A classwise PCA-based recognition of neural data for brain-computer interfaces.

Abstract

We present a simple, computationally efficient recognition algorithm that can systematically extract useful information from any large-dimensional neural datasets. The technique is based on classwise Principal Component Analysis, which employs the distribution characteristics of each class to discard non-informative subspace. We propose a two-step procedure, comprising of removal of sparse non-informative subspace of the large-dimensional data, followed by a linear combination of the data in the remaining subspace to extract meaningful features for efficient classification. Our method produces significant improvement over the standard discriminant analysis based methods. The classification results are given for iEEG and EEG signals recorded from the human brain.

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A Classwise PCA-based Recognition of Neural Data for
Brain-Computer Interfaces
Koel Das, Sergey Osechinskiy, and Zoran Nenadic
Abstract—We present a simple, computationally efficient
recognition algorithm that can systematically extract useful
information from any large-dimensional neural datasets. The
technique is based on classwise Principal Component Analysis,
which employs the distribution characteristics of each class
to discard non-informative subspace. We propose a two-step
procedure, comprising of removal of sparse non-informative
subspace of the large-dimensional data, followed by a linear
combination of the data in the remaining subspace to extract
meaningful features for efficient classification. Our method pro-
duces significant improvement over the standard discriminant
analysis based methods. The classification results are given for
iEEG and EEG signals recorded from the human brain.
I. INTRODUCTION
A brain-computer interface (BCI) is a communication link
between the brain and an external device. A class of BCIs
that are based solely on cognitive signals have received
increased attention recently [1], [2]. These systems could be
potentially useful for improving the quality of life of patients
with locked in syndrome and other individuals with severe
motor deficiencies. BCI typically consists of an assistive
(external) device and a set of algorithms that enable the
interaction of the brain and the device [3], [1]. BCIs can
be classified into two categories: invasive and non-invasive.
Non-invasive BCIs are typically realized by recording the
neural activity from the surface of the scalp by means of
electroencephalography (EEG) [1], [2] while invasive BCIs
require implantation of recording electrodes through surgical
procedure. In this article we will focus on both invasive and
non-invasive BCIs, although the technique we develop can
be potentially useful beyond BCI applications.
The main function of a BCI system is to analyze neural
patterns in real time and to utilize this information for
communication with the external devices (e.g. computers or
robots). To this end a training database is created, consisting
of multiple records of neural signals, conditioned upon
various cognitive classes (e.g. imagination of left vs. right
hand movements [4]). Future (unknown) intentions are then
decoded based on how well the corresponding neural signals
match the class-conditional signals in the training database.
EEG signals, commonly used in non-invasive BCI appli-
cations, are spatio-temporal and large-dimensional. Conse-
quently, their analysis is hindered with two major obstacles.
K. Das with the Department of Electrical Engineering and Computer Sci-
ence, University of California, Irvine, CA 92697, USA kdas@uci.edu
S.Osechinskiy is with the Department of Biomedical Engineering, Uni-
versity of California, Irvine, CA 92697, USA sosechin@uci.edu
Z. Nenadic is with the Department of Biomedical Engineering and the
Department of Electrical Engineering and Computer Science, University of
California, Irvine, CA 92697, USA znenadic@uci.edu
Firstly, the dimension of data, n, by far exceeds the number
of samples, nt, in the training database, giving rise to so-
called small sample size conditions [5]. Under the small
sample size conditions, the sample statistics of data can
be extremely poor, if not meaningless. More specifically,
the covariance matrices are highly singular. Secondly, large-
dimensional data necessarily translates to the curse of di-
mensionality, which presents challenges in handling and
manipulation of statistical data. In particular the inversion
and spectral decomposition of large matrices may not be
feasible with standard computer architectures.
For many BCI applications these challenges are tackled
heuristically. For example, a common approach is to separate
the processing in space and time [2], [4], without discussing
explicitly the consequences of this space-time separability
assumption. Furthermore, a vast majority of these studies
utilizes the spectral power of EEG signals (e.g. µ-band or β-
band [2], [4]), as low-dimensional features of interest. While
the power/frequency representation is physically intuitive, it
is unclear why these ad hoc features should have optimal pre-
dictive power. Several studies [6], [7], [8] report significantly
better decoding results with the use of other (more abstract)
features. Another common strategy, often used in conjunction
with the above, is to rank individual features (or recording
electrodes) according to some criterion. The feature set is
then constructed by concatenating a small number of features
on the top of hierarchy. This approach, however, ignores the
joint statistical properties of the features, which might result
in suboptimal decoding performance.
