Adaptive constrained independent vector analysis: An effective solution for analysis of large-scale medical imaging data

There is a growing need for flexible methods for the analysis of large-scale functional magnetic resonance imaging (fMRI) data for the estimation of global signatures that summarize the population while preserving individual-specific traits. Independent vector analysis (IVA) is a data-driven method that jointly estimates global spatio-temporal patterns from multi-subject fMRI data, and effectively preserves subject variability. However, as we show, IVA performance is negatively affected when the number of datasets and components increases especially when there is low component correlation across the datasets. We study the problem and its relationship with respect to correlation across the datasets, and propose an effective method for addressing the issue by incorporating reference information of the estimation patterns into the formulation, as a guidance in high dimensional scenarios. Constrained IVA (cIVA) provides an efficient framework for incorporating references, however its performance depends on a user-defined constraint parameter, which enforces the association between the reference signals and estimation patterns to a fixed level. We propose adaptive cIVA (acIVA) that tunes the constraint parameter to allow flexible associations between the references and estimation patterns, and enables incorporating multiple reference signals, without enforcing inaccurate conditions. Our results indicate that acIVA can reliably estimate high-dimensional multivariate sources from large-scale simulated datasets, when compared with standard IVA. It also successfully extracts meaningful functional networks from a large-scale fMRI dataset for which standard IVA did not converge. The method also efficiently captures subject-specific information, which is demonstrated through observed gender differences in spectral power, higher spectral power in males at low frequencies and in females at high frequencies, within the motor, attention, visual and default mode networks.

An ICA based approach for steady-state and transient analysis of task fMRI data: Application to study of thermal pain response

BACKGROUND

Data driven analysis methods such as independent component analysis (ICA) offer the advantage of estimating subject contributions when used in a second-level analysis. With the traditionally used regression-based methods this is achieved with a design matrix that has to be specified a priori.

NEW METHOD

We show that the ability of ICA to estimate subject contributions can be effectively used to perform steady-state as well as transient analysis of task functional magnetic resonance imaging (fMRI) data, which can help reveal important group differences.

RESULTS

We apply the method to steady-state and transient analysis of block designed thermal pain stimulated fMRI data, and identify distinct sex differences, in parts of the pain matrix: brain stem, thalamus, amygdala, frontal pole (FP), temporal pole (TP), operculum (second somatosensory cortex, SII), anterior insular (AI), dorsal anterior cingulate cortex (dACC), and default mode network (DMN). We also show that the identified regions have significant correlation with weekly exercise and anxiety. Using transient analysis, we identify regions (SII, AI, dACC, DMN) specific to female group showing difference mainly in the initial stages of the experiments.

COMPARISON WITH EXISTING METHOD

With exact same spatial components input in the second level, permutation analysis of linear models cannot identify any significant group difference. In addition, the proposed transient analysis cannot be realized if user is required to input a design matrix as is the case with regression-based analyses.

CONCLUSIONS

The proposed two-level ICA is an effective multi-variate analysis method for both steady-state and transient analysis of task data.

Extraction of Time-Varying Spatiotemporal Networks Using Parameter-Tuned Constrained IVA

Dynamic functional connectivity analysis is an effective way to capture the networks that are functionally associated and continuously changing over the scanning period. However, these methods mostly analyze the dynamic associations across the activation patterns of the spatial networks while assuming that the spatial networks are stationary. Hence, a model that allows for the variability in both domains and reduces the assumptions imposed on the data provides an effective way for extracting spatiotemporal networks. Independent vector analysis (IVA) is a joint blind source separation technique that allows for estimation of spatial and temporal features while successfully preserving variability. However, its performance is affected for higher number of datasets. Hence, we develop an effective two-stage method to extract time-varying spatial and temporal features using IVA, mitigating the problems with higher number of datasets while preserving the variability across subjects and time. The first stage is used to extract reference signals using group-independent component analysis (GICA) that are used in a parameter-tuned constrained IVA framework to estimate time-varying representations of these signals by preserving the variability through tuning the constraint parameter. This approach effectively captures variability across time from a large-scale resting-state fMRI data acquired from healthy controls and patients with schizophrenia and identifies more functionally relevant connections that are significantly different among healthy controls and patients with schizophrenia, compared with the widely used GICA method alone.

