# PPA: Principal Parcellation Analysis for Brain Connectomes and Multiple
Traits

Rongjie Liu  Meng Li  David B. Dunson  Department of Statistics, Florida State
University, Tallahassee, FL, USA  Department of Statistics, Rice University,
Houston, TX, USA  Department of Statistical Science, Duke University, Durham,
NC, USA

###### Abstract

Our understanding of the structure of the brain and its relationships with
human traits is largely determined by how we represent the structural
connectome. Standard practice divides the brain into regions of interest
(ROIs) and represents the connectome as an adjacency matrix having cells
measuring connectivity between pairs of ROIs. Statistical analyses are then
heavily driven by the (largely arbitrary) choice of ROIs. In this article, we
propose a novel tractography-based representation of brain connectomes, which
clusters fiber endpoints to define a data adaptive parcellation targeted to
explain variation among individuals and predict human traits. This
representation leads to Principal Parcellation Analysis (PPA), representing
individual brain connectomes by compositional vectors building on a basis
system of fiber bundles that captures the connectivity at the population
level. PPA reduces subjectivity and facilitates statistical analyses. We
illustrate the proposed approach through applications to data from the Human
Connectome Project (HCP) and show that PPA connectomes improve power in
predicting human traits over state-of-the-art methods based on classical
connectomes, while dramatically improving parsimony and maintaining
interpretability. Our PPA package is publicly available on GitHub, and can be
implemented routinely for diffusion image data.

###### keywords:

Brain networks; Brain Parcellation; Clustering; Human Connectome Project;
Structural Connectomics

††journal: arXiv

##  1 Introduction

Image-based brain parcellation is a fundamental tool for understanding brain
organization and function, in which the brain is divided into multiple non-
overlapping and interacting regions (Eickhoff et al., 2018). Modern
neuroimaging techniques enable the collection of whole-brain magnetic
resonance (MR) scans in large samples of individuals. Several large studies,
such as the Human Connectome Project (Van Essen et al., 2013), have collected
such data along with human behavioral and cognitive information, leading to a
surge of interest in relating structural brain networks with various human
traits (Glasser et al., 2016; Roine et al., 2019; Lin et al., 2020; Zhang et
al., 2019; Apkarian et al., 2020). A key issue in the understanding of brain
connectivity, as noted by Park and Friston (2013), is the way brain
connectivity is measured, represented, and modeled. In structural
connectomics, the brain connectome is typically defined as corresponding to
the collection of white matter fiber tracts. Classically, the intricate
spatial locations of all the fibers in the brain are summarized via an
adjacency matrix, with the cells containing summaries of connections between
pairs of regions of interest (ROIs) (O’Donnell et al., 2013). However, as we
argue and demonstrate in this paper, there are other ways to summarize the
brain connectome that have distinct advantages (Sporns et al., 2005; Toga et
al., 2012). A method does not need to focus on an adjacency matrix
representation for it to be an approach for analysis of brain connectomes.
This article contributes to the growing literature of structural connectomics
by proposing a new representation of brain connectivity.

Existing literature on brain parcellation and structural brain networks
typically represents structural brain connectivity via anatomical connectivity
that is estimated by tractography on diffusion-weighted images (Behrens et
al., 2003). The associated anatomical parcellation analysis (APA)—one of the
most popular approaches in connectivity-based parcellation analysis (Yao et
al., 2015)—obtains the connectivity map by calculating connectivities between
all pairs of regions in a predetermined anatomical parcellation scheme. Based
on the selected atlas, we may build the connectivity matrix by counting the
number of fibers passing between each pair of ROIs after fiber tracking. This
connectivity matrix can then be used as a matrix-valued predictor in
statistical analyses studying relationships with human traits (Zhang et al.,
2018; Wang et al., 2019; Lin et al., 2020; de Reus and Van den Heuvel, 2013).

However, such APA analyses require a particular anatomical ROI definition
(Eickhoff et al., 2015), and many different schemes are available involving
different numbers and locations of ROIs, such as the automated anatomical
labeling (AAL), automatic nonlinear imaging matching and anatomical labeling
(ANIMAL) atlases, and many others (Tzourio-Mazoyer et al., 2002; He et al.,
2007, 2008; Yao et al., 2015; Wang et al., 2019). Choosing which scheme to use
in practice is challenging. Several studies have reported impacts of different
atlases on brain networks (Zalesky et al., 2010; Messé, 2020), and evidence
suggests that not only the connectivity maps but also the inferences relating
connectomes to human traits are strongly sensitive to the parcellation
strategy.

An additional major issue is that APA leads to connectome representations
corresponding to high-dimensional adjacency matrices. While there is a growing
literature focused on statistical analysis of such replicated graph or network
data (Schiffler et al., 2017; Bansal et al., 2018), such methods are under-
developed and poorly understood relative to the rich literature on methods for
high-dimensional vector-valued predictors, and computationally efficient
methods that scale well in practice are lacking. For these reasons, it is
common to simply vectorize the upper-triangular part of the adjacency matrix
prior to statistical analysis. However, this fails to exploit the network
structure in performing dimension reduction (Wang et al., 2017; O’Malley and
Marsden, 2008; Hochberg et al., 2007), and can suffer from substantial loss of
efficiency and accuracy (Wang et al., 2019).

In this article, we propose a novel tractography-based representation of the
connectome, which clusters fiber endpoints to define a data adaptive
parcellation targeted to explain variation among individuals and predict human
traits. This leads to Principal Parcellation Analysis (PPA), representing
individual brain connectomes by compositional vectors building on a basis
system of fiber bundles that captures the connectivity at the population
level. Unlike APA connectomes, PPA connectomes do not rely on any anatomical
atlas or choosing ROIs a priori, thus reducing subjectivity and leading to a
substantially different representation of the connectome. This new
representation of structural brain connectomes facilitates statistical
analyses using well-established statistical methods designed for vector
data.The PPA representation provides an alternative to the current standard
ROI-based adjacency matrix representation in analyses studying how structural
connectomes vary across individuals, both randomly and in relation to
individual traits, and can accomplish these same inference goals at a fraction
of the cost for implementing APA-based counterparts. We illustrate the
proposed approach through applications to data from the Human Connectome
Project (HCP) and show that PPA connectomes, when combined with standard high-
dimensional regression methods, improve power in predicting human traits over
state-of-the-art methods based on classical connectomes, while dramatically
improving parsimony and maintaining interpretability. Our proposed method is
general for any data that consist of a collection of fibers. Our PPA package
is publicly available on GitHub, and can be implemented routinely for
diffusion image based parcellation analysis.

The rest of the paper is structured as follows. Section 2 introduces the
proposed parcellation approach and PPA formulation. In Section 3, we compare
PPA to state-of-the-art APA-based methods using HCP data, and focus on
prediction, visualization, and interpretability. Section 4 contains a
discussion.

##  2 Methods

###  2$\cdot$1 The PPA framework

Suppose that we observe structural MRI, diffusion MRI, and human traits from
$n$ individuals. The PPA pipeline, as illustrated in Figure 1, consists of
three modules: (i) reconstruction of fibers; (ii) representation of fibers and
unsupervised clustering; and (iii) high-dimensional supervised learning
adaptive to human traits. Each module of PPA encompasses a variety of choices,
equipping PPA with easy extensibility. We first describe each module of PPA
using initial default settings, followed by a discussion on extensions.

Figure. 1: Pipeline of tractography-based Principal Parcellation Analysis.

Let $\mathcal{F}=\\{f_{ik},k=1,\ldots,m_{i},i=1,\ldots,n\\}$, where $f_{ik}$
is the $k$-th fiber in the $i$-th individual’s brain, and $m_{i}$ is the total
number of fibers in the $i$-th subject. In addition, let $y_{i}(s)$ denote the
$s$th ‘trait’ of the $i$th individual with
${\mbox{\boldmath${y}$}}(s)=(y_{1}(s),\ldots,y_{n}(s))^{T}$ for
$s=1,\ldots,S$; traits can range from demographic characteristics, alcohol and
drug exposures, to scores on cognitive, psychological and behavioral
assessments.

