A situation where training and test samples follow different input distributions is called covariate shift. Under covariate shift, standard learning methods such as maximum likelihood estimation are no longer consistent-weighted variants according to the ratio of test and training input densities are consistent. Therefore, accurately estimating the density ratio, called the importance, is one of the key issues in covariate shift adaptation. A naive approach to this task is to first estimate training and test input densities separately and then estimate the importance by taking the ratio of the estimated densities. However, this naive approach tends to perform poorly since density estimation is a hard task particularly in high dimensional cases. In this paper, we propose a direct importance estimation method that does not involve density estimation. Our method is equipped with a natural cross validation procedure and hence tuning parameters such as the kernel width can be objectively optimized. Furthermore, we give rigorous mathematical proofs for the convergence of the proposed algorithm. Simulations illustrate the usefulness of our approach

There is a step of significant difficulty experienced by brain-computer interface (BCI) users when going from the calibration recording to the feedback application. This effect has been previously studied and a supervised adaptation solution has been proposed. In this paper, we suggest a simple unsupervised adaptation method of the linear discriminant analysis (LDA) classifier that effectively solves this problem by counteracting the harmful effect of nonclass-related nonstationarities in electroencephalography (EEG) during BCI sessions performed with motor imagery tasks. For this, we first introduce three types of adaptation procedures and investigate them in an offline study with 19 datasets. Then, we select one of the proposed methods and analyze it further. The chosen classifier is offline tested in data from 80 healthy users and four high spinal cord injury patients. Finally, for the first time in BCI literature, we apply this unsupervised classifier in online experiments. Additionally, we show that its performance is significantly better than the state-of-the-art supervised approach.

Identifying temporally invariant components in complex multivariate time series is key to understanding the underlying dynamical system and predict its future behavior. In this Letter, we propose a novel technique, stationary subspace analysis (SSA), that decomposes a multivariate time series into its stationary and nonstationary part. The method is based on two assumptions: (a) the observed signals are linear superpositions of stationary and nonstationary sources; and (b) the nonstationarity is measurable in the first two moments. We characterize theoretical and practical properties of SSA and study it in simulations and cortical signals measured by electroencephalography. Here, SSA succeeds in finding stationary components that lead to a significantly improved prediction accuracy and meaningful topographic maps which contribute to a better understanding of the underlying nonstationary brain processes.

The results presented in this mathematical study allow the computation of simple theoretical simulations and performance of real experimental paradigms. Additionally, they are nicely in line with previous results in the BCI literature.