Algorismes IRWLS. May 2015
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1 Algorsmes IRWLS May 2015
2 1 Introducton Ths report studes the Iteratvely Re-Weghted Least Squares (IRWLS) algorthm for fndng the Maxmum Lkelhood (ML) estmators of a Generalsed Lnear Model (GLM) coeffcents. The data s related to metal treatments. Four dfferent temperatures z, {1,, 4} are appled to metal ngots, and the result s evaluated wth a bnary varable whch explans whether the ngot s ready for rollng or not. Let x be the number of successes (treatments not ready) and n the number of gnots tested n the condtons of temperature z n trals x notready z heat Table 1: Data for the study In the frst part of the practce the maxmum lkelhood estmators are found assumng that our data follows a Bnomal dstrbuton. X B(n, p (β)), = 1,, 4, β = (β 0, β 1 ), p (β) = eβ0+β z 1 + e β0+β z For ths purpose, the optm functon of R s used. Ths functon mplements solvers for unconstraned optmzaton problems. Wth the transformaton k (x) = x/(1 + x) we have a problem wthout restrctons on the parameters. Dfferent solvers are tested, ncludng the dervatve-free Nelder-Mead method. In the second part of the report a dfferent model s assumed. A GLM of the posson famly s consdered wth a logarthmc lnk functon. X P osson(λ ), = 1,, 4 log(λ ) = β 0 + β 1 n + β 2 z, = 1,, 4 The algorthm IRWLS s derved from the Fsher Scorng method and mplemented n R. Its results are compared wth the functon glm to prove that the same result s attaned. 2 Bnomal Dstrbuton Assumpton 2.1 Lkelhood Functon It s known that for one Bnomal dstrbuton the probablty densty functon s f x = n p x (1 p ) n x The log-lkelhood functon of the four samples wth the p defned above altogether s 4 L = log n + x log(p ) + (n x ) log(1 p ) x x We can plot the contour of ths functon as t s 2-Dmensonal wth the followng R-code 2
3 1 p r n t. l o g l l < functon ( Beta0, Beta1, x1=0,x2=2,x3=7,x4=3, z1 =7, z2 =14, z3 =27, z4 =57,n1=55,n2=157, 2 n3=159,n4=16){ 3 log ( ( f a c t o r a l ( n1 )/( f a c t o r a l ( x1 ) f a c t o r a l ( n1 x1 ))))+ x1 log ( ( exp( Beta0+Beta1 z1 )/(1 4 +exp( Beta0+Beta1 z1 ) ) ) ) + ( n1 x1 ) log (1 (exp( Beta0+Beta1 z1 )/(1+exp( Beta0+Beta1 z1 ) ) ) ) 5 +log ( ( f a c t o r a l ( n2 )/( f a c t o r a l ( x2 ) f a c t o r a l ( n2 x2 ))))+ x2 log ( ( exp( Beta0+Beta1 z2 )/(1 6 +exp( Beta0+Beta1 z2 ) ) ) ) + ( n2 x2 ) log (1 (exp( Beta0+Beta1 z2 )/(1+exp( Beta0+Beta1 z2 ) ) ) ) 7 +log ( ( f a c t o r a l ( n3 )/( f a c t o r a l ( x3 ) f a c t o r a l ( n3 x3 ))))+ x3 log ( ( exp( Beta0+Beta1 z3 )/(1 8 +exp( Beta0+Beta1 z3 ) ) ) ) + ( n3 x3 ) log (1 (exp( Beta0+Beta1 z3 )/(1+exp( Beta0+Beta1 z3 ) ) ) ) 9 +log ( ( f a c t o r a l ( n4 )/( f a c t o r a l ( x4 ) f a c t o r a l ( n4 x4 ))))+ x4 log ( ( exp( Beta0+Beta1 z4 )/(1 10 +exp( Beta0+Beta1 z4 ) ) ) ) + ( n4 x4 ) log (1 (exp( Beta0+Beta1 z4 )/(1+exp( Beta0+Beta1 z4 ) ) ) ) 11 } # Plot o f the f u n c t o n 14 pb0 < seq( 6, 4,length=101) 15 pb1 < seq (0.05, 0.09, length=101) 16 l o g l l. B 0. B 1 < outer (pb0, pb1, functon (B0, B1){ p r n t. l o g l l (B0, B1 ) } ) 17 wndows ( ) 18 contour (pb0, pb1, l o g l l. B 0. B 1, nlevels =100) Fgure 1: Lkelhood functon A maxmum s observed at a pont around 5 and In order to use the functon optm the decson varables must be defned as a numerc vector. Hence, a slght modfcaton must be ntroduced: 1 l o g l l < functon ( Beta0Beta1, x1=0,x2=2,x3=7,x4=3, z1 =7, z2 =14, z3 =27, z4 =57,n1=55,n2= Beta0 = Beta0Beta1 [ 1 ] 3 Beta1 = Beta0Beta1 [ 2 ] ( log ( ( f a c t o r a l ( n1 )/( f a c t o r a l ( x1 ) f a c t o r a l ( n1 x1 ))))+ x1 log ( ( exp( Beta } Notce that the return s multpled by mnus one. Ths way we face a mnmzaton problem. The same result can be acheved passng an argument to the functon argument. 3
4 2.2 Functon Scores In order to derve the gradent of the log-lkelhood equaton we know that L 4 ( ) β = p β (1 p ) β x + (1 p ) n (1) β p 1 p 1 p Hence, we can defne the followng 2 4 matrx Let us also defne the 2 1 matrx N And the gradent can be stated as Mj = 1 p j N = p j + 1 p j (2) β 1 1 p j β 1 4 k=1 1 p k n k (3) 1 p k β 1 L = M x + N, t(x) = [x 1 x 2 x 3 x 4 ] (4) 1 2 g r a d. l l < functon ( Beta0Beta1, x1=0,x2=2,x3=7,x4=3, z1 =7, z2 =14, z3 =27, z4 =57,n1=55, n Beta0 = Beta0Beta1 [ 1 ] 4 Beta1 = Beta0Beta1 [ 2 ] 5 6 p1 = exp( Beta0+Beta1 z1 )/(1+exp( Beta0+Beta1 z1 ) ) 7 p2 = exp( Beta0+Beta1 z2 )/(1+exp( Beta0+Beta1 z2 ) ) 8 p3 = exp( Beta0+Beta1 z3 )/(1+exp( Beta0+Beta1 z3 ) ) 9 p4 = exp( Beta0+Beta1 z4 )/(1+exp( Beta0+Beta1 z4 ) ) d f f. p 1. B 0 = exp( Beta1 z1+beta0 )/(1+exp( Beta0+Beta1 z1 )) exp(2 Beta1 z d f f. p 2. B 0 = exp( Beta1 z2+beta0 )/(1+exp( Beta0+Beta1 z2 )) exp(2 Beta1 z d f f. p 3. B 0 = exp( Beta1 z3+beta0 )/(1+exp( Beta0+Beta1 z3 )) exp(2 Beta1 z d f f. p 4. B 0 = exp( Beta1 z4+beta0 )/(1+exp( Beta0+Beta1 z4 )) exp(2 Beta1 z d f f. p 1. B 1 = ( z1 exp( Beta1 z1+beta0 ) ) /(exp( Beta1 z1+beta0 )+1) ( z1 exp(2 B e d f f. p 2. B 1 = ( z2 exp( Beta1 z2+beta0 ) ) /(exp( Beta1 z2+beta0 )+1) ( z2 exp(2 B e d f f. p 3. B 1 = ( z3 exp( Beta1 z3+beta0 ) ) /(exp( Beta1 z3+beta0 )+1) ( z3 exp(2 B e d f f. p 4. B 1 = ( z4 exp( Beta1 z4+beta0 ) ) /(exp( Beta1 z4+beta0 )+1) ( z4 exp(2 B e M11 = 1/p1 d f f. p 1. B 0 + 1/(1 p1 ) d f f. p 1. B 0 22 M12 = 1/p2 d f f. p 2. B 0 + 1/(1 p2 ) d f f. p 2. B 0 23 M13 = 1/p3 d f f. p 3. B 0 + 1/(1 p3 ) d f f. p 3. B 0 24 M14 = 1/p4 d f f. p 4. B 0 + 1/(1 p4 ) d f f. p 4. B M21 = 1/p1 d f f. p 1. B 1 + 1/(1 p1 ) d f f. p 1. B 1 27 M22 = 1/p2 d f f. p 2. B 1 + 1/(1 p2 ) d f f. p 2. B 1 28 M23 = 1/p3 d f f. p 3. B 1 + 1/(1 p3 ) d f f. p 3. B 1 29 M24 = 1/p4 d f f. p 4. B 1 + 1/(1 p4 ) d f f. p 4. B N1 = 1/(1 p1 ) d f f. p 1. B 0 n1 1/(1 p2 ) d f f. p 2. B 0 n2 1/(1 p3 ) d f f. p 3. B N2 = 1/(1 p1 ) d f f. p 1. B 1 n1 1/(1 p2 ) d f f. p 2. B 1 n2 1/(1 p3 ) d f f. p 3. B M = matrx ( c (M11, M12, M13, M14, M21, M22, M23, M24), byrow=t, nrow=2) 35 N = matrx ( c (N1, N2),nrow=2) X = matrx ( c ( x1, x2, x3, x4 ),nrow=4) return( 1 (M% %X+N) ) 40 } 4
