easyX: a simple Python library for saving big data

Motivation

As a researcher, I am developing advanced machine learning approaches for analyzing cognitive states in human brains. During my Ph.D. study, I have developed the easy fMRI project as a toolbox for analyzing task-based fMRI datasets. This project has already been used in several advance academic studies, such as modeling consciousness (Michigan University), diagnosing children’s anxiety (University of Alberta), decoding visual stimuli (Oxford University), working memory, and decision making (University of Cambridge), etc.

When I had started the easy fMRI project, I selected MATLAB format for saving preprocessed datasets and the analysis results. On the one hand, this format cannot save data files with size 4GB+. On the other hand, fMRI datasets are massive — e.g., a dataset for movie stimuli can be more than 200GB. That was a significant disadvantage for the project. We had been developing advanced machine learning approaches — such as deep neural networks — that could provide an accurate model for analyzing big fMRI datasets. BUT, our toolbox could not save (or even load) a large dataset as a single file, even though you had enough hardware resources. We first came up with the idea of the “Easy Data” technique — i.e., we partition the complex data structure into a bunch of MATLAB files that could be controlled via a header file. There are two disadvantages to handle this solution. First, using a single dataset needs to load and concatenate all of those MATLAB files, which is not a time-efficient approach. Further, managing a bunch of files after running analysis was hard to trace — you need to be sure that the processed data will be correctly partitioned and stored, again!

We first tried to replace MATLAB files with some alternative libraries in Python — such as Pandas and HDF5. Pandas cannot save and load the complex structures the same as MATLAB format — Dataframe in Pandas is mostly designed for homogeneous matrices. Further, Pandas cannot efficiently handle large files.

HDF5 does not suffer Pandas issues. It has a time-efficient algorithm for saving and loading massive data files. However, it could only store homogeneous tensors. That was our motivation to create the “easyX project” based on the HDF5 data structure — that can handle massive data files with complex structures.

easyX library enables you to save a Python dictionary with a complex structure to a single HDF5 file. We have tested this library to save fMRI datasets with size 150GB+ — n.b., you need a computer with 155 GB memory. You can stack all of your data in the form of a Python dictionary with a complex structure — it could be nested dictionaries or nonhomogeneous tensors.

How easyX works?

easyX uses a simple procedure. This library saves homogeneous tensors by using the regular algorithm that is utilized for HDF5. It stores these tensors in an HDF5 group called “raw.” If the dictionary has other complex structures — such as another dictionary or nonhomogeneous tensors — easyX first dumps the bytes of data from memory and encodes them in a “base64” coding system. The encoded data will be stored as a vector in an HDF5 group called “binary.”

How to install?

You only need to copy the easyX.py to your Python project. You can use git command for downloading easyX:

git clone https://gitlab.com/myousefnezhad/easyx.git

Requirements

We have tested easyX on Python 3.7 and Python 3.8. You need to install the libraries from the requirements.txt:

pip install -r requirements.txt

This file indeed installs the following libraries numpy, pickle, codecs, and h5py.

How to use it?

You first copy easyX.py to the main folder of your project. Then, you will store all variables in the form of a dictionary in Python.

As an example, we have created a sample data as follows:

data = {"a": np.array([[1, 2, 5, 8], [2., 4, 1, 6]]),
		 "b": [[1], [2, 4]],
		 "c": [[1, 20], [7, 4]],
		 "d": "Hi There",
		 "e": ["A", "B"],
		 "f": [["a", "b"], ["c", "d"]],
		 "h": np.random.rand(100, 1000)
		}

Here, we have the Python dictionary data that includes different shapes of variables.

A) Saving a dictionary into a file

You can use following commands for saving a dictionary into a file:

# Import easyX Library
from easyX import easyX
# Create an object from easyX class
ezx = easyX()
# Change this one with the PATH you need to save your data
fname = "/tmp/a.ezx"  
# Here, `data` is the example dictionary, you may replace it with yours
ezx.save(data, fname=fname) 

B) Loading a data file into a dictionary

You can use following commands for loading a data file into a dictionary:

# Import easyX Library
from easyX import easyX
# Create an object from easyX class
ezx = easyX()
# Change this one with the PATH you need to save your data
fname = "/tmp/a.ezx"  
# Data will be recovered in the `data` dictionary
data = ezx.load(fname=fname) 

C) Loading the data structures (keys) from a data file into a dictionary

You can use following commands for loading the data structures (keys) from a data file into a dictionary:

# Import easyX Library
from easyX import easyX
# Create an object from easyX class
ezx = easyX()
# Change this one with the PATH you need to save your data
fname = "/tmp/a.ezx"
# Keys will be recovered in the `keys` dictionary
keys = ezx.load_keys(fname=fname)

How to uninstall easyX?