In Module (i), we reconstruct fibers using the recently proposed TractoFlow
method (Theaud et al., 2020); alternative fiber tracking algorithms can be
used instead without changing the subsequent steps in the PPA pipeline.
TractoFlow takes raw diffusion weighted images as the input, consists of 14
steps for the diffusion weighted image (DWI) processing and 8 steps for the T1
weighted image processing, and outputs classical diffusion imaging measures.
The outlier fiber tracts are detected and removed using the method proposed in
Garyfallidis et al. (2012). Module (i) is also a key step in estimating APA
connectomes.

In Module (ii), we formulate connectomes at the population level through basis
networks in the form of fiber bundles that represent groups/clusters of
streamlines, and represent individual connectomes via compositional vectors.
Let $\\{{\mbox{\boldmath${a}$}}_{ik}\\}_{k=1}^{m_{i}}$ and
$\\{{\mbox{\boldmath${b}$}}_{ik}\\}_{k=1}^{m_{i}}$ be the 3D coordinates of
two endpoints for each fiber from the $i$-th subject. Let $Z$ be a $6\times m$
matrix stacking all
$({\mbox{\boldmath${a}$}}_{ik}^{T},{\mbox{\boldmath${b}$}}_{ik}^{T})^{T}$ as
columns for $k=1,\ldots,m_{i}$ and $i=1,\ldots,n$, where $m=\sum_{i}m_{i}$ is
the total number of fibers from all subjects. We perform a cluster analysis at
the fiber level using the matrix $Z$, outputting a collection of partitions of
$\mathcal{F}$, denoted by
$\mathcal{A}_{K}=\\{{A}_{K}^{(1)},\ldots,{A}_{K}^{(K)}\\}$, where $K$ is the
number of clusters and each set ${A}_{K}^{(k)}$ can be interpreted as a fiber
bundle. The enormous number of fibers presents substantial computational
challenges in clustering; for example, traditional $K$-Means does not scale
well and requires large memory, leading to prohibitive computational cost. As
such, we adopt mini-batch $K$-Means (Sculley, 2010), which reduces the memory
use and converges to the optimal value orders of magnitude faster than the
full-batch $K$-Means. For a fiber and cluster center, we define their distance
as the minimum of their Euclidean distances considering two orderings of fiber
endpoints, accounting for fiber bi-directionality similarly to fiber flipping
(Garyfallidis et al., 2012). The number of clusters $K$ is an important
hyperparameter. Our experiments have provided guidance on good values of $K$
in practice. To automate the choice of $K$, we can rely on cross validation as
illustrated in Figure 6. Alternatively, we can decrease sensitivity to $K$ by
choosing multiple values in a multi-scale representation of the brain
connectome.

For a given $K$, an individual’s connectome can be represented by the
proportions of the individual’s fibers belonging to each of the inferred
population-level fiber bundles. In particular, the $i$th individual’s
connectome is represented via the $K$-dimensional compositional vector
${\mbox{\boldmath${\omega}$}}_{i}=\\{\omega_{i1},\ldots,\omega_{iK}\\}$, with
$\omega_{ik}$ the proportion of fibers belonging to the $k$th fiber bundle
$A_{K}^{(k)}$, for $k=1,\ldots,K$. The connectome data for all $n$ subjects is
then contained in the matrix
${\mbox{\boldmath${\omega}$}}=({\mbox{\boldmath${\omega}$}}_{1}^{T},\cdots,{\mbox{\boldmath${\omega}$}}_{n}^{T})^{T}$.
This provides a much simpler representation than the adjacency matrix-based
APA approach. Note that the framework does not require all clusters $K$ to be
present in all subjects since for certain individuals the number of fibers
belonging to a particular cluster can be zero.

In Module (iii), we relate the connectome ${\mbox{\boldmath${\omega}$}}_{i}$
to traits $y_{i}(s)$. For simplicity in interpretation, we initially focus on
trait-specific linear regression models:

${y}_{i}(s)=\beta_{0}(s)+\sum_{k=1}^{K-1}\omega_{ik}\beta_{k}(s)+\epsilon_{i}(s),$
(1)

where $\beta_{0}(s)$ is a trait specific intercept, which can be expanded to
include non-connectome covariates, and $\beta_{k}(s)$ ($k=1,\ldots,K$) is a
regression coefficient characterizing the relationship between the density of
connections in the $k$th fiber bundle and the $s$th trait. For a sufficiently
flexible specification, one may choose $K$ to be large in which case many of
the $\beta_{k}(s)$ coefficients are expected to be zero or close to zero.
Standard sparse learning methods can be used to estimate the coefficients
while learning the sparsity pattern. This yields a set of estimated non-zero
coefficients $\\{\hat{\beta}_{k}:k\in\mathcal{K}(s)\\}$, where
$\mathcal{K}(s)=\\{k_{1},\ldots,k_{m(s)}\\}\subset\\{1,\ldots,K-1\\}$ collects
the indices of the $m(s)$ fiber bundles having non-zero coefficients. We refer
to these fiber bundles as “active” for the $s$th trait. Active bundles impact
the response $y_{i}(s)$ via equation (1), while inactive bundles have no
impact on the response.

In our numerical experiments, we use LASSO (Tibshirani, 1996), one of the most
popular high-dimensional regression methods, as a representative example.
LASSO has been very widely studied and relatively efficient algorithms are
readily available. However, applying LASSO to the vectorized upper triangle
portion of APA connectome adjacency matrices produces less reliable estimation
and has worse predictive performance than an identical analysis using PPA
instead of APA connectomes. This is consistent with previous results
motivating complex statistical methods that take into account the graph
structure of the APA connectomes (Wang et al., 2019).

###  2$\cdot$2 Extensions of PPA

In Module (i), other fiber tracking algorithms alternatives to TractoFlow can
be considered, such as Euler Delta Crossings (EuDX) in Garyfallidis et al.
(2012) and Sparse Fascicle Model (SFM) in Rokem et al. (2015). We will compare
various fiber tracking algorithms in our analyses of HCP data. In Module (ii),
other clustering or factorization methods, including spectral clustering and
non-negative matrix factorization (NMF), can also be adopted. In addition to
the endpoints, the length and shape of the fibers may contain useful
information (Zhang et al., 2018), which can be incorporated in clustering
analyses. Module (iii) can be modified building on the rich literature on
high-dimensional supervised learning methods. Instead of LASSO, other sparse
shrinkage methods, such as elastic net (Zou and Hastie, 2005) and smoothly
clipped absolute deviation (SCAD) penalty (Fan and Li, 2001), can be used
without complication.

##  3 Human Connectome Project Data Analyses

In this section we use the HCP dataset to compare PPA-based methods with
state-of-the-art APA-based approaches using various human traits, demonstrate
how to choose $K$ in a data-driven manner, assess the robustness of PPA with
respect to the fiber tracking algorithms and regularization strategies, and
illustrate the interpretability of PPA.

###  3$\cdot$1 HCP Data Description

Data collection and sharing for this project was provided by the MGH-USC Human
Connectome Project (HCP; Principal Investigators: Bruce Rosen, M.D., Ph.D.,
Arthur W. Toga, Ph.D., Van J. Weeden, MD). HCP funding was provided by the
National Institute of Dental and Craniofacial Research (NIDCR), the National
Institute of Mental Health (NIMH), and the National Institute of Neurological
Disorders and Stroke (NINDS). HCP data are disseminated by the Laboratory of
Neuro Imaging at the University of Southern California.

