基于聚合法的正则化共模态空间CSPmatlab代码；脑电信号处理

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在脑电信号（EEG）处理领域，共模态空间（Common Spatial Pattern, CSP）是一种常用的数据分析技术，尤其在脑机接口（Brain-Computer Interface, BCI）中发挥着重要作用。CSP算法通过最大化两类不同状态之间的方差差异，提取出能够反映这两种状态之间差异的特征空间。在本资料包中，我们主要关注一种改进的CSP方法——基于聚合法的正则化共模态空间（Regularized Common Spatial Pattern, R-CSP）。标题中提到的"基于聚合法的正则化共模态空间CSP"是CSP的一种变体，它结合了正则化和聚类策略，旨在提高特征选择的稳定性和分类性能。正则化有助于避免过拟合，而聚合法可以将相似的特征聚集在一起，增强分类器的泛化能力。 `RegCsp.m` 是实现R-CSP算法的核心MATLAB代码，它可能包含以下部分： 1. 数据预处理：对原始脑电信号进行滤波、平均参考等预处理步骤，以减少噪声和提高信号质量。 2. CSP特征计算：应用CSP算法计算特征空间，包括计算协方差矩阵、进行特征值分解、选择最大化类间方差和最小化类内方差的特征向量。 3. 正则化处理：对CSP得到的特征矩阵进行正则化，如L2范数正则化，以限制特征权重的大小，防止过拟合。 4. 聚合策略：可能使用K-means或其他聚类算法，将相似的特征聚集到一起，形成更稳定的特征空间。 `demoR-CSP-Aggr.m` 是一个演示程序，用于展示如何使用`RegCsp.m`中的函数处理数据并进行分类。它可能包括加载数据、调用R-CSP函数、训练分类器以及评估性能的代码。 `FDA.m` 可能代表“Fisher Discriminant Analysis”（费雪判别分析），这是一个常见的分类器，常与CSP结合使用，以找到最佳的投影方向，最大化两类样本间的距离，从而提高分类性能。 `R-CSP-A_TBME2010.pdf` 是一篇相关的学术论文，可能详细介绍了R-CSP算法的理论背景、实现细节和实验结果。阅读这篇论文可以帮助深入理解R-CSP的工作原理及其在脑电信号处理中的应用优势。这个压缩包提供了实现R-CSP算法的MATLAB代码，以及一个示例演示和相关的学术文献，为研究者或工程师提供了一套完整的工具来处理和分析脑电信号，特别是对于那些希望提升BCI系统性能的人员来说，这是一个宝贵的资源。

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2936 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 57, NO. 12, DECEMBER 2010

Regularized Common Spatial Pattern With

Aggregation for EEG Classiﬁcation

in Small-Sample Setting

Haiping Lu

∗

, Member, IEEE, How-Lung Eng, Member, IEEE, Cuntai Guan, Senior Member, IEEE,

Konstantinos N. Plataniotis, Senior Member, IEEE, and Anastasios N. Venetsanopoulos, Fellow, IEEE

Abstract—Common spatial pattern (CSP) is a popular algo-

rithm for classifying electroencephalogram (EEG) signals in the

context of brain–computer interfaces (BCIs). This paper presents

a regularization and aggregation technique for CSP in a small-

sample setting (SSS). Conventional CSP is based on a sample-based

covariance-matrix estimation. Hence, its performance in EEG clas-

siﬁcation deteriorates if the number of training samples is small.

To address this concern, a regularized CSP (R-CSP) algorithm is

proposed, where the covariance-matrix estimation is regularized

by two parameters to lower the estimation variance while reducing

the estimation bias. To tackle the problem of regularization param-

eter determination, R-CSP with aggregation (R-CSP-A) is further

proposed, where a number of R-CSPs are aggregated to give an

ensemble-based solution. The proposed algorithm is evaluated on

data set IVa of BCI Competition III against four other competing

algorithms. Experiments show that R-CSP-A signiﬁcantly outper-

forms the other methods in average classiﬁcation performance in

three sets of experiments across various testing scenarios, with

particular superiority in SSS.

Index Terms—Aggregation, brain–computer interface (BCI),

common spatial pattern (CSP), electroencephalogram (EEG),

generic learning, regularization, small sample.

