kStatistics 1.0

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kStatistics

July 24, 2024

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

Introduction

This package is a Python implementation of the original R package kStatistics, written by E. Di Nardo and G. Guarino. This file aims at presenting the kStatistics functions and how to use them through examples. Further explanations on the algorithms are available here.

Contacts

[e-mail] (Maintainer) [e-mail]

Description

kStatistics is a package producing estimates of (joint) cumulants and (joint) cumulant products of a given dataset, using (multivariate) k-statistics and (multivariate) polykays, which are symmetric unbiased estimators. The procedures rely on a symbolic method arising from the classical umbral calculus and described in the following papers.

• (2008) Di Nardo E., Guarino G., Senato D. A unifying framework for k-statistics, polykays and their multivariate generalizations. Bernoulli 14, 440--468. here

• (2009) Di Nardo E., Guarino G., Senato D. A new method for fast computing unbiased estimators of cumulants. Statistics and Computing 19, 155--165. here

• (2011) Di Nardo E., Guarino G., Senato D. A new algorithm for computing the multivariate Faa di Bruno's formula. Appl. Math. Comp. 217, 6286--6295 here

In the package, a set of combinatorial tools are given useful in the construction of these estimations such as integer partitions, set partitions, multiset subdivisions or multi-index partitions, pairing and merging of multisets. In the package, there are also functions to recover univariate and multivariate cumulants from a sequence of univariate and multivariate moments (and vice-versa), using Faa di Bruno’s formula. Their evaluation is also provided when users specify in input numerical values for moments and/or cumulants. The function producing Faa di Bruno’s formula returns coefficients of exponential power series compositions such as $f[g(z)]$ with $f$ and $g$ both univariate, or $f[g(z_1,...,z_m)]$ with $f$ univariate and $g$ multivariate, or $f[g_1(z_1,...,z_m),...,g_n(z_1,...,z_m)]$ with $f$ and $g$ both multivariate. Let us recall that Faa di Bruno’s formula might also be employed to recover iterated (partial) derivatives of all these compositions. Lastly, using Faa di Bruno’s formula, some special families of polynomials are also generated, such as Bell polynomials, generalized complete Bell polynomials, partition polynomials and generalized partition polynomials.

For further applications of these functions, refer to the following paper:

• (2022) Di Nardo E., Guarino G. kStatistics: Unbiased Estimates of Joint Cumulant Products from the Multivariate Faà Di Bruno’s Formula. The R Journal 14(2) 208-228.

Presentation of the functions and their use

countP

Description

The function computes the multiplicity of a multi-index partition. Note that a multi-index partition corresponds to a subdivision of a multiset having the input multi-index as multiplicities.

Usage

countP( v=[1] )

Argument

v : list

Value

int, the multiplicity of the given item

Examples

countP([2,1])

Returns 3 which is the multiplicity of [1,2], partition of the integer 3, or of [[a],[a,a]], subdivision of the multiset [a,a,a].

3

countP([4,2])

Return 15 which is the multiplicity of [4,2], partition of the integer 6, or of [[a,a,a,a],[a,a]], subdivision of the multiset [a,a,a,a,a,a].

15

countP( [[2,0], [1,0], [1,0], [0,1], [0,2]] )

Return 18 which is the multiplicity of [[2,0], [1,0], [1,0], [0,1], [0,2]], partition of the multi-index (4,3), or of [[b],[b,b],[a],[a],[a,a]], subdivision of the multiset [a,a,a,a,b,b,b].

cum2mom

Description

The function computes a simple or a multivariate cumulant in terms of simple or multivariate moments.

Usage

cum2mom(n = 1)

Argument

n : integer or vector of integers.

Value

str, the expression of the cumulant in terms of moments

Examples

cum2mom(5)

Returns the simple cumulant k[5] in terms of the simple moments m[1],..., m[5].

24m[1]^5 - 60m[1]^3m[2] + 30m[1]m[2]^2 + 20m[1]^2m[3] - 10m[2]m[3] - 5m[1]m[4] + m[5]

cum2mom([3,1])

Returns the multivariate cumulant k[3,1] in terms of the multivariate moments m[i,j] for i=0,1,2,3 and j=0,1.

- 6m[0,1]m[1,0]^3 + 6m[0,1]m[1,0]m[2,0] - m[0,1]m[3,0] + 6m[1,0]^2m[1,1] - 3m[1,0]m[2,1] - 3m[1,1]m[2,0] + m[3,1]

eBellPol

Description

The function generates a complete or a partial exponential Bell polynomial.

Usage

eBellPol(n = 1, m = 0)

Argument

n : integer, the degree of the polynomial

m : integer, the fixed degree of each monomial in the polynomial

Value

str, the expression of the exponential Bell polynomial

Examples

eBellPol(5)

Returns the complete exponential Bell Polynomial for n=5, that is

(y1**5) + 10(y1**3)(y2) + 15(y1)(y2**2) + 10(y1**2)(y3) + 10(y2)(y3) + 5(y1)(y4) + (y5)

eBellPol(5,3)

Returns the partial exponential Bell Polynomial for n=5 and m=3, that is

15(y1)(y2**2) + 10(y1**2)(y3)

e_eBellPol

Description

The function evaluates a complete or a partial exponential Bell polynomial (output of the eBellPol function) when its variables are substituted with numerical values

Usage

e_eBellPol(n=1, m=0, v=None)

Argument

n : integer, the degree of the polynomial

m : integer, the fixed degree of each monomial in the polynomial

v : vector, the numerical values in place of the variables of the polynomial

Value

int, the value assumed by the polynomial.

Warnings

By default, the function returns the Stirling numbers of second kind.

Examples

e_eBellPol(5,3)

Returns S(5,3) = 25 (where S=Stirling number of second kind).

25

Or (same output)

e_eBellPol(5, 3, [1, 1, 1, 1, 1])

25

e_eBellPol(5)

Returns B5=52 (where B5 is the 5-th Bell number).

