| Title: | Unbiased Estimators for Cumulant Products and Faa Di Bruno's Formula |
|---|---|
| Description: | Tools for estimate (joint) cumulants and (joint) products of cumulants of a random sample using (multivariate) k-statistics and (multivariate) polykays, unbiased estimators with minimum variance. Tools for generating univariate and multivariate Faa di Bruno's formula and related polynomials, such as Bell polynomials, generalized complete Bell polynomials, partition polynomials and generalized partition polynomials. For more details see Di Nardo E., Guarino G., Senato D. (2009) <arXiv:0807.5008>, <arXiv:1012.6008>. |
| Authors: | Elvira Di Nardo <[email protected]>, Giuseppe Guarino <[email protected]> |
| Maintainer: | Giuseppe Guarino <[email protected]> |
| License: | GPL |
| Version: | 2.1.1 |
| Built: | 2024-11-09 06:22:36 UTC |
| Source: | CRAN |
kStatistics is a package producing estimates of (joint) cumulants and (joint) cumulant products
of a given dataset, using (multivariate) k-statistics and (multivariate) polykays, which are symmetric unbiased estimators. The
procedures rely on a symbolic method arising from the classical umbral calculus and described in the referred papers. In
the package, a set of combinatorial tools are given useful in the construction of these estimations such as integer partitions, set partitions, multiset subdivisions or
multi-index partitions, pairing and merging of multisets. In the package, there are also functions to recover univariate and
multivariate cumulants from a sequence of univariate and multivariate moments (and vice-versa), using Faa di Bruno's
formula. The function producing Faa di Bruno's formula returns coefficients of exponential power series compositions
such as f[g(z)] with f and g both univariate, or f[g(z1,...,zm)] with f univariate and
g multivariate, or f[g1(z1,...,zm),...,gn(z1,...,zm)] with f and g both multivariate. Let us
recall that Faa di Bruno's formula might also be employed to recover iterated (partial) derivatives of all these compositions.
Lastly, using Faa di Bruno's formula, some special families of polynomials are also generated, such as Bell polynomials,
generalized complete Bell polynomials, partition polynomials and generalized partition polynomials. Applications of these
polynomials are described in the referred papers.
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
Elvira Di Nardo <[email protected]>, Giuseppe Guarino <[email protected]>
Maintainer: Giuseppe Guarino <[email protected]>
C.A. Charalambides (2002) Enumerative Combinatoris, Chapman & Haii/CRC.
G. M. Constantine, T. H. Savits (1996) A Multivariate Faa Di Bruno Formula With Applications. Trans. Amer. Math. Soc. 348(2), 503-520.
E. Di Nardo (2016) On multivariable cumulant polynomial sequence with applications. Jour. Algebraic Statistics 7(1), 72-89. (download from https://arxiv.org/abs/1606.01004)
E. Di Nardo, G. Guarino, D. Senato (2008) An unifying framework for k-statistics, polykays and their generalizations. Bernoulli. 14(2), 440-468. (download from https://arxiv.org/pdf/math/0607623.pdf)
E. Di Nardo, G. Guarino, D. Senato (2008) Symbolic computation of moments of sampling distributions. Comp. Stat. Data Analysis. 52(11), 4909-4922. (download from https://arxiv.org/abs/0806.0129)
E. Di Nardo, G. Guarino, D. Senato (2009) A new method for fast computing unbiased estimators of cumulants. Statistics and Computing, 19, 155-165. (download from https://arxiv.org/abs/0807.5008)
E. Di Nardo, G. Guarino, D. Senato (2011) A new algorithm for computing the multivariate Faa di Bruno's formula. Appl. Math. Comp. 217, 6286-6295. (download from https://arxiv.org/abs/1012.6008)
E. Di Nardo, M. Marena, P. Semeraro (2020) On non-linear dependence of multivariate subordinated Levy processes. In press Stat. Prob. Letters (download from https://arxiv.org/abs/2004.03933)
P. McCullagh, J. Kolassa (2009) Scholarpedia, 4(3):4699. http://www.scholarpedia.org/article/Cumulants
A. Nijenhuis, H. Wilf. (1978) Combinatorial Algorithms for Computers and Calculators. Academic Press, Orlando FL, II edition.
R. P. Stanley (2012) Enumerative combinatorics. Vol.1. II edition. Cambridge Studies in Advanced Mathematics, 49. Cambridge University Press, Cambridge.
# Some of the most important functions: # Data assignment data1<-c(16.34, 10.76, 11.84, 13.55, 15.85, 18.20, 7.51, 10.22, 12.52, 14.68, 16.08, 19.43,8.12, 11.20, 12.95, 14.77, 16.83, 19.80, 8.55, 11.58, 12.10, 15.02, 16.83, 16.98, 19.92, 9.47, 11.68, 13.41, 15.35, 19.11) # Data assignment data2<-list(c(5.31,11.16),c(3.26,3.26),c(2.35,2.35),c(8.32,14.34),c(13.48,49.45), c(6.25,15.05),c(7.01,7.01),c(8.52,8.52),c(0.45,0.45),c(12.08,12.08),c(19.39,10.42)) # Return an estimate of the third cumulant of the random sample data1 with the indication # of which function has been employed # KS:[1] -1.44706 nPolyk(c(3), data1, TRUE) # Return an estimate of the product of the mean and the variance of the random sample data1 # with the indication of which function has been employed # PS:[1] 177.4233 nPolyk( list( c(2), c(1) ), data1, TRUE) # Return an estimate of the joint cumulant c[2,1] of the random sample data2 with the # indication of which function has been employed # KM:[1] -23.7379 nPolyk(c(2,1), data2, TRUE); # Return an estimate of the product of joint cumulants c[2,1]*c[1,0] of the random sample data2 # with the indication of which function has been employed # PM:[1] 48.43243 nPolyk( list( c(2,1), c(1,0) ), data2, TRUE) # Return all the subdivisions of a multiset with only one element of multiplicity 3 mkmSet(3) # Return all the subdivisions of a multiset with two elements, # having multiplicity respectively 2 and 1 mkmSet(c(2,1)) # OR (same output) mkmSet(c(2,1), FALSE) # Return the same output of the previous example but in a compact expression. mkmSet(c(2,1), TRUE) # Return the scompositions of the vector (1,0,1) in 2 vectors of 3 non-negative integers # such that their sum gives (1,0,1), that is # ([1,0,1],[0,0,0]) - ([0,0,0],[1,0,1]) - ([1,0,0],[0,0,1]) - ([0,0,1],[1,0,0]). # Note that the second value in each resulting vector is always zero. mkT(c(1,0,1),2) # OR (same output) mkT(c(1,0,1),2, FALSE) # Return the same output of the previous example but in a compact expression. mkT(c(1,0,1),2, TRUE) # Return all the partitions of the integer 4, that is # [1,1,1,1],[1,1,2],[1,3],[2,2],[4] intPart(4) # OR (same output) intPart(4, FALSE) # Return the same output of the previous example but in a compact expression. intPart(4, TRUE) # Faa di Bruno's formula (Univariate with Univariate Case) # The coefficient of z^2 in f[g(z)], that is f[2]g[1]^2 + f[1]g[2], where # f[1] is the coefficient of x in f(x) with x=g(z) # f[2] is the coefficient of x^2 in f(x) with x=g(z) # g[1] is the coefficient of z in g(z) # g[2] is the coefficient of z^2 in g(z) # MFB( c(2), 1 ) # Faa di Bruno's formula (Univariate with Multivariate Case) # The coefficient of z1 z2 in f[g(z1,z2)], that is f[1]g[1,1] + f[2]g[1,0]g[0,1] # where # f[1] is the coefficient of x in f(x) with x=g(z1,z2) # f[2] is the coefficient of x^2 in f(x) with x=g(z1,z2) # g[1,0] is the coefficient of z1 in g(z1,z2) # g[0,1] is the coefficient of z2 in g(z1,z2) # g[1,1] is the coefficient of z1z2 in g(z1,z2) # MFB( c(1,1), 1 ) # Faa di Bruno's formula (Multivariate with Multivariate Case) # The coefficient of z in f[g1(z),g2(z)], that is f[1,0]g1[1] + f[0,1]g2[1] where # f[1,0] is the coefficient of x1 in f(x1,x2) with x1=g1(z) and x2=g2(z) # f[0,1] is the coefficient of x2 in f(x1,x2) with x1=g1(z) and x2=g2(z) # g1[1] is the coefficient of z of g1(z) # g2[1] is the coefficient of z of g2(z) MFB( c(1), 2 ) # The numerical value of f[1]g[1,1] + f[2]g[1,0]g[0,1], that is the coefficient of z1z2 # in f[g1(z1,z2),g2(z1,z2)] output of MFB(c(1,1),1) when # f[1] = 5 and f[2] = 10 # g[0,1]=3, g[1,0]=6, g[1,1]=9 e_MFB(c(1,1),1, c(5,10), c(3,6,9)) # The multivariate cumulant k[3,1] in terms of the multivariate moments m[i,j] for i=0,1,2,3 # and j=0,1. cum2mom(c(3,1)) # The multivariate moment m[3,1] in terms of the multivariate cumulants k[i,j] for i=0,1,2,3 # and j=0,1. mom2cum(c(3,1)) # The partition polynomial F[5] pPart(5) # The general partition polynomial G[a1, a2; y1, y2], that is a2(y1^2) + a1(y2) gpPart(2) # The complete ordinary Bell Polynomial for n=5, that is # (y1^5) + 20(y1^3)(y2) + 30(y1)(y2^2) + 60(y1^2)(y3) + 120(y2)(y3) + 120(y1)(y4) + 120(y5) oBellPol(5,) # OR (same output) oBellPol(5,0) # The partial ordinary Bell polynomial for n=5 and m=3, that is # 30(y1)(y2^2) + 60(y1^2)(y3) oBellPol(5,3) # The complete exponential Bell Polynomial for n=5, that is # (y1^5) + 10(y1^3)(y2) + 15(y1)(y2^2) + 10(y1^2)(y3) + 10(y2)(y3) + 5(y1)(y4) + (y5) eBellPol(5) # OR (same output) eBellPol(5,0) # The partial exponential Bell Polynomial for n=5 and m=3, that is # 15(y1)(y2^2) + 10(y1^2)(y3) eBellPol(5,3) # The Stirling number of second kind S(5,3) = 25 e_eBellPol(5,3) # OR (same output) e_eBellPol(5,3,c(1,1,1,1,1)) # The Bell number B5 = 52 e_eBellPol(5) # OR (same output) e_eBellPol(5,0) # OR (same output) e_eBellPol(5,0,c(1,1,1,1,1) ) # The generalized complete Bell Polynomial for n=1, m=1 and g1=g, # that is (y^2)g[1]^2 + (y)g[2] # GCBellPol( c(2),1 ) # The generalized complete Bell Polynomial for n=1, m=2 and g1=g # that is 2(y^2)g[1,0]g[1,1] + (y^3)g[0,1]g[1,0]^2 + (y)g[2,1] + (y^2)g[0,1]g[2,0] # GCBellPol( c(2,1),1 ) # The generalized complete Bell Polynomial for n=2, m=2 and g1=g2=g # that is (y1)g[1,1] + (y1^2)g[0,1]g[1,0] + (y2)g[1,1] + (y2^2)g[0,1]g[1,0] + 2(y1)(y2) # g[0,1]g[1,0] # GCBellPol( c(1,1),2, TRUE ) # The polynomial 2(y^2)g[1,1]g[1,0] + (y^3)g[1,0]^2g[0,1] + (y)g[2,1] + (y^2)g[2,0]g[0,1], # output of GCBellPol( c(2,1),1 ), when g[0,1]=1, g[1,0]=2, g[1,1]=3, g[2,0]=4, g[2,1]=5, # that is 16(y^2) + 4(y^3) + 5(y) # e_GCBellPol(c(2,1),1,,c(1:5)) # # OR (same output) # e_GCBellPol(c(2,1),1,,c(1,2,3,4,5)) # # OR (same output) # e_GCBellPol( c(2,1),1,"g[0,1]=1, g[1,0]=2, g[1,1]=3, g[2,0]=4, g[2,1]=5" ) # The polynomial 2(y^2)g[1,1]g[1,0] + (y^3)g[1,0]^2g[0,1] + (y)g[2,1] + (y^2)g[2,0]g[0,1], # output of GCBellPol( c(2,1),1 ) when g[0,1]=1, g[1,0]=2, g[1,1]=3, g[2,0]=4, g[2,1]=5 and # y=7, that is 2191 # e_GCBellPol( c(2,1),1,c(7),c(1:5) ) # # OR (same output) # e_GCBellPol( c(2,1),1,"y=7, g[0,1]=1, g[1,0]=2, g[1,1]=3, g[2,0]=4, g[2,1]=5" )# Some of the most important functions: # Data assignment data1<-c(16.34, 10.76, 11.84, 13.55, 15.85, 18.20, 7.51, 10.22, 12.52, 14.68, 16.08, 19.43,8.12, 11.20, 12.95, 14.77, 16.83, 19.80, 8.55, 11.58, 12.10, 15.02, 16.83, 16.98, 19.92, 9.47, 11.68, 13.41, 15.35, 19.11) # Data assignment data2<-list(c(5.31,11.16),c(3.26,3.26),c(2.35,2.35),c(8.32,14.34),c(13.48,49.45), c(6.25,15.05),c(7.01,7.01),c(8.52,8.52),c(0.45,0.45),c(12.08,12.08),c(19.39,10.42)) # Return an estimate of the third cumulant of the random sample data1 with the indication # of which function has been employed # KS:[1] -1.44706 nPolyk(c(3), data1, TRUE) # Return an estimate of the product of the mean and the variance of the random sample data1 # with the indication of which function has been employed # PS:[1] 177.4233 nPolyk( list( c(2), c(1) ), data1, TRUE) # Return an estimate of the joint cumulant c[2,1] of the random sample data2 with the # indication of which function has been employed # KM:[1] -23.7379 nPolyk(c(2,1), data2, TRUE); # Return an estimate of the product of joint cumulants c[2,1]*c[1,0] of the random sample data2 # with the indication of which function has been employed # PM:[1] 48.43243 nPolyk( list( c(2,1), c(1,0) ), data2, TRUE) # Return all the subdivisions of a multiset with only one element of multiplicity 3 mkmSet(3) # Return all the subdivisions of a multiset with two elements, # having multiplicity respectively 2 and 1 mkmSet(c(2,1)) # OR (same output) mkmSet(c(2,1), FALSE) # Return the same output of the previous example but in a compact expression. mkmSet(c(2,1), TRUE) # Return the scompositions of the vector (1,0,1) in 2 vectors of 3 non-negative integers # such that their sum gives (1,0,1), that is # ([1,0,1],[0,0,0]) - ([0,0,0],[1,0,1]) - ([1,0,0],[0,0,1]) - ([0,0,1],[1,0,0]). # Note that the second value in each resulting vector is always zero. mkT(c(1,0,1),2) # OR (same output) mkT(c(1,0,1),2, FALSE) # Return the same output of the previous example but in a compact expression. mkT(c(1,0,1),2, TRUE) # Return all the partitions of the integer 4, that is # [1,1,1,1],[1,1,2],[1,3],[2,2],[4] intPart(4) # OR (same output) intPart(4, FALSE) # Return the same output of the previous example but in a compact expression. intPart(4, TRUE) # Faa di Bruno's formula (Univariate with Univariate Case) # The coefficient of z^2 in f[g(z)], that is f[2]g[1]^2 + f[1]g[2], where # f[1] is the coefficient of x in f(x) with x=g(z) # f[2] is the coefficient of x^2 in f(x) with x=g(z) # g[1] is the coefficient of z in g(z) # g[2] is the coefficient of z^2 in g(z) # MFB( c(2), 1 ) # Faa di Bruno's formula (Univariate with Multivariate Case) # The coefficient of z1 z2 in f[g(z1,z2)], that is f[1]g[1,1] + f[2]g[1,0]g[0,1] # where # f[1] is the coefficient of x in f(x) with x=g(z1,z2) # f[2] is the coefficient of x^2 in f(x) with x=g(z1,z2) # g[1,0] is the coefficient of z1 in g(z1,z2) # g[0,1] is the coefficient of z2 in g(z1,z2) # g[1,1] is the coefficient of z1z2 in g(z1,z2) # MFB( c(1,1), 1 ) # Faa di Bruno's formula (Multivariate with Multivariate Case) # The coefficient of z in f[g1(z),g2(z)], that is f[1,0]g1[1] + f[0,1]g2[1] where # f[1,0] is the coefficient of x1 in f(x1,x2) with x1=g1(z) and x2=g2(z) # f[0,1] is the coefficient of x2 in f(x1,x2) with x1=g1(z) and x2=g2(z) # g1[1] is the coefficient of z of g1(z) # g2[1] is the coefficient of z of g2(z) MFB( c(1), 2 ) # The numerical value of f[1]g[1,1] + f[2]g[1,0]g[0,1], that is the coefficient of z1z2 # in f[g1(z1,z2),g2(z1,z2)] output of MFB(c(1,1),1) when # f[1] = 5 and f[2] = 10 # g[0,1]=3, g[1,0]=6, g[1,1]=9 e_MFB(c(1,1),1, c(5,10), c(3,6,9)) # The multivariate cumulant k[3,1] in terms of the multivariate moments m[i,j] for i=0,1,2,3 # and j=0,1. cum2mom(c(3,1)) # The multivariate moment m[3,1] in terms of the multivariate cumulants k[i,j] for i=0,1,2,3 # and j=0,1. mom2cum(c(3,1)) # The partition polynomial F[5] pPart(5) # The general partition polynomial G[a1, a2; y1, y2], that is a2(y1^2) + a1(y2) gpPart(2) # The complete ordinary Bell Polynomial for n=5, that is # (y1^5) + 20(y1^3)(y2) + 30(y1)(y2^2) + 60(y1^2)(y3) + 120(y2)(y3) + 120(y1)(y4) + 120(y5) oBellPol(5,) # OR (same output) oBellPol(5,0) # The partial ordinary Bell polynomial for n=5 and m=3, that is # 30(y1)(y2^2) + 60(y1^2)(y3) oBellPol(5,3) # The complete exponential Bell Polynomial for n=5, that is # (y1^5) + 10(y1^3)(y2) + 15(y1)(y2^2) + 10(y1^2)(y3) + 10(y2)(y3) + 5(y1)(y4) + (y5) eBellPol(5) # OR (same output) eBellPol(5,0) # The partial exponential Bell Polynomial for n=5 and m=3, that is # 15(y1)(y2^2) + 10(y1^2)(y3) eBellPol(5,3) # The Stirling number of second kind S(5,3) = 25 e_eBellPol(5,3) # OR (same output) e_eBellPol(5,3,c(1,1,1,1,1)) # The Bell number B5 = 52 e_eBellPol(5) # OR (same output) e_eBellPol(5,0) # OR (same output) e_eBellPol(5,0,c(1,1,1,1,1) ) # The generalized complete Bell Polynomial for n=1, m=1 and g1=g, # that is (y^2)g[1]^2 + (y)g[2] # GCBellPol( c(2),1 ) # The generalized complete Bell Polynomial for n=1, m=2 and g1=g # that is 2(y^2)g[1,0]g[1,1] + (y^3)g[0,1]g[1,0]^2 + (y)g[2,1] + (y^2)g[0,1]g[2,0] # GCBellPol( c(2,1),1 ) # The generalized complete Bell Polynomial for n=2, m=2 and g1=g2=g # that is (y1)g[1,1] + (y1^2)g[0,1]g[1,0] + (y2)g[1,1] + (y2^2)g[0,1]g[1,0] + 2(y1)(y2) # g[0,1]g[1,0] # GCBellPol( c(1,1),2, TRUE ) # The polynomial 2(y^2)g[1,1]g[1,0] + (y^3)g[1,0]^2g[0,1] + (y)g[2,1] + (y^2)g[2,0]g[0,1], # output of GCBellPol( c(2,1),1 ), when g[0,1]=1, g[1,0]=2, g[1,1]=3, g[2,0]=4, g[2,1]=5, # that is 16(y^2) + 4(y^3) + 5(y) # e_GCBellPol(c(2,1),1,,c(1:5)) # # OR (same output) # e_GCBellPol(c(2,1),1,,c(1,2,3,4,5)) # # OR (same output) # e_GCBellPol( c(2,1),1,"g[0,1]=1, g[1,0]=2, g[1,1]=3, g[2,0]=4, g[2,1]=5" ) # The polynomial 2(y^2)g[1,1]g[1,0] + (y^3)g[1,0]^2g[0,1] + (y)g[2,1] + (y^2)g[2,0]g[0,1], # output of GCBellPol( c(2,1),1 ) when g[0,1]=1, g[1,0]=2, g[1,1]=3, g[2,0]=4, g[2,1]=5 and # y=7, that is 2191 # e_GCBellPol( c(2,1),1,c(7),c(1:5) ) # # OR (same output) # e_GCBellPol( c(2,1),1,"y=7, g[0,1]=1, g[1,0]=2, g[1,1]=3, g[2,0]=4, g[2,1]=5" )
The function computes the multiplicity of a multi-index partition. Note that a multi-index partition corresponds to a subdivision of a multiset having the input multi-index as multiplicities.
countP( v=c(1) )countP( v=c(1) )
v |
vector or list |
Consider the partitions of an integer, let's say 3, that are [1,1,1],[1,2],[3] such that
1+1+1=1+2=3. Consider the partitions of a set with cardinality 3, let's say [a1, a2, a3]
(I) |
[[a1], [a2], [a3]], [[a1],[a2,a3]], [[a2],[a1,a3]], [[a3],[a1,a2]], [[a1,a2,a3]]. |
The multiplicity of a partition of 3 is the number of partitions of [a1, a2, a3] in blocks having that
partition as cardinalities. In the example, we have
| a) | 1 partition in 3 blocks of cardinalities 1, that is
[[a1], [a2], [a3]] corresponding to [1,1,1] |
| b) | 1 partition in 1 block of cardinalities 3, that is
[[a1,a2,a3]] corresponding to [3] |
| c) | 3 partitions in 2 blocks of cardinalities 1 and 2 respectively, that is |
[[a1],[a2,a3]], [[a2],[a1,a3]], [[a3],[a1,a2]]. |
So running countP(c(1,2)) we get 3, running countP(c(1,1,1)) or countP(c(3)) we get 1.
The same device can be used to find the multiplicity of a subdivision. The subdivisions of a multiset are
obtained as follows: assume all distinct the elements of the multiset, determine all the corresponding set partitions
and then replace each element in each block with the original one. For example, consider the multiset [a, a, a]
having the integer 3 as multiplicity. Assuming all distinct the elements of the multiset, the partitions of [a1, a2, a3]
are given in (I). Now replace a1<-a, a2<-a, a3<-a and get
[[a], [a], [a]], [[a], [a,a]], [[a], [a,a]], [[a], [a,a]], [[a,a,a]] |
. Note that the partitions
[[a1,a2], [a3]], [[a1,a3], [a2]], [[a2,a3], [a1]] |
give rise to the same subdivision [[a,a], [a]]. Then the multiplicity of [[a,a], [a]] is 3. Therefore
the subdivisions of [a, a, a,] are
[[a], [a], [a]], [[a], [a,a]], [[a,a,a]]. |
The multiplicity of the subdivision [[a], [a,a]] is 3, as for the partition [1,2] of the integer 3.
The multiplicity of the subdivisions [[a], [a], [a]] and [[a,a,a]] is 1, as for the partitions
[1,1,1] and [3] of the integer 3 respectively. Thus running countP(c(1,2)) we get 3, that is
| a) | the multiplicity of the subdivision [[a,a], [a]] |
| b) | the number of partitions of the set [a1, a2, a3] in two blocks, one of cardinality 1 and the other |
| of cardinality 2 | |
| c) | the number of partitions of the set [a1, a2, a3] corresponding to the partition [1,2] of |
| the integer 3. |
Now consider the partition of a multi-index, let's say (4,2). One of the partitions of (4,2) is the matrix
(see the output of the mkmSet function)
0 1 1 2
(II) 1 0 0 1
as 1+1+2=4 and 1+1=2. The partition (II) corresponds to the subdivision [[b], [a], [a], [a,a,b]]
of the multiset [a,a,a,a,b,b] having the multi-index (4,2) as multiplicities. Indeed each column (i,j) gives
the instances i and j of a and b in a block of the subdivision. Running countP
(list(c(0,1), c(1,0), c(1,0), c(2,1))) we get 12. This multiplicity gives the number of partitions of the set [a1,a2,a3,
a4,b1,b2] corresponding to the subdivision [[b], [a], [a], [a,a,b]] after the
replacement
a1<-a, a2<-a, a3<-a, a4<-a, b1<-b, b2<-b.
Note that the input of the countP function does not necessarily be in lexicographic order. To find all the partitions
of a multi-index see the mkmSet function.
integer |
the multiplicity of the given item |
Called by the mkmSet and nPS functions in the kStatistics package.
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
E. Di Nardo, G. Guarino, D. Senato (2008) An unifying framework for k-statistics, polykays and their generalizations. Bernoulli. 14(2), 440-468. (download from https://arxiv.org/pdf/math/0607623.pdf)
mkmSet,
mCoeff,
nStirling2,
intPart,
ff
# Return 3 which is the multiplicity of [1,2], partition of the integer 3, or # of [[a],[a,a]], subdivision of the multiset [a,a,a] countP(c(1,2)) # Return 15 which is the multiplicity of [4,2], partition of the integer 6, or # of [[a,a,a,a],[a,a]], subdivision of the multiset [a,a,a,a,a,a] countP(c(4,2)) # Return 18 which is the multiplicity of # 0 0 1 1 2 # 1 2 0 0 0 # partition of the multi-index (4,3), or of [[b],[b,b],[a],[a],[a,a]], subdivision of # the multiset [a,a,a,a,b,b,b] countP( list(c(2,0), c(1,0), c(1,0), c(0,1),c(0,2)) )# Return 3 which is the multiplicity of [1,2], partition of the integer 3, or # of [[a],[a,a]], subdivision of the multiset [a,a,a] countP(c(1,2)) # Return 15 which is the multiplicity of [4,2], partition of the integer 6, or # of [[a,a,a,a],[a,a]], subdivision of the multiset [a,a,a,a,a,a] countP(c(4,2)) # Return 18 which is the multiplicity of # 0 0 1 1 2 # 1 2 0 0 0 # partition of the multi-index (4,3), or of [[b],[b,b],[a],[a],[a,a]], subdivision of # the multiset [a,a,a,a,b,b,b] countP( list(c(2,0), c(1,0), c(1,0), c(0,1),c(0,2)) )
The function computes a simple or a multivariate cumulant in terms of simple or multivariate moments.
cum2mom(n = 1)cum2mom(n = 1)
n |
integer or vector of integers |
Faa di Bruno's formula (the MFB function) gives the coefficients of the exponential formal power series
f[g()] where f and g are exponential formal power series too. Simple cumulants
are expressed in terms of simple moments using the Faa di Bruno's formula obtained from the MFB function in the case
"composition of univariate f with univariate g" with f[i]=(-1)^(i-1)*(i-1)!, g[i]=m[i]
for i from 1 to n and m[i] moments. Multivariate cumulants are expressed in terms
of multivariate moments using the Faa di Bruno's formula obtained from the MFB function in the case "composition
of univariate f with multivariate g". In such a case the coefficients of g are the multivariate moments.
string |
the expression of the cumulant in terms of moments |
The value of the first parameter is the same as the MFB function in the univariate with
univariate case composition and in the univariate with multivariate case composition.
This function calls the MFB function in the kStatistics package.
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
E. Di Nardo, G. Guarino, D. Senato (2008) An unifying framework for k-statistics, polykays and their generalizations. Bernoulli. 14(2), 440-468. (download from https://arxiv.org/pdf/math/0607623.pdf)
E. Di Nardo E., G. Guarino, D. Senato (2011) A new algorithm for computing the multivariate Faa di Bruno's formula. Appl. Math. Comp. 217, 6286–6295. (download from https://arxiv.org/abs/1012.6008)
P. McCullagh, J. Kolassa (2009) Scholarpedia, 4(3):4699. http://www.scholarpedia.org/article/Cumulants
# Return the simple cumulant k[5] in terms of the simple moments m[1],..., m[5]. cum2mom(5) # Return the multivariate cumulant k[3,1] in terms of the multivariate moments m[i,j] for # i=0,1,2,3 and j=0,1. cum2mom(c(3,1))# Return the simple cumulant k[5] in terms of the simple moments m[1],..., m[5]. cum2mom(5) # Return the multivariate cumulant k[3,1] in terms of the multivariate moments m[i,j] for # i=0,1,2,3 and j=0,1. cum2mom(c(3,1))
The function evaluates a complete or a partial exponential Bell polynomial
(output of the eBellPol function) when its variables are substituted with
numerical values.
e_eBellPol(n=1,m=0,v=c(rep(1,n)))e_eBellPol(n=1,m=0,v=c(rep(1,n)))
n |
integer, the degree of the polynomial |
m |
integer, the fixed degree of each monomial in the polynomial |
v |
vector, the numerical values in place of the variables of the polynomial |
The eBellPol function generates a complete or a partial exponential Bell polynomial
in the variables y[1],..., y[n-m+1]. The e_eBellPol function computes the value assumed
by this polynomial when its variables are substituted with numerical values.
numerical value |
the value assumed by the polynomial. |
By default, the function returns the Stirling numbers of second kind.
This function calls the eBellPol function in the kStatistics package.
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
C.A. Charalambides (2002) Enumerative Combinatoris, Chapman & Haii/CRC.
E. Di Nardo, G. Guarino, D. Senato (2008) An unifying framework for k-statistics, polykays and their generalizations. Bernoulli. 14(2), 440-468. (download from https://arxiv.org/pdf/math/0607623.pdf)
E. Di Nardo, G. Guarino, D. Senato (2008) Symbolic computation of moments of sampling distributions. Comp. Stat. Data Analysis. 52(11), 4909-4922. (download from https://arxiv.org/abs/0806.0129)
E. Di Nardo, G. Guarino, D. Senato (2011) A new algorithm for computing the multivariate Faa di Bruno's formula. Appl. Math. Comp. 217, 6286–6295. (download from https://arxiv.org/abs/1012.6008)
# Return S(5,3) = 25 (where S=Stirling number of second kind) e_eBellPol(5,3) # # OR (same output) # e_eBellPol(5,3,c(1,1,1,1,1)) # Return B5=52 (where B5 is the 5-th Bell number) e_eBellPol(5) # # OR (same output) # e_eBellPol(5,0) # # OR (same output) # e_eBellPol(5,0,c(1,1,1,1,1)) # Return s(5,3) = 35 (where s=Stirling number of first kind) e_eBellPol(5,3,c(1,-1,2,-6,24))# Return S(5,3) = 25 (where S=Stirling number of second kind) e_eBellPol(5,3) # # OR (same output) # e_eBellPol(5,3,c(1,1,1,1,1)) # Return B5=52 (where B5 is the 5-th Bell number) e_eBellPol(5) # # OR (same output) # e_eBellPol(5,0) # # OR (same output) # e_eBellPol(5,0,c(1,1,1,1,1)) # Return s(5,3) = 35 (where s=Stirling number of first kind) e_eBellPol(5,3,c(1,-1,2,-6,24))
The function evaluates a generalized complete Bell polynomial (output of the GCBellPol function) when its variables and/or its coefficients are substituted with
numerical values.
e_GCBellPol(pv = c(), pn = 0, pyc = c(), pc = c(), b = FALSE)e_GCBellPol(pv = c(), pn = 0, pyc = c(), pc = c(), b = FALSE)
pv |
vector of integers, the subscript of the polynomial |
pn |
integer, the number of variables |
pyc |
vector, the numerical values into the variables [optional], or the string with the direct assignment into the variables and/or the coefficients |
pc |
vector, the numerical values into the coefficients, [optional if |
b |
boolean, if |
The function GCBellPol returns the coefficient of the multivariate exponential formal power series
exp(y[1] g1(z1,...,zm) + ... + y[n] gn(z1,...,zm)), where y[1],...,y[n] are variables corresponding to the
subscript pv. The function e_GCBellPol allows us to substitute the coefficients of
the power series g1,...,gn and/or the variables y[1],...,y[n] with numerical values. These
values are passed to the e_GCBellPol function through the third and the fourth input
parameter. In the resulting expression, the y's and the g's are managed in lexicographic order.
