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I have recently been experimenting with using Axiom for maximum likelihood estimation,
borrowing code (with permission) from "Compact Numerical Methods" by Nash.

\begin{spad} )abbrev package OPTIM OptimNash ++ Author: M. Clements ++ Date Created: 24 August 2008 ++ Date Last Updated: ++ Basic Functions: ++ Related Constructors: ++ Also See: ++ AMS Classification: ++ Keywords: ++ References: ++ Description: ++ Numerical optimisation using methods from Nash ++ Permission granted by Nash to translate from his Pascal code to Axiom axisResults ==> Record(X: Vector Float, Fmin:Float, Lowerfun: Boolean) nmminResults ==> Record(X:Vector Float, Fmin:Float, Fail:Boolean) OptimNash: Exports == Implementation where Exports == with Rosen: Vector Float -> Float ++ Rosen(Bvec) returns the Rosen function nmmin: (Vector Float, (Vector Float -> Float), Float) -> nmminResults ++ nmmin(Bvec, fminfn, intol) returns values for the optimised function axissrch :(Vector Float, Vector Float -> Float) -> axisResults ++ axissrch(Bvec, fminfn) returns the axis search method demo: () -> nmminResults ++ demo() returns the rsults of a simple demo using the Rosen function demo2: () -> nmminResults ++ demo2() returns the results of a more complex demo --optimFun: (Expression Float,List Equation Expression Float) -> nmminResults Implementation == add --writeln(Classical starting point (-1.2,1)) Rosen(Bvec: Vector Float):Float == (Bvec(2)-Bvec(1)^2)^2*100.0+(1.0-Bvec(1))^2

--nmmin(vector([11.0, 1.0]), Rosen$OPTIM, -1.0)$OPTIM nmmin(Bvec: Vector Float, fminfn: Vector Float -> Float, intol: Float):nmminResults == Pcol ==> 27 Prow ==> 26 alpha ==> 1.0 beta ==> 0.5 gamma ==> 2.0 Calceps ==> 1.0e-35 big ==> 1.0e35 -- as per R/src/main/optim.c n : Integer := #Bvec action : String C : Integer calcvert : Boolean convtol : Float f : Float funcount : Integer H : Integer i : Integer j : Integer L : Integer notcomp : Boolean n1 : Integer oldsize : Float P : Matrix Float := new(Prow,Pcol,0.0)--[1..Prow,1..Pcol] shrinkfail: Boolean size : Float step : Float temp : Float trystep : Float VH,VL,VN : Float VR : Float x : Vector Float := Bvec -- ugly declaration --writeln(Nash Algorithm 19 version 2 1988-03-17) --writeln( Nelder Mead polytope direct search function minimiser) --writeln(confile,Nash Algorithm 19 version 2 1988-03-17) --writeln(confile, Nelder Mead polytope direct search function minimiser) fail := false (f,notcomp) := (fminfn(Bvec),false) if notcomp then --writeln(**** Function cannot be evaluated at initial parameters ****) --writeln(confile, -- **** Function cannot be evaluated at initial parameters ****) fail := true else --writeln(Function value for initial parameters = ,f) --writeln(confile,Function value for initial parameters = ,f) if intol<0.0 then intol := Calceps funcount := 1 convtol := intol*(abs(f)+intol) --writeln( Scaled convergence tolerance is ,convtol) --writeln(confile, Scaled convergence tolerance is ,convtol) n1 := n+1 C := n+2 P(n1,1) := f for i in 1..n repeat P(i,1) := Bvec(i) L := 1 size := 0.0 step := 0.0 for i in 1..n repeat if 0.1abs(Bvec(i))>step then step := 0.1abs(Bvec(i)) --writeln(Stepsize computed as ,step) --writeln(confile,Stepsize computed as ,step) for j in 2..n1 repeat action := "BUILD " for i in 1..n repeat P(i,j) := Bvec(i) trystep := step while P(j-1,j)=Bvec(j-1) repeat

