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<!DOCTYPE html>
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<title>Cluster Analysis in Dynamic Functional Network Connectivity</title>
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<div class="my-title">
The Role of Clustering in Dynamic Functional Network Connectivity for Schizophrenia Diagnosis
</div>
<div class="authors">
Bradley Baker, Dahim Choi, Anish Khazane, Jitendra Kumar, Eric Martin
</div>
<div class="markdown-body">
<div style="margin-bottom: 16em;">
<div id="toc_container" style="width: 30%; float: left; margin-left: 20%;">
<p class="toc_title">Contents</p>
<ul class="toc_list">
<li><a href="#introduction">Introduction</a></li>
<li><a href="#section-ii-data">Data</a></li>
<li><a href="#section-iii-methods">Methods</a></li>
<li><a href="#section-iv-results-discussion">Results & Discussion</a></li>
<li><a href="#section-v-conclusion">Conclusion</a></li>
</ul>
</div>
<div id="toc_container" style="width: 40%; float: right; margin-right: 10%;">
<p class="toc_title">Appendices</p>
<ul class="toc_list">
<li><a href="#gaussian_simulations">A: Simulation Details</a></li>
<li><a href="#elbow-criterion">B: Elbow Criterion</a></li>
<li><a href="#ucla-dataset-1">C: UCLA Data Results</a></li>
<li><a href="#gaussian-simulated-dataset-1">D: Simulated Data Results</a></li>
<li><a href="#initial-ucla-significant-comparisons">E: Statistical Analysis of Features</a></li>
</ul>
</div>
</div>
<h1 id="introduction">Introduction</h1>
<figure>
<img src="results/Overview_pipeline.png" />
<figcaption><b>Figure:</b> Overview of our contribution with this project. The standard pipeline for dFNC
uses
only KMeans clustering
as part of the pipeline, which introduces implicit assumptions on the organization of the FNC
feature-space. We
test four other clustering
algorithms, and evaluate how they produce FNC states, and how these states aid in schizophrenia
classification.
</figcaption>
</figure>
<p>Schizophrenia is a chronic and serious mental disorder which affects how a person thinks, feels, and
behaves.
Although there have been many studies about psychological and behavioral manifestations of schizophrenia,
neuroscientists have yet to determine a set of corresponding neurological biomarkers for this disorder. A
functional
magnetic resonance imaging (fMRI) can help determine non-invasive biomarkers for schizophrenia in brain
function<a href="#ref1">[1]</a><a href="#ref2">[2]</a> and one such fMRI analysis technique called dynamic
functional
network connectivity (dFNC)<a href="#ref3">[3]</a><a href="#ref4">[4]</a><a href="#ref10">[10]</a> uses
K-Means
clustering to characterize time-varying connectivity between functional networks. Researchers have worked on
finding
correlation between schizophrenia and dFNC<a href="#ref1">[1]</a><a href="#ref2">[2]</a><a href="#ref5">[5]</a><a
href="#ref6">[6]</a>. Indeed, dFNC features are well-established as features for
schizophrenia classification <a href="#ref14">[14]</a>.
</p>
<p>
Despite the utility of dFNC analysis, little work has been done analyzing the choice of clustering
algorithm,
which in most
applications is simply K-Means<a href="#ref7">[7]</a><a href="#ref9">[9]</a>.
It is possible that the reliance of K-Means assumptions (sphericality of clusters, reliability of
elbow-criterion,
etc)
have introduced bias into previous results using dFNC, and restricted both biological discovery, and
reliability
of
classifiers used for diagnosis.</p>
<p>
Therefore, in this project, we have
studied how modifying the
clustering technique in the dFNC pipeline can yield dynamic states from fMRI data, and how that choice
impacts the
accuracy of
classifying schizophrenia<a href="#ref8">[8]</a>.</p>
<p>We experimented with DBSCAN, Hiearcharial Clustering, Gaussian Mixture Models, and Bayesian Gaussian
Mixture
Models
clustering methods on subject connectivity matrices produced from fMRI data, and use each algorithm’s cluster
assignments as features for SVMs, MLP, Nearest Neighbor, and other supervised classification algorithms to
classify
schizophrenia.</p>
<p>Section II describes the fMRI data used in our experimentation, while Section III summarizes the
aforementioned
clustering and classification algorithms used in the pipeline. Section IV compares the accuracy of these
classifiers, along with presenting a series of charts that analyze the cluster assignments produced on the
fMRI
data.</p>
<h1 id="section-ii-data">Section II: Data</h1>
<p>All datasets used in this project are derivatives of fMRI data (functional magnetic resonance imaging),
which is
data measuring brain activity to track a patient’s thought processs over a predefined time period.</p>
<p> The BOLD signal in fMRI data tracks the movement of blood throughout a subject's brain during the scan
period.
