Techniques for Data Augmentation and Their Impact on Long-Range
Dependence and Applications
MARYAM GHANBARI1,2, WITOLD KINSNER1, NARIMAN SEPEHRI2
1Electrical and Computer Engineering Department,
University of Manitoba,
Winnipeg, MB,
CANADA
2Mechanical Engineering Department,
University of Manitoba,
Winnipeg, MB,
CANADA
Abstract: - Data augmentation is a common approach to enhance datasets for training machine learning models.
This study employs five distinct techniques to generate augmented datasets. Furthermore, eight measures are
applied to assess datasets both before and after augmentation techniques. A critical requirement is that any
augmentation should preserve the fundamental properties of the original dataset. The study reveals that certain
augmentation methods can disrupt the long-range dependence on Internet traffic data (ITD) with distributed
denial of service (DDoS) attacks (DDoS ITD). These DDoS ITDs originate from stochastic and bursty
environments, affecting the probability mass function (PMF) and data labeling.
Key-Words: - Long-range dependence, data augmentation, Internet traffic data, speech sound data, probability
mass function, dataset evaluation measures.
Received: April 9, 2024. Revised: October 11, 2024. Accepted: November 15, 2024. Published: December 23, 2024.
1 Introduction
This paper presents a study of data augmentation
when it is applied to processes with long-range
dependence to show that some augmentation
techniques may alter or even destroy the long-range
dependence. This paper is an extension of our
previous work presented at the IEEE International
Conference on Cognitive Informatics and Cognitive
Computing [1] to develop the foundation for our
research in this area. In this extended paper, we
delve deeper into the applications of long-range
dependence and include a more comprehensive set
of case studies.
Data augmentation is a commonly employed
method for enlarging and diversifying datasets
during the training of neural networks. The
advantage of data augmentation has been
demonstrated across various domains, including
image classification and data modeling, facilitating
the expansion of training samples, [2]. Bjerrum et
al. applied extended multiplicative scattering
(EMSC) to correct the datasets, as well as a spectral
data augmentation method to augment the datasets
using random variations in slope and offset, [3].
They used convolutional neural networks to extract
features from the data, and they demonstrated that
the combination of data augmentation and EMSC
was the best preprocessing method to enhance test
results.
A data augmentation technique is correct and
useful when the constraints of the dataset are
preserved. For example, by augmenting an image
dataset or an audio dataset, the properties of the
original and augmented dataset remain the same,
[2]. The principal requirement of any data
augmentation technique is that it should not alter the
constraints of a dataset. For example, when an
augmentation technique modifies the data
probability mass function (PMF) of an Internet
dataset, the properties of the dataset are modified.
Therefore, since the long-range dependence of the
Internet dataset is altered, the augmentation
technique is unsuitable.
This paper examines data augmentation for two
distinct categories of datasets: (i) Internet traffic
data with distributed denial of service (DDoS)
attacks and (ii) the Manitoba Speech Dataset with
standard voice sound collection, [1]. The study
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Volume 23, 2024
demonstrates that the speech dataset constraints can
be preserved using the selected data augmentation
methods, but these methods cannot preserve the
constraints of the Internet traffic data containing the
DDoS attacks (DDoS ITD), given its stochastic
nature. Techniques such as adding noise, mirroring,
squeezing, and expanding the DDoS ITD alter the
data's shape and PMF, indicating that conventional
augmentation methods may not be suitable for
DDoS ITD. Mono-scale, multi-scale, and poly-scale
measures are employed to assess the sensitivity of
these techniques to various factors by analyzing the
time series datasets before and after augmentation.
The structure of this paper is outlined as
follows. Section 2 provides a summary of the
datasets. Section 3 explains the methods of data
augmentation. Section 4 introduces the measures
utilized for dataset analysis. Sections 5 and 6
present the simulation outcomes and provide an
explanation and discussion of the results,
respectively. Section 7 discusses the potential
applications. Section 8 introduces an extension of
the research. Section 9 offers concluding remarks.
2 Description of the Datasets
This study uses two types of time series data sets.
Each complete dataset is referred to as an epoch
(TE), with individual stationary segments within
each epoch designated as a frame. The word
"window" is not used in this paper as it is reserved
to signify frame data modifications at the edges of
the frame through a window function (also known
as a tapering or apodization function) such as the
Hamming and Hann windows, often employed in
signal processing.
In this paper, we consider only discrete signals
that have been sampled from continuous (analog)
signals properly to represent the original signal. The
sampling frequency, fs, is adequate if it is strictly
above the Nyquist frequency, fs > fN = 2fc. For
narrowband signals, the critical cutoff frequency, fc,
is defined at the 3-dB drop in the log-log plot of the
frequency response of the signal. For broadband
signals originating from many self-affine processes,
the frequency is defined at a higher value, fc = fh,
where the log-log response reaches the noise level
of the signal.
2.1 Internet Traffic Data
The dataset employed for training and testing
purposes in this study was sourced from the trusted
Center for Applied Internet Data Analysis (CAIDA).
CAIDA collects a diverse range of real-time
network traffic from various parts of the world in
collaboration with research organizations,
governments, and commercial entities without
revealing their identities, [4].
