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Tikhonov Regularization Enhances Eeg-Based Spatial Filtering for Single-Trial Regression

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Tikhonov Regularization Enhances Eeg-Based Spatial Filtering for Single-Trial Regression Proceedings of the 7th Graz Brain-Computer Interface Conference 2017 DOI: 10.3217/978-3-85125-533-1-57 TIKHONOV REGULARIZATION ENHANCES EEG-BASED SPATIAL FILTERING FOR SINGLE-TRIAL REGRESSION A. Meinel1, F. Lotte2, M. Tangermann1 1Brain State Decoding Lab, Cluster of Excellence BrainLinks-BrainTools, Dept. of Computer Science, Albert-Ludwigs-University, Freiburg, Germany, 2Inria Bordeaux Sud-Ouest / LaBRI, Talence, France E-mail: [email protected] ABSTRACT: Robust methods for continuous brain state maximally different between classes. While the CSP decoding are of great interest for applications in the field algorithm proved very efficient and has become a gold of Brain-Computer Interfaces (BCI). When capturing standard in BCI, it is sensitive to noise, non-stationarity brain activity by an electroencephalogram (EEG), the and limited data. To address these limitations, various Source Power Comodulation (SPoC) algorithm enables regularized variants of CSP have been proposed making to compute spatial filters for the decoding of a con- this algorithm effectively more robust [5], [6], [7]. Typ- tinuous variable. However, as high-dimensional EEG ically, these approaches inject prior knowledge into the data generally suffer from low signal-to-noise ratio, CSP objective function, e.g. in the form of regularization the method reveals instabilities for small data sets and terms. The regularization approaches try to guide the is prone to overfitting. We introduce a framework for optimization process towards good solutions, despite applying Tikhonov regularization to SPoC by restricting noise and non-stationarities. the solution space of filters. Our findings show that However, not all BCIs are based on classification meth- additional trace normalization of covariance matrices ods. Several brain signal decoding problems require is a necessary prerequisite to tune the sensitivity of regression techniques to estimate continuous rather than the resulting algorithm. In an offline analysis on data discrete mental states. For instance, BCIs can be used of N = 18 subjects, the introduced trace normalized to estimate continuous workload levels [8] or reaction and Tikhonov regularized SPoC variant (NTR-SPoC) time from oscillatory activity [9]. As for classifica- outperforms standard SPoC for the majority of individ- tion techniques, regression models can also significantly uals. With this proof-of-concept study, a generalizable benefit from the use of spatial filters. Thus, Dahne¨ et regularization framework for SPoC has been established al. proposed the Source Power Comodulation (SPoC) which allows for implementing different regularization algorithm, which can be seen as an extension of CSP to strategies in the future. regression problems [10]. Indeed, SPoC aims at finding spatial filters such that the power of the filtered EEG INTRODUCTION signals maximally covaries with a continuous target variable. Designing electroencephalography (EEG)-based Brain- Computer Interfaces (BCIs) require translating EEG Due to similar mathematical formulations, CSP and signals into messages or commands for an applica- SPoC share a number of pros and cons. Both algorithms tion, e.g. by converting EEG activity recorded during can deliver informative oscillatory signal features but imagined hand movements into cursor movements [1]. are prone to noise, non-stationarity and limited data. In most current BCIs for communication and control, However, while robust variants of CSP have been pro- this is typically achieved using machine learning and posed based on regularization approaches [5], there are classification algorithms [2]. During online use, EEG no such robust variants for SPoC. Hence, this leaves signals are then assigned to a discrete set of classes SPoC with sub-optimal performances when used on (e.g. left or right hand imagined movements). noisy data such as those encountered outside laboratories A widely used component for effective classification for practical BCI use. In this paper, we aim at addressing of EEG signals is spatial filtering [3]. Addressing this limitation. In particular, we present a novel method volume conduction effects, spatial filter methods esti- to apply Tikhonov regularization to the existing SPoC mate sources, whose signals are more different between algorithm. We show how this regularization approach classes than signals obtained at the sensor level. The combined with appropriate normalization can indeed most popular spatial filter algorithm to classify oscilla- outperform the basic SPoC approach. We also illustrate tory EEG activity is the Common Spatial Pattern (CSP) the impact of various regularization parameters on the algorithm [3], [4]. It aims at finding filters such that resulting oscillatory components. the spatially filtered signals have a variance (and thus The reminder of this paper first presents in detail the a band power in narrow-band filtered signals) that is original SPoC algorithm and the regularized variant we Proceedings of the 7th Graz Brain-Computer Interface Conference 2017 DOI: 10.3217/978-3-85125-533-1-57 propose. Then it presents an evaluation of these two strategy [13], [14]. The penalty is expressed by a prior methods on real EEG data sets for motor performance that restricts the possible solution space. In this paper, prediction, before discussing the results. we assign a quadratic penalty term P (w) = w>1w = 2 kwk with 1 2 RNc×Nc stating the identity matrix. As MATERIALS AND METHODS this penalty scales with the spatial filter norm, solutions 1) Source Power Comodulation (SPoC): Supervised with small weights are preferred. Overall, the penalty spatial filtering algorithms are widely used in EEG- term is added to the denominator with a regularization BCI applications. Those filters represent a linear trans- parameter α leading to the following maximization formation to project the multi-variate EEG data to a problem: lower dimensional subspace. This work focuses on the w>Σ w J (w) = z (3) Source Power Comodulation algorithm (SPoC; [10]) P w>[(1 − α)Σ + α1]w which optimizes a spatial filter by solving a linear avg regression problem. This formulation is known as Tikhonov regularization In the following, x(t) 2 RNc describes the time course (TR, [15]) and has similarly been established for the of the multivariate bandpass-filtered EEG data acquired CSP algorithm [5]. Directly solving Eq. 3 refers to SPoC with Tikhonov regularization (TR-SPoC) in this paper. from Nc sensors. In accordance with the generative model of the EEG [11], a spatial filter w 2 RNc de- In case of an extreme regularization expressed by α = 1, scribes the linear projection of the sensor space data x(t) the Rayleigh quotient in Eq. 3 collapses to the one of to a one-dimensional source component s^(t) = w>x(t). the Principal Component Analysis (PCA, [16]). The Rayleigh quotient in Eq. 2 and 3 relates two Translating x(t) into segments of Ne single epochs Nc×Ns sample covariance matrices. In order to control for their x(e) 2 R with Ns sample points per epoch, then SPoC learns a spatial filter w such that the band- relative scaling, a normalization by the trace might be a 0 power Φ(e) = Var[^s(e)] of s^ has maximal covariance suitable strategy [4], [17], e.g. Σ (e) = Σ(e)=tr(Σ(e)). with a given epoch-wise univariate target variable z(e). Trace normalization will be applied to Σ(e) and Σavg Formally, this translates to maximizing the objective entering Eq. 3, but not upon Σz as the z-weighting function shall be maintained. This version of the algorithm will be stated as normalized Tikhonov regularization > J1(w) = Cov[Φ(e); z(e)] = w Σzw (1) of SPoC (NTR-SPoC). Applying the same scheme of by defining a z-weighted averaged covariance matrix trace normalization to the standard SPoC algorithm (Eq. 2), will be referred to as trace-normalized SPoC Σz := hΣ(e) z(e)i based on the trial-wise spatial co- −1 > (TN-SPoC). variance Σ(e) = (Ns − 1) x(e) x(e). h:i defines the average across N epochs. Furthermore, a norm con- e 3) Data Set for Offline Evaluation: To evaluate straint on w is applied by setting J (w) = Var[^s(e)] = 2 the introduced regularization algorithms, data of 18 w>Σ w =! 1 Σ = hΣ(e)i avg , where avg describes the subjects performing a visuomotor hand force task was averaged covariance matrix. Overall, this translates to used. The paradigm allowed to derive a trial-wise motor the Rayleigh quotient of the original SPoCλ formula- performance metric [18], [9]. Each subject completed tion [10]: one session with 400 trials. Within each trial, a ”get- J w>Σ w J(w) = 1 = z (2) ready” interval preceded a ”motor execution” phase J w>Σ w 2 avg which was initiated by a clear go-cue. EEG signals Technically, maximizing J(w) can be solved as a were used to predict the trial-wise reaction time (RT) generalized eigenvalue problem and returns a set of the motor task based on the time interval [-800, (j) fw gj=1;::;Nc of Nc spatial filters with j indexing the -50] ms prior to the go-cue. EEG activity was acquired rank which is determined in descending order of the by multichannel EEG amplifiers (BrainAmp DC, Brain eigenvalues and thereby according to the covariance. Products) with a sampling rate of 1 kHz from 63 passive Spatial filters allow for a visual interpretation Ag/AgCl electrodes (EasyCap) placed according to the by estimating the corresponding activity pattern extended 10-20 system. After preprocessing and outlier a = Σavgw as highlighted by Haufe et al. [12]. rejection following the methods described in [9], we restricted our analysis to oscillatory features within the 2) Tikhonov Regularization: The SPoC objective alpha-band frequency range of [8; 13] Hz. The bandpass function directly builds upon sample covariance matrices was realized applying a zero-phase butterworth filter of Σz and Σavg as stated by Eq. 2. Their estimation is 6th order.
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