These
changes in blood flow can be further analyzed to make conclusions about flow between different regions,
regions of
concentrated activity, and other conclusions, which can be useful for the diagnosis of mental disorders,
among
other
applications. The full 4-Dimensional volumes are often not used for classification in their raw form, and
instead
derivatives
such as Functional Network Connectivity states can be used for diagnosis.
</p>
<figure>
<img src="https://i.stack.imgur.com/lkXDb.png" />
<figcaption><b>Figure:</b> Visualization of a functional MRI time-series volume. Image from Chapter 2 of
"Principles of fMRI" by Wager and Lindquist <a href="#lindquist">[23]<a></figcaption>
</figure>
<table>
<thead>
<tr class="header">
<th>Data Set Name</th>
<th>Number of Subjects</th>
<th>Number of Healthy Controls</th>
<th>Number of Patients</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td><b>FBIRN</b></td>
<td>362</td>
<td>186</td>
<td>176</td>
</tr>
<tr class="even">
<td><b>UCLA</b></td>
<td>332</td>
<td>272</td>
<td>50</td>
</tr>
<tr class="odd">
<td><b>Simulations</b></td>
<td>300</td>
<td>150</td>
<td>150</td>
</tr>
</tbody>
</table>
<h3 id="gaussian-simulated-dataset">Gaussian Simulated Dataset</h3>
<p> We generated synthetic FNC data which were used for debugging code, and evaluating
our method. The details of the simulation are described in the appendix. </p>
<h3 id="fbirn-dataset">FBIRN Dataset</h3>
<p>We use derivatives from the Phase 3 Dataset of FBIRN (Functional Biomedical Infromatics Research Network
Data
Repository), which specifically focuses on brain activity maps from patients with schizophrenia. This
dataset
includes 186 healthy controls and 176 indivduals from schizophrenia from around the United States. Subject
participants in this dataset are between the ages of 18-62.</p>
<h3 id="ucla-dataset">UCLA Dataset</h3>
<p>We also use derivatives from the UCLA Consortium for Neuropsychiatric Phenomics archives, which includes
neuroimages for roughly 272 participants. The subject population consists of roughly 272 healthy controls,
as well
as participants with a diagnosis of schizophrenia (50 subjects). Subject participants in this dataset range
from
21
to 50 years</p>
<h1 id="section-iii-methods">Section III: Methods</h1>
<h2 id="overview">Overview</h2>
<p>
<figure>
<img width="50%" src="results/dfnc_pipeline(1).png?raw=True" />
<figcaption><b>Figure:</b> Pipeline overview of the full dFNC classification framework. In our analysis,
we
are interested in evaluating the role of the Clustering algorithm on the results.</figcaption>
</figure>
</p>
<h2 id="preprocessing">Preprocessing</h2>
<p>As is standard in Dynamic Functional Network Connectivity, Group Independent Component Analysis of subject
images
is used to compute a set of <span class="math inline"><em>M</em></span> statistically independent spatial
maps
from
the data, along with a set of <span class="math inline"><em>M</em></span> time-courses, which are used for
the
dFNC
analysis. For the FBIRN dataset, we used pre-computed ICA timecourses provided by Damaraju et al. 2014. These
timecourses were computed using Infomax ICA <a href="#ref11">[11]</a> with 100 components, which were then
manually selected based on biological relevance for a total of 47 remaining components.</p>
<p>For the UCLA data, because
no ICA derivatives were
readily available, we used Group Information Guided ICA <a href="#ref12">[12]</a> which is included as part
of the
GIFT NeuroImaging
Analysis Toolbox <a href="#ref15">[15]</a>. As a template for GIG-ICA, we used the NeuroMark template <a
href="#ref16">[16]</a>, which has been shown to provide
reliable estimations of components in many different settings <a href="#ref13">[13]</a>. This provides us
with 53
independent components for
the UCLA dataset.</p>
<h3 id="sliding-window-analysis">Sliding Window Analysis</h3>
<p>For all data-sets we chose a sliding window of size <span class="math inline"><em>W</em> = 22</span>,
following
the
precedent in Damaraju et al. 2014 <a href="#ref1">[1]</a>. Time-Series data <span
class="math inline"><em>X</em></span> with size <span class="math inline"><em>M</em> × <em>T</em></span>,
where
<span class="math inline"><em>M</em></span> is the
number of independent components, and <span class="math inline"><em>T</em></span> is the number of
timepoints are
used. Time series were normalized and convolved with a gaussian window with a kernel of size 0.015, again
following
the precedent in Damaraju et al. 2014 <a href="#ref1">[1]</a>. For each window on each time-series a number