P(j-1,j) := Bvec(j-1)+trystep trystep := trystep*10.0 size := size+trystep oldsize := size calcvert := true shrinkfail := false repeat if calcvert then for j in 1..n1 repeat if j~=L then for i in 1..n repeat Bvec(i) := P(i,j) (f,notcomp) := (fminfn(Bvec),false) if notcomp or f>=big then f := big funcount := funcount+1 P(n1,j) := f calcvert := false VL := P(n1,L) VH := VL H := L for j in 1..n1 repeat if j~=L then f := P(n1,j) if fVH then H := j VH := f if VH>VL+convtol then --str(funcount:5,tstr) --writeln(action,tstr,' ,VH, ,VL) --writeln(confile,action,tstr, ,VH, ',VL) VN := betaVL+(1.0-beta)VH for i in 1..n repeat temp := -P(i,H) for j in 1..n1 repeat temp := temp+P(i,j) P(i,C) := temp/n for i in 1..n repeat Bvec(i) := (1.0+alpha)P(i,C)-alphaP(i,H) (f,notcomp) := (fminfn(Bvec),false) if notcomp or f>=big then f := big funcount := funcount+1 action := "REFLECTION " VR := f if VRBvec(i)+(1-gamma)P(i,C) P(i,C) := Bvec(i) Bvec(i) := f (f,notcomp) := (fminfn(Bvec),false) if notcomp or f>=big then f := big funcount := funcount+1 if fP(i,H)+betaP(i,C) (f,notcomp) := (fminfn(Bvec),false) if notcomp or f>=big then f := big funcount := funcount+1 if f=VH then action := "SHRINK " calcvert := true size := 0.0 for j in 1..n1 repeat if j~=L then for i in 1..n repeat P(i,j) := beta*(P(i,j)-P(i,L))+P(i,L) size := size+abs(P(i,j)-P(i,L)) if sizePolytope size measure not decreased in shrink) --writeln(confile, -- Polytope size measure not decreased in shrink) shrinkfail := true if ((VH<=VL+convtol) or shrinkfail ) then break -- repeat loop --writeln(Exiting from Alg19.pas Nelder Mead polytope minimiser) --writeln(' ,funcount, function evaluations used') --writeln(confile,Exiting from Alg19.pas Nelder Mead polytope minimiser) --writeln(confile,' ,funcount, function evaluations used') fmin := P(n1,L) for i in 1..n repeat x(i) := P(i,L) if shrinkfail then fail := true return [x, fmin, fail]

axissrch(Bvec: Vector Float, fminfn: Vector Float -> Float): axisResults == cradius, eps, f, fplus, step, temp, tilt, fmin: Float i : Integer --notcomp : Boolean --writeln(alg20.pas -- axial search) --writeln(confile,alg20.pas -- axial search) Calceps : Float := 1.0e-35 big : Float := 1.0e35 -- as per R/src/main/optim.c n :Integer := #(Bvec)::NonNegativeInteger::Integer fmin := fminfn(Bvec) eps := Calceps eps := sqrt(eps) --writeln(' Axis':6,' Stepsize ':14,'function + ':14, -- 'function - :14, rad. of curv.:14, tilt') --writeln(confile,' Axis':6,' Stepsize ':14,'function + ':14, -- 'function - :14, rad. of curv.:14, tilt') lowerfn := false for i in 1..n repeat if (not lowerfn) then temp := Bvec(i) step := eps*(abs(temp)+eps) Bvec(i) := temp+step f := fminfn(Bvec) if (f>=big) then f := big --write(i:5,' ,step:12, ,f:12, ) --write(confile,i:5, ,step:12, ,f:12, ) if f<fmin then lowerfn := true if (not lowerfn) then fplus := f Bvec(i) := temp-step f := fminfn(Bvec) if f>=big then f := big --write(f:12, ) write(confile,f:12, ') if f temp := 0.5(fplus-f)/step fplus := 0.5(fplus+f-2.0fmin)/(stepstep)

if fplus~=0.0 then cradius := 1.0+temptemp cradius := cradiussqrt(cradius)/fplus else cradius := big tilt := 45.0*atan(temp)/atan(1.0) --write(cradius:12,' ,tilt:12) --write(confile,cradius:12, ',tilt:12) --writeln writeln(confile) [Bvec, f, lowerfn]

demo(): nmminResults == nmmin([-1.2, 1.0], Rosen, -1.0)