You only need to remove easyX.py from your project.

Do not forget feedback

I hope you will enjoy using easyX in your project. For support or feedback, you can also contact us: info@learningbymachine.com.

Reference

easyX website: https://gitlab.com/myousefnezhad/easyx

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An introduction to functional alignment for fMRI analysis

Previously, we talked about task-based fMRI analysis. In this post, we want to introduce one of the long-standing challenges in task-based fMRI analysis.

Introduction

Brain decoding, which is a conjunction between neuroscience and machine learning, extracts meaningful patterns (signatures) from neural activities of the human brain [1]. Most of the brain decoding approaches employed functional Magnetic Resonance Imaging (fMRI) technology for visualizing the brain activities because it can provide better spatial resolution in comparison with other imaging techniques [1—4].

fMRI can be used as a proxy to illustrate neural activities by analyzing the Blood Oxygen Level Dependent (BOLD) signals [1]. As one of the most popular supervised techniques in fMRI analysis, Multivariate Pattern (MVP) classification can map neural activities to distinctive brain tasks. MVP can generate a classification (cognitive) model, i.e., decision surfaces, in order to predict patterns associated with different cognitive states. This model can help us to figure out how the human brain works. MVP analysis has an extensive range of applications to seek novel treatments for mental diseases [1—6].

As a fundamental challenge in supervised fMRI studies, the generated MVP models must be generalized and validated across subjects. However, neuronal activities in a multi-subject fMRI dataset must be aligned to improve the performance of the final results. Technically, there are two different kinds of alignment techniques that can be used in harmony, i.e., anatomical alignment and functional alignment. The anatomical alignment as a general preprocessing step in fMRI analysis aligns the brain patterns by using anatomical features, which is extracted from structural MRI in the standard space — including Talairach or Montreal Neurological Institute (MNI). Nevertheless, the performance of anatomical alignment techniques is limited based on the shape, size, and spatial location of functional loci that differ across subjects. In contrast, the functional alignment can directly align the neural activities across subjects, which has been widely used in fMRI studies.

Hyperalignment

Figure 1. An example of Hyperalignment. Here, subjects watch three categories of visual stimuli. We learn transformation \mathbf{R}^{(i)} for each subject to map the original neural responses to a common space (G).

Most of the recent studies in functional alignment [1—6] have used Hyperalignment (HA) [1]. As Figure 1 depicted, HA refers to the functional alignment of multi-subject data, where shared space is generated from neural activities across subjects. Then, the mapped features can be utilized by MVP techniques in order to boost the performance of the classification analysis. In practice, HA applies a Generalized Canonical Correlation Analysis (GCCA) approach (aka, multi-set CCA) to temporally-aligned neural activities across subjects, where a unique time point must represent the same simulation for all subjects [6].

Let S be the number of subjects, V be the number of voxels (viewed as a 1D vector, even though it corresponds to a 3D volume), and T is the number of time-points in units of Time of Repetitions (TRs). The preprocessed brain image (neural responses) for \ell\text{-}th subject is defined as \mathbf{X}^{(\ell)}\in\mathbb{R}^{V \times T}\text{, }\ell = 1\text{:}S. We consider \mathbf{X}^{(i)} \sim \mathcal{N}(0,1) is normalized by zero-mean and unit-variance in the preprocessing step. Here, \mathbf{X}^{(i)} \text{, } i=1\text{:}S is also time synchronized to provide temporal alignment — i.e., each time point demonstrates the same stimuli for all subjects [6]. In fact, the columns of \mathbf{X}^{(i)} are aligned across subjects by utilizing HA methods. The original HA can be defined as follows where tr() is the trace function [6]:

\underset{\mathbf{R}^{(i)}, \mathbf{G}}{\min} \sum_{i=1}^{S}\| \mathbf{X}^{(i)}\mathbf{R}^{(i)} - \mathbf{G} \|_F^2, \Big(\mathbf{X}^{(\ell)}\mathbf{R}^{(\ell)} \Big)^\top \mathbf{X}^{(\ell)}\mathbf{R}^{(\ell)} = \mathbf{I},

where \ell = 1\dots S, and \mathbf{G} denotes the common space such that:

\mathbf{G} = \sum_{j=1}^{S} \mathbf{X}^{(j)}\mathbf{R}^{(j)}.