of
exemplar data-points were
selected by taking the local maximum of the variance from the smoothed time-series.</p>
<p>We computed correlation coefficients across the components, within each time series window to form the FNC
matrix
with entries <span class="math inline"><em>i</em></span> and
<span class="math inline"><em>j</em></span> given as
<figure>
<img style="float:left;padding-right:10px;padding-bottom:10px;" src="results/centered_equation_1.png?raw=True"
alt="Image" width="20%" />
</figure>
<h2 style="clear:left;"></h2>
<p>Through this process, we generate a total of <span
class="math inline"><em>N</em> × (<em>T</em> − <em>W</em>)</span> window instances, which are used as
the
input
for clustering.</p>
<h2 id="clustering-details">Clustering Details</h2>
<p>We implemented 5 different clustering algorithms as part of the dFNC pipeline:</p>
<ul>
<li>
<p><b>KMeans</b><a href="#kmeans"> [24.]</a></p>
</li>
<li>
<p><b>DBSCAN</b><a href="#dbscan"> [25.]</a></p>
</li>
<li>
<p><b>Gaussian Mixture Models</b> </p>
</li>
<li>
<p><b>Bayesian Gaussian Mixture Models</b></p>
</li>
<li>
<p><b>Agglomerative Hierarchical Clustering</b></p>
</li>
</ul>
<p>For each of these algorithms, we used them for both exemplar clustering, and clustering on the full data.
In
the
case of KMeans,
Gaussian Mixture Models, and Bayesian Gaussian Mixture Models, exemplar cluster-centers were used as input
for
initialization
in the second stage of clustering using the same algorithm. In the case of DBSCAN and Agglomerative
clustering,
the exemplar stage was only used for elbow criterion, and other hyper-parameter tuning. We provide a more
exhaustive analysis of our elbow-criteria experiments in Appendix B. </p>
<p>For GMMs and Bayesian GMMs, we elected to use hard-assignment for determining cluster membership, because
it
lent to simpler integration with other steps in the pipeline. In future work, we would like to try to use
the
more rich information provided by soft-assignment to somehow improve feature generation, which is further
discussed below.</p>
<h2 id="classification-details">Classification Details</h2>
<h3>Task Formulation</h3>
<p>Our chosen task is the binary classification of Schizophrenic Patients from Healthy Controls using
Functional Network Connectivity states. For patients with a schizophrenic diagnosis, we assign the label
<b>1</b>,
and for all other subjects, we assign the label <b>0</b>. Given the set of FNC features for all subjects,
our
goal is to find the classifier which provides the most accurate diagnosis.</p>
<h3 id="evaluation-metric">Evaluation Metric</h3>
<p>For our evaluation metric, we used the Area-Under the “Receiving Operating Characteric” (ROC) curve, or
<b>ROC-AUC</b>.
This metric, which is often used for binary classifiers, describes the probability that a given classifier
will
rank a randomly chosen positive instance higher than a randomly chosen negative instance.
The curve is creative by plotting the fraction of true positive out of the positive vs the fraction of
false pasitive out of the negatives over various thresholds <span class="math inline"><em>T</em></span>. In
general, a higher AUC indicates a
better
binary classification performance.
</p>
<h3 id="feature-generation">Feature Generation</h3>
<p>Classification over raw dynamic FNC states is often poor, and further feature generation is required.</p>
<p>We first experimented with the use of subject state-vectors for classification. Because each time point
for
each subject is assigned a state during clustering, we can generate a vector of <span
class="math inline"><em>T-W</em></span> state assignments
for each
subject to be used as features.
</p>
<p>Secondly, we follow the precedent in <a href="#ref14">[14]</a>, and compute means of clusters
for each class, and
use
these centers to form a regression matrix of size <span
class="math inline">2<em>k</em> × (<em>T</em> − <em>W</em>)</span>. For each subject, we then linearly
regress
out
these
cluster centers from each FNC window, and collect the beta coefficients. The mean of the <span
class="math inline"><em>β</em></span>-coefficients over all timepoints are then used as features for the
final
classification.</p>
<p>
We computed two-tailed T-Tests over each of the feature spaces used for classification, and also
over the centroids for each class prior to regression, in order to visualize statistically significant
features which may or may not improve the performance of the classifiers.