5 2.3 Optmzaton of ML functon In ths secton dfferent optmzaton algorthms are consdered for mnmzng the mnus logarthm of the lkelhood functon. The default solver s the Nelder-Mead algorthm, whch does not consder gradents but only functon evaluatons. For the other algorthms, the gradent s entered as a parameter. If nformaton of the optmzaton procedure s desred, the nput parameter trace s to be used. Wth the Hessan parameter set to TRUE a numerc approxmaton at the optmum pont s gven. When t comes to the Conjugate Gradent, there s a specfc command for choosng a step rule: type. The maxmum number of teratons can be enlarged wth the parameter maxt. All ntal ponts are set to (0, 0). Notce that the Hessan needs not to be computed for these methods, and a cheaper approxmaton s used nstead. In the prevous exercses, where de Fsher Scorng algorthm was studed, the Hessan matrx was derved analytcally for some cases. The general purpose optmzaton algorthms assume the Hessan to be dramatcally more expensve to compute than the gradent or the objectve functon. However, n convex problems wth an easy-to-compute Hessan matrx other algorthms may be more effcent. 1 2 X < data.frame ( matrx ( rep ( 0, 2 0 ),nrow=5)) 3 names(x) < c ( Nelder Mead, BFGS, CG, L BFGS B ) 4 rownames(x) < c ( counts f o b j., counts grad, s o l. B 0, s o l. B 1, o b j. v a l ) # a. 1 ) Nelder Mead 8 opt1 < optm( par = c ( 0, 0 ), fn = l o g l l, control = l s t ( trace =1)) 9 # a. 2 ) BFGS 10 opt2 < optm( par = c ( 0, 0 ), fn = l o g l l, gr = g r a d. l l, method = BFGS, h e s s a n=t) 11 # a. 3 ) CG 12 opt3 < optm( par = c ( 0, 0 ), fn = l o g l l, gr = g r a d. l l, method = CG, 13 control = l s t ( maxt =10000, type =3), h e s s a n=t) 14 # a. 4 ) L BFGS B 15 opt4 < optm( par = c ( 0, 0 ), fn = l o g l l, gr = g r a d. l l, method = L BFGS B, h e s s a n=t) # S o l u t o n > X 21 Nelder Mead BFGS CG L BFGS B 22 counts f o b j e counts grad NA e s o l. B e s o l. B e o b j. v a l e All the algorthms attan a smlar soluton. The Conjugate Gradent (CG) s not able to satsfy ts nternal optmalty tolerance and reaches the maxmum number of teratons. The Nelder-Mead method needs to evaluate the objectve functon 81 tmes. the BFGS method reduces ths number to 46 at the cost of evaluatng the gradent 14 tmes. When the gradent s known, dervatve-free methods are not recommended, especally when functons are non-convex. The BFGS varant evaluates 22 tmes both the gradent and the objectve functon. The selecton of the method depends on the computatonal cost of evaluatng the objectve functon and the gradent. 5