We use the same set of 1065 HCP subjects as in Wang et al. (2019), including
dMRI data along with human traits, downloaded from HCP 1200 Subjects Data
Release111https://www.humanconnectome.org/study/hcp-young-
adult/document/1200-subjects-data-release. Details about the dMRI data
acquisition and preprocessing can be found in Van Essen et al. (2012);
Sotiropoulos et al. (2013). For the human traits data, seven different scores
were selected: receptive vocabulary, oral reading, list sorting, flanker,
picture sequence memory, card sort, and processing speed. These scores can be
used to study human cognition. Note that although we use task-based scores in
this section, the proposed methods are broadly applicable for any measurement
reflecting human traits. All the scores are age-adjusted, and their details
can be found on the HCP
website222https://wiki.humanconnectome.org/display/PublicData/. A brief
description of each trait is also included in the Appendix for easy reference.

###  3$\cdot$2 Analysis using PPA and APA

PPA and APA provide distinct representations of human brain connectomes.
Performance in studying relationships between connectomes and traits depends
on the downstream analysis methods after the connectomes are obtained. As
such, we chose state-of-the-art methods developed under APA connectomes, and
adopted one of the most standard analysis methods, LASSO, under PPA
connectomes. Such comparisons give APA an advantage. We also implemented LASSO
for the vectorized APA connectomes.

We implemented PPA using the default choices in Section 2. In particular, we
used TractoFlow for fiber tracking, which depends on two main technologies:
Nextflow and Singularity (Kurtzer et al., 2017; Di Tommaso et al., 2017;
Garyfallidis et al., 2014; Tournier et al., 2019; Avants et al., 2009;
Jenkinson et al., 2012). We obtained around 2.8 billion fibers for the HCP
subjects. The fiber tracking results for two randomly selected subjects are
displayed in Figure 2. We clustered the fibers using mini-batch $K$-Means
(Sculley, 2010). We set the batch size to 1000 and varied the number of
clusters $K$ from 10 to 500 across $\\{10,50,100,200,300,400,500\\}$. For each
choice of $K$, we conducted analyses relating the PPA connectome to the trait
of interest. We can then assess, based on cross validation, which choice of
$K$ is best for that particular trait. For each $K$, we obtained $K$ fiber
bundles $A_{K}^{(k)}$ for $k=1,\ldots,K$, leading to PPA connectome
${\mbox{\boldmath${\omega}$}}_{i}$ for each individual $i=1,\ldots,n.$

Figures 3 and 4 show examples of the inferred fiber clusters ($K=10$) with
each color denoting one cluster. The same clusters, and corresponding colors,
are used for the different subjects, and some heterogeneity is apparent across
subjects. The numbers of fibers in each cluster are shown in Figure 5 for
these two subjects. The profile of these counts across clusters is similar for
the two subjects, but subject 2 has a considerably greater proportion of
fibers in cluster 10. A close inspection of the inferred clusters indicates
that the anatomical locations of fiber bundles produced by PPA often
correspond to known fascicles in the literature (Gupta et al. (2017); Shin et
al. (2019); Friederici and Gierhan (2013)). Taking the 10 clusters in Figure 4
as an example, cluster 2 and cluster 10 cover the inferior longitudinal
fasciculus; cluster 5 overlaps with corticospinal tract; cluster 3 and cluster
5 contain corpus callosum and uncinate fasciculus; cluster 8 and 9 contain
superior longitudinal fasciculus, arcuate fasciculus and inferior fronto-
occipital fasciculus (Friederici and Gierhan (2013)), which are language
function related ROIs.

Figure. 2: Examples of fiber tracking results for two randomly selected HCP subjects. |  |
---|---|---
|  |
(a) Axial | (b) Sagittal | (c) Coronal
Figure. 3: Fiber clusters ($K=10$) of two subjects (each row represent one subject) at three planes: (a) Axial, (b) Sagittal, and (c) Coronal. In each plane, each color denotes one cluster. |  |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---
Figure. 4: An illustration of $K=10$ clusters in one subject at 6 different angles of view (each row denotes one angle of view) where left to right: cluster 1 to cluster 10. |
---|---
(a) Subject 1 | (b) Subject 2
Figure. 5: The number of fibers in each cluster for the two subjects from
Figure 3.

For APA connectomes, one first chooses brain ROIs according to an atlas
template, and then selects a summary of connectivity between each pair of
ROIs, such as the number of connections. APA-based methods represent the $i$th
individual’s brain connectome as a $p\times p$ matrix $W_{i}$. Each cell of
this matrix contains a summary of the strength of connection between a pair of
brain ROIs; here, we use the number of fibers connecting the regions.
Different atlas templates lead to different connectivity matrices having
different dimensionality $p$. We used the same fiber tracking method
TractoFlow, and chose the HCP842 tractography atlas (Yeh et al., 2018), which
segments the brain into 80 regions; see Tables S1 and S2 in the Appendix for
descriptions of these 80 regions. Note that one can also obtain different
count values using different tractography algorithms (Knösche et al., 2015) as
well as many filtering approaches, e.g., scale-invariant feature transform
(SIFT) (Burger and Burge, 2016) to make streamline counting more quantitative.

For PPA connectomes, we used LASSO to fit Model (1); the hyperparameter in
LASSO was selected using cross validation. For APA connectomes, we implemented
two recently proposed methods: symmetric bilinear regression (SBL) (Wang et
al., 2019) and multi-graph principal component analysis (MultiGraphPCA)
(Winter et al., 2020). SBL investigates the relationship between human traits
and connectivity matrices through a symmetric bilinear regression model, and
MultiGraphPCA proposes a tensor network factorization method that links the
scale-specific brain structural connectivity matrices through a common set of
individual-specific scores, which are further used for human trait prediction.
Tuning in SBL and MultiGraphPCA followed the recommendations by the authors.
In particular, for SBL we use K = 14; gamma = 6.9; fullit = 50; maxit = 10000;
tol = 1e-6; Replicates = 5. There is a single tuning parameter $K$ in
MultiGraphPCA; we compare results for $K=\\{2,10,20,50,70,200,400,500\\}$. We
also implemented LASSO on the vectorized (only keeping upper-triangular
elements) connectivity matrix $W_{i}$.

In order to assess the robustness of the proposed method, we tested different
versions of PPA with respect to the fiber tracking algorithms and
regularization strategies. In particular, we adopted another two fiber
tracking algorithms in Module (i), Euler Delta Crossings or EuDX (Garyfallidis
et al., 2014), and local tracking with Sparse Fascicle Model or SFM (Rokem et
al., 2015). For the regularization strategy, we also consider elastic net (Zou
and Hastie, 2005), which combines the ${L_{1}}$ penalty of LASSO with the
${L_{2}}$ penalty of ridge regression (Hoerl and Kennard, 1970), with the goal
of simultaneous selection of correlated predictors.

Figures S1 and S2 show the comparison of various versions of PPA using
different fiber tracking algorithms and regularization strategies,
respectively. The results are similar to those shown above, and demonstrate
the robustness of the PPA method to the fiber tracking algorithm and
regularization approach.

