How useful are the different CAS languages for implementing numerical routines? Prompted by a comparison of R and C for implementing Fisher's exact test for 2x2 tables (http://fluff.info/blog/arch/00000172.htm), I thought that it would be interesting to implement this particular test in Spad, Boot, Reduce, Maxima and Common Lisp (see below). Each set of code was required to implement a univariate root finder and the hypergeometric distribution to calculate the p-value under different alternatives, together with the 95% confidence interval and the maximum likelihood estimator for the odds ratio. The reference implementation is R, where the code and output would be:
As a caveat: I have little experience with these programs. Any changes or improvements to the programs would be welcomed.
To summarise, all five languages (Spad, Boot, Reduce, Maxima and Common Lisp) provide arbitrary length integers and fractions, ensuring that the hypergeometric distribution was straightforward to implement. The Lisp and R implementations were very similar, which is not surprising, given that the two languages are closely related. In contrast, the nested functions in Spad seemed clumsy, with the requirement to use the #1 and #2 argument references (although this has changed recently in Fricas); I did, however, appreciate Spad's lexical scoping and facility to fall back to a symbolic analysis. An Aldor implementation may be cleaner than the Spad implementation. Boot's implementation was initially difficult, as I was unclear how to pass values to the nested functions with using function argument (which was not possible for the univariate root finder :-(). The use of the "$" prefix on derived variables seemed particularly clumsy. Finally, my version of Fricas:Boot defaulted to single precision floats, which caused problems with precision. Type specification in OpenAxiom:Boot and more recent versions of Fricas would negate the need for explicit coercion to double-floats. For Reduce, the lack of nested procedures in the algebraic mode made progress slow; importantly, the switch to symbolic mode made the implementation quite straightforward, with reasonably good debugging. The Maxima version was particularly short, given that Maxima already provides a root solver and the hypergeometric distribution.
This begs the question: when would one use any of these languages for mixed numerical/symbolic analysis? In my opinion, Boot is the least likely to be used, although it does play closely and well with Common Lisp (a la Reduce and Maxima). One could code for numerical analysis in Boot and Common Lisp - however Boot's lack of lexical scoping may be a detraction. Moreover, by my understanding, Lisp or Boot are unable to evaluate Spad or Axiom functions. Second, for an R user, Spad's type system seems fussy (and extremely elegant); I also found that debugging could be slow. The lack of ability to call Spad functions from Lisp or Boot is an important restriction, requiring that a large body of code in Lisp (or Fortran via f2cl), such as for optimisation, would need to be hand translated to Spad (see [SandBoxMLE]? for an example). Reduce and Maxima both provided fairly polished environments worthy of further consideration.
spad
)abbrev package TESTP TestPackage
R ==> Float
I ==> Integer
fisherRec ==> Record(PValue:R, CI:List R, Estimate:R)
TestPackage: with
ridder: (R->R,R,R) -> R
msign: (R,R) -> R
choose:(I, I)->Fraction I
--chooseNew:(Integer, Integer)->Fraction Integer
dhyper:(I, I, I, I)->Fraction Integer
phyper:(I, I, I, I, Boolean)->Fraction Integer
fisherTest:(I,I,I,I, String, R, Boolean, R)-> fisherRec
testTolerance:(R, R, R)->Boolean
test1: () -> Boolean
test2: () -> Boolean
test3: () -> Boolean
test4: () -> Boolean
test5: () -> Boolean
test6: () -> Boolean
test7: () -> Boolean
test8: () -> Boolean
test9: () -> Boolean
test10: () -> Boolean
alltests: () -> List Boolean
== add
import TrigonometricFunctionCategory -- for test1()
--import OrderedCompletion(Float) for plusInfinity()$OrderedCompletion(Float)
ridder(func, x1, x2) ==
eps:= 1.0e-16::R
maxit:= 30::Integer
--verbose:= false
fl:R := func x1
fh:R := func x2
xl:R := x1
xh:R := x2
ans:R := -1.11e30::R
xnew:R := 0.0e0::R
iterNum:= 0::Integer
if fl=0.0::R then return x1
else if fh=0.0::R then return x2
else if (fl*fh) > 0.0::R then error "Initial points are not either side of zero."
