MFB: Multivariate Faa di Bruno's formula

In kStatistics: Unbiased Estimators for Cumulant Products and Faa Di Bruno's Formula

Multivariate Faa di Bruno's formula

Description

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.

Usage

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

Arguments

v	vector of integers, the subscript of the coefficient
n	integer, the number of inner functions g's

Details

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.

Value

string

the expression of Faa di Bruno's formula

Warning

The value of the first parameter is the same as the mkmSet function

Note

Called by the e_MFB function in the kStatistics package. The routine uses the mkmSet function in the same package.

Author(s)

Elvira Di Nardo [e-mail],
Giuseppe Guarino [e-mail]

References

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)

See Also

mkmSet, e_MFB

Examples

#----------------------------------------#
# 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" )

kStatistics documentation built on June 8, 2022, 5:05 p.m.