In [6], Xu et al. proposed a regularized iterative approach for learning the common space (G) and the transformation matrices \mathbf{R}^{(i)}. Further, Lobert et al. developed a Kernel Hyperalignmenr for nonlinear fMRI analysis [5]. Recently, we also developed Deep Hyperalignment that can scale alignment techniques for large-scale analysis [4].

So, let’s look at the training and testing procedures. Based on the definition, we have a training set \mathbf{X}^{(\ell)}\in\mathbb{R}^{V \times T}\text{, }\ell = 1\text{:}S and a testing set \mathbf{\hat{X}}^{(\ell)}\in\mathbb{R}^{V \times T}\text{, }\ell = 1\text{:}\hat{S} — where \hat{S} is the number of subject in the testing set. In the training procedure, we first learn a common space \mathbf{G} and a set of transformation matrices \mathbf{R}^{(\ell)}\text{ for }\ell=1:S. Then, we generate a classification model by using \mathbf{X}^{(\ell)}\mathbf{R}^{(\ell)}\text{ for }\ell=1:S. In the testing stage, we learn the transformation matrices, where the shared space will not be updated anymore. We actually use following objective function:

\underset{\mathbf{\hat{R}}^{(i)}}{\min} \sum_{i=1}^{S}\| \mathbf{\hat{X}}^{(i)}\mathbf{\hat{R}}^{(i)} - \mathbf{G} \|_F^2,

Finally, the performance of the trained model can be evaluated by using the transformed testing features \mathbf{\hat{X}}^{(\ell)}\mathbf{\hat{R}}^{(\ell)}\text{ for }\ell=1:\hat{S}. This learning procedure is almost the same in most alignment techniques.

Shared Response Model (SRM)

Figure 2: Graphical model for SRM. Shaded nodes: observations, unshaded nodes: latent variables, and black squares: hyperparameters [7].

SRM is a probabilistic extension of the Hyperalignment [7] — i.e., SRM uses probabilistic CCA for generating the shared space (aka, common space). Let m be the number of subjects in a preprocessed fMRI dataset, d denotes the number of time points in TRs, and v is the number of voxels. fMRI time-series for i-th subject denotes by \mathbf{X}_{i} \in \mathbb{R}^{v \times d} for i=1:m. SRM’s objective function is to model each subject’s neural responses as \mathbf{X}_{i} = \mathbf{W}_{i}\mathbf{S} + \mathbf{E}_{i} , where \mathbf{W}_{i} \in \mathbb{R}^{v \times k} denotes a basis of topographies for subject i, k is the number of selected features, \mathbf{S} \in \mathbb{R}^{k \times d} is the shared space. SRM’s objective function for all subjects can be also written as follows: [7]

\underset{\mathbf{S,}\mathbf{W}_{i}}{\min} \sum_{i=1}^{m} \| \mathbf{X}_{i} - \mathbf{W}_{i}\mathbf{S} \|_F^2, \text{subject to } \mathbf{W}_{i}^{\top}\mathbf{W}_{i}=\mathbf{I}_k,

where \|. \|_F denotes the Frobenius norm, and \mathbf{I}_k is identity matrix in size k. Further, we calculate the shared space as follows:

\mathbf{S} = \frac{1}{m}\sum_{i=1}^{m} \mathbf{W}_{i}^{\top} \mathbf{X}_{i}.

Figure 2 shows the graphical model for SRM — where a probabilistic optimization approach is used to learn a shared space \mathbf{S} and the basis of topographies \mathbf{W}_{i}. In this figure, \mathbf{s}_{t} \in \mathbb{R}^{k} with covariance \mathbf{\Sigma}_{s} is a hyperparameter modeling the shared response at time t=1:d, \mathbf{x}_{it}\in \mathbb{R}^{v} denotes the observed pattern of voxel responses for the i-th subject at time t, \mathbf{\rho}^{2}_{i} is i-th subject independent hyperparameter, and \mathbf{\mu}_{i} denotes the subject specific mean. The final optimization procedure is explained in Section 3.1 of [7].

Supervised Hyperalignment (SHA)

Figure 3. Comparison of different HA algorithms for aligning neural activities [2].