--if (fl*fh) < 0.0 then
else repeat
xm:= 0.5::R *(xl+xh)
fm:= func xm
ss:= sqrt((fm*fm) - (fl*fh))
if ss =0.0::R then return ans
xnew:= xm + (((xm - xl) * (if (fl>fh) then 1.0::R else -1.0::R) * fm) / ss)
if abs(xnew-ans) <= eps then return ans
ans:= xnew
fnew:= func ans
if fnew=0.0::R then return ans
if msign(fm,fnew) ~= fm then
xl:= xm
fl:= fm
xh:= ans
fh:= fnew
else if msign(fl, fnew) ~= fl then
xh:= ans
fh:= fnew
else if msign(fh, fnew) ~= fh then
xl:= ans
fl:= fnew
iterNum:=iterNum+1::Integer
if iterNum >=maxit then
error "Maximum iterations exceeded"
--if verbose then FORMAT(true,"~,8f ~,8f ~,8f ~,8f~%", xl, xh, fl, fh)$Lisp
if abs(xh-xl) <= eps then return ans
msign(x, y) ==
(abs x) * (if y>0.0::R then 1.0::R else if y<0.0::R then -1.0::R else 0.0::R)
choose(n, x) ==
total:Fraction Integer := 1/1
for denom in 1..x repeat
total:=total*((n-denom+1)/(denom))::Fraction Integer
return total
--chooseNew(n, x) == product((n-i+1)::Fraction Integer/i::Fraction Integer,i=1..x)
dhyper(x, m, n, k) ==
choose(m, x) * choose(n, k - x) / choose(m + n, k)
phyper(x, m, n, k, lowerTail) ==
i:PositiveInteger
--total:Fraction Integer:=0/1
if lowerTail then
reduce("+",[dhyper(i, m, n, k) for i in 1..x])
else
reduce("+",[dhyper(i, m, n, k) for i in (x+1)..k])
fisherTest(a,b,c,d, alternative, OR, confInt, confLevel) ==
m:I := a+c -- first column
n:I := b+d -- second column
k:I := a+b -- first row
x00:I := a
lo:I := max(0, k-n)
hi:I := min(k, m)
support:List I := [i for i in lo..hi]
logdc:List R:= [log(dhyper(i, m, n, k)::R) for i in support]
doubleEps:R := 1.0e-50::R
plusInfinity:R := 1.0e6::R -- arbitrary
dnhyper:(R->List R) :=
ncp:R := #1
d:List R := [logdc(i)+log(ncp)*support(i)::R for i in 1..#logdc]
maxd:R := reduce(max,d)
d2:List R :=[exp(di-maxd) for di in d]
sumd2:R := reduce("+",d2)
[d2i/sumd2 for d2i in d2]
mnhyper:(R->R) :=
ncp:R := #1
if ncp=0.0::R then lo::R
--else if ncp=%plusInfinity then hi::R
else
d:List R := dnhyper(ncp)
reduce("+",[support(i)::R*d(i) for i in 1..#d])
pnhyper:((Integer,R,Boolean)->R) :=
q:I := #1
ncp:R := #2
upperTail:Boolean := #3
if ncp=1.0 then
if upperTail then phyper(q-1, m, n, k, false)::R
else phyper(q, m, n, k, true)::R
else if ncp=0.0 then
if upperTail then
if q<=lo then 1.0::R else 0.0::R
else if q>=lo then 1.0::R else 0.0::R
-- else if ncp=%plusInfinity then
-- if upperTail then
-- if q<=hi then 1.0::R else 0.0::R
-- else if q>= hi then 1.0::R else 0.0::R
else
d:List R := dnhyper(ncp)
if upperTail then
reduce("+",[d(i) for i in 1..#d | support(i)>=q])
else reduce("+",[d(i) for i in 1..#d | support(i)<=q])
mle:(I->R) :=
x:I := #1
if x=lo then 0.0::R
else if x=hi then plusInfinity
else
mu:R := mnhyper(1.0::R)
if mu>x::R then
f:(R->R) := mnhyper(#1) - x::R
ridder(f,0,1)
else if mu<x::R then
f:(R->R) := mnhyper(1/#1) - x::R
1/ridder(f,doubleEps,1.0::R)
else 1.0::R
ncpU:(I,R)->R :=
x:I := #1
alpha:R := #2
if x=hi then plusInfinity
else
p:R := pnhyper(x, 1.0::R, false)
if p<alpha then
f:(R->R) := pnhyper(x,#1,false) - alpha
ridder(f, 0.0::R, 1.0::R)
else if p>alpha then
f:(R->R) := pnhyper(x,1/#1,false) - alpha
1/ridder(f, doubleEps, 1.0::R)
else 1.0::R
ncpL:(Integer, R)->R :=
x:I := #1
alpha:R := #2
if x=lo then 0.0::R
else
p:R := pnhyper(x, 1, true)
if p>alpha then
f:(R->R) := pnhyper(x,#1,true) - alpha
ridder(f, 0,1)
else if p<alpha then
f:(R->R) := pnhyper(x,1/#1,true) - alpha
1/ridder(f, doubleEps,1.0::R)
else 1.0::R
pValue:R :=
if alternative="less" then pnhyper(x00, OR,false)
else if alternative="greater" then pnhyper(x00, OR,true)
else if alternative="two-sided" then
relErr:= 1+1.0e-7::R
dn:= dnhyper(OR)
dstar:= dn(x00-lo+1)*relErr
reduce("+",[di for di in dn | di<dstar])
else -1.0::R
cInterval:List R :=
if confInt then
if alternative="less" then [0.0::R, ncpU(x00, 1.0::R-confLevel)]
else if alternative="greater" then [ncpL(x00, 1.0::R-confLevel), plusInfinity]
else if alternative="two-sided" then
alpha:=(1-confLevel)/2
[ncpL(x00, alpha), ncpU(x00, alpha)]
else [-1.0::R,-1.0::R]
--else []
estimate:= mle(x00)
[pValue, cInterval, estimate]
testTolerance(x, y, atol) ==
if abs(x-y) <= atol then true else false
test1() ==
testTolerance(2*ridder(cos,0.0::R,2.0::R),pi()$Pi::R, 1.0e-18)
test2() == testTolerance(choose(100, 5)::R, 75287520::R, 0)
test3() == testTolerance(dhyper(5, 10, 7, 8)::R, 0.3628137::R, 1.0e-7)
test4() == testTolerance(log(dhyper(5, 10, 7, 8)::R),-1.013866::R, 1.0e-7)
test5() == testTolerance(phyper(5, 10, 7, 8, true)::R,0.7821884::R, 1.0e-7)
test6() == testTolerance(phyper(5, 10, 7, 8, false)::R,0.2178116::R, 1.0e-7)
test7() == testTolerance(fisherTest(10,10,10,20,"two-sided",1.0,true,0.95).PValue,
0.2575, 1.0e-3)
test8() == testTolerance(fisherTest(10,10,10,20,"two-sided",1.0,true,0.95).CI.1,
0.5383996, 1.0e-6)
test9() == testTolerance(fisherTest(10,10,10,20,"two-sided",1.0,true,0.95).CI.2,
7.4363242, 1.0e-4)
test10() == testTolerance(fisherTest(10,10,10,20,"two-sided",1.0,true,0.95).Estimate,
1.971640, 1.0e-4)
alltests() == [test1(), test2(), test3(), test4(), test5(), test6(),
test7(), test8(), test9(), test10()]
spad
Compiling FriCAS source code from file
/var/zope2/var/LatexWiki/303933172962875058-25px001.spad using
old system compiler.