</p>
<h3>Chosen Algorithms</h3>
<p>For supervised learning, we used a modified version of the <a
href="(https://github.com/alvarouc/polyssifier">Polyssifier classification
toolbox</a>,
which wraps the majority of the classification algorithms from SKlearn into an end-to-end framework. The
modified version
of the toolbox, which is being maintained by <a href="https://github.com/trendscenter/polyssifier">Brad
Baker</a>, . Within polyssifier, we implemented the following binary
classifiers:
</p>
<p>
<div style="column-count: 2;">
<ul>
<li>
<p>Multilayer Perceptron</p>
</li>
<li>
<p>K-Nearest Neighbors Classifier</p>
</li>
<li>
<p>Support Vector Machines</p>
</li>
<li>
<p>Decision Tree Classifier</p>
</li>
<li>
<p>Random Forests <a href="#random_forests">[17]</a></p>
</li>
<li>
<p>Extra Trees <a href="#extra_trees">[18]</p></a>
</li>
<li>
<p>Ada-Boost Decision Trees <a href="#adaboost">[19]</a></p>
</li>
<li>
<p>Bagging Decision Trees <a href="#bagging">[20]</a></p>
</li>
<li>
<p>Gradient Boosted Decision Trees <a href="#gradient_boost">[21]</a></p>
</li>
<li>
<p>Logistic Regression</p>
</li>
<li>
<p>Passive Aggressive Classifier <a href="#passive_aggresive">[22]</a></p>
</li>
<li>
<p>Perceptron</p>
</li>
<li>
<p>Naive Bayes</p>
</li>
<li>
<p>Voting Classifer</p>
</li>
</ul>
</p>
</div>
<h3 id="grid-search">Grid Search</h3>
<p>We performed a Grid-Search over all available parameters for each of the classifiers available in
Polyssifier.
For each set of training data used for cross-validation, we trained the grid-search separately, and took
the
parameters with the highest AUC.</p>
<p>For the sake of space, we have ommitted the details of the grid-search, but the results have been
included in
the
results section of the repo.
</p>
<h1 id="section-iv-results-discussion">Section IV: Results & Discussion</h1>
<p> For brevity, we only present results from modifying the dFNC pipeline with the aforementioned clustering
and classification algorithms on the FBIRN dataset in this section. We leave our results on the gaussian
simulated
and UCLA datasets in Appendix D and C respectively, if the reader is interested in viewing them. </p>
<h2 id="reporting-results">Evaluation Setup</h2>
<p>For all results, the Area Under Curve (AUC) metric was used since we were trying to classify two possible
outcomes. Patients were with either healthy or had schizophrenia. AUC scores near 0.50 meant that the
models
randomly diagnosed patients as healthy or schizophrenic. This represented the worst possible outcome. Any
model
with AUC scores between 0.40 and 0.60 was deemed a failed model. AUC scores near 1.0 meant that
diagnoses
were
accurate with few false positives (high specificity) and few false negatives (high recall). There were no
observed
models that completely reversed the diagnoses with AUC scores near 0.0.</p>
<p>Each classification algorithm was evaluated with <b>10-fold cross-validation</b>, with a grid-search
for classifier parameters performed separately on each validation set.
</p>
<h2 id="fbirn-dataset-1">Quantitative Analysis: AUC Analysis on FBIRN Dataset</h2>
<p>
We initially trained the supervised learning model on the unclustered FBIRN FNC matrices
in order to establish a baseline accuracy that we could compare our beta feature generation approach against.
All the classifiers failed to consistently achieve AUC scores outside of the 0.40-0.60 range suggesting
that all the models produced random diagnoses. This confimed our hypothesis that clustering and feature
reduction was paramount to succesful classfication of healthy and schizoprhenic patients. The baseline results
are displayed below.
</p>
<figure>
<img src="images/fbirn_pre_clus_accuracy.png?raw=true" width="25%" />
<figcaption><b>Figure: </b>AUC scores for 10-fold cross-validation of multiple classifiers using only the
concatenated FNC
matrices as features. The y axis is the AUC metric for each classifier, and the x-axis separates the
classifiers.</figcaption>
</figure>
<p>
After establishing our baseline results we trained the models on the FBIRN dataset using only cluster
assignments.
No beta features were generated to reduce the dimensionality of the training dataset. The results are displayed
below.
</p>
<figure>
<img width="100%" src="results/fbirn_assignments_accuracy.png?raw=true" />
<img style="width:30%" src="results/accuracy_legend.png?raw=true" />
<figcaption><b>Figure: </b> FBIRN Schizophrenia Classification AUC scores using subject state-vectors as
features. The y-axis
provides the AUC metric
and the x-axis separates the scores obtained over 10-fold cross-validation for each classifier.