We recently illustrated that the performance of HA methods might not be optimum for supervised fMRI analysis (i.e., MVP problems) because they mostly employed unsupervised GCCA techniques for aligning the neural activities across subjects [2, 3]. Therefore, we have developed Local Discriminant Hyperalignment (LDHA) [3] and then Supervised Hyperalignment (SHA) [2] for improving the alignment accuracy in the MVP problems. Although LDHA can improve the performances of both functional alignment and MVP analysis, its objective function cannot directly calculate a supervised shared space and still uses the classical unsupervised shared space [3]. Thus, it cannot provide stable performance and acceptable runtime for large datasets in real-world applications [2].

Figure 3 compares the main difference between unsupervised HA, LDHA, and SHA. As depicted in this figure, two subjects watch two photos of houses, as well as two photos of bottles — where [\mathbf{H1}, \mathbf{B1}, \mathbf{H2}, \mathbf{B2}], shows the sequence of stimuli (after temporal alignment). Here, the shared spaces can be calculated by employing different correlations between neural activities. Figure 3.a demonstrates that the unsupervised HA just maximizes the correlation between the voxels with the same position in the time series because it does not use the supervision information.

Figure 3.b illustrates the LDHA approach, where it utilizes the unsupervised shared space for the alignment problem. Indeed, the main issue in LDHA objective function is that the covariance matrices cannot decompose to the product of a symmetric matrix [2]. In order to calculate the shared space in LDHA, each pair of stimuli must be separately compared with each other, and the shared space is gradually updated in each comparison (see Algorithm 2 in [3]). Therefore, LDHA cannot use a generalized optimization approach (such as GCCA) and its time complexity is not efficient for large datasets.

As shown in Figure 3.c, SHA consists of two main steps:

  1. Generating a supervised shared space, where each stimulus is only compared with the shared space to align the neural activities;
  2. Calculating the mapped features in a single iteration.

The neural activities belong to \ell\text{-}th subject can be denoted by \mathbf{X}^{(\ell)} \in \mathbb{R}^{T\times V}\text{, } \ell=1\text{:}S (defined same as the Hyperalignment section) and the class labels that are denoted by \mathbf{Y}^{(\ell)}=\{{y}_{mn}^{(\ell)} \}\text{, }\mathbf{Y}^{(\ell)}\in\{0, 1\}^{L\times T}\text{, } m=1\text{:}L\text{, }n=1\text{:}T\text{, }L>1. Here, L is the number of classes (stimulus categories). In order to infuse supervision information to the HA problem, this medod defines a supervised term as follows:

\mathbf{K}^{(\ell)} \in \mathbb{R}^{L \times T} = \mathbf{Y}^{(\ell)}\mathbf{H},

where the normalization matrix \mathbf{H}\in\mathbb{R}^{T\times T} is denoted as follows:

\mathbf{H} = \mathbf{I}_{T} - \gamma{\mathbf{1}}_{T},

where {\mathbf{1}}_{T} \in {1}^{T\times T} denotes ones matrix in size T, and \gamma makes a trade-off between within-class and between-class samples. Objective function of SHA is defined as follows:

\underset{\mathbf{W}, \mathbf{R}^{(i)}}{\min}\{\sum_{i = 1}^{S} \| \mathbf{K}^{(i)}\mathbf{X}^{(i)}\mathbf{R}^{(i)} - \mathbf{W}\|^2_F\}, (\mathbf{R}^{(\ell)})^\top((\mathbf{K}^{(\ell)}\mathbf{X}^{(\ell)})^\top\mathbf{K}^{(\ell)}\mathbf{X}^{(\ell)} + \epsilon \mathbf{I}_V) \mathbf{R}^{(\ell)}=\mathbf{I}_{V},

where \ell=1\text{:}S, \epsilon is a regularization term, \mathbf{R}^{(\ell)} denotes the mapping matrices, and \mathbf{W} \in \mathbb{R}^{L\times V} is supervised shared space — such that:

\mathbf{W} = \frac{1}{S} \sum_{j=1}^{S} \mathbf{K}^{(j)}\mathbf{X}^{(j)}\mathbf{R}^{(j)}.