I. INTRODUCTION

OWADAYS, electroencephalography (EEG) signal clas-

siﬁcation is receiving increasing attention in the biomedi-

cal engineering community [1]. EEG captures the electric ﬁeld

generated by the central nervous system. Due to its simplicity,

inexpensiveness, and high temporal resolution, it is widely used

in noninvasive brain–computer interfaces (BCI) [2], [3], where

brain activity is translated into sequences of control commands

Manuscript received June 8, 2010; revised August 14, 2010; accepted

September 6, 2010. Date of publication September 30, 2010; date of current

version November 17, 2010. Asterisk indicates corresponding author.

∗

H. Lu is with the Institute for Infocomm Research, Agency for Science,

Technology and Research, Singapore 138632 (e-mail: hplu@ieee.org).

H.-L. Eng and C. Guan are with the Institute for Infocomm Research, Agency

for Science, Technology and Research, Singapore 138632 (e-mail: hleng@

i2r.a-star.edu.sg; ctguan@i2r.a-star.edu.sg).

K. N. Plataniotis is with the Department of Electrical and Computer En-

gineering, University of Toronto, Toronto, ON M5S 1A1, Canada (e-mail:

kostas@comm.utoronto.ca).

A. N. Venetsanopoulos is with the Department of Electrical and Computer

Engineering, Ryerson University, Toronto, ON M5B 2K3 Canada (e-mail:

tasvenet@gwemail.ryerson.ca).

Color versions of one or more of the ﬁgures in this paper are available online

at http://ieeexplore.ieee.org.

Digital Object Identiﬁer 10.1109/TBME.2010.2082540

that enable a subject, such as a disable person, to communicate

to a device, such as a computer, without using the peripheral

nervous system [2]. In noninvasive EEG-based BCI, the study

of motor imagery is of particular interest. It is measurable as po-

tential changes in EEG signals, the event-related desynchroniza-

tion/synchronization (ERD/ERS) patterns. EEG has also been

an important tool in epilepsy diagnosis [4] for seizure detection,

classiﬁcation, and localization.

EEG records brain activities as multichannel time series from

multiple electrodes placed on the scalp of a subject. However,

recorded multichannel EEG signals typically have very low

signal-to-noise ratio (SNR) [2], and they are not directly usable

in BCI applications. One of the most effective algorithms for

EEG signal classiﬁcation is the common spatial pattern (CSP)

algorithm, which extracts spatial ﬁlters that encode the most

discriminative information [5]–[8]. CSP was ﬁrst introduced

for the binary classiﬁcation of EEG trials in [5]. It is designed to

ﬁnd spatial projections that maximize the power/variance ratios

of the ﬁltered signals for two classes. Its calculation is through a

simultaneous diagonalization of the covariance matrices of two

classes. Usually, only the ﬁrst few most discriminative ﬁlters are

needed for classiﬁcation.

This paper focuses on EEG signal classiﬁcation in a small-

sample setting (SSS). There are two motivations for this prob-

lem. On one hand, this SSS problem often arises in practical

EEG signal-classiﬁcation problem, when there is only a small

training set with limited number of trials available. It should

be noted that although a large number of data points can be

sampled from a trial with sufﬁciently high frequency, these data

points are highly dependent. Generally, they are not representa-

tive enough for EEG signal classiﬁcation and a large number of

trials are still preferred for reliable classiﬁcation performance.

On the other hand, as the user usually has to perform a tedious

calibration measurement before starting the BCI feedback ap-

plications, one important objective in BCI research is to reduce

the number of training trials needed (and the time needed) for a

speciﬁc task [9]. Since the conventional CSP algorithm is based

on sample-based covariance-matrix estimation, the accuracy of

the estimation will be affected signiﬁcantly if there is only a

small training set.

The problem due to SSS in classiﬁcation is common in many

other applications. Regularization was ﬁrst introduced to tackle

the small-sample problem for linear and quadratic discriminant

analysis in the regularized discriminant analysis (RDA) [10].

It was pointed out in [10] that a small number of training

LU et al.: REGULARIZED COMMON SPATIAL PATTERN WITH AGGREGATION FOR EEG CLASSIFICATION IN SMALL-SAMPLE SETTING 2937

samples tends to result in a biased estimation of eigenvalues.

On the other hand, sample-based covariance estimates from

these poorly posed problems are usually highly unreliable. Two

regularization parameters were introduced by Friedman [10] to

account for these undesirable effects. Recently, the regulariza-

tion technique has been adopted to tackle small-sample prob-

lems in various applications such as face recognition [11]–[13]

and gait recognition [14].