52

Or (same output)

e_eBellPol(5,0)
52

Or (same output)

e_eBellPol(5, 0, [1, 1, 1, 1, 1])
52

e_eBellPol(5,3,[1,-1,2,-6,24])

Return s(5,3) = 35 (where s=Stirling number of first kind).

35

e_GCBellPol

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=[], pn=0, pyc=[], pc=[], b=False)

Argument

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

Value

string or numerical, the evaluation of the polynomial

Warnings

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

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( [2],1 )

when g[1]=3 and g[2]=4

e_GCBellPol( [2], 1, [], [3,4] )

9(y^2) + 4(y)

OR (same output)

e_GCBellPol( [2], 1, "g[1]=3,g[2]=4" )

9(y^2) + 4(y)

Check the assignments setting the boolean parameter equals to True, that is g[1]=3 and g[2]=4

e_GCBellPol( [2], 1, [], [3,4], True )

y=, g[1]=3, g[2]=4

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( [2], 1, [7], [3,4] )

469

OR (same output)

e_GCBellPol( [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( [2], 1, [7], [3,4], True )

y=7, g[1]=3, g[2]=4

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( [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)

import numpy as np

e_GCBellPol([2,1], 1, [], np.arange(1,6))

4(y^3) + 16.0(y^2) + 5(y)

Or (same output)

e_GCBellPol([2,1], 1, [], [1,2,3,4,5])

OR (same output)

e_GCBellPol( [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

import numpy as np

e_GCBellPol( [2,1], 1, [], np.arange(1,6), True )

y=, g[0,1]=1, g[1,0]=2, g[1,1]=3, g[2,0]=4, g[2,1]=5

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

GCBellPol( [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

import numpy as np

e_GCBellPol( [2,1], 1, [7], np.arange(1,6) )

2191

Or (same output)

e_GCBellPol( [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" )

2191

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

import numpy as np

e_GCBellPol( [2,1], 1, [7], np.arange(1,6) )

y=7, g[0,1]=1, g[1,0]=2, g[1,1]=3, g[2,0]=4, g[2,1]=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([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)

import numpy as np

e_GCBellPol( [1,1], 2, [], np.arange(1,7))

13.0(y1)(y2) + 2(y1^2) + 3(y1) + 20(y2^2) + 6(y2)

O (same output)

e_GCBellPol([1,1], 2, [], [1,2,3,4,5,6])

13.0(y1)(y2) + 2(y1^2) + 3(y1) + 20(y2^2) + 6(y2)

Or (same output)

e_GCBellPol( [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" )

13.0(y1)(y2) + 2(y1^2) + 3(y1) + 20(y2^2) + 6(y2)

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

import numpy as np

e_GCBellPol( [1,1], 2, [], np.arange(1,7), True )

y1=, y2=, g1[0,1]=1, g1[1,0]=2, g1[1,1]=3, g2[0,1]=4, g2[1,0]=5, g2[1,1]=6

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([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

import numpy as np

e_GCBellPol( [1,1], 2, [7,8], np.arange(1,7))

2175

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([1,1], 2, cVal)

2175

To recover which coefficients and variables are involved in the generalized complete Bell polynomial, run the

e_GCBellPol

function without any assignment.

e_GCBellPol([1, 1], 2)

The error message prints which coefficients and variables are involved, that is

ValueError: 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

ValueError: 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([1,1], 2, [1,2], [1,2,3,4,5,6], True)

3. trough 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 necessar

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([1,1], 2, cVal)

-2872.0

e_MFB

Description

The function evaluates the Faa di Bruno’s formula, output of the MFB function, when the coefficients of the exponential formal power series f and g1,...,gn in the composition f[g1(),...,gn()] are substituted with numerical values.

Usage

e_MFB(pv=[], pn=0, pf=[], pg=[], b=False)

Argument

pv : vector of integers, the subscript of Faa di Bruno’s formula

pn : integer, the number of the inner formal power series "g"

pf : vector, the numerical values in place of the coefficients of the outer formal power series "f" or the string with the direct assignments in place of the coefficients of both "f" and "g"

pg : vector, the numerical values in place of the coefficients of the inner formal power series "g" [Optional if pf is a string]

b : boolean

Value

numerical, the evaluation of Faa di Bruno’s formula

Warnings

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

Examples

The numerical value of f[1]g[1,1] + f[2]g[1,0]g[0,1], that is the coefficient of z1z2 in $f(g1(z1,z2),g2(z1,z2)))$, output of

MFB(c(1,1),1)

when f[1] = 5 and f[2] = 10, g[0,1]=3, g[1,0]=6, g[1,1]=9

e_MFB([1,1], 1, [5,10], [3,6,9])

225

Same as the previous example, with a string of assignments as third input parameter

e_MFB([1,1], 1, "f[1]=5, f[2]=10, g[0,1]=3, g[1,0]=6, g[1,1]=9")

225

Use the boolean parameter to verify the assignments to the coefficients of "f" and "g", that is f[1]=5, f[2]=10, g[0,1]=3, g[1,0]=6, g[1,1]=9

e_MFB([1,1], 1, [5,10], [3,6,9], True)

f[1]=5, f[2]=10, g[0,1]=3, g[1,0]=6, g[1,1]=9

To recover which coefficients are involved, run the function without any assignment. The error message recalls which coefficients are necessary, that is

e_MFB([1,1], 1)

ValueError: The third parameter must contain the 2 values of f: f[1] f[2]. The fourth parameter must contain the 3 values of g: g[0,1] g[1,0] g[1,1]

To assign correctly the values to the coefficients of "f" and "g" when the functions become more complex:

1. run

e_MFB([1,1], 2)

and get the errors with the indication of the involved coefficients of "f" and "g", that is

ValueError: The third parameter must contain the 5 values of f: f[0,1] f[0,2] f[1,0] f[2,0] f[1,1]. The fourth parameter must contain the 6 values of g: g1[0,1] g2[1,0] g1[1,0] g2[0,1] g1[1,1] g2[1,1]

