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Outlier Detection for Multi-Network Data

Pritam Dey, Zhengwu Zhang, David B. Dunson

arXiv:2205.06398v2stat.MEstat.AP

TL;DR

Poor neuroimaging reconstruction can yield outlying brain networks that influence analyses, while methods specifically designed to detect outliers across networks have been lacking. ODIN uses a hierarchical logistic model and subject-level influence measures to flag them. Simulations and a UK Biobank application show effective detection, and removing outliers can fundamentally change scientific conclusions.

  • Problem

    Reconstruction errors can create outlying brain networks that influence downstream analyses, and methods specifically designed for outlier detection in multi-network data have been lacking.

    Source: §1 · p. 2

  • Method

    ODIN fits a hierarchical logistic model and combines approximate leave-one-subject-out parameter influence with a measure of how extreme each subject’s model effects are.

    Source: §5 · p. 11

  • Results

    ODIN was fast and effective in detecting outliers in simulations and a large-dataset application, including 18,083 UK Biobank connectomes with moderate to extreme outliers.

    Source: §1 · p. 2, §5 · p. 12

  • Takeaways & Limitations

    Removing outliers can fundamentally alter scientific conclusions from brain network studies.

    Source: §4.3.3 · p. 11, §5 · p. 12

    The authors identify simultaneous outlier detection in structural and functional connectivity, and flexible modeling of weighted-edge distributions, as future extensions.

    Source: §5 · p. 12

Abstract

from arXiv · show

It has become routine in neuroscience studies to measure brain networks for different individuals using neuroimaging. These networks are typically expressed as adjacency matrices, with each cell containing a summary of connectivity between a pair of brain regions. There is an emerging statistical literature describing methods for the analysis of such multi-network data in which nodes are common across networks but the edges vary. However, there has been essentially no consideration of the important problem of outlier detection. In particular, for certain subjects, the neuroimaging data are so poor quality that the network cannot be reliably reconstructed. For such subjects, the resulting adjacency matrix may be mostly zero or exhibit a bizarre pattern not consistent with a functioning brain. These outlying networks may serve as influential points, contaminating subsequent statistical analyses. We propose a simple Outlier DetectIon for Networks (ODIN) method relying on an influence measure under a hierarchical generalized linear model for the adjacency matrices. An efficient computational algorithm is described, and ODIN is illustrated through simulations and an application to data from the UK Biobank. ODIN was successful in identifying moderate to extreme outliers. Removing such outliers can significantly change inferences in downstream applications.

1 Introduction

Neuroimaging reconstruction errors can produce outlying brain networks that influence downstream analyses, while existing informal checks and summary-based methods have important drawbacks. ODIN addresses the lack of methods designed for multi-network outlier detection by measuring each subject’s influence under a hierarchical logistic model.

1 Introduction

  • Reconstruction errors can be amplified through preprocessing, and gross errors may arise from subject movement or mistakes in complex reconstruction pipelines.The paper focuses on gross errors likely attributable to measurement, rather than small errors that are difficult to distinguish from biological variation.Source: §1 · p. 2
  • Outlying networks may distort connectome embeddings and obscure relationships between brain networks and human traits.Visual quality checks are subjective and time-consuming, while summary-statistic screening can miss outliers or remove non-outliers.Source: §1 · p. 2
  • ODIN fills a gap in multi-network outlier detection with a hierarchical logistic model and subject-level influence measures, and can incorporate covariates or extend to weighted networks.The model includes prior knowledge of brain anatomy and is designed to scale to datasets with tens of thousands of connectomes.Source: §1 · p. 2
  • ODIN is applied to 18,083 UK Biobank structural connectomes, with the aim of detecting outliers for data cleaning or down-weighting.The paper demonstrates outlier removal in simulations and the UK Biobank application.Source: §1 · pp. 2–3

2 Method

ODIN models binary brain networks with shared edge-specific baselines and subject-specific effects informed by ROI anatomy. It combines approximate leave-one-subject-out influence with distances between subject effects, then thresholds either measure to flag outliers.

