Partial signal space projection for artefact removal in MEG measurements: a theoretical analysis.

INSTITUTE OF PHYSICS PUBLISHING PHYSICS IN MEDICINE AND BIOLOGY
Phys. Med. Biol. 46 (2001) 2873–2887 PII: S0031-9155(01)22531-7
Partial signal space projection for artefact removal in
MEG measurements: a theoretical analysis
G Nolte1 and M S Ha¨ma¨la¨inen2
1 Department of Computer Science, University of New Mexico, Albuquerque, NM, USA
2 Brain Research Unit, Helsinki University of Technology, Helsinki, Finland
Received 5 March 2001, in final form 21 June 2001
Published 5 October 2001
Online at stacks.iop.org/PMB/46/2873
Abstract
Standard methods for artefact removal in MEG or EEG measurements consist
of rejection of either corrupted epochs or signal space projection (SSP). We
propose to combine the two methods by applying SSP only in corrupted
epochs and thus using both temporal and spatial information. This partial
signal space projection necessarily results in smaller variances for the source
localization. Formulae for dipole localization errors as a function of fraction of
corrupted epochs are derived and verified in Monte Carlo simulations of MEG
measurements corrupted with eye artefacts. A theoretical analysis of various
measuring devices, classes of artefact and locations of dipole of interest shows
that the proposed method leads to significant improvement for frontal signal
dipoles and for 30–80% corrupted epochs.
(Some figures in this article are in colour only in the electronic version)
1. Introduction
MEG measurements of evoked magnetic fields are usually corrupted by artefacts caused, e.g.,
by eye blinks or by eye movements. To remove the artefact bias in source estimation, several
rejection techniques have been suggested. One method is to detect time instances with artefacts
present and to discard or weight all corrupted epochs (de Beer et al 1995, MacCrimmon et al
1993, Bezerianos et al 1995). An alternative technique is to estimate the spatial structure
of the artefact and to remove its contribution to the data by signal space projection (SSP)
(Huotilainen et al 1993, Ha¨ma¨la¨inen 1995, Jousma¨ki and Hari 1996, Nolte and Curio 1999).
While the former method completely works in the temporal domain, the latter merely makes
use of spatial information. Here, we propose the natural combination of the two methods, i.e.,
to apply spatial projection only in corrupted epochs leaving as much information as possible
in the data. As a consequence of this partial signal space projection (pSSP), the localization
error will be smaller than that by any of the ‘standard’ methods alone. The subject of this
paper is the theoretical analysis of the gain of the proposed method and the numerical analysis
for a couple of examples motivated by realistic problems.
0031-9155/01/112873+15$30.00 © 2001 IOP Publishing Ltd Printed in the UK 2873