TESTP abbreviates package TestPackage
processing macro definition R ==> Float
processing macro definition I ==> Integer
processing macro definition fisherRec ==> Record(PValue: R,CI: List R,Estimate: R)
------------------------------------------------------------------------
initializing NRLIB TESTP for TestPackage
compiling into NRLIB TESTP
importing TrigonometricFunctionCategory
compiling exported ridder : (Float -> Float,Float,Float) -> Float
Time: 0.05 SEC.
compiling exported msign : (Float,Float) -> Float
Time: 0.03 SEC.
compiling exported choose : (Integer,Integer) -> Fraction Integer
Time: 0.04 SEC.
compiling exported dhyper : (Integer,Integer,Integer,Integer) -> Fraction Integer
Time: 0 SEC.
compiling exported phyper : (Integer,Integer,Integer,Integer,Boolean) -> Fraction Integer
Time: 0.03 SEC.
compiling exported fisherTest : (Integer,Integer,Integer,Integer,String,Float,Boolean,Float) -> Record(PValue: Float,CI: List Float,Estimate: Float)
Time: 0.20 SEC.
compiling exported testTolerance : (Float,Float,Float) -> Boolean
Time: 0 SEC.
compiling exported test1 : () -> Boolean
Time: 0.01 SEC.
compiling exported test2 : () -> Boolean
Time: 0 SEC.
compiling exported test3 : () -> Boolean
Time: 0.01 SEC.
compiling exported test4 : () -> Boolean
Time: 0.05 SEC.
compiling exported test5 : () -> Boolean
Time: 0.01 SEC.
compiling exported test6 : () -> Boolean
Time: 0.01 SEC.
compiling exported test7 : () -> Boolean
Time: 0.01 SEC.
compiling exported test8 : () -> Boolean
Time: 0.01 SEC.
compiling exported test9 : () -> Boolean
Time: 0.02 SEC.
compiling exported test10 : () -> Boolean
Time: 0.01 SEC.
compiling exported alltests : () -> List Boolean
Time: 0 SEC.
(time taken in buildFunctor: 0)
;;; *** |TestPackage| REDEFINED
;;; *** |TestPackage| REDEFINED
Time: 0 SEC.
Warnings:
[1] ridder: xh has no value
[2] ridder: xl has no value
[3] fisherTest: The conditional modes (List (Float)) and (Integer) conflict
Cumulative Statistics for Constructor TestPackage
Time: 0.49 seconds
finalizing NRLIB TESTP
Processing TestPackage for Browser database:
--->-->TestPackage((ridder (R (Mapping R R) R R))): Not documented!!!!
--->-->TestPackage((msign (R R R))): Not documented!!!!
--->-->TestPackage((choose ((Fraction I) I I))): Not documented!!!!
--->-->TestPackage((dhyper ((Fraction (Integer)) I I I I))): Not documented!!!!
--->-->TestPackage((phyper ((Fraction (Integer)) I I I I (Boolean)))): Not documented!!!!
--->-->TestPackage((fisherTest (fisherRec I I I I (String) R (Boolean) R))): Not documented!!!!
--->-->TestPackage((testTolerance ((Boolean) R R R))): Not documented!!!!
--->-->TestPackage((test1 ((Boolean)))): Not documented!!!!
--->-->TestPackage((test2 ((Boolean)))): Not documented!!!!
--->-->TestPackage((test3 ((Boolean)))): Not documented!!!!
--->-->TestPackage((test4 ((Boolean)))): Not documented!!!!
--->-->TestPackage((test5 ((Boolean)))): Not documented!!!!
--->-->TestPackage((test6 ((Boolean)))): Not documented!!!!
--->-->TestPackage((test7 ((Boolean)))): Not documented!!!!
--->-->TestPackage((test8 ((Boolean)))): Not documented!!!!
--->-->TestPackage((test9 ((Boolean)))): Not documented!!!!
--->-->TestPackage((test10 ((Boolean)))): Not documented!!!!
--->-->TestPackage((alltests ((List (Boolean))))): Not documented!!!!
--->-->TestPackage(constructor): Not documented!!!!
--->-->TestPackage(): Missing Description
------------------------------------------------------------------------
TestPackage is now explicitly exposed in frame initial
TestPackage will be automatically loaded when needed from
/var/zope2/var/LatexWiki/TESTP.NRLIB/codeUsing this code in Axiom:
axiom
-- test code is correct
alltests()
Type: List Boolean
axiom
-- show the example
fisherTest(10,10,10,20,"two-sided",1.0,true,0.95)
Type: Record(PValue
?: Float,CI: List Float,Estimate: Float)
The Boot translation was more fiddly - but, then again, I had never used Boot before.
boot
doubleFloat(x) == COERCE(x,'DOUBLE_-FLOAT)
DF(x) == COERCE(x,'DOUBLE_-FLOAT)
ridder(func, x1, x2) ==
--x2:=DF(x2)
eps:= DF(1.0e-16)
maxit:= 30
fl := DF(FUNCALL(func,x1))
fh := DF(FUNCALL(func,x2))
xl := x1
xh := x2
ans := DF(-1.11e20)
xnew := 0.0e0
iterNum:= 0
if fl=0.0 then return x1
else if fh=0.0 then return x2
else if (fl*fh) > 0.0 then error "Initial points are not either side of zero."