2. initialize f[0,1] f[0,2] f[1,0] f[1,1] f[2,0] with - for example - the first 5 integer numbers and do the same for g1[0,1] g1[1,0] g1[1,1] g2[0,1] g2[1,0] g2[1,1], that is

import numpy as np

e_MFB([1,1], 2, np.arange(1,6), np.arange(1,7), True)

f[0,1]=1, f[0,2]=2, f[1,0]=3, f[2,0]=4, f[1,1]=5, g1[0,1]=1, g2[1,0]=2, g1[1,0]=3, g2[0,1]=4, g1[1,1]=5, g2[1,1]=6

3. trough the boolean value True, recover the string f[0,1]=1, f[0,2]=2, f[1,0]=3, f[1,1]=4, f[2,0]=5, 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_MFB([1,1], 1, " ... ")

5. change the assignments if necessary

cfVal = "f[0,1]=2, f[0,2]=5, f[1,0]=13, f[1,1]=-4, f[2,0]=0"


cgVal = "g1[0,1]=-2.1, g1[1,0]=2, g1[1,1]=3.1, g2[0,1]=5, g2[1,0]=0, g2[1,1]=6.1"


cVal = cfVal + "," + cgVal


result = e_MFB((1, 1), 2, cVal)
print(result)

12.500000000000004

ff

Description

The function computes the descending (falling) factorial of a positive integer n with respect to a positive integer k less or equal to n.

Usage

ff( n=1, k )

Argument

n : integer

k : integer

Value

int, the descending factorial

Examples

ff(6,3)

120

GCBellPol

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=[], m=1, b=False)

Argument

nv : vector of integers, the subscript of the polynomial, corresponding to the powers of the product among z1, z2, ..., zm

m : integer, the number of z’s variables

b : boolean, TRUE if the inner formal power series "g" are all equal

Value

str, the expression of the polynomial

Warnings

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

Examples

Returns the generalized complete Bell Polynomial for n=1, m=1 and g1=g, that is (y^2)g[1]^2 + (y)g[2]

GCBellPol( [2], 1 )

(y**2)g[1]^2 + (y)g[2]

Returns 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( [2,1], 1 )

(y**3)g[0,1]g[1,0]^2 + (y**2)g[0,1]g[2,0] + 2(y**2)g[1,0]g[1,1] + (y)g[2,1]

Returns 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( [1,1], 2, True )

2(y1)(y2)g[0,1]g[1,0] + (y1**2)g[0,1]g[1,0] + (y1)g[1,1] + (y2**2)g[0,1]g[1,0] + (y2)g[1,1]

Returns 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( [1,1], 2 )

(y1)(y2)g1[0,1]g2[1,0] + (y1)(y2)g1[1,0]g2[0,1] + (y1**2)g1[0,1]g1[1,0] + (y1)g1[1,1] + (y2**2)g2[0,1]g2[1,0] + (y2)g2[1,1]

gpPart

Description

The function returns a general partition polynomial.

Usage

gpPart(n = 0)

Argument

n : integer

Value

str, the expression of the polynomial

Warnings

The value of the first parameter is the same as the MFB function in the univariate with univariate composition.

Examples

Return the general partition polynomial G[a1,a2; y1,y2], that is a2(y1^2) + a1(y2)

gpPart(2)

a2(y1**2) + a1(y2)

Return the general partition polynomial G[a1,a2,a3,a4,a5; y1,y2,y3,y4,y5], that is a5(y1^5) + 10a4(y1^3)(y2) + 15a3(y1)(y2^2) + 10a3(y1^2)(y3) + 10a2(y2)(y3) + 5a2(y1)(y4)

• a1(y5)

gpPart(5)

a5(y1**5) + 10a4(y1**3)(y2) + 15a3(y1)(y2**2) + 10a3(y1**2)(y3) + 10a2(y2)(y3) + 5a2(y1)(y4) + a1(y5)

intPart

Description

The function generates all possible (unique) decomposition of a positive integer n in the sum of positive integers less or equal to n.

Usage

intPart(n=0 ,vOutput = FALSE)

Argument

n : integer

vOutput : optional boolean parameter, if equal to TRUE the function produces a compact output that is easy to read.

Value

list, all the partitions of n

Examples

Return the partition of the integer 3, that is [1,1,1],[1,2],[3]

intPart(3)

[[1, 1, 1], [1, 2], [3]]

Return the partition of the integer 4, that is [1,1,1,1],[1,1,2],[1,3],[2,2],[4]

intPart(4)

[[1, 1, 1, 1], [1, 1, 2], [2, 2], [1, 3], [4]]

Or (same output)

intPart(4, False)

[[1, 1, 1, 1], [1, 1, 2], [2, 2], [1, 3], [4]]

Return the same output as the previous example but in a compact expression

intPart(4, TRUE)

[1, 1, 1, 1]
[1, 1, 2]
[2, 2]
[1, 3]
[4]
None

list2m

Description

The function returns the multiset representation of a vector or a list, in increasing order

Usage

list2m(v=[0])

Argument

v : single vector or list of vectors

Value

multiset, the list of multisets

Examples

Returns the list of multisets [[1],3], [[2],1] from the input vector (1,2,1,1)

list2m([1,2,1,1])

[[[1], 3], [[2], 1]]

Returns the list of multisets [[1,2],2], [[2,3],1] from the input [[1,2], [2,3], [1,2]]

list2m([[1,2], [2,3], [1,2]])

[[[1, 2], 2], [[2, 3], 1]]

list2Set

Description

Given a list, the function deletes the instances of an element in the list, leaving the order inalterated.

Usage

list2Set(v=[0])

Argument

v : single vector or list of vectors

Value

set, the sequence of distinct elements

Examples

Returns the vector [1,2,3,5,6]

list2Set([1,2,3,1,2,5,6])

[1, 2, 3, 5, 6]

Returns the list [[1,2], [10,11], [7,8]]

list2Set([[1,2], [1,2], [10,11], [1,2], [7,8]])

[[1, 2], [10, 11], [7, 8]]

m2Set

Description

The function returns the vectors (only counted once) of all the multi-index partitions output of the mkmSet function. These vectors correspond also to the blocks of the subdivisions of the multiset having the given multi-index as multeplicites.

