BCI

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
• Sampling principle
• VAE as PPCA encoder-decoder