MathAction statistical functions

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reading structures from a file ← ↑ → string conversions

statistical functions
last edited 10 years ago by test1

Anonymous shared the following library of statistical functions with us. Many thanks! (He added, that they are not checked...)

fricas
(1) -> )abbrev package STAT StatPackage
++ Description:
++ This package exports statistic utilities
StatPackage(S,A) : Exports == Implementation where 
 S: SetCategory
 A: Join(Collection(S), finiteAggregate)
 Exports == with
 if A has FiniteLinearAggregate(S) and S has OrderedSet then
 median: A -> S
 ++ median(a) median of a collection (ordering from OrderedSet)
 if S has Field then
 mean: A -> S
 ++ mean(a) arithmetic mean of a collection
 hmean: A -> S
 ++ hmean(a) harmonic mean of a collection
 moment: (A,Integer) -> S
 ++ moment(a,i) i nth moment of a collection
 variance: A -> S
 ++ variance(a) variance of a collection
 center: A -> A
 ++ center(a) center collection
 if A has shallowlyMutable then
 center!: A -> A
 ++ center!(a) center collection
 if S has RadicalCategory then
 gmean: A -> S
 ++ gmean(a) geometric mean of a collection
 stdev: A -> S
 ++ stdev(a) root mean square of a collection
 stdize: A -> A
 ++ stdize(a) standardize a collection 
 if A has shallowlyMutable then
 stdize!: A -> A
 ++ stdize!(a) standardize a collection 
 Implementation == add

 if A has FiniteLinearAggregate(S) and S has OrderedSet then
 median(m)==
 n := sort(m)
 i:= divide(#m,2)
 if (zero? i.remainder) then
 j:= elt(n,i.quotient+minIndex(m)-1)
 else
 j:= elt(n,i.quotient+minIndex(m))
 j

 if S has Field then
 if A has OneDimensionalArrayAggregate(S) then
 sums: A ->List S
 sums(m)==
 j:=0::S
 j2:=0::S
 for i in minIndex(m)..maxIndex(m) repeat
 h:= elt(m,i)
 j := j + h
 j2 := j2 + h*h
 [j,j2]
 variance(m)==
 i := center(m)
 s := sums(i) 
 k := (s.2 - s.1/#m::S)/(#m::S - 1::S)
 k
 if S has RadicalCategory then
 stdize(m)==
 i := mean(m)
 j := map(xx +-> xx-i, m)
 s := sums(j)
 k := sqrt((s.2 - s.1/#m::S)/(#m::S - 1::S))
 map(xx +-> (xx-i)/k,m)
 if A has shallowlyMutable then
 stdize!(m)==
 i := mean(m)
 j := map(xx +-> xx - i, m)
 s := sums(j)
 k := sqrt((s.2 - s.1/#m::S)/(#m::S - 1::S))
 map!(xx +-> (xx - i)/k, m)

 else
 variance(m)==
 i := center(m)
 j := reduce("+",i)
 j2 := reduce("+", map(xx +-> xx*xx,i))
 k := (j2 - j/#m::S)/(#m::S - 1::S)
 k
 if S has RadicalCategory then
 stdize(m)==
 me:= mean(m)
 i := map(xx +-> xx - me, m)
 j := reduce("+",i)
 j2 := reduce("+", map(xx +-> xx*xx,i))
 k := sqrt((j2 - j/#m::S)/(#m::S - 1::S))
 map(xx +-> (xx -me)/k,m)
 if A has shallowlyMutable then
 stdize!(m)==
 me:= mean(m)
 i := map(xx +-> xx - me,m)
 j := reduce("+",i)
 j2 := reduce("+", map(xx +-> xx*xx, i))
 k := sqrt((j2 - j/#m::S)/(#m::S - 1::S))
 map!(xx +->(xx - me)/k, m)