In this article, we present a computationally efficient,
locally adaptable, classification technique. Our method does
not assume space-time separability and it utilizes joint sta-
tistical properties of the features. The main idea behind
our technique is to identify and discard a useless (non-
informative) subspace in data. The recognition is then carried
out in the residual space, where the small sample size condi-
tions and the curse of dimensionality are no longer concerns.
While our method was developed and tested on human EEG
and intracranial EEG (iEEG) data, the technique is applicable
to any large-dimensional spatio-temporal biomedical data,
and in general to any large-dimensional statistical data.
II. CLASSWISE PRINCIPAL COMPONENT ANALYSIS
Under the small sample size conditions, a large portion
of the data space is sparse and carries very little or no
useful information. To obtain meaningful data statistics, this
irrelevant subspace must be discarded as noise, which is
typically accomplished by global dimensionality reduction
Proceedings of the 29th Annual International
Conference of the IEEE EMBS
Cité Internationale, Lyon, France
August 23-26, 2007.
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x*
ω1
ω2
S1
S2
S
(A)
0
0
0
0
x*
ω1
ω2
x1*
S1
(B)
0
0
x*
ω1
ω2
x2*
S2
(C)
Fig. 1. (A) PCA (dashed) vs. cPCA subspace for 2-class case, where the classes ω1and ω2are represented as Gaussian contours.(B) Test data x∗projected
on reduced subspace S1to produce test feature x∗
1. (C) x∗projected on S2to produce x∗
2
techniques, such as Principal Component Analysis (PCA)
or Independent Component Analysis (ICA). However, both
PCA and ICA rely on statistical properties of the common
data distribution, while the class-conditional statistics are ig-
nored. Therefore, it can be argued that global dimensionality
reduction techniques may be suboptimal for classification
purposes.
Large-dimensional data is commonly encountered in face
(image) recognition problems, where Linear Discriminant
Analysis (LDA) [5], and variants thereof [9], [10], have
been successfully used. However, our experience with face
recognition techniques applied to brain data were somewhat
disappointing [7]. Motivated by these limitations, we pro-
posed a modification of the direct LDA (DLDA) method [9],
by introducing a threshold [7], whose role was to regularize
the covariance matrix in a manner similar to the standard
shrinkage approach [11]. However, the optimum value of
the threshold was selected via internal cross-validation (CV),
which rendered the method computationally expensive. Fur-
thermore, LDA-based methods tend to overfit under the
small sample size conditions giving error rates which are
not generalizable.
In this article we exploit the strength of PCA as a di-
mensionality reduction technique, while preserving the class-
specific information to facilitate subsequent classification.
Our technique is based on classwise PCA (cPCA) and results
in a simple piecewise linear dimensionality reduction tech-
nique. Fig. 1 (A) illustrates the major difference between the
classical PCA method and our technique, applied to a binary
class case. In general, for c-class problems, our technique
generates as many as c subspaces {S1, S2, ··· , Sc}, which
approximate some nonlinear low-dimensional data manifold.
Presumably, a data point from the class ωiis best represented
in the local subspace Si, although this is not necessarily true
and further tests are required (see Section II-B). On the other
hand, PCA approximates data using a single low-dimensional
subspace, and is ignorant of any class-specific information
(e.g. curvature). The details of our algorithm will be given
for c=2, and the extension to an arbitrary number of classes
is straightforward.
A. Feature Extraction
Let ωi(i = 1,2) denote two classes with means µi, and
covariances Σi, and let x∗∈ Rnbe unknown (test) data to be
classified. In the first step, x∗is represented in 2 subspaces, S1
and S2(see Fig. 1), by means of the following transformation
x∗
i= FT
i(x∗−µi)
i = 1,2(1)
where the columns of Fi∈ Rn×m′i are taken as the basis
vectors of Si. The two classes are transformed in a similar
fashion [Fig. 1 (B)(C)]. In the simplest scenario, Fi= Vi,
where Vi∈ Rn×miconsists of the mi (mi to be chosen)
principal components of the class ωi. To account for classes
whose principal directions are nearly parallel, and hence the
projections of the two classes to Si are highly overlapped,
we propose to augment Fi with Vb∈ Rn×1, where Vb∝
µ1− µ2. This step ensures that class differences arising
from the two means are accounted for. For c-class cases,
Vb readily generalizes to a basis spanned by the columns
of the between-class-scatter matrix, commonly used in LDA
applications [11]. To keep all projections orthogonal, the
columns of Fi= [Vi|Vb] are orthonormalized through the
Gram-Schmidt procedure.