The CAIDA’s 2007 DDoS attack traffic was
used as network traffic data and contains TCP, UDP
Flood, SYN Flood, and ICMP (Ping) Flood packets.
A packet in network traffic data has the following
features: source IP address, destination IP address,
packet arrival time, packet length, and protocol. The
critical attribute called the packet arrival time series
signal is computed by calculating the difference
between a packet’s arrival time at t and its arrival
time at t-1. The distribution of packets with a
duration of 0.1 ms within each frame is illustrated in
Figure 1.
Fig. 1: The number of packets within a stationary
frame size
2.2 Speech Sound Data
For comparison with the CAIDA dataset, the
Manitoba Speech Dataset was utilized, obtained
from the University of Manitoba, [1]. This dataset
contains recordings of 44 words spoken by 12
female and 12 male volunteers. This study uses the
word “test” epoch as a speech sound dataset.
Figure 2 illustrates the plot of the "test" dataset,
sampled at a rate of 44.1 kilo samples per second
(kSps).
Fig. 2: Plot of the dataset for the word “test”
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3 Data Augmentation Techniques
This study employs the following five distinct
techniques to generate six augmented sets.
3.1 Mirroring Technique
The mirroring technique, also known as flipping,
involves reversing the order of the time series data
within a stationary frame. In this form of horizontal
flipping, the first sample in the series becomes the
last, while the last sample becomes the first. This
transformation effectively reverses the temporal
order, introducing a new perspective of the data
without altering the underlying magnitude or
amplitude patterns.
3.2 Time Stretching Technique
Time stretching is a resampling method that aims to
alter the temporal resolution of the time series data.
In this technique, the original time series is
upsampled by inserting additional data points
between existing samples. The new points are
generated by calculating the average of neighboring
values and placing them at even positions within the
data. Meanwhile, the existing points are shifted to
the right, increasing the overall length of the time
series. As illustrated in Figure 3, time stretching
preserves the general shape and trends of the
original signal.
Fig. 3: The upsampling technique applied to a time
series dataset
3.3 Squeezing Technique
The squeezing technique reduces the data resolution
by removing samples to downsample the time
series. It is an effective approach for minimizing the
size of the dataset while retaining key patterns and
trends. This technique can be accomplished in the
following two ways:
3.3.1 Downsampling
This technique selects data points located at odd
positions within the time series, as illustrated in
Figure 4.
Fig. 4: Downsampling technique applied to a time
series dataset, [1]
3.3.2 Wavelet Approximation Coefficients
Using the Daubechies wavelet transform 2 (db2)
basis function, this technique downsamples the data
by extracting approximate coefficients. These
coefficients are then downsampled within a
stationary frame, as illustrated in Figure 5.
Fig. 5: Downsampling technique applied to a time
series dataset using the approximate coefficients of
the dataset
3.4 Random Cut-and-Paste Technique
The random cut-and-paste technique generates
augmented data by randomly selecting a segment of
the original time series and appending it to the end.
This operation disrupts the natural order of the time
series, creating a synthetic sequence that may
combine different temporal patterns. By altering the
original data flow, this technique introduces new
transitions and relationships between segments,
challenging the model to learn more complex
temporal dependencies. The cut-and-paste approach
is useful when the dataset is limited, as it can create
a wide range of augmented samples from a single
time series.
3.5 Adding White Noise Technique
The addition of white noise involves injecting a
random noise component into the time series,
effectively creating a perturbed version of the
original data. The white noise is characterized by a
normal distribution with a mean of zero and a small
variance, typically set to 0.0001, to ensure minimal
distortion. This technique simulates random
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fluctuations and measurement errors that may occur
in real-world data.
4 Selecting Measures for the Analysis
of Datasets
In order to assess the sensitivity of the five
techniques to various factors, seven mono-scale,
multi-scale, and poly-scale measures are employed
to analyze the time series of the two datasets as
described below.
4.1 The Hurst Exponent Measure
The Hurst exponent (H) serves as a metric
(measure) for quantifying the degree of long-range
dependence present in a time series [5], thereby
detecting its existence. Additionally, the Hurst
exponent measures the smoothness of self-affine
processes. It ranges between 0 and 1, where
different values convey distinct characteristics of the
time series, [5]. As the H value approaches 1, the
level of persistence or long-range dependence
(LRD) increases. This implies that the signal's
behavior at a given time can be influenced by its
past values, a concept explained later in this paper.
A Hurst exponent close to 0.5 suggests a completely
random process or Brownian motion where there is
no LRD. In this case, the time series exhibits no
correlation, and the past values do not influence
future data points. Such behavior is typical of white
noise and purely stochastic processes, [5]. On the
other hand, an H value less than 0.5 indicates an
anti-persistence or strong negative correlation. In
this scenario, if the time series has an increasing
trend, it is likely to reverse and start decreasing, and
vice versa.
An N-dimensional object (a 2D signal in our
case) is said to be self-affine if its smaller fragments
are scaled-down versions of the entire object and if
the scaling factors are different in the N dimensions.
If the scaling factors are the same, the object is
called self-similar, [6]. Self-affinity refers to the
persistence of fractal patterns across various scales
of observation, while self-similarity denotes uniform
scaling behavior across all dimensions.
