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【信号分离】约束独立分量分析CICA算法实现信号盲分离附matlab代码.zip （8个子文件）

cICAdemo.m 2KB

genRectangleRef.m 1KB

remstd.m 593B

Neurocomputing-2008a.pdf 682KB

cICA.m 4KB

whiten.m 1KB

standarize.m 697B

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Neurocomputing 71 (2008) 1669–1679

Morphologically constrained ICA for extracting weak

temporally correlated signals

Zhi-Lin Zhang

a,b,

School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu 610054, China

Department of Computer Science and Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

Received 10 March 2006; received in revised form 9 October 2006; accepted 11 April 2007

Communicated by H. Valpola

Available online 24 April 2007

Abstract

Recently the constrained ICA (cICA) algorithm has been widely applied to many applications. But a crucial problem to the algorithm

is how to design a reference signal in advance, which should be closely related to the desired source signal. If the desired source signal is

very weak in mixed signals and there is no enough a priori information about it, the reference signal is difﬁcult to design. With some

detailed discussions on the cICA algorithm, the paper proposes a second-order statistics based approach to reliably ﬁnd suitable

reference signals for weak temporally correlated source signals. Simulations on synthetic data and real-world data have shown its validity

and usefulness.

Keywords: Constrained independent component analysis (cICA); Blind source extraction (BSE); Blind source separation (BSS); Reference signal;

Temporally correlated signal

1. Introduction

Temporally correlated signals widely exist in various

ﬁelds, such as biomedical engineering [3,6,8,9,19,21] and

ﬁnancial time series an alysis [7]. These signals are often

‘‘interesting’’ and important to us. For example, in

abnormal EEG analysis the quasi-periodic complex re-

sulted from periodic synchronous discharge is of impor-

tance to determine whether or not the subject suffers from

subacute sclerosing panencephalitis (SSPE) or other

diseases. In the non-invasive extraction of fetal electro-

cardiogram (FECG) [8,19,21], the FECG, which provides

information about fetal maturity, position of the fetus and

multiple pregnancies, also can be regarded as a quasi-

periodic signal.

Unfortunately, these valuable temporally correlated

source signals and other unwanted source signals are often

mixed in observed signals and often contaminated by noise.

To obtain the desired source signals, one powerful

technique is the blind source separation (BSS) [4,7], which

simultaneously separates all of the source signals. How-

ever, in many applications the number of sensors is often

large, which may result in heavy computational load and

cost lots of time, while the ‘‘interesting’’ source signals (the

desired ones) are few. For exampl e, in EEG or MEG we

obtain typically more than 64 sensor signals but only

several source signals (e.g. the periodically evoked brain

potentials) are considered interesting, and the rest are

considered to be interfering noise. For such applications it

is essential to develop reliable, robust and effective learning

algorithms whi ch enable us to extract only a small number

of temporally correlated source signals that are potentially

interesting and contain useful information [4,9,20–22].

The co nstrained ICA (cICA ) (also called ICA with

reference) [10–14] is a good can didate for extracting several

source signals from a large number of observed signals.

ARTICLE IN PRESS

www.elsevier.com/locate/neucom

doi:10.1016/j.neucom.2007.04.004



Corresponding author at: School of Computer Science and Engineer-

ing, University of Electronic Science and Technology of China, Chengdu

610054, China.

E-mail address: zlzhang@uestc.edu.cn.

It needs to ﬁrst construct a reference signal that should be

closely related to the desired source signal. The reference

signal is the one and only the one that is closest to the

desired source signal in terms of a closeness measure.

Therefore, the reference signal is of vital importance to

cICA. It has been pointed out [10,22] that both the shape

and the phase of the reference signal may strongly affect

the extraction results. Thus in some applications the

reference signal is difﬁcult to construct, at least in an

automatic manner. For example, in the non-invasive

extraction of FECG, the desired FECG is very weak so

that the accurate occurrence time and shape of each

complex of FECG are often not easy to obtain, especially

when the fetus is in early phase. Bes ides, an improperly

selected thres hold will result in the failure of cICA [14].

A feasible threshold depends on both the designed

reference signal and the closeness measure. Given the same

data set, different reference signals require different

suitable ranges of the threshold value. Similarly, different

closeness measures also determine different suitable ranges.

Lu and Rajapakse [14] suggest to use a small threshold

initially, and then gradually increase the threshold.

However, this method is not always feasible in practice.

Due to the above reasons, the applications of cICA are

limited to the ones where the reference signals are easy to

design or ﬁnd, such as the artifact removal of EEG [10,13].

In this paper, we pro pose a second-order statistics based

approach for designing suitable reference signals for

reliably extracting weak temporally correlated source

signals, which the original cICA often fails to extract.