2874 G Nolte and M S Ha¨ma¨la¨inen
As artefacts we have chosen the fields generated by eye blinks or eye movements. The
former was modelled by two equal vertical dipoles and the latter by two equal or by two
independent current dipoles. We want to emphasize that in real appications it is not necessary
to model the artefacts by current dipoles (one ‘merely’ has to measure the respective fields).
Instead, our choices are meant as examples to estimate the profit of the proposed method in
various cases. The specific details have to be figured out for each problem separately.
The calculation of localization errors using pSSP is done in linear approximation of the
forward calculation of the source model which is assumed to be a single dipole in a spherical
volume conductor. In section 2.1 we recall the principles of error analysis as far as they are
needed for the present problem. In section 2.2 we describe both standard artefact rejection
techniques and the proposed combination. The formulae for the parameter errors for pSSP as
a function of the fraction of corrupted epochs are derived in section 2.3. In section 2.4 we
recall and discuss the theoretical result and compare pSSP with the ‘standard’ techniques. In
section 3.1 we verify the analytical results. It will turn out that pSSP can lead to a significant
improvement if the impact of projection itself is large. In section 3.2 we present this impact
for various measuring devices, source locations and directions, and artefact spaces. Finally,
we discuss our results in section 4.
2. Theory
2.1. Principles of error estimation
The measured magnetic field at some fixed time instant is denoted as the N-dimensional vector
�Bm where N is the number of channels in the measuring system. To interpret the result one
(commonly) constructs a source model with M degrees of freedom with M � N. For MEG the
preferred model is the current dipole in a spherically symmetric volume conductor with M = 5
parameters: three parameters for the location and two tangential dipole moments (a radial
dipole is magnetically silent (Sarvas 1987)). If all parameters are given one can compute
the theoretical field �B th(�α) where �α denotes the M-dimensional parameter vector. Though
lengthy, the analytical solution for �B th(�α) is well known (Ilmoniemi et al 1985). To find
the parameters which best explain the measured magnetic field in the presence of noise one
performs a least-squares fit, i.e., one minimizes the Euclidian norm of the difference between
�Bm and �B th(�α)
|
�Bm − �B th( �α)|
!
= min. (1)
This procedure is optimal in a maximum likelihood sense if any deviance between �Bm and the
true field �B th( �αtrue) is due to Gaussian, spatially white noise �η such that
�Bm = �B th( �αtrue) + �η (2)
with
〈ηiηj 〉E = σ
2δij (3)
where 〈〉E denotes expectation value.
The case of spatially non-white noise can be reduced to the white case by pre-whitening the
data. In this paper, we derive the theory assuming that the noise �η is white. The modifications
according to a non-trivial covariance matrix of �η will be presented separately in appendix B.
Due to the noise the estimated source parameters in general contain an error which
we will estimate in linear approximation of the forward calculation (Nolte and Curio 1999,
Lu¨tkenho¨ner 1996, Yamazaki et al 1992, Stok 1987)
�B th( �α) = �B th( �αtrue + �αdiff) ≈ �B th( �αtrue) +D �αdiff (4)

Theoretical analysis of pSSP for artefact removal in MEG measurements 2875
where
D = ( �d1, . . . , �dM) (5)
is the Jacobian matrix with columns
�di =
∂
∂αi
�B th(�α)
|�α= �αtrue (6)
If αi is a dipole moment then, due to linearity, �di is a dipole field, and if αi is a coordinate then
�di is a quadrupole field (Nolte and Curio 1997).
Inserting (4) and (2) into (1) leads to3
�αdiff = D† �η (7)
where † denotes pseudoinverse (D† := (DT D)−1DT ). The covariance matrix of �αdiff is now
readily calculated to give
〈
αdiffi α
diff
j
〉
E
= (D†(D†)T )ij = (D
T D)−1ij . (8)
2.2. Artefact rejection techniques
2.2.1. Temporal domain. MEG measurements are typically corrupted by artefacts generated,
e.g., by eye blinks or eye movement. With the help of EOG electrodes it is possible to detect
the time instances when these artefacts occur and in subsequent data analysis one can simply
discard (de Beer et al 1995, MacCrimmon et al 1993) or weight (Bezerianos et al 1995) all
corrupted epochs. When one uses this method all spatial information about the artefact fields
are ignored.
2.2.2. Signal space projection. A different approach to artefact rejection is the SSP. In this
case one regards the measured field at a specific time instant as a superposition of a signal of
interest �B th( �α), K artefacts �si and noise �η
�Bm = �B th( �α) +
K
∑
i=1
λi �s
i + �η. (9)
It is assumed that the spatial patterns of the artefact fields are known up to the multiplicative
constants λi. Though this condition is mathematically very restrictive, these kinds of artefact
fields occur quite frequently. In fact, any kind of physiologically fixed generator can only vary
in its amplitude.
Using SSP for removal of eye artefacts is equivalent to the method proposed in Berg and
Scherg (1993) and Lins et al (1993). There, the authors include a K-dimensional linear model
for the artefact fields in the inverse problem, where K (≈3–5) is the number of significant
components of the artefact covariance matrix.
In the following we consider only one artefact field �s with |�s| = 1 for convenience. The
generalization to K artefacts will be presented as a result. In order to remove systematic errors
in estimating �α, one projects out the contribution of the artefact both in the measured field and
in the theoretical model; i.e., instead of (1) one minimizes
|P �Bm − P �B th( �α)|
!
= min (10)
with
P := id − S := id − �s�s T . (11)
When one uses this method all temporal information about the artefact fields are ignored.
3 Here and in the subsequent sections linear approximation of the forward calculation is implicitly understood and
the ‘≈’ sign is omitted.