A method to compare the discriminatory power of data-driven methods: Application to ICA and IVA

BACKGROUND

The widespread application of data-driven factorization-based methods, such as independent component analysis (ICA), to functional magnetic resonance imaging data facilitates the study of neural function and how it is disrupted by psychiatric disorders such as schizophrenia. While the increasing number of these methods motivates a comparison of their relative performance, such a comparison is difficult to perform on real fMRI data, since the ground truth is, relatively, unknown and the alignment of factors across different methods is impractical and imprecise.

NEW METHOD

We present a novel method, global difference maps (GDMs), to compare the results of different fMRI analysis techniques on real fMRI data, quantify their relative performances, and highlight the differences between the decompositions visually.

COMPARISON WITH EXISTING METHODS

We apply this method to compare the performances of two different factorization-based methods, ICA and its multiset extension independent vector analysis (IVA), for the analysis of fMRI data from 109 patients with schizophrenia and 138 healthy controls during the performance of three tasks.

RESULTS

Through this application of GDMs, we find that IVA can determine regions that are more discriminatory between patients and controls than ICA, though IVA is less effective at emphasizing regions found in only a subset of the tasks.

CONCLUSIONS

These results demonstrate that GDMs are an effective way to compare the performances of different factorization-based methods as well as regression-based analyses.

The role of diversity in data-driven analysis of multi-subject fMRI data: Comparison of approaches based on independence and sparsity using global performance metrics

Data-driven methods have been widely used in functional magnetic resonance imaging (fMRI) data analysis. They extract latent factors, generally, through the use of a simple generative model. Independent component analysis (ICA) and dictionary learning (DL) are two popular data-driven methods that are based on two different forms of diversity-statistical properties of the data-statistical independence for ICA and sparsity for DL. Despite their popularity, the comparative advantage of emphasizing one property over another in the decomposition of fMRI data is not well understood. Such a comparison is made harder due to the differences in the modeling assumptions between ICA and DL, as well as within different ICA algorithms where each algorithm exploits a different form of diversity. In this paper, we propose the use of objective global measures, such as time course frequency power ratio, network connection summary, and graph theoretical metrics, to gain insight into the role that different types of diversity have on the analysis of fMRI data. Four ICA algorithms that account for different types of diversity and one DL algorithm are studied. We apply these algorithms to real fMRI data collected from patients with schizophrenia and healthy controls. Our results suggest that no one particular method has the best performance using all metrics, implying that the optimal method will change depending on the goal of the analysis. However, we note that in none of the scenarios we test the highly popular Infomax provides the best performance, demonstrating the cost of exploiting limited form of diversity.

A window-less approach for capturing time-varying connectivity in fMRI data reveals the presence of states with variable rates of change

Functional connectivity during the resting state has been shown to change over time (i.e., has a dynamic connectivity). However, resting-state fluctuations, in contrast to task-based experiments, are not initiated by an external stimulus. Consequently, a more complicated method needs to be designed to measure the dynamic connectivity. Previous approaches have been based on assumptions regarding the nature of the underlying dynamic connectivity to compensate for this knowledge gap. The most common assumption is what we refer to as locality assumption. Under a locality assumption, a single connectivity state can be estimated from data that are close in time. This assumption is so natural that it has been either explicitly or implicitly embedded in many current approaches to capture dynamic connectivity. However, an important drawback of methods using this assumption is they are unable to capture dynamic changes in connectivity beyond the embedded rate while, there has been no evidence that the rate of change in brain connectivity matches the rates enforced by this assumption. In this study, we propose an approach that enables us to capture functional connectivity with arbitrary rates of change, varying from very slow to the theoretically maximum possible rate of change, which is only imposed by the sampling rate of the imaging device. This method allows us to observe unique patterns of connectivity that were not observable with previous approaches. As we explain further, these patterns are also significantly correlated to the age and gender of study subjects, which suggests they are also neurobiologically related.

