# Ovarian Cancer Diagnostics using Wavelet Packet Scaling Descriptors

Raymond J. Hinton, Jr.
Department of Statistics
Texas A&M University
College Station, TX 77843
<EMAIL_ADDRESS>
& Jihyun Byun
Department of Statistics
Texas A&M University
College Station, TX 77843
<EMAIL_ADDRESS>
Dixon Vimalajeewa
Department of Statistics
University of Nebraska - Lincoln
Lincoln, NE 68583
<EMAIL_ADDRESS>
& Brani Vidakovic
Department of Statistics
Texas A&M University
College Station, TX 77843
<EMAIL_ADDRESS>

###### Abstract

Detecting early-stage ovarian cancer accurately and efficiently is crucial for
timely treatment. Various methods for early diagnosis have been explored,
including a focus on features derived from protein mass spectra, but these
tend to overlook the complex interplay across protein expression levels. We
propose an innovative method to automate the search for diagnostic features in
these spectra by analyzing their inherent scaling characteristics. We compare
two techniques for estimating the self-similarity in a signal using the
scaling behavior of its wavelet packet decomposition. The methods are applied
to the mass spectra using a rolling window approach, yielding a collection of
self-similarity indexes that capture protein interactions, potentially
indicative of ovarian cancer. Then, the most discriminatory scaling
descriptors from this collection are selected for use in classification
algorithms. To assess their effectiveness for early diagnosis of ovarian
cancer, the techniques are applied to two datasets from the American National
Cancer Institute. Comparative evaluation against an existing wavelet-based
method shows that one wavelet packet-based technique led to improved
diagnostic performance for one of the analyzed datasets (95.67% vs. 96.78%
test accuracy, respectively). This highlights the potential of wavelet packet-
based methods to capture novel diagnostic information related to ovarian
cancer. This innovative approach offers promise for better early detection and
improved patient outcomes in ovarian cancer.

_Keywords_ self-similarity $\cdot$ wavelet transform $\cdot$ wavelet packets
$\cdot$ classification $\cdot$ ovarian cancer

##  1 Introduction

According to the American Cancer Society, ovarian cancer is expected to cause
13,270 deaths in 2023, making it the fifth leading cause of cancer-related
deaths among women in the United States. The lifetime risk of a woman
developing ovarian cancer is approximately 1 in 78, while her risk of dying
from the disease is about 1 in 108. Early detection is crucial in reducing
mortality risk, as it has a five-year survival rate of around 94% [1].
However, most cases of ovarian cancer are diagnosed at a more advanced stage
because early symptoms may not be noticeable and can resemble other less
serious conditions. Additionally, preventative screening has not been
effective in prospective randomized controlled trials. Only 18% of ovarian
cancer diagnoses are made at an early stage (Localized/Stage I), and the five-
year survival rates for advanced-stage diagnoses range from 72%
(Regional/Stage II-III) to 32% (Distant/Stage IV) [2]. Therefore, the
development of early-detection techniques for ovarian cancer is of utmost
importance.

Proteomic profiles used to detect ovarian cancer typically involve the
analysis of a wide range of proteins in biological samples, such as serum,
plasma, or tissue, to identify patterns or signatures that are indicative of
the presence or progression of ovarian cancer. These profiles can be generated
using various proteomic techniques, including mass spectrometry, immunoassays,
and protein microarrays. Mass spectrometry is the most frequently used
technique among these techniques [3]. For instance, MALDI-TOF MS (Matrix-
Assisted Laser Desorption/Ionization Time-of-Flight Mass Spectrometry) is
utilized to analyze serum or tissue samples for protein profiles associated
with ovarian cancer [4]. It allows for the detection of specific protein peaks
or patterns that may be associated with the disease. Existing studies
primarily involve identifying the expression of a combination of proteins or
protein patterns that can distinguish between ovarian cancer patients and
healthy individuals or those with other conditions. The goal is to improve the
accuracy and specificity of ovarian cancer detection and potentially aid in
early diagnosis.

Data analysis and bioinformatics tools play a critical role in ovarian cancer
diagnostics and research. They help researchers and clinicians process and
interpret large datasets to identify potential biomarkers, understand disease
mechanisms, and make diagnostic or therapeutic decisions. Among the commonly
used data analysis and bioinformatics tools for ovarian cancer diagnostics,
machine learning tools such as predictive models and classifiers have become
more popular with the growing interest in involvement of modern data science
and artificial intelligence. For instance, [5] proposes a novel approach for
identifying prognostic signatures in ovarian cancer patients using deep
learning to identify prognostic gene signatures using gene expression data
from multiple public databases. A machine learning model integrated into a
nanosensor is proposed to detect ovarian cancer from a blood test in [6],
while a distance novel variance-based modality is proposed to detect ovarian
cancer from protein mass spectra in [7]. References [8] and [9] review
existing and potential biomarkers for use in ovarian cancer diagnostics and
also discuss a range of opportunities for developing novel methods.

While data analysis and bioinformatics tools have significantly advanced our
ability to study and diagnose ovarian cancer, they are not without their
drawbacks and limitations. Some of the limitations of existing tools used to
detect ovarian cancer include interpretation challenges, biological noise,
standardization, clinical relevance, and dynamic nature of data. For instance,
[9] provides an overview of the current challenges associated with the
existing methods. Therefore, it is important to note that ongoing research in
proteomics continues to refine these approaches and discover new protein
biomarkers for ovarian cancer.

