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Inferring the finest pattern of mutual independence from data

G. Marrelec, A. Giron

TL;DR

It is shown that μ (X) can be obtained as the intersection of all elements of Delta when the data are independent and identically (i.i.d.) realizations of a multivariate normal distribution.

Abstract

For a random variable $X$, we are interested in the blind extraction of its finest mutual independence pattern $μ( X )$. We introduce a specific kind of independence that we call dichotomic. If $Δ( X )$ stands for the set of all patterns of dichotomic independence that hold for $X$, we show that $μ( X )$ can be obtained as the intersection of all elements of $Δ( X )$. We then propose a method to estimate $Δ( X )$ when the data are independent and identically (i.i.d.) realizations of a multivariate normal distribution. If $\hatΔ ( X )$ is the estimated set of valid patterns of dichotomic independence, we estimate $μ( X )$ as the intersection of all patterns of $\hatΔ ( X )$. The method is tested on simulated data, showing its advantages and limits. We also consider an application to a toy example as well as to experimental data.

Inferring the finest pattern of mutual independence from data

TL;DR

It is shown that μ (X) can be obtained as the intersection of all elements of Delta when the data are independent and identically (i.i.d.) realizations of a multivariate normal distribution.

Abstract

For a random variable , we are interested in the blind extraction of its finest mutual independence pattern . We introduce a specific kind of independence that we call dichotomic. If stands for the set of all patterns of dichotomic independence that hold for , we show that can be obtained as the intersection of all elements of . We then propose a method to estimate when the data are independent and identically (i.i.d.) realizations of a multivariate normal distribution. If is the estimated set of valid patterns of dichotomic independence, we estimate as the intersection of all patterns of . The method is tested on simulated data, showing its advantages and limits. We also consider an application to a toy example as well as to experimental data.
Paper Structure (32 sections, 5 theorems, 34 equations, 4 figures, 4 tables)

This paper contains 32 sections, 5 theorems, 34 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Mutual independence, partitions and lattices
  3. Notations
  4. Mutual independence and partitions
  5. A multiplicity of patterns
  6. The sublattice of mutual independence patterns
  7. Dichotomic independence
  8. Definition
  9. Relating mutual and dichotomic independence
  10. Inference
  11. Testing for the existence of one pattern of dichotomic independence
  12. Multiple comparison
  13. Summary of inference process
  14. Positive versus negative cases
  15. Simulation study
  16. ...and 17 more sections

Key Result

Proposition 1

Assume that $a_1 \mid \dots \mid a_k$ is a partition of $N$ such that $X_{ a_1 }$, …, $X_{ a_ k }$ are mutually independent. Let $b_1, \dots, b_l$ be disjoint subsets of $N$ such that no two $b_i$'s intersect the same $a_j$ (if $b_i \cap a_j \neq \emptyset$, then $b_{i'} \cap a_j = \emptyset$ for al

Figures (4)

  • Figure 1: Two examples of $\Pi ( X )$. Top: representation of $\Pi ( X )$ corresponding to $X = \{ X_1, X_2, X_3, X_4 \}$ such that $X_{ \{ 1, 2 \} }$, $X_3$ and $X_4$ are mutually independent, i.e., $\mu ( X ) = 1 2 \mid 3 \mid 4$. $\Pi ( X )$ is superimposed on $\Omega ( [4] )$ (in gray). Bottom: representation of $\Pi ( X )$ corresponding to $X = \{ X_1, X_2, X_3, X_4, X_5, X_6 \}$ such that $X_1$, $X_{ \{ 2, 3 \} }$, $X_{ \{4, 5 \} }$ and $X_6$ are mutually independent, i.e., $\mu ( X ) = 1 \mid 2 3 \mid 4 5 \mid 6$. Only $\Pi ( X )$ is represented.
  • Figure 2: Simulation study. (a) AUC. For a significance level $\alpha = 0.1$: (b) Sensitivity; (c) Specificity. (d) Ratio of patterns of mutual independence correctly detected. (a), (b) and (c) are boxplots (median and $[25\%,75\%]$ frequency interval).
  • Figure 3: Simulation study. Value of AUC as a function of the absolute value of the average correlation within blocks.
  • Figure 4: Real data. Top left: Boxplot (median and $[25\%,75\%]$ frequency interval) of $X$ over the 300 stimulations. Top right: Sample correlation matrix of $X$. Bottom left: Sample correlation of $X_7$, …, $X_{10}$ with all other variables.

Theorems & Definitions (8)

  • Proposition 1
  • Definition 1
  • Definition 2
  • Corollary 1
  • Theorem 1
  • Definition 3
  • Proposition 2
  • Theorem 2