Blind Source Separation for Unimodal and Multimodal Brain Networks: A Unifying Framework for Subspace Modeling

In the past decade, numerous advances in the study of the human brain were fostered by successful applications of blind source separation (BSS) methods to a wide range of imaging modalities. The main focus has been on extracting "networks" represented as the underlying latent sources. While the broad success in learning latent representations from multiple datasets has promoted the wide presence of BSS in modern neuroscience, it also introduced a wide variety of objective functions, underlying graphical structures, and parameter constraints for each method. Such diversity, combined with a host of datatype-specific know-how, can cause a sense of disorder and confusion, hampering a practitioner's judgment and impeding further development. We organize the diverse landscape of BSS models by exposing its key features and combining them to establish a novel unifying view of the area. In the process, we unveil important connections among models according to their properties and subspace structures. Consequently, a high-level descriptive structure is exposed, ultimately helping practitioners select the right model for their applications. Equipped with that knowledge, we review the current state of BSS applications to neuroimaging. The gained insight into model connections elicits a broader sense of generalization, highlighting several directions for model development. In light of that, we discuss emerging multi-dataset multidimensional (MDM) models and summarize their benefits for the study of the healthy brain and disease-related changes.

The role of diversity in complex ICA algorithms for fMRI analysis

BACKGROUND

The widespread use of data-driven methods, such as independent component analysis (ICA), for the analysis of functional magnetic resonance imaging data (fMRI) has enabled deeper understanding of neural function. However, most popular ICA algorithms for fMRI analysis make several simplifying assumptions, thus ignoring sources of statistical information, types of "diversity," and limiting their performance.

NEW METHOD

We propose the use of complex entropy rate bound minimization (CERBM) for the analysis of actual fMRI data in its native, complex, domain. Though CERBM achieves enhanced performance through the exploitation of the three types of diversity inherent to complex fMRI data: noncircularity, non-Gaussianity, and sample-to-sample dependence, CERBM produces results that are more variable than simpler methods. This motivates the development of a minimum spanning tree (MST)-based stability analysis that mitigates the variability of CERBM.

COMPARISON WITH EXISTING METHODS

In order to validate our method, we compare the performance of CERBM with the popular CInfomax as well as complex entropy bound minimization (CEBM).

RESULTS

We show that by leveraging CERBM and the MST-based stability analysis, we are able to consistently produce components that have a greater number of activated voxels in physically meaningful regions and can more accurately classify patients with schizophrenia than components generated using simpler models.

CONCLUSIONS

Our results demonstrate the advantages of using ICA algorithms that can exploit all inherent types of diversity for the analysis of fMRI data when coupled with appropriate stability analyses.

Sample-poor estimation of order and common signal subspace with application to fusion of medical imaging data

Due to their data-driven nature, multivariate methods such as canonical correlation analysis (CCA) have proven very useful for fusion of multimodal neurological data. However, being able to determine the degree of similarity between datasets and appropriate order selection are crucial to the success of such techniques. The standard methods for calculating the order of multimodal data focus only on sources with the greatest individual energy and ignore relations across datasets. Additionally, these techniques as well as the most widely-used methods for determining the degree of similarity between datasets assume sufficient sample support and are not effective in the sample-poor regime. In this paper, we propose to jointly estimate the degree of similarity between datasets and their order when few samples are present using principal component analysis and canonical correlation analysis (PCA-CCA). By considering these two problems simultaneously, we are able to minimize the assumptions placed on the data and achieve superior performance in the sample-poor regime compared to traditional techniques. We apply PCA-CCA to the pairwise combinations of functional magnetic resonance imaging (fMRI), structural magnetic resonance imaging (sMRI), and electroencephalogram (EEG) data drawn from patients with schizophrenia and healthy controls while performing an auditory oddball task. The PCA-CCA results indicate that the fMRI and sMRI datasets are the most similar, whereas the sMRI and EEG datasets share the least similarity. We also demonstrate that the degree of similarity obtained by PCA-CCA is highly predictive of the degree of significance found for components generated using CCA.