Self-affinity and long-range dependence in a
signal means that the signal not only exhibits fractal
patterns at different scales but also these patterns are
strongly correlated (or dependent) over an extended
period of time. These relationships can be
encountered in various real-world phenomena, such
as financial time series and Internet traffic, where
bursty operations and packet flows display similar
behavior. The ability to detect and measure these
patterns using the Hurst exponent is crucial for
analyzing the predictability and underlying
dynamics of time series data.
4.2 The Variance Fractal Dimension
Trajectory Measure
The variance (the second moment) of a self-affine
signal can be used to measure the power-law
behavior of the signal. To measure and analyze the
complexity of such broadband self-affine time series
signals (often characterized by the signal
stochasticity, non-stationarity, non-differentiability,
dynamic behavior, and long-range dependence), the
variance fractal dimension trajectory (VFDT)
measure has been proposed, [6]. The VFDT has
been refined to consider all data points, including
boundary points of time series signals, not solely
marginal points, [6].
When using the VFD with a time series, the
variance of the amplitude increments over each time
increment follows a power law relation with that
time increment, as given by [6].
2
2 1 2 1
var[ ( ) ( )] ~| | H
A t A t t t
(1)
where var denotes the variance function (the second
moment), A is the time series signal, H is the Hurst
exponent, and it represents the discrete time in a
discrete signal. As explained, the Hurst exponent
indicates the self-affine characteristics of the signal,
if any. Taking the logarithm of both sides of Eq. (1),
the Hurst exponent is calculated from a log-log plot
and is given by [6].
2
02
log [var( ) ]
1()
lim 2 log
n
n
A
Hn

(2)
where Δn denotes the scale in a discrete-time series
signal at which the variance is evaluated. Equation
(2) is used to analyze a time series signal in the time
domain by calculating the expansion of the time
series signal amplitude at different scales through its
variance [6].
The output of the VFD, denoted by
D
is used
as a measure of signal complexity and is obtained
from [6].
1D E H
(3)
where E is the Euclidean dimension and is E = 1 for
time series signals. Detailed descriptions of the real-
time VFDT algorithm and the second version of the
VFDT algorithm can be found in [6].
To validate the VFDT algorithm, the theoretical
VFDT value for white noise is
D
=2, while
D
=1
for a straight line, [6]. Thus, the VFDT values of a
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self-affine time series signal are bounded between
the two values. An example of the engineering
application of the H and the VFD is to facilitate
anomaly detection in Internet time series traffic.
4.3 Spectral Fractal Dimension Measure
The spectral fractal dimension (SFD) transforms a
self-affine time series self-affine signal into its
power spectrum density [6]. Operating in the
frequency domain [1], the SFD analyzes the
properties of frequencies underlying the time series
signals, [6]. This spectrum discovers inherent
characteristics of the time series frequencies [6]. For
the fractal time series signal, the power spectrum
density follows the spectral power law, as shown in
Eq. (4), [6].
1
( ) ~Pf f
(4)
where
is the power spectrum exponent. Taking
the logarithm of both sides of Eq. (4) yields a linear
relationship between the power spectrum and its
frequency, represented by a regression line. The
slope of this line, denoted by
, serves as the power
spectrum exponent of the DDoS ITD, as illustrated
in Figure 6.
Fig. 6: The power density spectrum of the DDoS
ITD and its slope
In Figure 6, the power density spectrum of the
DDoS ITD is plotted on a log-log scale. The figure
demonstrates the linear trend in the log-log plot,
with the slope
representing the power spectrum
exponent of the DDoS traffic. This exponent
signifies the self-affine relationship of the time
series. A slope of zero (
=0) indicates white noise,
suggesting no correlation within the time series and
implying purely random behavior. Consequently, an
increasing slope indicates an increasing correlation.
The analysis of the power density spectrum in
Figure 6 indicates strong long-range dependence
and persistent behavior. This observation is
consistent with the nature of DDoS attacks, where
packet bursts and correlated traffic patterns are
common due to the nature of the DDoS attack.
Complex self-affine time series, such as the
Internet time series data, may exhibit multiple
values due to their multifractal nature. The SFD
output, serving as a measure of signal complexity
denoted by
D
, is obtained from [6].
(3 )
2
DE

(5)
where E is the Euclidean embedding dimension, as
defined in Eq. (3). This is due to the following
relationship between the Hurst exponent and the
power spectrum exponent [6].
21H

(6)
4.4 Autocorrelation Function Measure
Correlation assesses both the strength and direction
of the linear relationship between two variables, [7].
Autocorrelation, on the other hand, approximates
the similarity between data points separated by
successive time intervals within a time series signal,
[8]. Equation (7a) presents the autocorrelation
function for an energy signal, while Eq. (7b) shows
the autocorrelation function for a power signal.
( ) ( ) ( )
xx n
r l x n x n l


(7a)
1
( ) ( ) ( )
lim 21
M
xx nM
M
r l x n x n l
M


(7b)
Autocorrelation reveals properties of a
stationary random process, [9]. The Fourier
transform is computed from a stationary time series
signal, allowing for the direct calculation of the
power density spectrum from the squared magnitude
of the Fourier transform. In the case of non-
stationary data, the Fourier transform can still be
calculated from a sequence of stationary frames.