###  3$\cdot$3 Predictive performance & parsimony

We implemented three methods under APA connectomes, which are coded as LASSO,
SBL, and MultiGraphPCA, and the version of PPA using Tractoflow fiber tracking
algorithm and LASSO regularization to analyze the 1065 HCP subjects. We
calculated the 5-fold cross validation mean squared error (MSE) to compare
their predictive performance for seven human traits. We also included the
performance of a null model, which only contains an intercept term and uses
the sample average of responses in the training set as the prediction; this
model assumes no significance of the connectome in explaining selected traits
and serves as a reference model.

|  |
---|---|---
|  |
(a) Receptive Vocabulary | (b) Oral Reading | (c) List Sorting
Figure. 6: Comparison of 5-fold cross validation MSE of trait prediction based on PPA and three APA-based methods (LASSO, SBL, and MultiGraphPCA) for three traits: (a) PicVocab, (b) ReadEng, and (c) ListSort. Second row is the bar-plot of MSE for APA and PPA based method for $K=400$. APA methods (SBL, LASSO, and MultiGraphPCA) are in blue and PPA method in cyan. The red horizontal line indicates the MSE of the null model, which is 230.92 in PicVocab; 219.02 in ReadEng; 174.66 in ListSort. |
---|---
(a) Flanker | (b) Picture Sequence Memory
|
(c) Card Sort | (d) Processing Speed
Figure. 7: Comparison of 5-fold cross validation MSE of trait prediction based
on PPA and three APA methods (LASSO, SBL, and MultiGraphPCA) for four traits:
(a) Flanker, (b) PicSeq, (c) CardSort, and (d) ProSpeed. Moreover, 5-fold
cross validation MSE under the null model is 101.0769 in Flanker; 272.0460 in
PicSeq; 97.7893 in CardSort; 402.0294 in ProcSpeed.

Figure 6 plots the MSEs for three traits. The upper row of Figure 6 shows that
the proposed PPA, using the simple LASSO method, clearly outperforms SBL,
LASSO, and MultiGraphPCA in most scenarios especially as the number of
clusters $K$ increases. For the connectivity matrix-based methods, e.g, SBL,
LASSO, and MultiGraphPCA, the performance does not depend on the number of
clusters $K$, which is a PPA-specific tuning parameter. The bottom row of
Figure 6 compares the MSEs of our PPA-based methods at $K=400$ to the selected
three APA methods, while the red horizontal line represents the performance of
the null model. Since all APA-based methods use TractoFlow, we first focus on
the PPA method using the same fiber tracking algorithm. In this case, MSEs of
PPA are smaller than the three APA-based methods, uniformly across the three
traits. In sharp contrast to the excellent performance of LASSO for the PPA
connectomes, LASSO predictions based on vectorized APA connectomes did no
better than the null model. SBL and MultiGraphPCA improve the MSEs over LASSO,
as a result of a better utilization of the network structure of APA
connectomes. The comparison of MSEs suggests that fundamentally changing the
connectome representation based on defining population fiber bundles can
perhaps lead to even bigger gains. Figure 6 shows that defining large numbers
of fiber bundles may lead to predictive gains.

For the other four traits (Flanker, PicSeq, CardSort, ProcSpeed), all methods
tend to give a MSE close to the null model (Figure 7), indicating limited
predictive power of structural connectivity for these traits. It is reassuring
that the proposed method is consistent with APA-based methods in these cases.
We remark that the lack of predictive power might be caused in part by a weak
relationship between these measured traits and actual innate abilities in the
test subjects.

While Figure 6 shows how MSE varies with $K$ for the three different fiber
tracking algorithms, TractoFlow, EuDX and SFM, Figure 8 plots the MSEs against
the number of active fibers to provide additional insight into the impact of
$K$. The number of active fibers is defined as the total number of fibers that
belong to the active fiber bundles selected by LASSO, i.e., it is
$\sum_{k\in\mathcal{K}(s)}|A_{K}^{(k)}|,$ where $|A_{K}^{(k)}|$ is the number
of fibers in the bundle $A_{K}^{(k)}$. Taking picture vocabulary test
(PicVocab) as an example, the best MSE is achieved when the number of active
fibers is around $0.8*10^{7}$. This optimal number varies from trait to trait,
but a U-shaped curve typically emerges with $K$ varied up to 500. For the
other traits in Figure 7, MSEs do not change much as we vary the number of
active fibers, which is expected as the MSE curve is flat when plotted against
$K$; we omit these curves here as they are redundant.

|  |
---|---|---
Figure. 8: Comparison of 5-fold cross validation MSE of trait prediction based
on PPA for three traits: (a) PicVocab, (b) ReadEng, and (c) ListSort.

Table 1 reports the number of selected parameters in PPA and APA-based
methods, which shows parsimony and effectiveness of PPA-based methods compared
to LASSO applied to APA connectomes and SBL. In particular, LASSO for APA
connectomes selects nearly zero active connections, which explains its poor
predictive performance in Figure 6. PPA selects substantially fewer non-zero
parameters than SBL; this combined with the better MSEs in Figure 6 shows the
effectiveness of PPA connectomes in representing key features of brain
networks predictive of traits. The last four traits show little to no signal
for any of the methods and selecting few if any features for these traits
seems appropriate.

| PicVocab | ReadEng | ListSort | Flanker | PicSeq | CardSort | ProcSpeed
---|---|---|---|---|---|---|---
PPA (K=50) | 23 | 20 | 12 | 11 | 15 | 0 | 1
PPA (K=100) | 38 | 33 | 19 | 3 | 29 | 3 | 2
PPA (K=200) | 64 | 47 | 12 | 7 | 6 | 0 | 0
PPA (K=300) | 53 | 31 | 2 | 3 | 1 | 0 | 0
PPA (K=400) | 19 | 20 | 18 | 18 | 15 | 2 | 1
PPA (K=500) | 56 | 39 | 22 | 1 | 3 | 0 | 1
LASSO+HCP842 | 2 | 1 | 0 | 0 | 11 | 0 | 1
SBL+HCP842 | 1134 | 1053 | 972 | 729 | 1134 | 810 | 1134
Table 1: Number of selected parameters in different methods. For
MultiGraphPCA, the number of parameters is set to be $K$ used in PPA.

To evaluate generalizability of our comparisons between PPA and APA and assess
robustness of APA methods, we tested multiple different versions of APA with
respect to the atlases and summary of connectivity used in defining the
connectivity matrices. In our analyses as we vary these two aspects when
implementing APA methods, the improved predictive performance of PPA over APA
that was shown earlier is consistently observed, indicating the way
connectomes are represented (PPA vs APA) appears to be a more important factor
in explaining the performance gain.

In particular, for all APA related methods (SBL, LASSO, MultiGraphPCA), we
checked three different atlases: HCP842, AAL2, and FreeSurferDKT. HCP842 is a
tractography atlas (Yeh et al., 2018), and we choose the settings used
previously in obtaining the results in Figures 6 and 7. This atlas provides
tractography of white matter with expert labeling and examination,
complementary to traditional histologically-based and voxel-based white matter
atlases. AAL2 stands for automated anatomical labeling atlas 2, providing an
alternative parcellation of the orbitofrontal cortex (Rolls et al., 2015).
Finally, FreeSurferDKT is an atlas manually labeled in the macroscopic anatomy
in magnetic resonance images by the Desikan–Killiany–Tourville (DKT) protocol
(Klein and Tourville, 2012). From the results in Figure S3, we can see that
changing the atlas has little impact on the predictive performance of APA
methods.

For analyses assessing sensitivity of the APA results to the summary of
connectivity between regions, we focused on the HCP842 atlas and tested three
different ways of calculating the connectivity matrix: “count”, “ncount”, and
“ncount2”. For each entry in the connectivity matrix, these three summaries
correspond to counting the number of tracts that pass two ROIs (“count”),
which is used in the previous comparison in Figures 6 and 7, normalizing the
count by the median length of the fibers (“ncount”), and multiplying the count
by the sum of the inverse of the length (“ncount2”). Figure S4 shows that the
performance of APA methods is robust to the summary used in calculating
connectivity matrices.

###  3$\cdot$4 Integrating PPA and atlas

The proposed PPA connectome does not rely on any tractography atlas in
defining the connectome or building a regression model for traits. This
reduces subjectivity in choosing ROIs and facilitates statistical analyses
with improved prediction and parsimony, as shown in the preceding sections. In
this section, we demonstrate another feature of PPA in terms of its
compatibility with traditional ROIs—as an ab initio, tractography-based
representation of connectomes, PPA can be integrated with any existing atlas
templates to borrow the ROI information encoded therein in a straightforward
manner. Such integration allows us to relate the interpretation of PPA results
to traditional ROIs. Through visualization, we find the proposed PPA leads to
interesting and interpretable findings.