Independent Vector Analysis for SSVEP Signal Enhancement, Detection, and Topographical Mapping

Steady state visual evoked potentials (SSVEPs) have been identified as an effective solution for brain computer interface (BCI) systems as well as for neurocognitive investigations. SSVEPs can be observed in the scalp-based recordings of electroencephalogram signals, and are one component buried amongst the normal brain signals and complex noise. We present a novel method for enhancing and improving detection of SSVEPs by leveraging the rich joint blind source separation framework using independent vector analysis (IVA). IVA exploits the diversity within each dataset while preserving dependence across all the datasets. This approach is shown to enhance the detection of SSVEP signals across a range of frequencies and subjects for BCI systems. Furthermore, we show that IVA enables improved topographic mapping of the SSVEP propagation providing a promising new tool for neuroscience and neurocognitive research.

Comparison of PCA approaches for very large group ICA

Large data sets are becoming more common in fMRI and, with the advent of faster pulse sequences, memory efficient strategies for data reduction via principal component analysis (PCA) turn out to be extremely useful, especially for widely used approaches like group independent component analysis (ICA). In this commentary, we discuss results and limitations from a recent paper on the topic and attempt to provide a more complete perspective on available approaches as well as discussing various issues to consider related to PCA for very large group ICA. We also provide an analysis of computation time, memory use, and number of dataloads for a variety of approaches under multiple scenarios of small and extremely large data sets.

Capturing subject variability in fMRI data: A graph-theoretical analysis of GICA vs. IVA

BACKGROUND

Recent studies using simulated functional magnetic resonance imaging (fMRI) data show that independent vector analysis (IVA) is a superior solution for capturing spatial subject variability when compared with the widely used group independent component analysis (GICA). Retaining such variability is of fundamental importance for identifying spatially localized group differences in intrinsic brain networks.

NEW METHODS

Few studies on capturing subject variability and order selection have evaluated real fMRI data. Comparison of multivariate components generated by multiple algorithms is not straightforward. The main difficulties are finding concise methods to extract meaningful features and comparing multiple components despite lack of a ground truth. In this paper, we present a graph-theoretical (GT) approach to effectively compare the ability of multiple multivariate algorithms to capture subject variability for real fMRI data for effective group comparisons. The GT approach is applied to components generated from fMRI data, collected from individuals with stroke, before and after a rehabilitation intervention.

COMPARISON WITH EXISTING METHOD

IVA is compared with widely used GICA for the purpose of group discrimination in terms of GT features. In addition, masks are applied for motor related components generated by both algorithms.

CONCLUSIONS

Results show that IVA better captures subject variability producing more activated voxels and generating components with less mutual information in the spatial domain than Group ICA. IVA-generated components result in smaller p-values and clearer trends in GT features.

Preserving subject variability in group fMRI analysis: performance evaluation of GICA vs. IVA

Independent component analysis (ICA) is a widely applied technique to derive functionally connected brain networks from fMRI data. Group ICA (GICA) and Independent Vector Analysis (IVA) are extensions of ICA that enable users to perform group fMRI analyses; however a full comparison of the performance limits of GICA and IVA has not been investigated. Recent interest in resting state fMRI data with potentially higher degree of subject variability makes the evaluation of the above techniques important. In this paper we compare component estimation accuracies of GICA and an improved version of IVA using simulated fMRI datasets. We systematically change the degree of inter-subject spatial variability of components and evaluate estimation accuracy over all spatial maps (SMs) and time courses (TCs) of the decomposition. Our results indicate the following: (1) at low levels of SM variability or when just one SM is varied, both GICA and IVA perform well, (2) at higher levels of SM variability or when more than one SMs are varied, IVA continues to perform well but GICA yields SM estimates that are composites of other SMs with errors in TCs, (3) both GICA and IVA remove spatial correlations of overlapping SMs and introduce artificial correlations in their TCs, (4) if number of SMs is over estimated, IVA continues to perform well but GICA introduces artifacts in the varying and extra SMs with artificial correlations in the TCs of extra components, and (5) in the absence or presence of SMs unique to one subject, GICA produces errors in TCs and IVA estimates are accurate. In summary, our simulation experiments (both simplistic and realistic) and our holistic analyses approach indicate that IVA produces results that are closer to ground truth and thereby better preserves subject variability. The improved version of IVA is now packaged into the GIFT toolbox (http://mialab.mrn.org/software/gift).