2 Method

  • Each network is represented by the unique edges of a symmetric binary adjacency matrix over V regions of interest, with L = V(V − 1)/2 edges.The model uses ROI hemisphere and lobe locations to encode subject-specific differences in edge connection probabilities.Source: §2 · pp. 3–4
  • The hierarchical logistic model gives each edge a flexible population baseline and models subject-specific deviations using hemisphere- and lobe-based effects.Ridge regularization is applied to the baseline edge parameters.Source: §2 · pp. 3–4
  • A modified Majorization–Minimization algorithm exploits the objective’s block arrowhead Hessian, with each estimation iteration scaling linearly in N and L.The algorithm is chosen for scalability in both sample size and number of edges.Source: §2 · p. 4
  • To estimate influence without refitting the model for every omitted subject, ODIN takes one Newton–Raphson step from the full-sample estimate after each subject is dropped.The resulting approximate parameter change is measured by Euclidean distance from the full-sample estimate.Source: §2 · pp. 4–5
  • ODIN supplements parameter-change influence with the Mahalanobis distance of each subject’s estimated effects from their across-subject mean, since IM1 alone may miss unusual subject effects.A network is flagged if either influence measure is large; thresholds are chosen at the elbow of the measure’s quantile plot using the kneedle algorithm.Source: §2 · p. 5
  • Influence calculation is linear in sample size N and quadratic in edge count L, and simulations show good separation of outliers from non-outliers.Figure 2 reports linear runtime for three computations and quadratic runtime for influence calculation versus L.Source: §2 · p. 5, Figure 2 · p. 6

3 Simulation Study

The simulations assess ODIN’s computational cost and outlier-detection performance under two data-generating schemes.

3.1 Computational Complexity

  • Each fitting iteration scales linearly in sample size N and edge count L, while influence calculation scales linearly in N and quadratically in L.The authors note that quadratic complexity in L is not prohibitive for popular atlases with moderate ROI counts.Source: §3.1 · p. 6

3.2 Performance in detecting outliers

  • Detection performance was evaluated on synthetic datasets containing outliers and non-outliers generated in two ways, using sensitivity and specificity.One scheme used model (1); the other used TN-PCA, a distinct data-generation approach.Source: §3.2 · p. 6, §3.2.2 · p. 7

3.2.1 Simulation from model (1)

  • Nearly 100% sensitivity and specificity were reported, with IM1 beginning to separate outliers at 7% edge flips in model-(1) simulations.The simulations used 500 networks with 70 ROIs and flipped a fixed share of edges in 10% of networks to create outliers.Source: §3.2.1 · pp. 6–7, Table 1 · p. 7, Figure 3 · p. 7

3.2.2 Simulation using Tensor Network PCA

  • ODIN distinguished the 10% noise-perturbed subjects from non-outliers across tested noise levels in TN-PCA simulations.The simulation embedded 500 non-outlier UK Biobank networks, perturbed selected embeddings with Gaussian noise, and reconstructed binary networks.Source: Table 2 · p. 8, §3.2.2 · p. 8

4 Application to UK Biobank data

In the UK Biobank application, ODIN flagged outlying networks, and removing them substantially changed some brain-connectivity inferences about cognitive traits. The three case studies show changes ranging from a near-complete reversal of the numeric-memory pattern to subtler differences for fluid intelligence.

4 Application to UK Biobank data

  • ODIN detected 1,931 outliers, about 10.6% of the 18,083 UK Biobank structural connectomes analyzed.The networks were represented using the Desikan atlas.Source: §4 · p. 8

4.1 Exploratory view of outliers vs. non-outliers

  • The exploratory comparison found that many detected outliers had far fewer or more connected edges than non-outliers.Figure 4 distinguishes outliers in blue from non-outliers in orange.Source: §4.1 · p. 8, Figure 4 · p. 9

4.2 Impact of sample size on outlier detection

  • Across varied stratified subsample sizes, ODIN re-detected most full-sample outliers while flagging very few full-sample non-outliers.Each subsample contained 10% full-sample outliers and 90% non-outliers.Source: §4.2 · p. 8, Table 3 · p. 9

4.3 Case study: Impact of removing outliers on a subsequent analysis using TN-PCA

  • The case study compared TN-PCA connectivity patterns for the full sample and the sample with outliers removed, across three cognitive traits.Networks were reconstructed from TN-PCA components, and edge-connectivity differences between high and low scorers were examined for numeric memory, symbol digit substitution, and fluid intelligence.Source: §4.3 · pp. 9–10

4.3.1 Numeric memory

  • After outlier removal, numeric-memory connectivity differences between high and low scorers changed almost completely and strongly suggested frontal-lobe differences.The authors relate the frontal-lobe pattern to prior work on short-term memory [Jacobsen, 1936, Funahashi et al., 1993, Pribram et al., 1952].Source: §4.3.1 · p. 10, Figure 5 · p. 10