Another advantage of the approach is that the threshold is

easy to set, ensuring the global convergence of cICA, since

the reference signal is morphologically close to the desired

signal. The rest of the paper is organized as follows. In

Section 2 the cICA algorithm is introduced, and some

important related issues are discussed in Section 3. In the

next section, an approach to obtain feasible reference

signals for extracting weak temporally correlated source

signals is proposed. Computer simulations are presented in

Section 5 and conclusions are drawn in Section 6.

2. The constrained independent component analysis

Let us consider unknown stochastic source signals s

ði ¼

1; ...; nÞ which are mutually independent, zero-mean and

unit-variance. It is assumed that the source signals are non-

Gaussian (at most one is Gaussian). The model for the

observed signals is

xðkÞ¼AsðkÞ, (1)

where k is the time index, A is an unknown non-singular

mixing matrix, sðkÞ¼½s

ðkÞ; ...; s

ðkÞ

and xð kÞ¼½x

ðkÞ;

...; x

ðkÞ

. Without knowing the source signals and the

mixing matrix, we want to recover a source signal from the

observed signals xðkÞ by the following linear transform:

yðkÞ¼w

xðkÞ¼w

AsðkÞ, (2)

where w is a column vector and yðkÞ is a recovered source

signal up to a scalar. To cope with ill-conditioned cases and

to make algorithms simpler and faster, before running

algorithms the whitening [4] is often used to trans form the

observed signals xðkÞ to

zðkÞ¼VxðkÞ, (3)

such that EfzðkÞzðkÞ

g¼I, where V is a whitening matrix.

A typical solution is given by

V ¼ D

1=2

, (4)

where D is the diagonal matrix of the eigenvalues of the

matrix EfxðkÞxðkÞ

g, and E is the matrix whose columns

are the corresponding unit-norm eigenvectors.

For extracting one independent source signal, a reliable

and ﬂexible contrast function is the one based on

negentropy [7], deﬁned by

JðyÞr½EfGðyÞg  EfGðvÞg

, (5)

where y ¼ w

z is the algorithm output, r is a positive

constant, v is a Gaussian variable with zero mean and unit

variance, and GðÞ can be any non-quadratic function, such

ðyÞ¼logðcoshða

yÞ=a

Þ, (6)

ðyÞ¼expða

=2Þ=a

, (7)

ðyÞ¼y

=4, (8)

where 1pa

p2anda

 1. The algorithm that maximizes

the contrast function (5) is called one-unit ICA [7].

However, maximization of (5) will give any of source

signals. That is to say, any source signal may be the output.

When one desires a speciﬁc source signal, he needs to use

some a priori information of the desired signal to modify

the one-unit algorithm, ensuring the output is necessarily

the desired one. To achieve this goal, the cICA is derived

[14] by ﬁrst deﬁning the following constrained contrast

function:

max JðwÞr½EfGðw

zgEfGðvÞg

ð9Þ

s:t: gðwÞ¼ðy; rÞxp0; hðwÞ¼Efy

g1 ¼ 0, ð10Þ

where the equality constr aint hðwÞ ensures that the contrast

function JðyÞ and the weight vector w are bounded. ðy; rÞ is

the closenes s measure between the extra cted signal y and

the reference signal r. Note that the desired source signal is

the one and only the one closest to the reference signal r,

satisfying the following inequality relationship:

ðw

T

z; rÞoðw

z; rÞp pðw

n1

z; rÞ, (11)

where the optimum vector w



corresponds to the desired

output and w

ði ¼ 1; ...; n  1Þ corresponds to other

unwanted source signals. The value of the threshold x in

(10) lies in ½ðw

T

z; rÞ;ðw

z; rÞÞ.

From the contrast function (9)–(10) Lu an d Rajapakse

derived the cICA algorithm [12–14]:

tþ1

¼ w

 ZR

1

, (12)

ARTICLE IN PRESS

Z.-L. Zhang / Neurocomputing 71 (2008) 1669 –16791670

¼ rEfzG

ðyÞg 

mEfzg

ðyÞg  lEfzyg, (13)

¼ rEfG

ðyÞg 

mEfg

ðyÞg  l, (14)

where t is the iteration count, R

¼ Efzz

g, r ¼

r  sign ðEfGðyÞg  EfGðvÞgÞ, G

ðyÞ and g

ðyÞ are the ﬁrst

derivatives of GðyÞ and gðyÞ with respect to y, and G

ðyÞ

and g

ðyÞ are the second derivatives. The opt imum

multipliers m



and l



are found by iteratively updating

them based on a gradient-ascent method:

¼ maxf0; m

t1

þ ggðw

t1

Þg, (15)

¼ l

t1

þ ghðw

t1

Þ. (16)

3. Some important issues of cICA

There are several important issues that should be

noticed. One issue is the value of the threshold x. If the

threshold value is too large, the cICA may converge to

other source signals since there may exist several source

signals whose closeness measures are also less than the

threshold. In contrast, if the threshold is too small, the

cICA cannot converge. So the threshold value should

be carefully selected. But the value is inﬂuenced by

some factors. One is the choice of the closeness ðy; rÞ.