This paper studies the regularization of the CSP algorithm in

SSS. A regularized CSP (R-CSP) algorithm is proposed to regu-

larize the covariance-matrix estimation in CSP extraction. Two

regularization parameters are adopted, as in [10]. The ﬁrst regu-

larization parameter controls the shrinkage of a subject-speciﬁc

covariance matrix toward a more generic covariance matrix to

lower the estimation variance. This is built upon the generic

learning principle in [15]. The second regularization parameter

controls the shrinkage of the sample-based covariance-matrix

estimation toward a scaled identity matrix to account for the bias

due to the limited number of samples. Furthermore, the problem

of regularization parameter determination needs to be addressed

for R-CSP. However, in SSS, the number of samples may not be

large enough for determining regularization parameters by the

commonly used cross-validation method [10]. Thus, the aggre-

gation strategy introduced in [12] for tensor object recognition

is adopted for regularization parameter determination in EEG

signal classiﬁcation through R-CSP, where a number of differ-

ently regularized CSPs are combined to give an ensemble-based

solution. The experimental evaluation is performed on data set

IVa from BCI Competition III. The proposed algorithm out-

performs four other competing CSP-based algorithms across a

wide range of testing scenarios with more advantage in SSS.

There are several other versions of regularized spatial ﬁlters

in the literature. The adaptive spatial ﬁlter in [16] replaced the

information used in training by aprioriinformation for more

robust performance by considering various artifacts in EEG sig-

nals. Heuristic parameter selection was used in [16]. The invari-

ant CSP in [17] incorporated neurophysiological prior knowl-

edge in covariance-matrix estimation to alleviate the nonsta-

tionarities in EEG signals, and the regularization parameter was

determined by cross validation. A method of logistic regression

with dual spectral regularization (LRDS) was introduced in [18]

for EEG classiﬁcation, where cross validation was employed for

parameter selection too. A composite CSP was proposed in [8]

to transfer information from other subjects to a subject of interest

with fewer training samples in order to boost the performance

in SSS. Ten values for the regularization parameter are tested,

with the average results reported. Lately, t he spatially regular-

ized CSP (SRCSP) [19] is proposed to include spatial apriori

in the learning process by penalizing spatially nonsmooth ﬁlters

with a regularization term. Another recent work that involves

regularization for EEG analysis is the regularized discriminative

framework in [20].

The main contributions of this work are as follows.

1) The introduction of an R-CSP algorithm for EEG sig-

nal classiﬁcation, which was ﬁrst reported in a prelimi-

nary version in [21]. It should be noted that while there

have been several approaches proposing variations of CSP

through more robust covariance-matrix estimation [8],

[16]–[18], [22], none has considered the effects of small

training-sample set on the eigenvalues of the covariance

matrices, as discussed previously. Thus, this paper com-

plements existing works on CSP extensions by addressing

this important problem frequently arising in practice. It

also has a positive impact in data collection effort, pro-

cessing efﬁciency, and memory/storage requirement in ap-

plications involving EEG signal classiﬁcation, since now

much fewer training samples are needed for the same level

of performance.

2) The proposal of an aggregation solution for the problem of

regularization parameter determination in R-CSP, where

the commonly-used cross-validation scheme [17], [18]

may not be applicable in SSS. This solution adopts the

principles introduced in uncorrelated multilinear discrim-

inant analysis [12], [23] for t ensor object recognition to

CSP extraction in EEG signal classiﬁcation. This is a sig-

niﬁcant further progress from the preliminary publication

in [21], where the regularization parameter determination

is not solved and only a feasibility study was provided.

In contrast, the composite CSP in [8] did not address the

problem of regularization parameter determination.

3) A detailed study on EEG signal classiﬁcation in SSS that

considers 2–120 trials per condition, including extreme

small number of trials (2–10) in contrast to the recent

literature [3] that considers 10–100 trials per condition.

This study consists of 1500 experiments in total in or-

der to study the statistical signiﬁcance of the obtained

results. Another two sets of experiments are carried out

for performance evaluation against four competing solu-

tions. Based on the simulations, insightful observations

have been made regarding t he proposed algorithm. This

is also a signiﬁcant development from the earlier publica-

tion [21].

The rest of this paper is organized as follows. Section II

presents the R-CSP algorithm for EEG signal classiﬁcation. In

Section III, the problem of regularization parameter determina-

tion is discussed, and an aggregation solution is formulated for

R-CSP. Section IV provides an experimental study of the EEG

signal-classiﬁcation problem in SSS and an evaluation of the

proposed algorithm. Finally, Section V draws the conclusion.

