Note
Go to the end to download the full example code.
Comparison of embeddings of covariance matrices¶
Comparison of several embeddings of a set of ERP covariance matrices extracted on MEG data: SE, LLE and t-SNE
Spectral Embedding (SE) is based on computing the low-dimensional representation that best preserves locality instead of local linearity in LLE [1].
Locally Linear Embedding (LLE) assumes that the local neighborhood of a matrix on the manifold can be well approximated by the affine subspace spanned by the k-nearest neighbors of the matrix and finds a low-dimensional embedding of the data based on these affine approximations.
t-SNE reduces SPD matrices into lower dimensional SPD matrices by computing conditional probabilities that represent similarities [2]. This fully Riemannian algorithm helps preserve the non-Euclidean structure of the data.
# Authors: Pedro Rodrigues <pedro.rodrigues01@gmail.com>,
# Gabriel Wagner vom Berg <gabriel@bccn-berlin.de>
# Thibault de Surrel <thibault.de-surrel@lamsade.dauphine.fr>
#
# License: BSD (3-clause)
import matplotlib.pyplot as plt
import mne
from mne import io
from mne.datasets import sample
from sklearn.model_selection import train_test_split
from pyriemann.estimation import XdawnCovariances
from pyriemann.utils.viz import plot_embedding
print(__doc__)
Set parameters and read data¶
data_path = str(sample.data_path())
raw_fname = data_path + "/MEG/sample/sample_audvis_filt-0-40_raw.fif"
event_fname = data_path + "/MEG/sample/sample_audvis_filt-0-40_raw-eve.fif"
tmin, tmax = -0., 1
event_id = dict(aud_l=1, aud_r=2, vis_l=3, vis_r=4)
# Setup for reading the raw data
raw = io.Raw(raw_fname, preload=True, verbose=False)
raw.filter(2, None, method="iir") # replace baselining with high-pass
events = mne.read_events(event_fname)
raw.info["bads"] = ["MEG 2443"] # set bad channels
picks = mne.pick_types(raw.info, meg=True, eeg=False, stim=False, eog=False,
exclude="bads")
# Read epochs
epochs = mne.Epochs(raw, events, event_id, tmin, tmax, proj=False,
picks=picks, baseline=None, preload=True, verbose=False)
X = epochs.get_data(copy=False)
y = epochs.events[:, -1]
Using default location ~/mne_data for sample...
0%| | 0.00/1.65G [00:00<?, ?B/s]
0%| | 179k/1.65G [00:00<15:43, 1.75MB/s]
0%| | 2.27M/1.65G [00:00<02:08, 12.9MB/s]
1%|▏ | 10.3M/1.65G [00:00<00:37, 43.6MB/s]
1%|▍ | 19.0M/1.65G [00:00<00:27, 60.4MB/s]
2%|▌ | 27.7M/1.65G [00:00<00:23, 70.0MB/s]
2%|▊ | 36.3M/1.65G [00:00<00:21, 75.7MB/s]
3%|█ | 45.0M/1.65G [00:00<00:20, 79.2MB/s]
3%|█▏ | 53.6M/1.65G [00:00<00:19, 81.4MB/s]