There is one further input boolean parameter: when equal to TRUE, the function prints the list of all
the assignments. See the examples for more details on the employment of this boolean parameter
when the coefficients and/or the variables of the polynomial are substituted with numerical values.
string or numerical |
the evaluation of the polynomial |
The value of the first parameter is the same as the mkmSet function.
Called by the GCBellPol function in the kStatistics package.
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
E. Di Nardo (2016) On multivariable cumulant polynomial sequence with applications. Jour. Algebraic Statistics 7(1), 72-89. (download from https://arxiv.org/abs/1606.01004)
E. Di Nardo, G. Guarino, D. Senato (2011) A new algorithm for computing the multivariate Faa di Bruno's formula. Appl. Math. Comp. 217, 6286–6295. (download from https://arxiv.org/abs/1012.6008)
E. Di Nardo, M. Marena, P. Semeraro (2020) On non-linear dependence of multivariate subordinated Levy processes. In press Stat. Prob. Letters (download from https://arxiv.org/abs/2004.03933)
#-------------------------------------------------------------------------------# # Evaluation of the generalized complete Bell polynomial with subscript 2 #-------------------------------------------------------------------------------# # # The polynomial (y^2)g[1]^2 + (y^1)g[2], output of GCBellPol( c(2),1 ), when # g[1]=3 and g[2]=4, that is 9(y^2) + 4(y) # e_GCBellPol( c(2),1,,c(3,4) ) # # OR (same output) # e_GCBellPol( c(2),1,"g[1]=3,g[2]=4" ) # Check the assignments setting the boolean parameter equals to TRUE, that is g[1]=3 # and g[2]=4 e_GCBellPol( c(2),1,,c(3,4),TRUE ) # The numerical value of (y^2)g[1]^2 + (y^1)g[2], output of GCBellPol( c(2),1 ), when # g[1]=3 and g[2]=4 and y=7, that is 469 # e_GCBellPol( c(2),1,c(7),c(3,4) ) # # OR (same output) # e_GCBellPol( c(2),1,"y=7, g[1]=3,g[2]=4" ) # Check the assignments setting the boolean parameter equals to TRUE, that is g[1]=3 # and g[2]=4 and y=7 e_GCBellPol( c(2),1,c(7),c(3,4),TRUE ) #-------------------------------------------------------------------------------# # Evaluation of the generalized complete Bell polynomial with subscript (2,1) #-------------------------------------------------------------------------------# # # The polynomial 2(y^2)g[1,1]g[1,0] + (y^3)g[1,0]^2g[0,1] + (y)g[2,1] + (y^2) # g[2,0]g[0,1], output of GCBellPol( c(2,1),1 ), when g[0,1]=1, g[1,0]=2, g[1,1]=3, # g[2,0]=4, g[2,1]=5, that is 16(y^2) + 4(y^3) + 5(y) # e_GCBellPol(c(2,1),1,,c(1:5)) # # OR (same output) # e_GCBellPol(c(2,1),1,,c(1,2,3,4,5)) # # OR (same output) # e_GCBellPol( c(2,1),1,"g[0,1]=1, g[1,0]=2, g[1,1]=3, g[2,0]=4, g[2,1]=5" ) # Check the assignments setting the boolean parameter equals to TRUE, that is # g[0,1]=1, g[1,0]=2, g[1,1]=3, g[2,0]=4, g[2,1]=5 e_GCBellPol( c(2,1),1,,c(1:5), TRUE ) # The numerical value of 2(y^2)g[1,1]g[1,0] + (y^3)g[1,0]^2g[0,1] + (y)g[2,1] + (y^2) # g[2,0]g[0,1], output of \code{\link{GCBellPol}}( c(2,1),1 ) when g[0,1]=1, g[1,0]=2, # g[1,1]=3, g[2,0]=4, g[2,1]=5 and y=7, that is 2191 # e_GCBellPol( c(2,1),1,c(7),c(1:5) ) # # OR (same output) # e_GCBellPol( c(2,1),1,"y=7, g[0,1]=1, g[1,0]=2, g[1,1]=3, g[2,0]=4, g[2,1]=5" ) # Check the assignments setting the boolean parameter equals to TRUE, that is # g[0,1]=1, g[1,0]=2, g[1,1]=3, g[2,0]=4, g[2,1]=5, y=7 e_GCBellPol( c(2,1),1,c(7),c(1:5) ) #-----------------------------------------------------------------------------------# # Evaluation of the generalized complete Bell Polynomial with subscript (1,1) #-----------------------------------------------------------------------------------# # The polynomial (y1)g1[1,1] + (y1^2)g1[1,0]g1[0,1] + (y2)g2[1,1] + (y2^2)g2[1,0] # g2[0,1] + (y1)(y2)g1[1,0]g2[0,1] + (y1)(y2)g1[0,1]g2[1,0], output of GCBellPol(c(1,1),2) # when g1[0,1]=1, g1[1,0]=2, g1[1,1]=3, g2[0,1]=4, g2[1,0]=5, g2[1,1]=6, that is # 3(y1) + 2(y1^2) + 6(y2) + 20(y2^2) + 13(y1)(y2) # e_GCBellPol( c(1,1),2,,c(1:6)) # # OR (same output) # e_GCBellPol(c(1,1),2,,c(1,2,3,4,5,6)) # # OR (same output) # e_GCBellPol( c(1,1),2,"g1[0,1]=1, g1[1,0]=2, g1[1,1]=3, g2[0,1]=4, g2[1,0]=5, g2[1,1]=6" ) # Check the assignments setting the boolean parameter equals to TRUE, that is # g1[0,1]=1, g1[1,0]=2, g1[1,1]=3, g2[0,1]=4, g2[1,0]=5, g2[1,1]=6 e_GCBellPol( c(1,1),2,,c(1:6), TRUE ) # The numerical value of (y1)g1[1,1] + (y1^2)g1[1,0]g1[0,1] + (y2)g2[1,1] + (y2^2)g2[1,0] # g2[0,1] + (y1)(y2)g1[1,0]g2[0,1] + (y1)(y2)g1[0,1]g2[1,0], output of GCBellPol(c(1,1),2) # when g1[0,1]=1, g1[1,0]=2, g1[1,1]=3, g2[0,1]=4, g2[1,0]=5, y1=7 and y2=8, that is 2175 e_GCBellPol( c(1,1),2,c(7,8),c(1:6)) # # OR (same output) # cVal<-"y1=7, y2=8, g1[0,1]=1, g1[1,0]=2, g1[1,1]=3, g2[0,1]=4, g2[1,0]=5,g2[1,1]=6" e_GCBellPol(c(1,1),2,cVal) # To recover which coefficients and variables are involved in the generalized complete # Bell polynomial, run the e_GCBellPol function without any assignment. # The error message prints which coefficients and variables are involved, that is # Error in e_GCBellPol(c(1, 1), 2) : # The third parameter must contain the 2 values of y: y1 y2. # The fourth parameter must contain the 6 values of g: # g1[0,1] g1[1,0] g1[1,1] g2[0,1] g2[1,0] g2[1,1] # To assign correctly the values to the coefficients and the variables: # 1) run e_GCBellPol(c(1, 1), 2) and get the errors with the indication of the involved # coefficients and variables, that is # The third parameter must contain the 2 values of y: y1 y2 # The fourth parameter must contain the 6 values of g: # g1[0,1] g1[1,0] g1[1,1] g2[0,1] g2[1,0] g2[1,1] # 2) initialize g1[0,1] g1[1,0] g1[1,1] g2[0,1] g2[1,0] g2[1,1] with - for example - the # first 6 integer numbers and do the same for y1 and y2, that is # e_GCBellPol(c(1,1),2, c(1,2), c(1,2,3,4,5,6), TRUE) # 3) trought the boolean value TRUE, recover the string y1=1, y2=1, g1[0,1]=1, g1[1,0]=2, # g1[1,1]=3, g2[0,1]=4, g2[1,0]=5, g2[1,1]=6 # 4) copy and past the string in place of "..." when run # e_GCBellPol(c(1,1),2,"...") # 5) change the assignments if necessary cVal<-"y1=10,y2=11,g1[0,1]=1.1,g1[1,0]=-2,g1[1,1]=3.2,g2[0,1]=-4,g2[1,0]=10,g2[1,1]=6" e_GCBellPol(c(1,1), 2,cVal)#-------------------------------------------------------------------------------# # Evaluation of the generalized complete Bell polynomial with subscript 2 #-------------------------------------------------------------------------------# # # The polynomial (y^2)g[1]^2 + (y^1)g[2], output of GCBellPol( c(2),1 ), when # g[1]=3 and g[2]=4, that is 9(y^2) + 4(y) # e_GCBellPol( c(2),1,,c(3,4) ) # # OR (same output) # e_GCBellPol( c(2),1,"g[1]=3,g[2]=4" ) # Check the assignments setting the boolean parameter equals to TRUE, that is g[1]=3 # and g[2]=4 e_GCBellPol( c(2),1,,c(3,4),TRUE ) # The numerical value of (y^2)g[1]^2 + (y^1)g[2], output of GCBellPol( c(2),1 ), when # g[1]=3 and g[2]=4 and y=7, that is 469 # e_GCBellPol( c(2),1,c(7),c(3,4) ) # # OR (same output) # e_GCBellPol( c(2),1,"y=7, g[1]=3,g[2]=4" ) # Check the assignments setting the boolean parameter equals to TRUE, that is g[1]=3 # and g[2]=4 and y=7 e_GCBellPol( c(2),1,c(7),c(3,4),TRUE ) #-------------------------------------------------------------------------------# # Evaluation of the generalized complete Bell polynomial with subscript (2,1) #-------------------------------------------------------------------------------# # # The polynomial 2(y^2)g[1,1]g[1,0] + (y^3)g[1,0]^2g[0,1] + (y)g[2,1] + (y^2) # g[2,0]g[0,1], output of GCBellPol( c(2,1),1 ), when g[0,1]=1, g[1,0]=2, g[1,1]=3, # g[2,0]=4, g[2,1]=5, that is 16(y^2) + 4(y^3) + 5(y) # e_GCBellPol(c(2,1),1,,c(1:5)) # # OR (same output) # e_GCBellPol(c(2,1),1,,c(1,2,3,4,5)) # # OR (same output) # e_GCBellPol( c(2,1),1,"g[0,1]=1, g[1,0]=2, g[1,1]=3, g[2,0]=4, g[2,1]=5" ) # Check the assignments setting the boolean parameter equals to TRUE, that is # g[0,1]=1, g[1,0]=2, g[1,1]=3, g[2,0]=4, g[2,1]=5 e_GCBellPol( c(2,1),1,,c(1:5), TRUE ) # The numerical value of 2(y^2)g[1,1]g[1,0] + (y^3)g[1,0]^2g[0,1] + (y)g[2,1] + (y^2) # g[2,0]g[0,1], output of \code{\link{GCBellPol}}( c(2,1),1 ) when g[0,1]=1, g[1,0]=2, # g[1,1]=3, g[2,0]=4, g[2,1]=5 and y=7, that is 2191 # e_GCBellPol( c(2,1),1,c(7),c(1:5) ) # # OR (same output) # e_GCBellPol( c(2,1),1,"y=7, g[0,1]=1, g[1,0]=2, g[1,1]=3, g[2,0]=4, g[2,1]=5" ) # Check the assignments setting the boolean parameter equals to TRUE, that is # g[0,1]=1, g[1,0]=2, g[1,1]=3, g[2,0]=4, g[2,1]=5, y=7 e_GCBellPol( c(2,1),1,c(7),c(1:5) ) #-----------------------------------------------------------------------------------# # Evaluation of the generalized complete Bell Polynomial with subscript (1,1) #-----------------------------------------------------------------------------------# # The polynomial (y1)g1[1,1] + (y1^2)g1[1,0]g1[0,1] + (y2)g2[1,1] + (y2^2)g2[1,0] # g2[0,1] + (y1)(y2)g1[1,0]g2[0,1] + (y1)(y2)g1[0,1]g2[1,0], output of GCBellPol(c(1,1),2) # when g1[0,1]=1, g1[1,0]=2, g1[1,1]=3, g2[0,1]=4, g2[1,0]=5, g2[1,1]=6, that is # 3(y1) + 2(y1^2) + 6(y2) + 20(y2^2) + 13(y1)(y2) # e_GCBellPol( c(1,1),2,,c(1:6)) # # OR (same output) # e_GCBellPol(c(1,1),2,,c(1,2,3,4,5,6)) # # OR (same output) # e_GCBellPol( c(1,1),2,"g1[0,1]=1, g1[1,0]=2, g1[1,1]=3, g2[0,1]=4, g2[1,0]=5, g2[1,1]=6" ) # Check the assignments setting the boolean parameter equals to TRUE, that is # g1[0,1]=1, g1[1,0]=2, g1[1,1]=3, g2[0,1]=4, g2[1,0]=5, g2[1,1]=6 e_GCBellPol( c(1,1),2,,c(1:6), TRUE ) # The numerical value of (y1)g1[1,1] + (y1^2)g1[1,0]g1[0,1] + (y2)g2[1,1] + (y2^2)g2[1,0] # g2[0,1] + (y1)(y2)g1[1,0]g2[0,1] + (y1)(y2)g1[0,1]g2[1,0], output of GCBellPol(c(1,1),2) # when g1[0,1]=1, g1[1,0]=2, g1[1,1]=3, g2[0,1]=4, g2[1,0]=5, y1=7 and y2=8, that is 2175 e_GCBellPol( c(1,1),2,c(7,8),c(1:6)) # # OR (same output) # cVal<-"y1=7, y2=8, g1[0,1]=1, g1[1,0]=2, g1[1,1]=3, g2[0,1]=4, g2[1,0]=5,g2[1,1]=6" e_GCBellPol(c(1,1),2,cVal) # To recover which coefficients and variables are involved in the generalized complete # Bell polynomial, run the e_GCBellPol function without any assignment. # The error message prints which coefficients and variables are involved, that is # Error in e_GCBellPol(c(1, 1), 2) : # The third parameter must contain the 2 values of y: y1 y2. # The fourth parameter must contain the 6 values of g: # g1[0,1] g1[1,0] g1[1,1] g2[0,1] g2[1,0] g2[1,1] # To assign correctly the values to the coefficients and the variables: # 1) run e_GCBellPol(c(1, 1), 2) and get the errors with the indication of the involved # coefficients and variables, that is # The third parameter must contain the 2 values of y: y1 y2 # The fourth parameter must contain the 6 values of g: # g1[0,1] g1[1,0] g1[1,1] g2[0,1] g2[1,0] g2[1,1] # 2) initialize g1[0,1] g1[1,0] g1[1,1] g2[0,1] g2[1,0] g2[1,1] with - for example - the # first 6 integer numbers and do the same for y1 and y2, that is # e_GCBellPol(c(1,1),2, c(1,2), c(1,2,3,4,5,6), TRUE) # 3) trought the boolean value TRUE, recover the string y1=1, y2=1, g1[0,1]=1, g1[1,0]=2, # g1[1,1]=3, g2[0,1]=4, g2[1,0]=5, g2[1,1]=6 # 4) copy and past the string in place of "..." when run # e_GCBellPol(c(1,1),2,"...") # 5) change the assignments if necessary cVal<-"y1=10,y2=11,g1[0,1]=1.1,g1[1,0]=-2,g1[1,1]=3.2,g2[0,1]=-4,g2[1,0]=10,g2[1,1]=6" e_GCBellPol(c(1,1), 2,cVal)
The function evaluates the Faa di Bruno's formula, output of the MFB
function, when the coefficients of the exponential formal power series f and g1,...,gn
in the composition f[g1(),...,gn()] are substituted with numerical values.
e_MFB(pv = c(), pn = 0, pf = c(), pg = c(), b = FALSE)e_MFB(pv = c(), pn = 0, pf = c(), pg = c(), b = FALSE)
pv |
vector of integers, the subscript of Faa di Bruno's formula |
pn |
integer, the number of the inner formal power series |
pf |
vector, the numerical values in place of the coefficients of the outer formal power series
|
pg |
vector, the numerical values in place of the coefficients of the inner formal power series |
b |
boolean |
The output of the MFB function is a coefficient of the exponential formal
power series compositions in the cases
| a) | univariate f with univariate g |
| b) | univariate f with multivariate g |
| c) | multivariate f with multivariates {gi}
|
The e_MFB function evaluates this coefficient when the coefficients of f and
{gi} are substituted with numerical values. These values are passed to the e_MFB
function trough the third and the fourth input parameter. There is one
further input boolean parameter: when equal to TRUE, the function prints the list of all
the assignments. See the examples for more details on how to use this boolean parameter when the expression
of the coefficients of f and {gi} becomes more complex.
numerical |
the evaluation of Faa di Bruno's formula |
The value of the first parameter is the same as the mkmSet function.