Usage

m2Set(v=[0])

Argument

v : sequence of type [[e1,e2,...], m1], [[f1,f2,...], m2],... with m1, m2,... multiplicities

Value

set, the sequence with distinct elements

Examples

M1 = mkmSet([2,1]) M1 is [ [ [[0, 1], [1, 0], [1, 0]], 1 ], [ [[0, 1], [2, 0]], 1 ], [ [[1, 0], [1, 1]], 2 ], [ [[2, 1]], 1 ] ] To print all the partitions of the multi-index (2,1) run

mkmSet([2,1], True)

[( 0 1 )( 1 0 )( 1 0 ), 1 ]
[( 0 1 )( 2 0 ), 1 ]
[( 1 0 )( 1 1 ), 2 ]
[( 2 1 ), 1 ]

Then m2Set(M1) returns the following set: [[0,1],[1,0],[2,0],[1,1],[2,1]]

m2Set( M1 )

[[0, 1], [1, 0], [2, 0], [1, 1], [2, 1]]

mCoeff

Description

Given a list containing vectors paired with numbers, the function returns the number paired with the vector matching the one passed in input.

Usage

mCoeff(v=None, L=None)

Argument

v : vector to be searched in the list

L : two-dimensional list: in the first there is a vector and in the second a number

Value

float, the number paired with the input vector

Examples

Run

mkmSet([3])

to get the list

L1 = [[[1,1,1],1], [[1,2],3], [[3],1]]

L1 = mkmSet([3])

Returns the number 3, which is the number paired with [1,2] in L1

mCoeff( [1,2], L1)

3

MFB

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=[], n=0)

Argument

v : vector of integers, the subscript of the coefficient

n : integer, the number of inner functions g’s

Value

str, the expression of Faa di Bruno’s formula

Warnings

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

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( [2], 1 )

f[2]g[1]^2 + f[1]g[2]

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( [3], 1 )

f[3]g[1]^3 + 3f[2]g[1]g[2] + f[1]g[3]

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( [1,1], 1 )

f[2]g[0,1]g[1,0] + f[1]g[1,1]

The coefficient of z1^2 z2 in f[g(z1,z2)]

MFB( [2,1], 1 )

f[3]g[0,1]g[1,0]^2 + f[2]g[0,1]g[2,0] + 2f[2]g[1,0]g[1,1] + f[1]g[2,1]

The coefficient of z1 z2 z3 in f[g(z1,z2,z3)]

MFB( [1,1,1], 1 )

f[3]g[0,0,1]g[0,1,0]g[1,0,0] + f[2]g[0,0,1]g[1,1,0] + f[2]g[0,1,0]g[1,0,1] + f[2]g[0,1,1]g[1,0,0] + f[1]g[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( [1], 2 )

f[1,0]g1[1] + f[0,1]g2[1]

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( [1,1], 2 )

f[1,1]g1[0,1]g2[1,0] + f[1,1]g1[1,0]g2[0,1] + f[2,0]g1[0,1]g1[1,0] + f[1,0]g1[1,1] + f[0,2]g2[0,1]g2[1,0] + f[0,1]g2[1,1]

The coefficient of z1 in f[g1(z1,z2),g2(z1,z2),g3(z1,z2)]

MFB( [1,0], 3 )

f[0,1,0]g2[1,0] + f[1,0,0]g1[1,0] + f[0,0,1]g3[1,0]

The coefficient of z1 z2 in f[g1(z1,z2),g2(z1,z2),g3(z1,z2)]

MFB( [1,1], 3 )

f[0,1,1]g2[0,1]g3[1,0] + f[1,0,1]g1[0,1]g3[1,0] + f[1,1,0]g1[0,1]g2[1,0] + f[0,1,1]g2[1,0]g3[0,1] + f[1,0,1]g1[1,0]g3[0,1] + f[1,1,0]g1[1,0]g2[0,1] + f[0,2,0]g2[0,1]g2[1,0] + f[0,1,0]g2[1,1] + f[2,0,0]g1[0,1]g1[1,0] + f[1,0,0]g1[1,1] + f[0,0,2]g3[0,1]g3[1,0] + f[0,0,1]g3[1,1]

The coefficient of z1^2 z2 in f[g1(z1,z2),g2(z1,z2)]

MFB( [2,1], 2 )

f[1,2]g1[0,1]g2[1,0]^2 + f[1,1]g1[0,1]g2[2,0] + f[2,1]g1[1,0]^2g2[0,1] + f[1,1]g1[2,0]g2[0,1] + 2f[1,2]g1[1,0]g2[0,1]g2[1,0] + 2f[1,1]g1[1,0]g2[1,1] + 2f[2,1]g1[0,1]g1[1,0]g2[1,0] + 2f[1,1]g1[1,1]g2[1,0] + f[3,0]g1[0,1]g1[1,0]^2 + f[2,0]g1[0,1]g1[2,0] + 2f[2,0]g1[1,0]g1[1,1] + f[1,0]g1[2,1] + f[0,3]g2[0,1]g2[1,0]^2 + f[0,2]g2[0,1]g2[2,0] + 2f[0,2]g2[1,0]g2[1,1] + f[0,1]g2[2,1]

The coefficient of z1^2 z2 in f[g1(z1,z2),g2(z1,z2),g3(z1,z2)]

MFB( [2,1], 3 )

Not displayed for readibility reasons

The coefficient of z1 z2 z3 in f[g1(z1,z2,z3),g2(z1,z2,z3),g3(z1,z2,z3)]

MFB( [1,1,1], 3 )

MFB2Set

Description

Secondary function useful for manipulating the result of the MFB function.