6 2.4 Covarance matrx estmaton As stated before, a numercal approxmaton to the Hessan can be obtaned wth optm. The nverse of ths matrx s an approxmaton to the covarance matrx. 1 solve ( opt2$ h e s s a n ) 2 # S o l u t o n 3 4 > solve ( opt2$ h e s s a n ) 5 [, 1 ] [, 2 ] 6 [ 1, ] [ 2, ] Ths result can be compared to the result obtaned usng the Fshers Informaton computed analytcally. It s easy to compute the Fsher s Informaton matrx as the varance of the scores functon. V ar(m x + N) = M V ar(x) t(m) + 0 (5) 1 F s h e r. n f < functon ( Beta0, Beta1, x1=0,x2=2,x3=7,x4=3, z1 =7, z2 =14, z3 =27, z4= (... ) 4 v.x1 < n1 p1 (1 p1 ) 5 v.x2 < n2 p2 (1 p2 ) 6 v.x3 < n3 p3 (1 p3 ) 7 v.x4 < n4 p4 (1 p4 ) 8 9 Var.X < dag ( c ( v.x1, v.x2, v.x3, v.x4 ) ) return (M% %Var.X% %t (M) ) # S o l u t o n > solve ( F s h e r. n f ( , ) ) 16 [, 1 ] [, 2 ] 17 [ 1, ] [ 2, ] As t was expected the Hessan provded wth the functon optm s a very good approxmaton to the analtcally derved Fshers nformaton matrx. Hence, t can be used to estmate the varance of the maxmum lkelhood estmators. 2.5 Bootstrap Covarance matrx estmaton In ths secton a parametrc bootstrap procedure s used to estmate the varance of the estmators wth a dervatve-free method. Our Data s assumed to follow a bnomal dstrbuton wth the parameters obtaned maxmzng the lkelhood functon. In each teraton the maxmum lkelhood estmator must be obtaned wth the dervatve-free algorthm as we do not have an analytc expresson. 1 # c ) e s t m a c o m t j a n a n t b o o t s t r a p parametrc 2 # assumm l e s p r o b a b l t a t s son l e s que hem t r o b a t ( nomes tenm l a mostra ) 3 B0 = X$L BFGS B [ 3 ] 4 B1 = X$L BFGS B [ 4 ] 5 6 s e t. s e e d (7234) 7 s. B = p1 < exp(b0+b1 7)/(1+exp(B0+B1 7 ) ) 10 p2 < exp(b0+b1 14)/(1+exp(B0+B1 1 4 ) ) 11 p3 < exp(b0+b1 27)/(1+exp(B0+B1 2 7 ) ) 12 p4 < exp(b0+b1 57)/(1+exp(B0+B1 5 7 ) ) 6
7 13 14 r e s. b < r e p l c a t e ( s.b, 15 { 16 # e s t m a c o per metode b o o t s t rap parametrc 17 x1.b < sum(rbnom ( 5 5, 1, p1 ) ) 18 x2.b < sum(rbnom ( 1 5 7, 1, p2 ) ) 19 x3.b < sum(rbnom ( 1 5 9, 1, p3 ) ) 20 x4.b < sum(rbnom ( 1 6, 1, p4 ) ) opt.b < optm( par = c ( 0, 0 ), fn = l o g l l, x1=x1.b, x2=x2.b, x3=x3.b, x4=x4.b ) 23 opt.b$par 24 } 25 ) mean( r e s. b [ 1, ] ) 28 mean( r e s. b [ 2, ] ) e s t m a t e. v a r. b < matrx ( rep ( 0, 4 ),nrow=2) 31 e s t m a t e. v a r. b [ 1, 1 ] < var ( r e s. b [ 1, ] ) 32 e s t m a t e. v a r. b [ 2, 2 ] < var ( r e s. b [ 2, ] ) 33 e s t m a t e. v a r. b [ 1, 2 ] < cov ( r e s. b [ 1, ], r e s. b [ 2, ] ) 34 e s t m a t e. v a r. b [ 2, 1 ] < cov ( r e s. b [ 1, ], r e s. b [ 2, ] ) # S o l u t o n 37 > e s t m a t e. v a r. b 38 [, 1 ] [, 2 ] 39 [ 1, ] [ 2, ] The covarance matrx obtaned s smlar to the result obtaned n the prevous secton. Bootstrap s a possblty of estmatng the varance of the maxmum lkelhood estmators f the gradent s unknown or very dffcult to compute. It has to be taken nto account that the number of functon evaluatons of the dervatve-free method s multpled by the number of bootstrap samples. 