Wavelets are mathematical functions that decompose a signal at different
frequencies with a temporal resolution that varies according to the frequency
scale. This can be useful for, broadly, uncovering complex patterns occurring
at different frequencies in a signal. Wavelet-based data analysis has been
used in many fields, including the processing and analysis of biomedical data
in bioinformatics [10, 11, 12, 13]. In the context of ovarian cancer research,
examples of wavelet-based techniques and tools that have been used include
feature extraction, such as in [14] and [7], where new predictor variables are
created based on the data that are then used in classification studies.
Reference [15] applied denoising methods to low-dose computed tomography
(LDCT) images to improve the quality of information obtained at safer levels
of radiation. In [16], a wavelet-based multiresolution analysis is used to
correct biases in whole-genome sequencing for identifying ovarian cancer-
related genetic variations. Wavelets offer several advantages for analyzing
biomedical data, such as the ability to summarize complex interplay between
different frequency components in a signal which can lead to simpler models
and thus improve interpretability. Wavelets are also a flexible tool since
different wavelet families and sizes can be chosen, and different levels of
analysis can be performed depending on the size of the signal and wavelet.
Overall, wavelet-based approaches can be valuable in characterizing the
complexity of biological systems relevant to ovarian cancer.

Wavelet packets produce a richer representation of the original signal by
extending the concept of the standard wavelet transform. By additionally
analyzing the detail (or residual) coefficients at the end of each step,
rather than only the signal approximation coefficients, wavelet packets can
provide better characterizations of certain types of signals, such as those
with significant high-frequency components. Wavelet packets have been used in
applications such as signal compression [17], bearing fault diagnosis [18],
analyzing variations in Earth’s magnetic field [19], and a wide variety of
other areas. In medicine, wavelet packets have particularly been applied to
analyzing electroencephalographic (EEG) signals [20] and also for medical
imaging such as ultrasounds [21]. Each level of a wavelet packet decomposition
is the same length as the original signal, meaning that the representation is
redundant. This leads to a choice of how to efficiently recover or otherwise
summarize the original signal.

In this study, we compare two wavelet packet-based techniques, [22] and [23],
for generating features from blood protein mass spectra and use these features
in classification algorithms to detect ovarian cancer. The potential of these
techniques to outperform traditional wavelet-based methods is also
demonstrated, highlighting their efficacy for this task. A broader
contribution of our paper is the use of wavelet packets in the context of
scaling. There are relatively few references that employ wavelet packets to
estimate Hurst exponents and other scaling indexes, compared to references of
their use in the regression, signal representation, and denoising contexts.
This is especially true for signals of sound, where wavelet packets produce
the most parsimonious representations [24].

The remainder of the paper is organized as follows: Section 2 gives an
overview of the motivating study and datasets used for the analysis. The
techniques used in this study, fundamentals of wavelet transform, and
background for measuring self-similarity in signals using wavelet transforms
are presented in Section 3. Section 4 provides data analysis procedures and
results, followed by a discussion in Section 5. Concluding remarks are in
Section 6.

##  2 Motivating Study

This study examines two ovarian cancer datasets obtained from the American
National Cancer Institute internet repository [25]. These datasets comprise
protein mass spectra derived from the surface-enhanced laser desorption-
ionization time-of-flight (SELDI-TOF) mass spectrometer, utilizing blood
samples collected from both ovarian cancer patients and healthy individuals.
The first dataset, referred to as Ovarian 4-3-02, was gathered using the weak
cation exchange (WCX2) protein chip and the PBI SELDI-TOF mass spectrometer.
It consists of 216 protein mass spectra, encompassing 100 cancer samples, 100
non-cancer samples, and 16 benign samples. The second dataset, known as
Ovarian 8-7-02, was also collected using the WCX2 protein chip, but with an
upgraded version of the PBI SELDI-TOF mass spectrometer. Consequently, Ovarian
8-7-02 differs significantly from Ovarian 4-3-02 in terms of the method used
to record the protein mass spectra. This dataset comprises protein mass
spectra collected from 162 women diagnosed with cancer and 91 women without a
cancer diagnosis. In both datasets, the protein mass spectra are represented
by the intensities of 15,153 peptides, defined by their mass-to-charge ratio
(m/z). Fig. 1 displays two sample protein mass spectra, one from the healthy
(Control) group and the other from the cancer (Case) group, selected from the
Ovarian 4-3-02 dataset. The $x$-axis represents the m/z values, while the
$y$-axis represents the spectral intensity. Additional information about these
datasets and the related studies can be found in [25, 26].

Figure 1: Examples of protein mass spectra from the Control and Case groups,
zoomed in to 0-5000 m/z to show detail. Note that the Case spectra exhibits
suppressed peaks in the 4000-5000 m/z region relative to the Control, while
both samples show complex differences in the 0-1000 m/z region. This
demonstrates the need for diagnostic techniques that look beyond individual
peak intensities.

##  3 Methods

This Section introduces the techniques used to extract discriminatory features
by assessing self-similar properties of protein mass spectra in the wavelet
domain. First, we provide a brief overview of wavelet transforms, wavelet
packets, and the notion of self-similarity. Then, we describe two wavelet
packet-based methods for assessing self-similarity. Appendices A-C provide
technical details about these techniques.

###  3.1 Wavelet Transform

Wavelet transforms (WTs) are extensively utilized tools in the field of signal
processing. When applied to a signal, they break down the signal into
localized contributions in both time and frequency/scale, resulting in a
hierarchical representation. This representation enables simultaneous analysis
at various resolutions or scales, facilitating the exploration of signal
properties that may not be evident in the domain of the original data
(acquisition domain).

