r=mean( R e s u l t s [ c u e n t a cromosoma >3 , c ( 4 , 5 , 6 ) ] )
return ( m e a n e r r o r ) }
GAestimationk f u l l<−function ( cromosoma ) {
cromosoma=t ( round ( cromosoma , d i g i t s =0))
datosGA = d a t o s
datosGak = matrix (nrow=nrow( d a t o s ) , ncol=(ncol ( d a t o s ) + 1 ) )
f o r ( i i n 1 : 3 ) { datosGak [ , i ]= d a t o s [ , i ] }
datosGak [ , 4 ] = t ( cromosoma )
colnames ( datosGak)=c ( ” O b s e r v a t i o n ” , ” Value ” , ” R e a l P o p u l a t i o n ” , ”Chromosome” )
tablacromosoma=as . numeric ( table ( cromosoma ) )
R e s u l t s=matrix (NA, nrow=max( cromosoma ) , ncol =7)
c u e n t a cromosoma=rep (NA,max( cromosoma ) )
f o r ( h i n 1 :max( cromosoma ) ) { c u e n t a cromosoma [ h]=sum( cromosoma==h ) }
f o r ( j i n 1 :max( cromosoma ) ) {
i f ( c u e n t a cromosoma [ j ] >3){
P o p u l a t i o n=datosGak [ cromosoma==j , 3 ]
Param= f i t d i s t r ( P o p u l a t i o n , d e n s f u n=” normal ” )
R e s u l t s [ j , 1 ] = ( c u e n t a cromosoma [ j ] ) /sum( c u e n t a cromosoma )
R e s u l t s [ j , 2 ] = coef ( Param ) [ 1 ] #p a r a m e t e r 1 (mu o r a l p h a )
R e s u l t s [ j , 3 ] = coef ( Param ) [ 2 ] # p a r a m e t e r 2 ( sigma o r b e t a ) R e s u l t s [ j , 4 ] = abs ( ParameterValue [ j , 2 ] − ( c u e n t a cromosoma [ j ] /
sum( c u e n t a cromosoma ) ) ) / ParameterValue [ j , 2 ]
R e s u l t s [ j , 5 ] = abs ( ParameterValue [ j , 3 ] − coef ( Param ) [ 1 ] ) / ParameterValue [ j , 3 ]
R e s u l t s [ j , 6 ] = abs ( ParameterValue [ j , 4 ] − coef ( Param ) [ 2 ] ) / ParameterValue [ j , 4 ]
}
}
m e a n e r r o r=mean( R e s u l t s [ c u e n t a cromosoma >3 , c ( 4 , 5 , 6 ) ] )
R e s u l t s [ , 7 ] = as . numeric ( table ( t ( cromosoma)==d a t o s [ , 2 ] ) ) [ 2 ] / sum( c u e n t a cromosoma )
B.1 Algorithm for number of populations known 83
return ( R e s u l t s ) }
GenerateData<−function ( n1 , D a t a S e t t i n g s ) {
C a l c u l a t e d W e i g h t s=rmultinom ( 1 , s i z e=n1 , prob=D a t a S e t t i n g s [ , 2 ] )
GeneratedData=c (rnorm( C a l c u l a t e d W e i g h t s [ 1 , 1 ] , mean=D a t a S e t t i n g s [ 1 , 3 ] , sd=D a t a S e t t i n g s [ 1 , 4 ] ) )
PopID=c ( rep ( 1 , C a l c u l a t e d W e i g h t s [ 1 , 1 ] ) )
f o r ( i i n 2 : k ) {
GeneratedData=c ( GeneratedData , rnorm( C a l c u l a t e d W e i g h t s [ i , 1 ] , mean=D a t a S e t t i n g s [ i , 3 ] , sd=D a t a S e t t i n g s [ i , 4 ] ) )
PopID=c ( PopID , rep ( i , C a l c u l a t e d W e i g h t s [ i , 1 ] ) ) }
Seq=seq ( 1 , n1 )
Datos=cbind ( Seq , PopID , GeneratedData )
rand<−sample (nrow( Datos ) )
D e s o r d e n a d o s=Datos [ rand , ]
return ( D e s o r d e n a d o s ) }