We align active fibers produced by PPA to an atlas. In particular, based on a
selected atlas, we build the connectivity matrix at the population level with
each matrix entry generated by counting the number of active fibers passing
between each pair of ROIs. Note that similarly to deriving the connectivity
matrix in APA, one can obtain many different summaries of connectivity between
two regions by using various filtering approaches such as SIFT and different
ways of normalization, with the count just one simple choice. We use the
HCP842 tractography atlas (Yeh et al., 2018) as in our implementation of APA-
based methods, which segments the brain into 80 regions. Refer to
http://brain.labsolver.org/diffusion-mri-templates/tractography for more
details about the 80 ROIs. For each human trait, we visualize the PPA-induced
connectivity matrix and anatomy of connections through DSI Studio (http://dsi-
studio.labsolver.org), a tractography software tool that maps brain
connections and correlates findings with traits. We adopt the default setting
in DSI Studio by thresholding matrix entries with a small number of connecting
tracks relative to the maximum connecting tracks in the connectivity matrix.
We use 0.5 as the threshold for this ratio.

According to the visualization plots in Figure 9-12, PPA discovers some
insightful connections of various anatomical regions that are related to the
human traits. Some interesting findings are listed as follows. For most human
traits, the primary pattern in the connectivity matrix does not vary much as
the number of clusters (fiber bundles) increases. When using the trait
PicVocab, the subgraph including connections among ROI 3
(Cortico_Striatal_Pathway_L), ROI 4 (Cortico_Striatal_Pathway_R), ROI 7
(Corticothalamic_Pathway_L), ROI 8 (Corticothalamic_Pathway_R), and ROI 44
(Corpus_Callosum), can be found in all the 4 different settings of number of
clusters, i.e., $K$=50,100,200,400 (see Figure 9). The trait ListSort that is
related to human working memory (Figure 10), also shows a common pattern in
the connectivity matrix across cluster settings, leading to a network with
shared nodes containing ROI 3 (Cortico_Striatal_Pathway_L), ROI 7
(Corticothalamic_Pathway_L), ROI 21 (Arcuate_Fasciculus_L), ROI 37 (U_Fiber_L)
and ROI 44 (Corpus_Callosum).

For language associated human traits (e.g., the trait PicVocab is related to
language and vocabulary comprehension while the trait ReadEng is related to
language and reading decoding), the significant regions are mainly located in
the left hemisphere (see Figure 9 and Figure 11). This finding indicates that
the left hemisphere is particularly important for language, which has been
consistently verified in clinical and experimental settings (Riès et al.,
2016).

We find that a particularly important ROI 44 (Corpus_Callosum) is detected for
most human traits; ROI 44 is consistently a prominent node in the identified
subgroup for most values of $K$ (see the second row in Figure 9-12). Corpus
Callosum is a large C shape white matter region, forming the floor of the
longitudinal fissure that separates the cerebral cortex into the left and
right hemispheres (Carpenter, 1985). This ROI is responsible for transmitting
sensory, motor, and cognitive signals between the hemispheres.

The Flanker task measures both a participant’s attention and inhibitory
control. According to the visualization plots in Figure 12, significant
regions identified are strongly lateralized to the right hemisphere. Right
frontal dominance for inhibitory motor control has become a commonly accepted
view (Swick et al., 2008; Garavan et al., 1999), indicating that our finding
is in agreement with existing literature. The visualization plots for the
remaining three human traits that we considered are reported in the appendix;
see Figures S6-S8.

|  |  |
---|---|---|---
|  |  |
|  |  |
(a) $K$=50 | (b) $K$=100 | (c) $K$=200 | (d) $K$=400
Figure. 9: Visualization for trait PicVocab: each column represents the visualization for a different number of clusters ($K$=50; $K$=100; $K$=200; $K$=400) in PPA; First row represents the connectivity matrix between any two ROI regions in the HCP842 tractography atlas. Second row shows anatomy of connections in an axial view. Third row shows anatomy of connections in a sagittal view. |  |  |
---|---|---|---
|  |  |
|  |  |
(a) $K$=50 | (b) $K$=100 | (c) $K$=200 | (d) $K$=400
Figure. 10: Visualization for trait ListSort: each column represents the visualization under different $K$ ($K$=50; $K$=100; $K$=200; $K$=400) in PPA; First row represents the connectivity matrix between any two ROI regions in HCP842 tractography atlas. Second row shows anatomy of connections in an axial view. Third row shows anatomy of connections in a sagittal view. |  |  |
---|---|---|---
|  |  |
|  |  |
(a) $K$=50 | (b) $K$=100 | (c) $K$=200 | (d) $K$=400
Figure. 11: Visualization for trait ReadEng: each column represents the visualization under different $K$ ($K$=50; $K$=100; $K$=200; $K$=400) in PPA; First row represents the connectivity matrix between any two ROI regions in HCP842 tractography atlas. Second row shows anatomy of connections in an axial view. Third row shows anatomy of connections in a sagittal view. |  |  |
---|---|---|---
|  |  |
|  |  |
(a) $K$=50 | (b) $K$=100 | (c) $K$=200 | (d) $K$=400
Figure. 12: Visualization for trait Flanker: each column represents the
visualization under different $K$ ($K$=50; $K$=100; $K$=200; $K$=400) in PPA;
First row represents the connectivity matrix between any two ROI regions in
HCP842 tractography atlas. Second row shows anatomy of connections in an axial
view. Third row shows anatomy of connections in a sagittal view.

##  4 Discussion

In this article, we propose a new tractography-based representation of brain
connectomes as an alternative to the widely used anatomical connectivity. This
representation leads to Principal Parcellation Analysis (PPA), where we
represent individual brain connectomes by compositional vectors building on a
basis system of fiber bundles that captures the connectivity at the population
level.

PPA reduces subjectivity of classical connectomes by eliminating the need to
choose atlases and ROIs a priori. Unlike the traditional connectomes where
data objects are of complex graph-structure and ultra-dimension, our PPA
connectomes can be analyzed using existing statistical tools for high-
dimensional vector-valued features. Our application to HCP data indicates that
PPA is robust to the specific choice of fiber tracking algorithm. We propose
an approach to integrate the parcellation produced by PPA with an atlas, so
that the results can be visualized and interpreted using traditional ROIs.

There are several interesting next directions building on our initial PPA
approach. Firstly, the methods used in each of the three modules can be
refined. For example, we can consider different fiber clustering algorithms
that take into account more than just the endpoint locations. Also, instead of
just applying LASSO within a linear regression model for the trait responses,
we can use more elaborate predictive algorithms and inferential approaches.
Particularly for large datasets, improved predictive accuracy may be possible
with flexible non-linear regression methods. We have focused on using the
proposed connectomes to analyze human traits via a regression model, and it is
interesting to consider other inference problems beyond explaining and
predicting task-based scores. For example, one may consider both structural
and functional connectivity (Tian and Zalesky, 2018) and utilize other
dynamical features (Kobeleva et al., 2021), with the proposed PPA connectomes
serving as a building block to represent structural connectivity.

## Data and code availability statement

The dMRI and human traits data from the original study are available from HCP:
https://www.humanconnectome.org/. Code for implementing PPA is freely
available online at https://github.com/xylimeng/PPA.

## Declaration of Competing Interest

The authors declare no conflict of interest.

## Credit authorship contribution statement

Rongjie Liu: Conceptualization, Methodology, Software, Writing- Original draft
preparation. Meng Li: Conceptualization, Methodology, Validation, Supervision,
Writing- Reviewing and Editing. David Dunson: Conceptualization, Methodology,
Validation, Writing- Reviewing and Editing.

