Revision d415ac4eba2a0f4dd170f58ced8f09afe73047ea authored by Mark Clements on 28 February 2023, 15:02:29 UTC, committed by cran-robot on 28 February 2023, 15:02:29 UTC
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<body>
<div id="content">
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#orgheadline8">1. Axiom/Fricas source file</a>
<ul>
<li><a href="#orgheadline1">1.1. General case</a></li>
<li><a href="#orgheadline2">1.2. Proportional hazards</a></li>
<li><a href="#orgheadline3">1.3. Proportional odds</a></li>
<li><a href="#orgheadline4">1.4. Probit</a></li>
<li><a href="#orgheadline5">1.5. Additive hazards</a></li>
<li><a href="#orgheadline6">1.6. Aranda-Ordaz</a></li>
<li><a href="#orgheadline7">1.7. Other links</a></li>
</ul>
</li>
</ul>
</div>
</div>
<div id="outline-container-orgheadline8" class="outline-2">
<h2 id="orgheadline8"><span class="section-number-2">1</span> Axiom/Fricas source file</h2>
<div class="outline-text-2" id="text-1">
</div><div id="outline-container-orgheadline1" class="outline-3">
<h3 id="orgheadline1"><span class="section-number-3">1.1</span> General case</h3>
<div class="outline-text-3" id="text-1-1">
<pre class="example">
(95) -> )cl p all
Compiled code for rule2 has been cleared.
Compiled code for G has been cleared.
(95) -> η := operator 'η
(95) η
Type: BasicOperator
(96) -> G := operator 'G
(96) G
Type: BasicOperator
(97) -> 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)
(97) {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))
(98) -> S := G(η(t,β))
(98) G(η(t,β))
Type: Expression(Integer)
(99) -> H := -log(S)
(99) - log(G(η(t,β)))
Type: Expression(Integer)
(100) -> h := D(H,t)
,
G (η(t,β))η (t,β)
,1
(100) - ------------------
G(η(t,β))
Type: Expression(Integer)
(101) -> [D(expr,β) for expr in [S,H,h]]
(101)
,
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))
(102) -> [rule1 expr for expr in [S,H,h]]
,
XD βG (X β)
(102) [G(X β),- log(G(X β)),- -----------]
G(X β)
Type: List(Expression(Integer))
(103) -> [rule1 D(expr,β) for expr in [S,H,h]]
(103)
,
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))
</pre>
</div>
</div>
<div id="outline-container-orgheadline2" class="outline-3">
<h3 id="orgheadline2"><span class="section-number-3">1.2</span> Proportional hazards</h3>
<div class="outline-text-3" id="text-1-2">
<pre class="example">
(104) -> G(x) == exp(-exp(x))
Type: Void
(105) -> S := G(η(t,β))
Compiling function G with type Expression(Integer) -> Expression(
Integer)
η(t,β)
- %e
(105) %e
Type: Expression(Integer)
(106) -> H := -log(S)
η(t,β)
(106) %e
Type: Expression(Integer)
(107) -> h := D(H,t)
η(t,β)
(107) %e η (t,β)
,1
Type: Expression(Integer)
(108) -> D(G(x),x)
Compiling function G with type Variable(x) -> Expression(Integer)
x
x - %e
(108) - %e %e
Type: Expression(Integer)
(109) -> [rule1 expr for expr in [S,H,h]]
X β
- %e X β X β
(109) [%e ,%e ,XD β %e ]
Type: List(Expression(Integer))
(110) -> [rule1 D(expr,β) for expr in [S,H,h]]
X β
X β - %e X β X β
(110) [- X %e %e ,X %e ,(X XD β + XD)%e ]
Type: List(Expression(Integer))
</pre>
</div>
</div>
<div id="outline-container-orgheadline3" class="outline-3">
<h3 id="orgheadline3"><span class="section-number-3">1.3</span> Proportional odds</h3>
<div class="outline-text-3" id="text-1-3">
<pre class="example">
(111) -> 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
(112) -> S := G(η(t,β))
Compiling function G with type Expression(Integer) -> Expression(
Integer)
1
(112) ------------
η(t,β)
%e + 1
Type: Expression(Integer)
(113) -> H := -log(S)
1
(113) - log(------------)
η(t,β)
%e + 1
Type: Expression(Integer)
(114) -> h := D(H,t)
η(t,β)
%e η (t,β)
,1
(114) ----------------
η(t,β)
%e + 1
Type: Expression(Integer)
(115) -> D(G(x),x)
Compiling function G with type Variable(x) -> Expression(Integer)
x
%e
(115) - -----------------
x 2 x
(%e ) + 2%e + 1
Type: Expression(Integer)
(116) -> [rule1 expr for expr in [S,H,h]]
X β
1 1 XD β %e
(116) [---------,- log(---------),----------]
X β X β X β
%e + 1 %e + 1 %e + 1
Type: List(Expression(Integer))
(117) -> [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
(117) [- ---------------------,---------,--------------------------------]
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))
</pre>
</div>
</div>
<div id="outline-container-orgheadline4" class="outline-3">
<h3 id="orgheadline4"><span class="section-number-3">1.4</span> Probit</h3>
<div class="outline-text-3" id="text-1-4">
<p>
Is there a more canonical approach in Axiom?