2876 G Nolte and M S Ha¨ma¨la¨inen
2.2.3. Partial signal space projection. The natural combination of classical artefact rejection
techniques and SSP is to use SSP only in corrupted epochs. The measurement now consists
of two parts which have to be treated differently. Here, we present and heuristically explain
the procedure to make optimal use of both information. A rigorous derivation as a maximum
likelihood estimator assuming Gaussian distributed noise will be given in appendix A.
Let N be the number of all epochs, A be the set of all NA corrupted epochs and
�Bml = �B
th( �αtrue) + �ηl be the measurement in epoch l. Then we construct
�Bp :=
1
N


∑
l �∈A
�Bml + P
∑
l∈A
�Bml

 (12)
which coincides with the standard average in non-artefact directions, while the average in
artefact direction gets an effective weight NA/N . So far this guarantees only that the epochs
are temporarily properly weighted with respect to each other: equal weight for allN epochs in
non-artefact directions, equal weight in allN −NA non-corrupted epochs in artefact direction
and zero weight in all NA corrupted epochs in artefact direction. The relative spatial weights
used for the actual dipole fit are less trivial and will be calculated below.
Defining a ‘partial projector’ as
Pβ = (1 − β)id + βP (13)
which is in fact not a projector since P 2β �= Pβ , (12) can be written as
�Bp = Pβ �B
th( �αtrue) + �ξ (14)
where β = NA/N is the fraction of corrupted epochs, and
�ξ =
1
N


∑
l �∈A
�ηl + P
∑
l∈A
�ηl

 (15)
is a noise vector with covariance matrix
C=〈 �ξ �ξ
T
〉E = σ
2(1 − β)id + σ 2βP = σ 2Pβ. (16)
Surprisingly, this result is not identical to the covariance matrix ofPβ �ηwhich erroneously
would have led to C = σ 2P 2β �= σ 2Pβ .
Note that the new averaged data �Bp contain a superposition of the signal field �B th( �αtrue)
and the projected signal field P �B th( �αtrue). The forward calculation has to be modified
accordingly,
�B th( �α)→ Pβ �B
th( �α) (17)
Although the original noise �η was assumed to be spatially white, the partial projection has
introduced a non-trivial covariance matrix: the data in artefact directions are noisier than the
ones in non-artefact directions since the projection has also destroyed information. According
to the maximum likelihood principle, a non-trivial covariance matrix has to be taken into
account by a pre-whitening, i.e., multiplying both the data and the theoretical model field by
the square root of the inverse covariance matrix before applying a least-squares fit. With (16)
the parameters �αtrueare then estimated by minimizing
∣
∣
∣
P−1/2β
(
�Bp − Pβ �B
th( �α)
)
∣
∣
∣
!
= min. (18)