While the above procedure typically yields Si of suffi-
ciently low dimension (m′
i≪ n), where the size of data
is no longer an obstacle, further improvements in terms
of classification accuracy are possible with simple feature
extraction techniques applied directly to the subspace Si. If
linear feature extraction techniques are used (e.g. LDA), the
mathematical formalism (1) remains the same, with mere
modifications in the definition of Fi. More specifically, Fi=
[Vi|Vb]Ti, where Ti∈ Rm′i×mis the feature extraction matrix
of the chosen method. In this article we use an information-
theoretic technique called Information Discriminant Analysis
(IDA), whose advantages over LDA and similar techniques
have been discussed at length in [12]. Unlike LDA, IDA
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has no constraints regarding the final dimension, m, of the
feature space. However, BCI data is generally so sparse
(small nt) that the choice of m is severely limited. In our
experience, working with 1 ≤ m ≤ 3, not only yields the
best performance, but also provides a safeguard against
overfitting [8], [7].
In summary, our feature extraction technique is a two-step
procedure. In the first step, a large-dimensional and mostly
sparse subspace of the original data space is discarded. In
the second step, the remaining data is linearly combined into
meaningful features for the purpose of classification.
B. Classification
Due to piecewise linear nature of our feature extraction
method, the test data x∗is represented in 2 feature subspaces
[Fig. 1 (B)(C)]. To complete the feature extraction process,
one of the subspaces must be eliminated. It turns out that
this question can be solved within a classification framework,
which is the ultimate goal of our technique. Therefore, the
formal completion of the feature extraction process can be
viewed as a bi-product of the classification process.
For simplicity, we will assume that the classes are Gaus-
sian with prior probabilities, P(ωi). A straightforward appli-
cation of the Bayes classifier at the first subspace yields
P(ωi1|x∗
1) =p(x∗
1|ωi1)P(ωi)
p(x∗
1)
i = 1,2 (2)
where p(x∗
posterior probabilities of the two classes in the first subspace.
Since (1) is an affine transformation, both classes remain
Gaussian, i.e. p(.|ωi1) ∼ N (FT
on the first subspace, x∗is assigned to the class ωkwith the
maximum posterior probability
1) = ∑2
i=1p(x∗
1|ωi1)P(ωi), and P(ωi1|x∗
1) are the
1(µi−µ1),FT
1ΣiF1). Based
k = argmax
i=1,2P(ωi1|x∗
1)
Similarly, x∗is transformed to the second subspace [see
Fig. 1 (C)], and its class membership, ωl, is determined as
follows:
l = argmax
i=1,2P(ωi2|x∗
2)
where
P(ωi2|x∗
2) =p(x∗
2|ωi2)P(ωi)
p(x∗
2)
i = 1,2
and p(x∗
p(.|ωi2) ∼ N (FT
Therefore, at each feature subspace, the test data, x∗, can
be associated with one of the classes. A final decision, x∗∈
ωg, is made by a direct comparison of the optimal class
assignments per individual subspaces, i.e.
2) is defined analogous to p(x∗
2(µi−µ2),FT
1). Also note that
2ΣiF2).
g = argmax
k,l[P(ωk1|x∗
1)
P(ωl2|x∗
2)]
III. EXPERIMENTAL RESULTS
The performance of our method was tested on a set of
iEEG data, adopted from Rizzuto et al. [6], and an EEG
dataset recorded in our lab. A brief account of the iEEG
experiments will be presented next. For more details, the
reader is referred to [6], [8], [7].
The iEEG signals were recorded from the human brain
during a standard memory reach task, consisting of 4 periods:
fixation, target, delay, and reach. The appearance of a ran-
dom target, either left or right of the fixation point, marked
the onset of the target period. Once the target disappeared,
the delay period started. The disappearance of the fixation
target marked the onset of the reach period. The duration
of each period was randomized and lasted between 1 and
1.3s, and the total number of electrodes implanted in both
hemispheres was 91. A total of 162 trials were recorded for
each period. The signals were amplified, sampled at 200Hz
and band-pass filtered. The goal of our study is to predict the
label of the trial (left vs. right) based on 1s of data during
target, delay and reach period. Note that data is a vector in
18200-dimensional space (n = 91×200).
