% Created 2016-11-26 Sat 19:04 \documentclass[11pt]{article} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{fixltx2e} \usepackage{graphicx} \usepackage{grffile} \usepackage{longtable} \usepackage{wrapfig} \usepackage{rotating} \usepackage[normalem]{ulem} \usepackage{amsmath} \usepackage{textcomp} \usepackage{amssymb} \usepackage{capt-of} \usepackage{hyperref} \author{Mark Clements} \date{\today} \title{} \hypersetup{ pdfauthor={Mark Clements}, pdftitle={}, pdfkeywords={}, pdfsubject={}, pdfcreator={Emacs 24.4.1 (Org mode 8.3.4)}, pdflang={English}} \begin{document} \tableofcontents \section{Axiom/Fricas source file} \label{sec:orgheadline8} \subsection{General case} \label{sec:orgheadline1} \begin{verbatim} (48) -> )cl p all Compiled code for rule2 has been cleared. Compiled code for G has been cleared. (48) -> η := operator 'η (48) η Type: BasicOperator (49) -> G := operator 'G (49) G Type: BasicOperator (50) -> rule1 := rule (D(η(t,β),β)==X; D(η(t,β),t)==XD*β; D(η(t,β),[t,β])==XD; η(t,β)==X*β) (50) {η (t,β) == X,η (t,β) == XD β,η (t,β) == XD,η(t,β) == X β} ,2 ,1 ,1,2 Type: Ruleset(Integer,Integer,Expression(Integer)) (51) -> S := G(η(t,β)) (51) G(η(t,β)) Type: Expression(Integer) (52) -> H := -log(S) (52) - log(G(η(t,β))) Type: Expression(Integer) (53) -> h := D(H,t) , G (η(t,β))η (t,β) ,1 (53) - ------------------ G(η(t,β)) Type: Expression(Integer) (54) -> [D(expr,β) for expr in [S,H,h]] (54) , G (η(t,β))η (t,β) , ,2 [G (η(t,β))η (t,β), - ------------------, ,2 G(η(t,β)) , ,, - G(η(t,β))G (η(t,β))η (t,β) - G(η(t,β))η (t,β)η (t,β)G (η(t,β)) ,1,2 ,1 ,2 + , 2 G (η(t,β)) η (t,β)η (t,β) ,1 ,2 / 2 G(η(t,β)) ] Type: List(Expression(Integer)) (55) -> [rule1 expr for expr in [S,H,h]] , XD βG (X β) (55) [G(X β),- log(G(X β)),- -----------] G(X β) Type: List(Expression(Integer)) (56) -> [rule1 D(expr,β) for expr in [S,H,h]] (56) , XG (X β) , [XG (X β), - --------, G(X β) ,, , 2 , - X XD β G(X β)G (X β) + X XD β G (X β) - XD G(X β)G (X β) ------------------------------------------------------------] 2 G(X β) Type: List(Expression(Integer)) \end{verbatim} \subsection{Proportional hazards} \label{sec:orgheadline2} \begin{verbatim} (57) -> G(x) == exp(-exp(x)) Type: Void (58) -> S := G(η(t,β)) Compiling function G with type Expression(Integer) -> Expression( Integer) η(t,β) - %e (58) %e Type: Expression(Integer) (59) -> H := -log(S) η(t,β) (59) %e Type: Expression(Integer) (60) -> h := D(H,t) η(t,β) (60) %e η (t,β) ,1 Type: Expression(Integer) (61) -> D(G(x),x) Compiling function G with type Variable(x) -> Expression(Integer) x x - %e (61) - %e %e Type: Expression(Integer) (62) -> [rule1 expr for expr in [S,H,h]] X β - %e X β X β (62) [%e ,%e ,XD β %e ] Type: List(Expression(Integer)) (63) -> [rule1 D(expr,β) for expr in [S,H,h]] X β X β - %e X β X β (63) [- X %e %e ,X %e ,(X XD β + XD)%e ] Type: List(Expression(Integer)) \end{verbatim} \subsection{Proportional odds} \label{sec:orgheadline3} \begin{verbatim} (64) -> G(x) == 1/(1+exp(x)) Compiled code for G has been cleared. 1 old definition(s) deleted for function or rule G Type: Void (65) -> S := G(η(t,β)) Compiling function G with type Expression(Integer) -> Expression( Integer) 1 (65) ------------ η(t,β) %e + 1 Type: Expression(Integer) (66) -> H := -log(S) 1 (66) - log(------------) η(t,β) %e + 1 Type: Expression(Integer) (67) -> h := D(H,t) η(t,β) %e η (t,β) ,1 (67) ---------------- η(t,β) %e + 1 Type: Expression(Integer) (68) -> D(G(x),x) Compiling function G with type Variable(x) -> Expression(Integer) x %e (68) - ----------------- x 2 x (%e ) + 2%e + 1 Type: Expression(Integer) (69) -> [rule1 expr for expr in [S,H,h]] X β 1 1 XD β %e (69) [---------,- log(---------),----------] X β X β X β %e + 1 %e + 1 %e + 1 Type: List(Expression(Integer)) (70) -> [rule1 D(expr,β) for expr in [S,H,h]] X β X β X β 2 X β X %e X %e XD (%e ) + (X XD β + XD)%e (70) [- ---------------------,---------,--------------------------------] X β 2 X β X β X β 2 X β (%e ) + 2%e + 1 %e + 1 (%e ) + 2%e + 1 Type: List(Expression(Integer)) \end{verbatim} \subsection{Probit} \label{sec:orgheadline4} Is there a more canonical approach in Axiom? \begin{verbatim} (71) -> Φ := operator 'Φ (71) Φ Type: BasicOperator (72) -> φ := operator 'φ (72) φ Type: BasicOperator (73) -> rule2 == rule D(Φ(x),x)==φ(x) Type: Void (74) -> G(x) == Φ(-x) Compiled code for G has been cleared. 