4%|█▍ | 62.2M/1.65G [00:00<00:19, 83.0MB/s]
4%|█▌ | 70.9M/1.65G [00:01<00:18, 84.1MB/s]
5%|█▊ | 79.4M/1.65G [00:01<00:18, 84.4MB/s]
5%|█▉ | 88.0M/1.65G [00:01<00:18, 84.9MB/s]
6%|██▏ | 96.5M/1.65G [00:01<00:18, 85.0MB/s]
6%|██▍ | 105M/1.65G [00:01<00:18, 85.1MB/s]
7%|██▌ | 114M/1.65G [00:01<00:18, 85.4MB/s]
7%|██▊ | 122M/1.65G [00:01<00:17, 85.7MB/s]
8%|███ | 131M/1.65G [00:01<00:17, 86.0MB/s]
8%|███▏ | 140M/1.65G [00:01<00:17, 86.1MB/s]
9%|███▍ | 148M/1.65G [00:01<00:17, 86.3MB/s]
9%|███▌ | 157M/1.65G [00:02<00:17, 86.3MB/s]
10%|███▊ | 166M/1.65G [00:02<00:17, 86.2MB/s]
11%|████ | 174M/1.65G [00:02<00:17, 86.2MB/s]
11%|████▏ | 183M/1.65G [00:02<00:17, 86.2MB/s]
12%|████▍ | 191M/1.65G [00:02<00:16, 86.3MB/s]
12%|████▌ | 200M/1.65G [00:02<00:16, 86.2MB/s]
13%|████▊ | 209M/1.65G [00:02<00:16, 86.2MB/s]
13%|████▉ | 217M/1.65G [00:02<00:16, 86.2MB/s]
14%|█████▏ | 226M/1.65G [00:02<00:16, 86.2MB/s]
14%|█████▍ | 235M/1.65G [00:02<00:16, 86.2MB/s]
15%|█████▌ | 243M/1.65G [00:03<00:16, 86.2MB/s]
15%|█████▊ | 252M/1.65G [00:03<00:16, 86.2MB/s]
16%|█████▉ | 260M/1.65G [00:03<00:16, 86.3MB/s]
16%|██████▏ | 269M/1.65G [00:03<00:16, 86.4MB/s]
17%|██████▍ | 278M/1.65G [00:03<00:15, 86.4MB/s]
17%|██████▌ | 286M/1.65G [00:03<00:15, 86.4MB/s]
18%|██████▊ | 295M/1.65G [00:03<00:15, 86.4MB/s]
18%|██████▉ | 304M/1.65G [00:03<00:15, 86.4MB/s]
19%|███████▏ | 312M/1.65G [00:03<00:15, 86.5MB/s]
19%|███████▍ | 321M/1.65G [00:03<00:15, 86.5MB/s]
20%|███████▌ | 330M/1.65G [00:04<00:15, 86.5MB/s]
20%|███████▊ | 338M/1.65G [00:04<00:15, 86.5MB/s]
21%|███████▉ | 347M/1.65G [00:04<00:15, 86.5MB/s]
22%|████████▏ | 356M/1.65G [00:04<00:14, 86.5MB/s]
22%|████████▍ | 364M/1.65G [00:04<00:14, 86.5MB/s]
23%|████████▌ | 373M/1.65G [00:04<00:14, 86.5MB/s]
23%|████████▊ | 382M/1.65G [00:04<00:14, 86.4MB/s]
24%|████████▉ | 390M/1.65G [00:04<00:14, 86.3MB/s]
24%|█████████▏ | 399M/1.65G [00:04<00:14, 86.4MB/s]
25%|█████████▎ | 408M/1.65G [00:04<00:14, 86.6MB/s]
25%|█████████▌ | 416M/1.65G [00:05<00:17, 70.5MB/s]
26%|█████████▋ | 424M/1.65G [00:05<00:16, 72.3MB/s]
26%|█████████▉ | 433M/1.65G [00:05<00:16, 76.0MB/s]
27%|██████████▏ | 441M/1.65G [00:05<00:15, 78.8MB/s]
27%|██████████▎ | 450M/1.65G [00:05<00:14, 80.9MB/s]
28%|██████████▌ | 458M/1.65G [00:05<00:14, 82.4MB/s]
28%|██████████▋ | 467M/1.65G [00:05<00:14, 83.5MB/s]
29%|██████████▉ | 476M/1.65G [00:05<00:13, 84.2MB/s]
29%|███████████▏ | 484M/1.65G [00:05<00:13, 84.8MB/s]
30%|███████████▎ | 493M/1.65G [00:05<00:13, 85.2MB/s]
30%|███████████▌ | 502M/1.65G [00:06<00:13, 85.6MB/s]
31%|███████████▋ | 510M/1.65G [00:06<00:13, 85.7MB/s]