The discrete wavelet transform (DWT), a popular variant of wavelet transforms,
has become a standard tool for analyzing complex data signals, particularly in
domains where discrete data is examined. DWTs are linear transforms that can
be represented using orthogonal matrices. However, the computational cost of
performing the DWT in matrix form increases as the length of the signal grows.
To address this, a fast algorithm based on filtering was proposed by Mallat
for computational efficiency [27]. This algorithm involves a series of
successive convolutions utilizing a wavelet-specific low-pass filter and its
mirrored high-pass filter. Operators $\mathcal{H}$ and $\mathcal{G}$ can be
constructed with each filter by combining them with a decimation step
(retaining every second coefficient of the convolution). By repeatedly
applying these operators to the signal (and then the resulting "approximation"
coefficients after the first level of decomposition), a multiresolution
representation of the signal is generated. The resulting representation
comprises a smooth approximation and a hierarchy of detail coefficients at
different resolutions (or scale indexes) and locations within the same
resolution, as show in Fig. 2. For detailed instructions on calculating these
coefficients, please refer to Appendix A. These coefficients provide a
description of the signal at different scales and locations, serving as a
valuable means of characterizing the signal’s multi-scale dynamics.

for tree = grow=south, l sep=0.0cm, s sep=0.0cm, minimum height=0.5cm, minimum
width=1cm, edge=->,line width=1pt, text centered, [$\mathbf{c}_{0,N/2^{0}}$
(Data), draw, text width=10.05cm, label=left:j=3 [$\mathbf{c}_{2,N/2^{1}}$,
draw, text width=4.87cm, edge label=node[midway,left]$\mathcal{H}$,
label=left:j=2 [$\mathbf{c}_{1,N/2^{2}}$, draw, text width=2.29cm, edge
label=node[midway,left]$\mathcal{H}$, label=left:j=1
[$\mathbf{c}_{0,N/2^{3}}$, draw, text width=1cm, fill=cyan, edge
label=node[midway,left]$\mathcal{H}$, label=left:j=0]
[$\mathbf{d}_{0,N/2^{3}}$, draw, text width=1cm, fill=lightgray, edge
label=node[midway,right]$\mathcal{G}$] ] [$\mathbf{d}_{1,N/2^{2}}$, draw, text
width=2.29cm, fill=lightgray, edge label=node[midway,right]$\mathcal{G}$ ] ]
[$\mathbf{d}_{2,N/2^{1}}$, draw, text width=4.87cm, fill=lightgray, edge
label=node[midway,right]$\mathcal{G}$ ] ] ]

Figure 2: Illustration of three levels of decomposition ($j=2,1,0$) of the
discrete wavelet transform of a signal of length $N=2^{J}(J\geq 3)$. At each
decomposition level $j$, $\mathbf{c}_{j,N/2^{J-j}}$ denotes smoothing
coefficients while $\mathbf{d}_{j,N/2^{J-j}}$ denotes the detail wavelet
coefficients. The second index for each vector indicates their length, to
emphasize that the number of detail coefficients is halved at each level.

###  3.2 Wavelet Packets

The wavelet packet decomposition (WPD) is an extension of the discrete wavelet
transform that yields a richer (yet redundant) representation of the original
signal, which can be arranged in a "table" or binary tree with nodes
containing coefficients. The extended analysis is potentially able to capture
certain patterns better than the DWT. However, a choice must be made as to
what representation from all those produced by the WPD best characterizes the
signal.

The wavelet packet table is created by applying the low- and high-pass
operators $\mathcal{H}$ and $\mathcal{G}$ to both the approximation and detail
coefficients, not just the approximation coefficients as in the DWT. This
creates $2^{J-j}$ sets, or nodes, of coefficients at each level $j$, where the
original signal is considered to be level $J$, the first level of
decomposition is level $J-1$, etc. The result is that each level of the
wavelet packet table has $N=2^{J}$ coefficients, whereas the number of
coefficients in the DWT is reduced by half at each level, i.e. $N/2^{J-j}$
(except for the final level which contains the approximation coefficients).
Fig. 3 shows an example of a wavelet packet tree. The gray nodes of the tree
correspond to the standard DWT, showing that it is a special case of the WPD.
The coefficients in the vector $\mathbf{w}_{0,0}$ (the DWT approximation
coefficients after three levels of decomposition) are the result of the
operator combination $\mathcal{HHH}$. As a further example, vector
$\mathbf{w}_{0,3}$ is a result of the operator combination $\mathcal{GGH}$ (in
this mathematical notation, the first operator is the last one that is
applied). More details about calculating the WPD can be found in Appendix B.

for tree = grow=south, l sep=0cm, s sep=0cm, minimum height=0.5cm, minimum
width=1cm, edge=->,line width=1pt, text centered [$\mathbf{w}_{3,0}$ (Data),
draw, text width=10.05cm, label=left:j=3 [$\mathbf{w}_{2,0}$, draw, text
width=4.87cm, edge label=node[midway,left]$\mathcal{H}$, label=left:j=2
[$\mathbf{w}_{1,0}$, draw, text width=2.29cm, edge
label=node[midway,left]$\mathcal{H}$, label=left:j=1 [$\mathbf{w}_{0,0}$,
draw, text width=1cm, edge label=node[midway,left]$\mathcal{H}$, fill=cyan,
label=left:j=0] [$\mathbf{w}_{0,1}$, draw, text width=1cm, edge
label=node[midway,right]$\mathcal{G}$, fill=lightgray] ] [$\mathbf{w}_{1,1}$,
draw, text width=2.29cm, edge label=node[midway,right]$\mathcal{G}$,
fill=lightgray [$\mathbf{w}_{0,2}$, draw, text width=1cm] [$\mathbf{w}_{0,3}$,
draw, text width=1cm, edge label=node[midway,right]$\mathcal{G}$] ] ]
[$\mathbf{w}_{2,1}$, draw, text width=4.87cm, edge
label=node[midway,right]$\mathcal{G}$, fill=lightgray [$\mathbf{w}_{1,2}$,
draw, text width=2.29cm [$\mathbf{w}_{0,4}$, draw, text width=1cm]
[$\mathbf{w}_{0,5}$, draw, text width=1cm] ] [$\mathbf{w}_{1,3}$, draw, text
width=2.29cm [$\mathbf{w}_{0,6}$, draw, text width=1cm] [$\mathbf{w}_{0,7}$,
draw, text width=1cm] ] ] ]

Figure 3: Diagram of the coefficient nodes of a wavelet packet tree. The
highlighted nodes represent the DWT, showing that the DWT is a subset of the
WPD. In this diagram, the second index indicates the number of the node within
the level.