Called from the MFB function in the kStatistics package.
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
E. Di Nardo, G. Guarino, D. Senato (2008) An unifying framework for k-statistics, polykays and their generalizations. Bernoulli. Vol. 14(2), 440-468. (download from http://www.elviradinardo.it/lavori1.html)
E. Di Nardo, G. Guarino, D. Senato (2008) Symbolic computation of moments of sampling distributions. Comp. Stat. Data Analysis Vol. 52(11), 4909-4922, (download from http://www.elviradinardo.it/lavori1.html)
E. Di Nardo, G. Guarino, D. Senato (2011) A new algorithm for computing the multivariate Faa di Bruno's formula. Appl. Math. Comp. 217, 6286–6295. (download from https://arxiv.org/abs/1012.6008)
# The numerical value of f[1]g[1,1] + f[2]g[1,0]g[0,1], that is the coefficient of z1z2 in # f(g1(z1,z2),g2(z1,z2))) output of MFB(c(1,1),1) when # f[1] = 5 and f[2] = 10 # g[0,1]=3, g[1,0]=6, g[1,1]=9 e_MFB(c(1,1),1, c(5,10), c(3,6,9)) # Same as the previous example, with a string of assignments as third input parameter e_MFB(c(1,1),1, "f[1]=5, f[2]=10, g[0,1]=3, g[1,0]=6, g[1,1]=9") # Use the boolean parameter to verify the assignments to the coefficients of "f" and "g", # that is f[1]=5, f[2]=10, g[0,1]=3, g[1,0]=6, g[1,1]=9 e_MFB(c(1,1),1, c(5,10), c(3,6,9), TRUE) # To recover which coefficients are involved, run the function without any assignment. # The error message recalls which coefficients are necessary, that is # e_MFB(c(1,1),1) # Error in e_MFB(c(1, 1), 1) : # The third parameter must contain the 2 values of f: f[1] f[2]. # The fourth parameter must contain the 3 values of g: g[0,1] g[1,0] g[1,1] # To assign correctly the values to the coefficients of "f" and "g" when the functions # become more complex: # 1) run e_MFB(c(1,1),2) and get the errors with the indication of the involved coefficients # of "f" and "g", that is # The third parameter must contain the 5 values of f: # f[0,1] f[0,2] f[1,0] f[1,1] f[2,0] # The fourth parameter must contain the 6 values of g: # g1[0,1] g1[1,0] g1[1,1] g2[0,1] g2[1,0] g2[1,1]" # 2) initialize f[0,1] f[0,2] f[1,0] f[1,1] f[2,0] with - for example - the first 5 integer # numbers and do the same for g1[0,1] g1[1,0] g1[1,1] g2[0,1] g2[1,0] g2[1,1], that is # e_MFB(c(1,1),2, c(1:5), c(1:6), TRUE) # 3) trought the boolean value TRUE, recover the string f[0,1]=1, f[0,2]=2, f[1,0]=3, f[1,1]=4, # f[2,0]=5, g1[0,1]=1, g1[1,0]=2, g1[1,1]=3, g2[0,1]=4, g2[1,0]=5, g2[1,1]=6 # 4) copy and past the string in place of " ... " when run # e_MFB(c(1,1),1," ... ") # 5) change the assignments if necessary cfVal<-"f[0,1]=2, f[0,2]=5, f[1,0]=13, f[1,1]=-4, f[2,0]=0" cgVal<-"g1[0,1]=-2.1, g1[1,0]=2,g1[1,1]=3.1, g2[0,1]=5, g2[1,0]=0, g2[1,1]=6.1" cVal<-paste0(cfVal,",",cgVal) e_MFB(c(1,1),2,cVal)# The numerical value of f[1]g[1,1] + f[2]g[1,0]g[0,1], that is the coefficient of z1z2 in # f(g1(z1,z2),g2(z1,z2))) output of MFB(c(1,1),1) when # f[1] = 5 and f[2] = 10 # g[0,1]=3, g[1,0]=6, g[1,1]=9 e_MFB(c(1,1),1, c(5,10), c(3,6,9)) # Same as the previous example, with a string of assignments as third input parameter e_MFB(c(1,1),1, "f[1]=5, f[2]=10, g[0,1]=3, g[1,0]=6, g[1,1]=9") # Use the boolean parameter to verify the assignments to the coefficients of "f" and "g", # that is f[1]=5, f[2]=10, g[0,1]=3, g[1,0]=6, g[1,1]=9 e_MFB(c(1,1),1, c(5,10), c(3,6,9), TRUE) # To recover which coefficients are involved, run the function without any assignment. # The error message recalls which coefficients are necessary, that is # e_MFB(c(1,1),1) # Error in e_MFB(c(1, 1), 1) : # The third parameter must contain the 2 values of f: f[1] f[2]. # The fourth parameter must contain the 3 values of g: g[0,1] g[1,0] g[1,1] # To assign correctly the values to the coefficients of "f" and "g" when the functions # become more complex: # 1) run e_MFB(c(1,1),2) and get the errors with the indication of the involved coefficients # of "f" and "g", that is # The third parameter must contain the 5 values of f: # f[0,1] f[0,2] f[1,0] f[1,1] f[2,0] # The fourth parameter must contain the 6 values of g: # g1[0,1] g1[1,0] g1[1,1] g2[0,1] g2[1,0] g2[1,1]" # 2) initialize f[0,1] f[0,2] f[1,0] f[1,1] f[2,0] with - for example - the first 5 integer # numbers and do the same for g1[0,1] g1[1,0] g1[1,1] g2[0,1] g2[1,0] g2[1,1], that is # e_MFB(c(1,1),2, c(1:5), c(1:6), TRUE) # 3) trought the boolean value TRUE, recover the string f[0,1]=1, f[0,2]=2, f[1,0]=3, f[1,1]=4, # f[2,0]=5, g1[0,1]=1, g1[1,0]=2, g1[1,1]=3, g2[0,1]=4, g2[1,0]=5, g2[1,1]=6 # 4) copy and past the string in place of " ... " when run # e_MFB(c(1,1),1," ... ") # 5) change the assignments if necessary cfVal<-"f[0,1]=2, f[0,2]=5, f[1,0]=13, f[1,1]=-4, f[2,0]=0" cgVal<-"g1[0,1]=-2.1, g1[1,0]=2,g1[1,1]=3.1, g2[0,1]=5, g2[1,0]=0, g2[1,1]=6.1" cVal<-paste0(cfVal,",",cgVal) e_MFB(c(1,1),2,cVal)
The function generates a complete or a partial exponential Bell polynomial.
eBellPol(n = 1, m = 0)eBellPol(n = 1, m = 0)
n |
integer, the degree of the polynomial |
m |
integer, the fixed degree of each monomial in the polynomial |
Faa di Bruno's formula gives the coefficients of the exponential formal power series composition
f[g()] obtained from the composition of the exponential formal power series f with g.
Complete exponential Bell polynomials in the variables y[1],...,y[n] are generated by setting
f[i]=1 and g[i]=y[i], for each i from 1 to n. Partial exponential Bell
polynomials are polynomials in the variables y[1],...,y[n-m+1] with fixed degree m
for each of the involved monomials. Partial exponential Bell polynomials are recovered from
Faa di Bruno's formula by setting g[i]=y[i] for each i from 1 to n and
f[i]=1 if i=m, f[i]=0 otherwise.
string |
the expression of the exponential Bell polynomial |
The value of the first parameter is the same as the MFB function in
the univariate with univariate composition.
This function calls the MFB function in the kStatistics package.
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
C.A. Charalambides (2002) Enumerative Combinatoris, Chapman & Haii/CRC.
E. Di Nardo, G. Guarino, D. Senato (2008) An unifying framework for k-statistics, polykays and their generalizations. Bernoulli. 14(2), 440-468. (download from https://arxiv.org/pdf/math/0607623.pdf)
E. Di Nardo, G. Guarino, D. Senato (2008) Symbolic computation of moments of sampling distributions. Comp. Stat. Data Analysis. 52(11), 4909-4922. (download from https://arxiv.org/abs/0806.0129)
E. Di Nardo, G. Guarino, D. Senato (2011) A new algorithm for computing the multivariate Faa di Bruno's formula. Appl. Math. Comp. 217, 6286–6295. (download from https://arxiv.org/abs/1012.6008)
# Return the complete exponential Bell Polynomial for n=5, that is # (y1^5) + 10(y1^3)(y2) + 15(y1)(y2^2) + 10(y1^2)(y3) + 10(y2)(y3) + 5(y1)(y4) + (y5) eBellPol(5) # OR (same output) eBellPol(5,0) # Return the partial exponential Bell Polynomial for n=5 and m=3, that is # 15(y1)(y2^2) + 10(y1^2)(y3) eBellPol(5,3)# Return the complete exponential Bell Polynomial for n=5, that is # (y1^5) + 10(y1^3)(y2) + 15(y1)(y2^2) + 10(y1^2)(y3) + 10(y2)(y3) + 5(y1)(y4) + (y5) eBellPol(5) # OR (same output) eBellPol(5,0) # Return the partial exponential Bell Polynomial for n=5 and m=3, that is # 15(y1)(y2^2) + 10(y1^2)(y3) eBellPol(5,3)
The function computes the descending (falling) factorial of a positive integer n
with respect to a positive integer k less or equal to n.
ff( n=1, k )ff( n=1, k )
n |
integer |
k |
integer |
Run ff(n, k) to get n(n-1)(n-2).....(n-k+1).
integer |
the descending factorial |
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
mkmSet,
countP,
nStirling2,
intPart,
mCoeff
# Return 6*5*4 = 120 ff(6,3)# Return 6*5*4 = 120 ff(6,3)
The function generates a generalized complete Bell polynomial, that is a coefficient
of the composition exp(y[1] g1(z1,...,zm) + ... + y[n] gn(z1,...,zm)), where y[1],...,y[n] are
variables. The input vector of integers identifies the subscript of the polynomial.
GCBellPol(nv = c(), m = 1, b = FALSE)GCBellPol(nv = c(), m = 1, b = FALSE)
nv |
vector of integers, the subscript of the polynomial, corresponding to the powers of the product
among |
m |
integer, the number of |
b |
boolean, |
The multivariate Faa di Bruno's formula, output of the MFB function, gives
a coefficient of the multivariate exponential power series obtained from the composition of
the multivariate exponential power series f(x1,...,xn) with xi=gi(z1,...,zm) for each
i from 1 to n. Now, set f(y[1],...,y[n];x1,...,xn)=exp(y[1] x1 + ... + y[n] xn).
In such a case, the coefficients are the generalized complete Bell polynomials,
see the referred papers. In particular, the GCBellPol function gives
the expression of these polynomials when n=1 or when n>1 and g1=...=gn=g or when
n>1 and g1, ..., gn are all different. See the e_GCBellPol function for
evaluating this polynomial when its variables y[1], ..., y[n] or/and its coefficients are substituted
with numerical values.
string |
the expression of the polynomial |
The value of the first parameter is the same as the mkmSet
function.
This function calls the MFB function in the kStatistics package.
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
G. M. Constantine, T. H. Savits (1996) A Multivariate Faa Di Bruno Formula With Applications. Trans. Amer. Math. Soc. 348(2), 503–520.
E. Di Nardo (2016) On multivariable cumulant polynomial sequence with applications. Jour. Algebraic Statistics 7(1), 72-89. (download from https://arxiv.org/abs/1606.01004)
E. Di Nardo, G. Guarino, D. Senato (2011) A new algorithm for computing the multivariate Faa di Bruno's formula. Appl. Math. Comp. 217, 6286–6295. (download from https://arxiv.org/abs/1012.6008)
E. Di Nardo, M. Marena, P. Semeraro (2020) On non-linear dependence of multivariate subordinated Levy processes. In press Stat. Prob. Letters (download from https://arxiv.org/abs/2004.03933)
# Return the generalized complete Bell Polynomial for n=1, m=1 and g1=g, # that is (y^2)g[1]^2 + (y)g[2] # GCBellPol( c(2),1 ) # Return the generalized complete Bell Polynomial for n=1, m=2 and g1=g, # 2(y^2)g[1,0]g[1,1] + (y^3)g[0,1]g[1,0]^2 + (y)g[2,1] + (y^2)g[0,1]g[2,0] # GCBellPol( c(2,1),1 ) # Return the generalized complete Bell Polynomial for n=2, m=2 and g1=g2=g, # (y1)g[1,1] + (y1^2)g[0,1]g[1,0] + (y2)g[1,1] + (y2^2)g[0,1]g[1,0] + 2(y1)(y2)g[0,1]g[1,0] # GCBellPol( c(1,1),2, TRUE ) # Return the generalized complete Bell Polynomial for n=2, m=2 and g1 different from g2, # that is (y1)g1[1,1] + (y1^2)g1[1,0]g1[0,1] + (y2)g2[1,1] + (y2^2)g2[1,0]g2[0,1] + # (y1)(y2)g1[1,0]g2[0,1] + (y1)(y2)g1[0,1]g2[1,0] # GCBellPol( c(1,1),2 )# Return the generalized complete Bell Polynomial for n=1, m=1 and g1=g, # that is (y^2)g[1]^2 + (y)g[2] # GCBellPol( c(2),1 ) # Return the generalized complete Bell Polynomial for n=1, m=2 and g1=g, # 2(y^2)g[1,0]g[1,1] + (y^3)g[0,1]g[1,0]^2 + (y)g[2,1] + (y^2)g[0,1]g[2,0] # GCBellPol( c(2,1),1 ) # Return the generalized complete Bell Polynomial for n=2, m=2 and g1=g2=g, # (y1)g[1,1] + (y1^2)g[0,1]g[1,0] + (y2)g[1,1] + (y2^2)g[0,1]g[1,0] + 2(y1)(y2)g[0,1]g[1,0] # GCBellPol( c(1,1),2, TRUE ) # Return the generalized complete Bell Polynomial for n=2, m=2 and g1 different from g2, # that is (y1)g1[1,1] + (y1^2)g1[1,0]g1[0,1] + (y2)g2[1,1] + (y2^2)g2[1,0]g2[0,1] + # (y1)(y2)g1[1,0]g2[0,1] + (y1)(y2)g1[0,1]g2[1,0] # GCBellPol( c(1,1),2 )
The function returns a general partition polynomial.
gpPart(n = 0)gpPart(n = 0)
n |
integer |
Faa di Bruno's formula gives the coefficients of the exponential formal power series composition
f[g()] obtained from the composition of the exponential formal power series f with g.
General partition polynomials in the variables y[1],...,y[n] are recovered from the Faa di Bruno's
formula (output of the MFB function) in the case "composition of univariate
f with univariate g" by setting f[i]=ai and g[i]=y[i], for i
from 1 to n.
string |
the expression of the polynomial |
The value of the first parameter is the same as the MFB function
in the univariate with univariate composition.
This function calls the MFB function in the kStatistics package.
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
C.A. Charalambides (2002) Enumerative Combinatoris, Chapman & Haii/CRC.
E. Di Nardo, G. Guarino, D. Senato (2011) A new algorithm for computing the multivariate Faa di Bruno's formula. Appl. Math. Comp. 217, 6286–6295. (download from https://arxiv.org/abs/1012.6008)
# Return the general partition polynomial G[a1,a2; y1,y2], that is a2(y1^2) + a1(y2) gpPart(2) # Return the general partition polynomial G[a1,a2,a3,a4,a5; y1,y2,y3,y4,y5], that is # a5(y1^5) + 10a4(y1^3)(y2) + 15a3(y1)(y2^2) + 10a3(y1^2)(y3) + 10a2(y2)(y3) + 5a2(y1)(y4) # + a1(y5) gpPart(5)# Return the general partition polynomial G[a1,a2; y1,y2], that is a2(y1^2) + a1(y2) gpPart(2) # Return the general partition polynomial G[a1,a2,a3,a4,a5; y1,y2,y3,y4,y5], that is # a5(y1^5) + 10a4(y1^3)(y2) + 15a3(y1)(y2^2) + 10a3(y1^2)(y3) + 10a2(y2)(y3) + 5a2(y1)(y4) # + a1(y5) gpPart(5)
The function generates all possible (unique) decomposition of a positive integer n in the
sum of positive integers less or equal to n.
intPart(n=0 ,vOutput = FALSE)intPart(n=0 ,vOutput = FALSE)
n |
integer |
vOutput |
optional boolean parameter, if equal to |
A partition of an integer n is a sequence of weakly increasing integers such
that their sum returns n. The intPart function generates all the partitions of a given integer
in increasing order.
list |
all the partitions of |
Called by the mkmSet function in the kStatistics package.