Theoretical analysis of pSSP for artefact removal in MEG measurements 2877
2.3. Calculation of localization error for pSSP
In the linear approximation of the forward calculation, (18) reads
∣
∣
∣
�η0 − P
1/2
β D �α
diff
∣
∣
∣
!
= min (19)
where �η0 = P
1/2
β
�ξ is a white noise vector with noise level σ . Analogous to (7) the above
equation is solved by
�αdiff =
(
DT PβD
)
−1
DT P 1/2β �η0 (20)
and leads to the parameter covariance matrix
〈 �αdiff( �αdiff)T 〉E = σ
2
(
DT PβD
)
−1
. (21)
In order to analyse the error increase due to pSSP, we will now compute the parameter
covariance matrix
(
DT PβD
)
−1 in terms of
(
DTD
)
−1
, the covariance matrix without artefacts.
The calculation follows the same line as the one in Nolte and Curio (1999), which, in fact,
corresponds to the special case β = 1.
From Pβ = id − βS it follows that
(
DT PβD
)
−1
= [id − β(DTD)−1DT SD]−1(DTD)−1. (22)
For further use we formally decompose D into its singular valuesD = U!V T and find
SD(DT D)−1DT S = |UT �s|2S = cos2 φS (23)
where φ is the angle between the artefact �s and the ‘solution space’, the space spanned by the
columns of D. Using (23) one can directly verify4 that
[id − β(DT D)−1DT SD]−1 = id + β
(
DTD
)
−1
DT SD
1 − β cos2 φ
(24)
and hence we arrive at
〈 �αdiff( �αdiff)T 〉E = σ
2
(
(DTD)−1 +
β
(
DTD
)
−1
DT SD
(
DTD
)
−1
1 − β cos2 φ
)
. (25)
A geometrical interpretation of (25) is available if one introduces the concept of covariant
and contravariant fields. Let us denote the vectors �di as covariant fields, then the contravariant
fields �d i (with an upper index) are defined to be the rows of the pseudoinverse of D. They fulfil
〈
�di, �dj
〉
= δij (26)
where 〈,〉 denotes scalar product. Covariant and contravariant vectors5 are very useful
whenever one deals with a non-orthonormal basis. Expanding, e.g., a vector �x, in an
orthonormal basis �ei as �x =
∑
i ai �ei , the coefficients ai are given by ai = 〈�x, �ei〉. If the
set (�ei) is not a complete basis the expansion is still the best fit in a least-squares sense. These
relations do not hold for a non-orthonormal set ( �di). Instead, with (26) it is readily seen that
the coefficients ai in an expansion �x =
∑
i ai �di are calculated as ai = 〈�x, �di〉, i.e., fitting a
linear parameter is equivalent to projecting the data onto the respective contravariant vector.
An illustration is given in figure 1 for a restricted example of a fit with two parameters,
one dipole moment and one location, corresponding in linear approximation to a dipole field
4 The derivation was done with the Neumann series.
5 These two sets of vectors are sometimes referred to as dual bases.

2878 G Nolte and M S Ha¨ma¨la¨inen
qΘ
artefact

quadrupole
contravariant
quadrupole

dipole
Φ
contravariant
dipole
Figure 1. Illustration of covariant and contravariant vectors. For a two-parameter model (one
dipole moment and one location), the linearized forward calculation consists of two vectors (dipole
and quadrupole) lying here in the x–y plane. The respective contravariant vectors lie in the same
plane. The angles ' and (q enter the formula for the localization error increase due to projecting
out the artefact.
and a quadrupole field. Since the dipole field is not orthogonal to the quadrupole field, the
necessity to estimate the former has an influence on the estimate of the latter. The location can
only be estimated from that part of the data which cannot be explained by the dipole moment.
This is implicitly taken into account by contravariant vectors which are, up to normalization,
defined to be that part of a basis vector which is orthogonal to all other basis vectors of the
considered space.
Returning to the error calculation, insertion of id = (DTD)−1DTD on the left-hand side
of the first term in (25) leads to
〈
αdiffi α
diff
j
〉
E
= σ 2
〈
�di,
(
id +
βS
1 − β cos2 φ
)
�dj
〉
(27)
and for the standard deviation of the ith parameter we get the final result
dev(αi) = σ | �di |
√
1 +
β cos2 θ i
1 − β cos2 φ
(28)
where θ i is the angle between the artefact and the contravariant field �di .
The result for a K-dimensional artefact will only be presented without derivation which
can, with the present modifications, be found in Nolte and Curio (1999). Let PA and PS be
orthogonal projection matrices onto the artefact space and the solution space, respectively.
One can show that the K eigenvectors �t k of PAPSPA with non-vanishing eigenvalue are an
orthonormal basis of the artefact space. In this basis the individual contributions to the final