The additional analysis with high-pass filters offers the potential to better
characterize high-frequency components in the original signal. However, since
each node in the tree contains $N/2^{J-j}$ coefficients at level $j$ (for an
original signal of length $N$), the wavelet packet table is redundant in the
sense that each level of the tree contains the same number of coefficients as
in the original signal. Therefore, we must use some different method when
compared to the DWT to extract information from these coefficients. In the
following sections, we describe two such methods – one from Wang et al. [22]
that estimates the Hurst exponent using a spectral slope similar to that for
the DWT, and another from Jones et al. [23] that chooses the best basis
representation from the wavelet packet table and estimates the Hurst exponent
using power law scaling of the best basis coefficients.

###  3.3 Self-similarity

Self-similarity is an omnipresent phenomenon that characterizes high frequency
time series obtained in different contexts: health, geoscience, economics, and
physics, to list a few. This behavior refers to the stochastic similarity in a
signal when viewed at different scales, an example of which is show in Fig. 4.
In the wavelet domain, self-similarity is quantified commonly by using the
wavelet spectra. The wavelet spectrum of a signal consists of wavelet log
energies (logarithm of the average of squared wavelet detail coefficients) as
a function of a resolution level.

In general, it is accepted that a given signal possesses the self-similar
property if its wavelet spectrum exhibits a regular decay with an increase of
resolution scales. The rate of the log energy decay (slope) is connected with
the Hurst exponent ($H\in[0,1]$), which characterizes the regularity of the
signal. To estimate the slope using the standard DWT, the log energy for each
level is calculated with corresponding wavelet detail coefficients. Then, the
linear regression of log energies to the scale index is found, and $H$ is
computed as $H=-(slope+1)/2$. The spectral slope theoretically ranges from
$-3$ to $-1$. Larger slopes ($>-2$, i.e., shallower) indicate a higher degree
of persistence (i.e., more regular/smooth signal), while smaller slopes
($<-2$, i.e., steeper) indicate a higher degree of anti-persistency and
intermittency. Thus, the spectral slope serves as a measure of signal
regularity, since signals with a Hurst exponent close to $1$ are more regular
(smooth), while signals with a small Hurst exponent are highly irregular.
Overall, the degree of regularity expressed by either spectral slope, or
equivalently by the Hurst exponent, represents an informative summary of a
complex and noisy signal for which standard statistical summaries (moments,
trends, etc.) may be irrelevant.

Figure 4: Example of a high-frequency signal at three different resolutions.
The signal exhibits similar properties and behaviors (e.g. mean, variance)
when examined at different resolutions. This quality is known as the self-
similar nature of the signal.

###  3.4 Wang et al. method

The [22] method is similar to the DWT-based method in that it estimates a
spectral slope for a signal using log energies. The difference is that the
former uses the richer wavelet packet decomposition, and thus utilizes more
detail coefficients at each level.

for tree = grow=south, l sep=0cm, s sep=0cm, minimum height=0.5cm, minimum
width=1cm, edge=->,line width=1pt, text centered [$\mathbf{w}_{3,0}$ (Data),
draw, text width=10.05cm, label=left:j=3 [$\mathbf{w}_{2,0}$, draw, text
width=4.87cm, edge label=node[midway,left]$\mathcal{H}$, label=left:j=2
[$\mathbf{w}_{1,0}$, draw, text width=2.29cm, label=left:j=1
[$\mathbf{w}_{0,0}$, draw, text width=1cm, label=left:j=0]
[$\mathbf{w}_{0,1}$, draw, text width=1cm, fill=lightgray, edge
label=node[midway,right]$\mathcal{G}$] ] [$\mathbf{w}_{1,1}$, draw, text
width=2.29cm, fill=lightgray, edge label=node[midway,right]$\mathcal{G}$
[$\mathbf{w}_{0,2}$, draw, text width=1cm] [$\mathbf{w}_{0,3}$, draw, text
width=1cm, fill=lightgray, edge label=node[midway,right]$\mathcal{G}$] ] ]
[$\mathbf{w}_{2,1}$, draw, text width=4.87cm, fill=lightgray, edge
label=node[midway,right]$\mathcal{G}$ [$\mathbf{w}_{1,2}$, draw, text
width=2.29cm, [$\mathbf{w}_{0,4}$, draw, text width=1cm] [$\mathbf{w}_{0,5}$,
draw, text width=1cm, fill=lightgray, edge
label=node[midway,right]$\mathcal{G}$] ] [$\mathbf{w}_{1,3}$, draw, text
width=2.29cm, fill=lightgray, edge label=node[midway,right]$\mathcal{G}$
[$\mathbf{w}_{0,6}$, draw, text width=1cm] [$\mathbf{w}_{0,7}$, draw, text
width=1cm, fill=lightgray, edge label=node[midway,right]$\mathcal{G}$] ] ] ]

Figure 5: Diagram of a wavelet packet tree with detail nodes highlighted.
These nodes result from applying a high-pass operator, $\mathcal{G}$, to their
parent nodes, as indicated with the labels. These are the nodes used for
calculating level-wise energies in Wang et al.’s method.

As described in Section 3.2, each level $j$ of a wavelet packet tree contains
$2^{J-j}$ nodes - half containing detail coefficients and the other half
containing approximation coefficients - that are the results of applying high-
pass and low-pass filters, respectively, to the coefficients in the parent
nodes. Thus, after the first level, each level contains multiple nodes with
detail coefficients, and the Wang et al. method uses all such detail nodes to
calculate an energy for the level. In the operator notation, these are any
nodes that result from an operation beginning with $\mathcal{G}$, which are
the shaded nodes in Fig. 5. The energy for individual detail nodes is
calculated similarly to the DWT-based method, and the energy for each level is
just the average of the detail node energies. This results in an energy for
each level, with which the wavelet spectra can be plotted and the spectral
slope can be estimated in the same way as the DWT method. The Hurst exponent
is estimated using $H=-(slope/2)$, as stated in [22].