A common choice is the mean square error (MSE) given

by ðy; rÞ¼Efðy  rÞ

g, and another is the correlation

ðy; rÞ¼Efyrg. Clearly, different choices require different

ranges of the threshold value. Another factor affecting the

threshold value is the choice of the reference signal r. If the

reference signal is very similar to the desired source signal,

then the threshold value should be very small so that the

algorithm can globally converge. Otherwise, the value

should be larger. Obviously, the suitable value in some

degree depends on the a priori knowledge about

the quantization of the closeness between the reference

signal and the closest source signal (i.e. the desired signal),

and of the closeness between the reference signal an d the

second closest source signal. Unfortunately, according to

our knowledge, there is no guide on how to set a suitable

value.

The second issue is the design of the reference signal.

James and Gibson [10] pointed out that the shape of the

reference signal may inﬂuence the output result in the

sense that source signals of slightly different morphology

may be extracted for different reference morphology. They

also pointed out that the phase of the reference signal

must be closely matched to that of the desired source

signal, or the occurrence time of each impulse of the

reference signal is consistent with that of the desired signal.

Until now, in most literature the main method is

constructing a simp le impulse signal by observing the

waveform of sensor signals and/or by exploiting strong a

priori information about the desired source signal

[10,12–14]. This method requires that the desired source

signal is strong enough to be observed in a sensor signal

and/or there is sufﬁcient a priori information available (e.g.

the morphology, the phase and the occurrence time of the

desired signal), which cannot be always satisﬁed. In

addition, sometimes one should use many similar reference

signals, each shifted by one sample to cover one expected

period of the desired source signal [10]. Thus, the method

sometimes may be trivial and results in increased computa-

tional load and time.

In the next section we will propose a valid approach to

ﬁnd a refer ence signal, whose phase and waveform is well

matched to the desired source signal. Thus a suitable

threshold value is easy to set, regardless of which closeness

measure is adopted.

4. The proposed approach for designing reference signals

In this section we propose an approach to design

suitable reference signals for extracting temporally corre-

lated weak source signals. Our idea is ﬁrst ﬁnding several

time delays at which the autocorrelation of the desired

source signal is maximized, and then using the time delays

to roughly extract the desired source signal. The roughly

extracted signal serves as the reference signal for the cICA

algorithm.

Suppose the desired source signal is non-Gaussian and

exhibits periodic behavior with period t

Without loss of

generality, we assume s

is the desired source signal,

satisfying the following relations:

p¼1

ðkÞs

ðk  l

()

40, (17)

p¼1

ðkÞs

ðk  l

()

¼ 0 8ja1, (18)

where s

are other source signals, l

ðp ¼ 1 ; ...; PÞ are

positive integers and P is the number of the time delays

(how to ﬁnd these time delays will be addressed later). Then

we propose the following objective function for coarsely

extracting s

max JðwÞ¼2E

p¼1

yðkÞyðk  l

()

¼ w

p¼1

ðR

ðl

ÞþR

ðl

()

w ð19Þ

s.t. kwk¼1, ð20Þ

where yðkÞ¼w

zðkÞ¼w

VxðkÞ and R

ðl

Þ¼EfzðkÞ

zðk  l

g. The reason for this formulation is that for

the desired signal s

, this averaged delayed autocorrelation

has a large positive value, while for other source signals this

ARTICLE IN PRESS

Here we mean that its waveform in a period is similar to its waveform

in another period. Thus, it is not a strictly periodic signal. The reason why

we make this assumption is that in practice the strictly periodic signal is

rare.

Z.-L. Zhang / Neurocomputing 71 (2008) 1669 –1679 1671

value is zero. It is easy to see [20] that maximization of (19)

under the constraint (20) is equivalent to ﬁnding the

normalized eigenvector corresponding to the maximal

eigenvalue of

p¼1

ðR

ðl

ÞþR

ðl

Þ. Thus we directly

have the following algorithm:

w ¼ EIG

p¼1

ðR

ðl

ÞþR

ðl

, (21)

where EIGð

p¼1

ðR

ðl

ÞþR

ðl

ÞÞ is the operator that

calculates the normalized eigenvector corresponding to the

maximal eigenvalue of

p¼1

ðR

ðl

ÞþR

ðl

Þ. Denote

w the converged solution of the algorithm (21). Then the

reference signal is given by

y ¼

If several desired source signals exhibit periodic behavior

with the same period t

, we still can obtain suitable

reference signals for each desired source signal. To see this,

suppose among the n source signals there are q ones, say

; s

; ...; s

, that have the same period t

. Let r

p¼1

ðkÞs

ðk  l

Þg ðb ¼ 1; ...; nÞ. Without loss of

generality, suppose r

4 4r

ðj ¼ q þ 1; ...; nÞ.