demo2(): nmminResults == --banner:=dr1920.pas -- driver for Nelder-Mead minimisation --startup --fminset(n,B,Workdata) {sets up problem and defines starting -- values of B} lowerfn:Boolean := false --{safety setting} B:Vector Float := [-1.2,1.0] --initial values mynmminResults :nmminResults myaxisResults : axisResults --n :Integer := #(B)::NonNegativeInteger::Integer repeat mytol:=-1.0 --Note: set the tolerance negative to indicate that --procedure must obtain an appropriate value. mynmminResults := nmmin(B,Rosen,mytol) --{minimise the function} --writeln --writeln(confile) --writeln( Minimum function value found =,Fmin) --writeln( At parameters) --writeln(confile, Minimum function value found =,Fmin) --writeln(confile, At parameters) --for i in 1..n repeat --{ --writeln(' B[',i,]=,X[i]) --writeln(confile,' B[',i,]=,X[i]) --} --{loop to write out parameters} B := mynmminResults.X myaxisResults := axissrch(B, Rosen) --{alg20.pas} lowerfn := myaxisResults.Lowerfun if lowerfn then B := myaxisResults.X --writeln(Lower function value found) --writeln(confile,Lower function value found) if (not lowerfn) then break return mynmminResults --flush(confile) close(confile) --if infname<>con then close(infile) --{dr1920.pas -- Nelder Mead minimisation with axial search} -- optimFun(e:Expression Float,initial:List Equation Expression Float):nmminResults == -- vars := [lhs(x) for x in initial] -- vals := [rhs(x) for x in initial]::Vector Float -- nmmin(vals, (x +-> eval(e, vars, x)), -1) \end{spad}

\begin{spad} )abbrev package SOMESTAT SomeStatisticsPackage SomeStatisticsPackage: Exports == Implementation where DF ==> DoubleFloat Exports ==> with pgamma:(Float,Float) -> Float ++ pgamma(y,p) returns the incomplete gamma function (Alg AS 147) erf:(Float) -> Float ++ erf(x) returns the error function erfc:(Float) -> Float ++ erfc(x) returns the complementary error function pnorm:(Float) -> Float ++ pnorm(x) returns the CDF for the normal(0,1) function pnorm:(Float,Float,Float) -> Float ++ pnorm(x,mean,sd) returns the CDF for a normal(mean,sd) function pchisq:(Float,Float) -> Float ppois:(Integer,Float) -> Float ppois2:(Integer,Float) -> Float rpois:(Float) -> Integer ++ rpois(lambda) returns a random Poisson variable with mean lambda rbinom:(Integer,Float) -> Integer ++ rbinom(n,p) returns a random binomial variable from n trials ++ and probability p mean:(Vector Float) -> Float mean:(List Float) -> Float mean:(Vector DF) -> DF mean:(List DF) -> DF var:(Vector Float) -> Float Beta:(Float,Float) -> Float betai:(Float,Float,Float) -> Float betacf:(Float,Float,Float) -> Float pbeta:(Float,Float,Float) -> Float ++ pbeta(x,a,b) returns the incomplete beta function betai(a,b,x) pt:(Float,Float) -> Float ++ pt(x,df) returns the CDF for a t distribution pbinom:(Float,Float,Float) -> Float ++ pbinom(k,n,p) returns the CDF for 0..k events given n trials with probability p pf:(Float,Float,Float) -> Float ++ pf(f,df1,df2) returns the CDF of the F distribution with df1 and df1 degrees of freedom Implementation ==> add import FloatSpecialFunctions import RandomFloatDistributions pgamma(y,p) == eps : Float := (10.0^(-digits()$Float+1))$Float g : Float := 0.0 if (y=0.0 or p=0.0) then return g if (y < 0.0 or p < 0.0) then error "Invalid arguments" a : Float := p + 1.0 f : Float := exp(p log(y) - logGamma(a) - y) if (f < eps) then return g c : Float := 1.0 g : Float := 1.0 a : Float := p while (c > eps g) repeat a := a+1.0 c := c(y / a) g := g+c g := gf return g erf(x:Float):Float == if x<0.0 then -erf(-x) else pgamma(x^2,0.5) erfc(x:Float):Float == 1-erf(x) pnorm(x:Float):Float == 0.5(1+erf(x/sqrt(2.0))) pnorm(x:Float,mu:Float,sigma:Float):Float == 0.5(1+erf((x-mu)/sigma/sqrt(2.0))) pchisq(x:Float,df:Float):Float == pgamma(x/2,df/2) ppois(y:Integer,lambda:Float):Float == if y<0 then return 0.0 reduce(_+,[exp(-lambda)lambda^yi/factorial(yi) for yi in 0..y])$List(Float) ppois2(y:Integer,lambda:Float):Float == 1-pgamma(lambda,y::Float+1) rpois(lambda:Float):Integer == cumP : Float := 0 x : Float := uniform01() for i in 0.. repeat cumP := cumP + exp(-lambda)lambda^i/factorial(i) if x1.0 then error "bad argument x in betai" bt := if x=0.0 or x=1.0 then 0.0 else x^a(1.0-x)^b/Beta(a,b) -- exp(logGamma(a+b)$FloatSpecialFunctions- -- logGamma(a)$FloatSpecialFunctions- -- logGamma(b)$FloatSpecialFunctions+ -- alog(x)+blog(1.0-x)) if x<(a+1.0)/(a+b+2.0) then return bt betacf(a,b,x)/a else return 1.0-btbetacf(b,a,1.0-x)/b betacf(a:Float,b:Float,x:Float):Float == itmax : Integer := 1000 eps : Float := (10.0^(-digits()$Float+1))$Float qab : Float := a+b qap : Float := a+1.0 qam : Float := a-1.0 c : Float := 1.0 d : Float := 1.0-qabx/qap d := 1.0/d h : Float := d for m in 1..itmax repeat em : Float := m::Float m2 := 2.0em aa : Float := m(b-em)x/((qam+m2)(a+m2)) d := 1.0+aad c := 1.0+aa/c d := 1.0/d h := hdc aa := -(a+em)(qab+em)x/((a+m2)(qap+m2)) d := 1.0+aad c := 1.0+aa/c d := 1.0/d del : Float := dc h := hdel if abs(del-1.0)f)) \end{spad}