</figcaption>
</figure>
<p>The results indicated that it was possible for some of the clustering algorithms to lift the AUC scores
to an
average of 0.70 for most supervised models with the exclusion of the Voting learner. The one exception was
DBSCAN
which failed to move the AUC from 0.50. Regardless, none of these models achieved a high enough score to
be used
in a clinical environment. It was surmised that the number of features in the datasets had to be reduced from
cluster
assignments over time to beta features in order for the supervised learning models to accurately diagnose
patients.
</p>
<p>
After performing beta features generation with clustering, the final results for the FBIRN dataset were
obtained and are displayed below.
</p>
<table style="width:50%;">
<colgroup>
<col style="width: 12%" />
<col style="width: 10%" />
<col style="width: 13%" />
<col style="width: 12%" />
<col style="width: 18%" />
<col style="width: 10%" />
<col style="width: 10%" />
<col style="width: 10%" />
</colgroup>
<thead>
<tr class="header">
<th>Clustering Algorithm</th>
<th>SVM</th>
<th>Multilayer Perceptron</th>
<th>Logistic Regression</th>
<th>Passive Aggressive Classifier</th>
<th>Perceptron</th>
<th>Random Forest</th>
<th>Extra Trees</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>kmeans</td>
<td><em>0.952 ± 0.036</em></td>
<td><em>0.92 ± 0.065</em></td>
<td><em>0.944 ± 0.039</em></td>
<td><em>0.945 ± 0.035</em></td>
<td><strong>0.902 ± 0.043</strong></td>
<td><em>0.871 ± 0.038</em></td>
<td><em>0.853 ± 0.04</em></td>
</tr>
<tr class="even">
<td>gmm</td>
<td><em>0.936 ± 0.054</em></td>
<td><em>0.946 ± 0.038</em></td>
<td><em>0.943 ± 0.038</em></td>
<td><em>0.929 ± 0.031</em></td>
<td><em>0.882 ± 0.04</em></td>
<td><strong>0.885 ± 0.022</strong></td>
<td><strong>0.874 ± 0.026</strong></td>
</tr>
<tr class="odd">
<td>bgmm</td>
<td><em>0.955 ± 0.037</em></td>
<td><em>0.932 ± 0.042</em></td>
<td><em>0.945 ± 0.038</em></td>
<td><em>0.939 ± 0.038</em></td>
<td><em>0.896 ± 0.074</em></td>
<td><em>0.86 ± 0.039</em></td>
<td><em>0.87 ± 0.056</em></td>
</tr>
<tr class="even">
<td>dbscan</td>
<td>0.883 ± 0.027</td>
<td>0.893 ± 0.031</td>
<td>0.892 ± 0.033</td>
<td>0.884 ± 0.027</td>
<td>0.828 ± 0.064</td>
<td>0.805 ± 0.064</td>
<td>0.806 ± 0.058</td>
</tr>
<tr class="odd">
<td>hierarchical</td>
<td><strong>0.957 ± 0.032</strong></td>
<td><strong>0.954 ± 0.038</strong></td>
<td><strong>0.953 ± 0.038</strong></td>
<td><strong>0.951 ± 0.032</strong></td>
<td><em>0.891 ± 0.098</em></td>
<td><em>0.881 ± 0.032</em></td>
<td><em>0.872 ± 0.048</em></td>
</tr>
</tbody>
</table>
<figure>
<img width="100%" src="results/fbirn_betas_accuracy.png?raw=True" />
<img style="width:30%" src="results/accuracy_legend.png?raw=true" />
<figcaption><b>Figure: </b> FBIRN Schizophrenia classification AUC scores using <span
class="math inline">\beta</span>-coefficients as
features. The y-axis
provides the AUC metric
and the x-axis separates the scores obtained over 10-fold cross-validation for each classifier.
</figcaption>
</figure>
<p>
The reduced number of training features increased the accuracy of all the learners with Support
Vector Machines achieving the highest accuracy across all clustering algorithms. Only DBCAN failed to get
above an average AUC score of 0.90 for support vector machines and Hierarchical clusterting obtained this
highest score of 0.957.