In the following we will show that we can use the

normalized eigenvector w

associated with the ith largest

eigenvalue of Ef

p¼1

zðkÞzð k  l

g to obtain the

reference signal for s

ði ¼ 1; ...; qÞ.

First, note that VA is an orthogonal matrix, since I ¼

Efzz

g¼VAEfss

and Efss

g¼I. Then consider

the following relation:

p¼1

zðkÞzðk  l

()

¼ VAE

p¼1

sðk Þsðk  l

()

, ð22Þ

which implies that all the eigenvalues of Ef

p¼1

zðkÞzð k 

g are those of Ef

p¼1

sðkÞsðk  l

g. Denote by w

the normalized eigenvector associated with the ith largest

eigenvalue of Ef

p¼1

zðkÞzðk  l

g.Wehave

p¼1

zðkÞzðk  l

()

¼ l

; i ¼ 1; ...; q, (23)

where l

is the ith largest eigenvalue. Since zðkÞ¼VAsðkÞ

and VA is an orthogonal matrix, (23) is further reduced to

p¼1

sðk Þsðk  l

()

ðA

¼ l

; i ¼ 1; ...; q, ð24Þ

implying that A

is the normalized eigenvector

associated with the ith largest eigenvalue l

of Ef

p¼1

sðkÞsðk  l

g. Suppose Ef

p¼1

ðkÞs

ðk  l

g¼

0 ðaab and a; b ¼ 1; ...; nÞ, then Ef

p¼1

sðk Þsðk  l

is a diagonal matrix. Therefore A

¼ e

, whose the ith

element is one while other elements are zero. Then we have

y ¼ w

z ¼ w

VAs ¼ e

s ¼ s

, which means that s

extracted by the vector w

. However, in practice the

assumption Ef

p¼1

ðkÞs

ðk  l

g¼0 ðiaj and i; j ¼

1; ...; nÞ cannot be met. One reason is that in real world

two source signals may be weakly correlated. Another

reason is concerned with the numerical errors. Practically

we use the empirical average over limited samples, instead

of the statistical expectation. As a result, even if two signals

satisfy the assumption, the calculated cross-correlation

over limited samples is generally non-zero (although we

have shown [20] that with sufﬁcient time delays the

numerical errors tend to zero, in practice we may not ﬁnd

so many suitable time delays). Due to these reasons,

 e

and thus y  s

. This means that the extracted

signal by w

is only a good reference signal for further

extracting s

by the cICA algorithm.

So, according to the analysis, if we ﬁnd the

matrix Ef

p¼1

zðkÞzðk  l

g has several large eigenva -

lues, say q eigenvalues that are obviously larger than zero,

then we can conclude that there may be q source signals

with the same period, and calculate the corresponding

eigenvectors to obtain reference signals for each source

signal.

If the desired source signal does not exhibit periodic

behavior, but only a general temporally correlated signal,

the above results should be slightly modiﬁed. Denote by s

the desired source signal and suppose it satisﬁes:

p¼1

ðkÞs

ðk  t

()

40, (25)

p¼1

ðkÞs

ðk  t

()

¼ 0 8ja1, (26)

where t

; t

; ...; t

are time delays and s

are other source

signals. Similarly, the objective function is

max JðwÞ¼2E

p¼1

yðkÞyð k  t

()

¼ w

p¼1

ðR

ðt

ÞþR

ðt

()

w ð27Þ

s.t. kwk¼1. ð28Þ

And the corresponding algorithm is given by

w ¼ EIG

p¼1

ðR

ðt

ÞþR

ðt

. (29)

Now we have an important issue that has not been

addressed, i.e. how to ﬁnd the suitable time delays, which

ARTICLE IN PRESS

Since s

; s

; ...; s

are not strictly periodic signals, we have

Efs

ðkÞs

ðk  l

Þgo1 and it is probable that Efs

ðkÞs

ðk  l

Þga

Efs

ðkÞs

ðk  l

Þg ðgahÞ. Therefore, the assumption that r

4 

ðj ¼ q þ 1; ...; nÞ is held in most cases.

We would like to draw the readers’ attention to some similar work

[3,14] that also showed second-order statistics based reference construc-

tion methods are insufﬁcient to recover independent source signals.

Z.-L. Zhang / Neurocomputing 71 (2008) 1669 –16791672

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