l i b r a r y (MASS) #For c o m p u t i n g t h e l i k e l i h o o d and f i t t i n g t h e d i s t r i b u t i o n
l i b r a r y ( g e n a l g ) #f o r t h e e v a l u a t i o n u s i n g GA
l i b r a r y ( m i x t o o l s ) #For t h e e v a l u a t i o n u s i n g EM
l i b r a r y ( m c l u s t ) #For t h e c l u s t e r i n g o f t h e d a t a
l i b r a r y ( l g a ) #For c o m p u t i n g t h e b i c
m o n i t o r <− function ( o b j ) {
minEval = min( o b j $ e v a l u a t i o n s ) ;
plot ( obj , t y p e=” h i s t ” ) ;
}
k=2
max expnumber=4
P a r a m e t e r s=matrix ( ncol =2 ,nrow=max expnumber ) #n & i t e r P a r a m e t e r s [ , 1 ] = c ( 3 0 , 5 0 , 1 0 0 , 2 0 0 )
P a r a m e t e r s [ , 2 ] = c ( rep ( 1 0 0 0 ,max expnumber ) )
D a t a S e t t i n g s= array (dim=c ( k , 4 ,max expnumber ) )
#p o p u l a t i o n number
D a t a S e t t i n g s [ , 1 , ] = c ( 1 , 2 , 1 , 2 , 1 , 2 , 1 , 2 )
#p o p u l a t i o n w e i g h t
D a t a S e t t i n g s [ , 2 , ] = c ( 0 . 0 5 , 0 . 9 5 , 0 . 0 5 , 0 . 9 5 , 0 . 0 5 , 0 . 9 5 , 0 . 0 5 , 0 . 9 5 )
#p a r a m e t e r 1
D a t a S e t t i n g s [ , 3 , ] = c ( 1 2 , 1 3 , 1 2 , 1 3 , 1 2 , 1 3 , 1 2 , 1 3 )
#Parameter 2
D a t a S e t t i n g s [ , 4 , ] = c ( 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 )
R e s u l t s C a s o 1=array (NA, dim=c ( k , 1 6 ,
max( P a r a m e t e r s [ , 2 ] ) , max expnumber ) )
f o r ( expnumber i n 1 :max expnumber ) {
n1=P a r a m e t e r s [ expnumber , 1 ]
i t e r =P a r a m e t e r s [ expnumber , 2 ]
f o r (m i n 1 : i t e r ) { ##p o n e r i t e r
s e t . s e e d=1+m
B.1 Algorithm for number of populations known 85
d a t o s<<−GenerateData ( n1 , D a t a S e t t i n g s [ , , expnumber ] )
G A l r e s u l t s = r b g a ( s t r i n g M i n=c ( rep ( 1 , n1 ) ) , s t r i n g M a x=c ( rep ( k , n1 ) ) , s u g g e s t i o n s=NULL,
p o p S i z e =200 , i t e r s =100 , mutationChance = 0 . 0 5 , e l i t i s m=NA,
e v a l F u n c=GAestimationk ,
s h o w S e t t i n g s=FALSE, v e r b o s e=FALSE)
GAl . r e s u l t s<−summary( G A l r e s u l t s , e c h o=TRUE)
s e p a r a r<−s t r s p l i t ( GAl . r e s u l t s , ” B e s t S o l u t i o n : ” )
s e p a r a r 2<−u n l i s t ( s t r s p l i t ( s e p a r a r [ [ 1 ] ] [ 2 ] , ” ” ) )
mejor . cromosoma<−round ( as . numeric ( s e p a r a r 2 [− length ( s e p a r a r 2 ) ] ) )
R e s u l t s .GA=GAestimationk f u l l ( mejor . cromosoma )
EM. r e s u l t= EM. r e s u l t s ( d a t o s [ , 3 ] , k , i t e r , expnumber )
#G e n e r a t e t h e r e s u l t s f o r ( i i n 1 : k ) {
R e s u l t s C a s o 1 [ i , 1 ,m, expnumber ]= D a t a S e t t i n g s [ i , 2 , expnumber ] R e s u l t s C a s o 1 [ i , 2 ,m, expnumber ]= D a t a S e t t i n g s [ i , 3 , expnumber ] R e s u l t s C a s o 1 [ i , 3 ,m, expnumber ]= D a t a S e t t i n g s [ i , 4 , expnumber ] R e s u l t s C a s o 1 [ i , 4 ,m, expnumber ]= R e s u l t s .GA[ i , 1 ]