31%|███████████▉ | 519M/1.65G [00:06<00:13, 85.9MB/s]
32%|████████████▏ | 527M/1.65G [00:06<00:13, 85.9MB/s]
32%|████████████▎ | 536M/1.65G [00:06<00:12, 86.0MB/s]
33%|████████████▌ | 545M/1.65G [00:06<00:12, 86.0MB/s]
33%|████████████▋ | 553M/1.65G [00:06<00:12, 86.1MB/s]
34%|████████████▉ | 562M/1.65G [00:06<00:12, 86.1MB/s]
35%|█████████████ | 570M/1.65G [00:06<00:12, 86.1MB/s]
35%|█████████████▎ | 579M/1.65G [00:06<00:12, 86.2MB/s]
36%|█████████████▌ | 588M/1.65G [00:07<00:12, 86.2MB/s]
36%|█████████████▋ | 596M/1.65G [00:07<00:12, 86.2MB/s]
37%|█████████████▉ | 605M/1.65G [00:07<00:12, 86.2MB/s]
37%|██████████████ | 614M/1.65G [00:07<00:12, 86.4MB/s]
38%|██████████████▎ | 622M/1.65G [00:07<00:11, 86.4MB/s]
38%|██████████████▌ | 631M/1.65G [00:07<00:11, 86.3MB/s]
39%|██████████████▋ | 640M/1.65G [00:07<00:11, 86.3MB/s]
39%|██████████████▉ | 648M/1.65G [00:07<00:11, 86.4MB/s]
40%|███████████████ | 657M/1.65G [00:07<00:11, 86.4MB/s]
40%|███████████████▎ | 666M/1.65G [00:07<00:11, 86.5MB/s]
41%|███████████████▍ | 674M/1.65G [00:08<00:11, 86.4MB/s]
41%|███████████████▋ | 683M/1.65G [00:08<00:11, 86.4MB/s]
42%|███████████████▉ | 691M/1.65G [00:08<00:11, 86.4MB/s]
42%|████████████████ | 700M/1.65G [00:08<00:11, 86.3MB/s]
43%|████████████████▎ | 709M/1.65G [00:08<00:10, 86.3MB/s]
43%|████████████████▍ | 717M/1.65G [00:08<00:10, 86.4MB/s]
44%|████████████████▋ | 726M/1.65G [00:08<00:10, 86.5MB/s]
44%|████████████████▉ | 735M/1.65G [00:08<00:10, 86.4MB/s]
45%|█████████████████ | 743M/1.65G [00:08<00:10, 86.5MB/s]
45%|█████████████████▎ | 752M/1.65G [00:08<00:10, 86.4MB/s]
46%|█████████████████▍ | 761M/1.65G [00:09<00:10, 86.5MB/s]
47%|█████████████████▋ | 769M/1.65G [00:09<00:10, 86.5MB/s]
47%|█████████████████▉ | 778M/1.65G [00:09<00:10, 86.4MB/s]
48%|██████████████████ | 787M/1.65G [00:09<00:10, 86.4MB/s]
48%|██████████████████▎ | 795M/1.65G [00:09<00:09, 86.4MB/s]
49%|██████████████████▍ | 804M/1.65G [00:09<00:09, 86.3MB/s]
49%|██████████████████▋ | 813M/1.65G [00:09<00:09, 86.4MB/s]
50%|██████████████████▉ | 821M/1.65G [00:09<00:09, 86.4MB/s]
50%|███████████████████ | 830M/1.65G [00:09<00:09, 86.5MB/s]
51%|███████████████████▎ | 838M/1.65G [00:09<00:09, 86.4MB/s]
51%|███████████████████▍ | 847M/1.65G [00:10<00:09, 86.4MB/s]
52%|███████████████████▋ | 856M/1.65G [00:10<00:09, 86.5MB/s]
52%|███████████████████▊ | 864M/1.65G [00:10<00:09, 86.5MB/s]
53%|████████████████████ | 873M/1.65G [00:10<00:09, 86.4MB/s]
53%|████████████████████▎ | 882M/1.65G [00:10<00:08, 86.3MB/s]
54%|████████████████████▍ | 890M/1.65G [00:10<00:08, 86.3MB/s]
