!" % ' ) ( # # & $ *! + , * - + + . ( ! % % 0 1 ( % 0 1 / + +* ",+ ! $ * + ! !' ( 2 3 "4 4 $ ' *+ 5+ ! "& $ % ( ( & 6 6 & & 7 # 6 # 7 8 9 :+* ! ; 7 , *+ < 4 % ! , , * * !" $ ( % = 7 ( = !+* . 4 4 4 & % 0 1 * ! 4 4 + . >% + % ( & 4 ? 0 1 @ % & % , A ";& <( 0= BCCD$ ) ) 4 ( ( 7 ( " " & ( $ $ ( " $ %& ! +, *. " $ %& 62 6 ' " ! E 6( , 4 %& = 1 0 FGGH$ 6( %& & I $( "5 % = " ! % & 4J $( 0.0 0.2 0.4 Penalty 0.6 0.8 1.0 Power family 0.0 0.2 0.4 0.6 0.8 1.0 0.8 1.0 Coefficient absolute value 0.0 0.2 0.4 Penalty 0.6 0.8 1.0 Generalized elastic net 0.0 0.2 0.4 0.6 I "B$ ' ' ! ( "F$ % "D$ I 4 ( % 4 4 6 "B$ + ' ( % / !' 4 1 4 % ( 4 E 4K ( + E ( 4 K * $ " , 4 8 ( = L+, * ( * 7 *+ " 1 + ' ) & $ 7 %& " $ 1 6 " $ ( & " ( $ ' 4 "B$ "F$ E 4 0 "' $ ( 6 " 6$ L+, * 1 E "M$ + 2, J % 6 2 & 6 % & & 2 + ( 4 ( 4 E ( * % " $ , 6 ' 4 4 % % 4 2 ' % 6 4 4 1 )4 ( & ( 4 )4 & % % " % % "& & 1 $( % $ % ( & ( *+ N 6 2 " " FGGF$ 6$2 & 0N' Elastic Net 1.9 Elastic Net 1.5 3 9 400 0.0 0.1 0.2 0.3 0.4 200 7 0 5 2 4 10 8 6 1 −200 8 10 1 6 Coefficients 200 7 0 4 −200 Coefficients 400 3 9 5 2 0.5 0.0 0.1 0.2 0.3 R − squared R − squared Elastic Net 1.25 Lasso 0.4 0.5 3 9 400 0.0 0.1 0.2 0.3 R − squared 0.4 0.5 200 7 0 5 2 4 10 8 6 1 −200 10 8 6 1 Coefficients 200 7 0 4 −200 Coefficients 400 3 9 5 2 0.0 0.1 0.2 0.3 R − squared 0.4 0.5 0.1 0.2 0.3 0.4 500 0 8 10 1 7 2 0.5 5 0.0 0.1 0.2 0.3 Elastic Net 0.25 Elastic Net 0.0 −500 2 5 0.1 0.2 0.3 R − squared 0.4 0.5 3 500 5 6 8 9 7 1 4 2 0 0.5 9 9 3 999 9 9 3 3 3 9 6 3 33 6 6 6 4 4 4 44 4 88 8 110 5 5 6 6 6 61 8 8 8 88 7 7 7 77 71 1 4 21 21 211 7 27 55 2 22 −500 8 10 1 7 Coefficients 500 0 1000 R − squared 9 3 6 4 0.0 0.4 R − squared 1000 0.0 Coefficients 9 3 6 4 −500 0 4 7 10 8 6 1 5 2 Coefficients 500 3 9 −500 Coefficients 1000 Elastic Net 0.5 1000 Lasso 55 5 5 0.0 0.1 0.2 0.3 R − squared 0.4 0.5 * # %4+ 4 2 % N 6 " 6$2 & 2 % 0N' Elastic Net 1.9 Elastic Net 1.5 7 0.05 0.10 0.15 0.4 0.2 7 0.20 0.00 0.05 0.10 0.15 Explained devience Explained devience Elastic Net 1.25 Lasso 1 8 4 0.6 9 0.4 5 2 3 6 1 8 4 7 0.00 0.05 0.10 0.15 Explained devience 0.20 0.20 0.2 0.2 0.4 5 2 3 6 Coefficients 0.6 9 0.0 Coefficients 1 4 8 0.0 0.00 5 2 3 6 0.0 8 Coefficients 0.2 0.0 1 4 −0.2 Coefficients 0.4 5 2 3 