R e s u l t s C a s o 1 [ i , 5 ,m, expnumber ]= R e s u l t s .GA[ i , 2 ] R e s u l t s C a s o 1 [ i , 6 ,m, expnumber ]= R e s u l t s .GA[ i , 3 ] R e s u l t s C a s o 1 [ i , 7 ,m, expnumber ]= R e s u l t s .GA[ i , 4 ] R e s u l t s C a s o 1 [ i , 8 ,m, expnumber ]= R e s u l t s .GA[ i , 5 ] R e s u l t s C a s o 1 [ i , 9 ,m, expnumber ]= R e s u l t s .GA[ i , 6 ] R e s u l t s C a s o 1 [ i , 1 0 ,m, expnumber ]= R e s u l t s .GA[ i , 7 ]
#P a ra m e t e r s u s i n g EM a l g o r i t h m
R e s u l t s C a s o 1 [ i , 1 1 ,m, expnumber ]=EM. r e s u l t $lambda [ i ]
R e s u l t s C a s o 1 [ i , 1 2 ,m, expnumber ]=EM. r e s u l t $mu [ i ] R e s u l t s C a s o 1 [ i , 1 3 ,m, expnumber ]=EM. r e s u l t $ sigma [ i ]
R e s u l t s C a s o 1 [ i , 1 4 ,m, expnumber ]=abs (EM. r e s u l t $lambda [ i ]−
D a t a S e t t i n g s [ i , 2 , expnumber ] ) / D a t a S e t t i n g s [ i , 2 , expnumber ] R e s u l t s C a s o 1 [ i , 1 5 ,m, expnumber ]=abs (EM. r e s u l t $mu [ i ]−
D a t a S e t t i n g s [ i , 3 , expnumber ] ) / D a t a S e t t i n g s [ i , 3 , expnumber ] R e s u l t s C a s o 1 [ i , 1 6 ,m, expnumber ]=abs (EM. r e s u l t $ sigma [ i ]−
D a t a S e t t i n g s [ i , 4 , expnumber ] ) / D a t a S e t t i n g s [ i , 4 , expnumber ]
}
print ( paste ( ” i t e r a c i o n ” ,m, ” d e l e x p e r i m e n t o ” , expnumber ) )
}
}
B.2. Algorithm for number of populations unknown
EM. r e s u l t s= function ( d a t o s 1 , k , i t e r , expnumber ) { t r y C a t c h (
normalmixEM ( d a t o s 1 , lambda = NULL, mu = NULL, sigma = NULL, k = k , sd . c o n s t r = NULL,
e p s i l o n = 1 e −08 , maxit = 1 0 0 , m a x r e s t a r t s =20 , v e r b = FALSE, f a s t=FALSE, ECM = FALSE,
arbmean = TRUE, a r b v a r = TRUE) ,
warning = function (w) {
l i s t ( lambda=rep (NA, k ) ,mu=rep (NA, k ) , sigma=rep (NA, k ) ) } ,
e r r o r = function ( e ) { l i s t ( lambda=rep (NA, k ) ,mu=rep (NA, k ) , sigma=rep (NA, k ) ) } )
}
GAestimationk<−function ( cromosoma ) {
cromosoma=t ( round ( cromosoma , d i g i t s =0))
datosGA = d a t o s
datosGak = matrix (nrow=nrow( d a t o s ) , ncol=(ncol ( d a t o s ) + 1 ) )
f o r ( i i n 1 : 3 ) { datosGak [ , i ]= d a t o s [ , i ] }
B.2 Algorithm for number of populations unknown 87
datosGak [ , 4 ] = t ( cromosoma )
colnames ( datosGak)=c ( ” O b s e r v a t i o n ” , ” Value ” , ” R e a l P o p u l a t i o n ” , ”Chromosome” )
ParameterValue=D a t a S e t t i n g s [ , , expnumber ]
tablacromosoma=table ( cromosoma )
R e s u l t s=matrix (NA, nrow=max( cromosoma ) , ncol =6)