54%|████████████████████▋ | 899M/1.65G [00:10<00:08, 86.3MB/s]
55%|████████████████████▊ | 908M/1.65G [00:10<00:08, 86.3MB/s]
55%|█████████████████████ | 916M/1.65G [00:10<00:08, 86.3MB/s]
56%|█████████████████████▎ | 925M/1.65G [00:10<00:08, 86.3MB/s]
56%|█████████████████████▍ | 934M/1.65G [00:11<00:08, 86.2MB/s]
57%|█████████████████████▋ | 942M/1.65G [00:11<00:08, 86.3MB/s]
58%|█████████████████████▊ | 951M/1.65G [00:11<00:08, 83.5MB/s]
58%|██████████████████████ | 959M/1.65G [00:11<00:08, 84.3MB/s]
59%|██████████████████████▎ | 968M/1.65G [00:11<00:08, 84.9MB/s]
59%|██████████████████████▍ | 977M/1.65G [00:11<00:07, 85.5MB/s]
60%|██████████████████████▋ | 985M/1.65G [00:11<00:07, 85.8MB/s]
60%|██████████████████████▊ | 994M/1.65G [00:11<00:07, 86.0MB/s]
61%|██████████████████████▍ | 1.00G/1.65G [00:11<00:07, 86.0MB/s]
61%|██████████████████████▋ | 1.01G/1.65G [00:11<00:07, 86.2MB/s]
62%|██████████████████████▊ | 1.02G/1.65G [00:12<00:07, 86.2MB/s]
62%|███████████████████████ | 1.03G/1.65G [00:12<00:07, 86.3MB/s]
63%|███████████████████████▏ | 1.04G/1.65G [00:12<00:07, 86.2MB/s]
63%|███████████████████████▍ | 1.05G/1.65G [00:12<00:07, 86.3MB/s]
64%|███████████████████████▌ | 1.05G/1.65G [00:12<00:06, 86.4MB/s]
64%|███████████████████████▊ | 1.06G/1.65G [00:12<00:06, 86.4MB/s]
65%|███████████████████████▉ | 1.07G/1.65G [00:12<00:06, 86.4MB/s]
65%|████████████████████████▏ | 1.08G/1.65G [00:12<00:06, 86.2MB/s]
66%|████████████████████████▍ | 1.09G/1.65G [00:12<00:06, 86.4MB/s]
66%|████████████████████████▌ | 1.10G/1.65G [00:13<00:07, 70.1MB/s]
67%|████████████████████████▊ | 1.11G/1.65G [00:13<00:07, 73.9MB/s]
67%|████████████████████████▉ | 1.11G/1.65G [00:13<00:06, 77.2MB/s]
68%|█████████████████████████▏ | 1.12G/1.65G [00:13<00:06, 79.7MB/s]
68%|█████████████████████████▎ | 1.13G/1.65G [00:13<00:06, 81.5MB/s]
69%|█████████████████████████▌ | 1.14G/1.65G [00:13<00:06, 82.7MB/s]
70%|█████████████████████████▋ | 1.15G/1.65G [00:13<00:06, 83.7MB/s]
70%|█████████████████████████▉ | 1.16G/1.65G [00:13<00:05, 84.3MB/s]
71%|██████████████████████████ | 1.17G/1.65G [00:13<00:05, 84.6MB/s]
71%|██████████████████████████▎ | 1.17G/1.65G [00:13<00:05, 84.6MB/s]
72%|██████████████████████████▍ | 1.18G/1.65G [00:14<00:05, 84.7MB/s]
72%|██████████████████████████▋ | 1.19G/1.65G [00:14<00:05, 84.8MB/s]
73%|██████████████████████████▊ | 1.20G/1.65G [00:14<00:05, 84.7MB/s]
73%|███████████████████████████ | 1.21G/1.65G [00:14<00:05, 84.8MB/s]
74%|███████████████████████████▎ | 1.22G/1.65G [00:14<00:05, 84.7MB/s]
74%|███████████████████████████▍ | 1.23G/1.65G [00:14<00:05, 84.5MB/s]