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
A. Nijenhuis, H. Wilf. (1978) Combinatorial Algorithms for Computers and Calculators. Academic Press, Orlando FL, II edition.
mkmSet,
mCoeff,
nStirling2,
countP,
ff
# Return the partition of the integer 3, that is # [1,1,1],[1,2],[3] intPart(3) # Return the partition of the integer 4, that is # [1,1,1,1],[1,1,2],[1,3],[2,2],[4] intPart(4) # OR (same output) intPart(4, FALSE) # Return the same output as the previous example but in a compact expression intPart(4, TRUE)# Return the partition of the integer 3, that is # [1,1,1],[1,2],[3] intPart(3) # Return the partition of the integer 4, that is # [1,1,1,1],[1,1,2],[1,3],[2,2],[4] intPart(4) # OR (same output) intPart(4, FALSE) # Return the same output as the previous example but in a compact expression intPart(4, TRUE)
The function returns the multiset representation of a vector or a list, in increasing order.
list2m( v=c(0) )list2m( v=c(0) )
v |
single vector or list of vectors |
Given a list in input, the list2m function returns a structure as
[[e1,e2,...], m1], [[f1,f2,...], m2],... where m1, m2,... are the instances
of c(e1,e2,...), c(f1,f2,...), ... in the input vector v.
multiset |
the list of multisets |
Called by the countP function in the kStatistics package.
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
D.E. Knuth (1998) The Art of Computer Programming. (3rd ed.) Addison Wesley.
# Return the list of multisets [[1],3], [[2],1] from the input vector (1,2,1,1) list2m(c(1,2,1,1 )) # Return the list of multisets [[1,2],2], [[2,3],1] from the input list (c(1,2),c(2,3),c(1,2)) list2m(list(c(1,2),c(2,3),c(1,2)))# Return the list of multisets [[1],3], [[2],1] from the input vector (1,2,1,1) list2m(c(1,2,1,1 )) # Return the list of multisets [[1,2],2], [[2,3],1] from the input list (c(1,2),c(2,3),c(1,2)) list2m(list(c(1,2),c(2,3),c(1,2)))
Given a list, the function deletes the instances of an element in the list, leaving the order inalterated.
list2Set(v=c(0))list2Set(v=c(0))
v |
single vector or list of vectors |
set |
the sequence of distinct elements |
Called by the list2m function in the kStatistics package.
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
# Return the vector c(1,2,3,5,6) list2Set(c(1,2,3,1,2,5,6)) # Return the list (c(1,2),c(10,11),c(7,8)) list2Set(list(c(1,2),c(1,2),c(10,11),c(1,2),c(7,8)))# Return the vector c(1,2,3,5,6) list2Set(c(1,2,3,1,2,5,6)) # Return the list (c(1,2),c(10,11),c(7,8)) list2Set(list(c(1,2),c(1,2),c(10,11),c(1,2),c(7,8)))
The function returns the vectors (only counted once) of all the multi-index
partitions output of the mkmSet function. These vectors correspond also to the blocks
of the subdivisions of the multiset having the given multi-index as multeplicites.
m2Set( v=c(0) )m2Set( v=c(0) )
v |
sequence of type |
Consider the multi-index (2,1). The partitions are
0 1 1 0 2 1 1 2
1 0 0 1 0 0 1 1
with multiplicities 1, 1, 2, 1 respectively. The m2Set function
deletes column repetitions, that is transforms the given list in
[[0,1],[1,0],[2,0],[1,1],[2,1]]
according to the order given in the input. In terms of subdivisions, suppose to consider
the multiset [a,a,b] with multiplicities (2,1). The subdivisions are
[[[b],[a],[a]],1], [[[a,a],[b]],1], [[[a],[a,b]],2], [[a,a,b],1]. |
The m2Set function deletes block repetitions, that is transforms the given list
in
[[b],[a],[a,a],[a,b],[a,a,b]] |
according to the order given in the input. See also the examples.
set |
the sequence with distinct elements |
Called by the nKM and nPM functions in the kStatistics package.
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
M1 <- mkmSet(c(2,1)) # M1 is # list( # list( list( c(0,1), c(1,0), c(1,0) ) ,1), # list( list( c(0,1), c(2,0) ) ,1), # list( list( c(1,0), c(1,1) ) ,2), # list( list( c(2,1) ) ,1), # ) # To print all the partitions of the multi-index (2,1) run mkmSet(c(2,1),TRUE) # [( 0 1 )( 1 0 )( 1 0 ), 1 ] # [( 0 1 )( 2 0 ), 1 ] # [( 1 0 )( 1 1 ), 2 ] # [( 2 1 ), 1 ] # # Then m2Set(M1) returns the following set: [[0,1],[1,0],[2,0],[1,1],[2,1]] # m2Set( M1 )M1 <- mkmSet(c(2,1)) # M1 is # list( # list( list( c(0,1), c(1,0), c(1,0) ) ,1), # list( list( c(0,1), c(2,0) ) ,1), # list( list( c(1,0), c(1,1) ) ,2), # list( list( c(2,1) ) ,1), # ) # To print all the partitions of the multi-index (2,1) run mkmSet(c(2,1),TRUE) # [( 0 1 )( 1 0 )( 1 0 ), 1 ] # [( 0 1 )( 2 0 ), 1 ] # [( 1 0 )( 1 1 ), 2 ] # [( 2 1 ), 1 ] # # Then m2Set(M1) returns the following set: [[0,1],[1,0],[2,0],[1,1],[2,1]] # m2Set( M1 )
Given a list containing vectors paired with numbers, the function returns the number paired with the vector matching the one passed in input.
mCoeff( v=NULL,L=NULL )mCoeff( v=NULL,L=NULL )
v |
vector to be searched in the list |
L |
two-dimensional list: in the first there is a vector and in the second a number |
The input variable L of the mCoeff function is a list
containing vectors and numbers. The input variable v of the mCoeff
function is one of vectors contained in the list. The function searches the vector
v in the list and returns the number which is paired with v in the list.
This function is useful in the construction of k-statistics but also to manage monomials
and their coefficients.
float |
the number paired with the input vector |
Called by the nPS, nKM and nPM functions
in the kStatistics package.
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
E. Di Nardo, G. Guarino, D. Senato (2008) An unifying framework for k-statistics, polykays and their generalizations. Bernoulli. 14(2), 440-468. (download from https://arxiv.org/pdf/math/0607623.pdf)
mkmSet,
countP,
nStirling2,
intPart,
ff
# Run mkmSet(c(3)) to get the list L1 = [[1,1,1],1], [[1,2],3], [[3],1] L1 <- mkmSet(c(3)) # Return the number 3, which is the number paired with [1,2] in L1 mCoeff( c(1,2), L1)# Run mkmSet(c(3)) to get the list L1 = [[1,1,1],1], [[1,2],3], [[3],1] L1 <- mkmSet(c(3)) # Return the number 3, which is the number paired with [1,2] in L1 mCoeff( c(1,2), L1)
The function returns the coefficient indexed by the integers i1,i2,... of an exponential
formal power series composition through the univariate or multivariate Faa di Bruno's formula.
MFB(v = c(), n = 0)MFB(v = c(), n = 0)
v |
vector of integers, the subscript of the coefficient |
n |
integer, the number of inner functions |
The MFB function computes a coefficient of an exponential formal power series composition:
| a) | univariate f with univariate g, that is f[g(z)], |
| b) | univariate f with multivariate g, that is f[g(z1,z2,...,zm)], |
| c) | multivariate f with multivariate g's, that is f[g1(z1,z2,...,zm),...,gn(z1,z2,...,gm)].
|
If i1 is the power of z1, i2 is the power of z2 and so on up to im power of zm, then
(i1,i2,....im) is the subscript of the output coefficient corresponding to the product z1^i1 z2^i2 ....zm^im.
Note that this coefficient gives also the (partial) derivative of order (i1,i2,...,im) of the composition of
the multivariate functions f and g's in terms of the partial derivatives of f and g's respectively.
See the e_MFB function, for evaluating this coefficient when the coefficients of
f and to the coefficients of g's are substituted with numerical values.
string |
the expression of Faa di Bruno's formula |
The value of the first parameter is the same as the mkmSet function
Called by the e_MFB function in the kStatistics package.
The routine uses the mkmSet function in the same package.
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
G. M. Constantine, T. H. Savits (1996) A Multivariate Faa Di Bruno Formula With Applications. Trans. Amer. Math. Soc. 348(2), 503-520.
E. Di Nardo, G. Guarino, D. Senato (2008) An unifying framework for k-statistics, polykays and their generalizations. Bernoulli. 14(2), 440-468. (download from https://arxiv.org/pdf/math/0607623.pdf)
E. Di Nardo, G. Guarino, D. Senato (2008) Symbolic computation of moments of sampling distributions. Comp. Stat. Data Analysis. 52(11), 4909-4922. (download from https://arxiv.org/abs/0806.0129)
E. Di Nardo E., G. Guarino, D. Senato (2011) A new algorithm for computing the multivariate Faa di Bruno's formula. Appl. Math. Comp. 217, 6286-6295. (download from https://arxiv.org/abs/1012.6008)
#----------------------------------------# # Univariate f with Univariate g # #----------------------------------------# # The coefficient of z^2 in f[g(z)], that is f[2]g[1]^2 + f[1]g[2], where # f[1] is the coefficient of x in f(x) with x=g(z) # f[2] is the coefficient of x^2 in f(x) with x=g(z) # g[1] is the coefficient of z in g(z) # g[2] is the coefficient of z^2 in g(z) # MFB( c(2), 1 ) # The coefficient of z^3 in f[g(z)], that is f[3]g[1]^3 + 3f[2]g[1]g[2] + f[1]g[3] # MFB( c(3), 1 ) #----------------------------------------# # Univariate f with Multivariate g # #----------------------------------------# # The coefficient of z1 z2 in f[g(z1,z2)], that is f[1]g[1,1] + f[2]g[1,0]g[0,1] # where # f[1] is the coefficient of x in f(x) with x=g(z1,z2) # f[2] is the coefficient of x^2 in f(x) with x=g(z1,z2) # g[1,0] is the coefficient of z1 in g(z1,z2) # g[0,1] is the coefficient of z2 in g(z1,z2) # g[1,1] is the coefficient of z1 z2 in g(z1,z2) # MFB( c(1,1), 1 ) # The coefficient of z1^2 z2 in f[g(z1,z2)] # MFB( c(2,1), 1 ) # The coefficient of z1 z2 z3 in f[g(z1,z2,z3)] # MFB( c(1,1,1), 1 ) #----------------------------------------------------------------# # Multivariate f with Univariate/Multivariate g1, g2, ..., gn # #----------------------------------------------------------------# # The coefficient of z in f[g1(z),g2(z)], that is f[1,0]g1[1] + f[0,1]g2[1] where # f[1,0] is the coefficient of x1 in f(x1,x2) with x1=g1(z) and x2=g2(z) # f[0,1] is the coefficient of x2 in f(x1,x2) with x1=g1(z) and x2=g2(z) # g1[1] is the coefficient of z of g1(z) # g2[1] is the coefficient of z of g2(z) MFB( c(1), 2 ) # The coefficient of z1 z2 in f[g1(z1,z2),g2(z1,z2)], that is # f[1,0]g1[1,1] + f[2,0]g1[1,0]g1[0,1] + f[0,1]g2[1,1] + f[0,2]g2[1,0]g2[0,1] + # f[1,1]g1[1,0]g2[0,1] + f[1,1]g1[0,1]g2[1,0] where # f[1,0] is the coefficient of x1 in f(x1,x2) with x1=g1(z1,z2) and x2=g2(z1,z2) # f[0,1] is the coefficient of x2 in f(x1,x2) with x1=g1(z1,z2) and x2=g2(z1,z2) # g1[1,1] is the coefficient of z1z2 in g1(z1,z2) # g1[1,0] is the coefficient of z1 in g1(z1,z2) # g1[0,1] is the coefficient of z2 in g1(z1,z2) # g2[1,1] is the coefficient of z1 z2 in g2(z1,z2) # g2[1,0] is the coefficient of z1 in g2(z1,z2) # g2[0,1] is the coefficient of z2 in g1(z1,z2) MFB( c(1,1), 2 ) # The coefficient of z1 in f[g1(z1,z2),g2(z1,z2),g3(z1,z2)] MFB( c(1,0), 3 ) # The coefficient of z1 z2 in f[g1(z1,z2),g2(z1,z2),g3(z1,z2)] MFB( c(1,1), 3 ) # The coefficient of z1^2 z2 in f[g1(z1,z2),g2(z1,z2)] MFB( c(2,1), 2 ) # The coefficient of z1^2 z2 in f[g1(z1,z2),g2(z1,z2),g3(z1,z2)] MFB( c(2,1), 3 ) # The previous result expressed in a compact form for (m in unlist(strsplit( MFB(c(2,1),3), " + ", fixed=TRUE)) ) cat( m,"\n" ) # The coefficient of z1 z2 z3 in f[g1(z1,z2,z3),g2(z1,z2,z3),g3(z1,z2,z3)] MFB( c(1,1,1), 3 ) # The previous result expressed in a compact form for (m in unlist(strsplit( MFB(c(1,1,1),3), " + ", fixed=TRUE)) ) cat( m,"\n" )#----------------------------------------# # Univariate f with Univariate g # #----------------------------------------# # The coefficient of z^2 in f[g(z)], that is f[2]g[1]^2 + f[1]g[2], where # f[1] is the coefficient of x in f(x) with x=g(z) # f[2] is the coefficient of x^2 in f(x) with x=g(z) # g[1] is the coefficient of z in g(z) # g[2] is the coefficient of z^2 in g(z) # MFB( c(2), 1 ) # The coefficient of z^3 in f[g(z)], that is f[3]g[1]^3 + 3f[2]g[1]g[2] + f[1]g[3] # MFB( c(3), 1 ) #----------------------------------------# # Univariate f with Multivariate g # #----------------------------------------# # The coefficient of z1 z2 in f[g(z1,z2)], that is f[1]g[1,1] + f[2]g[1,0]g[0,1] # where # f[1] is the coefficient of x in f(x) with x=g(z1,z2) # f[2] is the coefficient of x^2 in f(x) with x=g(z1,z2) # g[1,0] is the coefficient of z1 in g(z1,z2) # g[0,1] is the coefficient of z2 in g(z1,z2) # g[1,1] is the coefficient of z1 z2 in g(z1,z2) # MFB( c(1,1), 1 ) # The coefficient of z1^2 z2 in f[g(z1,z2)] # MFB( c(2,1), 1 ) # The coefficient of z1 z2 z3 in f[g(z1,z2,z3)] # MFB( c(1,1,1), 1 ) #----------------------------------------------------------------# # Multivariate f with Univariate/Multivariate g1, g2, ..., gn # #----------------------------------------------------------------# # The coefficient of z in f[g1(z),g2(z)], that is f[1,0]g1[1] + f[0,1]g2[1] where # f[1,0] is the coefficient of x1 in f(x1,x2) with x1=g1(z) and x2=g2(z) # f[0,1] is the coefficient of x2 in f(x1,x2) with x1=g1(z) and x2=g2(z) # g1[1] is the coefficient of z of g1(z) # g2[1] is the coefficient of z of g2(z) MFB( c(1), 2 ) # The coefficient of z1 z2 in f[g1(z1,z2),g2(z1,z2)], that is # f[1,0]g1[1,1] + f[2,0]g1[1,0]g1[0,1] + f[0,1]g2[1,1] + f[0,2]g2[1,0]g2[0,1] + # f[1,1]g1[1,0]g2[0,1] + f[1,1]g1[0,1]g2[1,0] where # f[1,0] is the coefficient of x1 in f(x1,x2) with x1=g1(z1,z2) and x2=g2(z1,z2) # f[0,1] is the coefficient of x2 in f(x1,x2) with x1=g1(z1,z2) and x2=g2(z1,z2) # g1[1,1] is the coefficient of z1z2 in g1(z1,z2) # g1[1,0] is the coefficient of z1 in g1(z1,z2) # g1[0,1] is the coefficient of z2 in g1(z1,z2) # g2[1,1] is the coefficient of z1 z2 in g2(z1,z2) # g2[1,0] is the coefficient of z1 in g2(z1,z2) # g2[0,1] is the coefficient of z2 in g1(z1,z2) MFB( c(1,1), 2 ) # The coefficient of z1 in f[g1(z1,z2),g2(z1,z2),g3(z1,z2)] MFB( c(1,0), 3 ) # The coefficient of z1 z2 in f[g1(z1,z2),g2(z1,z2),g3(z1,z2)] MFB( c(1,1), 3 ) # The coefficient of z1^2 z2 in f[g1(z1,z2),g2(z1,z2)] MFB( c(2,1), 2 ) # The coefficient of z1^2 z2 in f[g1(z1,z2),g2(z1,z2),g3(z1,z2)] MFB( c(2,1), 3 ) # The previous result expressed in a compact form for (m in unlist(strsplit( MFB(c(2,1),3), " + ", fixed=TRUE)) ) cat( m,"\n" ) # The coefficient of z1 z2 z3 in f[g1(z1,z2,z3),g2(z1,z2,z3),g3(z1,z2,z3)] MFB( c(1,1,1), 3 ) # The previous result expressed in a compact form for (m in unlist(strsplit( MFB(c(1,1,1),3), " + ", fixed=TRUE)) ) cat( m,"\n" )
Secondary function useful for manipulating the result of the MFB function.