c u e n t a cromosoma=rep (NA,max( cromosoma ) )
f o r ( h i n 1 :max( cromosoma ) ) { c u e n t a cromosoma [ h]=sum( cromosoma==h ) }
f o r ( j i n 1 :max( cromosoma ) ) {
i f ( c u e n t a cromosoma [ j ] >3){
P o p u l a t i o n=datosGak [ cromosoma==j , 3 ]
Param= f i t d i s t r ( P o p u l a t i o n , d e n s f u n=” normal ” )
R e s u l t s [ j , 1 ] = c u e n t a cromosoma [ j ] /sum( c u e n t a cromosoma )
R e s u l t s [ j , 2 ] = coef ( Param ) [ 1 ] #p a r a m e t e r 1 (mu o r a l p h a )
R e s u l t s [ j , 3 ] = coef ( Param ) [ 2 ] # p a r a m e t e r 2 ( sigma o r b e t a )
R e s u l t s [ j , 4 ] = abs ( ParameterValue [ j , 2 ] −
( c u e n t a cromosoma [ j ] /sum( c u e n t a cromosoma ) ) ) / ParameterValue [ j , 2 ]
R e s u l t s [ j , 5 ] = abs ( ParameterValue [ j , 3 ] − coef ( Param ) [ 1 ] ) / ParameterValue [ j , 3 ]
R e s u l t s [ j , 6 ] = abs ( ParameterValue [ j , 4 ] − coef ( Param ) [ 2 ] ) / ParameterValue [ j , 4 ]
}
}
m e a n e r r o r=mean( R e s u l t s [ c u e n t a cromosoma >3 , c ( 4 , 5 , 6 ) ] )
return ( m e a n e r r o r ) }
GAestimationk unknown<−function ( cromosoma ) {
cromosoma=t ( round ( cromosoma , d i g i t s =0))
datosGA = d a t o s
datosGak = matrix (nrow=nrow( d a t o s ) , ncol=(ncol ( d a t o s ) + 1 ) )
f o r ( i i n 1 : 3 ) { datosGak [ , i ]= d a t o s [ , i ] }
datosGak [ , 4 ] = t ( cromosoma )
tablacromosoma=as . numeric ( table ( cromosoma ) )
R e s u l t s=matrix (NA, nrow=max( cromosoma ) , ncol =1)
c u e n t a cromosoma=rep (NA,max( cromosoma ) )
f o r ( h i n 1 :max( cromosoma ) ) { c u e n t a cromosoma [ h]=sum( cromosoma==h ) }
f o r ( i i n 1 :max( cromosoma ) ) {
i f ( c u e n t a cromosoma [ i ] >3){
P o p u l a t i o n=datosGak [ cromosoma==i , 3 ]
Param= f i t d i s t r ( P o p u l a t i o n , d e n s f u n=” normal ” )
R e s u l t s [ i , 1 ] = Param$ l o g l i k
}
}
R e s u l t a d o=−1∗sum( R e s u l t s [ c u e n t a cromosoma > 3 , 1 ] )
B.2 Algorithm for number of populations unknown 89
return ( R e s u l t a d o ) }
N c l u s t= function ( d a t o s 1 , k ) { t r y C a t c h ( gap ( d a t o s 1 , K=k , B=100) ,
warning = function (w) { l i s t ( n c l u s t=NA)
} ,
e r r o r = function ( e ) { l i s t ( n c l u s t=NA) } )
}
GenerateData<−function ( n1 , D a t a S e t t i n g s ) {
GeneratedData=c (rnorm( C a l c u l a t e d W e i g h t s [ 1 , 1 ] , mean=D a t a S e t t i n g s [ 1 , 3 ] , sd=D a t a S e t t i n g s [ 1 , 4 ] ) ) PopID=c ( rep ( 1 , C a l c u l a t e d W e i g h t s [ 1 , 1 ] ) )
f o r ( i i n 2 : k ) {
GeneratedData=c ( GeneratedData , rnorm( C a l c u l a t e d W e i g h t s [ i , 1 ] , mean=D a t a S e t t i n g s [ i , 3 ] , sd=D a t a S e t t i n g s [ i , 4 ] ) )
