Memory and Processing Efficient Formula for Moving Variance
Calculation in EEG and EMG Signal Processing
Mario Michael Krell
1
, Marc Tabie
1
, Hendrik W¨ohrle
2
and Elsa Andrea Kirchner
1,2
1
University of Bremen, Robotics Lab, Bremen, Germany
2
German Research Center for Artificial Intelligence (DFKI), Robotics Innovation Center, Bremen, Germany
Keywords:
Variance, Mean, EMG, Electromyogram, EEG, Electroencephalogram, BCI.
Abstract:
Adaptation of human-machine interaction devices by means of physiological data requires online analysis.
We introduce new update formulas for otherwise time-demanding calculations of window based current mean
and variance of the signal. Those were required for efficient realtime time series data processing. We discuss
the formulas with the help of synthetic data. They differ from existing incremental calculations due to a
decremental component, because of samples leaving the window of observation. An example application for
EMG-based movement onset prediction is presented.
1 INTRODUCTION
For the improvement of human-machine interaction
(HMI), many approaches make use of physiologi-
cal data, e.g., gaze, electroencephalogram (EEG) or
electromyogram (EMG) (Zander et al., 2010; Kirch-
ner et al., 2013). Any data recorded during inter-
action has to be analyzed online to allow an online
adaptation and improvement of the interface itself,
e.g., by self-correction based on error potential de-
tection (Zander et al., 2010), of the control of a ma-
chine or robotic device (Folgheraiter et al., 2012), or
of the working condition of the user, e.g., adaption
to high workload (Kohlmorgen et al., 2007). Recent
approaches make use of hybrid brain-computer inter-
faces (BCIs) (Pfurtscheller et al., 2010) which support
the human by detecting changes in different physi-
ological data streams in parallel and by this do im-
prove reliability and speed of the interface. Taking
into account the growing demands in analysis of sev-
eral data streams in parallel and that HMIs should
even adapt before an interaction is carried out (Fol-
gheraiter et al., 2012), the speed of data processing
within the interface is critical. Further, for mobile
support the analysis of physiological data must be en-
abled by means of mobile HMI devices (W¨ohrle et al.,
2013). Hence, new processing methods have to be
applicable to small, low-cost processing devices with
typically low memory resources. For these applica-
tions, specialized signal processing hardware like a
Field Programmable Gate Array (FPGA) system is
a reasonable choice. For future HMI support, it is
therefore required to review and improveexisting data
analysis algorithms regarding their applicability for
the analysis of multiple data streams on mobile de-
vices. In this paper, a new update formula for means
and variance calculation was developed, which meets
the before explained requirements of mobile process-
ing.
Typically, one of the first processing steps in on-
line analysis of physiological data is to calculate the
mean and variance. Both can be used in two ways:
as features or for an online standardization. A stan-
dardization might be needed, e.g., for EEG process-
ing, to compensate for conductivity changes at the
electrodes between human body and sensors or to re-
duce noise and artifact influence. The moving vari-
ance can be used as a feature for movement detec-
tion in HMI using EMG. Furthermore, in (Tabie and
Kirchner, 2013) it is described how the variance in
combination with an adaptive threshold can also be
used to predict upcoming movements based on EMG
signal analysis. This can be used to improve the con-
trol of an exoskeleton, as described in (Kirchner et al.,
2013), where prediction times of 95ms (28 Standard
Error) on average could be achieved. An important
requirement for online estimations of variance and
mean is to update these values with each incoming
sample and give the current up-to-date estimate which
reflects their changing nature.
Earlier work handled this challenge only partially
as described in the following. An incremental ap-
41
Krell M., Tabie M., Wöhrle H. and Kirchner E..
Memory and Processing Efficient Formula for Moving Variance Calculation in EEG and EMG Signal Processing.
DOI: 10.5220/0004633800410045
In Proceedings of the International Congress on Neurotechnology, Electronics and Informatics (NEUROTECHNIX-2013), pages 41-45
ISBN: 978-989-8565-80-8
Copyright
c
 2013 SCITEPRESS (Science and Technology Publications, Lda.)