1 old definition(s) deleted for function or rule G Type: Void (75) -> S := G(η(t,β)) Compiling function G with type Expression(Integer) -> Expression( Integer) (75) Φ(- η(t,β)) Type: Expression(Integer) (76) -> H := -log(S) (76) - log(Φ(- η(t,β))) Type: Expression(Integer) (77) -> h := D(H,t) , Φ (- η(t,β))η (t,β) ,1 (77) -------------------- Φ(- η(t,β)) Type: Expression(Integer) (78) -> rule2 D(G(x),x) Compiling function G with type Variable(x) -> Expression(Integer) Compiling body of rule rule2 to compute value of type RewriteRule( Integer,Integer,Expression(Integer)) (78) - φ(- x) Type: Expression(Integer) (79) -> [rule1 rule2 expr for expr in [S,H,h]] XD β φ(- X β) (79) [Φ(- X β),- log(Φ(- X β)),-------------] Φ(- X β) Type: List(Expression(Integer)) (80) -> [rule1 rule2 D(expr,β) for expr in [S,H,h]] (80) X φ(- X β) [- X φ(- X β), ----------, Φ(- X β) , 2 - X XD β Φ(- X β)φ (- X β) + X XD β φ(- X β) + XD Φ(- X β)φ(- X β) -------------------------------------------------------------------] 2 Φ(- X β) Type: List(Expression(Integer)) \end{verbatim} \subsection{Additive hazards} \label{sec:orgheadline5} \begin{verbatim} (81) -> G(x) == exp(-x) Compiled code for G has been cleared. 1 old definition(s) deleted for function or rule G Type: Void (82) -> S := G(η(t,β)) Compiling function G with type Expression(Integer) -> Expression( Integer) - η(t,β) (82) %e Type: Expression(Integer) (83) -> H := -log(S) (83) η(t,β) Type: Expression(Integer) (84) -> h := D(H,t) (84) η (t,β) ,1 Type: Expression(Integer) (85) -> D(G(x),x) Compiling function G with type Variable(x) -> Expression(Integer) - x (85) - %e Type: Expression(Integer) (86) -> [rule1 expr for expr in [S,H,h]] - X β (86) [%e ,X β,XD β] Type: List(Expression(Integer)) (87) -> [rule1 D(expr,β) for expr in [S,H,h]] - X β (87) [- X %e ,X,XD] Type: List(Expression(Integer)) \end{verbatim} \subsection{Aranda-Ordaz} \label{sec:orgheadline6} \begin{verbatim} (88) -> G(x) == exp(-log(θ*exp(x)+1)/θ) Compiled code for G has been cleared. 1 old definition(s) deleted for function or rule G Type: Void (89) -> S := G(η(t,β)) Compiling function G with type Expression(Integer) -> Expression( Integer) η(t,β) log(θ %e + 1) - ------------------- θ (89) %e Type: Expression(Integer) (90) -> H := -log(S) η(t,β) log(θ %e + 1) (90) ------------------- θ Type: Expression(Integer) (91) -> h := D(H,t) η(t,β) %e η (t,β) ,1 (91) ---------------- η(t,β) θ %e + 1 Type: Expression(Integer) (92) -> D(G(x),x) Compiling function G with type Variable(x) -> Expression(Integer) x log(θ %e + 1) - -------------- x θ %e %e (92) - --------------------- x θ %e + 1 Type: Expression(Integer) (93) -> [rule1 expr for expr in [S,H,h]] X β log(θ %e + 1) - ---------------- X β X β θ log(θ %e + 1) XD β %e (93) [%e ,----------------,-----------] θ X β θ %e + 1 Type: List(Expression(Integer)) (94) -> [rule1 D(expr,β) for expr in [S,H,h]] (94) X β log(θ %e + 1) - ---------------- X β θ X β X %e %e X %e [- ---------------------------, -----------, X β X β θ %e + 1 θ %e + 1 X β 2 X β XD θ (%e ) + (X XD β + XD)%e ----------------------------------] 2 X β 2 X β θ (%e ) + 2θ %e + 1 Type: List(Expression(Integer)) \end{verbatim} \subsection{Other links} \label{sec:orgheadline7} The development for other links would be similar. \end{document}