II. R

EGULARIZED CSP FOR EEG SIGNAL CLASSIFICATION

This section presents the R-CSP algorithm for classiﬁcation of

EEG signals. Regularized covariance-matrix estimation is used

in R-CSP by employing the regularization technique introduced

in [10] and the generic learning principle in [15]. The EEG

classiﬁcation scheme of R-CSP follows that in the conventional

CSP algorithm [5].

A. Sample-Based Covariance Matrix in CSP

In CSP-based EEG signal classiﬁcation, a matrix E of size

N × T is used to represent a single N -channel EEG trial, with T

samples in each channel for a single trial. The sample covariance

matrix of a trial E is normalized with the total variance as [5]

2938 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 57, NO. 12, DECEMBER 2010

follows:

S =

tr(EE

)

(1)

where the superscript “T ” denotes the transpose of a matrix, and

tr(·) denotes the trace of a matrix. This paper considers only

binary classiﬁcation problems and the two classes are indexed

by c = {1, 2}. For simplicity, it is assumed that M trials are

available for training in each class for a subject of interest,

indexed by m as E

(c,m)

, m =1, 2,...,M. Hence, each trial has

a covariance matrix S

(c,m)

and the average spatial covariance

matrix is then calculated for each class as [5] follows:



m =1

(c,m)

,c= {1, 2}. (2)

The discriminative spatial patterns in CSP are extracted based

on this sample-based covariance-matrix estimation. When there

are only a small number of training trials, such an estimation

problem could be poorly posed [10] and the estimated param-

eters could be highly unreliable, giving rise to high variance.

Moreover, the low SNR for EEG signals makes the estimation

variance even higher.

B. Regularized Covariance-Matrix Estimation in SSS

Regularization technique has been proved to be effective in

tackling the small-sample problem. It works by biasing the esti-

mates away from their sample-based values toward more “phys-

ically plausible” values [10], which reduces the variance of the

sample-based estimates while tending to increase bias. This

bias-variance trade-off is commonly controlled by one or more

regularization parameters [10].

As in [10], the proposed R-CSP calculates the regularized

average spatial covariance matrix for each class as follows:

(β,γ)=(1− γ)

(β)+

tr[

(β)] · I (3)

where β and γ are two regularization parameters (0 ≤ β,γ ≤ 1),

and I is an N × N identity matrix.

(β) comprises covariance

matrices for the trials from the speciﬁc subject as well as generic

trials from other subjects. It is deﬁned as follows:

(β)=

(1 − β) · S

+ β ·

(1 − β) · M + β ·

(4)

where S

is the sum of the sample covariance matrices for all

M training trials in class c



m =1

(c,m)

(5)

and

is the sum of the sample covariance matrices for

generic training trials {E

(c, ˆm )

} from other subjects in class c



ˆm=1

(c, ˆm )

. (6)

In these deﬁnitions, S

(c,m)

and S

(c, ˆm )

are the normalized sample

covariance matrix deﬁned in (1).

The term

introduced in (4) aims to reduce the variance in

the covariance-matrix estimation, and it tends to produce more

reliable results. This is built upon the idea of generic learning

for the one-training-sample case in face recognition [15], and

it also embodies the same principle as that in [8] and [17].

For the classiﬁcation of EEG signals from a particular subject,

the proposed training process constructs the regularization term

using corresponding EEG trials collected from some other

subjects, i.e., generic trials from the population. When there are

S subjects available in total, each with M trials for each class,

M =(S − 1) × M .

The rationales of the regularization scheme in (4) follow those

in [10]. The ﬁrst regularization parameter β controls the degree

of shrinkage of the training-sample covariance-matrix estimates

to the pooled estimate, which is to reduce the variance of the

estimates. The second regularization parameter γ controls the

degree of shrinkage toward a multiple of the identity matrix,

with the average eigenvalue of

(β) as the multiplier. This

second shrinkage has the effect of decreasing the larger eigen-

values while increasing the smaller ones. This is because the

sample-based estimates in (1) tend to bias the eigenvalues in the

opposite direction, especially in SSS [10]. Thus, γ is to coun-

teract such bias due to the limited number of samples. From

the aforementioned, the conventional CSP can be considered

as a special case of R-CSP, i.e., when β = γ =0. In addition,

the composite CSP introduced in [8] could be considered as a

special case of R-CSP with γ =0.