</p>
<hr />
<h2 id="fbirn-data-visualizations">Qualitative Analysis: FNC State and Clustering Visualization Analysis</h2>
<p> In the following section, we display both 2D and 3D clustering visualizations for KMeans, GMMs, BGMMs,
DBSCAN, and Hiearcharial Clustering after performing PCA on the
FBIRN dataset. We mark the centroids of each clustering algorithm with a bolded <b> X </b>, and take the average
of points of each cluster in the DBSCAN visualizations to represent its cluster's centers. </p>
<p> We also present FNC state diagrams that analyze how each clustering method clustered data related to healthy
patients and schizophrenic patients differently. We hope these diagrams can help neuroscience researchers find
interesting biomarkers for further work. </p>
<p> Lastly, we present clustering visualizations and FNC State diagrams for the UCLA and gaussian simulated dataset in
Appendix C and D respectively. </p>
<h3> Kmeans Clustering </h3>
<figure>
<img width="98%" src="results/kmeans_fbirn_betas/kmeans_fbirn_states.png?raw=true" />
<figcaption><b>Figure: </b> the FBIRN FNC states detected for each class using K-Means clustering. The top
row
are the
states for healthy controls, and the bottom row are states for schizophrenic patients. These results
follow
closely with the results from Damaraju et al. 2014 [14], which indicates our pipeline works.
</figcaption>
</figure>
<figure><img style="width: 49%" src="results/kmeans_fbirn_betas/kmeans_fbirn_visualization_3d.png?raw=true" /> <img
style="width: 49%" src="results/kmeans_fbirn_betas/kmeans_fbirn_visualization.png?raw=true" /><br />
<figcaption><b>Figure: </b>Visualization of FBIRN cluster-assignments to FNC windows for PCA-reduced
windows in
2-D
and 3-D. Each color corresponds to one of the <span class="math inline"><em>k</em> = 5</span> states.
</figcaption>
</figure>
<hr />
<h3> GMM Clustering </h3>
<figure><img width="98%" src="results/gmm_fbirn_betas/gmm_fbirn_states.png?raw=true" />
<figcaption><b>Figure: </b> the FBIRN FNC states detected for each class using GMM clustering. The top row
are
the states for healthy controls, and the bottom row are states for schizophrenic patients. We notice
that
states 2 and
3 resemble states 3 and 4 from KMeans, and state 1 resembles state 2 from kmeans. State 0 and State 4
most closely resemble state 0 from KMeans; however, both show greater negative connectivity in modules
where it is not detected by KMeans.</figcaption>
</figure>
<figure><img style="width: 49%" src="results/gmm_fbirn_betas/gmm_fbirn_visualization_3d.png?raw=true" /> <img
style="width: 49%" src="results/gmm_fbirn_betas/gmm_fbirn_visualization.png?raw=true" /><br />
<figcaption><b>Figure: </b>Visualization of FBIRN cluster-assignments to FNC windows for PCA-reduced
windows in
2-D
and 3-D. Each color corresponds to one of the <span class="math inline"><em>k</em> = 5</span> states. We see
that cluster 4 here is
not detected by KMeans, perhaps indicating that the GMM has detected a new state in the FBIRN
data, which is still useful for schizophrenia diagnosis.
</figcaption>
</figure>
<hr />
<h3> Bayesian GMM </h3>
<figure><img width="98%" src="results/bgmm_fbirn_betas/bgmm_fbirn_states.png?raw=true" />
<figcaption>The Fbirn FNC states detected for each class using Bayesian GMM clustering. The states closely
resemble those detected with
standard GMM clustering.
</figcaption>
</figure>
<figure><img style="width: 49%" src="results/bgmm_fbirn_betas/bgmm_fbirn_visualization_3d.png?raw=true" /> <img
style="width: 49%" src="results/bgmm_fbirn_betas/bgmm_fbirn_visualization.png?raw=true" /><br />
<b>Figure: </b>Visualization of clusters with BGMM clustering in 2-d and 3-d with FBirn Data. We notice that a
similar clustering
partition occurs for the Bayesian GMM as for the standard GMM.</figure>
<hr />
<h3> DBSCAN Clustering </h3>
<figure><img width="98%" src="results/dbscan_fbirn_betas/dbscan_fbirn_states.png?raw=true" />
<figcaption><b>Figure: </b>The two states detected by DBSCAN
clustering on the FBIRN data. Although the resulting classification performance using these states is quite
low (~0.80 AUC), we notice that patterns of
connectivity detected do closely resemble other states detected with KMeans, GMM, and bGMM, indicating that
these may represent "super"-clusters,
which alone do not provide a good characterization of classes, but which still may be useful for further
analysis.</figcaption>
</figure>
<figure><img style="width: 49%" src="results/dbscan_fbirn_betas/dbscan_fbirn_visualization_3d.png?raw=true" /> <img
style="width: 49%" src="results/dbscan_fbirn_betas/dbscan_fbirn_visualization.png?raw=true" /><br />
<b>Figure: </b> Visualization of clusters with DBSCAN clustering in 2-d and 3-d with FBIRN Data. As seen
in the plots above, DBSCAN fails to find more than 2 clusters, even labeling a large majority of the data
labels as "noise" (-1 label). Accordingly, the clusterer performs poorly on FBIRN data, but
does receive a lift after beta coefficient features are created for its clustering assignments.