Usage

MFB2Set(sExpr="")

Argument

sExpr : the output of the MFB function

Value

set, a set

Examples

Run

MFB([3], 1)

to generate f[3]g[1]^3 + 3f[2]g[1]g[2] + f[1]g[3] Convert the output of the MFB(c(3),1) into a vector using

import numpy as np

np.array(MFB2Set(MFB([3], 1)))

The result is the following:

[['1' '1' 'f' '3' '1']
 ['1' '1' 'g' '1' '3']
 ['2' '3' 'f' '2' '1']
 ['2' '1' 'g' '1' '1']
 ['2' '1' 'g' '2' '1']
 ['3' '1' 'f' '1' '1']
 ['3' '1' 'g' '3' '1']]

mkmSet

Description

The function returns all the partitions of a multi-index, that is a vector of non-negative integers. Note that these partitions correspond to the subdivisions of a multiset having the input multi-index as multiplicities.

Usage

mkmSet(vPar=None, vOutput=False)

Argument

vPar : vector of non-negative integers

vOutput : optional boolean variable. If equal to TRUE, the function produces a compact output that is easy to read.

Value

list, two-dimensional list: in the first there is the partition, while in the second there is its multiplicity

Examples

Returns [ [[1,1,1],1], [[1,2],3], [[3],1] ]

3 is the multiplicity of a multiset with 3 elements all equal

mkmSet([3])

[[[1, 1, 1], 1], [[1, 2], 3], [[3], 1]]

Returns [[[[0, 1], [1, 0], [1, 0]], 1], [[[0, 1], [2, 0]], 1], [[[1, 0], [1, 1]], 2], [[[2, 1]], 1]]

(2,1) is the multiplicity of a multiset with 2 equal elements and a third distinct element

mkmSet([2,1])

[[[[0, 1], [1, 0], [1, 0]], 1], [[[0, 1], [2, 0]], 1], [[[1, 0], [1, 1]], 2], [[[2, 1]], 1]]

Or (same output)

mkmSet([2,1], False)

[[[[0, 1], [1, 0], [1, 0]], 1], [[[0, 1], [2, 0]], 1], [[[1, 0], [1, 1]], 2], [[[2, 1]], 1]]

Returns the same output of the previous example but in a compact form.

mkmSet([2,1], True)

[(0 1) (1 0) (1 0), 1 ]
[(0 1) (2 0), 1 ]
[(1 0) (1 1), 2 ]
[(2 1), 1 ]


None

mkT

Description

Given a multi-index, that is a vector of non-negative integers and a positive integer n, the function returns all the lists $(v_1,...,v_n)$ of non-negative integer vectors, with the same lenght of the multiindex and such that $v=v_1+...+v_n$.

Usage

mkT(v=[], n=0, vOutput=False)

Argument

v : vector of integers

n : integer, number of addends

vOutput : optional boolean variable. If equal to TRUE, the function produces a compact output that is easy to read.

Value

list, the list of n vectors $(v_1,...,v_n)$

Warnings

The vector in the first variable must be not empty and must contain all non-negative integers. The second parameter must be a positive integer

Examples

Returns the scompositions of the vector (1,1) in 2 vectors of 2 non-negative integers such that their sum is (1,1), that is

([1,1],[0,0]) - ([0,0],[1,1]) - ([1,0],[0,1]) - ([0,1],[1,0])

mkT([1,1], 2)

[[[0, 1], [1, 0]], [[1, 0], [0, 1]], [[1, 1], [0, 0]], [[0, 0], [1, 1]]]

Or (same output)

mkT(|1,1], 2, False)

[[[0, 1], [1, 0]], [[1, 0], [0, 1]], [[1, 1], [0, 0]], [[0, 0], [1, 1]]]

Returns the scompositions of the vector (1,0,1) in 2 vectors of 3 non-negative integers such that their sum gives (1,0,1), that is

([1,0,1],[0,0,0]) - ([0,0,0],[1,0,1]) - ([1,0,0],[0,0,1]) - ([0,0,1],[1,0,0]).

Note that the second value in each resulting vector is always zero.

mkT([1,0,1], 2)

[[[0, 0, 1], [1, 0, 0]], [[1, 0, 0], [0, 0, 1]], [[1, 0, 1], [0, 0, 0]], [[0, 0, 0], [1, 0, 1]]]

Or (same output)

mkT([1,0,1], 2, False)

[[[0, 0, 1], [1, 0, 0]], [[1, 0, 0], [0, 0, 1]], [[1, 0, 1], [0, 0, 0]], [[0, 0, 0], [1, 0, 1]]]

Returns the same output of the previous example but in a compact form.

mkT([1,0,1], 2, True)

[( 0 0 1 )( 1 0 0 )]
[( 1 0 0 )( 0 0 1 )]
[( 1 0 1 )( 0 0 0 )]
[( 0 0 0 )( 1 0 1 )]
None

Returns the scompositions of the vector (1,1,1) in 3 vectors of 3 non-negative integers such that their sum gives (1,1,1). The result is given in a compact form.

for m in mkT([1, 1, 1], 3):
    for n in m:
        print(n, end=" - ")
    print()

mom2cum

Description

The function compute a simple or a multivariate moment in terms of simple or multivariate cumulants.

Usage

mom2cum(n=1)

Argument

n : integer or vector of integers

Value

str : the expression of the moment in terms of cumulants

Warnings

The value of the first parameter is the same as the MFB function in the univariate with univariate case composition and in the univariate with multivariate case composition.

Examples

Returns the simple moment m[5] in terms of the simple cumulants k[1],...,k[5].

mom2cum(5)

k[1]^5 + 10k[1]^3k[2] + 15k[1]k[2]^2 + 10k[1]^2k[3] + 10k[2]k[3] + 5k[1]k[4] + k[5]

Returns the multivariate moment m[3,1] in terms of the multivariate cumulants k[i,j] for i=0,1,2,3 and j=0,1.

mom2cum([3,1])

k[0,1]k[1,0]^3 + 3k[0,1]k[1,0]k[2,0] + k[0,1]k[3,0] + 3k[1,0]^2k[1,1] + 3k[1,0]k[2,1] + 3k[1,1]k[2,0] + k[3,1]

mpCart

Description

Given two lists with elements of the same type, the function returns a new list whose elements are the joining of the two original lists, except for the last elements, which are multiplied.

