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Effect of parametric variation of center frequency and bandwidth of morlet wavelet
transform on time-frequency analysis of event-related potentials
© Springer Nature Singapore Pte Ltd. 2018
Accepted version (Final draft)
Zhang, Guanghui; Tian, Lili; Chen, Huaming; Li, Peng; Ristaniemi, Tapani; Wang,
Huili; Li, Hong; Chen, Hongjun; Cong, Fengyu
Zhang, G., Tian, L., Chen, H., Li, P., Ristaniemi, T., Wang, H., Li, H., Chen, H., & Cong, F. (2017).
Effect of parametric variation of center frequency and bandwidth of morlet wavelet transform
on time-frequency analysis of event-related potentials. In Y. Jia, J. Du, & W. Zhang (Eds.), CISC
2017 : Proceedings of 2017 Chinese Intelligent Systems Conference (pp. 693-702). Springer
Nature Singapore Pte Ltd.. Lecture Notes in Electrical Engineering, 459.
https://doi.org/10.1007/978-981-10-6496-8_63
2017
Effect of Parametric Variation of Center Frequency and
Bandwidth of Morlet Wavelet Transform on Time-
frequency Analysis of Event-related Potentials
Guanghui Zhang
1,2
, Lili Tian
3
, Huaming Chen
1
, Peng Li
4
, Tapani Ristaniemi
2
,
Huili Wang
3
, Hong Li
4
, Hongjun Chen
3
, Fengyu Cong
1,2
Abstract. Time-frequency (TF) analysis of event-related potentials (ERPs) using
Complex Morlet Wavelet Transform has been widely applied in cognitive
neuroscience research. It has been widely suggested that the center frequency (fc) and
bandwidth (σ) should be considered in defining the mother wavelet. However, the
issue how parametric variation of fc and σ of Morlet wavelet transform exerts
influence on ERPs time-frequency results has not been extensively discussed in
previous research. The current study, through adopting the method of Complex
Morlet Continuous Wavelet Transform (CMCWT), aims to investigate whether time-
frequency results vary with different parametric settings of fc and σ. Besides, the
nonnegative canonical polyadic decomposition (NCPD) is used to further confirm the
differences manifested in time-frequency results. Results showed that different
parametric settings may result in divergent time-frequency results, including the
corresponding time-frequency representation (TFR) and topographical distribution.
Furthermore, no similar components of interest were obtained from different TFR
results by NCPD. The current research, through highlighting the importance of
parametric setting in time-frequency analysis of ERP data, suggests that different
parameters should be attempted in order to get optimal time-frequency results.
Keywords: Complex Morlet Wavelet Transform, event-related potentials, center
frequency, bandwidth, time-frequency representation.
Guanghui Zhang, Lili Tian, Huaming Chen, Peng Li, Tapani Ristaniemi, Huili Wang, Hong Li,
Hongjun Chen (), Fengyu Cong()
1.Department of Biomedical Engineering, Faculty of Electronic Information and Electrical
Engineering, Dalian University of Technology, 116024, Dalian, China.
E-mail: cong@dlut.edu.cn
2. Department of Mathematical Information Technology, University of Jyväskylä, 40014, Jyväskylä,
Finland
3. School of Foreign Languages, Dalian University of Technology, 116024, Dalian, China
E-mail: chenhj@dlut.edu.cn
4. College of Psychology and Sociology, Shenzhen University, 518060, Shenzhen, China
2
1 Introduction
Electroencephalogram (EEG) has been extensively applied in cognitive neuroscience
research. EEG, according to different experimental paradigms and external stimuli,
can be divided into three categories: spontaneous EEG [1], event-related potentials
(ERP) [2], and ongoing EEG [3]. The main methods employed in ERP data
processing are as the following: 1) Time-domain analysis, 2) Frequency-domain
analysis and 3) Time-frequency analysis [4-8]. As ERP signals are non-stationary and
time-varying, neither the time-domain nor the frequency-domain analysis can be used
to effectively reveal the time-frequency information of ERP data. Time-frequency
analysis, by focusing on the time-varying features of ERP components, is conducted
to transform a one-dimensional time signal into a two-dimensional time-frequency
density function, which aims to reveal the number of frequency components and how
each component varies over time.