MFB2Set(sExpr="")MFB2Set(sExpr="")
sExpr |
the output of the |
set |
a set |
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
# Run MFB(c(3),1) to generate f[3]g[1]^3 + 3f[2]g[1]g[2] + f[1]g[3] # Convert the output of the MFB(c(3),1) into a vector using # MFB2Set(MFB(c(3),1)). The result is the following: # "1" "1" "f" "3" "1" # "1" "1" "g" "1" "3" # "2" "3" "f" "2" "1" # "2" "1" "g" "1" "1" # "2" "1" "g" "2" "1" # "3" "1" "f" "1" "1" # "3" "1" "g" "3" "1" MFB2Set(MFB(c(3),1))# Run MFB(c(3),1) to generate f[3]g[1]^3 + 3f[2]g[1]g[2] + f[1]g[3] # Convert the output of the MFB(c(3),1) into a vector using # MFB2Set(MFB(c(3),1)). The result is the following: # "1" "1" "f" "3" "1" # "1" "1" "g" "1" "3" # "2" "3" "f" "2" "1" # "2" "1" "g" "1" "1" # "2" "1" "g" "2" "1" # "3" "1" "f" "1" "1" # "3" "1" "g" "3" "1" MFB2Set(MFB(c(3),1))
The function returns all the partitions of a multi-index, that is a vector of non-negative integers. Note that these partitions correspond to the subdivisions of a multiset having the input multi-index as multiplicities.
mkmSet(vPar = NULL, vOutput = FALSE)mkmSet(vPar = NULL, vOutput = FALSE)
vPar |
vector of non-negative integers |
vOutput |
optional boolean variable. If equal to |
The mkmSet function finds all the vectors, different from the zero vector, whose
sum (in column) is equal to the vector (in column) of nonnegative integers given in input. When the input
vector is just an integer, let's say n, the function returns the partitions of n.
Each partition is paired with the number of set partitions having that partition as their class.
For example, if n=3 the output is
[[1,1,1],1], [[1,2],3], [[3],1], |
where 1+1+1=1+2=3. From this output, the subdivisions of a multiset with multiplicity 3 can be
recovered. For example, the subdivisions of [a,a,a] are [[a], [a], [a]] corresponding to
[1,1,1], [[a], [a,a]] corresponds to [1,2] and [[a,a,a]] corresponds to
[3]. When the input vector is a multi-index, the function returns all the partitions of the
multi-index. For example, if the input is (2,1) then the function returns
0 1 1 0 2 1 1 2
1 0 0 1 0 0 1 1
with multiplicities 1, 1, 2, 1 respectively, which corresponds to the output (the columns become rows)
of the mkmSet function when the flag variable vOutput is set equal to TRUE
[( 0 1 )( 1 0 )( 1 0 ), 1 ] |
[( 0 1 )( 2 0 ), 1 ] |
[( 1 0 )( 1 1 ), 2 ] |
[( 2 1 ), 1 ]
|
From this output, the subdivisions of a multiset with multiplicity (2,1) can be easily recovered.
For example the previous partitions correspond to the following subdivisions of the multiset
[a,a,b]
[[b], [a], [a]] |
[[b], [a,a]] |
[[a], [a,b]] |
[[a,a,b]]
|
The mkmSet function is the core of the kStatistics package. The strategy to find
all the partitions of a multi-index is described in the refereed papers. To find the multiplicities of the
multi-index partitions see the countP function.
list |
two-dimensional list: in the first there is the partition, while in the second there is its multiplicity |
Called by the nKS, nKM, nPS and nPM functions in
the kStatistics package.
In the output list, the sum of all multiplicities is the Bell Number whereas the sum of all
multiplicities of the partitions with the same lenght is the Stirling number of the second kind.
For example, mkmSet(4, TRUE) returns
[( 1 )( 1 )( 1 )( 1 ), 1 ] |
[( 1 )( 1 )( 2 ), 6 ] |
[( 2 )( 2 ), 3 ] |
[( 1 )( 3 ), 4 ] |
[( 4 ), 1 ]
|
Observe that 3 + 4 = 7 = S(4,2), where 4 is the input integer, 2 is the length of the
partitions [1,3] and [2,2] and S(i,j) denotes the Stirling numbers of the second
kind, see also the nStirling2 function. Similarly, we have 1 = S(4,4) = S(4,1)
and 6 = S(4,3). Note that 1 + 6 + 4 + 3 + 1 = 15 = Bell(4) which is the number of
partitions of the integer 4.
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
E. Di Nardo, G. Guarino, D. Senato (2008) An unifying framework for k-statistics, polykays and their generalizations. Bernoulli. 14(2), 440-468. (download from https://arxiv.org/pdf/math/0607623.pdf)
E. Di Nardo, G. Guarino, D. Senato (2008) Symbolic computation of moments of sampling distributions. Comp. Stat. Data Analysis. 52(11), 4909-4922. (download from https://arxiv.org/abs/0806.0129)
mCoeff,
countP,
nStirling2,
intPart,
ff
# Return [ [[1,1,1],1], [[1,2],3], [[3],1] ] # 3 is the multiplicity of a multiset with 3 elements all equal mkmSet(3) # Return [ [[1,1],[1,0],2], [[1,0],[1,0],[0,1],1],[[2,0],[0,1],1], [[2,1],1] ] # (2,1) is the multiplicity of a multiset with 2 equal elements and a third distinct element mkmSet(c(2,1)) # OR (same output) mkmSet(c(2,1), FALSE) # Returns the same output of the previous example but in a compact form. mkmSet(c(2,1), TRUE)# Return [ [[1,1,1],1], [[1,2],3], [[3],1] ] # 3 is the multiplicity of a multiset with 3 elements all equal mkmSet(3) # Return [ [[1,1],[1,0],2], [[1,0],[1,0],[0,1],1],[[2,0],[0,1],1], [[2,1],1] ] # (2,1) is the multiplicity of a multiset with 2 equal elements and a third distinct element mkmSet(c(2,1)) # OR (same output) mkmSet(c(2,1), FALSE) # Returns the same output of the previous example but in a compact form. mkmSet(c(2,1), TRUE)
Given a multi-index, that is a vector of non-negative integers and a positive integer n, the function returns all
the lists (v1,...,vn) of non-negative integer vectors, with the same lenght of the multi-index and such that v=v1+...+vn.
mkT(v = c(), n = 0, vOutput = FALSE)mkT(v = c(), n = 0, vOutput = FALSE)
v |
vector of integers |
n |
integer, number of addends |
vOutput |
optional boolean variable. If equal to |
From the input vector v of non-negative integers, which represents the multi-index,
the function produces all the
lists of n vectors (v1,...,vn) of non-negative integers, including the zero vector, having the same lenght of
v and such that their sum gives v. Note that two lists are different if they contain the same vectors but permuted.
list |
the list of |
The vector in the first variable must be not empty and must contain all non-negative integers. The second parameter must be a positive integer.
Called by the MFB function in the kStatistics package.
The routine uses the mkmSet function in the same package.
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
Di Nardo E., Guarino G., Senato D. (2011) A new algorithm for computing the multivariate Faa di Bruno's formula. Appl. Math. Comp. 217, 6286-6295. (download from https://arxiv.org/abs/1012.6008)
# Return the scompositions of the vector (1,1) in 2 vectors of 2 non-negative integers # such that their sum is (1,1), that is # ([1,1],[0,0]) - ([0,0],[1,1]) - ([1,0],[0,1]) - ([0,1],[1,0]) mkT(c(1,1),2) # OR (same output) mkT(c(1,1),2,FALSE) # Return the scompositions of the vector (1,0,1) in 2 vectors of 3 non-negative integers # such that their sum gives (1,0,1), that is # ([1,0,1],[0,0,0]) - ([0,0,0],[1,0,1]) - ([1,0,0],[0,0,1]) - ([0,0,1],[1,0,0]). # Note that the second value in each resulting vector is always zero. mkT(c(1,0,1),2) # OR (same output) mkT(c(1,0,1),2, FALSE) # Return the same output of the previous example but in a compact form. mkT(c(1,0,1),2, TRUE) # Return the scompositions of the vector (1,1,1) in 3 vectors of 3 non-negative integers # such that their sum gives (1,1,1). The result is given in a compact form. for (m in mkT(c(1,1,1),3)) {for (n in m) cat(n," - "); cat("\n")}# Return the scompositions of the vector (1,1) in 2 vectors of 2 non-negative integers # such that their sum is (1,1), that is # ([1,1],[0,0]) - ([0,0],[1,1]) - ([1,0],[0,1]) - ([0,1],[1,0]) mkT(c(1,1),2) # OR (same output) mkT(c(1,1),2,FALSE) # Return the scompositions of the vector (1,0,1) in 2 vectors of 3 non-negative integers # such that their sum gives (1,0,1), that is # ([1,0,1],[0,0,0]) - ([0,0,0],[1,0,1]) - ([1,0,0],[0,0,1]) - ([0,0,1],[1,0,0]). # Note that the second value in each resulting vector is always zero. mkT(c(1,0,1),2) # OR (same output) mkT(c(1,0,1),2, FALSE) # Return the same output of the previous example but in a compact form. mkT(c(1,0,1),2, TRUE) # Return the scompositions of the vector (1,1,1) in 3 vectors of 3 non-negative integers # such that their sum gives (1,1,1). The result is given in a compact form. for (m in mkT(c(1,1,1),3)) {for (n in m) cat(n," - "); cat("\n")}
The function compute a simple or a multivariate moment in terms of simple or multivariate cumulants.
mom2cum(n = 1)mom2cum(n = 1)
n |
integer or vector of integers |
Faa di Bruno's formula (the MFB function) gives the coefficients of the exponential formal
power series f[g()] where f and g are exponential formal power series too. Simple moments
are expressed in terms of simple cumulants using the Faa di Bruno's formula obtained from the MFB function
in the case "composition of univariate f with univariate g" with f[i]=1, g[i]=k[i]
for each i from 1 to n and k[i] cumulants. Multivariate moments are expressed in terms
of multivariate cumulants using the Faa di Bruno's formula obtained from the MFB function in
the case "composition of univariate f with multivariate g". In such a case the coefficients of
g are the multivariate cumulants.
string |
the expression of the moment in terms of cumulants |
The value of the first parameter is the same as the MFB function in the univariate with
univariate case composition and in the univariate with multivariate case composition.
This function calls the MFB function in the kStatistics package.
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
E. Di Nardo, G. Guarino, D. Senato (2008) An unifying framework for k-statistics, polykays and their generalizations. Bernoulli. 14(2), 440-468. (download from https://arxiv.org/pdf/math/0607623.pdf)
E. Di Nardo E., G. Guarino, D. Senato (2011) A new algorithm for computing the multivariate Faa di Bruno's formula. Appl. Math. Comp. 217, 6286-6295. (download from https://arxiv.org/abs/1012.6008)
P. McCullagh, J. Kolassa (2009) Scholarpedia, 4(3):4699. http://www.scholarpedia.org/article/Cumulants
# Return the simple moment m[5] in terms of the simple cumulants k[1],...,k[5]. mom2cum(5) # Return the multivariate moment m[3,1] in terms of the multivariate cumulants k[i,j] for # i=0,1,2,3 and j=0,1. mom2cum(c(3,1))# Return the simple moment m[5] in terms of the simple cumulants k[1],...,k[5]. mom2cum(5) # Return the multivariate moment m[3,1] in terms of the multivariate cumulants k[i,j] for # i=0,1,2,3 and j=0,1. mom2cum(c(3,1))
Given two lists with elements of the same type, the function returns a new list whose elements are the joining of the two original lists, except for the last elements, which are multiplied.
mpCart( M1 = NULL, M2 = NULL )mpCart( M1 = NULL, M2 = NULL )
M1 |
list of vectors |
M2 |
list of vectors |
The input of the mpCart function are two lists. Each list might contain multiple lists
of two vectors: the first vector contains multisets whose elements are of the same type (integers or vectors
with the same lenght), the second vector is a number (for example a multiplicity if the multiset is a
subdivision). The mpCart function generates a new list of two vectors: the first is
obtained by joining the first vectors in the two input lists, the second is just the product
of the numbers in the second vectors. See the examples.
list |
the list with the joined input lists |
Called by the function nPM in the package kStatistics.
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
E. Di Nardo, G. Guarino, D. Senato (2008) An unifying framework for k-statistics, polykays and their generalizations. Bernoulli. 14(2), 440-468. (download from https://arxiv.org/pdf/math/0607623.pdf)
E. Di Nardo, G. Guarino, D. Senato (2008) Symbolic computation of moments of sampling distributions. Comp. Stat. Data Analysis. 52(11), 4909-4922. (download from https://arxiv.org/abs/0806.0129)
E. Di Nardo, G. Guarino, D. Senato (2009) A new method for fast computing unbiased estimators of cumulants. Statistics and Computing, 19, 155-165. (download from https://arxiv.org/abs/0807.5008)
A <- list( list( list(c(1),c(2) ),c(-1)), list(list(c(3)),c(1)) ) # where # -1 is the multiplicative factor of list(c(1),c(2) ) # 1 is the multiplicative factor of list(c(3)) B <- list( list( list(c(5)),c(7))) # where 7 is the multiplicative factor of list(c(5)) # Return [[[1],[2],[5]], -7] , [[[3],[5]], 7] mpCart(A,B) A <- list( list( list( c(1,0),c(1,0) ), c(-1)), list( list( c(2,0)), c(1) )) # where # - 1 is the multiplicative factor of list( c(1,0),c(1,0) ) # 1 is the multiplicative factor of list( c(2,0) ) B <- list( list( list( c(1,0)), c(1)) ) # where 1 is the multiplicative factor of list( c(1,0)) # Return [[[1,0],[1,0],[1,0]], -1], [[[2,0],[1,0]],1] mpCart(A,B)A <- list( list( list(c(1),c(2) ),c(-1)), list(list(c(3)),c(1)) ) # where # -1 is the multiplicative factor of list(c(1),c(2) ) # 1 is the multiplicative factor of list(c(3)) B <- list( list( list(c(5)),c(7))) # where 7 is the multiplicative factor of list(c(5)) # Return [[[1],[2],[5]], -7] , [[[3],[5]], 7] mpCart(A,B) A <- list( list( list( c(1,0),c(1,0) ), c(-1)), list( list( c(2,0)), c(1) )) # where # - 1 is the multiplicative factor of list( c(1,0),c(1,0) ) # 1 is the multiplicative factor of list( c(2,0) ) B <- list( list( list( c(1,0)), c(1)) ) # where 1 is the multiplicative factor of list( c(1,0)) # Return [[[1,0],[1,0],[1,0]], -1], [[[2,0],[1,0]],1] mpCart(A,B)
Given a multivariate data sample, the function returns an estimate of a joint (or multivariate) cumulant with a fixed order.
nKM( v = NULL, V = NULL)nKM( v = NULL, V = NULL)
v |
vector of integers |
V |
vector of a multivariate data sample |
For a sample of i.i.d. random vectors, multivariate k-statistics are unbiased estimators of
the population joint cumulants with minimum variance and are expressed in terms of power sum symmetric polynomials
in the random vectors of the sample. See the referred papers to read more about these estimators.