6 9 0.6 9 7 0.00 0.05 0.10 0.15 Explained devience 0.20 Lasso Elastic Net 0.5 0.6 8 4 7 0.05 0.10 0.15 0.4 0.2 1 4 8 7 0.20 0.00 0.05 0.10 0.15 Explained devience Explained devience Elastic Net 0.25 Elastic Net 0.0 0.2 0.4 5 6 2 3 −0.2 0.0 1 4 Coefficients 0.6 9 0.20 0.2 0.4 0.6 0.8 0.00 5 6 2 3 0.0 0.0 1 9 5 6 2 3 1 4 8 8 7 0.00 0.05 0.10 0.15 Explained devience 0.20 −0.2 Coefficients 9 −0.2 Coefficients 0.4 5 2 3 6 0.2 Coefficients 0.6 9 7 0.00 0.05 0.10 0.15 Explained devience 0.20 ( % O 6 2 K ! " " # # # 500 200 100 50 20 Non zero coefficients 10 5 2 1 0.0 0.2 0.4 0.6 0.8 R − squared B C2 B P2 B H2 B G " 2 & % $2 G H2 G D2 G F2 G B2 1.0 * # % O K ! 7 $ $ 6 500 100 Non zero coefficients 50 10 5 1 0.0 0.2 0.4 0.6 0.8 Explained devience B C2 B P2 B H2 B G " 2 & % $2 G P2 G H2 G D2 G G 1.0 E 4 " $ ( % O 6 2 &% , 4 ; < %4 ( ' ( 1 ! 1 ! 2 ! ! 2 "B H$ " 0$ Corr = 0.4 50 100 Variables entered 0.6 0.4 0 0.0 0.2 Model error 150 0.8 200 1.0 Corr = 0.4 step 0.0 0.1 0.2 0.5 lasso step elnet 1.5 0.0 0.2 0.5 lasso elnet 1.5 Corr = 0.4, a > 0 0.1 0.2 Model error 0.6 0.4 0.2 0.0 0.0 Model error 0.3 0.8 0.4 1.0 Corr = 0.0 0.1 step 0.0 0.1 0.2 0.5 lasso elnet 1.5 step 0.0 0.1 0.2 0.5 lasso elnet 1.5 Corr = 0.4 25 15 0.6 0 0.2 5 0.4 Model error 0.8 Corr = 0.4 0.2 bridge lasso 0.0 0.1 lasso elnet 1.5 20 15 5 10 0.6 0 0.2 Model error 0.5 Corr = 0.0 1.0 Corr = 0.0 0.2 0.1 bridge lasso 0.0 0.1 0.5 lasso elnet 1.5 Corr = 0.4, a > 0 30 0.20 10 20 0.15 0 0.10 Model error 40 Corr = 0.4, a > 0 0.2 elnet 1.5 bridge lasso 0.0 0.1 0.2 0.5 lasso elnet 1.5 O "B$ 6" $ " "F$ % & "D$ $ % % " & $( & &% 4 0 4 % Corr = 0.4, Post Variables entered 0.6 0 0.0 0.2 50 0.4 Model error 100 0.8 150 1.0 Corr = 0.4, Post step 0.0 0.1 0.2 0.5 lasso elnet 1.5 step 0.0 0.2 0.5 lasso elnet 1.5 lasso elnet 1.5 Corr = 0.4, a > 0, Post 0.0 0.2 0.0 0.2 0.1 0.4 Model error 0.6 0.3 0.8 1.0 0.4 Corr = 0, Post Model error 0.1 step 0.0 0.1 0.2 0.5 lasso elnet 1.5 step 0.0 0.1 0.2 0.5 ! * "B$ & " ! $ 7 "F$ & 4 "D$ 14 & "H$ 6 & "Q$ & & = 4 4 % &% & 0 ( ' 4 & E 4 = ( 6 % 1 4( 4 6 6 2 % 4 ,% 0 6 2 % 4 I ! " # 6$ ( R % BG7 6 ( S% E " % & 6$ 1 4 ! " $ +*/ 4 (##111O %#TA4 # 0 #' %#TA4 # #' + 4 (##111O 0
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