--if (fl*fh) < 0.0 then
else repeat
xm:= 0.5 *(xl+xh)
fm:= FUNCALL(func,xm)
ss:= SQRT((fm*fm) - (fl*fh))
if ss =0.0 then return ans
xnew:= xm + (((xm - xl) * (if (fl>fh) then 1.0 else -1.0) * fm) / ss)
if ABS(xnew-ans) <= eps then return ans
ans:= xnew
fnew:= DF(FUNCALL(func,ans))
if fnew=0.0 then return ans
if msign(fm,fnew) ^= fm then
xl:= xm
fl:= fm
xh:= ans
fh:= fnew
else if msign(fl, fnew) ^= fl then
xh:= ans
fh:= fnew
else if msign(fh, fnew) ^= fh then
xl:= ans
fl:= fnew
iterNum:=iterNum+1
if iterNum >=maxit then
error "Maximum iterations exceeded"
--if verbose then FORMAT(true,"~,8f ~,8f ~,8f ~,8f~%", xl, xh, fl, fh)$Lisp
if ABS(xh-xl) <= eps then return ans
msign(x, y) ==
(ABS x) * (if y>0.0 then 1.0 else if y<0.0 then -1.0 else 0.0)
choose(n, x) ==
total := 1
for denom in 1..x repeat
total:=total*(n-denom+1)/denom
return total
--chooseNew(n, x) == product((n-i+1)::Fraction Integer/i::Fraction Integer,i=1..x)
dhyper(x, m, n, k) ==
DF(choose(m, x) * choose(n, k - x)) / choose(m + n, k)
-- reduce(func,list) ==
-- value := list.0
-- for i in 1..(#list-1) repeat
-- value:=FUNCALL(func,value,list.i))
-- value
phyper(x, m, n, k, lowerTail) ==
--total:Fraction Integer:=0/1
if lowerTail then
+/[dhyper(i, m, n, k) for i in 1..x]
else
+/[dhyper(i, m, n, k) for i in (x+1)..k]
dnhyper(ncp,logdc,support) ==
d := [DF(logdc.i+LOG(ncp)*support.i) for i in 0..(#logdc-1)]
maxd := APPLY(FUNCTION(MAX),d)
d2 :=[EXP(di-maxd) for di in d]
sumd2 := +/d2
[d2i/sumd2 for d2i in d2]
testTolerance(x, y, atol) ==
if ABS(x-y) <= atol then true else false
test1() == testTolerance(2*ridder('COS,0.0,2.0),3.1415926535897932385, 1.0e-7)
test2() == testTolerance(choose(100, 5), 75287520, 0)
test3() == testTolerance(dhyper(5, 10, 7, 8), 0.3628137, 1.0e-7)
test4() == testTolerance(LOG(dhyper(5, 10, 7, 8)),-1.013866, 1.0e-7)
test5() == testTolerance(phyper(5, 10, 7, 8, true),0.7821884, 1.0e-7)
test6() == testTolerance(phyper(5, 10, 7, 8, false),0.2178116, 1.0e-7)
fisherTest(a,b,c,d, alternative, OR, confInt, confLevel) == main where
main() ==
$m := a+c -- first column
$n := b+d -- second column
$k := a+b -- first row
$x00 := a
$lo := MAX(0, $k-$n)
$hi := MIN($k, $m)
$support := [i for i in $lo..$hi]
$logdc := [LOG(dhyper(i, $m, $n, $k)) for i in $support]
$doubleEps := 1.0e-10
$plusInfinity := 1.0e10
pvalue :=
if alternative='"less" then pnhyper($x00, OR,false)
else if alternative='"greater" then pnhyper($x00, OR,true)
else if alternative='"two-sided" then
relErr:= 1+1.0e-7
d:= dnhyper(OR,$logdc,$support)
dstar:= ELT(d,$x00-$lo)*relErr
+/[di for di in d | di<dstar]
else -1.0 -- no match
estimate :=
if $x00=$lo then 0
-- else if $x00=hi then return($plusInfinity)
else
mu:= mnhyper(1)
if mu>$x00 then ridder(FUNCTION(f1),0,1)
else if mu<$x00 then 1/ridder(FUNCTION(f2),$doubleEps,1)
else 1
interval :=
if confInt then
$alpha := 1 - confLevel
if alternative='"less" then [0, ncpU($x00)]
else if alternative='"greater" then [ncpL($x00), $plusInfinity]
else if alternative='"two-sided" then
$alpha :=(1-confLevel)/2.0
[ncpL($x00), ncpU($x00)]
else [-1,-1]
else [-2,-2]
[pvalue,interval,estimate]
pnhyper (q,ncp,upperTail) ==
if ncp=1 then
if upperTail then phyper(q-1, $m, $n, $k, false)
else phyper(q, $m, $n, $k, true)
else if ncp=0 then
if upperTail then
if q<=$lo then 1 else 0
else if q>=$lo then 1 else 0
-- else if ncp=$plusInfinity then
-- if upperTail then
-- if q<=hi then return(1) else return(0)
-- else if q>= hi then return(1) else return(0)
else
d:= dnhyper(ncp, $logdc, $support)
if upperTail then
+/[d.i for i in 0..(#d-1) | $support.i>=q]
else +/[d.i for i in 0..(#d-1) | $support.i<=q]
mnhyper(ncp) ==
if ncp=0.0 then $lo
--if ncp=$plusInfinity then return(hi::R)
else
d := dnhyper(ncp,$logdc,$support)
+/[si*di for di in d for si in $support]
f1(u) == mnhyper(u) - $x00
f2(u) == mnhyper(1/u) - $x00
ncpU x ==
--if x=$hi then $plusInfinity
p:= pnhyper(x, 1.0, false)
if p<$alpha then
ridder(FUNCTION(fu1),0.0,1.0)
else if p>$alpha then
1/ridder(FUNCTION(fu2), $doubleEps,1)
else 1
fu1 u == pnhyper($x00,u,false) - $alpha
fu2 u == pnhyper($x00,1/u,false) - $alpha
ncpL x ==
if x=$lo then 0
else
p:= pnhyper(x, 1, true)
if p>$alpha then ridder(FUNCTION(fl1), 0,1)
else if p<$alpha then 1/ridder(FUNCTION(fl2), $doubleEps,1)
else 1
fl1 u == pnhyper($x00,u,true) - $alpha
fl2 u == pnhyper($x00,1/u,true) - $alpha
test7() == fisherTest(10,10,10,20,'"two-sided",1,true,0.95)
alltests() == [test1(), test2(), test3(), test4(), test5(), test6()]
boot
Value = T
; (DEFUN |fisherTest,fl2| ...) is being compiled.
