Functional data analysis for BCI and biomedical signals

Vadim, 2023

Brain-computer interfaces require a sophisticated forecasting model. This model fits heterogeneous data. The signals come from ECoG, ECG, fMRI, hand and eye movements, and audio-video sources. The model must reveal hidden dependencies in these signals and establish relations between brain signals and limb motions. My research focuses on the constriction of BCI forecasting models. The main challenges of the research are phase space construction, dimensionality reduction, manifold learning, heterogeneous modeling, and knowledge transfer. Since the measured data are stochastic and contain errors, I actively use and develop Bayesian model selection methods. These methods infer criteria to optimize model structure and parameters. They aim to select an accurate and robust BCI model.

Brain signals and dimensionality reduction

Intracranial electroencephalography (iEEG) signals are tensors or time-related tensor fields. They have several indexes for physical space, time, and frequency. The multi-index structure of time series causes redundancy of space features and multi-correlation. It turns out to increase the complexity of the model and obtain unstable forecasts. I address the dimensionality reduction problem for high-dimensional data. The essential methods are tensor decomposition and high-order singular value decomposition. To reveal hidden dependencies in data, we proposed a feature selection method. It minimizes multi-correlation in the source space and maximizes the relation between source and target spaces. This solution shrinks the number of model parameters tenfold and stabilizes the forecast. To boost the quality of approximation I plan to investigate dimensionality reduction for nonlinear models in discrete time: the stacks of autoencoders and recurrent neural networks, and in the continuous time: the neural ODEs.

The BCI models are the signal decoding models. These models are a special class that includes canonical correlation analysis for multivariate and tensor variables. I plan to study the problem of model selection to restore hidden dependencies in the source and target spaces. For example, limb movements cause the target dependencies and multi-correlation in the target space. We proposed to reduce the dimension by projecting the source and target in the latent space. Linear and non-linear available methods for matching predictive models in spaces of high dimensions. Recently we proposed a feature selection algorithm for linear models and tested it on ECoG signals. I plan to develop this algorithm for tensor dimensionality reduction. The base method to compare is High-order Partial Least Squares. An exemplary problem is manifold learning. A continuous tensor field defines the manifold. It is a solution to neural PDEs. One has to find an optimal dimensionality of the manifold.