75%|███████████████████████████▋ | 1.23G/1.65G [00:14<00:04, 84.5MB/s]
75%|███████████████████████████▊ | 1.24G/1.65G [00:14<00:04, 84.5MB/s]
76%|████████████████████████████ | 1.25G/1.65G [00:14<00:04, 84.5MB/s]
76%|████████████████████████████▏ | 1.26G/1.65G [00:14<00:04, 84.5MB/s]
77%|████████████████████████████▍ | 1.27G/1.65G [00:15<00:04, 84.6MB/s]
77%|████████████████████████████▌ | 1.28G/1.65G [00:15<00:04, 84.7MB/s]
78%|████████████████████████████▊ | 1.29G/1.65G [00:15<00:04, 84.6MB/s]
78%|████████████████████████████▉ | 1.29G/1.65G [00:15<00:04, 84.6MB/s]
79%|█████████████████████████████▏ | 1.30G/1.65G [00:15<00:04, 84.7MB/s]
79%|█████████████████████████████▎ | 1.31G/1.65G [00:15<00:04, 84.7MB/s]
80%|█████████████████████████████▌ | 1.32G/1.65G [00:15<00:03, 84.7MB/s]
80%|█████████████████████████████▋ | 1.33G/1.65G [00:15<00:03, 84.6MB/s]
81%|█████████████████████████████▉ | 1.34G/1.65G [00:15<00:03, 84.6MB/s]
81%|██████████████████████████████ | 1.34G/1.65G [00:15<00:03, 84.6MB/s]
82%|██████████████████████████████▎ | 1.35G/1.65G [00:16<00:03, 84.7MB/s]
82%|██████████████████████████████▍ | 1.36G/1.65G [00:16<00:03, 84.8MB/s]
83%|██████████████████████████████▋ | 1.37G/1.65G [00:16<00:03, 84.7MB/s]
83%|██████████████████████████████▊ | 1.38G/1.65G [00:16<00:03, 84.8MB/s]
84%|███████████████████████████████ | 1.39G/1.65G [00:16<00:03, 84.4MB/s]
84%|███████████████████████████████▏ | 1.40G/1.65G [00:16<00:03, 84.5MB/s]
85%|███████████████████████████████▍ | 1.40G/1.65G [00:16<00:02, 84.5MB/s]
85%|███████████████████████████████▌ | 1.41G/1.65G [00:16<00:02, 84.4MB/s]
86%|███████████████████████████████▊ | 1.42G/1.65G [00:16<00:02, 84.5MB/s]
86%|███████████████████████████████▉ | 1.43G/1.65G [00:16<00:02, 84.5MB/s]
87%|████████████████████████████████▏ | 1.44G/1.65G [00:17<00:02, 84.5MB/s]
87%|████████████████████████████████▎ | 1.45G/1.65G [00:17<00:02, 84.4MB/s]
88%|████████████████████████████████▌ | 1.45G/1.65G [00:17<00:02, 84.4MB/s]
89%|████████████████████████████████▋ | 1.46G/1.65G [00:17<00:02, 84.4MB/s]
89%|████████████████████████████████▉ | 1.47G/1.65G [00:17<00:02, 84.2MB/s]
90%|█████████████████████████████████▏ | 1.48G/1.65G [00:17<00:02, 84.2MB/s]
90%|█████████████████████████████████▎ | 1.49G/1.65G [00:17<00:01, 84.3MB/s]
91%|█████████████████████████████████▌ | 1.50G/1.65G [00:17<00:01, 84.4MB/s]
91%|█████████████████████████████████▋ | 1.51G/1.65G [00:17<00:01, 84.5MB/s]
92%|█████████████████████████████████▉ | 1.51G/1.65G [00:17<00:01, 84.7MB/s]
92%|██████████████████████████████████ | 1.52G/1.65G [00:18<00:01, 84.8MB/s]