;; The variable |$x00| is undefined.
;; The compiler will assume this variable is a global.
;; The variable |$alpha| is undefined.
;; The compiler will assume this variable is a global.
; (DEFUN |fisherTest,ncpL| ...) is being compiled.
;; The variable |$lo| is undefined.
;; The compiler will assume this variable is a global.
;; The variable |$doubleEps| is undefined.
;; The compiler will assume this variable is a global.
; (DEFUN |fisherTest,mnhyper| ...) is being compiled.
;; The variable |$logdc| is undefined.
;; The compiler will assume this variable is a global.
;; The variable |$support| is undefined.
;; The compiler will assume this variable is a global.
; (DEFUN |fisherTest,pnhyper| ...) is being compiled.
;; The variable |$m| is undefined.
;; The compiler will assume this variable is a global.
;; The variable |$n| is undefined.
;; The compiler will assume this variable is a global.
;; The variable |$k| is undefined.
;; The compiler will assume this variable is a global.
; (DEFUN |fisherTest| ...) is being compiled.
;; The variable |$hi| is undefined.
;; The compiler will assume this variable is a global.
;; The variable |$plusInfinity| is undefined.
;; The compiler will assume this variable is a global.
Value = 17616
Using this code in Axiom:
axiom
alltests()$Lisp
axiom
fisherTest(10,10,10,20,"two-sided",1,true,0.95::SF)$Lisp
For Reduce (which was also my first Reduce program):
symbolic;
nil
on rounded;
nil | reduce |
symbolic procedure msign(x, y);
(abs x) * (if y>0.0 then 1.0 else if y<0.0 then -1.0 else 0.0);
msign | reduce |
%% Numerical root finding using Ridders method
%% (Exit criteria hacked: how can one return from a repeat .. until statement?)
symbolic procedure ridders(func, x1, x2);
begin scalar eps, maxit, fl, fh, xl, xh, ans, xnew, iterNum, fnew;
eps:= 1.0e-12;
maxit:= 100;
fl := funcall(func, x1);
fh := funcall(func, x2);
xl := x1;
xh := x2;
ans := -1.0e30;
xnew := 0.0e0;
iterNum := 0;
if (fl*fh) > 0.0 then rederr "Initial points are not either side of zero.";
if fl=0.0 then x1
else if fh=0.0 then x2
%if (fl*fh) < 0.0 then
else repeat begin scalar xm, fm, ss;
xm:= 0.5*(xl+xh);
fm:= funcall(func, xm);
ss:= sqrt((fm*fm) - (fl*fh));
%if ss =0.0 then return ans;
xnew:= xm + (((xm - xl) * (if (fl>fh) then 1.0 else -1.0) * fm) / ss);
%if abs(xnew-ans) <= eps then return ans;
ans:= xnew;
fnew:= funcall(func, ans);
%write(fnew);
%if fnew=0.0 then return ans;
if msign(fm,fnew) neq fm then begin;
xl:= xm;
fl:= fm;
xh:= ans;
fh:= fnew;
end
else if msign(fl, fnew) neq fl then begin;
xh:= ans;
fh:= fnew;
end
else if msign(fh, fnew) neq fh then begin;
xl:= ans;
fl:= fnew;
end;
iterNum:=iterNum+1;
if iterNum >=maxit then rederr "Maximum iterations exceeded";
%if verbose then write xl, xh, fl, fh;
end until abs(fnew)<eps or abs(xh-xl) <= eps;
return ans
end;
ridders | reduce |
symbolic procedure choose(n, x);
begin scalar total, denom;
total := 1.0;
for denom:=1:x do <<
total:=total/denom*(n-denom+1) >>;
return total
end;
choose | reduce |
symbolic procedure dhyper(x, m, n, k);
choose(m, x) * choose(n, k - x) / choose(m + n, k);
dhyper
procedure phyper(x, m, n, k, lowerTail);
if lowerTail then
for i:=1:x sum dhyper(i, m, n, k)
else
for i:=(x+1):k sum dhyper(i, m, n, k);
phyper | reduce |
symbolic procedure testTolerance(x, y, atol);
if abs(x-y) <= atol then t else nil;
testtolerance | reduce |
symbolic procedure test1(); testTolerance(2*ridders(function(cos),0,2),
cdr reval(algebraic pi),1e-8);
test1
symbolic procedure test2(); testTolerance(choose(100, 5), 75287520, 0);
test2
symbolic procedure test3(); testTolerance(dhyper(5, 10, 7, 8), 0.3628137, 1.0e-7);
test3
symbolic procedure test4(); testTolerance(log(dhyper(5, 10, 7, 8)),-1.013866, 1.0e-7);
test4
symbolic procedure test5(); testTolerance(phyper(5, 10, 7, 8, t),0.7821884, 1.0e-7);
test5
symbolic procedure test6(); testTolerance(phyper(5, 10, 7, 8, nil),0.2178116, 1.0e-7);
test6
symbolic procedure testSet1(); {test1(), test2(), test3(), test4(), test5(), test6()};
testset1 | reduce |
testSet1();
(t t t t t t) | reduce |
%%
symbolic procedure fisherTest(a,b,c,d,alternative,oddsratio,confLevel);
begin scalar m, n, k, x00, lo, hi, doubleEps,
dnhyper,mnhyper,pnhyper,mle,ncpu,ncpl,