</p>
<pre class="example">
(118) -> Φ := operator 'Φ
(118) Φ
Type: BasicOperator
(119) -> φ := operator 'φ
(119) φ
Type: BasicOperator
(120) -> rule2 == rule D(Φ(x),x)==φ(x)
Type: Void
(121) -> G(x) == Φ(-x)
Compiled code for G has been cleared.
1 old definition(s) deleted for function or rule G
Type: Void
(122) -> S := G(η(t,β))
Compiling function G with type Expression(Integer) -> Expression(
Integer)
(122) Φ(- η(t,β))
Type: Expression(Integer)
(123) -> H := -log(S)
(123) - log(Φ(- η(t,β)))
Type: Expression(Integer)
(124) -> h := D(H,t)
,
Φ (- η(t,β))η (t,β)
,1
(124) --------------------
Φ(- η(t,β))
Type: Expression(Integer)
(125) -> 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))
(125) - φ(- x)
Type: Expression(Integer)
(126) -> [rule1 rule2 expr for expr in [S,H,h]]
XD β φ(- X β)
(126) [Φ(- X β),- log(Φ(- X β)),-------------]
Φ(- X β)
Type: List(Expression(Integer))
(127) -> [rule1 rule2 D(expr,β) for expr in [S,H,h]]
(127)
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))
</pre>
</div>
</div>
<div id="outline-container-orgheadline5" class="outline-3">
<h3 id="orgheadline5"><span class="section-number-3">1.5</span> Additive hazards</h3>
<div class="outline-text-3" id="text-1-5">
<pre class="example">
(128) -> G(x) == exp(-x)
Compiled code for G has been cleared.
1 old definition(s) deleted for function or rule G
Type: Void
(129) -> S := G(η(t,β))
Compiling function G with type Expression(Integer) -> Expression(
Integer)
- η(t,β)
(129) %e
Type: Expression(Integer)
(130) -> H := -log(S)
(130) η(t,β)
Type: Expression(Integer)
(131) -> h := D(H,t)
(131) η (t,β)
,1
Type: Expression(Integer)
(132) -> D(G(x),x)
Compiling function G with type Variable(x) -> Expression(Integer)
- x
(132) - %e
Type: Expression(Integer)
(133) -> [rule1 expr for expr in [S,H,h]]
- X β
(133) [%e ,X β,XD β]
Type: List(Expression(Integer))
(134) -> [rule1 D(expr,β) for expr in [S,H,h]]
- X β
(134) [- X %e ,X,XD]
Type: List(Expression(Integer))
</pre>
</div>
</div>
<div id="outline-container-orgheadline6" class="outline-3">
<h3 id="orgheadline6"><span class="section-number-3">1.6</span> Aranda-Ordaz</h3>
<div class="outline-text-3" id="text-1-6">
<pre class="example">
(135) -> 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
(136) -> S := G(η(t,β))
Compiling function G with type Expression(Integer) -> Expression(
Integer)
η(t,β)
log(θ %e + 1)
- -------------------
θ
(136) %e
Type: Expression(Integer)
(137) -> H := -log(S)
η(t,β)
log(θ %e + 1)
(137) -------------------
θ
Type: Expression(Integer)
(138) -> h := D(H,t)
η(t,β)
%e η (t,β)
,1
(138) ----------------
η(t,β)
θ %e + 1
Type: Expression(Integer)
(139) -> D(G(x),x)
Compiling function G with type Variable(x) -> Expression(Integer)
x
log(θ %e + 1)
- --------------
x θ
%e %e
(139) - ---------------------
x
θ %e + 1
Type: Expression(Integer)
(140) -> [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
(140) [%e ,----------------,-----------]
θ X β
θ %e + 1
Type: List(Expression(Integer))
(141) -> [rule1 D(expr,β) for expr in [S,H,h]]
(141)
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))
</pre>
</div>
</div>
<div id="outline-container-orgheadline7" class="outline-3">
<h3 id="orgheadline7"><span class="section-number-3">1.7</span> Other links</h3>
<div class="outline-text-3" id="text-1-7">
<p>
The development for other links would be similar.
</p>
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Mark Clements</p>
<p class="date">Created: 2016-11-26 Sat 19:04</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div>
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