Clustering analysis
Contents
Clustering analysis#
Within this section, the first step of the analyses workflow, the k-means clustering, is shown, explained and set up in a way that it can be rerun and reproduced. In short, we employed the Neurosynth database to create a meta-analytic functional profile for each voxel in the auditory cortex and derived different cluster solutions from 2 to 15 parcels. We then evaluated all cluster solutions to find the best number of parcels given the data which were then used for subsequent analyses, that is functional co-activation and functional profiles. The analysis was divided into the following sections:
Neurosynth database
Clustering analyses
Meta-analytic functional profiles
Neurosynth’s magic
Cluster evaluation
We provide more detailed information and explanations in the respective sections. As mentioned before, you can rerun the analyses either by downloading this page as a Jupyter Notebook (via the download button at the top) or interactively via a cloud instance through the amazing mybinder project (the little rocket at the top). We recommend the latter to spare installation and conda environment related problems. One more thing… Please note, that the command that performs the clustering is computationally very expensive and would never run via the resources available through binder. We provide all cluster solutions (in volume and surface format) via the project’s OSF page. However, we included the code and explanations depicting how this steps were run. The code that will download the cluster solutions is included in the respective sections.
First let’s import some basic necessities:
from os.path import join as opj
import seaborn as sns
import plotly.graph_objects as go
from IPython.core.display import display, HTML
from plotly.offline import init_notebook_mode, plot
from nilearn.plotting import plot_anat, plot_roi
from matplotlib.colors import ListedColormap
from matplotlib import colorbar, colors
from matplotlib.patches import Patch
from matplotlib.lines import Line2D
import numpy as np
from neurosynth import Dataset
from neurosynth import decode, network
from neurosynth.base import dataset, mask
from neurosynth.analysis.cluster import magic
from sklearn import metrics
from sklearn import cluster as sk_cluster
from sklearn.metrics import pairwise_distances
from sklearn import decomposition as sk_decomp
from neurosynth.analysis.cluster import Clusterable
from sklearn.datasets import make_blobs
from sklearn.cluster import KMeans
from sklearn.metrics import silhouette_score
from sklearn.model_selection import GridSearchCV
import seaborn as sns
import pandas as pd
from matplotlib import pyplot as plt
import random
We will also define some paths for our prerequisites and outcomes:
prereq_path = 'prerequisites'
results_path = 'parcellations'
Next, we will set up the Neurosynth database.
Neurosynth database#
The Neurosynth database is a large-scale repository targeted for meta-analyses, containing thousands of peak activations across the entire range of neuroscientific literature. Here, we utilize the latest version, v0.7 including over 14k studies. As everything is open and built around a great API, we can easily download the Neurosynth files:
neurosynth_data_dir = 'prerequisites'
dataset.download(neurosynth_data_dir, unpack=True)
Downloading the latest Neurosynth files: https://github.com/neurosynth/neurosynth-data/blob/master/current_data.tar.gz?raw=true bytes: 1