support,d2,logdc,alpha,
pvalue,cinterval;
m := a+c; % first column
n := b+d; % second column
k := a+b; % first row
x00 := a;
lo := max(0, k-n);
hi := min(k, m);
support := for i:=lo:hi collect i;
logdc := for each i in support collect log(dhyper(i, m, n, k));
doubleEps := 1.0e-10;
dnhyper := function(lambda ncp;
begin scalar maxd, sumd2, d, d2;
d := for i:=1:length(logdc) collect nth(logdc,i)+log(ncp)*nth(support,i);
maxd := apply(function(max),d);
d2 :=for each di in d collect exp(di-maxd);
sumd2 := for each i in d2 sum i;
return for each d2i in d2 collect d2i/sumd2
end);
mnhyper := function(lambda ncp;
begin scalar d, value;
value := if ncp=0.0 then lo
%else if ncp equal 'plusInfinity then hi
else begin;
d := funcall(dnhyper,ncp);
return for i:=1:length(d) sum nth(support,i)*nth(d,i)
end;
return value
end);
pnhyper := function(lambda(q,ncp,upperTail);
if ncp=1.0 then
(if upperTail then phyper(q-1, m, n, k, nil)
else phyper(q, m, n, k, t))
else if ncp=0.0 then
(if upperTail then
(if q<=lo then 1.0 else 0.0)
else (if q>=lo then 1.0 else 0.0))
else if ncp equal 'plusInfinity then
(if upperTail then
(if q<=hi then 1.0 else 0.0)
else (if q>= hi then 1.0 else 0.0))
else
begin scalar d, value;
d := funcall(dnhyper,ncp);
value := if upperTail then
for i:=1:length(d) sum (if nth(support,i)>=q then nth(d,i) else 0)
else for i:=1:length(d) sum (if nth(support,i)<=q then nth(d,i) else 0);
return value
end);
ncpL := function(lambda alpha;
if x00=lo then 0
else begin scalar p,value,f1,f2;
f1 := function(lambda y; funcall(pnhyper,x00,y,t) - alpha);
f2 := function(lambda y; funcall(pnhyper,x00,1/y,t) - alpha);
p := funcall(pnhyper,x00, 1.0, t);
value := if p>alpha then
ridders(f1, doubleEps, 1) % zero bound caused problems
else if p<alpha then 1/ridders(f2, doubleEps, 1)
else 1;
return value;
end);
ncpU := function(lambda alpha;
if x00=hi then 'plusInfinity
else begin scalar p,value,f1,f2;
f1 := function(lambda y; funcall(pnhyper,x00,y,nil) - alpha);
f2 := function(lambda y; funcall(pnhyper,x00,1/y,nil) - alpha);
p := funcall(pnhyper,x00, 1.0, nil);
value := if p<alpha then ridders(f1, 0, 1)
else if p>alpha then 1/ridders(f2, doubleEps, 1)
else 1;
return value;
end);
pvalue :=
if alternative equal 'less then funcall(pnhyper,x00,oddsratio,nil)
else if alternative equal 'greater then funcall(pnhyper,x00,oddsratio,t)
else if alternative equal 'twosided then
begin scalar relerr,dstar,dn;
relErr:= 1+1.0e-7;
dn:= funcall(dnhyper,oddsratio);
dstar:= nth(dn,x00-lo+1)*relErr;
return for each di in dn sum (if di<dstar then di else 0)
end
else -1;
cInterval :=
if alternative equal 'less then
{0.0, funcall(ncpU,1-confLevel)}
else if alternative equal 'greater then
{funcall(ncpL,1-confLevel), 'plusInfinity}
else if alternative equal 'twosided then
{funcall(ncpL,(1-confLevel)/2), funcall(ncpU,(1-confLevel)/2)}
else {-1,-1}; % no match
mle :=
if x00=lo then 0.0
else if x00=hi then 'plusInfinity
else
begin scalar mu, value, f1, f2;
f1 := function(lambda y; funcall(mnhyper,y) - x00);
f2 := function(lambda y; funcall(mnhyper,1/y) - x00);
mu := funcall(mnhyper,1.0);
value := if mu>x00 then ridders(f1,0,1)
else if mu<x00 then
1/ridders(f2,doubleEps,1.0)
else 1.0;
return value;
end;
return {pvalue,cinterval,mle};
end;
fishertest | reduce |
fishertest(10,10,10,20,'twosided,1,0.95);
(0.25754924281098 (0.53839938167829 7.4363408387441) 1.9716269432603)
fishertest(10,10,10,20,'less,1,0.95);
(0.92948113166106 (0.0 6.1438085831682) 1.9716269432603)
fishertest(10,10,10,20,'greater,1,0.95);
(0.18830137576992 (0.6459937819479 plusinfinity) 1.9716269432603) | reduce |
symbolic procedure testSet2;
begin scalar fit1;
fit1 := fishertest(10,10,10,20,'twosided,1,0.95);
return {testTolerance(first(fit1), 0.2575, 1.0e-3),
testTolerance(caadr(fit1),0.5383996, 1.0e-6),
testTolerance(cadadr(fit1),7.4363242, 1.0e-4),
testTolerance(third(fit1),1.971640, 1.0e-4)}
end;
testset2 | reduce |
testSet2();
(t t t t) | reduce |
The Maxima implementation was fairly brief:
maxima
| (%i2) | 
|
maxima
| (%i4) | 
|
maxima
| (%i6) | 
|
maxima
(%i7) fishertest(a,b,c,d,alternative,oddsratio,confLevel):=block(
[m,n,k,x00,lo,hi,doubleEps,dnhyper,mnhyper,pnhyper,mle,ncpu,
