kStatistics (version 2.0)

MFB: Faa di Bruno's formula

Description

The routine generates the coefficients of the composition of (exponential) formal power series provided by the univariate and multivariate Faa di Bruno's formula indexed by the integers i1,i2,....

Usage

MFB(v = c(), n = 0)

Arguments

v

vector of integers, the subscript of the coefficient

n

integer, the number of inner functions "g"

Value

string

the expression of Faa di Bruno's formula

Warning

The value of the first parameter is the same as the mkmSet function, i.e. the number of assigned blocks.

Details

The function computes the coefficients of the composition of (exponential) formal power series: 1) Univariate with Univariate Case: f(g(z),z,z...) 2) Univariate with Multivariate Case: f(g(z1,z2,...),z1,z1,..,z2,z2...,.. ) 3) Multivariate with Multivariate Case: f(g1(z1,z2,..),g2(z1,z2,..),..), z1,z1,...,z2,z2,..,..) If i1 is the number of z1, i2 is the number of z2 and so on, then (i1,i2, ...) is the index of the output coefficient corresponding to the product z1^i1 z2^i2 .... Note that this coefficient is also related to the (partial) derivative of order (i1,i2, ...) of the composition of (exponential) formal power series evaluated at z1=0, z2=0, and so on. See the function e_MFB, for evaluating these coefficients by assigning numerical values to the coefficients of f() and to the coefficients of g1(),g2(),...

References

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 http://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 http://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 http://arxiv.org/abs/1012.6008)

See Also

mkmSet, e_MFB

Examples

# NOT RUN {
#----------------------------------------#
# 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 t in f(t) with t=g(z) 
# f[2] is the coefficient of t^2 in f(t) with  t=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 t in f(t) with t=g(z1,z2)
# f[2]   is the coefficient of t^2 in f(t) with t=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 Multivariate g1, g2, ...   #
#-------------------------------------------------#

# 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 t1 in f(t1,t2) with t1=g1(z) and t2=g2(z)
# f[0,1] is the coefficient of t2 in f(t1,t2) with t1=g1(z) and t2=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 t1 in f(t1,t2) with t1=g1(z1,z2) and t2=g2(z1,z2)
# f[0,1]  is the coefficient of t2 in f(t1,t2) with t1=g1(z1,z2) and t2=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" )

# }