A statistically motivated framework for simulation of stochastic data fusion models applied to multimodal neuroimaging

Multimodal fusion is becoming more common as it proves to be a powerful approach to identify complementary information from multimodal datasets. However, simulation of joint information is not straightforward. Published approaches mostly employ limited, provisional designs that often break the link between the model assumptions and the data for the sake of demonstrating properties of fusion techniques. This work introduces a new approach to synthetic data generation which allows full-compliance between data and model while still representing realistic spatiotemporal features in accordance with the current neuroimaging literature. The focus is on the simulation of joint information for the verification of stochastic linear models, particularly those used in multimodal data fusion of brain imaging data. Our first goal is to obtain a benchmark ground-truth in which estimation errors due to model mismatch are minimal or none. Then we move on to assess how estimation is affected by gradually increasing model discrepancies toward a more realistic dataset. The key aspect of our approach is that it permits complete control over the type and level of model mismatch, allowing for more educated inferences about the limitations and caveats of select stochastic linear models. As a result, impartial comparison of models is possible based on their performance in multiple different scenarios. Our proposed method uses the commonly overlooked theory of copulas to enable full control of the type and level of dependence/association between modalities, with no occurrence of spurious multimodal associations. Moreover, our approach allows for arbitrary single-modality marginal distributions for any fixed choice of dependence/association between multimodal features. Using our simulation framework, we can rigorously challenge the assumptions of several existing multimodal fusion approaches. Our study brings a new perspective to the problem of simulating multimodal data that can be used for ground-truth verification of various stochastic multimodal models available in the literature, and reveals some important aspects that are not captured or are overlooked by ad hoc simulations that lack a firm statistical motivation.

Dynamic changes of spatial functional network connectivity in healthy individuals and schizophrenia patients using independent vector analysis

Recent work on both task-induced and resting-state functional magnetic resonance imaging (fMRI) data suggests that functional connectivity may fluctuate, rather than being stationary during an entire scan. Most dynamic studies are based on second-order statistics between fMRI time series or time courses derived from blind source separation, e.g., independent component analysis (ICA), to investigate changes of temporal interactions among brain regions. However, fluctuations related to spatial components over time are of interest as well. In this paper, we examine higher-order statistical dependence between pairs of spatial components, which we define as spatial functional network connectivity (sFNC), and changes of sFNC across a resting-state scan. We extract time-varying components from healthy controls and patients with schizophrenia to represent brain networks using independent vector analysis (IVA), which is an extension of ICA to multiple data sets and enables one to capture spatial variations. Based on mutual information among IVA components, we perform statistical analysis and Markov modeling to quantify the changes in spatial connectivity. Our experimental results suggest significantly more fluctuations in patient group and show that patients with schizophrenia have more variable patterns of spatial concordance primarily between the frontoparietal, cerebellar and temporal lobe regions. This study extends upon earlier studies showing temporal connectivity differences in similar areas on average by providing evidence that the dynamic spatial interplay between these regions is also impacted by schizophrenia.

Independent component analysis for brain FMRI does indeed select for maximal independence

A recent paper by Daubechies et al. claims that two independent component analysis (ICA) algorithms, Infomax and FastICA, which are widely used for functional magnetic resonance imaging (fMRI) analysis, select for sparsity rather than independence. The argument was supported by a series of experiments on synthetic data. We show that these experiments fall short of proving this claim and that the ICA algorithms are indeed doing what they are designed to do: identify maximally independent sources.

Multisubject independent component analysis of fMRI: a decade of intrinsic networks, default mode, and neurodiagnostic discovery

Since the discovery of functional connectivity in fMRI data (i.e., temporal correlations between spatially distinct regions of the brain) there has been a considerable amount of work in this field. One important focus has been on the analysis of brain connectivity using the concept of networks instead of regions. Approximately ten years ago, two important research areas grew out of this concept. First, a network proposed to be "a default mode of brain function" since dubbed the default mode network was proposed by Raichle. Secondly, multisubject or group independent component analysis (ICA) provided a data-driven approach to study properties of brain networks, including the default mode network. In this paper, we provide a focused review of how ICA has contributed to the study of intrinsic networks. We discuss some methodological considerations for group ICA and highlight multiple analytic approaches for studying brain networks. We also show examples of some of the differences observed in the default mode and resting networks in the diseased brain. In summary, we are in exciting times and still just beginning to reap the benefits of the richness of functional brain networks as well as available analytic approaches.

