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last edited 9 years ago by test1 |
Edit detail for statistical functions revision 1 of 2
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Time: 2007/11/13 00:16:27 GMT-8 |
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| Note: transferred from axiom-developer.org | ||
changed: - Anonymous shared the following library of statistical functions with us. Many thanks! (He added, that they are not checked...) \begin{axiom} )abbrev package STAT StatPackage ++ Description: ++ This package exports statistic utilities StatPackage(S,A) : Exports == Implementation where S: SetCategory A: Collection(S) with (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(#1-i,m) s := sums(j) k := sqrt((s.2 - s.1/#m::S)/(#m::S - 1::S)) map((#1-i)/k,m) if A has shallowlyMutable then stdize_!(m)== i := mean(m) j := map(#1-i,m) s := sums(j) k := sqrt((s.2 - s.1/#m::S)/(#m::S - 1::S)) map_!((#1-i)/k,m) else variance(m)== i := center(m) j := reduce("+",i) j2 := reduce("+", map(#1*#1,i)) k := (j2 - j/#m::S)/(#m::S - 1::S) k if S has RadicalCategory then stdize(m)== me:= mean(m) i := map(#1-me,m) j := reduce("+",i) j2 := reduce("+", map(#1*#1,i)) k := sqrt((j2 - j/#m::S)/(#m::S - 1::S)) map((#1-me)/k,m) if A has shallowlyMutable then stdize_!(m)== me:= mean(m) i := map(#1-me,m) j := reduce("+",i) j2 := reduce("+", map(#1*#1,i)) k := sqrt((j2 - j/#m::S)/(#m::S - 1::S)) map_!((#1-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(1::S/#1,m) j := reduce("+", i) ((#m::S) / j) moment(m,n)== n = 0 => 1 n = 1 => mean(m) i := map(#1**n, m) j := reduce("+",i) (j / (#m::S)) center(m)== i := mean(m) map(#1-i,m) if A has shallowlyMutable then center_!(m)== i := mean(m) map_!(#1-i,m) m if S has RadicalCategory then gmean(m)== i := reduce("*", m) nthRoot(i,#m) stdev(m)== sqrt(variance(m)) )abbrev package IDSTAT IntegralDomainStatPackage ++ Description: ++ This package exports statistic utilities over IntegralDomain IntegralDomainStatPackage(S,A) : Exports == Implementation where S: IntegralDomain A: Collection(S) with (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(#1**n, m) j := reduce("+",i) j / #m::S )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 Lisp 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-j,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) \end{axiom} And \begin{axiom} a:= ["a","c","d","g","z","a","d","g","f"] b:= [1,7,8,9,2,4,6,17,18,14,32] median a median b mean b hmean b moment(b,1) moment(b,2) moment(b,3) variance b stdev b gmean b center b stdize b -- beware, the following is not mathematically justified since -- median use ordering from OrderedSet (and not OrderedRing) c := [3+7*%i,5+8*%i,8+3*%i,4+5*%i] median c \end{axiom}
Anonymous shared the following library of statistical functions with us. Many thanks! (He added, that they are not checked...)