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[1610547200.00%16113664 [1611366400.00%16121856 [1612185600.00%16130048 [1613004800.00%16138240 [1613824000.00%16146432 [1614643200.00%16154624 [1615462400.00%16162816 [1616281600.00%16171008 [1617100800.00%16179200 [1617920000.00%16187392 [1618739200.00%16195584 [1619558400.00%16203776 [1620377600.00%16211968 [1621196800.00%16220160 [1622016000.00%16228352 [1622835200.00%16236544 [1623654400.00%16244736 [1624473600.00%16252928 [1625292800.00%16261120 [1626112000.00%16269312 [1626931200.00%16277504 [1627750400.00%16285696 [1628569600.00%16293888 [1629388800.00%16302080 [1630208000.00%16310272 [1631027200.00%16318464 [1631846400.00%16326656 [1632665600.00%16334848 [1633484800.00%16343040 [1634304000.00%16351232 [1635123200.00%16359424 [1635942400.00%16367616 [1636761600.00%16375808 [1637580800.00%16384000 [1638400000.00%16392192 [1639219200.00%16400384 [1640038400.00%16408576 [1640857600.00%16416768 [1641676800.00%16424960 [1642496000.00%16433152 [1643315200.00%16441344 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After downloading the database, we need to initiate the Dataset:
neurosynth_dataset = Dataset(opj(prereq_path, 'database.txt'))
and add the features:
neurosynth_dataset.add_features(opj(prereq_path,'features.txt'))
In order to prevent having to run these steps every time, we will save the final dataset as pkl so that we can just load it:
neurosynth_dataset.save(opj(prereq_path,'neurosynth_dataset.pkl'))
neurosynth_dataset = Dataset.load(opj(prereq_path,'neurosynth/neurosynth_dataset.pkl'))
We now have a Neurosynth database/set instance we can interact with, it contains the information of included peak activations and corresponding studies among other things:
neurosynth_dataset.activations.head(n=10)
| id | doi | x | y | z | space | peak_id | table_id | table_num | title | authors | year | journal | i | j | k | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 9065511 | NaN | 38.0 | -48.0 | 49.0 | MNI | 548691 | 28698 | 1. | Environmental knowledge is subserved by separa... | Aguirre GK, D'Esposito M | 1997 | The Journal of neuroscience : the official jou... | 60 | 39 | 26 |
| 1 | 9065511 | NaN | -4.0 | -70.0 | 50.0 | MNI | 548692 | 28698 | 1. | Environmental knowledge is subserved by separa... | Aguirre GK, D'Esposito M | 1997 | The Journal of neuroscience : the official jou... | 61 | 28 | 47 |
| 2 | 9065511 | NaN | -34.0 | -52.0 | 60.0 | MNI | 548693 | 28698 | 1. | Environmental knowledge is subserved by separa... | Aguirre GK, D'Esposito M | 1997 | The Journal of neuroscience : the official jou... | 66 | 37 | 62 |
| 3 | 9065511 | NaN | -23.0 | 15.0 | 67.0 | MNI | 548694 | 28698 | 1. | Environmental knowledge is subserved by separa... | Aguirre GK, D'Esposito M | 1997 | The Journal of neuroscience : the official jou... | 70 | 70 | 56 |
| 4 | 9065511 | NaN | -23.0 | -20.0 | 68.0 | MNI | 548695 | 28698 | 1. | Environmental knowledge is subserved by separa... | Aguirre GK, D'Esposito M | 1997 | The Journal of neuroscience : the official jou... | 70 | 53 | 56 |
| 5 | 9065511 | NaN | 42.0 | -47.0 | -19.0 | MNI | 548696 | 28698 | 1. | Environmental knowledge is subserved by separa... | Aguirre GK, D'Esposito M | 1997 | The Journal of neuroscience : the official jou... | 26 | 40 | 24 |
| 6 | 9065511 | NaN | 25.0 | -35.0 | -8.0 | MNI | 548697 | 28698 | 1. | Environmental knowledge is subserved by separa... | Aguirre GK, D'Esposito M | 1997 | The Journal of neuroscience : the official jou... | 32 | 46 | 32 |
| 7 | 9065511 | NaN | -25.0 | -45.0 | -2.0 | MNI | 548698 | 28698 | 1. | Environmental knowledge is subserved by separa... | Aguirre GK, D'Esposito M | 1997 | The Journal of neuroscience : the official jou... | 35 | 40 | 58 |
| 8 | 9065511 | NaN | 52.0 | -62.0 | 14.0 | MNI | 548699 | 28698 | 1. | Environmental knowledge is subserved by separa... | Aguirre GK, D'Esposito M | 1997 | The Journal of neuroscience : the official jou... | 43 | 32 | 19 |
| 9 | 9065511 | NaN | 21.0 | -81.0 | 28.0 | MNI | 548700 | 28698 | 1. | Environmental knowledge is subserved by separa... | Aguirre GK, D'Esposito M | 1997 | The Journal of neuroscience : the official jou... | 50 | 22 | 34 |
As said before: easy and straightforward. With that we already have everything we need to start our analyses, beginning with the creation of a meta-analytic functional profile for each voxel in the auditory cortex.
