Statistics in the tangent space

What you’ll learn: Compute individual statistics in the the tangent space, and pinpoint functional connectivity differences in an individual from a reference group.

Author: Dhaif BEKHA

Retrieve the example dataset

In this section of the tutorials, we will need a set of brain signals from two distinct group of subjects. You will work on pre-computed set of times series extracted from 72 brain regions atlas. The first group, called controls is a set of brain signals from healthy seven years old children, and the second group called patients, is a set of brain signals from seven years old children who have suffered a stroke. You can download the times series dictionary storing those signals here. Finally, we will plot some results on a glass brain, and we will need the nodes coordinates of the atlas regions in which the signals were extracted. You can download the atlas, and the corresponding labels. As usual, we will suppose that all needed files are in your home directory.

Modules import

from pathlib import Path
import os
from conpagnon.utils.folders_and_files_management import load_object
from conpagnon.computing.compute_connectivity_matrices import tangent_space_projection
from conpagnon.data_handling import atlas
import numpy as np
from matplotlib.backends import backend_pdf
import matplotlib.pyplot as plt
from nilearn.connectome import vec_to_sym_matrix
from nilearn.plotting import plot_connectome
import networkx as nx

Load the functional atlas

We load the atlas that we already used in multiple section of the tutorials section. Please, refer to the very first section were we manipulated the exact same atlas.

# Fetch the path of the home directory
home_directory = str(Path.home())

# Atlas set up
atlas_folder = home_directory
atlas_name = 'atlas.nii'
labels_text_file = os.path.join(home_directory, 'atlas_labels.csv')
colors = ['navy', 'sienna', 'orange', 'orchid', 'indianred', 'olive',
          'goldenrod', 'turquoise', 'darkslategray', 'limegreen', 'black',
          'lightpink']
# Number of regions in each user defined networks
networks = [2, 10, 2, 6, 10, 2, 8, 6, 8, 8, 6, 4]
# Atlas path
# Read labels regions files
atlas_nodes, labels_regions, labels_colors, n_nodes = atlas.fetch_atlas(
    atlas_folder=atlas_folder,
    atlas_name=atlas_name,
    network_regions_number=networks,
    colors_labels=colors,
    labels=labels_text_file,
    normalize_colors=True)

A brief theoretical background

In this section we will you the theoretical basis and mathematical development to help you understand the advantages of using the tangent space metric in functional connectivity. We encourage you to dive in more depth in the subject by reading the references we select here and here !

Problematic: Detecting functional connectivity difference

The statistical analyses of functional connectivity based on connectivity matrices aim to find differences between subject in the same group, or in two different group for example. So the first difficulty is to find a sensible enough parametrization of those functional connectivity matrices. It is common to use the correlation as the base metric, and compare correlation coefficients across subjects. This simple procedure can be modelled by the following linear model, for a subject \(s\), the correlation matrix \(\Sigma^{s}\) is:

\[\Sigma^{s} = \Sigma_{0} + d\Sigma^{s}\]

where \(\Sigma_{0}\) is the mean covariance matrix of the whole group, and \(d\Sigma^{s}\) encode the subject specific contribution in functional connectivity. Based on correlation metric, \(d\Sigma^{s}\) can be very difficult to model, drown in the natural dependence that inherently exist between the functional connectivity coefficients in a correlation matrix. In the following sub-section, we detailed one solution based on the projection of those correlation matrices in another space.

The projection in the tangent space

The main problem to mitigate is the natural inter-dependence of functional connectivity coefficient that arise with correlation-like metric. Indeed, mathematically, those matrices belong to the symmetric positive definite space, that is, for any vector \(v\), and \(\Sigma\) a correlation matrix, then we have that \(v^{T}\Sigma v > 0\). Considering two correlation matrices \(\Sigma_{1}\) and \(\Sigma_{2}\), we can alleviate this problem by projecting onto the tangent space \(\Sigma_{2}\) at the point \(\Sigma_{1}\). Back to our functional connectivity, let’s consider \(\Sigma^{s}\) the subject correlation matrix of subject \(s\), \(\Sigma_{0}\) the mean correlation matrix of the whole group, and finally \(d\Sigma^{s}\), the subject-specific contribution we have to compute:

\[\Sigma^{s} = \Sigma_{0}^{1/2} exp(d\Sigma^{s}) \Sigma_{0}^{1/2}\]

The \(d\Sigma^{s}\) matrix, is the tangent space matrix of the subject \(s\). It is reasonable to consider that \(d\Sigma^{s}\) is very small, \(||d\Sigma^{s}||_{2} << 1\), so that the above equation simplify in:

\[\Sigma^{s} = \Sigma_{0}^{1/2} (I_{n} + d\Sigma^{s}) \Sigma_{0}^{1/2}\]

Using the above equation, we can finally compute \(d\Sigma^{s}\):

\[d\Sigma^{s} = \Sigma_{0}^{-1/2} \Sigma^{s} \Sigma_{0}^{-1/2} - I_{n}\]

Giving \(\Sigma_{0}\), the projection matrix point, we can now compute for every subject, at that point, the tangent space functional connectivity matrix.