Phase Ambiguity Correction and Visualization Techniques for Complex-Valued ICA of Group fMRI Data

Analysis of functional magnetic resonance imaging (fMRI) data in its native, complex form has been shown to increase the sensitivity both for data-driven techniques, such as independent component analysis (ICA), and for model-driven techniques. The promise of an increase in sensitivity and specificity in clinical studies, provides a powerful motivation for utilizing both the phase and magnitude data; however, the unknown and noisy nature of the phase poses a challenge. In addition, many complex-valued analysis algorithms, such as ICA, suffer from an inherent phase ambiguity, which introduces additional difficulty for group analysis. We present solutions for these issues, which have been among the main reasons phase information has been traditionally discarded, and show their effectiveness when used as part of a complex-valued group ICA algorithm application. The methods we present thus allow the development of new fully complex data-driven and semi-blind methods to process, analyze, and visualize fMRI data.We first introduce a phase ambiguity correction scheme that can be either applied subsequent to ICA of fMRI data or can be incorporated into the ICA algorithm in the form of prior information to eliminate the need for further processing for phase correction. We also present a Mahalanobis distance-based thresholding method, which incorporates both magnitude and phase information into a single threshold, that can be used to increase the sensitivity in the identification of voxels of interest. This method shows particular promise for identifying voxels with significant susceptibility changes but that are located in low magnitude (i.e. activation) areas. We demonstrate the performance gain of the introduced methods on actual fMRI data.

Modulations of functional connectivity in the healthy and schizophrenia groups during task and rest

Connectivity analysis using functional magnetic resonance imaging (fMRI) data is an important area, useful for the identification of biomarkers for various mental disorders, including schizophrenia. Most studies to date have focused on resting data, while the study of functional connectivity during task and the differences between task and rest are of great interest as well. In this work, we examine the graph-theoretical properties of the connectivity maps constructed using spatial components derived from independent component analysis (ICA) for healthy controls and patients with schizophrenia during an auditory oddball task (AOD) and at extended rest. We estimate functional connectivity using the higher-order statistical dependence, i.e., mutual information among the ICA spatial components, instead of the typically used temporal correlation. We also define three novel topological metrics based on the modules of brain networks obtained using a clustering approach. Our experimental results show that although the schizophrenia patients preserve the small-world property, they present a significantly lower small-worldness during both AOD task and rest when compared to the healthy controls, indicating a consistent tendency towards a more random organization of brain networks. In addition, the task-induced modulations to topological measures of several components involving motor, cerebellum and parietal regions are altered in patients relative to controls, providing further evidence for the aberrant connectivity in schizophrenia.

Order Selection of the Linear Mixing Model for Complex-valued FMRI Data

Functional magnetic resonance imaging (fMRI) data are originally acquired as complex-valued images, which motivates the use of complex-valued data analysis methods. Due to the high dimension and high noise level of fMRI data, order selection and dimension reduction are important procedures for multivariate analysis methods such as independent component analysis (ICA). In this work, we develop a complex-valued order selection method to estimate the dimension of signal subspace using information-theoretic criteria. To correct the effect of sample dependence to information-theoretic criteria, we develop a general entropy rate measure for complex Gaussian random process to calibrate the independent and identically distributed (i.i.d.) sampling scheme in the complex domain. We show the effectiveness of the approach for order selection on both simulated and actual fMRI data. A comparison between the results of order selection and ICA on real-valued and complex-valued fMRI data demonstrates that a fully complex analysis extracts more information about brain activation.

Decomposing the brain: components and modes, networks and nodes

Smith and colleagues recently presented a temporal independent component analysis (tICA) decomposition of resting-state functional MRI data. Compared to the widely used spatial ICA (sICA), tICA better allows for a brain region to engage in multiple, independent interactions with other regions and will potentially offer new insights into brain function.