 if A has FiniteLinearAggregate(S) and S has OrderedSet then
 median(m)==
 n := sort(m)
 min := minIndex(m)
 i:= divide(#m,2)
 if (zero? i.remainder) then
 j:= (elt(n,i.quotient+min-1)+elt(n,i.quotient+min))/2::S
 else
 j:=elt(n,i.quotient+min)
 j
 mean(m)==
 i := reduce("+", m)
 (i / (#m::S))

 hmean(m)==
 i := map(xx +-> 1::S/xx, m)
 j := reduce("+", i)
 ((#m::S) / j)

 moment(m,n)==
 n = 0 => 1
 n = 1 => mean(m)
 i := map(xx +-> xx^n, m)
 j := reduce("+",i)
 (j / (#m::S))

 center(m)==
 i := mean(m)
 map(xx +-> xx - i, m)

 if A has shallowlyMutable then
 center!(m)==
 i := mean(m)
 map!(xx +-> xx - i, m)
 m
 if S has RadicalCategory then
 gmean(m)==
 i := reduce("*", m)
 nthRoot(i,#m)

 stdev(m)==
 sqrt(variance(m))
fricas
)abbrev package IDSTAT IntegralDomainStatPackage
++ Description:
++ This package exports statistic utilities over IntegralDomain
IntegralDomainStatPackage(S,A) : Exports == Implementation where 
 S: IntegralDomain
 A: Join(Collection(S), finiteAggregate)
 Exports == with
 mean: A -> Fraction(S)
 ++ mean(a) arithmetic mean of a collection
 moment: (A,NonNegativeInteger) -> Fraction(S)
 ++ moment(m,n) nth moment of a collection

 Implementation == add
 mean(m)==
 i := reduce("+", m)
 i / #m::S
 moment(m,n)==
 n = 0 => 1
 n = 1 => mean(m)
 i := map(xx +-> xx^n, m)
 j := reduce("+",i)
 j / #m::S
fricas
)abbrev package MSTAT MatrixStatPackage
++ Description:
++ This package exports statistic utilities on two dimensional arrays
++ Function are applied column by column
MatrixStatPackage(S,Row,Col,M) : Exports == Implementation where 
 S: SetCategory
 Row : FiniteLinearAggregate S
 Col : FiniteLinearAggregate S
 M : TwoDimensionalArrayCategory(S,Row,Col)
 Exports == with
 if Col has shallowlyMutable then
 if S has OrderedSet then
 median: M -> Col
 ++ median(a) median of a two dimensional array (ordering from OrderedSet)
 if S has Field then
 mean: M -> Col
 ++ mean(a) arithmetic mean of a two dimensional array
 hmean: M -> Col
 ++ hmean(a) harmonic mean of a two dimensional array
 variance: M -> Col
 ++ variance(a) variance of a two dimensional array
 moment: (M,Integer) -> Col
 ++ moment(a,i) i nth moment of a two dimensional array
 center: M -> M
 ++ center(a) center two dimensional array
 center!: M -> M
 ++ center!(a) center two dimensional array
 if M has MatrixCategory(S,Row,Col) then
 cov: M -> M
 ++ cov(a): covariance matrix
 if S has RadicalCategory then
 gmean: M -> Col
 ++ gmean(a) geometric mean of a two dimensional array
 stdev: M -> Col
 ++ stdev(a) root mean square of a two dimensional array
 stdize: M -> M
 ++ stdize(a) standardize two dimensional array
 stdize!: M -> M
 ++ stdize!(a) standardize two dimensional array
 if M has MatrixCategory(S,Row,Col) then
 cor: M -> M
 ++ cor(a): correlation matrix