Theoretical analysis of pSSP for artefact removal in MEG measurements 2879
parameter covariance matrix decouple from each other. With Sk := �t k(�tk)T the generalizations
to (28) and (29) read
〈
αdiffi α
diff
j
〉
E
= σ 2
〈
�di,
(
id +
K
∑
k=1
βSk
1 − β cos2 φk
)
�dj
〉
(29)
and
dev(αi) = σ | �di |
√
√
√
√1 +
K
∑
k=1
β cos2 θ ik
1 − β cos2 φk
(30)
where φk (θ ik ) is the angle between the k th artefact vector and the solution space ( �di).
2.4. Comparison of methods
The relevant term in the calculated error using pSSP is the factor by which the error increases
relative to the artefact-free case. Let us in general denote this factor by
f (β) =
dev(αi)
dev0(αi)
(31)
where dev(αi) is the error of the ith parameter after removing the artefacts with a specific
technique, and dev0(αi) is the error in the artefact-free measurement. The index i on f (β) has
been omitted.
For pSSP this factor can be read from (28)
f pSSP(β) =
√
1 +
β cos2 θ i
1 − β cos2 φ
. (32)
We recall that β is the fraction of corrupted epochs and that φ (θ i) is the angle between
the artefact and the solution space (ith contravariant solution vector). These angles become
smaller the more similar the artefact is to relevant features of the possible solutions.
The result for pSSP has to be compared with standard techniques using temporal or
spatial information alone. If corrupted epochs are completely removed from the data, the error
increase is merely determined by the number of remaining epochs:
f temp(β) =
√
1
1 − β
. (33)
Obviously, the error diverges as β → 1 since then there are no data left to perform a source
localization.
If SSP is applied in its standard form the artefacts are projected out in all epochs, and
hence
f SSP(β) = f pSSP(1) =
√
1 +
cos2 θ i
sin2 φ
. (34)
The error increase using pSSP is necessarily smaller than that found using temporal or
spatial information alone (apart from the trivial cases β = 0 or θ i = π/2). The question
arises whether the additional gain is worth the effort. For very few (many) artefacts, using
only temporal (spatial) information is sufficient as there is only little unused information left.
Hence, only for a medium fraction of corrupted epochs, pSSP is likely to give significant
improvement over standard methods. This will be confirmed by simulations below.

2880 G Nolte and M S Ha¨ma¨la¨inen
We proposed to combine temporal and spatial information by a logical ‘and’ operation:
discard data only if they correspond spatially and temporarily to artefacts. For the derivation
of resulting errors, we assumed that the occurrence and spatial pattern (up to an overall scale)
of the artefacts are known. However, a corrupted epoch might be missed and there might be an
error in the estimation of the artefact pattern. These misestimations lead to additional errors
which have not been analysed here. If one isn’t sure about the proper spatial and temporal
detection of artefacts, one might also consider to discard data if they coincide temporarily or
spatially with assumed artefact behaviour. This means that one removes all corrupted epochs
and applies SSP on the remaining data. The error increase is given by f SSP corrected by the
reduced number of used epochs resulting in
f or(β) =
f SSP(β)
√
1 − β
=
√
√
√
√
1 + cos2 θi
sin2 φ
1 − β
. (35)
f or(β) is necessarily larger than the error increase of the standard techniques. However, this
method might be useful if the artefacts are large and an improper removal can cause severe
mislocalization.
3. Examples
3.1. Simulations
In order to validate the formulae for error increase due to pSSP, we performed a series of
Monte-Carlo simulations and compared the results to the theory in section 2. As artefacts
we have chosen the fields corresponding to eye blinks and eye movements. The former was
modelled by two equal vertical dipoles placed within in the right and left eye, respectively,
resulting in a one-dimensional artefact space. For the forward calculation we assumed a
spherical volume conductor throughout the paper. Eye movements were modelled by two
tangential, but otherwise arbitrary,dipoles placed in the two eyes resulting in a two-dimensional
or four-dimensional artefact space if the eyes move dependently or independently, respectively.
In general it is not clear what dimensionality one should assign to eye movement; e.g., in Elbert
et al (1985) it was suggested to use a three-dimensional model.
In this section we choose the signal dipoles to be placed in the frontal lobe with dipole
directions being (a) rostral and (b) lateral. Eye artefacts were modelled by a two-dimensional
sub-space corresponding to dependent eye movement in arbitrary direction. The simulations
were carried out for a 61-channel whole head magnetometer system.
For the coordinate system we chose the x-axis to point from the left to the right ear, the
y-axis to point from back to front and the z-axis to point in the vertical direction upwards.
The centre was chosen to coincide with the centre of the spherical volume conductor6. In this
coodinate system the eye dipoles have coordinates (±3.3, 7.8,−5.4)T (in cm) and the frontal
and hand-cortex dipoles are assumed to be located at (−4.4, 3.5,−2)T and (3.8, 3.3, 2.0)T
(in cm) respectively.
In figure 2 we show the mean localization error of 1000 simulations for each value of the
fraction of corrupted epochs. The results are always in good agreement with the theoretical
findings. We also included the theoretical curves for pure temporal artefact rejection (diverging
curve) and for the pure SSP (constant curve). We see that pSSP always results in the smallest
error increase. The error increase is different for different components. Here, the largest
6 This is the coordinate system in which the locations and directions of the sensors of the Neuromag systems are
typically provided.