Now, let's try this.

\begin{axiom} nmminResults ==> Record(X:Vector Float, Fmin:Float, Fail:Boolean) optimFun(e:Expression Float,initial:List Equation Expression Float):nmminResults == vars := [lhs(x) for x in initial] vals := [rhs(x) for x in initial]::Vector Float nmmin(vals, ((x:Vector Float):Float +-> eval(e, vars, x)), -1)$OPTIM

-- some examples dpois := exp(-lambda)*lambda^y/Gamma(y+1) negll1 := -log(eval(dpois, y=10.0)) optimFun(negll1, [lambda=5.0]) optimFun((b-a^2)^2*100.0+(1.0-a)^2, [a=11.0,b=1.0]) x := [xi::Float for xi in 1..5] Y := [10, 15, 20, 25, 31] negll1 := reduce(+, [eval(-log(dpois),[y=Y(i),lambda=exp(alpha+beta*x(i))]) for i in 1..#Y]); out1 := optimFun(negll1, [alpha=2, beta=0]) -- R: summary(glm(c(10, 15, 20, 25, 31) ~ I(1:5), family=poisson))

mleResult ==> Record(coef:Matrix Float,vars:List Symbol, AIC:Float) mle(negll:Expression Float, initial:List Equation Expression Float):mleResult == vars := [lhs(x) for x in initial]::List Symbol vals := [rhs(x) for x in initial]::Vector Float out : nmminResults := nmmin(vals, (x +-> eval(negll, vars, x)), -1)$OPTIM estimate := out.X covMatrix := inverse(matrix([[eval(D(D(negll,x),y),vars,estimate) for y in vars] for x in vars])) se := [sqrt(covMatrix(i,i)) for i in 1..nrows(covMatrix)] zscore := [estimate(i)/se(i) for i in 1..nrows(covMatrix)] pvalues := [2(1-pnorm(zscore(i)::Float)$SOMESTAT) for i in 1..nrows(covMatrix)] AIC := out.Fmin2+2*#(out.X) coef := transpose(matrix([estimate,se,zscore,pvalues])) return [coef,vars,AIC]

mle(negll1, [alpha=2.0, beta=0.0]) negll2 := -log(eval(dpois, y=10.0)) mle(negll2, [lambda=5.0]) \end{axiom}

As an aside: can we calculate an elasticity from logistic regression?

\begin{axiom} Y:=exp(alpha+betakPk)/(1+exp(alpha+betakPk)) D(Y,Pk)/Y D(Y,Pk)/Y/(1-Y) \end{axiom}


Some or all expressions may not have rendered properly, because Axiom returned the following error:
Error: export FRICAS=/usr/local/lib/fricas/target/x86_64-unknown-linux; export ALDORROOT=/usr/local/aldor/linux/1.1.0; export PATH=$ALDORROOT/bin:$PATH; export HOME=/var/zope2/var/LatexWiki; ulimit -t 600; export LD_LIBRARY_PATH=/usr/local/lib/fricas/target/x86_64-unknown-linux/lib; LANG=en_US.UTF-8 $FRICAS/bin/FRICASsys < /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/7245897146845666469-25px.axm
/bin/sh: /usr/local/lib/fricas/target/x86_64-unknown-linux/bin/FRICASsys: not found


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You can't use `macro parameter character #' in horizontal mode. l.166 n : Integer := # Bvec

