kStatistics (version 2.0)

GCBellPol: Generalized Complete Bell Polynomial

Description

The function returns generalized complete Bell polynomials, that are coefficients of exp(y1 g1(z1,...,zm) + ... + yn gn(z1,...,zm)), where y1,...,yn are the variables. The input vector of integers identifies the subscripts of the polynomial coefficients.

Usage

GCBellPol(n = 0, m = 1, b = FALSE)

Arguments

n

vector of integers, the subscript of the polynomial, corresponding to the powers of z1, z2, ..., zm

m

integer, the number of z's variables

b

boolean, TRUE if g1=g2=...=gn=g

Value

string

expression of the generalized complete Bell polynomial

Warning

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

Details

The multivariate Faa di Bruno's formula output of the MFB function gives the coefficients of the multivariate exponential power series obtained from the composition of the multivariate exponential power series f(t1,...,tn) with ti=gi(z1,...,zm) for each i from 1 to n. Choose f(t1,...,tn)=exp(y1 t1 + ... + yn tn). The coefficients obtained through multivariate Faa di Bruno's formula are the generalized complete Bell polynomials. In particular, the function GCBellPol gives the expression of these polynomials when n=1 or n> 1 with g1=...=gn=g or n>1 with g1, ..., gn all different. Among the various applications, we point out the cumulant polynomial sequences. Cumulant polynomials allow us to compute moments and cumulants of multivariate Levy processes and subordinated multivariate Levy processes, multivariate compound Poisson processes with applications in photocounting, multivariable Sheffer polynomial sequences, see the referred papers. See the function e_GCBellPol for evaluating the generalized complete Bell polynomials by assigning numerical values to the variables and their coefficients.

References

Constantine G. M., Savits T. H. (1996) A Multivariate Faa Di Bruno Formula With Applications. Trans. Amer. Math. Soc. 348(2), 503--520.

E. Di Nardo E. (2016) On multivariable cumulant polynomial sequence with applications. Jour. Algebraic Statistics 7(1), 72-89. (download from http://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 http://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

mkmSet, MFB, e_GCBellPol

Examples

# NOT RUN {
# 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, 
# corresponding to the cumulant polinomial indexed by (2,1), 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 )


# Return the generalized complete Bell Polynomial for n=2, m=2 and g1=g2=g, 
# corresponding to the cumulant polinomial in y1+y2 indexed by (1,1), 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 )

# 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 )

# }