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)abbrev package STAT StatPackage ++ Description: ++ This package exports statistic utilities StatPackage(S,A) : Exports == Implementation where S: SetCategory A: Collection(S) with (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(#1-i, m) s := sums(j) k := sqrt((s.2 - s.1/#m::S)/(#m::S - 1::S)) map((#1-i)/k, m) if A has shallowlyMutable then stdize_!(m)== i := mean(m) j := map(#1-i, m) s := sums(j) k := sqrt((s.2 - s.1/#m::S)/(#m::S - 1::S)) map_!((#1-i)/k, m)
else variance(m)== i := center(m) j := reduce("+",i) j2 := reduce("+", map(#1*#1, i)) k := (j2 - j/#m::S)/(#m::S - 1::S) k if S has RadicalCategory then stdize(m)== me:= mean(m) i := map(#1-me, m) j := reduce("+", i) j2 := reduce("+", map(#1*#1, i)) k := sqrt((j2 - j/#m::S)/(#m::S - 1::S)) map((#1-me)/k, m) if A has shallowlyMutable then stdize_!(m)== me:= mean(m) i := map(#1-me, m) j := reduce("+", i) j2 := reduce("+", map(#1*#1, i)) k := sqrt((j2 - j/#m::S)/(#m::S - 1::S)) map_!((#1-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(1::S/#1,m) j := reduce("+", i) ((#m::S) / j)
moment(m,n)== n = 0 => 1 n = 1 => mean(m) i := map(#1**n, m) j := reduce("+", i) (j / (#m::S))
center(m)== i := mean(m) map(#1-i,m)
if A has shallowlyMutable then center_!(m)== i := mean(m) map_!(#1-i,m) m if S has RadicalCategory then gmean(m)== i := reduce("*", m) nthRoot(i, #m)
stdev(m)== sqrt(variance(m))
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)abbrev package IDSTAT IntegralDomainStatPackage ++ Description: ++ This package exports statistic utilities over IntegralDomain IntegralDomainStatPackage(S,A) : Exports == Implementation where S: IntegralDomain A: Collection(S) with (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(#1**n, m) j := reduce("+", i) j / #m::S
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)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 Lisp 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$Lisp) j := minColIndex(m) - 1 for i in j+1..maxColIndex(m) repeat setelt(ret, i-j, moment(column(m, i), pow)$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/1059333137717529116-25px.001.spad
using old system compiler.
STAT abbreviates package StatPackage
******** Spad syntax error detected ********
Expected: |)|
The prior line was:
125> n = 1 => mean(m)
The current line is:
126> i := map(#1**n,
The number of valid tokens is 1.
The prior token was #S(TOKEN
:SYMBOL |#1|
:TYPE ARGUMENT-DESIGNATOR
:NONBLANK 126)
The current token is #S(TOKEN :SYMBOL POWER :TYPE KEYWORD :NONBLANK 126)And
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a:= ["a","c", "d", "g", "z", "a", "d", "g", "f"]
![\label{eq1}\left[ \mbox{\tt "a"}, \: \mbox{\tt "c"}, \: \mbox{\tt "d"}, \: \mbox{\tt "g"}, \: \mbox{\tt "z"}, \: \mbox{\tt "a"}, \: \mbox{\tt "d"}, \: \mbox{\tt "g"}, \: \mbox{\tt "f"}\right]](images/5867032578980925005-16.0px.png "\label{eq1}\left[ \mbox{\tt \"a\"}, \: \mbox{\tt \"c\"}, \: \mbox{\tt \"d\"}, \: \mbox{\tt \"g\"}, \: \mbox{\tt \"z\"}, \: \mbox{\tt \"a\"}, \: \mbox{\tt \"d\"}, \: \mbox{\tt \"g\"}, \: \mbox{\tt \"f\"}\right]") | (1) |
Type: List(String)
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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]](images/4622356956878012728-16.0px.png "\label{eq2}\left[ 1, \: 7, \: 8, \: 9, \: 2, \: 4, \: 6, \:{17}, \:{18}, \:{14}, \:{32}\right]") | (2) |
Type: List(PositiveInteger?)
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median a
There are no library operations named median Use HyperDoc Browse or issue )what op median to learn if there is any operation containing " median " in its name.
Cannot find a definition or applicable library operation named median with argument type(s) List(String)
Perhaps you should use "@" to indicate the required return type,or "$" to specify which version of the function you need. median b
There are no library operations named median Use HyperDoc Browse or issue )what op median to learn if there is any operation containing " median " in its name.
Cannot find a definition or applicable library operation named median with argument type(s) List(PositiveInteger)
Perhaps you should use "@" to indicate the required return type,or "$" to specify which version of the function you need. mean b
There are no library operations named mean Use HyperDoc Browse or issue )what op mean to learn if there is any operation containing " mean " in its name.
