GCBellPol {kStatistics} | R Documentation |
Generalized Complete Bell Polynomial
Description
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.
Usage
GCBellPol(nv = c(), m = 1, b = FALSE)
Arguments
nv |
vector of integers, the subscript of the polynomial, corresponding to the powers of the product
among |
m |
integer, the number of |
b |
boolean, |
Details
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.
Value
string |
the expression of the polynomial |
Warning
The value of the first parameter is the same as the mkmSet
function.
Note
This function calls the MFB
function in the kStatistics
package.
Author(s)
Elvira Di Nardo elvira.dinardo@unito.it,
Giuseppe Guarino giuseppe.guarino@rete.basilicata.it
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 (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)
See Also
Examples
# 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 )