e_GCBellPol: Evaluation of Generalized Complete Bell Polynomials
In kStatistics: Unbiased Estimators for Cumulant Products and Faa Di Bruno's Formula
e_GCBellPol R Documentation
Evaluation of Generalized Complete Bell Polynomials
Description
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.
Usage
e_GCBellPol(pv = c(), pn = 0, pyc = c(), pc = c(), b = FALSE)
Arguments
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 pyc is
a string]
b
boolean, if TRUE the function prints the list of all the assignments
Details
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.
Value
string or numerical
the evaluation of the polynomial
Warning
The value of the first parameter is the same as the mkmSet function.
Note
Called by the GCBellPol function in the kStatistics package.
Author(s)
Elvira Di Nardo elvira.dinardo@unito.it,
Giuseppe Guarino giuseppe.guarino@rete.basilicata.it
References
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
mkmSet,
MFB,
GCBellPol
Examples
#-------------------------------------------------------------------------------#
# 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)
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| e_GCBellPol | R Documentation |
Evaluation of Generalized Complete Bell Polynomials
Description
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.
Usage
e_GCBellPol(pv = c(), pn = 0, pyc = c(), pc = c(), b = FALSE)
Arguments
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 |
Details
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.
Value
string or numerical |
the evaluation of the polynomial |
Warning
The value of the first parameter is the same as the mkmSet function.
Note
Called by the GCBellPol function in the kStatistics package.
Author(s)
Elvira Di Nardo elvira.dinardo@unito.it,
Giuseppe Guarino giuseppe.guarino@rete.basilicata.it
References
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
mkmSet,
MFB,
GCBellPol
Examples
#-------------------------------------------------------------------------------#
# 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)
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