Wang et al. showed that this method produced estimates of the Hurst exponent
with lower absolute error for simulated fractal Brownian motion (fBm) signals
at most values of $H$ when compared to DWT and other methods. We performed
similar experiments using the DWT and Wang et al. methods that included
calculating the standard deviation of the Hurst exponent estimates.
Specifically, $n=1000$ fBm signals of length $N=2^{10}=1024$ were simulated
for several values of $H$. The full wavelet packet tree with
$j_{0}=\log_{2}N=10$ levels was calculated with the Haar wavelet. The DWT
wavelet spectra was calculated using the first detail nodes in each level, and
the Wang et al. method wavelet spectra was calculated as described above. The
Hurst exponents were calculated for each method as described above, and the
averages and standard deviations of the estimates across the simulations are
shown in Table 1. The bias of the Wang et al. estimator is lower than that for
the DWT-based method in five out of nine of the tested $H$ values, and the
standard deviation of the $H$ estimates is lower for the Wang et al. method
for all $H$ values. Simulations with the Jones et al. method (described in the
next section) showed some of the lowest overall standard deviations. However,
most of these were coupled with the highest overall bias. Therefore, among
these three methods, the Wang et al. method may be a good compromise between
bias and variance.

H | DWT | Wang et al. | Jones et al.
---|---|---|---
(mean $\pm$ st. dev.) | (mean $\pm$ st. dev.) | (mean $\pm$ st. dev.)
0.1 | -0.0303 $\pm$ 0.0970 |  0.1049 $\pm$ 0.0248 | 0.1306 $\pm$ 0.0352
0.2 | 0.1211 $\pm$ 0.1001 |  0.2071 $\pm$ 0.0385 | 0.2171 $\pm$ 0.0426
0.3 | 0.2498 $\pm$ 0.1017 | 0.3103 $\pm$ 0.0488 |  0.3086 $\pm$ 0.0510
0.4 | 0.3712 $\pm$ 0.1031 | 0.4148 $\pm$ 0.0577 |  0.3930 $\pm$ 0.0519
0.5 |  0.4870 $\pm$ 0.1091 | 0.5202 $\pm$ 0.0663 | 0.4460 $\pm$ 0.0461
0.6 |  0.6000 $\pm$ 0.1126 | 0.6256 $\pm$ 0.0748 | 0.5176 $\pm$ 0.0477
0.7 |  0.7079 $\pm$ 0.1180 | 0.7288 $\pm$ 0.0822 | 0.6312 $\pm$ 0.0601
0.8 |  0.8068 $\pm$ 0.1217 | 0.8251 $\pm$ 0.0857 | 0.7603 $\pm$ 0.0769
0.9 | 0.8922 $\pm$ 0.1125 |  0.9069 $\pm$ 0.0818 | 0.9080 $\pm$ 0.1016
Table 1: Simulation results from estimating the Hurst exponent for known
fractal Brownian motion (fBm) signals. The closest average estimates and
lowest standard deviation for each $H$ are in bold. The Haar wavelet was used
with the DWT-based and Wang et al. methods, and the SYMM4 wavelet was used
with the Jones et al. method. Ten levels of WPD were performed for the DWT-
based and [22] methods, and nine were performed for the [23] method.

###  3.5 Jones et al. method

The method by [23] uses a specially chosen subset of the wavelet packet
coefficients to estimate the Hurst exponent according to a power law scaling
relationship.

There are many valid bases contained in the wavelet packet library, so it is
desirable to define some criteria with which to choose the best basis. Any
particular basis may contain wavelet packets which do not match the original
signal well, but are otherwise not entirely negligible. We could aim to
concentrate the amount of information described with well-matching packets,
and minimize the information described with poorly-matching packets. Thus, we
seek to minimize an _information cost functional_ that is "large when
coefficients are roughly the same size and small when all but a few
coefficients are negligible" [28]. Entropy is an example of such a cost
functional and is the one used in the method by Jones et al. The best basis is
constructed by selecting a subset of nodes which overall minimize the entropy
(while still forming an orthonormal basis). There are a large number of
potential bases, but the algorithm proposed by [28] has a complexity that is
proportional only to the number of nodes in the tree. Further details about
selecting the best basis can be found in Appendix D.

Jones et al. propose that best basis wavelet packet coefficients can be
described with a power law relationship – Korcak’s Law – which measures the
scaling behavior of a quantity. More information about the law can be found in
[29] and a history in [30]. Specifically, the law relates the number of
objects that are larger than some threshold size to that threshold size. In
the context of wavelet packet coefficients, Jones et al. propose using

$N_{r}(A>a)\approx ka^{-\delta}\text{,}$ (1)

with the sorted absolute coefficient values as $a$, an index over the sorted
absolute coefficients as $N_{r}$, and $(1+H)$ as $\delta$, the slope exponent.
Then, a log-log plot of the sorted absolute coefficient values versus the
index can be used to estimate a slope with a linear regression fit and thus
estimate the Hurst exponent as $H=|\delta+1|.$

One motivation for using the best basis to estimate the Hurst exponent in this
fashion can be found by comparing this to the Maximum Modulus approach [31]
that is used when calculating scaling exponents with Continuous Wavelet
Transforms (CWTs). In that context, the evaluation goes along paths ("snakes")
defined by the most energetic wavelet coefficients. The decay of energy in the
snakes along the scale is used to estimate regular scaling. The Jones et al.
method insists on the “best” basis in a wavelet packet decomposition for
related reasons. The best basis determined by minimizing the additive entropy
measure results in the most disbalanced decomposition, in the sense that most
of the coefficients are non-energetic and that most of the energy is contained
in a few wavelet packet coefficients.

As described in more detail Section 4, the blood protein spectra signals were
divided into smaller sections, as in [7], and each of these methods were used
to estimate slopes for the sections to be used as features for classification.