Clustering analyses#
The clustering analyses was divided into three steps: (1) creating a meta-analytic functional profile for each voxel in the auditory cortex, (2) derivation of cluster solutions and (3) cluster evaluation. While (1) is actually automatically conducted as part of (2), we will still showcase it in more detail to enable a better understanding of what was going on and enable the cluster evaluation in (3). The general aim of this part of the analyses workflow was to
first, evaluate if
data-driven parcellationbased onlarge-scale meta-analytic datawould resemble theparcellationsderived and suggested in prior research work based on a wide range of markers, stimuli, paradigms and analyses methodssecond, find the best
parcellationgiven the data to utilize it for subsequent analyses
Clustering analyses - meta-analytic functional profiles#
The basis for the clustering analyses, as well as all others, was a meta-analytic functional profile for each voxel in the auditory cortex. It entailed the co-activation of auditory cortex voxels with the rest of the brain by means of a correlation between their activation patterns across studies. Please note that activation pattern here referred to a binary vector entailing either 1or 0 depending on if a voxel fell within 10 mm of an activation focus reported in a given study. To obtain this meta-analytic functional profile, we thus need to define a mask of the auditory cortex to indicate which voxels we would like to include. We created and explained this mask in the previous step and therefore will only plot it here so that we can easily check it:
ac_mask = opj(prereq_path, 'ac_rois/tpl-MNI152NLin6Sym_res_02_desc-ACdilated_mask.nii.gz')
#suppress contour related warnings
import warnings
warnings.filterwarnings("ignore")
sns.set_style('white')
plt.rcParams["font.family"] = "Arial"
fig=plt.figure(num=None, figsize=(12, 6))
check_mask = plot_anat(cut_coords=[-55, -34, 7], figure=fig, draw_cross=False)
check_mask.add_contours(ac_mask,
levels=[0], colors='black')
plt.text(-215, 90, 'Auditory cortex mask', fontsize=30)
Ah yeah, that was what it looked like. Using the Clusterable class of Neurosynth we can utilize it to create the above mentioned binary vector for each auditory cortex voxel from the neurosynth dataset we set up before. With a focus on stability and robustness we indicated that at a given voxel should be included in at least 80 studies in order to be part of the analysis. An important thing we need to do before is to reset the masker of our dataset as we would otherwise run into version incompatibility issues with nibabel and neurosynth.
neurosynth_dataset.masker = mask.Masker('/opt/miniconda/envs/neurosynth-ac/lib/python3.6/site-packages/neurosynth/resources/MNI152_T1_2mm_brain.nii.gz')
Ok, but now we are ready.
auditory_cortex_roi = Clusterable(neurosynth_dataset,
opj(prereq_path, 'ac_rois/tpl-MNI152NLin6Sym_res_02_desc-ACdilated_mask.nii.gz'),
min_studies=80)
What we get in return is a Clusterable object that contains our wanted binary vector:
auditory_cortex_roi.data
array([[0., 0., 0., ..., 0., 0., 0.],
[0., 0., 0., ..., 0., 0., 0.],
[0., 0., 0., ..., 0., 0., 0.],
...,
[0., 0., 1., ..., 0., 0., 0.],
[0., 0., 1., ..., 0., 0., 0.],
[0., 0., 1., ..., 0., 0., 0.]])
As we are working with large scale analyses here, it is a good idea to check the dimensions of our binary vector. We will also save it to be able to easily reload it later on:
np.save('prerequisites/auditory_cortex_data.npy', auditory_cortex_roi.data)
auditory_cortex_roi.data.shape
(9111, 14371)
Large scale is right, eh? We see that we have a voxels (4635) by studies (14371) array containing 0 and 1 across studies for each auditory cortex voxel. As outlined above, the plan was to correlate this binary vector with the rest of the brain to obtained the meta-analytic functional profile. However, seeing how large this array already is and knowing that the brain contains more than 100k voxels we would definitely run into computational problems going further, especially concerning resources. We therefore decided to reduce the dimensionality of the whole brain data to 100 components via a Principal component analysis. To do so, we at first need to define a binary vector as done before, but this time without providing a mask to restrict it to certain voxels. We will however keep the “voxel active in at least 80 studies” requirement:
reference = Clusterable(neurosynth_dataset, min_studies=80)
This already took longer to compute, let’s check the dimensions:
reference.data.shape
(186629, 14371)
See? Dimension reduction definitely seems like a good idea given that we have 186629 voxels by 14371 studies. Luckily, neurosynth includes functionality to perform this operation. All we need to do is to define the number of components we would like to obtain (100), the number of features in our binary vector (14371) and the dimension reduction approach (PCA):
n_components=100
n_feat = reference.data.shape[1]
reduce_reference = sk_decomp.PCA(n_components)
We can now apply these settings within the transform function of our whole brain binary vector to achieve the wanted dimension reduction. Please note that this step is computationally expensive and might not run via the interactive online instance (if it does, it will take a long time). We therefore also provide the resulting components so that the subsequent steps can be run. Thus, if you do not want to try to run this command or do not want to wait, please skip the next line and continue with the np.load('prerequistes/reference_data.npy') step.