4.3.2 Symbol digit substitution

  • For symbol digit substitution, removing outliers significantly changed the inferred connectivity differences between the highest- and lowest-scoring participants.The comparison uses the 10% lowest and 10% highest scorers.Source: §4.3.2 · p. 10, Figure 6 · p. 11

4.3.3 Fluid intelligence

  • For fluid intelligence, outlier removal had a subtler impact on connectivity differences than in the other two trait examples.In both analyses, fluid intelligence appeared highly related to frontal-lobe connections, consistent with literature.Source: §4.3.3 · pp. 10–11, Figure 7 · p. 11

4.3 Case study: Impact of removing outliers on a subsequent analysis using TN-PCA

  • Across three cognitive-trait case studies, removing outliers could fundamentally alter scientific conclusions about brain-network relationships.The fluid-intelligence example showed a subtler impact than the numeric-memory and symbol digit substitution examples.Source: §4.3.3 · p. 11

5 Discussion

ODIN detects outlying binary brain networks using influence measures from a hierarchical logistic model. Simulations and a UK Biobank application show it can identify outliers, whose removal can change downstream scientific conclusions; extensions to weighted networks and joint structural-functional detection remain future work.

5 Discussion

  • For binary adjacency matrices, ODIN combines a hierarchical logistic model with influence from deleting each subject and extremity of subject-specific parameters, then thresholds the measures to classify outliers.The method is designed for detection rather than direct inference with the logistic model.Source: §5 · p. 11
  • ODIN was fast and effective at detecting outliers in simulations and a large-dataset application, and removing them could fundamentally change scientific conclusions.The application used UK Biobank data.Source: §5 · p. 12
  • ODIN could be extended to weighted adjacency matrices using a generalized linear model, such as negative binomial models for fiber counts or Gaussian models for continuous weights.The proposed extension would also make the method applicable to weighted functional connectivity.Source: §5 · p. 12
  • Joint outlier detection for structural and functional connectivity, and methods for weight distributions not well characterized by parametric GLMs, are identified as future work.Source: §5 · p. 12

Data Availability

The article’s underlying data are held by UK Biobank and may be shared on request, subject to its permission.

Data Availability

  • Underlying data are available on request to the corresponding author, with UK Biobank’s permission.Source: Data Availability · p. 12

SUPPLEMENT TO OUTLIER DETECTION FOR MULTI-NETWORK DATA

The supplement provides additional material for the paper and identifies its authors and affiliations.

SUPPLEMENT TO OUTLIER DETECTION FOR MULTI-NETWORK DATA

S1 Mathematical Proofs and Results

The supplement establishes matrix results used in estimation and in the leave-one-subject-out influence calculation, and gives the parameter-estimation algorithm.

S1 Mathematical Proofs and Results

  • A block-matrix lemma supplies inverse formulas using a Schur complement, and its proof relies on matrix multiplication and algebra.Source: S1 Mathematical Proofs and Results · pp. 14–15
  • The estimation derivation forms a joint parameter vector and its gradient and Hessian; the block structure allows updates that avoid repeatedly computing with large matrices.The supplement notes that the large-matrix computation in each iteration is the matrix multiplication for updating Z.Source: S1 Mathematical Proofs and Results · pp. 15–16
  • The MM algorithm takes binary adjacency matrices and ROI hemisphere and lobe memberships as inputs, initializes parameters, and iteratively updates them until its stopping condition is met.It returns estimates of Z and the subject-specific β_i parameters.Source: S1 Mathematical Proofs and Results · p. 16

S1.2 Proof of Proposition 1

S2 Thresholding influence measures for outlier detection

Most subjects have low influence-measure values, while a handful may have high values. A quantile plot’s sharp increase at the upper quantiles provides an elbow that can be used as a threshold.

  • Most subjects have very low influence-measure values, while only a handful may have high values.Source: S2 Thresholding influence measures for… · p. 17
  • The quantile plot remains relatively flat across most lower quantiles before increasing sharply at higher quantiles.Source: Figure S1 · p. 17
  • The plot’s elbow, where the increase begins, can serve as a threshold and is identified using the kneedle algorithm [Satopaa et al., 2011].Source: Figure S1 · p. 17

S3 Additional description of the regions of interest for the circular plots

The circular plots arrange dots around a circular region, with each dot representing a region of interest. Color and labels identify each ROI’s lobe and hemisphere and its Desikan atlas position.

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