The Fourier transform of the autocorrelation
sequence is generally valid for non-stationary data,
enabling the calculation of the power of the time
series. Consequently, the Fourier transform of the
autocorrelation sequence is interpreted as the
frequency distribution of the signal’s power,
representing the power density spectrum, [9]. For
example, analyzing autocorrelation in voice datasets
aids in pitch detection, while analyzing Internet
traffic flow datasets facilitates anomaly detection.
4.5 Long-Range Dependence Measure
In statistical analysis, dependence signifies any
relation and association between two variables.
LRD describes the extent to which a time series
signal is influenced by its past values over an
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extended period. Unlike short-term dependence,
where correlations diminish quickly, LRD implies
that significant correlations persist even as the time
lag increases, indicating memory and persistence
within the time series, [10], [11]. The magnitude of
long-range dependence is typically measured using
the autocorrelation function, which measures the
correlation of a time series with its lagged versions.
In instances where a time series signal exhibits
long-range dependence, its autocorrelation decays in
a hyperbolic manner [11], whereas signals with
short-term dependence experience exponential
decay [12]. Consequently, the autocorrelation
distribution of a time series signal with long-range
dependence tends to be more dispersed compared to
that of a time series with short-term dependence
[11], [13]. LRD is a key indicator of the
effectiveness of data augmentation. Because the
autocorrelation function decays hyperbolically, the
power density spectrum for a signal with long-term
dependence shows an increase without the
frequency ever reaching zero, [11].
Long-range dependence indicates that past
events can influence future events over an extended
period, [14]. For example, Internet traffic patterns,
particularly in scenarios like DDoS ITD, tend to be
bursty and exhibit LRD [14]. Another example is
Call Holding Time (CHT), which occurs in a
stochastic (random) environment and follows a
heavy-tailed distribution while also showing LRD.
In telecommunications, CHT, or the duration of a
phone call, varies randomly due to factors such as
user behavior, network conditions, and the type of
call. Processes characterized by long-range
dependence, such as CHT and Internet traffic, often
exhibit self-affinity. As a result of this LRD,
periods of high activity are likely to be followed by
similar high activity periods, while low activity
periods tend to be followed by low activity periods.
Both CHT and Internet traffic follow a heavy-tailed
distribution, meaning that extreme values (such as
unusually very long or very short call durations or
traffic levels) are common and support the presence
of bursty patterns.
The power spectrum exponent indicates the
presence of long-range dependence within a time
series signal [11] and can be determined using
Eq. (6). The time series represents white noise when
the slope
is zero, indicating no correlation and,
therefore, no long-range dependence. As the slope
increases, so does the correlation, subsequently
enhancing the long-range dependence of the time
series signal.
4.6 Zero Crossing (ZC) Measure
The zero crossing (ZC) measure quantifies how
many times a signal's magnitude crosses a specified
threshold value, such as zero, within a given interval
[1]. When the threshold value is set to zero, the ZC
indicates the rate of transitions between positive and
negative mathematical signs of the signal within that
interval. Zero crossing is effective in identifying
edges and sudden changes within a time series
signal and is related to the lowest frequency
component of the signal [15], which can be utilized
for feature extraction. The zero crossing is
calculated through the following equation [1], [16].
[| sgn[ ( )] sgn[ ( 1)] |] [ ]
n
ZC x m x m w n m

(8)
where
sgn[]
denotes the mathematical sign function
as given by [1], [16].
1, ( )
sgn[ ( )]
1, ( )
x n threshold
xn x n threshold


(9)
and
[]w
denotes a frame containing a stationary
segment of a time series signal, defined by [16].
1, 0 1
[] 2
0,
nN
wn N
otherwise
(10)
This measure is computed in the time domain
and can be performed in real-time. The ZC rate is
useful for distinguishing speech from noise and for
determining the start and the end of speech
segments, [16].
4.7 Turns Count (TC) Measure
The turns count (TC) is used to extract features from
a time series signal based on changes in slope
direction rather than zero crossings, [15]. A turn is
counted each time the slope of the signal changes its
sign, [1], [15]. This technique evaluates signals by
identifying the number of spikes present within the
signal [15] and is related to the highest frequency
component of the signal in the frame. A turn is
computed from [1].
&( ) ( 1) ( 1) ( 2)
i
tr x n x n x n x n
(11a)
&( ) ( 1) ( 1) ( 2)
i
tr x n x n x n x n
(11b)
where
i
tr
represents the turn occurring within a time
interval such as a stationary-frame interval, and x(n)
denotes an input time series signal. The total turn
count in the frame is given by [1].
1i
i
TC tr
(12)
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5 Simulation and Results
Table 1 and Table 2in Appendix present the
statistical properties and outcomes related to eight
mono-scale, multi-scale, and poly-scale assessments
of both the DDoS ITD and the “test” dataset before
and after the implemented data augmentation.
Subsequent discussions will explain the findings of
the simulated measure proposed.