reference = reference.transform(reduce_reference, transpose=True)
How does our whole brain binary vector look now?
reference.data.shape
(100, 14371)
Great, we get the expected output: while the number of features, i.e. studies remained the same (14371) we successfully reduced the whole brain binary vector from 186629 voxels to 100 components. We are going to save this reduced form so that we can reload it easily later on and also share it.
np.save('prerequisites/reference_data.npy', reference.data)
We now have both parts we need to compute the meta-analytic functional profile: the binary vector for each auditory cortex voxel and the 100 component whole brain binary vector. Using scikit-learn’s pairwise distance function we can now compute the correlation between the activation pattern of each auditory cortex voxel and whole brain component
reference = np.load('prerequisites/reference_data.npy')
auditory_cortex_roi = np.load('prerequisites/auditory_cortex_data.npy')
ac_wb_distances = pairwise_distances(auditory_cortex_roi.data, reference.data,
metric='correlation')
If everything worked as planned, the result should be an auditory cortex voxel (4635) by whole brain component (100) matrix:
ac_wb_distances.shape
(9111, 100)
Fantastic! Lets plot the meta-analytic functional profile to get an idea of what it looks like:
sns.set(font="Arial")
sns.set_context('paper')
grid_kws = {"height_ratios": (.9, .01), "hspace": .1}
f, (ax, cbar_ax) = plt.subplots(2, figsize=(12,20), gridspec_kw=grid_kws)
ax = sns.heatmap(ac_wb_distances, ax=ax,
cbar_ax=cbar_ax,
cbar_kws={"orientation": "horizontal", "label": "Correlation"},
cmap='viridis', yticklabels=False, xticklabels=False)
ax.set_xlabel("PCA components (n=100)", fontsize=30, fontweight="bold")
ax.set_ylabel("auditory cortex voxels (n=9111)", fontsize=30, fontweight="bold")
ax.set_title('Auditory cortex - whole brain (PCA) \n correlation matrix', fontsize=29, fontweight="bold")
cbar_ax.tick_params(labelsize=20)
cbar_ax.xaxis.label.set_size(30)
cbar_ax.xaxis.label.set_fontname("Arial")
We will also save it to ease up later analysis steps:
np.save('prerequisites/ac_wb_distances.npy', ac_wb_distances)
This meta-analytic functional profile can be now be used to perform a clustering analysis which will group the auditory cortex voxels based on their profile similarity. As mentioned before, this step was conducted using neurosynth’s magic function.
Neurosynth’s Magic#
We already mentioned it, but just to be sure: the meta-analytic functional profile is actually automatically computed through the magic function which then also subsequently performs the clustering. Thus, we do not provide the outputs of the previous step, but the parameters we used to derive the meta-analytic functional profile: the neurosynth dataset, the auditory cortex mask and the voxel inclusion criterion. Additionally, we need to set an output directory, specifying where our parcellations should be saved and the number of cluster, indicating in how many clusters the auditory cortex voxels should be divided into. Following cluster solutions proposed in prior research work, we decided to derive different of cluster solutions that ranged from 2 to 15 clusters mirroring frequently utilized and proposed parcellations. We thus pack the magic function in a small for-loop to run it for each number of clusters we want to obtain. As other commands before, this one will also not run via the interactive online instance (therefore, please skip the next line). We thus provide the derived cluster solution within this resource.
