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Efficient computation of complementary set partitions, with applications to an extension and estimation of generalized cumulants

Elvira Di Nardo, Giuseppe Guarino

TL;DR

This work addresses the computational bottleneck in enumerating complementary set partitions, which are central to generalized cumulants. It introduces a novel two-blocks partition method that reduces complexity and enables non-symbolic implementations (e.g., in $R$), outperforming graph-based and symbolic approaches. The authors extend generalized cumulants to repeated variables via multisets and multi-index partitions, deriving a closed-form expression that expresses these cumulants as linear combinations of multivariate cumulants with clear combinatorial coefficients. They further develop efficient estimators for generalized cumulants using polykays and dummy-variable techniques, and demonstrate practical performance through computational results and an R implementation, broadening the applicability of generalized cumulant analysis in multivariate settings.

Abstract

This paper develops new combinatorial approaches to analyze and compute special set partitions, called complementary set partitions, which are fundamental in the study of generalized cumulants. Moving away from traditional graph-based and algebraic methods, a simple and fast algorithm is proposed to list complementary set partitions based on two-block partitions, making the computation more accessible and implementable also in non-symbolic programming languages like R. Computational comparisons in Maple demonstrate the efficiency of the proposal. Additionally the notion of generalized cumulant is extended using multiset subdivisions and multi-index partitions to include scenarios with repeated variables and to address more sophisticated dependence structures. A formula is provided that expresses generalized multivariate cumulants as linear combinations of multivariate cumulants, weighted by coefficients that admit a natural combinatorial interpretation. Finally, the introduction of dummy variables and specialized multi-index partitions enables an efficient procedure for estimating generalized multivariate cumulants with a substantial reduction in data power sums involved.

Efficient computation of complementary set partitions, with applications to an extension and estimation of generalized cumulants

TL;DR

This work addresses the computational bottleneck in enumerating complementary set partitions, which are central to generalized cumulants. It introduces a novel two-blocks partition method that reduces complexity and enables non-symbolic implementations (e.g., in ), outperforming graph-based and symbolic approaches. The authors extend generalized cumulants to repeated variables via multisets and multi-index partitions, deriving a closed-form expression that expresses these cumulants as linear combinations of multivariate cumulants with clear combinatorial coefficients. They further develop efficient estimators for generalized cumulants using polykays and dummy-variable techniques, and demonstrate practical performance through computational results and an R implementation, broadening the applicability of generalized cumulant analysis in multivariate settings.

Abstract

This paper develops new combinatorial approaches to analyze and compute special set partitions, called complementary set partitions, which are fundamental in the study of generalized cumulants. Moving away from traditional graph-based and algebraic methods, a simple and fast algorithm is proposed to list complementary set partitions based on two-block partitions, making the computation more accessible and implementable also in non-symbolic programming languages like R. Computational comparisons in Maple demonstrate the efficiency of the proposal. Additionally the notion of generalized cumulant is extended using multiset subdivisions and multi-index partitions to include scenarios with repeated variables and to address more sophisticated dependence structures. A formula is provided that expresses generalized multivariate cumulants as linear combinations of multivariate cumulants, weighted by coefficients that admit a natural combinatorial interpretation. Finally, the introduction of dummy variables and specialized multi-index partitions enables an efficient procedure for estimating generalized multivariate cumulants with a substantial reduction in data power sums involved.
Paper Structure (21 sections, 12 theorems, 49 equations, 1 table)

This paper contains 21 sections, 12 theorems, 49 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background on complementary set partitions
  3. Swapping property.
  4. Intersection matrices.
  5. Generalized cumulants.
  6. Methods to list complementary set partitions
  7. Connected graphs.
  8. Stafford's algorithm.
  9. The new two-blocks partition method
  10. Generalized multivariate cumulants
  11. Complementary $\vec{{\bm 1}}_n$-partitions
  12. The algorithm NullSpace: csp_NS.
  13. Generalized cumulant.
  14. Labelling rules and transformations of $\vec{{\bm 1}}_n$-partitions
  15. Applications
  16. ...and 6 more sections

Key Result

Theorem 3.1

mccullagh1984tensor$\pi, \tilde{\pi} \in \Pi_n$ are complementary if and only if ${\mathcal{G}}(\pi) \oplus {\mathcal{G}}(\tilde{\pi})$ is connected.

Theorems & Definitions (50)

  • Definition 2.1
  • Example 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 3.1
  • Theorem 3.1
  • Example 3.1
  • Example 3.2
  • Theorem 3.2
  • proof
  • ...and 40 more