5.1 Analysis of Stationarity
Weak stationarity in a dataset is characterized by the
first two moments falling within specified ranges,
typically within a 95% confidence interval, [6]. The
DDoS ITD demonstrated weak stationarity, as
indicated by the trajectory of the mean falling within
the range of 0.000782 to 0.003041 and the variance
trajectory within the range of 1.64E–06 to 8.21E–
06, given a minimum frame size of 8192. The
skewness trajectory ranged from 1.076 to 2.94,
while the kurtosis trajectory remained at 1.8. The
trajectories of the mean and variance are presented
in Figure 7, while the trajectories of skewness and
kurtosis are illustrated in Figure 8.
Recall that the mean represents the typical value
of a set of data, while variance quantifies how
spread out the data is from this mean, [7]. Skewness
signals if the distribution leans towards the right or
left, [17]. Kurtosis measures the thickness of the
distribution's tail, with positive values indicating
distributions with heavier tails.
Fig. 7: The trajectories of the mean and variance
within a stationary frame of samples related to the
DDoS ITD
Fig. 8: The trajectories of skewness and kurtosis
within a stationary frame of samples related to the
DDoS ITD
The “test” speech dataset exhibited weak
stationarity due to the minimum window size of
512, resulting in the mean trajectory ranging from–
0.0087 to 0.0049 and the variance trajectory ranging
from 2.48E–06 to 0.0091. Similarly, the skewness
trajectory ranged from –2.09 to 1.55, and the
kurtosis trajectory ranged from 1.56 to 31.03. The
trajectories of the mean and variance are presented
in Figure 9, while the trajectories of skewness and
kurtosis are illustrated in Figure 10.
Fig. 9: The trajectories of mean and variance within
a stationary frame of samples related to the “test”
dataset
Fig. 10: The trajectories of skewness and kurtosis
within a stationary frame of samples related to the
“test” dataset
5.2 Probability Mass Function Analysis
To fit a probability distribution to the time series of
both datasets, the maximum likelihood estimation
method and polynomial regression are employed,
[18].
For the DDoS ITD, the probability mass function
(PMF) reveals a Lévy distribution, where the
location parameter (µ) is 0.0019223631, and the
scale parameter (σ) is –0.0004226877 as given by:
2( )
3/ 2
( ) . , , 0
2 ( )
x
e
f x x
x


(13)
These parameters are determined through
maximum likelihood estimation. The DDoS ITD’s
heavy tail becomes apparent when contrasting its
Lévy distribution with a normal distribution. The
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presence of self-affinity in the delay of packets
contributes to the heavy tail distribution, [13]. The
on-off queuing model, where “on” represents active
data transmission and “off” represents no
transmission, generates the traffic model. If the
durations of “on-off” periods exhibit heavy-tailed
characteristics, it leads to long-range dependence,
[5]. Figure 11 illustrates the PMF of the DDoS ITD.
Fig. 11: The Probability Mass Function (PMF) of
the DDoS ITD
The PMF of the “test” speech dataset conforms
to a distorted (skewed and flattened) normal
distribution, characterized by a mean (µ) of
0.090909091 and a standard deviation (σ) of
0.028493958. These values are obtained through
maximum likelihood estimation. Figure 12
illustrates the PMF of the “test” dataset.
Fig. 12: The Probability Mass Function (PMF) of
the “test” dataset
5.3 The Hurst Exponent Analysis
In this study, the DDoS ITD exhibited Hurst
exponent (2) values ranging from 0.0063 to 0.0289,
as illustrated in Figure 13. Similarly, the “test”
dataset demonstrated Hurst exponent values ranging
from 0.0137 to 0.8520, as illustrated in Figure 14.
Fig. 13: The trajectory of the Hurst exponent for the
DDoS ITD
Fig. 14: The trajectory of the Hurst exponent for the
“test” dataset
5.4 Variance Fractal Dimension Trajectory
Analysis
The VFD algorithm output for the DDoS ITD
ranged from 1.9711 to 1.9937 using the non-
overlapping frame version, as shown in Figure 15.
Similarly, for the “test” dataset, the VFD ranged
from 1.1480 to 1.9863, as illustrated in Figure 16.
Fig. 15: The trajectory of the VFD for the DDoS
ITD
Fig. 16: The trajectory of the VFD for the “test”
dataset
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To validate the VFD algorithm results, a
uniformly distributed white-noise time series with a
mean of zero, variance of 0.08, and an epoch size of
220 was generated. This white noise within the
same frame of 512 samples of stationary time series
data is shown in Figure 17.
Fig. 17: Uniformly distributed white noise is
illustrated within a 512-element frame
For the white noise epoch, the VFD output
using the non-overlapping frame version ranged
from 1.955 to 1.995, with an absolute error of –
0.0241 and an absolute error of –4.15%, confirming
the accuracy of the algorithm. Figure 18 illustrates
the VFDT of this white noise signal with zero
overlapping frames for the epoch.
Fig. 18: The VFDT of the uniform distribution
white noise signal
5.5 Zero Crossing and Turns Count
Analysis
In this study, the DDoS ITD shows 1250 zero-
crossing, and 5349 turns count within a frame size
of 8192, while the “test” dataset exhibits 145 zero-
crossing and 192 turns count within a frame size of
512.
