Matched Networks¶

We will use the one sample discriminability test from hyppo to assess how well each kernel detects dissimilarity between labels. The discriminability index is a measure of whether a data acquisition method is more discriminable than others, and the one sample test determines whether the discriminability of the sample differs from random chance [Bridgeford et al., 2019].

First, we load in 32 networks that were generated from MRI scans of mice brains, provided by Graspologic. The mice came from 4 different genotypes, and there were 8 mice per genotype.

Load Data¶

# HIDE CELL

# load the full mouse dataset
from graspologic.datasets import load_mice
mice = load_mice()

# Stack all adjacency matrices in a 3D numpy array
graphs = np.array(mice.graphs)
print(graphs.shape)

# initialize labels
labels = {}
for i in np.arange(0, 332):
    labels[i] = i

(32, 332, 332)

Now we generate dissimilarity matrices based on each of the kernels outlined in the kernels notebook in the Introduction. Note that by default, we only show results for the Density kernel, the Edge weight kernel (using the \(L^1\)-norm), the Laplacian spectral kernel, and the Omnibus embedding kernel, but all results can be expanded for viewing.

Every dissimilarity matrix is visualized by the heatmap function from Graspologic, and are sorted by their labels. We also normalize the dissimilarity matrix by dividing it by its maximum value, scaling the matrix from 0 to 1.

Global Properties¶

Density¶

# HIDE CODE
scaled_density_dissim = calculate_dissim(graphs, method="density", norm=None, normalize=True)

ax = heatmap(scaled_density_dissim, context="talk", inner_hier_labels=mice.labels)
ax.figure.set_facecolor('w')

Average Edge Weight¶

# HIDE CELL
scaled_avgedgeweight_dissim = calculate_dissim(graphs, method="avgedgeweight", norm=None, normalize=True)

ax = heatmap(scaled_avgedgeweight_dissim, context="talk", inner_hier_labels=mice.labels)
ax.figure.set_facecolor('w')

Average of the Adjacency Matrix¶

# HIDE CELL
scaled_avgadjmat_dissim = calculate_dissim(graphs, method="avgadjmatrix", norm=None, normalize=True)

ax = heatmap(scaled_avgadjmat_dissim, context="talk", inner_hier_labels=mice.labels)
ax.figure.set_facecolor('w')

Laplacian Spectral Distance¶

Each network is transformed using pass-to-ranks, and the L2 norm of the difference between the sorted eigenspecta is calculated.

# HIDE CODE
scaled_lap_dissim = laplacian_dissim(graphs, transform='pass-to-ranks', metric='l2', normalize=True)

ax = heatmap(scaled_lap_dissim, context="talk", inner_hier_labels=mice.labels)
ax.figure.set_facecolor('w')

Node-wise Properties¶

Node Degrees - L1 Norm¶

# HIDE CELL
scaled_nodedeg_dissim_l1 = calculate_dissim(graphs, method="degree", norm="l1", normalize=True)

ax = heatmap(scaled_nodedeg_dissim_l1, context="talk", inner_hier_labels=mice.labels)
ax.figure.set_facecolor('w')

Node Degrees - L2 Norm¶

# HIDE CELL
scaled_nodedeg_dissim_l2 = calculate_dissim(graphs, method="degree", norm="l2", normalize=True)

ax = heatmap(scaled_nodedeg_dissim_l2, context="talk", inner_hier_labels=mice.labels)
ax.figure.set_facecolor('w')

Node Strength - L1 Norm¶

# HIDE CELL
scaled_nodestr_dissim_l1 = calculate_dissim(graphs, method="strength", norm="l1", normalize=True)

ax = heatmap(scaled_nodestr_dissim_l1, context="talk", inner_hier_labels=mice.labels)
ax.figure.set_facecolor('w')

Node Strength - L2 Norm¶

# HIDE CELL
scaled_nodestr_dissim_l2 = calculate_dissim(graphs, method="strength", norm="l2", normalize=True)

ax = heatmap(scaled_nodestr_dissim_l2, context="talk", inner_hier_labels=mice.labels)
ax.figure.set_facecolor('w')

Edge-wise Properties¶

Edge weights - L1 Norm¶

# HIDE CODE
scaled_edgeweight_dissim_l1 = calculate_dissim(graphs, method="edgeweight", norm="l1", normalize=True)

ax = heatmap(scaled_edgeweight_dissim_l1, context="talk", inner_hier_labels=mice.labels)
ax.figure.set_facecolor('w')

Edge Weights - L2 Norm¶

# HIDE CELL
scaled_edgeweight_dissim_l2 = calculate_dissim(graphs, method="edgeweight", norm="l2", normalize=True)

ax = heatmap(scaled_edgeweight_dissim_l2, context="talk", inner_hier_labels=mice.labels)
ax.figure.set_facecolor('w')

Omnibus Embedding¶

Here we use omnibus embedding in order to provide a comparison to a well-established kernel.

# HIDE CODE
from graspologic.embed import OmnibusEmbed

# embed using Omni
embedder = OmnibusEmbed(n_elbows=4)
omni_embedding = embedder.fit_transform(graphs)

# create dissimilarity matrix
omni_matrix = np.zeros((len(graphs), len(graphs)))
for i, embedding1 in enumerate(omni_embedding):
    for j, embedding2 in enumerate(omni_embedding):
        dist = np.linalg.norm(embedding1 - embedding2, ord="fro")
        omni_matrix[i, j] = dist

# plot heatmap
scaled_omni_dissim = omni_matrix / np.max(omni_matrix)

ax = heatmap(scaled_omni_dissim, context="talk", inner_hier_labels=mice.labels)
ax.figure.set_facecolor('w')

NDD II - Investigating Kernels

Matched Networks

Contents