ncpl,support,d2,logdc,alpha,pvalue,cinterval],m:c+a,n:d+b,
k:b+a,x00:a,lo:max(0,k-n),hi:min(k,m),
support:makelist(i,i,lo,hi),
logdc:create_list(log(pdf_hypergeometric(i,m,n,k)),i,
support),doubleEps:1.0E-10,
dnhyper:lambda([ncp],
block([maxd,sumd2,d,d2],
d:makelist(log(ncp)*support[i]+logdc[i],
i,1,length(logdc)),
maxd:apply(max,d),
d2:create_list(exp(di-maxd),di,d),
sumd2:lsum(i,i,d2),
create_list(d2i/sumd2,d2i,d2))),
mnhyper:lambda([ncp],
block([d,value],
if equal(ncp,0) then lo
else (if equal(ncp,inf) then hi
else block([],
d:dnhyper(ncp),
sum(
support[i]
*d[i],i,1,
length(
d)))))),
pnhyper:lambda([q,ncp,upperTail],
if equal(ncp,1)
then (if upperTail
then 1
-cdf_hypergeometric(q-1,m,n,k)
else cdf_hypergeometric(
q,m,n,k))
else (if equal(ncp,0)
then (if upperTail
then (if q <= lo
then 1
else 0)
else (if q >= lo
then 1
else 0))
else (if equal(ncp,inf)
then (if upperTail
then (if
q <= hi
then 1
else 0)
else (if
q >= hi
then 1
else 0))
else block(
[d],d:dnhyper(ncp),
if upperTail
then sum(
if support[i] >= q
then d[i]
else 0,i,1,
length(d))
else sum(
if support[i] <= q
then d[i]
else 0,i,1,
length(d)))))),
ncpL:lambda([alpha],
if equal(x00,lo) then 0
else block([p],p:pnhyper(x00,1,true),
if p > alpha
then find_root(
lambda([y],
pnhyper(x00,y,true)
-alpha),0,1)
else (if p < alpha
then 1
/find_root(
lambda([y],
pnhyper(
x00,1/y,
true)
-alpha),
doubleEps,1)
else 1))),
ncpU:lambda([alpha],
if x00 = hi then inf
else block([p],p:pnhyper(x00,1,false),
if p < alpha
then find_root(
lambda([y],
pnhyper(x00,y,false)
-alpha),0,1)
else (if p > alpha
then 1
/find_root(
lambda([y],
pnhyper(
x00,1/y,
false)
-alpha),
doubleEps,1)
else 1))),
pvalue:if alternative = 'less
then pnhyper(x00,oddsratio,false)
else (if alternative = 'greater
then pnhyper(x00,oddsratio,true)
else (if alternative = 'twosided
then block([relerr,dstar,dn],
relErr
:9.9999999999999995E-8
+1,
dn
:dnhyper(
oddsratio),
dstar
:dn[1-lo+x00]
*relErr,
lsum(
if di < dstar
then di
else 0,di,dn))
else -1)),
cInterval:if alternative = 'less then [0,ncpU(1-confLevel)]
else (if alternative = 'greater
then [ncpL(1-confLevel),inf]
else (if alternative = 'twosided
then [
ncpL((1-confLevel)/2),
ncpU((1-confLevel)/2)]
else [-1,-1])),
mle:if equal(x00,lo) then 0
else (if equal(x00,hi) then inf
else block([mu],mu:mnhyper(1),
if mu > x00
then find_root(
lambda([y],mnhyper(y)-x00),
0,1)
else (if mu < x00
then 1
/find_root(
lambda([y],
mnhyper(
1/y)
-x00),
doubleEps,1)
else 1))),
[pvalue,cInterval,mle])
(%i8) display2d:true
maxima
| (%i10) | 
|
maxima
| (%i12) | 
|
maxima
| (%i14) | 
|
maxima
| (%i16) |
|
maxima
(%i17) testTolerance(x,y,atol):=if abs(x-y) <= atol then true
maxima
(%i19) testSet():=block([fit1],fit1:fishertest(10,10,10,20,'twosided,1,0.95),
[testTolerance(fit1[1],0.2575,0.001),
testTolerance(fit1[2][1],0.5383996,9.9999999999999995E-7),
testTolerance(fit1[2][2],7.4363242,1.0E-4),
testTolerance(fit1[3],1.97164,1.0E-4)])
(%i20) display2d:true
maxima
| (%i22) | 
|
The implementation in Common Lisp was a more direct translation of the R code:
lisp
;; from cl-statistics.lisp
(defun safe-exp (x)
"Eliminates floating point underflow for the exponential function.
Instead, it just returns 0.0d0"
(setf x (coerce x 'double-float))
(if (< x (log least-positive-double-float))
0.0d0
(exp x)))
(defun ridder (func x1 x2 &key (eps 1.0d-16) (maxit 30) (verbose nil))
(let (
(fl (funcall func x1))
(fh (funcall func x2))
(xl x1)
(xh x2)
(ans -1.11d30)
(xnew 0.0d0)
(iter-num 0)
)
(cond
((= fl 0) x1)
((= fh 0) x2)
((> (* fl fh) 0.0d0)
(error "Functions of the start points are not either side of zero."))
((< (* fl fh) 0.0d0)
(loop
(let* (
(xm (* 0.5d0 (+ xl xh)))
(fm (funcall func xm))
(ss (sqrt (- (* fm fm) (* fl fh))))
)
(if (= ss 0.0d0) (return ans))
(setf xnew (+ xm (/ (* (- xm xl) (if (> fl fh) 1.0d0 -1.0d0) fm) ss)))
(if (<= (abs (- xnew ans)) eps) (return ans))
(setf ans xnew fnew (funcall func ans))
(if (= fnew 0.0d0) (return ans))
(cond ((not (= (msign fm fnew) fm))
(setf xl xm fl fm xh ans fh fnew))
((not (= (msign fl fnew) fl))
(setf xh ans fh fnew))
((not (= (msign fh fnew) fh))
(setf xl ans fl fnew)))
(incf iter-num)
(if (>= iter-num maxit)
(return (values nil "Maximum iterations exceeded"))) ;; (error)?