Thus, for the input multivariate sample data, running
nKM( c(r, s, ...), data) with fixed order v=(r, s, ...) returns an estimate of the joint cumulant
k[r, s, ...] of the population distribution.
float |
the value of the multivariate k-statistics |
The size of each data vector must be equal to the length of the vector passed trough the first input variable.
Called by the master nPolyk function in the kStatistics package.
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
E. Di Nardo, G. Guarino, D. Senato (2008) An unifying framework for k-statistics, polykays and their generalizations. Bernoulli. 14(2), 440-468. (download from https://arxiv.org/pdf/math/0607623.pdf)
E. Di Nardo, G. Guarino, D. Senato (2008) Symbolic computation of moments of sampling distributions. Comp. Stat. Data Analysis. 52(11), 4909-4922. (download from https://arxiv.org/abs/0806.0129)
E. Di Nardo, G. Guarino, D. Senato (2009) A new method for fast computing unbiased estimators of cumulants. Statistics and Computing, 19, 155-165. (download from https://arxiv.org/abs/0807.5008)
P. McCullagh, J. Kolassa (2009), Scholarpedia, 4(3):4699. http://www.scholarpedia.org/article/Cumulants
# Data assignment data1<-list(c(5.31,11.16),c(3.26,3.26),c(2.35,2.35),c(8.32,14.34),c(13.48,49.45), c(6.25,15.05),c(7.01,7.01),c(8.52,8.52),c(0.45,0.45),c(12.08,12.08),c(19.39,10.42)) # Return an estimate of the joint cumulant k[2,1] nKM(c(2,1),data1) # Data assignment data2<-list(c(5.31,11.16,4.23),c(3.26,3.26,4.10),c(2.35,2.35,2.27), c(4.31,10.16,6.45),c(3.1,2.3,3.2),c(3.20, 2.31, 7.3)) # Return an estimate of the joint cumulant k[2,2,2] nKM(c(2,2,2),data2) # Data assignment data3<-list(c(5.31,11.16,4.23,4.22),c(3.26,3.26,4.10,4.9),c(2.35,2.35,2.27,2.26), c(4.31,10.16,6.45,6.44),c(3.1,2.3,3.2,3.1),c(3.20, 2.31, 7.3,7.2)) # Return an estimate of the joint cumulant k[2,1,1,1] nKM(c(2,1,1,1),data3)# Data assignment data1<-list(c(5.31,11.16),c(3.26,3.26),c(2.35,2.35),c(8.32,14.34),c(13.48,49.45), c(6.25,15.05),c(7.01,7.01),c(8.52,8.52),c(0.45,0.45),c(12.08,12.08),c(19.39,10.42)) # Return an estimate of the joint cumulant k[2,1] nKM(c(2,1),data1) # Data assignment data2<-list(c(5.31,11.16,4.23),c(3.26,3.26,4.10),c(2.35,2.35,2.27), c(4.31,10.16,6.45),c(3.1,2.3,3.2),c(3.20, 2.31, 7.3)) # Return an estimate of the joint cumulant k[2,2,2] nKM(c(2,2,2),data2) # Data assignment data3<-list(c(5.31,11.16,4.23,4.22),c(3.26,3.26,4.10,4.9),c(2.35,2.35,2.27,2.26), c(4.31,10.16,6.45,6.44),c(3.1,2.3,3.2,3.1),c(3.20, 2.31, 7.3,7.2)) # Return an estimate of the joint cumulant k[2,1,1,1] nKM(c(2,1,1,1),data3)
Given a data sample, the function returns an estimate of a cumulant with a fixed order.
nKS( v = NULL, V = NULL)nKS( v = NULL, V = NULL)
v |
integer or one-dimensional vector |
V |
vector of a data sample |
For a sample of i.i.d. random variables, k-statistics are unbiased estimators with minimum variance of the population
cumulants and are expressed in terms of power sum symmetric polynomials in the random variables
of the sample. See the referred papers to read more about these estimators. Thus, for the input
sample data, running nKS(v,data) or nKS(c(v),data)
returns an estimate of the v-th cumulant of the population distribution.
float |
the value of the k-statistics |
Called by the master nPolyk function in the kStatistics package.
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
E. Di Nardo, G. Guarino, D. Senato (2008) An unifying framework for k-statistics, polykays and their generalizations. Bernoulli. 14(2), 440-468. (download from https://arxiv.org/pdf/math/0607623.pdf)
E. Di Nardo, G. Guarino, D. Senato (2008) Symbolic computation of moments of sampling distributions. Comp. Stat. Data Analysis. 52(11), 4909-4922. (download from https://arxiv.org/abs/0806.0129)
E. Di Nardo, G. Guarino, D. Senato (2009) A new method for fast computing unbiased estimators of cumulants. Statistics and Computing, 19, 155-165. (download from https://arxiv.org/abs/0807.5008)
P. McCullagh, J. Kolassa (2009), Scholarpedia, 4(3):4699. http://www.scholarpedia.org/article/Cumulants
# Data assignment data<-c(16.34, 10.76, 11.84, 13.55, 15.85, 18.20, 7.51, 10.22, 12.52, 14.68, 16.08, 19.43,8.12, 11.20, 12.95, 14.77, 16.83, 19.80, 8.55, 11.58, 12.10, 15.02, 16.83, 16.98, 19.92, 9.47, 11.68, 13.41, 15.35, 19.11) # Return an estimate of the cumulant of order 7 nKS(7, data) # Return an estimate of the cumulant of order 1, that is the mean (R command: mean(data)) nKS(1, data) # Return an estimate of the cumulant of order 2, that is the variance (R command: var(data)) nKS(2, data) # Return an estimate of the skewness (R command: skewnes(data) in the library "moments") nKS(3, data)/sqrt(nKS(2, data))^3 # Return an estimate of the kurtosis (R command: kurtosis(data) in the library "moments") nKS(4, data)/nKS(2, data)^2 + 3# Data assignment data<-c(16.34, 10.76, 11.84, 13.55, 15.85, 18.20, 7.51, 10.22, 12.52, 14.68, 16.08, 19.43,8.12, 11.20, 12.95, 14.77, 16.83, 19.80, 8.55, 11.58, 12.10, 15.02, 16.83, 16.98, 19.92, 9.47, 11.68, 13.41, 15.35, 19.11) # Return an estimate of the cumulant of order 7 nKS(7, data) # Return an estimate of the cumulant of order 1, that is the mean (R command: mean(data)) nKS(1, data) # Return an estimate of the cumulant of order 2, that is the variance (R command: var(data)) nKS(2, data) # Return an estimate of the skewness (R command: skewnes(data) in the library "moments") nKS(3, data)/sqrt(nKS(2, data))^3 # Return an estimate of the kurtosis (R command: kurtosis(data) in the library "moments") nKS(4, data)/nKS(2, data)^2 + 3
The function returns all possible different permutations of objects in a list or in a vector.
nPerm(L = c())nPerm(L = c())
L |
List/Vector |
In order to manage permutations of numbers or vectors, the standard permutation process is applied.
list |
all the permutations of |
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
C. A. Charalambides (2002) Enumerative Combinatoris, Chapman & Haii/CRC.
# permutations of 1,2,3 nPerm( c(1,2,3) ) # permutations of 1,2,1 (two elements are equal) nPerm( c(1,2,1) ) # permutations of the words "Alice", "Bob","Jack" nPerm( c("Alice", "Bob","Jack") ) # permutations of the vectors c(0,1), c(2,3), c(7,3) nPerm( list(c(0,1), c(2,3), c(7,3)) )# permutations of 1,2,3 nPerm( c(1,2,3) ) # permutations of 1,2,1 (two elements are equal) nPerm( c(1,2,1) ) # permutations of the words "Alice", "Bob","Jack" nPerm( c("Alice", "Bob","Jack") ) # permutations of the vectors c(0,1), c(2,3), c(7,3) nPerm( list(c(0,1), c(2,3), c(7,3)) )
Given a multivariate data sample, the function returns an estimate of a product of joint cumulants with fixed orders.
nPM( v = NULL, V = NULL)nPM( v = NULL, V = NULL)
v |
list of integer vectors |
V |
vector of a multivariate data sample |
Multivariate polykays or multivariate generalized k-statistics are unbiased estimators of
joint cumulant products with minimum variance. See the referred papers to read more about these estimators. Multivariate polykays
are usually expressed in terms of power sum symmetric polynomials in the random vectors of the sample.
Thus, for the input multivariate sample data, running
nPM( list( c(r1, s1, ...), c(r1, s2, ...),.. ), data) returns an estimate of the product
k[r1, s1,....]*k[r2, s2, ...]*... where k[r1, s1,....], k[r2, s2, ...], ... are
the joint cumulants of the population distribution.
float |
the estimate of the multivariate polykay |
The size of each data vector must be equal to the length of the vector passed trough the first input variable. The vectors in the list must have the same length.
Called by the master nPolyk function in the kStatistics package.
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
E. Di Nardo, G. Guarino, D. Senato (2008) An unifying framework for k-statistics, polykays and their generalizations. Bernoulli. 14(2), 440-468. (download from https://arxiv.org/pdf/math/0607623.pdf)
E. Di Nardo, G. Guarino, D. Senato (2008) Symbolic computation of moments of sampling distributions. Comp. Stat. Data Analysis. 52(11), 4909-4922. (download from https://arxiv.org/abs/0806.0129)
E. Di Nardo, G. Guarino, D. Senato (2009) A new method for fast computing unbiased estimators of cumulants. Statistics and Computing, 19, 155-165. (download from https://arxiv.org/abs/0807.5008)
P. McCullagh, J. Kolassa (2009), Scholarpedia, 4(3):4699. http://www.scholarpedia.org/article/Cumulants
# Data assignment data1<-list(c(5.31,11.16),c(3.26,3.26),c(2.35,2.35),c(8.32,14.34),c(13.48,49.45), c(6.25,15.05),c(7.01,7.01),c(8.52,8.52),c(0.45,0.45),c(12.08,12.08),c(19.39,10.42)) # Return an estimate of the product k[2,1]*k[1,0], where k[2,1] and k[1,0] are the # cross-correlation of order (2,1) and the marginal mean of the population distribution # respectively nPM( list( c(2,1), c(1,0) ), data1) # Data assignment data2<-list(c(5.31,11.16,4.23),c(3.26,3.26,4.10),c(2.35,2.35,2.27), c(4.31,10.16,6.45),c(3.1,2.3,3.2),c(3.20, 2.31, 7.3)) # Return an estimate of the product k[2,0,1]*k[1,1,0], where k[2,0,1] and k[1,1,0] # are joint cumulants of the population distribution nPM( list( c(2,0,1), c(1,1,0) ), data2)# Data assignment data1<-list(c(5.31,11.16),c(3.26,3.26),c(2.35,2.35),c(8.32,14.34),c(13.48,49.45), c(6.25,15.05),c(7.01,7.01),c(8.52,8.52),c(0.45,0.45),c(12.08,12.08),c(19.39,10.42)) # Return an estimate of the product k[2,1]*k[1,0], where k[2,1] and k[1,0] are the # cross-correlation of order (2,1) and the marginal mean of the population distribution # respectively nPM( list( c(2,1), c(1,0) ), data1) # Data assignment data2<-list(c(5.31,11.16,4.23),c(3.26,3.26,4.10),c(2.35,2.35,2.27), c(4.31,10.16,6.45),c(3.1,2.3,3.2),c(3.20, 2.31, 7.3)) # Return an estimate of the product k[2,0,1]*k[1,1,0], where k[2,0,1] and k[1,1,0] # are joint cumulants of the population distribution nPM( list( c(2,0,1), c(1,1,0) ), data2)
The master function executes one of the functions to compute simple k-statistics (nKS),
multivariate k-statistics (nKM), simple polykays (nPS) or multivariate polykays (nPM).
nPolyk( L = NULL, data = NULL, bhelp=NULL )nPolyk( L = NULL, data = NULL, bhelp=NULL )
L |
vector of orders |
data |
vector of a (univariate or multivariate) sample data |
bhelp |
|
The master function analizes the first two input variables and recalls one of the nKS, nKM,
nPS or nPM functions in the kStatistics package.
Given a sample data:
simple k-statistics are computed using nPolyk(c(r), data)) or nPolyk(list(c(r)), data))
multivariate k-statistics are computed using nPolyk(c(r, s), data)) or nPolyk( list(c(r, s)),
data))
simple polykays are computed using nPolyk(list(c(r),c(s)...),data))
multivariate polykays are computed using nPolyk(list(c(r1, r2,...),c(s1, s2,...),...),data))
float |
the estimate of the (joint) cumulant or of the (joint) cumulant product |
The dimension of the vector with the sample data depends on the first parameter.
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
E. Di Nardo, G. Guarino, D. Senato (2008) An unifying framework for k-statistics, polykays and their generalizations. Bernoulli. 14(2), 440-468. (download from https://arxiv.org/pdf/math/0607623.pdf)
E. Di Nardo, G. Guarino, D. Senato (2008) Symbolic computation of moments of sampling distributions. Comp. Stat. Data Analysis. 52(11), 4909-4922. (download from https://arxiv.org/abs/0806.0129)
E. Di Nardo, G. Guarino, D. Senato (2009) A new method for fast computing unbiased estimators of cumulants. Statistics and Computing, 19, 155-165. (download from https://arxiv.org/abs/0807.5008)
P. McCullagh, J. Kolassa (2009), Scholarpedia, 4(3):4699. http://www.scholarpedia.org/article/Cumulants
# Data assignment data1<-c(16.34, 10.76, 11.84, 13.55, 15.85, 18.20, 7.51, 10.22, 12.52, 14.68, 16.08, 19.43,8.12, 11.20, 12.95, 14.77, 16.83, 19.80, 8.55, 11.58, 12.10, 15.02, 16.83, 16.98, 19.92, 9.47, 11.68, 13.41, 15.35, 19.11) # Display "KS:[1] -1.44706" which indicates the type of subfunction (nKS) called by # the master function nPolyk and gives the estimate of the third cumulant nPolyk(c(3),data1, TRUE) # Display "[1] -1.44706" (without the indication of the employed subfunction) nPolyk(c(3),data1, FALSE) # Display "PS:[1] 177.4233" which indicates the type of subfunction (nPS) called by # the master function nPolyk and gives the estimate of the product between the # variance k[2] and the mean k[1] nPolyk( list( c(2), c(1) ),data1,TRUE) # Data assignment data2<-list(c(5.31,11.16),c(3.26,3.26),c(2.35,2.35),c(8.32,14.34),c(13.48,49.45), c(6.25,15.05),c(7.01,7.01),c(8.52,8.52),c(0.45,0.45),c(12.08,12.08),c(19.39,10.42)) # Display "KM:[1] -23.7379" which indicates the type of subfunction (nKM) called by # the master function nPolyk and gives the estimate of k[2,1] nPolyk(c(2,1),data2,TRUE) # Display "PM:[1] 48.43243" which indicates the type of subfunction (nPM) called by # the master function nPolyk and gives the estimate of k[2,1]*k[1,0] nPolyk( list( c(2,1), c(1,0) ),data2,TRUE)# Data assignment data1<-c(16.34, 10.76, 11.84, 13.55, 15.85, 18.20, 7.51, 10.22, 12.52, 14.68, 16.08, 19.43,8.12, 11.20, 12.95, 14.77, 16.83, 19.80, 8.55, 11.58, 12.10, 15.02, 16.83, 16.98, 19.92, 9.47, 11.68, 13.41, 15.35, 19.11) # Display "KS:[1] -1.44706" which indicates the type of subfunction (nKS) called by # the master function nPolyk and gives the estimate of the third cumulant nPolyk(c(3),data1, TRUE) # Display "[1] -1.44706" (without the indication of the employed subfunction) nPolyk(c(3),data1, FALSE) # Display "PS:[1] 177.4233" which indicates the type of subfunction (nPS) called by # the master function nPolyk and gives the estimate of the product between the # variance k[2] and the mean k[1] nPolyk( list( c(2), c(1) ),data1,TRUE) # Data assignment data2<-list(c(5.31,11.16),c(3.26,3.26),c(2.35,2.35),c(8.32,14.34),c(13.48,49.45), c(6.25,15.05),c(7.01,7.01),c(8.52,8.52),c(0.45,0.45),c(12.08,12.08),c(19.39,10.42)) # Display "KM:[1] -23.7379" which indicates the type of subfunction (nKM) called by # the master function nPolyk and gives the estimate of k[2,1] nPolyk(c(2,1),data2,TRUE) # Display "PM:[1] 48.43243" which indicates the type of subfunction (nPM) called by # the master function nPolyk and gives the estimate of k[2,1]*k[1,0] nPolyk( list( c(2,1), c(1,0) ),data2,TRUE)
Given a data sample, the function returns an estimate of a product of cumulants with fixed orders.
nPS( v = NULL, V = NULL)nPS( v = NULL, V = NULL)
v |
vector of integers |
V |
vector of a data sample |
Simple polykays or generalized k-statistics are unbiased estimators of cumulant products
with minimum variance.