 Implementation == add
 import StatPackage(S,Col)
 if Col has shallowlyMutable then
 colReduce: (Col -> S, M) -> Col
 colReduce(fx,m)==
 ret : Col := new(ncols(m),NIL$Lisp)
 j := minColIndex(m) - 1
 for i in j+1..maxColIndex(m) repeat
 setelt!(ret,i-j,fx(column(m,i)))
 ret
 if S has OrderedSet then
 median(m) ==
 ret := colReduce(median$StatPackage(S,Col),m)
 ret
 if S has Field then
 mean(m)==
 ret := colReduce(mean$StatPackage(S,Col),m)
 ret
 hmean(m)==
 ret := colReduce(hmean$StatPackage(S,Col),m)
 ret
 variance(m)==
 ret := colReduce(variance$StatPackage(S,Col),m)
 ret
 moment(m,pow)==
 ret : Col := new(ncols(m),NIL$Lisp)
 j := minColIndex(m) - 1
 for i in j+1..maxColIndex(m) repeat
 setelt!(ret,i,moment(column(m,i),pow)$StatPackage(S,Col))
 ret
 center(m)==
 ret : M := new(nrows(m),ncols(m),NIL$Lisp)
 j := minColIndex(m) - 1
 for i in j+1..maxColIndex(m) repeat
 setColumn!(ret,i-j,center!(column(m,i))$StatPackage(S,Col))
 ret
 center!(m)==
 j := minColIndex(m) - 1
 for i in j+1..maxColIndex(m) repeat
 setColumn!(m,i-j,center!(column(m,i))$StatPackage(S,Col))
 m
 if M has MatrixCategory(S,Row,Col) then
 cov(m)==
 ret := center(m)
 transpose(ret/(nrows(m)-1)::S) * ret

 if S has RadicalCategory then
 gmean(m)==
 colReduce(gmean$StatPackage(S,Col),m)
 stdev(m)==
 colReduce(stdev$StatPackage(S,Col),m)
 stdize(m)==
 ret : M := new(nrows(m),ncols(m),NIL$Lisp)
 j := minColIndex(m) - 1
 for i in j+1..maxColIndex(m) repeat
 setColumn!(ret,i-j,stdize!(column(m,i))$StatPackage(S,Col))
 ret
 stdize!(m)==
 j := minColIndex(m) - 1
 for i in j+1..maxColIndex(m) repeat
 setColumn!(m,i-j,stdize!(column(m,i))$StatPackage(S,Col))
 m
 if M has MatrixCategory(S,Row,Col) then
 cor(m)==
 ret := stdize(m)
 (transpose(ret/((nrows(m)-1)::S)) * ret)
fricas
Compiling FriCAS source code from file 
 /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/172717192283825040-25px.001.spad
 using old system compiler.
 STAT abbreviates package StatPackage 
------------------------------------------------------------------------
 initializing NRLIB STAT for StatPackage 
 compiling into NRLIB STAT 
****** Domain: A already in scope
****** Domain: A already in scope
augmenting A: (FiniteLinearAggregate S)
****** Domain: S already in scope
augmenting S: (OrderedSet)
 compiling exported median : A -> S
Time: 0.04 SEC.

****** Domain: S already in scope
augmenting S: (Field)
****** Domain: A already in scope
augmenting A: (OneDimensionalArrayAggregate S)
 compiling local sums : A -> List S
Time: 0 SEC.

 compiling exported variance : A -> S
Time: 0 SEC.

****** Domain: S already in scope
augmenting S: (RadicalCategory)
 compiling exported stdize : A -> A
Time: 0 SEC.

 compiling exported stdize! : A -> A
Time: 0 SEC.

 compiling exported variance : A -> S
Time: 0 SEC.

****** Domain: S already in scope
augmenting S: (RadicalCategory)
 compiling exported stdize : A -> A
Time: 0 SEC.

****** Domain: A already in scope
augmenting A: (shallowlyMutable)
 compiling exported stdize! : A -> A
Time: 0 SEC.

****** Domain: A already in scope
augmenting A: (FiniteLinearAggregate S)
****** Domain: S already in scope
augmenting S: (OrderedSet)
 compiling exported median : A -> S
Time: 0 SEC.

 compiling exported mean : A -> S
Time: 0 SEC.

 compiling exported hmean : A -> S
Time: 0 SEC.

 compiling exported moment : (A,Integer) -> S
Time: 0 SEC.

 compiling exported center : A -> A
Time: 0 SEC.