5.6 Power Density Spectrum Analysis
The power density spectrum of both the DDoS ITD
and the “test” datasets are illustrated as log-log plots
in Figure 19 and Figure 20, respectively. The power
density spectrum has a slope of 1.9216 for the
DDoS ITD and 1.4727 for the “test” dataset when
plotted against frequency. However, according to
Eq. (6), the power spectrum exponent
is 0.9728
for the DDoS ITD and 1.6095 for the “test” dataset,
respectively.
Fig. 19: A log-log plot showing the power density
spectrum against frequency, along with the
corresponding slope for the DDoS ITD
Fig. 20: A log-log plot showing the power density
spectrum against frequency, along with the
corresponding slope for the “test” dataset
5.7 Autocorrelation Function Analysis
The autocorrelation for the DDoS ITD and the “test”
dataset is illustrated in Figure 21 and Figure 22,
respectively.
Fig. 21: The DDoS ITD’s autocorrelation at
different lags in the first frame
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Fig. 22: The “test” dataset’s autocorrelation at
different lags in the first frame
6 Results and Discussion
This study reveals that data augmentation affects the
constraints of the DDoS ITD, whereas the
constraints of the “test” dataset remained
unchanged.
Typically, a Lévy distribution exhibits infinite
mean and variance with undefined skewness and
kurtosis. Also, in a Lévy distribution, inflection
points vary in magnitude, making it impossible to
determine the sigma to find the variance.
Consequently, these statistical tools are inadequate
for feature extraction. However, the PMF serves as a
convenient statistical tool for displaying data
distribution to check whether the shape of the data
has changed as a result of the augmentation.
In this study, the PMF of the DDoS ITD
followed a Lévy distribution with finite values for
its first four moments. Although these values were
finite, they were appropriate since the DDoS ITD
data had a limited stationary frame size, ensuring
the data magnitude within this frame remained
finite.
The augmentation techniques that violate the
fundamental characteristics of the data also violate
the constraints of the time series signals. However,
the constraints are altered when the augmented
DDoS ITD does not conform to the Lévy
distribution. For example, employing the stretching
technique for data augmentation results in a
polynomial distribution in the PMF of the DDoS
ITD. Similarly, utilizing the squeezing technique
leads to a Pareto distribution in the PMF, while
employing the random cut-and-paste technique
results in a Lévy distribution. These distributions are
illustrated in Figure 23, Figure 24, Figure 25, Figure
26, and Figure 27, respectively.
Fig. 23: After applying the stretching technique, the
PMF of the augmented DDoS ITD results in a
polynomial of degree 9 distribution. The
coefficients for this polynomial are [4.7e-09, –4.5e-
07, 1.8e-05, –0.0004, 0.0062, –0.0548, 0.3054, –
1.0178, 1.7902, –1.0285]
Fig. 24: After applying the squeezing technique
(downsampling), the PMF of the augmented DDoS
ITD results in a Pareto distribution with parameters:
Shape equal to 1.71292, Scale equal to 0.00601753
and Threshold equal to 0.
Fig. 25: After applying the random cut-and-paste
technique, the PMF of the augmented DDoS ITD
results in a Lévy distribution with parameters µ
equal to 0.07692308 and σ equal to 0.006554275
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Fig. 26: After applying the mirroring flipping
horizontally technique, the PMF of the augmented
DDoS ITD results in a Lévy distribution with
parameters µ equal to 0.0588451 and σ equal to
0.00346275
Fig. 27: After applying the adding noise technique,
the PMF of the augmented DDoS ITD results in a
normal distribution with parameters µ equal to
0.067 and σ equal to 0.008234
Flipping and random cut-and-paste of the DDoS
ITD resulted in similar PMF patterns, whereas
stretching, squeezing, and adding noise did not yield
similar outcomes. With the horizontal flipping
technique, the tagging (labeling) of normal or
anomalous data output cannot be maintained since
the tag can be altered. Data output was labeled
according to the packet count within a 0.1 ms
interval, with fewer than 40 packets considered
normal and more than 40 packets considered
anomalous. The horizontal flipping technique
changed the data output tag and made this
augmentation technique impractical for machine
learning model training, [2]. Moreover, the random
cut-and-paste technique is not suitable for
adequately expanding small datasets due to their
non-stationary nature; thus, it is only viable for large
datasets where augmentation is unnecessary due to a
large number of data points. Furthermore, the data
output tag can be modified.
The PMF distributions of the “test” dataset
augmented with stretching, squeezing, and random
cut-and-paste techniques result in normal
distributions, as illustrated in Figure 28, Figure 29,
Figure 30, Figure 31 and Figure 32, respectively.
In voice datasets, flipping, stretching,
squeezing, and random cut-and-paste techniques
yielded similar distributions, specifically normal
distribution.