</figure>
<hr />
<h3>Agglomerative Hierarchical Clustering</h3>
<figure><img width="98%" src="results/hierarchical_fbirn_betas/hierarchical_fbirn_states.png?raw=true" />
<figcaption><b>Figure: </b>The FNC states detected for the FBIRN data using hierarchical clustering.
We observe that although some similarity in connectivity is shared between these states and the states
resulted from KMeans, the centers are still quite different than the standard states detected for this data
set. Since the
resulting states provide the best classification of the data, further neurological interpretation may confirm
that these
connectivity patterns are useful biomarkers for schizophrenia diagnosis.</figcaption>
</figure>
<figure><img style="width: 49%"
src="results/hierarchical_fbirn_betas/hierarchical_fbirn_visualization_3d.png?raw=true" />
<img style="width: 49%"
src="results/hierarchical_fbirn_betas/hierarchical_fbirn_visualization.png?raw=true" /><br />
<figcaption><b>Figure: </b>Visualization of clusters with Hierarchical clustering in 2-d and 3-d with FBirn
Data. We observe that the
partitioning of the clusters is not clear under PCA reduction, like it is for the other clustering algorithms.
Because hierarchical
clustering provides the best states for producing classification features, this may mean it is better
partitioning distinctive
states in the higher-dimensional space.</figcaption>
</figure>
<p> As a summary of the results posted above, we see that 4 out of 5 tested clustering methods (KMeans,
Hiearcharial
Clustering, GMM, BGMM) are able to find 4-5 distinct clusters within the FBIRN data, as evidence by their
respective 2D and 3D
visualizations shown in this section. Interestingly, each of these clustering algorithms
produced clusters that are fairly segmented, which indicates that applying clustering on fMRI data is valuable
for finding features that more easily separable in classification. DBSCAN only appears to find 2 clusters, with one of them being marked
as "noise," and accordingly produces features that perform worse in comparison to other clustering algorithms
on the Schizophrenia prediction task. </p>
<p> In terms of the Schizophrenia prediction task, Hiearcharial Clustering appears to produce assignments that
give
the greatest lift to all classifiers, with SVMs benefitting the most from its cluster assignments. However, the
a few of the other clustering algorithms also perform strongly with GMMs and BGMMs performing comparatively,
KMeans slightly behind,
and DBSCAN performing slightly worse in all experiments. </p>
<p> More biological analysis is required to make any concrete recommendations based off these results, but our
results
appear to signal that the <b> choice of clustering and classification algorithm in the dFNC pipeline does make
a significant difference in the quality of the output states and diagnosis prediction accuracy. </b> </p>
<h1 id="section-v-conclusion">Section V: Conclusion</h1>
<p>
<ul>
<li>
<p>
<b>Cluster-Based features greatly improve classification</b> performance for the
schizophrenia classification task.</p>
</li>
<li>
<p>Classification of dFNC states <b>performs comporably when using GMM, Bayesian GMM, Hierarchical
Clustering,
and
KMeans</b></p>
</li>
<li>
<p><b>GMM, bGMM, and Hierarchical clustering all detect states not detected with KMeans</b>, which
provide
either comparable or better classification of schizophrenia, indicating these states may have
biological significance for diagnosis.</p>
</li>
<li>
<p><b>Hierarchical Clustering provides the highest classification results</b> for FBIRN, and GMM/bGMM
provide
the
highest results for
UCLA data, while also providing slightly different states than are detected with KMeans. These states
may be
useful for further
neurological analysis.</p>
</li>
<li>
<p>DBSCAN's sensitivity to parameters requires exhaustive hyper-parameter searching, without clear
improvements
in performance, but because states collected by DBSCAN in FBIRN data provided decent results (~80%
accuracy),
they may provide
some biological insight.</p>
</li>
<li>
<p>As established in literature <a href="#ref14">[14]</a>, SVM classifiers perform most reliably for
classifying dFNC states, which
is
consistent for both real data sets, and all clustering algorithms.</p>
</li>
</ul>
</p>
<h2>Further Work</h2>
<p>
<ul>
<li>
<p>Further neurological analysis of detected states is required for more thorough interpretation of
results</p>
</li>
<li>
<p>Elbow criterion for different clustering algorithms revealed some inconsistency in the use of <span
class="math inline"><em>k</em> = 5</span>
for
the
number of states. Applying the full classification pipeline to other numbers of states may provide
further
insight
into states, and improve classification even further. </p>
</li>
<li>
<p>More advanced clustering and classification algorithms using Deep Learning may provide significant
lifts
classification. Particularly the use of Variational AutoEncoders for clustering seems a particularly
promising
avenue of future investigation.</p>
</li>
</ul>
</p>
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<h1 id="appendices">Appendices</h1>
<i> <h2 id="gaussian_simulations"> Appendix A: Gaussian Simulation Details </h2> </i>
<i> <p>We derive our own synthetic dataset in order to test out our clusterers and classifiers on a simulated
dataset
for
sanity checking their implementation.</p>
<p>The data set was generated by a seed set of <span class="math inline"><em>M</em> = 50</span> source
gaussian
signals, which were organized into <span class="math inline"><em>k</em> = 5</span> “connectivity states”.