Usage

mpCart(m1=None, m2=None)

Argument

M1 : list of vectors

M2 : list of vectors

Value

list, the list with the joined input lists

Examples

A = [[[ [1], [2] ], -1],[[ [3] ], 1]]

where

-1 is the multiplicative factor of [[1],[2]]

1 is the multiplicative factor of [[3]]

B = [[[ [5] ], 7]]

where 7 is the multiplicative factor of [[5]]

Return [[[1],[2],[5]], -7] , [[[3],[5]], 7]

mpCart(A,B)

[[[[1], [2], [5]], -7], [[[3], [5]], 7]]

A = [[[ [1, 0], [1, 0] ], -1], [[ [2, 0] ], 1]]

where

• 1 is the multiplicative factor of [[1,0],[1,0]]

1 is the multiplicative factor of [[2,0]]

B = [[[ [1, 0] ], 1]]

where 1 is the multiplicative factor of [[1,0]]

Return [[[1,0],[1,0],[1,0]], -1], [[[2,0],[1,0]],1]

mpCart(A,B)

[[[[1, 0], [1, 0], [1, 0]], -1], [[[2, 0], [1, 0]], 1]]

nKM

Description

Given a multivariate data sample, the function returns an estimate of a joint (or multivariate) cumulant with a fixed order

Usage

nKM(v=None, V=None)

Argument

v : vector of integers

V : vector of a multivariate data sample

Value

float, the value of the multivariate k-statistics

Warnings

The size of each data vector must be equal to the length of the vector passed trough the first input variable.

Examples

Data assignment

data1 = [
    [5.31, 11.16], [3.26, 3.26], [2.35, 2.35], [8.32, 14.34], [13.48, 49.45],
    [6.25, 15.05], [7.01, 7.01], [8.52, 8.52], [0.45, 0.45], [12.08, 12.08], [19.39, 10.42]
]

Returns an estimate of the joint cumulant k[2,1]

nKM([2,1], data1)

-23.737902999999278

Data assignment

data2 = [
    [5.31, 11.16, 4.23], [3.26, 3.26, 4.10], [2.35, 2.35, 2.27],
    [4.31, 10.16, 6.45], [3.1, 2.3, 3.2], [3.20, 2.31, 7.3]
]

Returns an estimate of the joint cumulant k[2,2,2]

nKM([2,2,2], data2)

678.1045339247212

Data assignment

data3 = [
    [5.31, 11.16, 4.23, 4.22], [3.26, 3.26, 4.10, 4.9], [2.35, 2.35, 2.27, 2.26],
    [4.31, 10.16, 6.45, 6.44], [3.1, 2.3, 3.2, 3.1], [3.20, 2.31, 7.3, 7.2]
]

Returns an estimate of the joint cumulant k[2,1,1,1]

nKM([2,1,1,1], data3)

32.053913213909254

nKS

Description

Given a data sample, the function returns an estimate of a cumulant with a fixed order.

Usage

nKS(v=None, V=None)

Argument

v : integer or one-dimensional vector

V : vector of a data sample

Value

float, the value of the k-statistics

Examples

data = [
    16.34, 10.76, 11.84, 13.55, 15.85, 18.20, 7.51, 10.22, 12.52, 14.68, 16.08,
    19.43, 8.12, 11.20, 12.95, 14.77, 16.83, 19.80, 8.55, 11.58, 12.10, 15.02,
    16.83, 16.98, 19.92, 9.47, 11.68, 13.41, 15.35, 19.11
]

nKS(7, data)

Returns an estimate of the cumulant of order 7

1322.8183753490448

nKS(1, data)

Returns an estimate of the cumulant of order 1, that is the mean

14.021666666666672

nKS(2, data)

Returns an estimate of the cumulant of order 2, that is the variance

12.650069540229737

nKS(3, data) / (nKS(2, data) ** 0.5) ** 3

Returns an estimate of the skewness

-0.03216229420531556

nKS(4, data) / nKS(2, data) ** 2 + 3

Returns an estimate of the kurtosis

2.114707796531899

nPerm

Description

The function returns all possible different permutations of objects in a list or in a vector

Usage

nPerm(L=[])

Argument

L : list

Value

list, all the permutations of L

Examples

permutations of 1,2,3

nPerm( [1,2,3] )

[[3, 1, 2], [1, 3, 2], [1, 2, 3], [3, 2, 1], [2, 3, 1], [2, 1, 3]]

permutations of 1,2,1 (two elements are equal)

nPerm( [1,2,1] )

[[1, 1, 2], [1, 2, 1], [2, 1, 1]]

permutations of the words "Alice", "Bob","Jack"

nPerm( ["Alice", "Bob","Jack"] )

[['Jack', 'Alice', 'Bob'], ['Alice', 'Jack', 'Bob'], ['Alice', 'Bob', 'Jack'], ['Jack', 'Bob', 'Alice'], ['Bob', 'Jack', 'Alice'], ['Bob', 'Alice', 'Jack']]

permutations of the vectors [0,1], [2,3], [7,3]

nPerm( [[0,1], [2,3], [7,3]] )

[[[7, 3], [0, 1], [2, 3]], [[0, 1], [7, 3], [2, 3]], [[0, 1], [2, 3], [7, 3]], [[7, 3], [2, 3], [0, 1]], [[2, 3], [7, 3], [0, 1]], [[2, 3], [0, 1], [7, 3]]]

nPM

Description

Given a multivariate data sample, the function returns an estimate of a product of joint cumulants with fixed orders.

