Zhengshi Yang^{1}, Xiaowei Zhuang^{1}, Christopher Bird^{1}, Karthik Sreenivasan^{1}, Virendra Mishra^{1}, Sarah J Banks^{1}, and Dietmar Cordes^{1,2}

^{1}Cleveland Clinic Lou Ruvo Center for Brain Health, Las Vegas, NV, United States, ^{2}University of Colorado, Boulder, CO, United States

### Synopsis

Principal component analysis is commonly used in data fusion for
dimension reduction prior to performing fusion analysis. However, PCA does not
address that a large proportion of voxels may be irrelevant to extract joint information
in data fusion. We implemented sparse PCA to suppress irrelevant voxels while simultaneously
reducing the data dimension. Results show that introducing sparsity to data
fusion provides better group discrimination.

### Introduction

Data fusion techniques have been proposed in the recent past to leverage
complementary information of multiple modalities. Many fusion studies used principal component analysis (PCA) for
dimension reduction prior to associating modalities to overcome the problem of
having more voxels than observations [1]. PCA, however, does not address that a
large proportion of voxels might be irrelevant for a fusion analysis. In our
study, we integrated sparse regularization into PCA and developed a technique
to control the sparsity level. This sparse PCA (sPCA) method was then applied
together with canonical correlation analysis (CCA) to fuse resting-state fMRI
data and structural data acquired from normal controls (NC) and subjects with mild
cognitive impairment (MCI). ### Method

*Subjects:* Both structural and
resting-state functional MRI data from 22 age- and gender-matched MCI subjects (F/M=10/12,
age=74.72±4.02 ys, all amyloid positive) and 22 NC
(F/M=12/10, age=74.61±6.02 ys) were obtained from the ADNI database (*http://adni.loni.usc.edu/*).
Standard preprocessing steps, including slice-timing correction, realignment,
coregistration, and normalization, were performed for fMRI data. Fast eigenvector
centrality mapping (ECM) [2] was performed on each subject, yielding individual
voxel-wise whole brain centrality maps as a measure of functional connectivity.
Voxel-based morphometry (VBM) using SPM12 was applied on T1 structural data to create
statistical VBM maps. *sPCA+CCA:*
The schematic diagram of the sparse PCA+CCA fusion method is shown in Fig.1. sPCA is performed on ECM and VBM maps
separately to generate a dimension-reduced data set $$$Y_r$$$. The input data to sPCA has a dimension
of subjects x voxels. Since the
goal of implementing sparsity is to suppress irrelevant voxels,* **L*_{1} sparsity penalty term over voxels $$$||v||_1<=c$$$ is added to the conventional PCA but not over subjects. The objective
function in sPCA is formulated as Eq.1. Considering that conventional PCA has the *L*_{2} penalty, the penalty in sPCA is the elastic penalty and
thus sPCA has a unique solution even when the number of voxels is larger than
the number of subjects. sPCA is solved by an alternating iteration method [3]. An
imputation method is developed to select the sparsity level and the number of
principal components. In detail, one tenth of elements are randomly removed
from the data matrix, then sPCA is applied on the imputed matrix to optimize
sparsity parameters. The number of principal components is determined by using the
Akaike information criterion [4]. Then
CCA is applied on $$$Y_r$$$ to maximize the canonical correlation
between imaging modalities. Finally, the spatial maps corresponding to $$$A_r$$$ can be obtained as $$$C_r=A_r^+X_r$$$. *Methods comparison:* sPCA+CCA was compared with previously
introduced fusion methods including PCA+CCA [5], parallel ICA [6] and sparse CCA
(sCCA). Two sample t-tests with unequal variances were applied on each
component of modulation profiles $$$A_r$$$. Receiver
operating characteristic (ROC) method was performed on $$$A_r$$$ to determine group separation for each
component and the area under ROC ($$$AUC$$$) was calculated.### Results

The principal components
having largest variance from sPCA and PCA are shown in Fig.2. No threshold is
applied for both ECM maps (Fig.2a) and VBM maps (Fig.2b). In sPCA, around 70%
of voxels have been specified as zero at the optimal sparsity level. As can be
seen in Fig.2, sPCA keeps high-weight voxels in the default mode network but
removes most low-weight voxels. Fig.3 and Fig.4 show the most significant
group-discriminant ECM and VBM z-score maps from sPCA+CCA, PCA+CCA, parallel
ICA and sCCA with threshold $$$|z|>2$$$. The clusters in the spatial
maps passing corrected $$$p<0.05$$$ are listed in Table1. In the ECM dataset, only
the component from sPCA+CCA is significant (corrected $$$p=0.018, AUC=0.76$$$). In the VBM dataset, the
component from sPCA+CCA is the most significant among these four fusion methods
and there is no significant component in sCCA (sPCA+CCA: corrected $$$p=2.2 \times 10^{-4}, AUC=0.83$$$; PCA+CCA: corrected $$$p=7.5 \times 10^{-3}, AUC=0.79$$$; parallel ICA: corrected $$$p=1.9 \times 10^{-3}, AUC=0.78$$$). Overall, sPCA+CCA has the strongest statistical power in discriminating
MCI from NC in terms of $$$AUC$$$. ### Discussion

We have proposed a sPCA
method to eliminate irrelevant voxels and developed an imputation method to
control the signal sparsity level and number of principal components. Furthermore,
sPCA was applied together with CCA to fuse fMRI and T1 images. Compared with
three state-of-the-art fusion methods, the components obtained from sPCA+CCA in
both modalities are most group-discriminative. The default mode network in ECM
and hippocampus in VBM found by sPCA+CCA corroborate with other studies [7, 8]. ### Conclusion

sPCA is a powerful method
for sparse regularization and dimensionality reduction. In addition, sPCA is
completely data-driven and self-adaptive without user intervention.### Acknowledgements

The study is
supported by the National Institutes of Health (grant number 1R01EB014284 and
P20GM109025). Data collection and sharing for this project was funded by the ADNI
(National Institutes of Health Grant U01 AG024904) and DOD ADNI (Department of
Defense award number W81XWH-12-2-0012). ### References

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