for cluster_size in range(1,16):
magic(dataset, roi_mask=opj(prereq_path, 'ac_rois/tpl-MNI152NLin6Sym_res_02_desc-AC_mask.nii.gz'),
min_studies_per_voxel=80, output_dir=results_path, n_clusters=cluster_size)
Using a small for-loop again, we can now visually inspect the obtained cluster solutions:
for n_cluster in range(2,16):
plt.rcParams["font.family"] = "Arial"
fig=plt.figure(num=None, figsize=(12, 6))
cmap = sns.cubehelix_palette(n_cluster+1, start=-.2, rot=.6, reverse=True, as_cmap=True)
plot_roi('parcellations/tpl-MNI152NLin6Sym_atlas-nsac_desc-%s.nii.gz' %n_cluster,
draw_cross=False, cmap=cmap, alpha=0.8, cut_coords=[-55, -34, 7], figure=fig)
plt.text(-205,85, '%s cluster solution' %str(n_cluster), fontsize=30)
While this already looks great and is highly informative, we need to investigate the obtained clusters further. Given that we want to conduct a complementary set of analyses in the next analysis steps and we do not want to run everything for each cluster solution, selecting only “the best” sounds like a great idea. But how would one do that? The next section provides a possible approach, which is also the one that was used within the project.
Cluster evaluation#
In order to select “the best cluster solution” a lot of things need to be considered. In fact, the pure question might be ill-posed as “best” depends on a variety of factors such as the data at hand, the research questions, etc. . Additionally, the selection should be guided by some sort of measure that quantifies things. Bringing these aspects together, we decided to utilize the silhouette score to evaluate the obtained cluster solutions in an objective and quantitative manner. The silhouette score provides a measure of within-cluster cohesion of a given cluster solution by relating the mean intracluster Euclidean distanace (a) and the Euclidean distance between a smaple and the nearest cluster that does not entail the sample (b) via the following equation:
\[(b-a)/max(a,b)\]
Thus, cluster solutions that minimize the average distance between voxels within each cluster receive higher scores. To obtain the silhouette scores, we will basically redo the clustering for all solutions (2-15) and compute the silhouette score for each. As there is always some uncertainty, we will run a 10-fold cross-validation grid-search for each cluster solution to assess the stability of their corresponding silhouette scores. At first, we make sure we have our AC voxel meta-analytic functional profiles:
ac_wb_distances = np.load('prerequisites/ac_wb_distances.npy')
Next, we define a scoring to use within the grid-search, i.e. the silhouette score as implemented in scikit-learn:
def scorer(clustering, X_test):
return silhouette_score(X_test, clustering.predict(X_test))
As there will be cross-validation folds and therefore some randomness, we set the random seed to make things reproducible:
random.seed(0)
With that we are ready to define our grid-search. Here we are going to use scikit-learn’s GridSearchCV function and specify our KMeans with ++ initiation, 10 time repetition on varying centroid seeds and the cluster solutions we want to obtain. Additionally, we set our scoring function and the number of cross-validation folds:
grid = GridSearchCV(
KMeans(init='k-means++', n_init=10),
{"n_clusters": [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]},
cv=10,
scoring=scorer,
refit=False,
n_jobs=5,
verbose=1
)
We can now fit it to the AC voxel meta-analytic functional profiles and wait for a bit:
grid.fit(ac_wb_distances)
Fitting 10 folds for each of 14 candidates, totalling 140 fits
GridSearchCV(cv=10, estimator=KMeans(), n_jobs=5,
param_grid={'n_clusters': [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,
14, 15]},
refit=False, scoring=<function scorer at 0x7fab95fac730>,
verbose=1)
Instead of providing direct outputs, the things we are interested in, i.e. silhouette scores per cluster solution across cross-validation folds, are part of the fitted GridSearchCV. To make things easier, we will pack everything into a DataFrame:
silhouette_scores = pd.DataFrame(grid.cv_results_)
silhouette_scores.set_index("param_n_clusters", inplace=True)
silhouette_scores = silhouette_scores.loc[:, [f"split{i}_test_score" for i in range(n_splits)]]
silhouette_scores = silhouette_scores.stack().reset_index().rename(columns={0: "silhouette"})
How do things look now? Let’s check DataFrame:
silhouette_scores.head()
| param_n_clusters | level_1 | silhouette | |
|---|---|---|---|
| 0 | 2 | split0_test_score | 0.221925 |
| 1 | 2 | split1_test_score | 0.229948 |
| 2 | 2 | split2_test_score | 0.167363 |
| 3 | 2 | split3_test_score | 0.119612 |
| 4 | 2 | split4_test_score | 0.109091 |
Great, that looks about right. In order to provide an informative overview and more insights, we will define a small plotting function that will plot the silhouette scores across cross-validation folds against the cluster solutions. To make the most of these resources, we will not only support a static plot via seaborn, but also an interactive plot via plotly. This will be set via the interactive parameter of the function (defaults to True, thus interactive). Please note, that we will create a black and white plot in order to not convey any false impression concerning the cluster solutions via a colormap.