PopID=c ( PopID , rep ( i , C a l c u l a t e d W e i g h t s [ i , 1 ] ) ) }
Seq=seq ( 1 , n1 )
Datos=cbind ( Seq , PopID , GeneratedData )
rand<−sample (nrow( Datos ) )
D e s o r d e n a d o s=Datos [ rand , ]
return ( D e s o r d e n a d o s ) }
l i b r a r y (MASS) #For c o m p u t i n g t h e l i k e l i h o o d and f i t t i n g t h e d i s t r i b u t i o n
l i b r a r y ( g e n a l g ) #f o r t h e e v a l u a t i o n u s i n g GA
l i b r a r y ( m i x t o o l s ) #For t h e e v a l u a t i o n u s i n g EM
l i b r a r y ( m c l u s t ) #For t h e c l u s t e r i n g o f t h e d a t a
B.2 Algorithm for number of populations unknown 91
R e s u l t s C a s o 1=array (NA, dim=c (max( P a r a m e t e r s [ , 2 ] ) , 3 , max expnumber ) ) a=Sys . time ( )
f o r ( expnumber i n 1 :max expnumber ) {
n1=P a r a m e t e r s [ expnumber , 1 ]
i t e r =P a r a m e t e r s [ expnumber , 2 ]
f o r (m i n 1 : i t e r ) { ##p o n e r i t e r
s e t . s e e d=1+m
d a t o s<<−GenerateData ( n1 , D a t a S e t t i n g s [ , , expnumber ] )
R e s u l t s C a s o 1 [m, 1 , expnumber ]=k
k em c a l c u l a t e d=N c l u s t ( d a t o s [ , 3 ] , k+3)
R e s u l t s C a s o 1 [m, 2 , expnumber ]=k em c a l c u l a t e d $ n c l u s t
G A l r e s u l t s = r b g a ( s t r i n g M i n=c ( rep ( 1 , n1 ) ) , s t r i n g M a x=c ( rep ( k+3 , n1 ) ) , s u g g e s t i o n s=NULL,
p o p S i z e =200 , i t e r s =100 , mutationChance = 0 . 0 5 , e l i t i s m=NA,
e v a l F u n c=GAestimationk unknown , s h o w S e t t i n g s=FALSE, v e r b o s e=FALSE)
GAl . r e s u l t s<−summary( G A l r e s u l t s , e c h o=TRUE)
s e p a r a r<−s t r s p l i t ( GAl . r e s u l t s , ” B e s t S o l u t i o n : ” )
s e p a r a r 2<−u n l i s t ( s t r s p l i t ( s e p a r a r [ [ 1 ] ] [ 2 ] , ” ” ) )
mejor . cromosoma<−round ( as . numeric ( s e p a r a r 2 [− length ( s e p a r a r 2 ) ] ) )
c u e n t a cromosoma=rep (NA,max( mejor . cromosoma ) )
f o r ( h i n 1 :max( mejor . cromosoma ) ) { c u e n t a cromosoma [ h]=
sum( mejor . cromosoma==h ) }
R e s u l t s C a s o 1 [m, 3 , expnumber ]= length ( c u e n t a cromosoma [ c u e n t a cromosoma > 3 ] )
print ( paste ( ” i t e r a c i o n ” ,m, ” d e l e x p e r i m e n t o ” , expnumber ) )
}
}
b=Sys . time ( ) print ( b−a )
References
[1] A. Lijoi, R.H. M. ; Pr¨unster., I.: Controlling the Reinforcement in Bayesian Non-Parametric Mixture Models. En: Journal of the Royal Statistical Society. Series B (Statistical Methodology) 64 (2007), Nr. 4, p. 715–740
[2] Adele Cutler, Olga I. Cordero-Bra˜na: Minimum Hellinger Distance Estimation for Finite Mixture Models. En: Journal of the American Statistical Association 91 (1996), Nr. 436, p. 1716–1723. – ISSN 01621459