In 1996, Tallon-Baudry et al. introduced the Morlet wavelet for time-frequency
analysis of ERP data [9]. Since then, the Morlet wavelet has been widely applied by
researchers in conducting time-frequency analysis, with its citations over 1100 times
(From the Google scholar). However, a synthesis of previous research showed that in
most cases the value of 𝐾 is fixed (e.g.,𝐾 = 7) [9-12], therefore leaving the issue
whether parametric variation of fc and σ has an impact on time-frequency results
unresolved. This study is devoted to investigation of the issue.
2 Method
2.1 Data Description
The data was collected to investigate whether a short delay in presenting an outcome
affects brain activity. For the detailed information of experimental procedure, readers
can refer to Wang et al. research [13]. Twenty-two undergraduates and graduate
students participated in the experiment as volunteers. All the participants, aged from
18 to 24, were right-handed with normal or corrected-to-normal vision and no one
was reported to have neurological or psychological disorders. EEG was recorded
using a 64-channel system (Brian Products GmbH, Gilching, Germany) with
reference on the left mastoid. The vertical and horizontal electrooculogram (EOG)
was recorded from electrodes placed above and below the right eye and on the outer
canthi of the left and right eyes respectively. Electrode impedance was maintained
below 10k Ohm. The EEG and EOG were sampled continuously at 500Hz with 0.01-
100Hz bandpass filtering.
2.2 Complex Morlet Wavelet Transform
The CMCWT method, based on the Complex Morlet Wavelets, was adopted for time-
frequency analysis in the present study.
3
If x(t) is a discrete sequence of length T, the definition of the Continuous Wavelet
Transform (CWT) can be expressed as follows:
𝑋
(
𝑎,𝑏
)
=
1
|
𝑎
|
𝑥
(
𝑡
)
𝛷
𝑡 −𝑏
𝑎
.
(1)
In the above formula, x(t) represents the signal to be transformed; a refers to the
scaling and b the time location or shifting parameters; 𝛷
(
𝑡
)
stands for the mother
wavelet. In this study, the Complex Morlet Wavelets is defined as the mother wavelet
[9-12]:
𝛷
(
𝑡,𝑓𝑐
)
=
1
√
𝜋𝜎
𝑒
𝑒
. (2)
According to the above formula, a Gaussian shape respectively in the time and
frequency domain around its 𝑓
can be obtained.
A wavelet family is characterized by a constant ratio:
𝐾 =
𝑓
𝜎
= 2𝜋𝜎𝑓
. (3)
In this formula, 𝜎
=
1
2𝜋𝜎
, K should be greater than 5 [9].
Taken together, this method (CMCWT) can be described as below:
CMCWT
(
𝑡,𝑓
)
=
|
𝛷
(
𝑡,𝑓
)
∗ 𝑥
(
𝑡
)|
. (4)
In the above formula, ‘*’ refers to convolution.
2.3 Nonnegative Canonical Polyadic Decomposition
Nonnegative Canonical Polyadic Decomposition (NCPD) has been widely applied to
study time-frequency representation (TFR) of EEG [14, 15]. For example, given a
third-order tensor including the modes of time, frequency and space, X ∈ ℛ
×
×
,
the NCPD can be defined:
X = 𝑡
∘ 𝑓
∘ 𝑠
+ 𝐸 = 𝑋
+ 𝐸 = X
+ 𝐸 ≈ X
.
(
5
)
In this formula, the symbol ‘∘’ denotes the outer product of vectors. The 𝑡
, 𝑓
, and 𝑠
correspond to the temporal component ⋕ 𝑟, the spectral component ⋕ 𝑟, and the
spatial component ⋕ 𝑟, and the three components reveal the properties of the multi-
domain properties of an ERP in the time, frequency and space domains [14].