For the EEG experiments, a similar set-up was used, with 2
periods: fixation and target. The EEG signals were acquired
using an EEG cap (Electro-Cap International, Eaton, OH)
with 6 electrodes, and the signals were amplified, band-pass
filtered and sampled at 200Hz (Biopac Systems, Goleta,
CA). The number of trials (left+right) was nt= 140 per
session, and there were 3 such sessions. Our goal is to predict
the label of the trial (left vs. right) based on 1s of data during
target period.
The performance of our method was assessed through
leave-one-out CV as explained below. A single trial (out of
nt) was selected for testing, and the remaining trials were
designated for training. This procedure was repeated nttimes,
each time choosing a different sample as a test trial. Within
a single fold of CV the following steps were performed:
1) cPCA was applied to the training data. Principal com-
ponents with an eigenvalue smaller than 1% of the
total variance (trace of the covariance matrix for each
class), were discarded. This effectively determines Vi,
and in turn mi (see Section II-A). The basis Vbwas
calculated, and the transformation matrix Fi= [Vi|Vb]
was found.
2) The training data was transformed according to (1),
and the feature extraction matrices, Ti∈ Rm′i×1, were
obtained using IDA on each of the subspaces. This
resulted in two 1-dimensional (1-D) feature subspaces.
Note that the full transformation matrix can be for-
mally written as Fi= [Vi|Vb]Ti.
3) The testdata, x∗, was transformed to the 2 feature
subspaces, and its posteriors, P(ωji|x∗
were calculated. Class membership was determined as
per classification rule, explained in Section II-B
i) (i, j = 1,2),
The overall performance is estimated by dividing the number
of correctly classified trials by nt.
The performance of our method was compared to those
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of DLDA [9] and threshold-based DLDA methods [7]. All
decoding results for DLDA and threshold-based DLDA were
based on the quadratic classifier, and are comparable to our
classification strategy (Section II-B).
TABLE I
THE PERFORMANCES (%) OF CPCA, DLDA AND THRESHOLD-BASED
DLDA (T-B DLDA) FOR IEEG DATA, DURING target, delay AND reach
PERIOD. (TOP) UNCONSTRAINED, (BOTTOM) CONSTRAINED DATA.
Period
n
TimeDLDAT-B DLDAcPCA+IDA
target
delay
reach
18200
18200
18200
all
all
all
70.37
58.02
66.05
72.22
58.64
66.05
79.01
61.73
70.99
target
10920
1920
3200
6400
9600
150:750
150:750
72.22
86.42
83.95
85.80
65.43
80.25
87.06
87.65
92.59
71.60
83.33
85.80
88.27
87.65
73.46
all
all
all
delay
TABLE II
THE PERFORMANCES (%) OF CPCA, DLDA AND THRESHOLD-BASED
DLDA FOR EEG DATA, DURING target PERIOD
Session
n
Time DLDA T-B DLDAcPCA +IDA
1
1
1200
180
all52.14
60.71
54.29
62.86
75.71
90.00
100:250
2
2
1200
180
all50.71
55.00
52.86
55.71
62.14
82.14
100:250
3
3
1200
180
all57.24
60.14
58.69
61.59
75.35
90.58
100:250
The estimated classification rates for the iEEG and EEG
datasets are shown in Tables I and II, respectively. Our
cPCA-based classification method outperforms the other
methods in almost all cases by a significant margin. In
some cases, our method produces an improvement of around
50% over DLDA-based methods, using both unconstrained
and constrained data. Usually the first 100 − 150 ms of
the target period can be discarded due to visual processing
delay [13]. Similarly, the electrodes can be constrained based
on brain areas of interest. Various combinations of space time
constraints have been reported in Tables I and II.
In particular, with the EEG datasets, DLDA-based meth-
ods fail to capture the class information, yielding nearly
chance-level classification rates, while our method produces
significant improvement. Note that the features of our method
have been limited to 1-D subspace. This constraint was im-
posed for comparison purposes, since LDA-based techniques
operate on at most (c−1)-D subspaces (c = 2 here). By
allowing features of larger dimension (e.g. m = 2,3), the
classification rates of our technique were slightly better. The
superior performance of our method can be attributed to the
fact that LDA-based methods are susceptible to the noise,
which explains their good performance on less noisy tasks,
such as face recognition.