We then show that supervised shared space can be calculated directly as follows: [2]

\underset{\mathbf{W}, \mathbf{R}^{(i)}}{\min}\{\sum_{i = 1}^{S} \|\mathbf{K}^{(i)}\mathbf{X}^{(i)}\mathbf{R}^{(i)} - \mathbf{W} \|^2_F\} \equiv \underset{\mathbf{W}}{\min}\{\text{tr}(\mathbf{W}^\top\mathbf{U}\mathbf{W})\}, \text{subject to }\mathbf{R}^{(\ell)}=((\mathbf{K}^{(\ell)}\mathbf{X}^{(\ell)})^\top\mathbf{K}^{(\ell)}\mathbf{X}^{(\ell)} + \epsilon\mathbf{I}_V\Big)^{-1}\big(\mathbf{K}^{(\ell)}\mathbf{X}^{(\ell)})^\top\mathbf{W}, \mathbf{U} = \sum_{\ell=1}^{S}\mathbf{I}_{L} - \mathbf{K}^{(\ell)}\mathbf{X}^{(\ell)}((\mathbf{K}^{(\ell)}\mathbf{X}^{(\ell)})^\top\mathbf{K}^{(\ell)}\mathbf{X}^{(\ell)}+\epsilon\mathbf{I_{V}})^{-1}\mathbf{K}^{(\ell)}\mathbf{X}^{(\ell)})^\top.

Here, \mathbf{W} is the right eigenvectors of \mathbf{U} [2, 4]. Further, the unsupervised shared space for the testing stage is calculated as follows: [2]

\mathbf{G} = \frac{1}{S}\Big(\sum_{\ell=1}^{S}\mathbf{W}^T\mathbf{K}^{(\ell)}\Big)^\top.

Indeed, the testing phase for SHA is the same as unsupervised HA. The only difference between SHA and unsupervised HAs lies in the procedure of generating the shared space in the training phase.

Conclusion

One of the main challenges in fMRI studies, especially Multivariate Pattern (MVP) analysis, is using multi-subject datasets. On the one hand, the multi-subject analysis is necessary to estimate the validity of the generated results across subjects. On the other hand, analyzing multi-subject fMRI data requires accurate functional alignment between neuronal activities of different subjects for improving the performance of the final results. Hyperalignment (HA) is one of the most significant functional alignment methods, which can be formulated as a CCA problem for aligning neural activities of different subjects to a common/shared space. HA techniques can use different optimization solutions for generating an adequate shared space — classic CCA (in HA), probabilistic CCA (in SRM), and supervised methods (in LDHA and SHA). In the future, we will describe the related math background for alignment techniques and explain some challenging issues that may happen during the analysis.

References

  1. Decoding Neural Representational Spaces Using Multivariate Pattern Analysis. DOI: https://doi.org/10.1146/annurev-neuro-062012-170325
  2. Supervised Hyperalignment for multi-subject fMRI data alignment. DOI: 10.1109/TCDS.2020.2965981
  3. Local Discriminant Hyperalignment for multi-subject fMRI data alignment. Link: https://www.aaai.org/ocs/index.php/AAAI/AAAI17/paper/download/14347/13743
  4. Deep Hyperalignment. Link: https://papers.nips.cc/paper/6758-deep-hyperalignment.pdf
  5. Kernel Hyperalignment. Link: https://papers.nips.cc/paper/4592-kernel-hyperalignment.pdf
  6. Regularized hyperalignment of multi-set fMRI data. DOI: 10.1109/SSP.2012.6319668
  7. A Reduced-Dimension fMRI Shared Response Model. Link: https://papers.nips.cc/paper/5855-a-reduced-dimension-fmri-shared-response-model
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Task-based fMRI analysis

The past two decades have seen significant advances in computational neuroscience [1]. Functional neuroimaging is one of the prevalent advanced technologies that are used in most brain studies. Developing methods for analyzing neural responses involves many modalities of measurement in various species. Some of these modalities are single-unit recording, electrocorticography (ECoG), electro– and magnetoencephalography (EEG and MEG), and functional magnetic resonance imaging (fMRI). Most human brain studies employed fMRI data — a non-invasive brain imaging technique with unprecedented spatiotemporal resolution [1].

fMRI is an imaging technology that uses an indirect measure of oxygen levels in the brain, referred to as Blood Oxygen Level Dependent (BOLD) signals, to estimate the neural activities associated with brain cells [2]. fMRI is a 4D data — it is a time series of 3D brain snapshots. Each 3D snapshot includes a set of voxels (3D pixels) that show the level of Oxygen usage in corresponding loci. This spatiotemporal resolution of fMRI data needs to be analyzed by using advanced machine learning techniques.