[3] Agha, M. ; Ibrahim, M. T.: Algorithm AS 203: Maximum Likelihood Estimation of Mixtures of Distributions. En: Journal of the Royal Statistical Society. Series C (Applied Statistics) 33 (1984), Nr. 3, p. 327–332
[4] Benaglia, Tatiana ; Chauveau, Didier ; Hunter, David R. ; Young, Derek: mix-tools: An R Package for Analyzing Finite Mixture Models. En: Journal of Statistical Software 32 (2009), Nr. 6, p. 1–29
[5] Beran, Rudolf: Minimum Hellinger Distance Estimates for Parametric Models. En:
The Annals of Statistics 5 (1977), Nr. 3, p. 445–463. – ISSN 00905364
[6] Crawford, Sybil L.: An Application of the Laplace Method to Finite Mixture Distri-butions. En: Journal of the American Statistical Association 89 (1994), Nr. 425, p. pp.
259–267. – ISSN 01621459
[7] Czarn, A. ; MacNish, C. ; Vijayan, K. ; Turlach, B. ; Gupta, R.: Statistical exploratory analysis of genetic algorithms. En: IEEE Transactions on evolutionary computation 8 (2004), Nr. 14, p. 405–421
[8] Dacunha-Castelle, D. ; Gassiat, E.: The Estimation of the Order of a Mixture Model. En: Bernoulli 3 (1997), Nr. 3, p. 279–299
[9] Denning, P.J.: The science of computing: Genetic algorithms. En: American Scientist 80 (1992), Nr. 1, p. 12–14
[10] Dey, D.K.: Estimation of Scale Parameters in Mixture Distributions. En: The Canadian Journal of Statistics / La Revue Canadienne de Statistique 18 (1990), Nr. 2, p. 171–178
[11] E. Susko, J. D. K. ; Chen., J.: Constrained Nonparametric Maximum-Likelihood Estimation for Mixture Models. En: The Canadian Journal of Statistics / La Revue Canadienne de Statistique 26 (1998), Nr. 4, p. 601–617
[12] Estrada, J. ; Camacho, J.A ; Restrepo, M.T. ; Parra, C.M.: Par´ametros antro-pom´etricos de la poblaci´on laboral colombiana 1995 (acopla95). En: Rev. Fac. Nac.
Salud P´ublica 15 (1988), Nr. 2, p. 112–139
[13] Fouskakis, D. ; Draper, D.: Stochastic optimization: a review. En: International Statistical Review / Revue Internationale de Statistique 70, Nr. 3
[14] Gallegos, M. ; Ritter, G.: Trimmed ML Estimation of Contaminated Mixtures. En:
Sankhy¯a: The Indian Journal of Statistics, Series A 71 (2009), Nr. 2, p. 164–220 [15] Glover, Fred: Tabu Search—Part I. En: ORSA Journal on Computing 1 (1989), Nr.
3, p. 190–206
[16] Harrington, Justin: lga: Tools for linear grouping analysis (LGA), 2012. – R package version 1.1-1
[17] Haupt, R.L. ; Haupt., S.E.: Practical Genetic Algorithms. Wiley, 2004
[18] Jingjing WU, Rohana J. K.: On minimum Hellinger distance estimation. En: The Canadian Journal of Statistics / La Revue Canadienne de Statistique 37 (2009), Nr. 4, p. 514–533. – ISSN 03195724
[19] K. Do, P. M. ; Tang., F.: A Bayesian Mixture Model for Differential Gene Expression.
En: Journal of the Royal Statistical Society. Series C (Applied Statistics) 54 (2005), Nr.