See the referred papers to read more about these estimators. Simple polykays are usually expressed
in terms of power sum symmetric polynomials in the i.i.d. random variables of the sample. Thus,
for the input sample data, running nPS(c(i,j,...),data) returns an estimate of
the product k[i]*k[j]*... with k[i], k[j], ... the cumulants of the population
distribution and v=(i,j,...) their fixed orders.
float |
the estimate of the polykay |
Called by the master nPolyk function in the kStatistics package.
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
E. Di Nardo, G. Guarino, D. Senato (2008) An unifying framework for k-statistics, polykays and their generalizations. Bernoulli. 14(2), 440-468. (download from https://arxiv.org/pdf/math/0607623.pdf)
E. Di Nardo, G. Guarino, D. Senato (2008) Symbolic computation of moments of sampling distributions. Comp. Stat. Data Analysis. 52(11), 4909-4922. (download from https://arxiv.org/abs/0806.0129)
E. Di Nardo, G. Guarino, D. Senato (2009) A new method for fast computing unbiased estimators of cumulants. Statistics and Computing, 19, 155-165. (download from https://arxiv.org/abs/0807.5008)
P. McCullagh, J. Kolassa (2009), Scholarpedia, 4(3):4699. http://www.scholarpedia.org/article/Cumulants
# Data assignment data<-c(16.34, 10.76, 11.84, 13.55, 15.85, 18.20, 7.51, 10.22, 12.52, 14.68, 16.08, 19.43,8.12, 11.20, 12.95, 14.77, 16.83, 19.80, 8.55, 11.58, 12.10, 15.02, 16.83, 16.98, 19.92, 9.47, 11.68, 13.41, 15.35, 19.11) # Return an estimate of the product k[2]*k[1], where k[1] and k[2] are the mean and # the variance of the population distribution respectively nPS(c(2,1), data)# Data assignment data<-c(16.34, 10.76, 11.84, 13.55, 15.85, 18.20, 7.51, 10.22, 12.52, 14.68, 16.08, 19.43,8.12, 11.20, 12.95, 14.77, 16.83, 19.80, 8.55, 11.58, 12.10, 15.02, 16.83, 16.98, 19.92, 9.47, 11.68, 13.41, 15.35, 19.11) # Return an estimate of the product k[2]*k[1], where k[1] and k[2] are the mean and # the variance of the population distribution respectively nPS(c(2,1), data)
The function computes the Stirling number of the second kind.
nStirling2( n, k )nStirling2( n, k )
n |
integer |
k |
integer less or equal to |
The Stirling number of the second kind S(n,k) is equal to the number of ways to split a
set of cardinality n into k nonempty subsets. For example, if the set is [a,b,c,d], then
the partitions in 2 blocks are: [[a], [bcd]], [[b], [acd]], [[c], [abd]], [[d],[abc]] with
cardinalities (1,3) and [ab, cd], [ac, bd], [ad, bc] with cardinalities (2,2). Then S(4,2)
is equal to 7. Note that (1,3) and (2,2) are also the partitions of the integer 4 in 2 parts.
integer |
the Stirling number of the second kind |
Called by the nKS and nKM functions in the kStatistics package.
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
R. P. Stanley (2012) Enumerative combinatorics. Vol.1. II edition. Cambridge Studies in Advanced Mathematics, 49. Cambridge University Press, Cambridge.
mkmSet,
mCoeff,
intPart,
countP,
ff
# Return the number of ways to split a set of 6 objects into 2 nonempty subsets nStirling2(6,2)# Return the number of ways to split a set of 6 objects into 2 nonempty subsets nStirling2(6,2)
The function generates a complete or a partial ordinary Bell polynomial.
oBellPol(n = 1, m = 0)oBellPol(n = 1, m = 0)
n |
integer, the degree of the polynomial |
m |
integer, the fixed degree of each monomial in the polynomial |
Faa di Bruno's formula gives the coefficients of the exponential formal power series obtained
from the composition f[g()] of the exponential formal power series f with g. The
partial ordinary Bell polynomials B[n,m] can be expressed
in the terms of the partial exponential Bell polynomials B(n,m)(y[1],...,y[n-m+1]) using the
following formula:
B[n,m](y[1],...,y[n-m+1])=k!/n!B(n,m)(y[1],...,y[n-m+1]). |
The complete ordinary Bell polynomials are given by B[n]=B[n,1]+B[n,2]+...B[n,n], where
B[n,m] is the partial ordinary Bell polynomial of order (n,m) for m from 1
to n.
string |
the expression of the polynomial |
The value of the first parameter is the same as the MFB function in the univariate with
univariate composition.
This function calls the MFB function in the kStatistics package.
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
C.A. Charalambides (2002) Enumerative Combinatoris, Chapman & Haii/CRC.
E. Di Nardo, G. Guarino, D. Senato (2011) A new algorithm for computing the multivariate Faa di Bruno's formula. Appl. Math. Comp. 217, 6286-6295. (download from https://arxiv.org/abs/1012.6008)
# Return the complete ordinary Bell Polynomial for n=5, that is # (y1^5) + 20(y1^3)(y2) + 30(y1)(y2^2) + 60(y1^2)(y3) + 120(y2)(y3) + 120(y1)(y4) + 120(y5) oBellPol(5) # # OR (same output) # oBellPol(5,0) # Return the partial ordinary Bell polynomial for n=5 and m=3, that is # 30(y1)(y2^2) + 60(y1^2)(y3) oBellPol(5,3)# Return the complete ordinary Bell Polynomial for n=5, that is # (y1^5) + 20(y1^3)(y2) + 30(y1)(y2^2) + 60(y1^2)(y3) + 120(y2)(y3) + 120(y1)(y4) + 120(y5) oBellPol(5) # # OR (same output) # oBellPol(5,0) # Return the partial ordinary Bell polynomial for n=5 and m=3, that is # 30(y1)(y2^2) + 60(y1^2)(y3) oBellPol(5,3)
The function returns the cartesian product between vectors.
pCart( L )pCart( L )
L |
vectors in a list |
The pCart function pairs any element of the first vector with any element of
the second vector, iteratively, if there are more than two vectors in input. Repetitions are allowed.
See examples.
list |
the list with the cartesian product |
Called by the nPS function in the kStatistics package.
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
D. E. Knuth (1998) The Art of Computer Programming. (3rd ed.) Addison Wesley.
A <- c(1,2) B <- c(3,4,5) # Return the cartesian product [[1,3],[1,4],[1,5],[2,3],[2,4],[2,5]] pCart( list( A, B) ) L1<-list( c(1,1), c(2)) L2<-list( c(5,5), c(7) ) # Return the cartesian product [[1,1],[5,5]], [[1,1],[7]], [[2],[5,5]], [[2],[7]] # and assign the result to L3 L3<-pCart ( list(L1, L2) ) # Return the cartesian product between L3 and [7]. # The result is [[1,1],[5,5],[7]], [[1,1],[7],[7]], [[2],[5,5],[7]], [[2],[7],[7]] pCart ( list(L3, c(7)) )A <- c(1,2) B <- c(3,4,5) # Return the cartesian product [[1,3],[1,4],[1,5],[2,3],[2,4],[2,5]] pCart( list( A, B) ) L1<-list( c(1,1), c(2)) L2<-list( c(5,5), c(7) ) # Return the cartesian product [[1,1],[5,5]], [[1,1],[7]], [[2],[5,5]], [[2],[7]] # and assign the result to L3 L3<-pCart ( list(L1, L2) ) # Return the cartesian product between L3 and [7]. # The result is [[1,1],[5,5],[7]], [[1,1],[7],[7]], [[2],[5,5],[7]], [[2],[7],[7]] pCart ( list(L3, c(7)) )
The function returns the value of the power sum symmetric polynomial, with fixed degrees and in one or more sets of variables, when the variables are substituted with the input lists of numerical values.
powS(vn = NULL, lvd = NULL)powS(vn = NULL, lvd = NULL)
vn |
vector of integers (the powers of the indeterminates) |
lvd |
list of numerical values in place of the variables |
Given the lists of numerical values (x[1],x[2],...), (y[1],y[2],...), (z[1],z[2],...), ...
in the input parameter lvd and the integers (n,m,j,...) in the input parameter vn,
the powS function returns the value of
(x[1]^n)*(y[1]^m)*(z[1]^j)*...+(x[2]^n)*(y[2]^m)*(z[2]^j)*+....
integer |
the value of the polynomial |
Called by the nKS, nKM, nPS and nPM functions
in the kStatistics package.
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
E. Di Nardo, G. Guarino, D. Senato (2008) Symbolic computation of moments of sampling distributions. Comp. Stat. Data Analysis. 52(11), 4909-4922. (download from https://arxiv.org/abs/0806.0129)
# Return 1^3 + 2^3 + 3^3 = 36 powS(c(3), list(c(1),c(2),c(3))) # Return (1^3 * 4^2) + (2^3 * 5^2) + (3^3 * 6^2) = 1188 powS(c(3,2),list(c(1,4),c(2,5),c(3,6)))# Return 1^3 + 2^3 + 3^3 = 36 powS(c(3), list(c(1),c(2),c(3))) # Return (1^3 * 4^2) + (2^3 * 5^2) + (3^3 * 6^2) = 1188 powS(c(3,2),list(c(1,4),c(2,5),c(3,6)))
The function generates the partition polynomial of degree n, whose
coefficients are the number of partitions of n into k parts for k
from 1 to n.
pPart(n = 0)pPart(n = 0)
n |
integer, the degree of the polynomial |
Faa di Bruno's formula gives the coefficients of the exponential formal
power series obtained from the composition f[g()] of the exponential formal power
series f and g. The partition polynomial F[n] of degree n
is obtained using the Faa di Bruno's formula, output of the MFB function,
in the case "composition of univariate f with univariate g" with f[i]=1/n!,
g[i]^k=(i!)^k*k!*y^k for i and k from 1 to n. Note the
symbolic substitution of g[i], as the power of g[i] appears
in the substitution. This function is an example of application of Faa di Bruno's formula
and the symbolic calculus with two indexes.
string |
the expression of the polynomial |
The value of the first parameter is the same as the MFB function
in the univariate with univariate case composition.
This function calls the MFB function in the kStatistics package.
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
E. Di Nardo E., G. Guarino, D. Senato (2011) A new algorithm for computing the multivariate Faa di Bruno's formula. Appl. Math. Comp. 217, 6286-6295. (download from https://arxiv.org/abs/1012.6008)
# Return the partition polynomial F[5] pPart(5) # Return the partition polynomial F[11] and its evaluation when y=7 # s<-pPart(11) # run the command s<-paste0("1",s) # add the coefficient to the first term (fixed command) s<-gsub(" y","1y",s) # replace the variable y without coefficient (fixed command) s<-gsub("y", "*7",s) # assignment y = 7 eval(parse(text=s)) # evaluation of the expression (fixed command)# Return the partition polynomial F[5] pPart(5) # Return the partition polynomial F[11] and its evaluation when y=7 # s<-pPart(11) # run the command s<-paste0("1",s) # add the coefficient to the first term (fixed command) s<-gsub(" y","1y",s) # replace the variable y without coefficient (fixed command) s<-gsub("y", "*7",s) # assignment y = 7 eval(parse(text=s)) # evaluation of the expression (fixed command)
The function returns the product between polynomials without constant term.
pPoly( L = NULL)pPoly( L = NULL)
L |
lists of the coefficients of the polynomials |
vector |
the coefficients of the polynomial output of the product |
Called by the nKS and nKM functions in the kStatistics package.
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
# c(1,-3) are the coefficients of (x-3x^2), c(2) is the coefficient of 2x # Return c(0, 2,-6), coefficients of 2x^2-6x^3 =(x-3x^2)*(2x) pPoly(list(c(1,-3), c(2))) # c(0,3,-2) are the coefficients of 3x^2-2x^3, c(0,2,-1) are the coefficients of (2x^2-x^3) # Return c(0,0,0,6,-7,2), coefficients of 6x^4-7x^5+2x^6=(3x^2-2x^3)*(2x^2-x^3) pPoly(list(c(0,3,-2),c(0,2,-1)))# c(1,-3) are the coefficients of (x-3x^2), c(2) is the coefficient of 2x # Return c(0, 2,-6), coefficients of 2x^2-6x^3 =(x-3x^2)*(2x) pPoly(list(c(1,-3), c(2))) # c(0,3,-2) are the coefficients of 3x^2-2x^3, c(0,2,-1) are the coefficients of (2x^2-x^3) # Return c(0,0,0,6,-7,2), coefficients of 6x^4-7x^5+2x^6=(3x^2-2x^3)*(2x^2-x^3) pPoly(list(c(0,3,-2),c(0,2,-1)))
The function converts a set into a string.
Set2expr(v = NULL )Set2expr(v = NULL )
v |
Set |
string |
the string |
Called by the MFB and MFB functions in the kStatistics package
being useful for manipulating the result before its print.
Elvira Di Nardo [email protected],
Giuseppe Guarino [email protected]
# To print 6f[3]^2g[2]^5 run Set2expr( list(c("1","2","f","3","2"),c("1","3","g","2","5"))) # Run MFB(c(3),1) to recover f[3]g[1]^3 + 3f[2]g[1]g[2] + f[1]g[3] # Run S<-MFB2Set(MFB(c(3),1)) to convert the output of MFB(c(3),1) into a vector. # The result is # "1" "1" "f" "3" "1" # "1" "1" "g" "1" "3" # "2" "3" "f" "2" "1" # "2" "1" "g" "1" "1" # "2" "1" "g" "2" "1" # "3" "1" "f" "1" "1" # "3" "1" "g" "3" "1" # To set f[2]=1, run S[[3]][4]<-"" and S[[3]][3]<-"". # Then run Set2expr(S) to recover # f[3]g[1]^3 + 3g[1]g[2] + f[1]g[3] # S<-MFB2Set(MFB(c(3),1)) S[[3]][4]<-"" S[[3]][3]<-"" Set2expr(S)# To print 6f[3]^2g[2]^5 run Set2expr( list(c("1","2","f","3","2"),c("1","3","g","2","5"))) # Run MFB(c(3),1) to recover f[3]g[1]^3 + 3f[2]g[1]g[2] + f[1]g[3] # Run S<-MFB2Set(MFB(c(3),1)) to convert the output of MFB(c(3),1) into a vector. # The result is # "1" "1" "f" "3" "1" # "1" "1" "g" "1" "3" # "2" "3" "f" "2" "1" # "2" "1" "g" "1" "1" # "2" "1" "g" "2" "1" # "3" "1" "f" "1" "1" # "3" "1" "g" "3" "1" # To set f[2]=1, run S[[3]][4]<-"" and S[[3]][3]<-"". # Then run Set2expr(S) to recover # f[3]g[1]^3 + 3g[1]g[2] + f[1]g[3] # S<-MFB2Set(MFB(c(3),1)) S[[3]][4]<-"" S[[3]][3]<-"" Set2expr(S)