BCI

ERP data

Event-related potential, see ERP and P300 is triggered by a stimulus and registered over EEG monitor electrodes after the stimulus onset and before it.

The listed EEG datasets are expected to be of a maximum number of events to test the models. So the idea of collecting these data is to analyze a model on various data.

The ERP refers to the classification problem. The stimulus belongs to a class label from a given set. The event, starting from the stimulus onset, is a classification object. The Object description is the time series from tmin to tmax over the electrodes. Usually, tmax = 1200 ms and tmin = -300 ms. So the data set is a 3-way design matrix \(X\) and a binary vector \(y\). Sometimes the design matrix is 2-way after an EEG feature extraction procedure.

• MNE tutorials example datasets, MEG sample
  • ERP Core dataset

Brain-Computer Interfaces and Functional Data Analysis

This course is under construction. It enlightens fundamental mathematical concepts of brain signal analysis.

Each class combines five parts:

1. Comprehensive introduction
2. Practical example with code and homework
3. Algebraic part of modeling
4. Statistical part of modeling
5. Join them in Hilbert (or any convenient) space
6. Quiz for the next part (could be in the beginning) to show the theory to catch up

Linear models

SSA, SVD, PCA

• non-parametric phase space Hankel matrix
• convoluion ?
• forecasting with SSA

Acceleroneter data

• Energy

Tensor product and spectral decomposition

• vector, covector, dot product
• linear operator
• in Euclidean and (Hilbert space with useful example) dot product=bilinear form
• bilinear form
• factorization
• spectral decomposition
• SVD
•  ??? SVD in Hilbert space

Why do we go from Eucledian to Hilbert space? Was: a vector as a number of measurements. Now it is a finite number of samples. Then it is a distribution of samples. The distribution is a point in the Hilbert space. We can make an inner product and tensor product of two and more distributions. Machine learning: given samples, multivariate distribution can be represented as a (direct?) sum of elements' tensor products.

PPCA

• PPCA
• How to tell stochastic from deterministic variable? Are expectation and variance deterministic?
• Recap: joint and conditional distribution, marginalization.
• Sampling principle
• VAE as PPCA encoder-decoder