## Acknowledgements

We thank Lu Wang for helpful discussions on implementing the SBL method, and
Yerong Li for helping with numerical experiments in an earlier version of PPA.
This work was partially funded by grant DMS-2015569 from the National Science
Foundation and grant R01MH118927 of the National Institute of Mental Health of
the United States National Institutes of Health.

## References

  * Apkarian et al. (2020) V Apkarian, C Bonin Pinto, J Bielefeld, R Jabakhanji, D Reckziegel, and J Griffith.  Neural and genetic bases for human ability traits.  _Frontiers in Human Neuroscience_ , 14:565, 2020.
  * Avants et al. (2009) B Avants, N Tustison, and G Song.  Advanced normalization tools (ANTS).  _Insight Journal_ , 2:1–35, 2009.
  * Bansal et al. (2018) K Bansal, J Nakuci, and S Muldoon.  Personalized brain network models for assessing structure–function relationships.  _Current Opinion in Neurobiology_ , 52:42–47, 2018.
  * Behrens et al. (2003) T Behrens, H Johansen-Berg, M Woolrich, S Smith, C Wheeler-Kingshott, P Boulby, G Barker, E Sillery, K Sheehan, O Ciccarelli, A Thompson, J Brady, and P Matthews.  Non-invasive mapping of connections between human thalamus and cortex using diffusion imaging.  _Nature Neuroscience_ , 6(7):750–757, 2003.
  * Burger and Burge (2016) W Burger and M Burge.  Scale-invariant feature transform (sift).  In _Digital Image Processing_ , pages 609–664. Springer, 2016.
  * Carpenter (1985) M Carpenter.  _Study Guide and Self-examination Review for Core Text of Neuroanatomy_.  Williams & Wilkins, 1985.
  * de Reus and Van den Heuvel (2013) M de Reus and M Van den Heuvel.  The parcellation-based connectome: limitations and extensions.  _Neuroimage_ , 80:397–404, 2013.
  * Di Tommaso et al. (2017) P Di Tommaso, M Chatzou, E Floden, P Barja, E Palumbo, and C Notredame.  Nextflow enables reproducible computational workflows.  _Nature Biotechnology_ , 35(4):316, 2017.
  * Eickhoff et al. (2015) S Eickhoff, B Thirion, G Varoquaux, and D Bzdok.  Connectivity-based parcellation: Critique and implications.  _Human Brain Mapping_ , 36(12):4771–4792, 2015\.
  * Eickhoff et al. (2018) S Eickhoff, T Yeo, and S Genon.  Imaging-based parcellations of the human brain.  _Nature Reviews Neuroscience_ , 19(11):672–686, 2018.
  * Fan and Li (2001) J Fan and R Li.  Variable selection via nonconcave penalized likelihood and its oracle properties.  _Journal of the American Statistical Association_ , 96(456):1348–1360, 2001.
  * Friederici and Gierhan (2013) Angela D. Friederici and Sarah ME Gierhan.  The language network.  _Current Opinion in Neurobiology_ , 23(2):250–254, Apr 2013.
  * Garavan et al. (1999) H Garavan, T Ross, and E Stein.  Right hemispheric dominance of inhibitory control: an event-related functional MRI study.  _Proceedings of the National Academy of Sciences_ , 96(14):8301–8306, 1999.
  * Garyfallidis et al. (2012) E Garyfallidis, M Brett, M Correia, G Williams, and I Nimmo-Smith.  Quickbundles, a method for tractography simplification.  _Frontiers in Neuroscience_ , 6:175, 2012.
  * Garyfallidis et al. (2014) E Garyfallidis, M Brett, B Amirbekian, A Rokem, S Van Der Walt, M Descoteaux, and I Nimmo-Smith.  Dipy, a library for the analysis of diffusion MRI data.  _Frontiers in Neuroinformatics_ , 8:8, 2014.
  * Glasser et al. (2016) M Glasser, S Smith, D Marcus, J Andersson, E Auerbach, T Behrens, T Coalson, M Harms, M Jenkinson, S Moeller, E Robinson, S Sotiropoulos, J Xu, E Yacoub, K Ugurbil, and D Van Essen.  The Human Connectome Project’s neuroimaging approach.  _Nature Neuroscience_ , 19(9):1175–1187, 2016\.
  * Gupta et al. (2017) Vikash Gupta, Sophia I. Thomopoulos, Faisal M. Rashid, and Paul M. Thompson.  Fibernet: An ensemble deep learning framework for clustering white matter fibers.  _bioRxiv_ , 2017.  doi: 10.1101/141036.
  * He et al. (2007) Y He, Z Chen, and A Evans.  Small-world anatomical networks in the human brain revealed by cortical thickness from MRI.  _Cerebral Cortex_ , 17(10):2407–2419, 2007.
  * He et al. (2008) Y He, Z Chen, and A Evans.  Structural insights into aberrant topological patterns of large-scale cortical networks in Alzheimer’s disease.  _Journal of Neuroscience_ , 28(18):4756–4766, 2008.
  * Hochberg et al. (2007) Y Hochberg, A Ljungqvist, and Y Lu.  Whom you know matters: Venture capital networks and investment performance.  _The Journal of Finance_ , 62(1):251–301, 2007\.
  * Hoerl and Kennard (1970) Arthur E Hoerl and Robert W Kennard.  Ridge regression: applications to nonorthogonal problems.  _Technometrics_ , 12(1):69–82, 1970.
  * Jenkinson et al. (2012) M Jenkinson, C Beckmann, T Behrens, M Woolrich, and S Smith.  FSL.  _Neuroimage_ , 62(2):782–790, 2012.
  * Klein and Tourville (2012) Arno Klein and Jason Tourville.  101 labeled brain images and a consistent human cortical labeling protocol.  _Frontiers in neuroscience_ , 6:171, 2012.
  * Knösche et al. (2015) T Knösche, A Anwander, M Liptrot, and T Dyrby.  Validation of tractography: comparison with manganese tracing.  _Human brain mapping_ , 36(10):4116–4134, 2015\.
  * Kobeleva et al. (2021) X Kobeleva, A López-González, M Kringelbach, and G Deco.  Revealing the relevant spatiotemporal scale underlying whole-brain dynamics.  _bioRxiv_ , pages 2020–09, 2021.
  * Kurtzer et al. (2017) G Kurtzer, V Sochat, and M Bauer.  Singularity: Scientific containers for mobility of compute.  _PLOS ONE_ , 12(5):e0177459, 2017.
  * Lin et al. (2020) Y Lin, S Baete, X Wang, and F Boada.  Mapping brain–behavior networks using functional and structural connectome fingerprinting in the HCP dataset.  _Brain and Behavior_ , 10(6):e01647, 2020.
  * Messé (2020) A Messé.  Parcellation influence on the connectivity-based structure–function relationship in the human brain.  _Human Brain Mapping_ , 41(5):1167–1180, 2020\.
  * O’Donnell et al. (2013) L O’Donnell, A Golby, and C Westin.  Fiber clustering versus the parcellation-based connectome.  _NeuroImage_ , 80:283–289, 2013.
  * O’Malley and Marsden (2008) J O’Malley and P Marsden.  The analysis of social networks.  _Health Services and Outcomes Research Methodology_ , 8(4):222–269, 2008.