Each
of
the 5 connectivity states is a block-diagonal matrix with a block of size <span
class="math inline"><em>M</em>/<em>k</em></span> which is located on the <span
class="math inline"><em>i</em> * <em>M</em>/<em>k</em> + <em>M</em>/<em>k</em></span>th entry with a
size of
<span class="math inline"><em>i</em> * <em>M</em>/<em>k</em></span>. For each state, the set the pairs of
relevant
source signals to come from the same distribution, so that their correlation without additional additive
noise
is
1.
</p>
<p>For each subject, we generate a source signal of sime <span
class="math inline"><em>M</em> × <em>T</em> − <em>W</em></span> where <span class="math inline"><em>T</em></span>
is the length of the time-course, and <span class="math inline"><em>W</em></span> is the size of the
sliding
window.
Over each window, we randomly select which state the window belongs to, drawing from a prior distribution
of
state
probabilities for a subject’s class, <span class="math inline"><em>P</em><sub><em>c</em>, <em>k</em></sub></span>.
Additionally, for each class we simulate a transition probability <span
class="math inline"><em>Q</em><sub><em>c</em>, <em>k</em></sub></span> for each state, with higher
probabilities
meaning a higher chance of transitioning out of that state into some other state. Finally, we restrict the
simulation such that transitions can only occur after a state has existed for a <span
class="math inline"><em>W</em></span>-many timepoints. This restrictions allows us to assert with some
certainty
that each of the seed states will occur within some window for that subject.</p>
<p>For each class, we simulate baseline signal noise from a normal distribution, which the relevant source
signal
being added to that baseline.</p>
<p>Formally a single subjects signal within a window of size W is given as:</p>
<p align="center"><br /><span
class="math display"><em>X</em><sub><em>i</em></sub> = 𝒩(<em>θ</em><sub><em>c</em></sub>) + 𝒩(<em>θ</em><sub><em>k</em></sub>)</span><br />
</p>
<p>where <span class="math inline"><em>θ</em><sub><em>c</em></sub></span> and <span
class="math inline"><em>θ</em><sub><em>k</em></sub></span> are the parameters for the baseline class
signal
for
class <span class="math inline"><em>c</em></span> and the source signal for state <span
class="math inline"><em>k</em></span> respectively. Because all source signals for state <span
class="math inline"><em>k</em></span> will have the same distribution, their correlation will be high,
providing
the same block-matrix correlation for the dFNC analysis.</p>
<p>The probability for entering from state <span class="math inline"><em>k</em></span> into a new state
<span class="math inline"><em>k</em>′</span> at a timepoint <span class="math inline"><em>T</em></span>,
given that
the
last transition occured <span class="math inline"><em>W</em></span> time-points ago, is given as the joint
probability of the prior distribution for the class ever entering into state <span
class="math inline"><em>k</em>′</span>, as well as the probability of transitioning out of state <span
class="math inline"><em>k</em></span>. i.e. <br />
<p align="center">
<span
class="math display"><em>Z</em><sub><em>c</em>, <em>k</em>, <em>k</em>′</sub>(<em>P</em><sub><em>c</em>, <em>k</em>′</sub>, <em>Q</em><sub><em>c</em>, <em>k</em></sub>)</span>
</p><br />
</p>
<p>The addition of noise, and the fact that windows are created with a size <span
class="math inline"><em>W</em></span>
at each timepoint means that there is a high chance for bleed-over between
the actual states detected for an individual subject.</p>
<p>For example the following connectivity matrices were computed from randomly selected subjects from each
class,
for
each state:</p>
<table style="width: 98%">
<thead>
<tr class="header">
<th style="text-align: center;">Class</th>
<th style="text-align: center;">State 0</th>
<th style="text-align: center;">State 1</th>