3, p. 627–624
[20] Lahoz-Beltra, R. ; Perales-Gravan., C.: A survey of nonparametric tests for the statistical analysis of evolutionary computational experiments. En: International Journal Information Theories and Application 17 (2010), Nr. 1, p. 49–61
[21] McCrea, Rachel S. ; Morgan, Byron J. T. ; Cole, Diana J.: Age-dependent mixture models for recovery data on animals marked at unknown age. En: Journal of the Royal Statistical Society. Series C (Applied Statistics) 62 (2013), Nr. 1, p. pp. 101–113. – ISSN 00359254
[22] McLachlan, G.J. ; Basford., K.E.: Mixture models: inference and applications to clustering. Marcel Dekker, 1988
[23] N. Metropolis, M.N. Rosenbluth A.H. T. ; Teller., E.: Equation of state calculation by fast computing machines. En: Journal of Chemical Physics 21 (1953), Nr. 6, p. 1087–
1091
References 95
[24] Nemec, James ; NEMEC, Amanda F. L.: Mixture models for studying stellar popula-tions. I. Univariate mixture models, parameter estimation, and the number of discrete population components. En: Publications of the Astronomical Society of the Pacific 103 (1991), Nr. 659, p. pp. 95–121. – ISSN 00046280
[25] Quintana, Fernando A. ; Newton, Michael A.: Computational Aspects of Non-parametric Bayesian Analysis with Applications to the Modeling of Multiple Binary Sequences. En: Journal of Computational and Graphical Statistics 9 (2000), Nr. 4, p.
pp. 711–737. – ISSN 10618600
[26] R Core Team: R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing, 2014
[27] R. Tibshirani, G. W. ; Hastie, T.: Estimate the number of clusters in a data set via the gap statistic. En: Journal of Royal Statistical Society B 63, Nr. 2
[28] Reschenhofer, E.: The Bimodality Principle. En: Journal of Statistics Education 9 (2001), Nr. 1
[29] Reynolds, J.H. ; Templin., W.D.: Comparing Mixture Estimates by Parametric Bootstrapping Likelihood Ratios. En: Journal of Agricultural, Biological, and Environ-mental Statistics 9 (2004), Nr. 1, p. 54–74
[30] S. Chatterjee, M. L. ; Lynch, L.A.: Genetic algorithms and their statistical appli-cation: an introduction. En: Computational Statics and Data Analysis 22 (1996), Nr.
6, p. 219–234
[31] Santner, T. ; Williams, B. ; Notz, W.: The Design and Analysis of Computer Experiments. Springer, 2003. – ISBN 0387954201
[32] Scrucca., L.: GA: A Package for Genetic Algorithms in R. En: Journal of Statistical Software 53 (2013), Nr. 4, p. 1–37
[33] Silver., EA.: An overview of heuristic solution methods. En: Journal of the Operational Research Society 55 (2004), p. 936–956
[34] Snee., R.D.: Techniques for the analysis of mixture data. En: Technometrics 15 (1973), Nr. 3, p. 517–528
[35] Stephens, Matthew: Dealing with Label Switching in Mixture Models. En: Journal of the Royal Statistical Society. Series B (Statistical Methodology) 62 (2000), Nr. 4, p.
pp. 795–809. – ISSN 13697412
[36] Tolvi., J.: Genetic algorithms for outlier detection and variable selection in linear regression models. En: Soft Computing 8 (2004), p. 527–533
[37] Venables, W. N. ; Ripley, B. D.: Modern Applied Statistics with S. Fourth. New York : Springer, 2002. – ISBN 0-387-95457-0
[38] West., M.: Approximating Posterior Distributions by Mixture. En: Journal of the Royal Statistical Society. Series B (Methodological) 55 (1993), Nr. 2, p. 409–422 [39] Willighagen, Egon ; Ballings, Michel: genalg: R Based Genetic Algorithm, 2015. –
R package version 0.2.0
[40] Zhu, M. ; Chipman., H.: Darwinian evolution in parallel universes: A parallel genetic algorithm for variable selection. En: Technometrics 48, Nr. 4