Theoretical analysis of pSSP for artefact removal in MEG measurements 2881
0.5
1
1.5
2
f(β
)
lateral dipole, radial coord. lateral dipole, longit. coord. lateral dipole, transv. coord.
0 0.5 1
0.5
1
1.5
2
f(β
)
β
rostral dipole, radial coord.
0 0.5 1
β
rostral dipole, longit. coord.
0 0.5 1
β
rostral dipole, transv. coord.
Figure 2. Dipole localization errors, normalized to the errors for artefact-free measurements, for
radial, longitudinal and transversal dipole coordinates for a frontal dipole pointing in lateral (top)
or rostral (bottom) direction. Eye artefacts were modelled by two coupled dipoles resulting in a
two-dimensional artefact space. Simulations (crosses) of pSSP are in good agreement with the
theoretical prediction (solid curve). The respective errors are always smaller than by using SSP
(dashed–dotted) or temporal artefact rejection (dotted) alone.
error increase is found in the transversal direction, i.e., tangential but orthogonal to the dipole
direction. We found that this is usually, but not necessarily, the case. Eventually, the best
standard rejection technique is worse by more than 60% compared to pSSP which corresponds
to a factor 2.5 in the measuring time.
3.2. Impact of projection
Partial projection requires reliable results both in the temporal and in the spatial domain of the
artefacts. We recommend this method only if a significant improvement of the localization
error can be expected. For this the fraction of corrupted epochs has to assume a moderate value
(between 30 and 80%), and the projection has to affect localization accuracy significantly. To
analyse the latter we computed the relative error increase f SSP for the standard SSP which
we recall to be independent of β. For pSSP one has 1 � f pSSP(β) � f SSP: a significant
advantage of pSSP over SSP can only be achieved if f SSP is significantly larger than 1.
As measuring devices we chose standard Neuromag whole head systems. These systems
consist of 61 (102) sensor locations measuring the planar gradient in two directions of the
essentially radial component of the magnetic field at each location. The larger system also
measures the ‘radial’ component as magnetometer systems directly. For our numerical