Automatic identification of functional clusters in FMRI data using spatial dependence

In independent component analysis (ICA) of functional magnetic resonance imaging (fMRI) data, extracting a large number of maximally independent components provides a detailed functional segmentation of brain. However, such high-order segmentation does not establish the relationships among different brain networks, and also studying and classifying components can be challenging. In this study, we present a multidimensional ICA (MICA) scheme to achieve automatic component clustering. In our MICA framework, stable components are hierarchically grouped into clusters based on higher order statistical dependence--mutual information--among spatial components, instead of the typically used temporal correlation among time courses. The final cluster membership is determined using a statistical hypothesis testing method. Since ICA decomposition takes into account the modulation of the spatial maps, i.e., temporal information, our ICA-based approach incorporates both spatial and temporal information effectively. Our experimental results from both simulated and real fMRI datasets show that the use of spatial dependence leads to physiologically meaningful connectivity structure of brain networks, which is consistently identified across various ICA model orders and algorithms. In addition, we observe that components related to artifacts, including cerebrospinal fluid, arteries, and large draining veins, are grouped together and encouragingly distinguished from other components of interest.

On entropy rate for the complex domain

We derive the entropy rate formula for a complex Gaussian random process by using a widely linear model. The resulting expression is general and applicable to both circular and noncircular Gaussian processes, since any second-order stationary process can be modeled as the output of a widely linear system driven by a circular white noise. Furthermore, we demonstrate application of the derived formula to an order selection problem. We extend a scheme for independent and identically distributed (i.i.d.) sampling to the complex domain to improve the estimation performance of information-theoretic criteria when samples are correlated. We show the effectiveness of the approach for order selection for simulated and actual functional magnetic resonance imaging (fMRI) data that are inherently complex valued.

Joint Blind Source Separation by Multi-set Canonical Correlation Analysis

In this work, we introduce a simple and effective scheme to achieve joint blind source separation (BSS) of multiple datasets using multi-set canonical correlation analysis (M-CCA) [1]. We first propose a generative model of joint BSS based on the correlation of latent sources within and between datasets. We specify source separability conditions, and show that, when the conditions are satisfied, the group of corresponding sources from each dataset can be jointly extracted by M-CCA through maximization of correlation among the extracted sources. We compare source separation performance of the M-CCA scheme with other joint BSS methods and demonstrate the superior performance of the M-CCA scheme in achieving joint BSS for a large number of datasets, group of corresponding sources with heterogeneous correlation values, and complex-valued sources with circular and non-circular distributions. We apply M-CCA to analysis of functional magnetic resonance imaging (fMRI) data from multiple subjects and show its utility in estimating meaningful brain activations from a visuomotor task.

Canonical Correlation Analysis for Feature-Based Fusion of Biomedical Imaging Modalities and Its Application to Detection of Associative Networks in Schizophrenia

Typically data acquired through imaging techniques such as functional magnetic resonance imaging (fMRI), structural MRI (sMRI), and electroencephalography (EEG) are analyzed separately. However, fusing information from such complementary modalities promises to provide additional insight into connectivity across brain networks and changes due to disease. We propose a data fusion scheme at the feature level using canonical correlation analysis (CCA) to determine inter-subject covariations across modalities. As we show both with simulation results and application to real data, multimodal CCA (mCCA) proves to be a flexible and powerful method for discovering associations among various data types. We demonstrate the versatility of the method with application to two datasets, an fMRI and EEG, and an fMRI and sMRI dataset, both collected from patients diagnosed with schizophrenia and healthy controls. CCA results for fMRI and EEG data collected for an auditory oddball task reveal associations of the temporal and motor areas with the N2 and P3 peaks. For the application to fMRI and sMRI data collected for an auditory sensorimotor task, CCA results show an interesting joint relationship between fMRI and gray matter, with patients with schizophrenia showing more functional activity in motor areas and less activity in temporal areas associated with less gray matter as compared to healthy controls. Additionally, we compare our scheme with an independent component analysis based fusion method, joint-ICA that has proven useful for such a study and note that the two methods provide complementary perspectives on data fusion.