proach is given in (Welford, 1962) where the change
in the “sum of squares of the deviations of the val-
ues about their mean” (short: sum of squares, S) is
calculated. In this approach, all recorded samples are
used to estimate the mean and variance of the total
signal. The expected changes over time, which we are
interested in, are not monitored by this method. Thus,
we added a decremental component to this approach,
which enables the method to observe these changes.
In (Chan et al., 1983) an algorithm for parallel com-
putation of the sum of squares problem is given. This
approach is very useful for fast offline calculations but
requires extra memory and calculation resources on a
mobile device. Formulas for an incremental approach
are given as well and compared but no decremental
approach is used. To allow forgetting of samples, it is
suggested in (West, 1979) to use the weighted mean
and variance. This approach would not be efficient
enough because it requires two calculation steps.
To fulfill the requirements of mobile processing,
we calculate the necessary formulas for mean and
variance in Section 2. In Section 3 we give some
empirical results showing the benefits of these new
formulas. In Section 4 we present an experiment for
EMG movement detection.
2 PROPOSED METHOD
In this section, the approach given in (Welford, 1962)
is adapted to obtain the required relation for mean and
variance. The mean m
0
and the variance
S
0
n
are calcu-
lated with n samples x
0
, ..., x
n−1
, the updated m
1
and
S
1
n
with x
1
, ..., x
n
. S
0
and S
1
are used for the sum of
squares before and after the update. They have to be
finally divided by n to get the variance. Thus, x
n
is
the incrementally added sample and x
0
is the sample,
which has to be forgotten in each calculation step.
There are two equivalent formulas for the mean:
m
1
=m
0
+
1
n
(x
n
− x
0
) and (1)
m
1
n =m
0
n+ (x
n
− x
0
)
To avoid rounding errors, induced by division, the
sum of samples nm
1
should be used for calculations.
If the real mean is needed, this value has to be di-
vided by the constant n. Next, the update formula for
the mean 1 can be used to calculate the variance S
1
:
S
0
=
n−1
∑
i=0
(x
i
− m
0
)
2
(2)
S
1
=
n
∑
i=1
(x
i
− m
1
)
2
= −(x
0
− m
1
)
2
+ (x
n
− m
1
)
2
+
n−1
∑
i=0
(x
i
− m
1
)
2
.
(3)
The last part of equation (3) is separately transformed
using the equations (2) and (1):
n−1
∑
i=0
(x
i
− m
1
)
2
=
n−1
∑
i=0

(x
i
− m
0
) −
1
n
(x
n
− x
0
)

2
(4)
=
n−1
∑
i=0

(x
i
− m
0
)
2
−
2
n
(x
n
− x
0
)(x
i
− m
0
) +
(x
n
− x
0
)
2
n
2

(5)
=S
0
− 2(x
n
− x
0
)
1
n
n−1
∑
i=0
x
i
−
1
n
n−1
∑
i=0
m
0
!
+
1
n
n−1
∑
i=0
(x
n
− x
0
)
2
n
(6)
=S
0
− 2(x
n
− x
0
)(m
0
− m
0
) +
(x
n
− x
0
)
2
n
(7)
=S
0
+
(x
n
− x
0
)
2
n
. (8)
The result is now substituted into equation (3) and m
1
is replaced using equation (1):
S
1
=S
0
+
(x
n
− x
0
)
2
n
− (x
0
− m
1
)
2
+ (x
n
− m
1
)
2
(9)
=S
0
+
(x
n
− x
0
)
2
n
− (x
2
0
− 2m
1
x
0
+ m
2
1
) +(x
2
n
− 2m
1
x
n
+ m
2
1
) (10)
=S
0
+
(x
n
− x
0
)
2
n
+ (x
2
n
− x
2
0
) − 2m
1
(x
n
− x
0
) (11)
=S
0
+ (x
n
− x
0
)

x
n
− x
0
n
+ (x
n
+ x
0
) − 2m
1

(12)
=S
0
+ (x
n
− x
0
)

n+1
n
x
n
+
n− 1
n
x
0
− 2m
1

(13)
=S
0
+ (x
n
− x
0
)

n+1
n
x
n
+
n− 1
n
x
0
− 2m
0
−2
1
n
(x
n
− x
0
)

(14)
=S
0
+ (x
n
− x
0
)

n− 1
n
x
n
+
n+1
n
x
0
− 2m
0

.
(15)
NEUROTECHNIX2013-InternationalCongressonNeurotechnology,ElectronicsandInformatics
42