93%|██████████████████████████████████▎ | 1.53G/1.65G [00:18<00:01, 84.8MB/s]
93%|██████████████████████████████████▍ | 1.54G/1.65G [00:18<00:01, 84.9MB/s]
94%|██████████████████████████████████▋ | 1.55G/1.65G [00:18<00:01, 85.0MB/s]
94%|██████████████████████████████████▊ | 1.56G/1.65G [00:18<00:01, 85.1MB/s]
95%|███████████████████████████████████ | 1.56G/1.65G [00:18<00:01, 82.2MB/s]
95%|███████████████████████████████████▏ | 1.57G/1.65G [00:18<00:01, 69.4MB/s]
96%|███████████████████████████████████▍ | 1.58G/1.65G [00:18<00:00, 73.2MB/s]
96%|███████████████████████████████████▌ | 1.59G/1.65G [00:18<00:00, 76.5MB/s]
97%|███████████████████████████████████▊ | 1.60G/1.65G [00:19<00:00, 78.9MB/s]
97%|███████████████████████████████████▉ | 1.61G/1.65G [00:19<00:00, 80.7MB/s]
98%|████████████████████████████████████▏| 1.62G/1.65G [00:19<00:00, 82.0MB/s]
98%|████████████████████████████████████▎| 1.62G/1.65G [00:19<00:00, 83.1MB/s]
99%|████████████████████████████████████▌| 1.63G/1.65G [00:19<00:00, 83.9MB/s]
99%|████████████████████████████████████▋| 1.64G/1.65G [00:19<00:00, 84.3MB/s]
100%|████████████████████████████████████▉| 1.65G/1.65G [00:19<00:00, 84.7MB/s]
0%| | 0.00/1.65G [00:00<?, ?B/s]
100%|█████████████████████████████████████| 1.65G/1.65G [00:00<00:00, 7.34TB/s]
Download complete in 52s (1576.2 MB)
Filtering raw data in 1 contiguous segment
Setting up high-pass filter at 2 Hz
IIR filter parameters
---------------------
Butterworth highpass zero-phase (two-pass forward and reverse) non-causal filter:
- Filter order 8 (effective, after forward-backward)
- Cutoff at 2.00 Hz: -6.02 dB
Extract Xdawn covariance matrices¶
nfilter = 4
xdwn = XdawnCovariances(estimator="scm", nfilter=nfilter)
split = train_test_split(X, y, train_size=0.25, random_state=42)
Xtrain, Xtest, ytrain, ytest = split
covs = xdwn.fit(Xtrain, ytrain).transform(Xtest)
Spectral Embedding (SE)¶
plot_embedding(covs, ytest, metric="riemann", embd_type="Spectral",
normalize=True)
plt.show()
Locally Linear Embedding (LLE)¶
plot_embedding(covs, ytest, metric="riemann", embd_type="LocallyLinear",
normalize=False)
plt.show()
TNSE¶
plot_embedding(covs, ytest, metric="riemann", embd_type="TSNE",
normalize=False, max_iter=50)
plt.show()
/home/docs/checkouts/readthedocs.org/user_builds/pyriemann/checkouts/v0.10/pyriemann/optimization/positive_definite.py:284: UserWarning: Convergence not reached. Try increasing max_iter.
warnings.warn("Convergence not reached. Try increasing max_iter.")
References¶
Total running time of the script: (1 minutes 0.360 seconds)