  * Park and Friston (2013) H Park and K Friston.  Structural and functional brain networks: from connections to cognition.  _Science_ , 342(6158), 2013.
  * Riès et al. (2016) S Riès, N Dronkers, and R Knight.  Choosing words: left hemisphere, right hemisphere, or both? Perspective on the lateralization of word retrieval.  _Annals of the New York Academy of Sciences_ , 1369(1):111, 2016.
  * Roine et al. (2019) T Roine, B Jeurissen, D Perrone, J Aelterman, W Philips, J Sijbers, and A Leemans.  Reproducibility and intercorrelation of graph theoretical measures in structural brain connectivity networks.  _Medical Image Analysis_ , 52:56–67, 2019.
  * Rokem et al. (2015) A Rokem, J Yeatman, F Pestilli, K Kay, A Mezer, S Van Der Walt, and B Wandell.  Evaluating the accuracy of diffusion MRI models in white matter.  _PLOS ONE_ , 10(4):e0123272, 2015.
  * Rolls et al. (2015) Edmund T Rolls, Marc Joliot, and Nathalie Tzourio-Mazoyer.  Implementation of a new parcellation of the orbitofrontal cortex in the automated anatomical labeling atlas.  _Neuroimage_ , 122:1–5, 2015.
  * Schiffler et al. (2017) P Schiffler, J Tenberge, H Wiendl, and S Meuth.  Cortex parcellation associated whole white matter parcellation in individual subjects.  _Frontiers in Human Neuroscience_ , 11:352, 2017.
  * Sculley (2010) D Sculley.  Web-scale k-means clustering.  In _Proceedings of the 19th International Conference on World Wide Web_ , pages 1177–1178. ACM, 2010.
  * Shin et al. (2019) Jiwon Shin, Jared Rowley, Rasheda Chowdhury, Pierre Jolicoeur, Denise Klein, Christophe Grova, Pedro Rosa-Neto, and Eliane Kobayashi.  Inferior longitudinal fasciculus’ role in visual processing and language comprehension: A combined meg-dti study.  _Frontiers in Neuroscience_ , 13, 2019.  doi: 10.3389/fnins.2019.00875.
  * Sotiropoulos et al. (2013) S Sotiropoulos, S Jbabdi, J Xu, J Andersson, S Moeller, E Auerbach, M Glasser, M Hernandez, G Sapiro, M Jenkinson, D Feinberg, E Yacoub, C Lenglet, D Van Essen, K Ugurbil, T Behrens, and WU-Minn HCP Consortium.  Advances in diffusion MRI acquisition and processing in the Human Connectome Project.  _Neuroimage_ , 80:125–143, 2013.
  * Sporns et al. (2005) O Sporns, G Tononi, and R Kötter.  The human connectome: A structural description of the human brain.  _PLOS Computational Biology_ , 1(4), 09 2005.
  * Swick et al. (2008) D Swick, V Ashley, and U Turken.  Left inferior frontal gyrus is critical for response inhibition.  _BMC Neuroscience_ , 9(1):1–11, 2008.
  * Theaud et al. (2020) G Theaud, J Houde, A Boré, F Rheault, F Morency, and M Descoteaux.  Tractoflow: A robust, efficient and reproducible diffusion MRI pipeline leveraging Nextflow & Singularity.  _Neuroimage_ , page 116889, 2020.
  * Tian and Zalesky (2018) Y Tian and A Zalesky.  Characterizing the functional connectivity diversity of the insula cortex: Subregions, diversity curves and behavior.  _NeuroImage_ , 183:716–733, 2018.
  * Tibshirani (1996) R Tibshirani.  Regression shrinkage and selection via the lasso.  _Journal of the Royal Statistical Society: Series B (Statistical Methodological)_ , 58(1):267–288, 1996.
  * Toga et al. (2012) A Toga, K Clark, P Thompson, D Shattuck, and J Van Horn.  Mapping the human connectome.  _Neurosurgery_ , 71(1):1–5, 2012.
  * Tournier et al. (2019) J Tournier, R Smith, D Raffelt, R Tabbara, T Dhollander, M Pietsch, D Christiaens, B Jeurissen, C Yeh, and A Connelly.  Mrtrix3: A fast, flexible and open software framework for medical image processing and visualisation.  _Neuroimage_ , 202:116137, 2019.
  * Tzourio-Mazoyer et al. (2002) N Tzourio-Mazoyer, B Landeau, D Papathanassiou, F Crivello, O Etard, N Delcroix, B Mazoyer, and M Joliot.  Automated anatomical labeling of activations in SPM using a macroscopic anatomical parcellation of the MNI MRI single-subject brain.  _Neuroimage_ , 15(1):273–289, 2002.
  * Van Essen et al. (2012) D Van Essen, K Ugurbil, E Auerbach, D Barch, T Behrens, R Bucholz, A Chang, L Chen, M Corbetta, S Curtiss, S Della Penna, D Feinberg, M Glasser, N Harel, A Heath, L Larson-Prior, D Marcus, G Michalareas, S Moeller, R Oostenveld, S Petersen, F Prior, B Schlaggar, S Smith, A Snyder, J Xu, E Yacoub, and WU-Minn HCP Consortium.  The Human Connectome Project: a data acquisition perspective.  _Neuroimage_ , 62(4):2222–2231, 2012.
  * Van Essen et al. (2013) D Van Essen, S Smith, D Barch, T Behrens, E Yacoub, K Ugurbil, and Wu-Minn HCP Consortium.  The WU-Minn Human Connectome Project: an overview.  _Neuroimage_ , 80:62–79, 2013.
  * Wang et al. (2017) L Wang, D Durante, R Jung, and D Dunson.  Bayesian network–response regression.  _Bioinformatics_ , 33(12):1859–1866, 2017.
  * Wang et al. (2019) L. Wang, Z. Zhang, and D. Dunson.  Symmetric bilinear regression for signal subgraph estimation.  _IEEE Transactions on Signal Processing_ , 67(7):1929–1940, 2019.
  * Winter et al. (2020) S Winter, Z Zhang, and D Dunson.  Multi-scale graph principal component analysis for connectomics.  _arXiv preprint arXiv:2010.02332_ , 2020.
  * Yao et al. (2015) Z Yao, B Hu, Y Xie, P Moore, and J Zheng.  A review of structural and functional brain networks: small world and atlas.  _Brain Informatics_ , 2(1):45–52, 2015.
  * Yeh et al. (2018) F Yeh, S Panesar, D Fernandes, A Meola, M Yoshino, J Fernandez-Miranda, J Vettel, and T Verstynen.  Population-averaged atlas of the macroscale human structural connectome and its network topology.  _Neuroimage_ , 178:57–68, 2018.
  * Zalesky et al. (2010) A Zalesky, A Fornito, I Harding, L Cocchi, M Yücel, C Pantelis, and E Bullmore.  Whole-brain anatomical networks: does the choice of nodes matter?  _Neuroimage_ , 50(3):970–983, 2010.
  * Zhang et al. (2018) Z Zhang, M Descoteaux, J Zhang, G Girard, M Chamberland, D Dunson, A Srivastava, and H Zhu.  Mapping population-based structural connectomes.  _Neuroimage_ , 172:130–145, 2018.
  * Zhang et al. (2019) Z Zhang, G Allen, H Zhu, and D Dunson.  Tensor network factorizations: Relationships between brain structural connectomes and traits.  _Neuroimage_ , 197:330–343, 2019.
  * Zou and Hastie (2005) H Zou and T Hastie.  Regularization and variable selection via the elastic net.  _Journal of the Royal Statistical Society: Series B (Statistical Methodology)_ , 67(2):301–320, 2005.