****** Domain: A already in scope
augmenting A: (shallowlyMutable)
 compiling exported center! : A -> A
Time: 0 SEC.

****** Domain: S already in scope
augmenting S: (RadicalCategory)
 compiling exported gmean : A -> S
Time: 0 SEC.

 compiling exported stdev : A -> S
Time: 0 SEC.

****** Domain: A already in scope
augmenting A: (FiniteLinearAggregate S)
****** Domain: S already in scope
augmenting S: (OrderedSet)
****** Domain: A already in scope
augmenting A: (shallowlyMutable)
****** Domain: S already in scope
augmenting S: (Field)
****** Domain: A already in scope
augmenting A: (shallowlyMutable)
****** Domain: S already in scope
augmenting S: (Field)
****** Domain: S already in scope
augmenting S: (RadicalCategory)
****** Domain: S already in scope
augmenting S: (Field)
****** Domain: S already in scope
augmenting S: (RadicalCategory)
****** Domain: S already in scope
augmenting S: (Field)
(time taken in buildFunctor: 0)
Time: 0 SEC.

 Cumulative Statistics for Constructor StatPackage
 Time: 0.06 seconds

 finalizing NRLIB STAT 
 Processing StatPackage for Browser database:
--------constructor---------
--------(median (S A))---------
--------(mean (S A))---------
--------(hmean (S A))---------
--------(moment (S A (Integer)))---------
--------(variance (S A))---------
--------(center (A A))---------
--------(center! (A A))---------
--------(gmean (S A))---------
--------(stdev (S A))---------
--------(stdize (A A))---------
--------(stdize! (A A))---------
; compiling file "/var/aw/var/LatexWiki/STAT.NRLIB/STAT.lsp" (written 13 AUG 2025 12:34:22 AM):

; wrote /var/aw/var/LatexWiki/STAT.NRLIB/STAT.fasl
; compilation finished in 0:00:00.072
------------------------------------------------------------------------
 StatPackage is now explicitly exposed in frame initial 
 StatPackage will be automatically loaded when needed from 
 /var/aw/var/LatexWiki/STAT.NRLIB/STAT

 IDSTAT abbreviates package IntegralDomainStatPackage 
------------------------------------------------------------------------
 initializing NRLIB IDSTAT for IntegralDomainStatPackage 
 compiling into NRLIB IDSTAT 
****** Domain: A already in scope
 compiling exported mean : A -> Fraction S
Time: 0 SEC.

 compiling exported moment : (A,NonNegativeInteger) -> Fraction S
Time: 0 SEC.

(time taken in buildFunctor: 0)
Time: 0 SEC.

 Cumulative Statistics for Constructor IntegralDomainStatPackage
 Time: 0 seconds

 finalizing NRLIB IDSTAT 
 Processing IntegralDomainStatPackage for Browser database:
--------constructor---------
--------(mean ((Fraction S) A))---------
--------(moment ((Fraction S) A (NonNegativeInteger)))---------
; compiling file "/var/aw/var/LatexWiki/IDSTAT.NRLIB/IDSTAT.lsp" (written 13 AUG 2025 12:34:22 AM):

; wrote /var/aw/var/LatexWiki/IDSTAT.NRLIB/IDSTAT.fasl
; compilation finished in 0:00:00.008
------------------------------------------------------------------------
 IntegralDomainStatPackage is now explicitly exposed in frame initial

 IntegralDomainStatPackage will be automatically loaded when needed 
 from /var/aw/var/LatexWiki/IDSTAT.NRLIB/IDSTAT

 MSTAT abbreviates package MatrixStatPackage 
------------------------------------------------------------------------
 initializing NRLIB MSTAT for MatrixStatPackage 
 compiling into NRLIB MSTAT 
 importing StatPackage(S,Col)
****** Domain: Col already in scope
augmenting Col: (shallowlyMutable)
 compiling local colReduce : (Col -> S,M) -> Col
Time: 0 SEC.

