PCA & multivariate signal processing, applied to neural data

Know the several interpretations of the word “dimension,” and the definition used in this course.

Understand the different interpretations of “source,” know which definition is used in this course, and appreciate the importance of source separation.

Mechanisms of source mixing, the idea and importance of unmixing, and some key source separation terminology.

Reducing dimensionality and separating sources are very different. This video explains why.

Know the difference between linear and nonlinear filters, and why linear filters are appropriate in many areas of neuroscience.

Want to know whether you can apply source separation methods to your data? Watch this video to find out!

Learn about the concepts, assumptions, and representations of correlations and covariances.

Learn the element-wise and matrix equations for covariances. Have some pointers for what to look for when viewing covariance matrices.

Create covariance matrices of simulated data.

Create covariance matrices of real EEG data.

See why covariance matrices must be symmetric.

See the theory implemented in MATLAB.

The "quadratic form" provides a way of understanding and visualizing the beautiful geometry of a covariance matrix.

Create and visualize the quadratic form of a covariance matrix in MATLAB.

Learn the objective and math of PCA.

Build geometric intuition of PCA using dots and surfaces.

Have step-by-step instructions for computing a PCA on multichannel data.

Follow the instructions to perform a PCA!

Build geometric intuition of PCA using dots and surfaces.

See a mathematical proof that all PCs (covariance eigenvectors) are orthogonal.

Learn how to interpret the eigenvalues.

Perform PCA of simulated EEG data.

Perform PCA of real EEG data.

Make sure you can use the pca() function.

Understand what happens when you don't mean-center data before computing a covariance matrix.

Understand why singular value decomposition of a data matrix will give the same results as eigendecomposition of a covariance matrix.

Demonstration of the equivalence of SVD and eigendecomposition for PCA.

Learn how to use PCA to observe the state-space of a system.

Demonstration of 2D state-space analysis in real EEG data.

PCA on phase-locked vs. total signal.

Learn why PCA is not good for source separation in multivariate neural datasets.

Conceptual idea of why and how to generalize PCA for an appropriate and general source separation algorithm.

Build geometric intuition of GED using the quadratic form of covariance matrices.

Learn the math (linear algebra) that underlies the solution to GED-based source separation.

Understand why the spatial filter is applied to the data, while the spatial pattern is interpreted.

Understand the origin of eigenvector sign indeterminancy, and how to "fix" the component sign.

See how GED can separate two independent sources in simulated EEG data.

A discussion of how to construct the two covariance matrices.

See GED in action in task-related real EEG data!

Use GED to create optimal spatial filters for narrowband spectral activity.

PCA and GED can be combined for compression/source separation, which is good for large-scale datasets.

Apply your knowledge about two-stage compression/separation to real EEG data!

Increase spatial precision by reducing large dimensions using (partial) whitening.

Apply your knowledge about ZCA and two-stage compression/separation to real EEG data!

Consider the consequences of covariance non-stationarities on source separation results.

Demonstrate the effects of covariance changes on resulting source projection topographies.

Learn the math and theory of shrinkage regularization.

Shrinkage regularization applied in MATLAB. 

How to let the data tell you how much to regularize.

Don't worry, you didn't break the universe.

See a demonstration of GED and factor analysis (with default settings) on source separation.

What happens in your brain when you turn on a strobe light?

Learn about the key analysis motivations for a spatial filter for SSVEP.

The RESS analysis pipeline in words and pictures.

See RESS applied in action on real EEG data.

Theoretical/conceptual overview of ICA

See ICA in a simple example in a small dataset.

Independent components analysis in simulated EEG data.

ICs are supposed to be non-Gaussian. What do they really look like?!

Learn what "overfitting" means and what implications it has for source separation (hint: it's not all bad!).

Use overfitting in an unbiased way by creating and testing the spatial filters in different ways.

Avoid overfitting biases by applying the spatial filters to different data than from which they were created.

Learn about inferential statistics using permutation testing to compute an effect size (z) and p-value of eigenvalues.