## Appendix

In this Appendix we present (1) the description of seven traits, (2) Table S1
and Table S2 for region names with their corresponding region numbers in
HCP842 tractography atlas, (3) Figures S1 and S2 for comparing various
versions of PPA using different fiber tracking algorithms and regularization
strategies, respectively, (4) Figures S3 and S4 for the predictive performance
of various versions of APA using different atlases and metrics in calculating
the connectivity matrix, respectively, (5) Figure S5 for the comparison of
5-fold cross validation MSE of trait prediction against the number of active
fibers based on PPA with the other two fiber tracking options EuDX and SFM for
three traits: PicVocab, ReadEng and ListSort, and (6) Figure S6–S8 for
visualization for the other three traits: PicSeq, CardSort, ProcSpeed.

The brief description of each trait is listed below.

  * 1.

Receptive vocabulary score (PicVocab). This score comes from a picture
vocabulary test, in which respondents are presented with an audio recording of
a word and four photographic images on the computer screen and are asked to
select the picture that most closely matches the meaning of the word.

  * 2.

Oral reading score (ReadEng). This is from an oral reading recognition test.
In this test, respondents were scored on their ability in reading and
pronouncing letters and words accurately.

  * 3.

List sorting score (ListSort). This test assesses working memory and requires
each participant to sequence different visually- and orally- presented
stimuli: pictures of different foods and animals with both sound clips and
written names. Concretely, participants are asked to either order items by
size or report items in size order.

  * 4.

Flanker score (Flanker). This test measures both a subject’s attention and
inhibitory control. In this test, participants are asked to focus on a given
stimulus while inhibiting attention to stimuli flanking it. Sometimes the
middle stimulus is pointing in the same direction as the flankers (congruent)
and sometimes in the opposite direction (incongruent).

  * 5.

Picture sequence memory score (PicSeq). This test is a measure developed for
the assessment of episodic memory. It involves recalling increasingly lengthy
series of illustrated objects and activities that are presented in a
particular order on the computer screen. The participants are asked to recall
the sequence of pictures that is demonstrated over two learning trials, where
sequence length varies from 6-18 pictures, depending on age.

  * 6.

Card sort score (CardSort). This test is a measure of cognitive flexibility.
Two target pictures are presented that vary along two dimensions (e.g., shape
and color). Participants are asked to match a series of bivalent test pictures
to the target pictures, first according to one dimension and then, after a
number of trials, according to the other dimension.

  * 7.

Processing speed score (ProcSpeed). This test measures speed of processing by
asking participants to discern whether two side-by-side pictures are the same
or not. Participants’ raw score is the number of items correct in a 90-second
period. The items are designed to be simple to most purely measure processing
speed.

No. | Region Name | No. | Region Name
---|---|---|---
1 | Acoustic_Radiation_L | 2 | Acoustic_Radiation_R
3 | Cortico_Striatal_Pathway_L | 4 | Cortico_Striatal_Pathway_R
5 | Cortico_Spinal_Tract_L | 6 | Cortico_Spinal_Tract_R
7 | Corticothalamic_Pathway_L | 8 | Corticothalamic_Pathway_R
9 | Fornix_L | 10 | Fornix_R
11 | Frontopontine_Tract_L | 12 | Frontopontine_Tract_R
13 | Occipitopontine_Tract_L | 14 | Occipitopontine_Tract_R
15 | Optic_Radiation_L | 16 | Optic_Radiation_R
17 | Parietopontine_Tract_L | 18 | Parietopontine_Tract_R
19 | Temporopontine_Tract_L | 20 | Temporopontine_Tract_R
21 | Arcuate_Fasciculus_L | 22 | Arcuate_Fasciculus_R
23 | Cingulum_L | 24 | Cingulum_R
25 | Extreme_Capsule_L | 26 | Extreme_Capsule_R
27 | Frontal_Aslant_Tract_L | 28 | Frontal_Aslant_Tract_R
29 | Inferior_Fronto_Occipital_Fasciculus_L | 30 | Inferior_Fronto_Occipital_Fasciculus_R
31 | Inferior_Longitudinal_Fasciculus_L | 32 | Inferior_Longitudinal_Fasciculus_R
33 | Middle_Longitudinal_Fasciculus_L | 34 | Middle_Longitudinal_Fasciculus_R
35 | Superior_Longitudinal_Fasciculus_L | 36 | Superior_Longitudinal_Fasciculus_R
37 | U_Fiber_L | 38 | U_Fiber_R
39 | Uncinate_Fasciculus_L | 40 | Uncinate_Fasciculus_R
Table S1: 80 regions’ names in HCP842 tractography atlas No. | Region Name | No. | Region Name
---|---|---|---
41 | Vertical_Occipital_Fasciculus_L | 42 | Vertical_Occipital_Fasciculus_R
43 | Anterior_Commissure | 44 | Corpus_Callosum
45 | Posterior_Commissure | 46 | Cerebellum_L
47 | Cerebellum_R | 48 | Inferior_Cerebellar_Peduncle_L
49 | Inferior_Cerebellar_Peduncle_R | 50 | Middle_Cerebellar_Peduncle
51 | Superior_Cerebellar_Peduncle | 52 | Vermis
53 | Central_Tegmental_Tract_L | 54 | Central_Tegmental_Tract_R
55 | Dorsal_Longitudinal_Fasciculus_L | 56 | Dorsal_Longitudinal_Fasciculus_R
57 | Lateral_Lemniscus_L | 58 | Lateral_Lemniscus_R
59 | Medial_Lemniscus_L | 60 | Medial_Lemniscus_R
61 | Medial_Longitudinal_Fasciculus_L | 62 | Medial_Longitudinal_Fasciculus_R
63 | Rubrospinal_Tract_L | 64 | Rubrospinal_Tract_R
65 | Spinothalamic_Tract_L | 66 | Spinothalamic_Tract_R
67 | CNII_L | 68 | CNII_R
69 | CNIII_L | 70 | CNIII_R
71 | CNIV_L | 72 | CNIV_R
73 | CNV_L | 74 | CNV_R
75 | CNVII_L | 76 | CNVII_R
77 | CNVIII_L | 78 | CNVIII_R
79 | CNX_L | 80 | CNX_R
Table S2: cont. 80 regions’ names in HCP842 tractography atlas

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Figure. S1: Comparison of 5-fold cross validation MSE of trait prediction
based on PPA with three fiber tracking algorithms: TractoFlow, EuDX, and SFM.

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Figure. S2: Comparison of 5-fold cross validation MSE of trait prediction
based on PPA with two regularization strategies: LASSO and elastic net.

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Figure. S3: Comparison of 5-fold cross validation MSE of trait prediction
based on APA with three atlases: HCP842, AAL2, and FreeSurferDKT.

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Figure. S4: Comparison of 5-fold cross validation MSE of trait prediction based on APA with three metrics in calculating the connectivity matrix: count, ncount, and ncount2. |  |
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(a) EuDX for Receptive Vocabulary | (b) EuDX for Oral Reading | (c) EuDX for List Sorting
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(d) SFM for Receptive Vocabulary | (e) SFM for Oral Reading | (f) SFM for List Sorting
Figure. S5: The comparison of 5-fold cross validation MSE of trait prediction
along with the number of active fiber bundles based on PPA with the other two
fiber tracking options: EuDX ((a)-(c)) and SFM ((d)-(f)) for three traits:
PicVocab, ReadEng and ListSort.

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(a) $K$=50 | (b) $K$=100 | (c) $K$=200 | (d) $K$=400
Figure. S6: Visualization for trait PicSeq: each column represents the
visualization under different $K$ ($K$=50; $K$=100; $K$=200; $K$=400) in PPA;
First row represents the connectivity matrix between any two ROI regions in
HCP842 tractography atlas. Second row shows anatomy of connections in an axial
view. Third row shows anatomy of connections in a sagittal view.

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(a) $K$=50 | (b) $K$=100 | (c) $K$=200 | (d) $K$=400
Figure. S7: Visualization for trait CardSort: each column represents the
visualization under different $K$ ($K$=50; $K$=100; $K$=200; $K$=400) in PPA;
First row represents the connectivity matrix between any two ROI regions in
HCP842 tractography atlas. Second row shows anatomy of connections in an axial
view. Third row shows anatomy of connections in a sagittal view.

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(a) $K$=50 | (b) $K$=100 | (c) $K$=200 | (d) $K$=400
Figure. S8: Visualization for trait ProcSpeed: each column represents the
visualization under different $K$ ($K$=50; $K$=100; $K$=200; $K$=400) in PPA;
First row represents the connectivity matrix between any two ROI regions in
HCP842 tractography atlas. Second row shows anatomy of connections in an axial
view. Third row shows anatomy of connections in a sagittal view.