IV. CONCLUSIONS AND FUTURE WORKS
Using classwise PCA, we have developed a novel clas-
sification technique for large-dimensional data. The method
is particularly suited for noisy measurements, arising in the
imaging of brain’s electrical activity (e.g. EEG, iEEG). We
hypothesize that the technique will be a useful analysis tool
for any large-dimensional biomedical data, and in general,
for any data hindered with the small sample size conditions.
Our current research efforts are directed toward the validation
of our technique on a variety of large-dimensional recogni-
tion problems.
The major weakness of our technique is that it does not
scale favorably with the number of classes, c. In particular
both the number of possible feature subspaces, and the
dimension of the intermediate subspaces, Si, increase with
c. However, this scaling is linear, as opposed to frequently
used pairwise criteria [14], [15], which scale quadratically
with c. Moreover, within a single subspace, our feature
extraction is essentially linear, and reduces to simple matrix
manipulations, which can be implemented efficiently.
V. ACKNOWLEDGMENTS
The authors would like to thank Daniel S. Rizzuto and
Richard A. Andersen for providing iEEG data.
REFERENCES
[1] G. Pfurtscheller, C. Neuper, G. R. Muller, B. Obermaier, G. Krausz,
A. Schlogl, R. Scherer, B. Graimann, C. Keinrath, D. Skliris, G. Supp
M. Wortz, and C. Schrank. Graz-BCI: State of the art and clinical
applications. IEEE T. Neur. Sys. Reh., 11(2):177–180, 2003.
[2] J. R. Wolpaw and D. J. McFarland. Control of a two-dimensional
movement signal by a noninvasive brain-computer interface in humans.
P. Natl. Acad. Sci. USA, 101(51):17849–17854, 2004.
[3] J.R. Wolpaw, N. Birbaumer, D.J McFarland, G. Pfurtscheller, and T.M.
Vaughan. Brain-computer interfaces for communication and control.
Clin. Neurophysiol., 6(113):767–791, 2002.
[4] G. Pfurtscheller, C. Neuper, D. Flotzinger, and M. Pregenzer. EEG-
based discrimination between imagination of right and left hand
movement. Electroen. Clin. Neuro., 103(6):642–651, 1997.
[5] K. Fukunaga. Inroduction to Statistical Pattern Recognition. Academic
Press, 1990.
[6] D.S. Rizzuto, A.N. Mamelak, W.W. Sutherling, I. Fineman, and R.A.
Andersen. Spatial selectivity in human ventrolateral prefrontal cortex.
Nat. Neurosci., 8:415–417, 2005.
[7] Koel Das, Joerg Meyer, and Zoran Nenadic. Analysis of large-scale
brain data for brain-computer interfaces. In Proceedings of the 28th
Annual International Conference of the IEEE Engineering in Medicine
and Biology Society, pages 5731–5734, 2006.
[8] Z. Nenadic, D.S. Rizzuto, R.A. Andersen, and J.W. Burdick. Advances
in cognitive neural prosthesis: Recognition of neural data with an
information-theoretic objective. In G. Dornhege, J.R. Millan, T. Hin-
terberger, D.J. McFarland, and K.-R. Muller, editors, Toward Brain
Computer Interfacing, number ISBN: 9780262042444. The MIT Press,
2007. (in press).
[9] H. Yu and H. Yang. A direct lda algorithm for high-dimensional
data - with application to face recognition. Patter Recognition Letters,
34(10):2067–2070, 2001.
[10] L.-F. Chen, H.-Y. Mark Liao, M.-T. Ko, J.-C. Lin, and G.-J. Yu. A new
lda-based face recognition system which can solve the small sample
size problem. Pattern Recognition, 33(10):1713–1726, 2000, Pages.
[11] R. O. Duda, P. E. Hart, and D. G. Stork. Pattern Classification. Wiley-
Interscience, 2001.
[12] Zoran Nenadic. Information discriminant analysis: Feature extraction
with an information-theoretic objective. IEEE T. Pattern Anal., 2007(in
press).
[13] S. Thorpe, D. Fize, and C. Marlot. Speed of processing in the human
visual system. Nature, 381(6582):520–522, 1995.
[14] M. Loog and R.P.W. Duin. Linear dimensionality reduction via a
heteroscedastic extension of lda: the chernoff criterion. IEEE T. Pattern
Anal., 26:732–739, 2004.
[15] P. A. Devijver and J. Kittler.
approach. Prentice Hall International, Englewood Cliffs, N.J., 1982.
Pattern recognition: a statistical
6522