Usage

nPM(v=None, V=None)

Argument

v : list of integer vectors

V : vector of a multivariate data sample

Value

float, the estimate of the multivariate polykay

Examples

Data asignment

data1 = [
    [5.31, 11.16], [3.26, 3.26], [2.35, 2.35], [8.32, 14.34], [13.48, 49.45],
    [6.25, 15.05], [7.01, 7.01], [8.52, 8.52], [0.45, 0.45], [12.08, 12.08], [19.39, 10.42]
]

Returns an estimate of the product k[2,1]*k[1,0], where k[2,1] and k[1,0] are the cross-correlation of order (2,1) and the marginal mean of the population distribution respectively

nPM([[2, 1], [1, 0]], data1)

48.43242806555327

Data asignment

data2 = [
    [5.31, 11.16, 4.23], [3.26, 3.26, 4.10], [2.35, 2.35, 2.27],
    [4.31, 10.16, 6.45], [3.1, 2.3, 3.2], [3.20, 2.31, 7.3]
]

Returns an estimate of the product k[2,0,1]*k[1,1,0], where k[2,0,1] and k[1,1,0] are joint cumulants of the population distribution

nPM([[2, 0, 1], [1, 1, 0]], data2)

-0.9858162208170143

nPolyk

Description

The master function executes one of the functions to compute simple k-statistics (nKS), multivariate k-statistics (nKM), simple polykays (nPS) or multivariate polykays (nPM).

Usage

nPolyk(L=None, data=None, bhelp=None)

Argument

L : vector of orders

data : vector of a (univariate or multivariate) sample data

bhelp : boolean

Value

float, the estimate of the (joint) cumulant or of the (joint) cumulant product

Examples

Data assignment

data1 = [
    16.34, 10.76, 11.84, 13.55, 15.85, 18.20, 7.51, 10.22, 12.52, 14.68, 16.08,
    19.43, 8.12, 11.20, 12.95, 14.77, 16.83, 19.80, 8.55, 11.58, 12.10, 15.02,
    16.83, 16.98, 19.92, 9.47, 11.68, 13.41, 15.35, 19.11
]

Displays "KS: -1.4470595032807978" which indicates the type of subfunction (nKS) called by the master function nPolyk and gives the estimate of the third cumulant

nPolyk([3], data1, True)

KS:
-1.4470595032807978

Displays " -1.4470595032807978" (without the indication of the employed subfunction)

nPolyk([3], data1, False)

-1.4470595032807978

Displays "PS: 177.42329372003013" which indicates the type of subfunction (nPS) called by the master function nPolyk and gives the estimate of the product between the variance k[2] and the mean k[1]

nPolyk([[2], [1]], data1, True)

177.42329372003013

Data assignment

data2 = [
    [5.31, 11.16], [3.26, 3.26], [2.35, 2.35], [8.32, 14.34], [13.48, 49.45],
    [6.25, 15.05], [7.01, 7.01], [8.52, 8.52], [0.45, 0.45], [12.08, 12.08], [19.39, 10.42]
]

Displays "KM: -23.737902999999278" which indicates the type of subfunction (nKM) called by the master function nPolyk and gives the estimate of k[2,1]

nPolyk([2, 1], data2, True)

KM:
-23.737902999999278

Displays "PM: 48.43242806555327" which indicates the type of subfunction (nPM) called by the master function nPolyk and gives the estimate of k[2,1]*k[1,0]

nPolyk([[2, 1], [1, 0]], data2, True)

PM:
48.43242806555327

nPS

Description

Given a data sample, the function returns an estimate of a product of cumulants with fixed orders.

Usage

nPS(v=None, V=None)

Argument

v : vector of integers

V : vector of a data sample

Value

float, the estimate of the polykay

Examples

Data assignment

data = [
    16.34, 10.76, 11.84, 13.55, 15.85, 18.20, 7.51, 10.22, 12.52, 14.68, 16.08,
    19.43, 8.12, 11.20, 12.95, 14.77, 16.83, 19.80, 8.55, 11.58, 12.10, 15.02,
    16.83, 16.98, 19.92, 9.47, 11.68, 13.41, 15.35, 19.11
]

Returns an estimate of the product k[2]*k[1], where k[1] and k[2] are the mean and the variance of the population distribution respectively

nPS([2, 1], data)

177.42329372003013

nStirling2

Description

The function computes the Stirling number of the second kind.

Usage

nStirling2(n, k)

Argument

n : integer

k : integer less or equal to n

Value

int, the Stirling number of the second kind

Examples

Returns the number of ways to split a set of 6 objects into 2 nonempty subsets

nStirling2(6,2)

31

oBellPol

Description

The function generates a complete or a partial ordinary Bell polynomial.

Usage

oBellPol(n=1, m=0)

Argument

n : integer, the degree of the polynomial

m : integer, the fixed degree of each monomial in the polynomial

Value

str, the expression of the polynomial

Warning

The value of the first parameter is the same as the MFB function in the univariate with univariate composition.

Examples

Returns the complete ordinary Bell Polynomial for n=5, that is

(y1^5) + 20(y1^3)(y2) + 30(y1)(y2^2) + 60(y1^2)(y3) + 120(y2)(y3) + 120(y1)(y4) + 120(y5)

oBellPol(5)

1/120*( 120(y1**5) + 480(y1**3)(y2) + 360(y1)(y2**2) + 360(y1**2)(y3) + 240(y2)(y3) + 240(y1)(y4) + 120.0(y5) )

Or (same output)

oBellPol(5,0)

1/120*( 120(y1**5) + 480(y1**3)(y2) + 360(y1)(y2**2) + 360(y1**2)(y3) + 240(y2)(y3) + 240(y1)(y4) + 120.0(y5) )

Returns the partial ordinary Bell polynomial for n=5 and m=3, that is

30(y1)(y2^2) + 60(y1^2)(y3)

oBellPol(5,3)

1/120*( 360(y1)(y2**2) + 360(y1**2)(y3) )

pCart

Description

The function returns the cartesian product between vectors.