3 Posson GLM assumpton 3.1 Algorthm deducton The densty functon of a Posson dstrbuton s Then, the lkelyhood functon can be derved as And the log-lkelhood functon L = f x l = = λx x! (6) e λ n ( λ x ) log x! = e λ λ x x! e λ (7) x log(λ ) log(x!) λ (8) If we substtute L = x (B 0 + B 1 n + B 2 z ) log(x!) e B0+B1 n+b2 z (9) 7
8 We then derve n order to get the gradent L B 1 = L B 2 = L B 0 = x x n x z e B0+B1 n+b2 z = e B0+B1 n+b2 z n = e B0+B1 n+b2 z z = x λ (10) x n x z λ n (11) λ z (12) Ths gradent can be expressed n a matrx form as x λ 1 L = n 1 n 2 n 3 n 4 x 2 λ 2 = X [x λ] (13) x z 1 z 2 z 3 z 4 3 λ 3 x 4 λ 4 And the Fsher s Informaton matrx λ λ I x = var( L) = X X t = X W X t (14) 0 0 λ λ 4 Now we formulate an teraton of the Fsher Scorng method B0 k+1 B1 k+1 B k+1 2 = And after some algebra and groupng terms we obtan When we have a model of the form B k 0 B k 1 B k 2 + (X W Xt ) 1 X W (W 1 (x λ)) (15) (X W X t ) 1 (X W ) ((X t B) + (W 1 (x λ))) (16) Y = X β + ε (17) where the varance of the error s not constant t s called Weghted Least Squares. The maxmum lkelyhood estmator of β n those models s β ML = (X t W X t ) 1 X t W Y (18) Hence, f we make a varable change we can solve an teraton of the Fsher Scorng algorthm solvng a WLS Z = ((X t B) + (W 1 (x λ))) (19) z = B 0 + n B 1 + z B 2 + x λ λ (20) 8
9 3.2 Algorthm mplementaton In ths secton the algorthm s mplemented n R followng the defntons of the prevous secton. 1 # Algorsme 2 B 0 < NULL 3 B 1 < NULL 4 B 2 < NULL 5 6 B 0 [ 1 ] = 1 7 B 1 [ 1 ] = B 2 [ 1 ] = n. t e r = lambda < matrx ( 0,nrow=n. t e r, ncol=4) 13 z < matrx ( 0,nrow=n. t e r, ncol=4) 14 v < matrx ( 0,nrow=n. t e r, ncol=4) for ( k n 1 : n. t e r ){ lambda [ k, ] = exp(b 0 [ k]+n B 1 [ k]+ z t B 2 [ k ] ) 19 z [ k, ] = B 0 [ k ] + B 1 [ k ] n + B 2 [ k ] z t + ( x lambda [ k, ] ) /lambda [ k, ] 20 v [ k, ] = lambda [ k, ] z p < z [ k, ] 23 v p < v [ k, ] 24 model < lm( z p n+z t, weghts = v p ) B 0 [ k+1] = model$ c o e f f c e n t s [ 1 ] 27 B 1 [ k+1] = model$ c o e f f c e n t s [ 2 ] 28 B 2 [ k+1] = model$ c o e f f c e n t s [ 3 ] 29 } 3.3 Algorthm executon 1 # S o l u t o n o f the algorthm 2 > lambda 3 [, 1 ] [, 2 ] [, 3 ] [, 4 ] 4 [ 1, ] [ 2, ] [ 3, ] [ 4, ] [ 5, ] [ 6, ] [ 7, ] [ 8, ] [ 9, ] [ 1 0, ] > cbnd (B 0,B 1,B 2) 16 B 0 B 1 B 2 17 [ 1, ] [ 2, ] [ 3, ] [ 4, ] [ 5, ] [ 6, ] [ 7, ]
10 24 [ 8, ] [ 9, ] [ 1 0, ] # r e s u l t a t s de l a f u n c o glm 29 glm( x n + z t, famly = posson ) # S o l u t o n > glm( x n + z t, famly=posson ) C o e f f c e n t s : 36 ( I n t e r c e p t ) n z t After 8 teratons an optmum s attaned. The result s the same wth our mplementaton and wth the functon glm of R. 10
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