2882 G Nolte and M S Ha¨ma¨la¨inen
S1, lateral
S1, rostral
front., lateral
front., rostral
Mag., 61 chan.
Mag., 102 chan.
Grad., 122 chan.
Grad., 204 chan.
1
1.2
1.4
1.6
Eye blinks
fS
SP
Figure 3. Impact of SSP on the error of the transversal coordinate for various whole head
gradiometer and magnetometer systems and for signal dipoles placed in the frontal lobe or in
the hand cortex. The artefacts were modelled by two coupled vertical dipoles placed in the eyes
resulting in a one-dimensional artefact space.
examples we also assumed that the smaller system can measure the ‘radial’ components
themselves. Finally, splitting the whole number of sensors into gradiometer and magnetometer
sensors, we have four different systems: (a) a 61-channel magnetometer, (b) a 102-channel
magnetometer, (c) a 122-channel planar gradiometer and (d) a 204-channel planar gradiometer.
The calculations for a combined system of magnetometer and gradiometer channels require a
careful treatment of the noise. Gradiometer channels typically have a smaller signal but also
a smaller noise level. Assuming spatially white noise for the whole system would lead to an
unrealistic domination of the magnetometer channels.
In figure 3 we show the result for eye blinks for various systems and for various signal
dipoles placed in the frontal lobe and the (lateral) hand cortex denoted by ‘S1’. The dipole
directions were chosen to be rostral and lateral. In figures 4 and 5 we show the corresponding
results for two-dimensional and four-dimensional eye movements, respectively.
The largest increase of localization error due to projection was found for frontal signal
dipoles using a magnetometer system with ‘few’ channels and for the four-dimensional artefact
space. For our examples the error increase was typically much larger for the 102-magnetometer
system than that for the 122-gradiometer system, although the difference in the number of
channels is relatively small. However, it should be noted that we calculated the relative errors.
The absolute error without artefacts is significantly larger for gradiometer systems which
corresponds to spatial high pass filtering.

Theoretical analysis of pSSP for artefact removal in MEG measurements 2883
S1, lateral
S1, rostral
front., lateral
front., rostral
Mag., 61 chan.
Mag., 102 chan.
Grad., 122 chan.
Grad., 204 chan.
1
1.2
1.4
1.6
1.8
2
2D Eye movement
fS
SP
Figure 4. Same as figure 2 with artefacts modelled by two coupled dipoles placed in the eyes and
pointing in an arbitrary direction resulting in a two-dimensional artefact space.
4. Discussion
We proposed to combine temporal and spatial information about artefacts by projecting out
artefact contributions only in corrupted epochs. This method always results in a smaller
localization error than using temporal or spatial information alone. The improvement is large
if the fraction of corrupted epochs assumes a moderate value (about 30–80%) and if projection
removes important information of the signal, e.g. if the artefacts are caused by eye movement
and the signal dipole is frontal. For signal dipoles being relatively far away from the artefact
dipoles, projection is harmless and the improvement due to pSSP is probably not worth the
effort. However, the precise gain of the method depends on the measuring device and on the
kind of artefact.
In this paper we assumed spatially uncorrelated noise. Correlation was only considered
as being induced by the partial projection itself. Though correlated noise can be reduced
to the spatially white case by pre-whitening, the explicit results may differ significantly.
Correlated noise effectively reduces the number of available channels. The information
contained in neighbouring channels is largely redundant: source localization then ‘lives’ on
remote channels. Though not explicitly studied, we assume that in this case SSP will affect
localization accuracy more severely and the relative gain of pSSP will be more pronounced.
We always assumed that the temporal and spatial information is valid. If misestimations
are likely both in the temporal and spatial domain one might also consider to combine temporal
and spatial information by projecting out the artefacts in all epochs and by taking out corrupted
epochs completely. The resulting localization has a larger variance than any standard technique
but eventually removes a severe bias caused by very large artefacts. If, on the other hand, the

2884 G Nolte and M S Ha¨ma¨la¨inen
S1, lateral
S1, rostral
front., lateral
front., rostral
Mag., 61 chan.
Mag., 102 chan.
Grad., 122 chan.
Grad., 204 chan.
1
1.5
2
2.5
3
3.5
4D Eye movement
fS
SP
Figure 5. Same as figure 2 with artefacts modelled by two independent dipoles placed in the eyes
and pointing in an arbitrary direction resulting in a four-dimensional artefact space.
artefact fields are small it might be preferable to keep them since, in general, the increase of
variance due to artefact-rejection techniques can be larger than the bias due to the artefacts.
In summary, pSSP can lead to significant improvement of dipole localization accuracy.
Its specific performance depends on the problem under study.
Acknowledgments
We thank Riitta Salmelin for providing us with anatomical coordinates and George Bissias for
careful reading of the manuscript. The study was supported in part by Neuro-Birch II and by
the National Foundation for Functional Brain Imaging (USA).
Appendix A
Here, we derive (18) from the maximum likelihood principle. Defining the averages of the
data in the corrupted and non-corrupted epochs as
�BmA =
1
NA
∑
l∈A
�Bml (36)
and
�BmB =
1
N − NA
∑
l �∈A
�Bml (37)