Usage

pCart( L )

Argument

L : list of lists

Value

list, the list with the cartesian product

Examples

A = [1, 2]
B = [3, 4, 5]

Returns the cartesian product [[1,3],[1,4],[1,5],[2,3],[2,4],[2,5]]

pCart([A, B])

[[1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5]]

L1 = [[1, 1], [2]]
L2 = [[5, 5], [7]]

Return the cartesian product [[1,1],[5,5]], [[1,1],[7]], [[2],[5,5]], [[2],[7]] and assign the result to L3

L3 = pCart([L1, L2])
print(L3)

[[[1, 1], [5, 5]], [[1, 1], [7]], [[2], [5, 5]], [[2], [7]]]

Returns the cartesian product between L3 and [7].

The result is [[1,1],[5,5],[7]], [[1,1],[7],[7]], [[2],[5,5],[7]], [[2],[7],[7]]

result = pCart([L3, [7]])
print(result)

[[[1, 1], [5, 5], [7]], [[1, 1], [7], [7]], [[2], [5, 5], [7]], [[2], [7], [7]]]

powS

Description

The function returns the value of the power sum symmetric polynomial, with fixed degrees and in one or more sets of variables, when the variables are substituted with the input lists of numerical values.

Usage

powS(vn=None, lvd=None)

Arguments

vn : vector of integers (the powers of the indeterminates)

lvd : list of numerical values in place of the variables

Value

int, the value of the polynomial

Examples

Returns 1^3 + 2^3 + 3^3 = 36

powS([3], [[1], [2], [3]])

36

Returns (1^3 * 4^2) + (2^3 * 5^2) + (3^3 * 6^2) = 1188

powS([3, 2], [[1, 4], [2, 5], [3, 6]])

1188

pPart

Description

The function generates the partition polynomial of degree n, whose coefficients are the number of partitions of n into k parts for k from 1 to n

Usage

pPart(n=0)

Argument

n : integer, the degree of the polynomial

Value

str, the expression of the polynomial

Warning

The value of the first parameter is the same as the MFB function in the univariate with univariate case composition.

Examples

Returns the partition polynomial F[5]

pPart(5)

y^5 + y^4 + 2y^3 + 2y^2 + y

pPoly

Description

The function returns the product between polynomials without constant term.

Usage

pPoly(L=None)

Argument

L : lists of the coefficients of the polynomials

Value

list, the coefficients of the polynomial output of the product

Examples

[1,-3] are the coefficients of (x-3x^2), [2] is the coefficient of 2x

Returns [0, 2,-6], coefficients of 2x^2-6x^3 =(x-3x^2)*(2x)

pPoly([ [1, -3], [2] ])

[0, 2, -6]

[0,3,-2] are the coefficients of 3x^2-2x^3, [0,2,-1] are the coefficients of (2x^2-x^3)

Return [0,0,0,6,-7,2], coefficients of 6x^4-7x^5+2x^6=(3x^2-2x^3)*(2x^2-x^3)

pPoly([ [0, 3, -2], [0, 2, -1] ])

[0, 0, 0, 6, -7, 2]

Set2expr

Description

The function converts a set into a string.

Usage

Set2expr(v=None)

Argument

v : set

Value

str, the string

Examples

To print 6f[3]^2g[2]^5 run

Set2expr( [["1","2","f","3","2"],["1","3","g","2","5"]])

6.0f[3]^2g[2]^5

Evaluate

Description

Calculate the value of a given cumulant (from known moments) or moment (from known cumulants).

Usage

evaluate(s=None, S=None)

Argument

s : a string containing the values of the moments or cumulants

S : the string to evaluate

Value

float, the numerical value of the cumulant/moment

Examples

Run

cum2mom(5)

to get the cumulant of order 5, that is

24m[1]^5 - 60m[1]^3m[2] + 30m[1]m[2]^2 + 20m[1]^2m[3] - 10m[2]m[3] - 5m[1]m[4] + m[5]

Then, to calculate the value of the above cumulant given the moments up to order 5

m[1]=1, m[2]=2, m[3]=3, m[4]=4, m[5]=5.

run

evaluate('m[1]=1, m[2]=2, m[3]=3, m[4]=4, m[5]=5', '24m[1]^5 - 60m[1]^3m[2] + 30m[1]m[2]^2 + 20m[1]^2m[3] - 10m[2]m[3] - 5m[1]m[4] + m[5]')

The output is

9.0

To get the same output, run

evaluate('m[1]=1, m[2]=2, m[3]=3, m[4]=4, m[5]=5', cum2mom(5))

9.0

When a value is missing, an error occurs:

print(evaluate('m[1]=1, m[2]=2, m[3]=3, m[5]=5', cum2mom(5)))

ValueError: Undefined variables in expression: ['m[4]']

The function is also usable for mom2cum. For example run

mom2cum(4)

to get

k[1]^4 + 6k[1]^2k[2] + 3k[2]^2 + 4k[1]k[3] + k[4]

To evaluate the above expression with

k[1] = 2, k[2] = 4, k[3] = 6, k[4] = 8

run

evaluate('k[1] = 2, k[2] = 4, k[3] = 6, k[4] = 8', 'k[1]^4 + 6k[1]^2k[2] + 3k[2]^2 + 4k[1]k[3] + k[4]')

216.0

Or (same output)

evaluate('k[1] = 2, k[2] = 4, k[3] = 6, k[4] = 8', mom2cum(4))

216.0

and if a value is missing

evaluate('k[1] = 2, k[2] = 6, k[4] = 8', 'k[1]^4 + 6k[1]^2k[2] + 3k[2]^2 + 4k[1]k[3] + k[4]')

ValueError: Undefined variables in expression: ['k[3]']

Finally, if one tries to evaluate cum2mom giving in input cumulant values instead of moment values, the function produces the following error:

evaluate('k[1] = 2, k[2] = 4, k[3] = 6, k[4] = 8', cum2mom(5))

ValueError: Trying to evaluate cum2mom with values of cumulants

or the other way round

evaluate('m[1]=1, m[2]=2, m[3]=4, m[4]=3', mom2cum(4))

ValueError: Trying to evaluate mom2cum with values of moments