****** Domain: S already in scope
augmenting S: (OrderedSet)
 compiling exported median : M -> Col
Time: 0 SEC.

****** Domain: S already in scope
augmenting S: (Field)
 compiling exported mean : M -> Col
Time: 0.01 SEC.

 compiling exported hmean : M -> Col
Time: 0 SEC.

 compiling exported variance : M -> Col
Time: 0 SEC.

 compiling exported moment : (M,Integer) -> Col
Time: 0 SEC.

 compiling exported center : M -> M
Time: 0 SEC.

 compiling exported center! : M -> M
Time: 0 SEC.

****** Domain: M already in scope
augmenting M: (MatrixCategory S Row Col)
 compiling exported cov : M -> M
Time: 0 SEC.

****** Domain: S already in scope
augmenting S: (RadicalCategory)
 compiling exported gmean : M -> Col
Time: 0 SEC.

 compiling exported stdev : M -> Col
Time: 0 SEC.

 compiling exported stdize : M -> M
Time: 0 SEC.

 compiling exported stdize! : M -> M
Time: 0 SEC.

****** Domain: M already in scope
augmenting M: (MatrixCategory S Row Col)
 compiling exported cor : M -> M
Time: 0 SEC.

****** Domain: Col already in scope
augmenting Col: (shallowlyMutable)
****** Domain: M already in scope
augmenting M: (MatrixCategory S Row Col)
****** Domain: S already in scope
augmenting S: (Field)
****** Domain: Col already in scope
augmenting Col: (shallowlyMutable)
****** Domain: M already in scope
augmenting M: (MatrixCategory S Row Col)
****** Domain: S already in scope
augmenting S: (Field)
****** Domain: S already in scope
augmenting S: (RadicalCategory)
****** Domain: Col already in scope
augmenting Col: (shallowlyMutable)
****** Domain: S already in scope
augmenting S: (Field)
****** Domain: Col already in scope
augmenting Col: (shallowlyMutable)
****** Domain: S already in scope
augmenting S: (Field)
****** Domain: S already in scope
augmenting S: (RadicalCategory)
****** Domain: Col already in scope
augmenting Col: (shallowlyMutable)
****** Domain: S already in scope
augmenting S: (OrderedSet)
(time taken in buildFunctor: 0)
Time: 0 SEC.

 Cumulative Statistics for Constructor MatrixStatPackage
 Time: 0.03 seconds

 finalizing NRLIB MSTAT 
 Processing MatrixStatPackage for Browser database:
--------constructor---------
--------(median (Col M))---------
--------(mean (Col M))---------
--------(hmean (Col M))---------
--------(variance (Col M))---------
--------(moment (Col M (Integer)))---------
--------(center (M M))---------
--------(center! (M M))---------
--------(cov (M M))---------
--------(gmean (Col M))---------
--------(stdev (Col M))---------
--------(stdize (M M))---------
--------(stdize! (M M))---------
--------(cor (M M))---------
; compiling file "/var/aw/var/LatexWiki/MSTAT.NRLIB/MSTAT.lsp" (written 13 AUG 2025 12:34:22 AM):

; wrote /var/aw/var/LatexWiki/MSTAT.NRLIB/MSTAT.fasl
; compilation finished in 0:00:00.024
------------------------------------------------------------------------
 MatrixStatPackage is now explicitly exposed in frame initial 
 MatrixStatPackage will be automatically loaded when needed from 
 /var/aw/var/LatexWiki/MSTAT.NRLIB/MSTAT