In general, fMRI can be captured in two different ways — viz., rest-state, and task-based. In a rest-state fMRI (rs-fMRI), subjects do not do any cognitive task during the imaging procedure. rs-fMRI can be used to compare brain networks between subjects with mental disease and control ones — such as rest network, default network, etc. rs-fMRI has several applications for prognosis and diagnosis of mental diseases — such as Alzheimer’s disease, Autism, etc.

Figure 1. An example of task-based fMRI data, where subject watches visual stimuli and the neural responses are showed in temporal cortex.

As Figure 1 depicted, task-based fMRI enables us to study the human brain when subjects are pursuing tasks. Here, task refers to the cognitive task performed by the subject during an fMRI experiment — e.g., watching photos, making decisions, etc. [2].

Figure 2. An example of a classification model for fMRI analysis. We learn a model by using training data (with labels) to predict testing data (without labels) [1].

Task-based fMRI can be analyzed with two different perspectives — viz, predictive methods, and similarity analysis [3]. As Figure 2 shows, predictive approaches such as Multivariable Pattern (MVP) Classification learns a model by using a training-set — including, a subset of neural activities and their corresponding responses (aka, labels). This trained-model enables us to predict the labels for testing data — i.e., the new neural responses that are unseen during the learning procedure [1—3]. As an example, consider a visual task, where we are using an fMRI machine to capture the neural activities of five subjects as they are watching photos (stimuli). Here, this experiment is considering three categories of visual stimuli — e.g., watching photos of human faces, photos of chairs, photos of shoes. Note that all subjects are watching the same groups of visual stimuli. However, each stimulus — such as the specific face seen (perhaps Mr. Smith), or the particular shoe (maybe the black one) — can be different across subjects. A classification model can use the neural responses belonging to Subjects 1—4 to learn a model and then use this model and the neural responses of Subject 5 to determine when Subject 5 watched each of those visual stimuli — i.e., faces, shoes, and chairs [1, 3].

MVP classification techniques have a black-box approach for analyzing the neural responses. They cannot show which sub-regions are significantly involved with an accurate prediction. For instance, a classification can be used to predict Alzheimer’s disease based on the neural activities, but it cannot explain how much the cognitive processes are similar (or different) between groups of patients and controls, or even which brain regions generate distinguish neural signatures across different cognitive states [1, 4].

Figure 3. An example of fMRI similarity analysis between human brains and monkey brains for a common set of stimuli [5].

Similarity analysis is one of the popular techniques that can be used to explain similarities (or differences) between different cognitive states [4, 5]. This process first applies a similarity analysis method — such as Representational Similarity Analysis (RSA) — to generate a unique neural signature for each category of stimuli — e.g., one signature for any face watched by any subject, and another signature for any photo of shoes watched by any subject, etc. Neural signature refers to a vector that identifies which parts of the brain are involved with a category of stimuli. The generated neural signatures can be compared by using different similarity metrics — such as correlation — for understanding different cognitive states or demonstrating the neurological process both within and
between brain regions.

In summary, fMRI enables us to ask what information is represented in a region of the human brain and how that information is encoded, instead of asking what is the function of a region [1, 6]. It can be used to find novel treatments for mental diseases or even to create a new generation of the user interface [1, 6].

References

  1. Decoding Neural Representational Spaces Using Multivariate Pattern Analysis. DOI: https://doi.org/10.1146/annurev-neuro-062012-170325
  2. Supervised Hyperalignment for multi-subject fMRI data alignment. DOI: 10.1109/TCDS.2020.2965981
  3. Multi-Objective Cognitive Model: a supervised approach for multi-subject fMRI analysis. DOI: 10.1007/s12021-018-9394-9 
  4. A Bayesian method for reducing bias in neural representational similarity analysis. Link: https://papers.nips.cc/paper/6131-a-bayesian-method-for-reducing-bias-in-neural-representational-similarity-analysis.pdf
  5. Matching categorical object representations in inferior temporal cortex of man and monkey. DOI: https://dx.doi.org/10.1016%2Fj.neuron.2008.10.043
  6. Local Discriminant Hyperalignment for multi-subject fMRI data alignment. Link: https://www.aaai.org/ocs/index.php/AAAI/AAAI17/paper/download/14347/13743
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