Tip

The computation of the tangent space matrices is implemented in the Nilearn library, that you may already use, in the previous tutorial in ConPagnon. Please, see the official documentation of the Nilearn package for more information. In this implementation, you must give the whole stack of all times series from all groups to compute the tangent space matrices. In ConPagnon, we implement a more flexible version, where you can give a reference point, and the points to project, in the function conpagnon.computing.compute_connectivity_matrices.tangent_space_projection() function

In the next section of this tutorial, we will use the tangent space metric to compute a test statistic, that pinpoint for each patient in the patients group, the functional connectivity differences from the mean correlation matrices of the control group.

Subject specific statistic in the tangent space

In this section, we will adapt the algorithm describe by Gael Varoquaux et al in this reference. With the same notation as above, we will first compute \(\Sigma_{0}\), which which is the mean correlation matrix of the controls group. Then, for each patient \(s\), we will project the correlation matrix \(\Sigma^{s}\) at \(\Sigma_{0}\) to compute the tangent space connectivity matrix, \(d\Sigma^{s}\). We also compute the tangent matrices for the controls group. Finally, for each patient, we will compute a one sample t-test between \(d\Sigma^{s}\) and \({ d\Sigma_{0}^{1} ... d\Sigma_{0}^{N} }\), the tangent matrices of the controls group. To assess the significance of each p-value, for each subject, we generate the null distribution for each coefficient by bootstrapping of the reference group.. The user have to set the size of the bootstrapped sample \(m\), the number of time we repeat the bootstrapping process, and the level of significance \(\alpha\).

# Load the times series dictionary in your
# home directory
times_series_dictionary = load_object(
    full_path_to_object=os.path.join(home_directory, 'times_series.pkl'))

# Retrieve the groups in the study: it's
# simply the keys of the dictionary
groups = list(times_series_dictionary.keys())

# Name of the projected group
projected_group_name = "patients"

# Name of the reference group
reference_group_name = "controls"

# Subjects list in each group:
reference_group = list(times_series_dictionary[reference_group_name].keys())
subjects_to_project = list(times_series_dictionary[projected_group_name].keys())

# Stack the times series for each group
reference_group_time_series = np.array([times_series_dictionary[reference_group_name][s]['time_series']
                                        for s in reference_group])
group_to_project_time_series = np.array([times_series_dictionary[projected_group_name][s]['time_series']
                                         for s in subjects_to_project])

# Number of bootstrap
m = 10000
# Size of the bootstrapped sample
size_subset_reference_group = 15
# Level of Type-I error rate
alpha = 0.01

# Compute the tangent space projection, followed
# by the one sample t-test and estimation of the
# null distribution.
tangent_space_projection_dict = tangent_space_projection(
    reference_group=reference_group_time_series,
    group_to_project=group_to_project_time_series,
    bootstrap_number=m,
    bootstrap_size=size_subset_reference_group,
    output_directory=home_directory,
    verif_null=False,
    statistic='t',
    correction_method="fdr_bh",
    alpha=alpha)

Out:

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Note

The null distribution \(H_{i,j}\) for functional connectivity coefficient \(i,j\) is estimated by leave-one-out inside the bootstrapped sample. We compute the mean matrix of all subject in the reference group, and project all subject in the sample, onto the tangent space at that mean matrix. Finally, we perform a one sample t-test between the leftout subject and the mean group in the tangent space.

We stored all the useful results (the null distribution, the statistic for each subjects….) in the variable tangent_space_projection_dict. We will first retrieve the corrected p-value and corresponding t-statistic for each subject, and then plot it on a glass brain. We will use the atlas that you previously downloaded in the first section of this tutorial.

# Corrected p values for each projected subject
p_values_corrected = tangent_space_projection_dict['p_values_corrected']
# Tangent Connectivity matrices for each projected subject
group_to_project_tangent_matrices = tangent_space_projection_dict['group_to_project_tangent_matrices']
# Reference group mean correlation matrices
reference_group_tangent_mean = tangent_space_projection_dict['reference_group_tangent_mean']
# output statistic for each projected subject
group_to_project_stats = tangent_space_projection_dict['group_to_project_stats']

# Initialize a empty adjacency matrix for plotting purpose,
# if for a subject no significant nodes are to be found.
empty_adjacency_matrix = np.zeros(shape=(n_nodes, n_nodes))