###  3.6 Feature Extraction

A direct feature extraction method has been used in many studies to analyze
the aforementioned ovarian cancer datasets [32]. This method uses the whole
mass spectra to select discriminatory features with large differences in mean
values between Cases and Controls characterized by Fisher’s criterion,

$F=\frac{(\mu_{case}-\mu_{control})^{2}}{\sigma_{case}^{2}+\sigma_{control}^{2}}$
(2)

where $\mu_{case}$ and $\mu_{control}$ are the arithmetic mean for the
intensities at each mass-to-charge ratio in the mass spectra for the Case and
Control groups, respectively, and $\sigma_{case}^{2}$ and
$\sigma_{control}^{2}$ are the sample variances of the corresponding
intensities over the mass-to-charge ratios in the mass spectra for the Case
and Control groups. This measure is computed for all 15,153 mass-to-charge
ratios in the mass spectra. The mass-to-charge ratios with the highest $F$
values are then selected for subsequent classification purposes. Prior studies
indicate that using ten mass-to-charge ratios with the largest $F$ values can
achieve good classification performance [32].

This study, however, uses an evolutionary spectra-based approach for deriving
discriminatory features from the mass spectra. The evolutionary spectra-based
method uses a moving data window to select features. That is, discriminatory
features are selected from a data window while moving the data window along
the mass spectra with a specific step size. Thus, this approach enables
capturing more localized features in the mass spectra while still accounting
for intensities of mass-to-charge-ratios. Spectral slopes of the wavelet
spectra of those data windows are extracted as discriminatory descriptors.
Depending on the step size used to move the data window along the mass
spectra, those descriptors could be extracted from overlapping (or non-
overlapping) data windows. This study uses overlapping data windows, aiming to
extract as many discriminatory features as possible from the mass spectra.

Fisher’s criterion (2) is then used to select a set of the most informative
data windows in the mass spectra that well discriminate the Cases and
Controls. To compute the $F$ value for each data window, the mean and variance
($\mu$ and $\sigma^{2}$) of slopes of each data window corresponding to the
Case and Control groups are used. Then, a set of data windows that correspond
to the highest $F$ values are selected as the most informative discriminatory
features.

##  4 Ovarian Cancer Spectra Classification

This section evaluates the performance of the proposed modalities in detecting
ovarian cancer. We compare classification accuracy using the DWT, Wang et al.,
and Jones et al. methods for the Ovarian 4-3-02 and Ovarian 8-7-02 datasets.

###  4.1 Data Preparation

The following data preparation process was performed in MATLAB (computing
scaling descriptors) and Python (train/test splitting and feature selection).

  * (1)

Within a data window of length 1024, the wavelet packet tree is computed for
each of the protein mass spectra within this window (for the DWT-based and
Wang et al. methods, $J=10$ levels of decomposition with the Haar wavelet were
used; for the Jones et al. method, $J=9$ levels with the Symmlet wavelet with
four vanishing moments (SYMM4) were used).

  * (2)

The slopes are estimated for each method using the WPD coefficients obtained
in Step (1). See 4.2 for more details about specific steps taken for each
method.

  * (3)

Wavelet-based features are selected in the following manner:

    * (a)

Steps (1) and (2) are repeated while shifting the data window by 500 data
points along the protein mass spectra. To cover nearly all of the 15,153 mass-
to-charge ratios, $\lfloor(15153-1024)/500+1\rfloor=29$ overlapping windows
are required and cover 15,024 mass-to-charge ratios (i.e., resulting in 29
spectral slope features). The largest 129 mass-to-charge ratios are not
included in the analysis. Two feature matrices are then formed using the
estimated slopes, and they are denoted as $X_{N_{1}\times 29}^{control}$ and
$X_{N_{2}\times 29}^{case}$, where $N_{1}$ and $N_{2}$ is the number of
spectra in the Control and Case groups, respectively.

    * (b)

To select most significant classifying features, Fisher’s criterion ($F$)
given in Equation (2) is computed, and the $p$ features with the largest $F$
values from the feature matrices are then selected for classification
purposes. This results in smaller feature matrices $X_{N_{1}\times
p}^{control}$ and $X_{N_{2}\times p}^{case}$ where $p\leq 29$.

    * (c)

Response matrices are created by assigning labels 1 and 0 for Cases and
Controls respectively ($y_{N_{1}\times 1}^{control}$ and $y_{N_{2}\times
1}^{case}$).

###  4.2 Computing Scaling Descriptors

  * (1)

DWT-based method: The DWT-based wavelet spectra was calculated using the
second node in each level of the WPD from Step (1), since these nodes are just
the detail coefficients for the respective levels of the DWT. See Section 3.3
for details about calculating slopes from wavelet spectra.

The wavelet spectra may deviate from linearity at the coarsest and/or finest
levels when using real data, in which case choosing a subset of levels will
improve estimation of the spectral slope. In order to determine which levels
to use for this study, four observations were randomly selected from the Case
and Control groups each from each dataset, and then the wavelet spectra plot
was examined for the chosen observations across each data window (i.e.
$4\times 2\times 29$ plots per dataset). The randomly chosen observations for
Ovarian 4-3-02 were: Control - 101, 130, 142, 172; Case - 10, 15, 19, 34. For
Ovarian 8-7-02, they were: Control - 199, 200, 211, 223; Case - 5, 34, 107,
142. The spectra were assessed visually to determine what subset of levels for
each group showed the least deviation.

After examining all data windows, a subset of levels was chosen to give the
best coverage across all windows for each group. A breakdown of the specific
levels chosen for each group and dataset is given in Appendix C. Only the
chosen levels were used for estimating slopes in Step (2).

  * (2)

Wang et al. method: The Wang et al. method wavelet spectra was calculated
using the detail nodes from the WPD from Step (1). See Appendix C for details
about calculating slopes using the Wang et al. method. As in the DWT-based
method, only a subset of the levels was used to estimate the slopes. The same
technique was used with the same randomly chosen observations, and the chosen
levels are listed in Appendix C.