Finally the formula is multiplied with the constant n:
nS
1
= nS
0
+(x
n
− x
0
)((n− 1)x
n
+ (n+ 1)x
0
− 2nm
0
).
(16)
For the calculation nS
0
and nm
0
as well as the last
n samples have to be stored. Storing these samples
might be also required for other signal processing
steps like baseline correction, normalization and ar-
tifact correction. If the used samples are integers,
the update can be calculated accurately without any
rounding errors. This is usually the case for EEG
and EMG signal processing. For the calculation with
floating point or fixed point numbers, the multiplica-
tion with (x
n
− x
0
) might be problematic. To finally
divide by n or n
2
is not problematic, for two reasons:
First of all, for EMG movement detection, we use the
variance as a filter, hence, it does not matter if it is
scaled by n
2
and second we fixed n and therefore can
calculate
1
n
and
1
n
2
beforehand with the required accu-
racy.
3 EMPIRICAL RESULTS
In this section, we compare the proposed method and
the naiveway of calculating the variance for a window
of fixed size with n samples. In the proposed formula,
the complexity is O(k) where k is the size of the time
series. Compared to O(nk) for the naive way, here the
complexity also depends on the window size since the
variances have to be completely recalculated for each
new sample.
In order to verify this and to point out the problem
with the naive way, we performed tests on artificial
data. The data, passed to the two algorithms, was a
sinusoidal wave with a frequency of 5 Hz sampled
with 5 kHz for 1 minute (300000 samples). With both
approaches, the moving variance for each window us-
ing 8 different sizes (2, 10, 50, 100, 150, 200, 250, 300
samples) was calculated with 10 repetitions for each
window size. The total execution time was measured.
Results are shown in Table 1.
All calculations were performed on an Apple
iMac with an Intel Core i5 CPU @2.7 GHz and 16 GB
of RAM running Mac OS X 10.7.5. The implemen-
tation was performed in Python using version 2.6.8.
The results show that for small window sizes up to
10 samples, the naive calculation of the variance out-
performs the proposed method. With increasing win-
dow sizes, the computational time needed increases,
too. For a size of 300samples the time exceeds the
length of the time series by 10.6s making an online
processing with this approach impossible. For the
proposed method, the results show that its calculation
time is independent of the window sizes.
4 MOVEMENT PREDICTION
WITH EMG
In a previous study, we could show that EMG sig-
nals can be used to reliably predict voluntary move-
ments of the right human arm. We recorded EMG
data from the right arm of 8 healthy, right-handed and
normal or corrected-to-normalvision male subjects of
age 26.0 ± 3.3. For each subject 6 runs containing
40 movements each were recorded. We investigated
two movements speeds, slow and fast, and therefore 3
sets each, were recorded for every subject. For further
details on the experiments please refer to (Tabie and
Kirchner, 2013). The variance was used as non linear
filter for preprocessing the EMG signals. The prepro-
cessed signals were passed to an adaptive threshold
for the detection of movement onsets. The adaptive
threshold is given as
T
n
(t) = µ
n
+ pσ
n
, (17)
where µ and σ are the mean and the standard devia-
tion, for a window of n samples ending at time point t,
respectively, and p is a sensitivity factor. The variance
filter was calculated similar
V
m
(t) = σ
m
2
, (18)
where σ
2
m
is the variance for a window of m samples
ending at time point t. The window sizes where op-
timized and resulted in values n = 20000 (adaptive
threshold) and m = 100 (variance filter).
We analyzed the needed time for each run of this
study for the here presented approach and the naive
one. The results are shown in Table 2. The du-
rations for each run varied in a range from 5.3 to
12.3minutes. The needed computational times for
our approach varied in range from 1.0 to 2.3 minutes
and for the naive one from 404.8 to 944.3minutes
(≈ 6.7 to 15.7 hours). Thus, the naive approach is
not capable of online EMG processing for movement
prediction in contrast to our approach. The long dura-
tions for the naive approach can be explained by the
large window size (20000 samples) for the standard
deviation in the adaptive threshold.
The presented results were calculated on the basis
of results explained in Section 3. From those results
we derived a formula to calculate the computational
time for each method based on the window size and
the length of a set. Even though the standard devia-
tion used in the adaptive threshold is not exactly the
variance its calculation is basically the same since it
is the square root of the variance. Therefore, it is fea-
sible to determine the needed time by looking at the
time of the variance calculation.
MemoryandProcessingEfficientFormulaforMovingVarianceCalculationinEEGandEMGSignalProcessing
43