The effects of the adopted regularization scheme are illus-

trated in Figs. 1 and 2 with some typical examples. In the ﬁgures,

the ﬁrst 20 largest eigenvalues of a typical average spatial co-

variance matrix are shown in descending order with magnitudes

in log scale. Fig. 1 depicts the eigenvalue distribution without

regularization and with a regularization β =0.3 (γ =0) for ﬁve

randomly selected training sets with M =10for the same class

of a particular subject. It is observed that the variance of the

eigenvalues are much higher when there is no regularization by

β. Fig. 2 simply shows the eigenvalue distribution without regu-

larization and with a regularization γ =0.1 (β =0)foraﬁxed

training set with M =10. It can be seen that the regularization

by γ decreases the relative magnitudes of the larger eigenvalues

over those of the smaller eigenvalues, which reduces the bias

due the small number of training samples in turn.

C. R-CSP Feature Extraction and Classiﬁcation

With the formulation of the regularized covariance-matrix

estimation in SSS, feature extraction in R-CSP follows that in

the classical CSP method [5]. The regularized composite spatial

covariance is formed and factorized as follows:

Σ(β,γ)=

(β,γ)+

(β,γ)=

(7)

where

U denotes the matrix of eigenvectors, and

Λ denotes the

diagonal matrix of corresponding eigenvalues. This paper adopts

the convention that the eigenvalues are sorted in descending

order. The full projection matrix is then formed as follows:

−1/2

(8)

LU et al.: REGULARIZED COMMON SPATIAL PATTERN WITH AGGREGATION FOR EEG CLASSIFICATION IN SMALL-SAMPLE SETTING 2939

Fig. 1. Illustration of the effects of the regularization parameter β on ﬁve

random training sets with M =10for the same class of a particular subject. (a)

Eigenvalue distribution of a typical average covariance matrix without regular-

ization by β. (b) Eigenvalue distribution of a typical average covariance matrix

with regularization β =0.3.

where

B denotes the matrix of eigenvectors for the whitened

spatial covariance matrix

−1/2

(β,γ)

−1/2

To get the most discriminative patterns, the ﬁrst and last α

columns of

are retained to form an N × Q matrix

W with

Q =2α. In R-CSP feature extraction, an input trial E is ﬁrst

projected as follows:

X =

E. (9)

A Q-dimensional feature vector

y is then constructed from the

variance of the rows of

X as follows:

ˆy

= log



var(

)



q=1

var(

)



(10)

where ˆy

denotes the qth component of

denotes the qth

row of

X, and var(

) denotes the variance of

Finally, R-CSP classiﬁcation in this paper employs the

Fisher’s discriminant analysis (FDA) followed by the simple

nearest-neighbor classiﬁer (NNC). FDA solves for a projec-

tion v to maximize the ratio of the between-class s catter to the

Fig. 2. Illustration of the effects of regularization parameter γ on a ﬁxed

training set: the eigenvalue distribution of a typical average spatial covariance

matrix with and without regularization by γ.

Fig. 3. Two examples showing the variation of classiﬁcation accuracy (coded

as the gray levels in the displayed checkerboard) for 121 pairs of regularization

parameters β and γ. The pair resulting in the highest classiﬁcation accuracy is

marked with a black star.

within-class scatter

FDA

= arg max

(11)

where Ψ

and Ψ

are the between-class scatter matrix and

the within-class scatter matrix [24] for the features

y in (10),

respectively. This problem can be solved as a generalized eigen-

value problem [25] and the discriminant feature vector z

obtained as follows:

z = v

FDA

y. (12)

In NNC classiﬁcation, the nearest neighbor is found as μ

∗

arg min

z − z

, where z

is the feature vector for the μth

training trial, μ =1, 2,...,2M, and ·is the Euclidean norm

for vectors. The class label of the μ

∗

th training sample c

∗

then assigned to the test trial E.

Fig. 3 gives two examples on the variation of classiﬁcation

accuracy for 121 pairs of regularization parameters β and γ.The

classiﬁcation accuracy is coded as the gray l evels (white for the

highest and black for the lowest) in the displayed checkerboard.

A black star is used to mark the highest classiﬁcation accuracy

in each example. The effectiveness of both β and γ is observed

in the ﬁgure. At the same time, it could be seen that the classi-

ﬁcation accuracy could be sensitive to parameter settings, and

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