  * (3)

Jones et al. method: After calculating the WPD in Step (1), the best basis
nodes were identified as described in Appendix D using the non-normalized
Shannon entropy as the cost function. The absolute values of the best basis
coefficients were placed in a single vector, $C$, and sorted, and an index
vector from 1 to 1024, $N_{I}$, was generated. No values from $C$ were
discarded. The slope of $\log C$ vs. $\log N_{I}$ was calculated.

###  4.3 Classification Models and Performance

First, the Wilcoxon rank-sum test was used to verify that the estimated slopes
for the Case and Control groups from the DWT, Wang et al., and Jones et al.
methods were highly significantly different ($p<0.001$). Since Ovarian 8-7-02
has $N_{1}=91$ Controls and $N_{2}=162$ Cases, 91 of the 162 Cases were
randomly selected to match the number of Controls prior to randomly
determining the training and test sets to avoid bias due to imbalanced group
sizes.

Next, classification performance was evaluated using scikit-learn [33] in
Python with the following four algorithms and parameters: logistic regression
(L2 penalty, C = 1.0 and limited-memory BFGS solver), support vector machine
(radial basis function kernel and C = 1.0), K-nearest neighbors (5 neighbors,
equal weights and standard Euclidean distance), and random forest (Gini
impurity and 100 trees).

For model fitting, 67% of the samples were randomly selected and used for
training; the samples in the remaining rows were used for testing. To
determine the optimal number of classifying features, model training with
randomly sampled data was repeated 1,000 times, and average accuracy over the
repetitions was evaluated over different numbers of features sorted by
Fisher’s criterion values. The use of ten features for each algorithm was
determined after observing that additional features did not offer significant
accuracy improvements for the best-performing algorithms as indicated in Fig.
6. Next, using the ten most significant features identified for each method
and dataset, the average test set classification accuracy was evaluated for
10,000 different training and testing splits.

Figure 6: Average SVM training and test set accuracy over 1,000 repetitions
over 1-29 predictors for the Ovarian 8-7-02 dataset

For the Ovarian 4-3-02 dataset, the classification accuracies achieved with
the DWT and wavelet packet-based methods are similar, with the DWT-based
method performing slightly better overall (Table 2). For the Ovarian 8-7-02
dataset, the features generated by the Wang et al. method achieved slightly
better classification accuracies than those from the DWT-based method. The
features generated with the Jones et al. method resulted in lower
classification accuracies in both datasets. Hypotheses for the three methods’
performance, future research areas, and biomarkers that are captured by this
study’s generated features will be discussed in the following section.

Dataset | Model | DWT | Wang et al. | Jones et al.
---|---|---|---|---
Ovarian 4-3-02 | Logistic Regression | 81.83 | 79.24 | 79.14
Support Vector Machine | 83.07 | 80.38 | 81.05
K-Nearest Neighbors | 82.48 | 80.12 | 79.05
| Random Forest | 81.93 | 77.65 | 80.54
Ovarian 8-7-02 | Logistic Regression | 94.90 | 96.48 | 79.17
Support Vector Machine | 95.67 | 96.78 | 79.89
K-Nearest Neighbors | 92.05 | 94.44 | 81.19
| Random Forest | 91.74 | 94.22 | 80.82
Table 2: Average test set accuracy over 10,000 training and testing iterations
with the ten most significant features with respect to Fisher’s criterion for
the DWT, [22], and [23] methods.

##  5 Discussion

This study underscores the significance of incorporating wavelet packets and
accounting for scaling properties in protein mass spectra, as it contributes
to enhanced ovarian cancer diagnosis efficacy. In the following subsections,
we discuss the specific advantages of using wavelet-based methods and wavelet
packets for ovarian cancer diagnosis in particular. We also review biomarkers
that can be automatically captured by the proposed descriptors.

###  5.1 Wavelet Packets

The method by Wang et al. may be well-suited for generating discriminatory
features for datasets like Ovarian 8-7-02, in which the range of Hurst
exponent estimates is small. As shown in Fig. 7, the Hurst exponents estimated
using the Wang et al. method cover a smaller range for the Ovarian 8-7-02
dataset (mostly between 0.6 to 0.9) compared to the Ovarian 4-3-02 dataset
(spanning the range from 0 to 1). In our simulations (see Table 1), although
the DWT-based method showed the lowest bias in this range of Hurst exponents,
it also had the largest variability. Conversely, while the method by Jones et
al. had the lowest variability, it also had the worst bias. The Wang et al.
method may provide accurate enough estimates of the self-similarity while also
having low enough variability to separate signals with subtle differences in
this smaller range. Since wavelet packets create a redundant representation of
a signal, the method of Wang et al. takes advantage of this by calculating the
average energy of all the detail coefficients at each level. In the DWT-based
method, the number of available detail coefficients is halved at each level.
This may explain why, between these two wavelet spectra-based methods, the
Wang et al. method estimates the Hurst exponent with lower variability, since
it uses a larger number of coefficients in the level-wise energy calculations.

Figure 7: Distribution of the estimated Hurst exponent within each signal
window for the Ovarian 4-3-02 and 8-7-02 datasets using the [22] method. The
overall range of the estimated $H$ values is smaller for the Ovarian 8-7-02
dataset.

The wavelet packet-based methods employed in this study inherit many of the
advantages of standard wavelet-based methods, such as requiring minimal
preprocessing of the data and offering the ability to capture co-expression of
proteins. As described in [7], previous studies which extract features from
complex protein mass spectra employ preprocessing steps such as baseline
correction, peak alignment, normalization, and denoising. The method by Wang
et al. does not require any preprocessing steps beyond those which would
already be employed for the DWT-based method (primarily, choosing levels that
are used to estimate the rate of energy decay in the wavelet spectra). The
former method is also simple to implement, since it is conceptually similar
and merely extends the DWT-based method to the case where there are multiple
sets of coefficients at each level. As for capturing co-expression: groups of
biomarkers have been studied for their ability to diagnose ovarian cancer in
combination. For example, [34] used the levels of five proteins as features in
a logistic regression model. In such studies, the individual proteins are
manually selected, whereas measurements of self-similarity automatically
describe intricate patterns within a signal, including how signal intensities
relate across different indices. Thus, wavelet packets can offer a more
automated approach for capturing these complex relationships. Another
advantage of wavelet packets is that the features generated with them do not
require additional features in order to achieve high classification accuracy.
For example, [7] demonstrated that DWT-based features led to an improvement in
classification accuracy when combined with standard intensity measurements.
However, models built exclusively with DWT-based features achieved less than
90% accuracy even when up to ten features were included.

