math.tex
% 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}