Cannot find a definition or applicable library operation named mean with argument type(s) List(PositiveInteger)
Perhaps you should use "@" to indicate the required return type,or "$" to specify which version of the function you need. hmean b
There are no library operations named hmean Use HyperDoc Browse or issue )what op hmean to learn if there is any operation containing " hmean " in its name.
Cannot find a definition or applicable library operation named hmean with argument type(s) List(PositiveInteger)
Perhaps you should use "@" to indicate the required return type,or "$" to specify which version of the function you need. moment(b, 1)
There are no exposed library operations named moment but there is one unexposed operation with that name. Use HyperDoc Browse or issue )display op moment to learn more about the available operation.
Cannot find a definition or applicable library operation named moment with argument type(s) List(PositiveInteger) PositiveInteger
Perhaps you should use "@" to indicate the required return type,or "$" to specify which version of the function you need. moment(b, 2)
There are no exposed library operations named moment but there is one unexposed operation with that name. Use HyperDoc Browse or issue )display op moment to learn more about the available operation.
Cannot find a definition or applicable library operation named moment with argument type(s) List(PositiveInteger) PositiveInteger
Perhaps you should use "@" to indicate the required return type,or "$" to specify which version of the function you need. moment(b, 3)
There are no exposed library operations named moment but there is one unexposed operation with that name. Use HyperDoc Browse or issue )display op moment to learn more about the available operation.
Cannot find a definition or applicable library operation named moment with argument type(s) List(PositiveInteger) PositiveInteger
Perhaps you should use "@" to indicate the required return type,or "$" to specify which version of the function you need. variance b
There are no library operations named variance Use HyperDoc Browse or issue )what op variance to learn if there is any operation containing " variance " in its name.
Cannot find a definition or applicable library operation named variance with argument type(s) List(PositiveInteger)
Perhaps you should use "@" to indicate the required return type,or "$" to specify which version of the function you need. stdev b
There are no library operations named stdev Use HyperDoc Browse or issue )what op stdev to learn if there is any operation containing " stdev " in its name.
Cannot find a definition or applicable library operation named stdev with argument type(s) List(PositiveInteger)
Perhaps you should use "@" to indicate the required return type,or "$" to specify which version of the function you need. gmean b
There are no library operations named gmean Use HyperDoc Browse or issue )what op gmean to learn if there is any operation containing " gmean " in its name.
Cannot find a definition or applicable library operation named gmean with argument type(s) List(PositiveInteger)
Perhaps you should use "@" to indicate the required return type,or "$" to specify which version of the function you need. center b
There are 1 exposed and 1 unexposed library operations named center having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse,or issue )display op center to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named center with argument type(s) List(PositiveInteger)
Perhaps you should use "@" to indicate the required return type,or "$" to specify which version of the function you need. stdize b
There are no library operations named stdize Use HyperDoc Browse or issue )what op stdize to learn if there is any operation containing " stdize " in its name.
Cannot find a definition or applicable library operation named stdize with argument type(s) List(PositiveInteger)
Perhaps you should use "@" to indicate the required return type,or "$" to specify which version of the function you need.
-- beware,the following is not mathematically justified since -- median use ordering from OrderedSet (and not OrderedRing) c := [3+7*%i, 5+8*%i, 8+3*%i, 4+5*%i]
![\label{eq3}\left[{3 +{7 \ i}}, \:{5 +{8 \ i}}, \:{8 +{3 \ i}}, \:{4 +{5 \ i}}\right]](images/7973644425097409714-16.0px.png "\label{eq3}\left[{3 +{7 \ i}}, \:{5 +{8 \ i}}, \:{8 +{3 \ i}}, \:{4 +{5 \ i}}\right]") | (3) |
Type: List(Complex(Integer))
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median c
There are no library operations named median Use HyperDoc Browse or issue )what op median to learn if there is any operation containing " median " in its name.
Cannot find a definition or applicable library operation named median with argument type(s) List(Complex(Integer))
Perhaps you should use "@" to indicate the required return type,or "$" to specify which version of the function you need.