Theoretical analysis of pSSP for artefact removal in MEG measurements 2885
then, assuming Gaussian noise, the probability to measure �BmA and �B
m
B given a source with
parameters �α reads
P
(
�BmA, �B
m
B, �B
th( �α)
)
= c exp
(
−
1
2
(
[
�BmA − �B
th( �α)
]T
C−1A
[
�BmA − �B
th( �α)
]
+
[
�BmB − �B
th( �α)
]T
C−1B
[
�BmB − �B
th( �α)
]
))
(38)
where CA and CB are the covariance matrices of the averaged noise (including artefacts) of
the corrupted and non-corrupted data set, respectively, and c is an irrelevant normalization
constant.
We assume that the artefact-free noise averaged over all epochs
�η =
1
N
N
∑
l=1
�ηl (39)
has covariance matrix
〈�η �η T 〉E = σ
2id (40)
where id is the identity matrix. Hence, the average over artefact-free epochs, consisting of
only N −NA data vectors, has a noise covariance matrix
CB =
N
N − NA
σ 2id =
σ 2
1 − β
id (41)
with an inverse
C−1B =
1 − β
σ 2
id. (42)
The NA corrupted epochs in addition contain the covariance matrix of the artefacts.
Projecting out the artefact space is equivalent to implicitly assuming that the noise is infinite
within the artefact space, which will be modelled by an artefact covariance matrix ˆC with
infinite noise level σˆ ,
ˆC = σˆ 2S (43)
where S defines the artefact space (see (11)). The full covariance matrix in the corrupted
domain reads
CA =
σ 2
β
id + σˆ 2S (44)
and, using S 2 = S, it is readily seen by insertion that
C−1A =
β
σ 2
+
(
β
σ 2 + σˆ 2β
−
β
σ 2
)
S (45)
which in the limit σˆ → ∞ converges to
C−1A =
β
σ 2
(1 − S) =
β
σ 2
P. (46)
In order to simplify the probability we make the ansatz
P
(
�BmA, �B
m
B, �B
th( �α)
)
= c˜ exp
(
−
1
2σ 2
|
�F −G �B th( �α)|2
)
(47)
where c˜ does not depend on �α and hence is irrelevant for maximization of the probability with
respect to �α. Inserting C−1A and C
−1
B into (38) it follows, from comparing (38) with (47), that
G2 = (1 − β)id + βP = Pβ (48)
GT F = βP �BmA + (1 − β) �B
m
B =
�Bp. (49)

2886 G Nolte and M S Ha¨ma¨la¨inen
Using GT =G we finally find that maximizing the probability is equivalent to minimizing
∣
∣
∣
P−1/2β �B
p
− P 1/2β �B
th
∣
∣
∣
2
(50)
which is equivalent to (18).
Appendix B
Here we will, in short, address the question of non-white noise �η with
�η �η T = C. (51)
This case can be solved by a preceeding pre-whitening of the data, the model field and the
artefact-vector
�Bml → C
−1/2
�Bml (52)
�B th( �α)→ C−1/2 �B th( �α) (53)
�s →
C−1/2�s
|C−1/2�s|
. (54)
Accordingly, the projector also transforms as
P → id −
C−1/2�s �s T C−1/2
�s T C−1�s
. (55)
All results remain formally the same. However, the angles now have a different meaning
as they denote the geometrical relationships between the transformed solution and the artefact
space.
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