(if verbose (format t "~,8f ~,8f ~,8f ~,8f~%" xl xh fl fh))
(if (<= (abs (- xh xl)) eps) (return ans))))))))
(defun msign (x y)
(* (abs x) (cond ((> y 0.0d0) 1.0d0) ((< y 0.0d0) -1.0d0) (t 0.0d0))))
;;(- (* (ridder #'cos 0.0d0 2.0d0) 2.0d0) pi)
(defun choose (n x)
(loop
for denom from 1 to x
and numerator from n downto (- n (1- x))
and total = 1 then (* total (/ numerator denom))
finally (return total)))
(defun dhyper (x m n k &key (log nil))
(let ((val
(/ (* (choose m x) (choose n (- k x))) (choose (+ m n) k))))
(if log
(log (coerce val 'double-float))
val)))
(defun phyper (x m n k &key (lower-tail t))
(if lower-tail
(loop for i from 1 to x summing (dhyper i m n k))
(loop for i from (1+ x) to k summing (dhyper i m n k))))
(defun fisher-test (x &key (alternative 'two-sided) (or 1.0d0)
(conf-int t)
(conf-level 0.95d0) (uniroot #'ridder))
"Fisher's exact test for a 2x2 integer array.
This is a hand translation of R's fisher.test()
making use of CL's large integers for the hypergeometric distribution"
(let* ((m (loop for i upto 1 summing (aref x i 0)))
(n (loop for i upto 1 summing (aref x i 1)))
(k (loop for i upto 1 summing (aref x 0 i)))
(x00 (aref x 0 0)) ; cf replacing x by (aref x 0 0)
(lo (max 0 (- k n)))
(hi (min k m))
(support (loop for i from lo to hi collect i))
(log-dc (loop for i in support
collect (dhyper i m n k :log t)))
(double-eps 1.0d-50))
(labels
((dnhyper (ncp)
(setf ncp (coerce ncp 'double-float))
(let* ((d (loop for i in log-dc and j in support
collect (+ i (* (log ncp) j))))
(max-d (apply #'max d))
(d2 (loop for i in d collect
(safe-exp (- i max-d)))) ;; NB: safe-exp used here
(sum-d2 (reduce #'+ d2)))
(loop for i in d2 collect (/ i sum-d2))))
(mnhyper (ncp)
(cond
((= ncp 0) lo)
((equal ncp 'infinity) hi)
(t (loop for i in support and j in (dnhyper ncp)
summing (* i j)))))
(pnhyper (q ncp &key (upper-tail nil))
(cond
((= ncp 1)
(if upper-tail
(coerce (phyper (1- x00) m n k :lower-tail nil) 'double-float)
(coerce (phyper x00 m n k) 'double-float)))
((= ncp 0)
(if upper-tail
(if (<= q lo) 1 0)
(if (>= q lo) 1 0)))
((equal ncp 'infinity)
(if upper-tail
(if (<= q hi) 1 0)
(if (>= q hi) 1 0)))
(t
(let ((d (dnhyper ncp)))
(if upper-tail
(loop for d-i in d and support-i in support
when (>= support-i q)
summing d-i)
(loop for d-i in d and support-i in support
when (<= support-i q)
summing d-i))))))
(mle (x)
(cond
((= x lo) 0)
((= x hi) 'infinity)
(t
(let ((mu (mnhyper 1)))
(cond
((> mu x)
(funcall uniroot (lambda (u) (- (mnhyper u) x))
0 1))
((< mu x)
(/ (funcall uniroot (lambda (u) (- (mnhyper (/ u)) x))
double-eps 1)))
(t 1))))))
(ncp-u (x alpha)
(and (= x hi) 'infinity)
(let ((p (pnhyper x 1)))
(cond
((< p alpha)
(funcall uniroot (lambda (u) (- (pnhyper x u) alpha))
0 1))
((> p alpha)
(/ (funcall uniroot (lambda (u) (- (pnhyper x (/ u)) alpha))
double-eps 1)))
(t 1))))
(ncp-l (x alpha)
(and (= x lo) 0)
(let ((p (pnhyper x 1 :upper-tail t)))
(cond
((> p alpha)
(funcall uniroot (lambda (u) (- (pnhyper x u :upper-tail t) alpha))
0 1))
((< p alpha)
(/ (funcall uniroot (lambda (u) (- (pnhyper x (/ u) :upper-tail t) alpha))
double-eps 1)))
(t 1)))))
(let ((p-value
(ecase alternative
(less (pnhyper x00 or))
(greater (pnhyper x00 or :upper-tail t))
(two-sided
(let* ((relErr (1+ 1.0d-7))
(d (dnhyper or))
(dstar (* (elt d (- x00 lo)) relErr)))
(loop for di in d when (< di dstar) summing di)))))
(c-interval
(if conf-int
(ecase alternative
(less (list 0 (ncp-u x00 (- 1 conf-level))))
(greater (list (ncp-l x00 (- 1 conf-level)) 'infinity))
(two-sided
(let ((alpha (/ (- 1 conf-level) 2)))
(list (ncp-l x00 alpha) (ncp-u x00 alpha)))))
nil))
(estimate (mle x00)))
(values p-value c-interval estimate)))))
;;(fisher-test #2a((10 10) (10 20)))
lisp
; (DEFUN RIDDER ...) is being compiled.
;; The variable FNEW is undefined.
;; The compiler will assume this variable is a global.
Value = 17624
With output:
