math.org
* Axiom/Fricas source file
** General case
#+begin_src axiom :exports results :results output
)cl p all
η := operator 'η
G := operator 'G
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)
S := G(η(t,β))
H := -log(S)
h := D(H,t)
[D(expr,β) for expr in [S,H,h]]
[rule1 expr for expr in [S,H,h]]
[rule1 D(expr,β) for expr in [S,H,h]]
#+end_src
#+RESULTS:
#+begin_example
(1) -> )cl p all
(1) -> η := operator 'η
(1) η
Type: BasicOperator
(2) -> G := operator 'G
(2) G
Type: BasicOperator
(3) -> 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)
(3) {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))
(4) -> S := G(η(t,β))
(4) G(η(t,β))
Type: Expression(Integer)
(5) -> H := -log(S)
(5) - log(G(η(t,β)))
Type: Expression(Integer)
(6) -> h := D(H,t)
,
G (η(t,β))η (t,β)
,1
(6) - ------------------
G(η(t,β))
Type: Expression(Integer)
(7) -> [D(expr,β) for expr in [S,H,h]]
(7)
,
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))
(8) -> [rule1 expr for expr in [S,H,h]]
,
XD βG (X β)
(8) [G(X β),- log(G(X β)),- -----------]
G(X β)
Type: List(Expression(Integer))
(9) -> [rule1 D(expr,β) for expr in [S,H,h]]
(9)
,
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_example
** Proportional hazards
#+begin_src axiom :exports results :results output
G(x) == exp(-exp(x))
S := G(η(t,β))
H := -log(S)
h := D(H,t)
D(G(x),x)
[rule1 expr for expr in [S,H,h]]
[rule1 D(expr,β) for expr in [S,H,h]]
#+end_src
#+RESULTS:
#+begin_example
(10) -> G(x) == exp(-exp(x))
Type: Void
(11) -> S := G(η(t,β))
Compiling function G with type Expression(Integer) -> Expression(
Integer)
η(t,β)
- %e
(11) %e
Type: Expression(Integer)
(12) -> H := -log(S)
η(t,β)
(12) %e
Type: Expression(Integer)
(13) -> h := D(H,t)
η(t,β)
(13) %e η (t,β)
,1
Type: Expression(Integer)
(14) -> D(G(x),x)
Compiling function G with type Variable(x) -> Expression(Integer)
x
x - %e
(14) - %e %e
Type: Expression(Integer)
(15) -> [rule1 expr for expr in [S,H,h]]
X β
- %e X β X β
(15) [%e ,%e ,XD β %e ]
Type: List(Expression(Integer))
(16) -> [rule1 D(expr,β) for expr in [S,H,h]]
X β
X β - %e X β X β
(16) [- X %e %e ,X %e ,(X XD β + XD)%e ]
Type: List(Expression(Integer))
#+end_example
** Proportional odds
#+begin_src axiom :exports results :results output
G(x) == 1/(1+exp(x))
S := G(η(t,β))
H := -log(S)
h := D(H,t)
D(G(x),x)
[rule1 expr for expr in [S,H,h]]
[rule1 D(expr,β) for expr in [S,H,h]]
#+end_src
#+RESULTS:
#+begin_example
(17) -> 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
(18) -> S := G(η(t,β))
Compiling function G with type Expression(Integer) -> Expression(
Integer)
1
(18) ------------
η(t,β)
%e + 1
Type: Expression(Integer)
(19) -> H := -log(S)
1
(19) - log(------------)
η(t,β)
%e + 1
Type: Expression(Integer)
(20) -> h := D(H,t)
η(t,β)
%e η (t,β)
,1
(20) ----------------
η(t,β)
%e + 1
Type: Expression(Integer)
(21) -> D(G(x),x)
Compiling function G with type Variable(x) -> Expression(Integer)
x
%e
(21) - -----------------
x 2 x
(%e ) + 2%e + 1
Type: Expression(Integer)
(22) -> [rule1 expr for expr in [S,H,h]]
X β
1 1 XD β %e
(22) [---------,- log(---------),----------]
X β X β X β
%e + 1 %e + 1 %e + 1
Type: List(Expression(Integer))
(23) -> [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
(23) [- ---------------------,---------,--------------------------------]
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_example
** Probit
Is there a more canonical approach in Axiom?
#+begin_src axiom :exports results :results output
Φ := operator 'Φ
φ := operator 'φ
rule2 == rule D(Φ(x),x)==φ(x)
G(x) == Φ(-x)
S := G(η(t,β))
H := -log(S)
h := D(H,t)
rule2 D(G(x),x)
[rule1 rule2 expr for expr in [S,H,h]]
[rule1 rule2 D(expr,β) for expr in [S,H,h]]
#+end_src
#+RESULTS:
#+begin_example
(24) -> Φ := operator 'Φ
(24) Φ
Type: BasicOperator
(25) -> φ := operator 'φ
(25) φ
Type: BasicOperator
(26) -> rule2 == rule D(Φ(x),x)==φ(x)
Type: Void
(27) -> G(x) == Φ(-x)
Compiled code for G has been cleared.
1 old definition(s) deleted for function or rule G
Type: Void
(28) -> S := G(η(t,β))
Compiling function G with type Expression(Integer) -> Expression(
Integer)
(28) Φ(- η(t,β))
Type: Expression(Integer)
(29) -> H := -log(S)
(29) - log(Φ(- η(t,β)))
Type: Expression(Integer)
(30) -> h := D(H,t)
,
Φ (- η(t,β))η (t,β)
,1
(30) --------------------
Φ(- η(t,β))
Type: Expression(Integer)
(31) -> 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))
(31) - φ(- x)
Type: Expression(Integer)
(32) -> [rule1 rule2 expr for expr in [S,H,h]]
XD β φ(- X β)
(32) [Φ(- X β),- log(Φ(- X β)),-------------]
Φ(- X β)
Type: List(Expression(Integer))
(33) -> [rule1 rule2 D(expr,β) for expr in [S,H,h]]
(33)
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_example
** Additive hazards
#+begin_src axiom :exports results :results output
G(x) == exp(-x)
S := G(η(t,β))
H := -log(S)
h := D(H,t)
D(G(x),x)
[rule1 expr for expr in [S,H,h]]
[rule1 D(expr,β) for expr in [S,H,h]]
#+end_src
#+RESULTS:
#+begin_example
(34) -> G(x) == exp(-x)
Compiled code for G has been cleared.
1 old definition(s) deleted for function or rule G
Type: Void
(35) -> S := G(η(t,β))
Compiling function G with type Expression(Integer) -> Expression(
Integer)
- η(t,β)
(35) %e
Type: Expression(Integer)
(36) -> H := -log(S)
(36) η(t,β)
Type: Expression(Integer)
(37) -> h := D(H,t)
(37) η (t,β)
,1
Type: Expression(Integer)
(38) -> D(G(x),x)
Compiling function G with type Variable(x) -> Expression(Integer)
- x
(38) - %e
Type: Expression(Integer)
(39) -> [rule1 expr for expr in [S,H,h]]
- X β
(39) [%e ,X β,XD β]
Type: List(Expression(Integer))
(40) -> [rule1 D(expr,β) for expr in [S,H,h]]
- X β
(40) [- X %e ,X,XD]
Type: List(Expression(Integer))
#+end_example
** Aranda-Ordaz
#+begin_src axiom :exports results :results output
G(x) == exp(-log(θ*exp(x)+1)/θ)
S := G(η(t,β))
H := -log(S)
h := D(H,t)
D(G(x),x)
[rule1 expr for expr in [S,H,h]]
[rule1 D(expr,β) for expr in [S,H,h]]
#+end_src
#+RESULTS:
#+begin_example
(41) -> 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
(42) -> S := G(η(t,β))
Compiling function G with type Expression(Integer) -> Expression(
Integer)
η(t,β)
log(θ %e + 1)
- -------------------
θ
(42) %e
Type: Expression(Integer)
(43) -> H := -log(S)
η(t,β)
log(θ %e + 1)
(43) -------------------
θ
Type: Expression(Integer)
(44) -> h := D(H,t)
η(t,β)
%e η (t,β)
,1
(44) ----------------
η(t,β)
θ %e + 1
Type: Expression(Integer)
(45) -> D(G(x),x)
Compiling function G with type Variable(x) -> Expression(Integer)
x
log(θ %e + 1)
- --------------
x θ
%e %e
(45) - ---------------------
x
θ %e + 1
Type: Expression(Integer)
(46) -> [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
(46) [%e ,----------------,-----------]
θ X β
θ %e + 1
Type: List(Expression(Integer))
(47) -> [rule1 D(expr,β) for expr in [S,H,h]]
(47)
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_example
** Other links
The development for other links would be similar.