For the same multi-channel EEG data, different parameters of CMCWT may
produce different TFR (indeed, third-order tensors in this study) in terms of visual
inspection. Then, the application of NCPD on those tensors can assist to investigate
whether the similar components of interest can be extracted from different tensors
resulting different TFR parameters of the same EEG data. For the detailed
4
information of the number of extracted components for each mode, and the criteria of
selecting multi-domain features, readers can refer to Cong et al. research [15].
3 Data Processing and Analysis
The ERP data were pre-processed in MATLAB and EEGLAB [16], including the
following steps: a 50Hz notch filter to remove line noise, a low-pass filtering of
100Hz, segmentation of the filtered continuous EEG into single trials (each trial was
extracted offline from 200ms pre-stimulus onset to 1000ms post-stimulus onset),
baseline correction, artifact rejection and averaging.
In CMCWT analysis, the frequency range was set from 1 to 30Hz, respectively
in 0.1 Hz step ( fc = 9,10, respectively), in 0.2Hz step ( fc = 5,6,7,8,9,10,
respectively), in 0.3Hz step ( fc = 3,4,5,6,7,8,9,10, respectively), in 0.4Hz step ( fc =
3,4,5,6,7, 8,9,10, respectively), in 0.5Hz step ( fc = 2,3,4,5,6,7,8,9,10, respectively), in
0.6Hz step ( fc = 2,3, 4,5,6,7,8,9,10, respectively), in 0.7Hz ( fc = 2,3,4,5,6,7,8,910,
respectively), in 0.8Hz step ( fc = 2,3,4,5,6,7,8,9,10, respectively), in 0.9Hz step ( fc =
1,2, 3,4,5,6,7,8, 9,10, respectively) and in 1Hz step ( fc = 1,2,3,4,5,6,7,8,9,10,
respectively). All the above parametric settings met the requirement of constant ratio
(greater than 5).
To further investigate whether parametric variation of fc and σ has an impact on
time-frequency results, four steps are carried out in the following sequence:
(1) Select a typical topographical distribution of TFR results as the template.
When 𝜎
= 1, the value of fc can be respectively set as 1,2,3,4,5,6,7,8,9 and 10. The
topographical distribution of 𝑓𝑐
= 4 is finally chosen as the template
T
(𝜎
,𝑓𝑐
) in terms of the prior knowledge of the ERP of interest.
(2) Define a 𝑓𝑐
, calculate the Correlation Coefficients (CCs) between the
template (Y
) and each spatial component 𝑠
(𝜎
,𝑓𝑐
) obtained by NCPD (R
components were extracted in each mode), which can be described as:
Y
(
𝜎
,𝑓𝑐
,r
)
= 𝜌𝑠
(𝜎
,𝑓𝑐
),𝑇
(𝜎
,𝑓𝑐
). (6)
In the above formula, r = 1,2,⋯ ,21,n = 1,2,⋯,10. Subsequently, the maximal CC
is chosen as:
q
(
𝜎
,𝑓𝑐
)
= max𝑌
(
𝜎
,𝑓𝑐
,1
)
,𝑌
(
𝜎
,𝑓𝑐
,2
)
,⋯,𝑌
(
𝜎
,𝑓𝑐
,𝑅
)
. (7)
Then, the corresponding r
th
components with the maximum CC were obtained.
(3) Based on the obtained components of each dimension and their
corresponding TFR results, we need to judge whether the TFR results of different
parameters are similar or not. The TFR results are different when the fc is respectively
set as 1 and 9. Besides, the corresponding 16
th
components of 𝑓𝑐
= 1 and 1
th
components of 𝑓𝑐
= 9 are similar in the spacial dimension, but not the temporal and
spectral dimension (as shown in the first and third row of Figure 4).
(4) With the same procedure mentioned above, we can analyze the results of
other σ and fc parameters to explore potential differences in the time-frequency
results.