Overfull \hbox (27.39969pt too wide) in paragraph at lines 155--223 []\T1/cmr/m/n/12 --nmmin(vector([11.0, 1.0]), Rosen$\OML/cmm/m/it/12 OPTIM; \OM S/cmsy/m/n/12 ^^@\OT1/cmr/m/n/12 1\OML/cmm/m/it/12 :\OT1/cmr/m/n/12 0)$\T1/cmr/ m/n/12 OPTIM nm-min(Bvec: Vec-tor Float, fminfn: Vec-tor Float -> Float, in-tol : Float):nmminResults [1] You can't use `macro parameter character #' in horizontal mode. l.333 n :Integer := # (Bvec)::NonNegativeInteger::Integer You can't use `macro parameter character #' in horizontal mode. l.388 --n :Integer := # (B)::NonNegativeInteger::Integer [2] Missing $ inserted. <inserted text> $ l.464 eps : Float := (10.0^ (-digits()$Float+1))$Float Missing $ inserted. <inserted text> $ l.494 cumP := cumP + exp(-lambda)*lambda^ i/factorial(i) Missing $ inserted. <inserted text> $ l.504 mean(x:Vector Float):Float == reduce(_ +,x)/#x You can't use `macro parameter character #' in math mode. l.504 ...n(x:Vector Float):Float == reduce(_+,x)/# x You can't use `macro parameter character #' in math mode. l.505 mean(x:List Float):Float == reduce(_+,x)/# x You can't use `macro parameter character #' in math mode. l.506 mean(x:Vector DF):DF == reduce(_+,x)/# x You can't use `macro parameter character #' in math mode. l.507 mean(x:List DF):DF == reduce(_+,x)/# x You can't use `macro parameter character #' in math mode. l.510 reduce(_+,[(x(i)-meanx)^2 for i in 1..# x])$List(Float)/(#x-1) You can't use `macro parameter character #' in horizontal mode. l.510 ...-meanx)^2 for i in 1..#x])$List(Float)/(# x-1) Missing $ inserted. <inserted text> $ l.527 eps : Float := (10.0^ (-digits()$Float+1))$Float Missing $ inserted. <inserted text> $ l.557 \end{spad} \newpage

Overfull \hbox (11.6583pt too wide) in paragraph at lines 422--557 \OML/cmm/m/it/12 Float \OT1/cmr/m/n/12 == \OML/cmm/m/it/12 ify < \OT1/cmr/m/n/1 2 0\OML/cmm/m/it/12 thenreturn\OT1/cmr/m/n/12 0\OML/cmm/m/it/12 :\OT1/cmr/m/n/1 2 0\OML/cmm/m/it/12 reduce\OT1/cmr/m/n/12 ([]\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 [\OML/cmm/m/it/12 exp\OT1/cmr/m/n/12 (\OMS/cmsy/m/n/12 ^^@\OML/cmm/m/it/12 lam bda\OT1/cmr/m/n/12 ) \OMS/cmsy/m/n/12 ^^C \OML/cmm/m/it/12 lambda[]i=factorial\ OT1/cmr/m/n/12 (\OML/cmm/m/it/12 yi\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 foryiin\OT 1/cmr/m/n/12 0\OML/cmm/m/it/12 ::y\OT1/cmr/m/n/12 ])$\T1/cmr/m/n/12 List(Float) ppois2(y:Integer,lambda:Float):Float

Overfull \hbox (65.58339pt too wide) in paragraph at lines 422--557 \OML/cmm/m/it/12 meanx \OT1/cmr/m/n/12 : \OML/cmm/m/it/12 Float \OT1/cmr/m/n/12 := \OML/cmm/m/it/12 mean\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 x\OT1/cmr/m/n/12 )\O ML/cmm/m/it/12 reduce\OT1/cmr/m/n/12 ([]\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 [(\O ML/cmm/m/it/12 x\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 i\OT1/cmr/m/n/12 ) \OMS/cmsy/ m/n/12 ^^@ \OML/cmm/m/it/12 meanx\OT1/cmr/m/n/12 )[]\OML/cmm/m/it/12 foriin\OT1 /cmr/m/n/12 1\OML/cmm/m/it/12 ::x\OT1/cmr/m/n/12 ])$\T1/cmr/m/n/12 List(Float)/ (x-1) --Gamma(x:Float):Float == Gamma(x)$\OML/cmm/m/it/12 FloatSpecialFunctions Beta\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 a \OT1/cmr/m/n/12 : [3] (/usr/share/texmf-texlive/tex/latex/base/t1cmtt.fd)

LaTeX Warning: Characters dropped after `\end{axiom}' on input line 593.

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[4] [5] (./3789055917543086017-16.0px.aux) ) (see the transcript file for additional information) Output written on 3789055917543086017-16.0px.dvi (5 pages, 21476 bytes). Transcript written on 3789055917543086017-16.0px.log.




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