Fig. 28: After applying the stretching technique, the
PMF of the augmented “test” dataset results in a
normal distribution with parameters µ equal to
0.090909091 and σ equal to 0.011071521
Fig. 29: After applying the squeezing technique
(downsampling), the PMF of the augmented “test”
dataset results in a normal distribution with
parameters µ equal to 0.090909091 and σ equal to
0.032889586
Fig. 30: After applying the random cut-and-paste
technique, the PMF of the augmented “test” dataset
results in a normal distribution with parameters µ
equal to 0.076923077 and σ equal to 0.006554275
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Fig. 31: After applying the mirroring flipping
horizontally technique, the PMF of the augmented
“test” dataset results in a normal distribution with
parameters µ equal to 0.09091 and σ equal to
0.01415
Fig. 32: After applying the squeezing technique
(wavelet approximation coefficients), the PMF of
the augmented “test” dataset results in a normal
distribution with parameters: µ equal to -
0.000743937 and σ equal to 0.001609512
7 Discussion on the Applications
The first application of detecting input-data
properties, like long-range dependence, is a
validation of the augmentation process. The second
application of detecting input-data properties, like
long-range dependence, is using proper machine
learning tools to analyze the input data for
extracting features, classification, or regression.
Detecting the existence of long-range dependence in
the input data is important to selecting or designing
various structures of neural networks. If input data
do not have long-range dependence, a regular neural
network or a multiscale neural network structure can
be sufficient for analyzing the data. If there is long-
range dependence within the input data, designing a
poly-scale neural network structure could be
particularly useful. Consequently, a poly-scale
analysis algorithm is recommended for designing
the architecture of the poly-scale neural network
structure, [19], [20], [21]. In contrast with the
multiscale analysis, where there is no correlation
between the outcomes at different scales, a poly-
scale analysis measures input data at various scales,
and its outcome requires all the scales to be used
simultaneously [6]. Thus, the hidden feature in the
input data could be extracted using the poly-scale
neural network. The design of such a poly-scale
neural network is addressed in [19].
Applications of data augmentation include all
the deep learning and machine learning architectures
[22], particularly in the convolutional neural
networks (CNNs) models, whenever there are
limited sizes of quality data and to improve the
model’s robustness and performance. It has been
used in healthcare [23] and autonomous vehicles to
expand the range of scenarios for self-driving cars
and drones. It has also been used in natural
language processing (NLP) to improve its
performance by synonym augmentation, word
embedding, character swap, and random insertion
and deletion, [24]. Automatic speech recognition
has also benefited significantly from data
augmentation. Image processing and computer
vision have also used data augmentation
extensively. The traditional limitations of data
augmentation include biases, [25]. Our paper
identifies another serious limitation of data
augmentation for data with long-term dependence.
This insight highlights the need to refine
augmentation practices in modern communications,
enhancing the reliability and robustness of models
used for network monitoring and security
applications, [26], [27].
Moreover, this study shows some data
augmentation methods and measures that can be
used for other applications, such as particle filters.
When using particle filters or sample-based methods
in general, researchers need sampling. Instead of
random sampling, one effective approach is
employing a data augmentation method, such as
wavelet approximation and detail coefficients. As a
result, the area that has more information can be
exaggerated. Then, the area of the input signal with
more information has a larger amplitude.
Conversely, areas without any information exhibit
lower amplitude in the augmented data. Finally, the
sampling process can be launched from the
approximation coefficients area where its
corresponding detail coefficients have high values.
One application of data augmentation
techniques that preserves LRD is the effective
management of big data in communications. By
maintaining the fundamental properties of the
original datasets, augmented data can enhance the
performance of predictive models, anomaly
detection systems, and traffic management
strategies. This leads to more accurate analysis,
improved decision-making, and greater operational
efficiency in handling the complex, bursty, and
highly dynamic nature of communications data.
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Finally, the Hurst exponent and the VFDT can
also be used to sample particle filters. In normal
data, the boundaries of VFDT are different from the
boundaries of anomalous data. Therefore, to extract
more detailed information from the corresponding
raw data, it is crucial to find the trajectory of the
VFD, as this will help identify points that belong to
anomalous boundaries, providing more insight into
the underlying patterns. Therefore, more samples
from this signal area with VFD and anomalous
boundaries can be obtained.
8 Extension of the Work
In this paper, seven measures are considered to
analyze the impact of data augmentation on datasets.
To expand the analysis of the dataset, other
measures can also be considered, such as the
discrete wavelet transform (DWT), principal
component analysis (PCA), the distance between a
packet’s arrival time with the adjacent packet, and
the average of the distance between a packet’s
arrival time with the last five adjacent packets.
These measures can be utilized to examine the time
series signals of both pre and post-data
augmentation.
For future work, exploring the combined use of
these additional measures could provide deeper
insights into the effectiveness of data augmentation
techniques, particularly for datasets originating from
stochastic environments like Internet traffic data.
9 Concluding Remarks
This study investigated the impact of five
augmentation techniques on the long-range
dependence of the DDoS ITD and the "test" dataset.
It highlights how these techniques can disrupt the
LRD of certain time series datasets by altering their
PMF.
While none of the augmentation methods
preserved the original data constraints of the DDoS
ITD, as indicated by the changes in PMF, the
augmented "test" dataset closely mirrored the
original distribution, following a normal distribution
and thus preserving the constraints. Consequently,
the proposed augmentation techniques may be
suitable for audio time series data but not for
Internet time series data that originate from bursty
operations.