Table 1: Execution time of the two methods for different window sizes. Time is given in seconds.
window size 2 10 50 100 150 200 250 300
time of our approach 5.5 5.5 5.6 5.5 5.5 5.5 5.5 5.6
standard deviation 0.03 0.02 0.1 0.08 0.05 0.05 0.04 0.07
time of naive approach 3.1 4.9 14.3 25.1 36.6 47.4 59.2 70.6
standard deviation 0.03 0.03 0.4 0.2 0.2 0.2 0.9 0.8
Table 2: Duration of sets (Run) recorded in (Tabie and Kirchner, 2013), and needed processing time with the presented
approach (Our) and the naive approach (Naive) in minutes, respectively.
Set Subject 1 2 3 4 5 6 7 8
1 slow Run 8.2 8.2 9.7 9.3 8.2 10.7 7.9 8.2
Our 1.5 1.5 1.8 1.7 1.5 2 1.5 1.5
Naive 629.1 631 745 711.2 632.2 823.3 608.5 627.6
2 slow Run 7.7 7.4 9.5 8.5 8.2 9.1 8.1 7.9
Our 1.4 1.4 1.7 1.6 1.5 1.7 1.5 1.4
Naive 589.6 571 726.9 649.2 632.4 699.3 618.3 602.6
3 slow Run 7.1 7.3 9.4 9.2 7.7 7.9 7.4 7.7
Our 1.3 1.3 1.7 1.7 1.4 1.5 1.4 1.4
Naive 544 558.7 720 703.8 591.2 609 569.5 589.6
1 fast Run 6.4 5.6 6.2 5.9 5.4 6.9 8.7 5.4
Our 1.2 1 1.1 1.1 1 1.3 1.6 1
Naive 488.7 426.1 475.9 454.9 414.3 527.4 664.4 414
2 fast Run 6.3 5.9 6.7 11.3 4.9 6.6 12.3 6.6
Our 1.2 1.1 1.2 2 0.9 1.2 2.3 1.2
Naive 481 448.9 515.1 865.2 373.8 506.8 944.3 505
3 fast Run 5.3 5.6 6.7 8.2 5.3 6.5 11.1 5.9
Our 1 1 1.2 1.5 1 1.2 2.1 1.1
Naive 410.1 428.8 511.9 627.7 404.8 500.9 852.8 455.9
5 CONCLUSIONS
In this paper, we motivated the necessity of moving
mean and moving variance calculations for realtime
muscle movement onset detection using EMG signals
and for EEG data normalization. Formulas for an effi-
cient calculation of mean and variance are derived in
detail. The advantage of these new equations is shown
in comparison with the naive approach. The result-
ing algorithm works in realtime and could be simply
integrated into an FPGA to allow mobile adaptation
of HMI systems by online analysis of physiological
data. Furthermore, it was proven that the moving vari-
ance, when applied as a filter for EMG data, enables
a very fast online movement detection. If there are
other applications needing higher central moments
(e.g., skewness), the same calculation scheme could
be applied. This would result in update formulas with
higher polynomial order.
ACKNOWLEDGEMENTS
This work was funded by the Federal Ministry of Eco-
nomics and Technology (BMWi, grant no. 50 RA
1012 and 50 RA 1011).
REFERENCES
Chan, T. F., Golub, G. H., and LeVeque, R. J. (1983). Al-
gorithms for computing the sample variance: Analy-
sis and recommendations. The American Statistician,
37(3):pp. 242–247.
Folgheraiter, M., Jordan, M., Straube, S., Seeland, A., Kim,
S. K., and Kirchner, E. A. (2012). Measuring the im-
provement of the interaction comfort of a wearable ex-
oskeleton. International Journal of Social Robotics.
Kirchner, E. A., Albiez, J., Seeland, A., Jordan, M., and
Kirchner, F. (2013). Towards assistive robotics for
home rehabilitation. In Chimeno, M. F., Sol´e-Casals,
J., Fred, A., and Gamboa, H., editors, In Proceedings
of the 6th International Conference on Biomedical
NEUROTECHNIX2013-InternationalCongressonNeurotechnology,ElectronicsandInformatics
44

Electronics and Devices (BIODEVICES-13), pages
168–177, Barcelona. SciTePress.
Kohlmorgen, J., Dornhege, G., Braun, M., Blankertz, B.,
Mueller, K.-R., Curio, G., Hagemann, K., Bruns, A.,
Schrauf, M., and Kincses, W. (2007). Improving
human performance in a real operating environment
through real-time mental workload detection, pages
409–422. MIT Press.
Pfurtscheller, G., Allison, B., Brunner, C., Bauernfeind,
G., Solis-Escalante, T., Scherer, R., Zander, T. O.,
Mueller-Putz, G., Neuper, C., and Birbaumer, N.
(2010). The hybrid BCI. Front. Neurosci., 4(30):1–
11.
Tabie, M. and Kirchner, E. A. (2013). EMG onset detection
- comparison of different methods for a movement
prediction task based on EMG. In Proceedings of the
International Conference on Bio-inspired Systems and
Signal Processing, pages 242–247. SciTePress.
Welford, B. P. (1962). Note on a method for calculating cor-
rected sums of squares and products. Technometrics,
4(3):pp. 419–420.
West, D. H. D. (1979). Updating mean and variance
estimates: an improved method. Commun. ACM,
22(9):532–535.
W¨ohrle, H., Teiwes, J., Kirchner, E. A., and Kirchner,
F. (2013). A framework for high performance em-
bedded signal processing and classification of psy-
chophysiological data. In APCBEE Procedia. Inter-
national Conference on Biomedical Engineering and
Technology (ICBET-2013), 4th, May 19-20, Kopen-
hagen, Denmark. Elsevier.
Zander, T. O., Kothe, C., Jatzev, S., and Gaertner, M.
(2010). Enhancing human-computer interaction with
input from active and passive brain-computer inter-
faces. BrainComputer Interfaces, pages 181–199.
MemoryandProcessingEfficientFormulaforMovingVarianceCalculationinEEGandEMGSignalProcessing
45