# We will plot only the first five patients glass brain
with backend_pdf.PdfPages(os.path.join(home_directory, 'tangent_space_statistic_report')) as pdf:
    for subject in range(len(subjects_to_project))[0:5]:
        # compute node degree for each subject
        # based on the surviving connection
        group_to_project_significant_edges = vec_to_sym_matrix(p_values_corrected[subject, ...] < alpha,
                                                               diagonal=np.zeros(n_nodes))
        patient_adjacency_matrices = nx.from_numpy_array(group_to_project_significant_edges)
        degrees = np.array([val for (node, val) in patient_adjacency_matrices.degree()])*40
        # plot corrected connection
        if np.unique(group_to_project_significant_edges).size == 1:
            plot_connectome(
                adjacency_matrix=empty_adjacency_matrix,
                node_coords=atlas_nodes,
                node_color=labels_colors,
                title='{}'.format(
                    subjects_to_project[subject]),
                colorbar=False,
                node_kwargs={'edgecolor': 'black', 'alpha': 1})
            plt.tight_layout()
            pdf.savefig()
            plt.show()
        else:
            plot_connectome(
                adjacency_matrix=vec_to_sym_matrix(np.multiply(p_values_corrected[subject, ...] < alpha,
                                                               group_to_project_tangent_matrices[subject, ...]),
                                                   diagonal=np.zeros(n_nodes)),
                node_coords=atlas_nodes,
                node_color=labels_colors,
                title='{}'.format(
                    subjects_to_project[subject]),
                colorbar=True,
                node_size=degrees,
                node_kwargs={'edgecolor': 'black', 'alpha': 1},
                edge_threshold=None)
            plt.tight_layout()
            pdf.savefig()
            plt.show()

    plot_connectome(adjacency_matrix=reference_group_tangent_mean,
                    node_coords=atlas_nodes,
                    node_color=labels_colors, edge_threshold='80%',
                    title='{} tangent mean'.format(reference_group_name),
                    colorbar=True)
    plt.tight_layout()
    pdf.savefig()
    plt.show()

• 
• 
• 
• 
• 
• 

Out:

/home/dhaif/anaconda3/envs/conpagnon/lib/python3.7/site-packages/nilearn/plotting/displays.py:1752: FutureWarning: elementwise comparison failed; returning scalar instead, but in the future will perform elementwise comparison
  if node_color == 'auto':
/media/dhaif/Samsung_T5/Work/Programs/ConPagnon/examples/04_Advanced_Statistical_Analyses/plot_individual_statistics_in_the_tangent_space.py:315: UserWarning: This figure includes Axes that are not compatible with tight_layout, so results might be incorrect.
  plt.tight_layout()
/media/dhaif/Samsung_T5/Work/Programs/ConPagnon/examples/04_Advanced_Statistical_Analyses/plot_individual_statistics_in_the_tangent_space.py:317: UserWarning: Matplotlib is currently using agg, which is a non-GUI backend, so cannot show the figure.
  plt.show()
/media/dhaif/Samsung_T5/Work/Programs/ConPagnon/examples/04_Advanced_Statistical_Analyses/plot_individual_statistics_in_the_tangent_space.py:331: UserWarning: This figure includes Axes that are not compatible with tight_layout, so results might be incorrect.
  plt.tight_layout()
/media/dhaif/Samsung_T5/Work/Programs/ConPagnon/examples/04_Advanced_Statistical_Analyses/plot_individual_statistics_in_the_tangent_space.py:333: UserWarning: Matplotlib is currently using agg, which is a non-GUI backend, so cannot show the figure.
  plt.show()
/media/dhaif/Samsung_T5/Work/Programs/ConPagnon/examples/04_Advanced_Statistical_Analyses/plot_individual_statistics_in_the_tangent_space.py:340: UserWarning: This figure includes Axes that are not compatible with tight_layout, so results might be incorrect.
  plt.tight_layout()
/media/dhaif/Samsung_T5/Work/Programs/ConPagnon/examples/04_Advanced_Statistical_Analyses/plot_individual_statistics_in_the_tangent_space.py:342: UserWarning: Matplotlib is currently using agg, which is a non-GUI backend, so cannot show the figure.
  plt.show()

We only plot a couple of glass brain in the above plot. As you can see, the way functional connectivity is impacted differ patients to patients. You see that the number of time a certain node is impacted is also different. The projection in the tangent space allow us then, to explore at the individual level the difference in functional connectivity in a subject regarding to a mean reference group.

Note

In the above glass brain, the size of the nodes are proportional to the number of times they make significant link with other nodes. The edge between two nodes, are red if the connectivity coefficient are positive and blue this coefficient is negative.

Depending on what you are studying, with our implementation of the projection of the tangent space, you can easily choose another reference group. For example, we can imagine that inside the only patients population, there is a homogeneous sub-group of patients regarding one or multiple parameters that can serve as a reference to study the remaining patients subjects.