This study also demonstrates how LRD can
serve as a tool to assess the reliability and validity of
the augmentation process. Moreover, this research
illustrates the real-time applicability of using PMF
to evaluate the suitability of data augmentation
techniques. Furthermore, the PMF validation
approach is adaptable to various time series datasets,
offering significant convenience and advantageous
in determining whether to expand input data for
machine learning purposes.
Future work could investigate hybrid
augmentation methods that integrate domain-
specific constraints. This approach aims to enhance
the robustness and applicability of augmented
datasets in modern communications, particularly for
tasks such as network anomaly detection and
predictive traffic modeling.
Acknowledgement:
The National Science Foundation, the US
Department of Homeland Security, and CAIDA
Members provided funding for CAIDA’s Internet
Traces. Additionally, Sina Sedigh supported the
University of Manitoba Speech Dataset.
Declaration of Generative AI and AI-assisted
Technologies in the Writing Process
Maryam Ghanbari used ChatGPT and Grammarly to
only edit this paper, enhancing its clarity and
grammatical accuracy.
The authors reviewed and edited the content and
take full responsibility for the content of this
publication.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Conceptualization, M.G., and W.K.; Data Curation,
M.G.; Investigation, M.G; Writing—original draft
preparation, M.G.; Writing—review and editing,
M.G., W.K. and N.S.; Supervision, W.K.; Project
Administration, W.K., and N.S.; funding
acquisition, W.K., N.S.; All authors have read and
agreed to the published version of the manuscript.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
This research was partially funded by the Natural
Sciences and Engineering Research Council of
Canada (NSERC). Grant number: RGPIN-2018-
05352.
Conflict of Interest
The authors declare no conflicts of interest.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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APPENDIX
Table 1. The statistical data of the DDoS ITD and the results of eight mono-scale, multi-scale, and poly-scale
measures both before and after data augmentations for the initial stationary frame of trajectories
Raw Data
Flipping
Horizontally
(Mirroring)
Stretching
Squeeze
Downsampling
Squeeze
Wavelet
transform
Random
Cut-and-
Paste
Adding
Noise
Mean
0.0029
0.0029
0.0027
0.0014
0.0041
0.0030
0.0029
Median
0.0020
0.0020
0.0023
0
0.0038
0.0021
0.0029
Mode
5.0000e–06
5.0000e–06
6.0000e–06
0
–0.0014
5.0000e–06
–0.0340
Variance
7.7418e–06
7.7418e–06
4.4139e–06
5.9432 e–06
6.4685e–06
8.6087e–06
1.1161e–04
Standard
deviation
0.0028
0.0028
0.0021
0.0024
0.0025
0.0029
0.0106
Skewness
1.2928
1.2928
1.0344
2.1488
0.4927
1.2740
0.0804
Kurtosis
1.2582
1.2582
0.9338
4.6103
–0.2094
1.1205
3.8383e–04
Autocorrelation
(lag 0)
–3.4694e–18
–3.4694e–18
0
5.2042e–18
3.0095e–07
8.5698e–07
–1.0536e–05
VFD
1.9931
1.9931
1.9836
1.9880
1.9891
1.9944
1.9864
Hurst exponent
0.0069
0.0069
0.0164
0.0120
0.0109
0.0056
0.0136
Slope (
)
1.0138
1.0138
1.0327
1.0240
1.0218
1.0111
1.0272
Zero crossing
1250
1250
1008
598
3154
1228
4129
Turns count
5349
5349
5516
2846
6017
5240
5472
Table 2. The statistical information of the word “test” dataset and the results of eight mono-scale, multi-scale,
and poly-scale measures both before and after data augmentations for the initial stationary frame of trajectories
Raw Data
Flipping
Horizontally
(Mirroring)
Stretching
Squeeze
Downsampling
Squeeze
Wavelet
transform
Random
Cut-and-
Paste
Adding
Noise
Mean
–0.0051
–0.0052
–0.0060
–0.0025
–0.0066
–0.0055
–0.0054
Median
–0.0055
–0.0055
–0.0050
0
–0.0078
–0.0013
–0.0073
Mode
–0.0429
–0.0429
–0.0050
0
–0.1583
–0.0119
–0.1361
Variance
0.0015
0.0015
0.0017
7.3658e–04
0.0026
0.0013
0.0016
Standard
deviation
0.0383
0.382
0.0410
0.0271
0.0508
0.0358
0.0397
Skewness
0.0917
0.0890
0.0834
0.0068
0.0525
0.0151
0.1513
Kurtosis
0.3912
0.4106
0.2173
2.7957
0.6188
0.2191
0.4643
Autocorrelation
(lag 0)
0.0010
0.0010
–0.0003
0
–0.0024
–4.0610e–05
0.0025
VFD
1.9412
1.9412
1.9273
1.9867
1.9835
1.9007
1.9767
Hurst exponent
0.0588
0.0588
0.0727
0.0133
0.165
0.0993
0.0233
Slope (
)
1.1177
1.1117
1.1455
1.0267
1.0330
1.1987
1.0466
Zero crossing
145
145
139
123
250
114
159
Turns count
192
192
418
150
290
171
226
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