def plot_cluster_silhouette(interactive=True):
if interactive is False:
sns.set_style('white')
plt.rcParams["font.family"] = "Arial"
plt.figure(figsize=(14,8))
ax = sns.boxplot(x="param_n_clusters", y="silhouette", data=silhouette_scores, color='white')#, palette=sns.cubehelix_palette(15, start=-.2, rot=.6, reverse=True))
sns.swarmplot(x="param_n_clusters", y="silhouette", data=silhouette_scores, color=".25", edgecolor="white", linewidth=0.2)
plt.setp(ax.artists, edgecolor = 'k', facecolor='w')
plt.setp(ax.lines, color='k')
plt.title('Cluster solution evaluation', fontsize=30, fontweight="bold")
plt.xlabel("number of clusters", fontsize=30)
plt.ylabel("silhouette score", fontsize=30)
plt.tick_params(labelsize=25)
sns.despine(offset=5, trim=True)
else:
fig = go.Figure()
for i, tmp_df in silhouette_scores.groupby('param_n_clusters'):
fig.add_trace(go.Box(x=tmp_df['silhouette'].values, name=str(i), marker_color = 'black'))
fig.update_traces(boxpoints='all', jitter=0)
fig.update_traces(orientation='h') # horizontal box plots
fig.update_layout(template='plotly_white',
title="Silhouette score per cluster solution", title_x=0.5,
autosize=False,
width=700,
height=1000)
fig.update_xaxes(title_text='Silhouette score (across 10 folds)')
fig.update_yaxes(title_text='Cluster solution (n clusters)', tickvals = list(range(2,16)))
fig.update_layout(showlegend=False)
init_notebook_mode(connected=True)
plot(fig, filename = 'plots/silhouette_scores.html')
display(HTML('plots/silhouette_scores.html'))
Coolio, so how do our cluster solutions perform with regard to their silhouette score?
plot_cluster_silhouette(interactive=True)
Ok ok, we see a lot of things going on here. Given these outcomes and the available information, we decided to base our decision which cluster solution to investigate further on three aspects: first, the highest mean silhouette score across folds; second, the least variance across folds; and third, the least outliers. Bringing these things together, we decided on the 5 cluster solution. However, if folks are interested in the other solutions, the code and information provided here should allow to run the conducted analyses for other cluster solutions comparably easy. For now, let’s check the solution utilized here in more detail and provide some names for our clusters (please note that these names are meant to ease up referring to certain clusters and supply some rough information regarding their location):
plt.rcParams["font.family"] = "Arial"
fig= plt.figure(figsize=(12, 6))
cmap_plot = ListedColormap(sns.cubehelix_palette(5, start=-.2, rot=.6, reverse=True))
plot_roi('parcellations/tpl-MNI152NLin6Sym_atlas-nsac_desc-5.nii.gz', cmap=cmap_plot, draw_cross=False, figure=fig)
legend_elements = [Patch(facecolor=cmap_plot(0), label='1/PAC'),
Patch(facecolor=cmap_plot(1), label='2/STG'),
Patch(facecolor=cmap_plot(2), label='3/PT'),
Patch(facecolor=cmap_plot(3), label='4/PP'),
Patch(facecolor=cmap_plot(4), label='5/A4_A5')]
plt.legend(handles=legend_elements, loc='lower center', bbox_to_anchor=(-0.5, -0.2),
frameon=False, ncol=5, title='Cluster values', prop={'size': 16}, title_fontsize=20)
plt.text(-205,85, '5 cluster solution', fontsize=30)
Fantastic! Please note, that by default all subsequent analysis descriptions refer to the chosen cluster solution exclusively.