And

fricas
a:= ["a","c","d","g","z","a","d","g","f"]
\label{eq1}\begin{array}{@{}l} \displaystyle \left[ \verb#"a"# , \: \verb#"c"# , \: \verb#"d"# , \: \verb#"g"# , \: \verb#"z"# , \: \right. \ \ \displaystyle \left.\verb#"a"# , \: \verb#"d"# , \: \verb#"g"# , \: \verb#"f"# \right] (1)
Type: List(String)
fricas
b:= [1,7,8,9,2,4,6,17,18,14,32]
\label{eq2}\left[ 1, \: 7, \: 8, \: 9, \: 2, \: 4, \: 6, \:{17}, \:{18}, \:{14}, \:{32}\right] (2)
Type: List(PositiveInteger?)
fricas
median a
\label{eq3}\verb#"d"# (3)
Type: String
fricas
median b
\label{eq4}8 (4)
Type: PositiveInteger?
fricas
mean b
\label{eq5}\frac{118}{11} (5)
Type: Fraction(Integer)
fricas
hmean b
\label{eq6}\frac{125664}{28705} (6)
Type: Fraction(Integer)
fricas
moment(b,1)
\label{eq7}\frac{118}{11} (7)
Type: Fraction(Integer)
fricas
moment(b,2)
\label{eq8}\frac{2084}{11} (8)
Type: Fraction(Integer)
fricas
moment(b,3)
\label{eq9}\frac{48130}{11} (9)
Type: Fraction(Integer)
fricas
variance b
\label{eq10}\frac{900}{11} (10)
Type: Fraction(Integer)
fricas
stdev b
\label{eq11}\frac{30}{\sqrt{11}} (11)
Type: AlgebraicNumber?
fricas
gmean b
\label{eq12}2 \ {\root{11}\of{1619352}} (12)
Type: AlgebraicNumber?
fricas
center b
\label{eq13}\begin{array}{@{}l} \displaystyle \left[ -{\frac{107}{11}}, \: -{\frac{41}{11}}, \: -{\frac{30}{1 1}}, \: -{\frac{19}{11}}, \: -{\frac{96}{11}}, \right. \ \ \displaystyle \left.\: -{\frac{74}{11}}, \: -{\frac{52}{11}}, \:{\frac{69}{1 1}}, \:{\frac{80}{11}}, \:{\frac{36}{11}}, \: \right. \ \ \displaystyle \left.{\frac{234}{11}}\right] (13)
Type: List(Fraction(Integer))
fricas
stdize b
\label{eq14}\begin{array}{@{}l} \displaystyle \left[ -{\frac{{107}\ {\sqrt{11}}}{330}}, \: -{\frac{{41}\ {\sqrt{1 1}}}{330}}, \: -{\frac{\sqrt{11}}{11}}, \: -{\frac{{19}\ {\sqrt{1 1}}}{330}}, \right. \ \ \displaystyle \left.\: -{\frac{{16}\ {\sqrt{11}}}{55}}, \: -{\frac{{37}\ {\sqrt{1 1}}}{165}}, \: -{\frac{{26}\ {\sqrt{11}}}{165}}, \: \right. \ \ \displaystyle \left.{\frac{{23}\ {\sqrt{11}}}{110}}, \:{\frac{8 \ {\sqrt{11}}}{3 3}}, \:{\frac{6 \ {\sqrt{11}}}{55}}, \:{\frac{{39}\ {\sqrt{11}}}{5 5}}\right] (14)
Type: List(AlgebraicNumber?)
fricas
c := [3+7*%i,5+8*%i,8+3*%i,4+5*%i]
\label{eq15}\left[{3 +{7 \ i}}, \:{5 +{8 \ i}}, \:{8 +{3 \ i}}, \:{4 +{5 \ i}}\right] (15)
Type: List(Complex(Integer))
fricas
mean c
\label{eq16}\frac{{20}+{{23}\ i}}{4} (16)
Type: Fraction(Complex(Integer))




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