###  5.2 Biomarkers

Ovarian cancer, a complex disease with multiple potential biomarkers, has been
the subject of extensive research for diagnostic purposes. Mass spectrometry
is a widely used technique for identifying potential biomarkers, with the
mass-to-charge (m/z) ratio being a critical parameter [35]. In this study, we
explored the capacity of the proposed scaling descriptors to identify and
capture m/z ratios associated with key biomarkers.

Fig. 8 visually presents the m/z windows corresponding to the top ten most
significant scaling descriptors, identified with Fisher’s criterion, for the
DWT, Wang et al. and Jones et al. methods. Integer labels in the highlighted
windows correspond to each of the 29 windows used to cover the mass-to-charge
ratios (e.g., "1" corresponds to the first window, with m/z indices from 1 to
1024, while "2" corresponds to the second window, with indices from 501 to
1524; see the "Data Preparation" section above).

Figure 8: Protein mass spectra for an individual Case sample from Ovarian
8-7-02. Highlighted ranges of mass-to-charge ratios reflect the ranges for
which spectral descriptors are determined to be significant (determined by
Fisher’s criterion) in classifying ovarian cancer using three wavelet-based
methods, with integer labels within ranges corresponding to each of the 29
windows.

Of note, the scaling descriptors successfully capture m/z regions associated
with several key biomarkers indicative of ovarian cancer presence. For
example, transgelin 2, which is a potential biomarker, corresponds to an m/z
value of 2345.19 in the protein mass spectra and is encompassed within window
11. Table 3 provides a summary of the biomarkers contained within these
windows [36, 37, 38, 39]. In essence, the proposed scaling descriptors offer
an automated means of identifying several previously recognized biomarkers, as
well as new mass-to-charge ratios that hold promise as crucial biomarkers for
ovarian cancer detection. This demonstrates the valuable potential of the
scaling descriptors in advancing our understanding and diagnosis of ovarian
cancer.

Window | m/z Indices | m/z Values | Methods | Biomarkers
---|---|---|---|---
6 | 2,501 - 3,524 | 543.93 - 1,080.39 | DWT, Wang et al., | Keratin 2a
Jones et al.
8 | 3,501 - 4,524 | 1,066.33 - 1,780.99 | Jones et al. | Complement component 4A
preproprotein
Glycosyltransferase-like 1B
11 | 5,001 - 6,024 | 2,176.53 - 3,158.48 | DWT | Transthyretin
Inter--trypsin inhibitor
heavy chain H4
Transgelin 2
19 | 9,001 - 10,024 | 7,053.15 - 8,747.88 | Wang et al. | Apolipoprotein A2
20 | 9,501 - 10,524 | 7,858.69 - 9,642.52 | DWT, Jones et al. | Apolipoprotein A2
Table 3: Potential ovarian cancer biomarkers that coincide with feature
windows from this study that were identified as significant by Fisher’s
criterion.

###  5.3 Drawbacks

A drawback common to methods involving a “rolling window,” and in particular
to all of the methods demonstrated here, is the subjective choice of the data
window size, which could change or generate different features and thus
influence the performance of the resulting models. A larger window size would
allow the individual scaling descriptors to characterize the interplay between
more protein expressions, but conversely could lead to an averaging effect
that smooths out subtle patterns. On the other hand, a smaller window size may
allow for finer characterization of localized patterns in the signals, but
could quickly lead to an overabundance of features relative to the available
samples. The length of the data windows is also related to the number of
levels of DWT or WPD that can be carried out, which affects the number of
points available for estimating the spectral slope in the DWT and Wang et al.
methods. The authors in [7] found that 1024 points per window worked well in
related studies, and so this choice of window size was adopted here.

###  5.4 Classification Accuracy

As stated in [25], the Ovarian 8-7-02 dataset was collected with an upgraded
instrument, a potential reason for the higher performance of the
classification algorithms on this dataset as compared to Ovarian 4-3-02. The
Jones et al. method yielded several highly correlated features, as shown in
Fig. 9. This likely contributes to overfitting of the classifiers, evidenced
by the divergence between testing and training accuracies for this method (see
Fig. 6), thus leading to lower test classification accuracy than that of the
DWT and Wang et al. methods.

As it is important to combine several instruments for cancer testing, this
paper provides a rapid and robust quantitative measure to strengthen existing
ovarian cancer diagnostic procedures. Although the accuracy rates could be
argued to be relatively low for Ovarian 4-3-02, even classifiers that perform
only slightly better than chance can improve diagnostic accuracy when added to
a battery of other independent testing modalities.

Figure 9: Correlation coefficient matrix for features from the Jones et al.
method for Ovarian 8-7-02 data indicate that a significant number of feature
pairs show moderate to strong correlation

.

##  6 Conclusion

This paper introduces a novel set of scaling descriptors rooted in the self-
similarity of protein mass spectra and evaluates their performance in ovarian
cancer diagnostics. The self-similar characteristics of data signals are
examined using two wavelet packet-based techniques applied to protein mass
spectra. We then explore the ability of these descriptors to improve ovarian
cancer detection by using them as features in several classification
algorithms.

This study has shown that wavelet packet-based descriptors using the Wang et
al. method can improve classification accuracy over the DWT method, and also
can achieve performance approaching that of existing methods that are based on
a combination of both wavelet and direct mass-to-charge ratio-based
descriptors [7]. For both datasets, classification accuracy falls within the
range of 79% to 97% for the proposed wavelet packet-based features, and 82% to
96% for the DWT-based features. Since early detection of ovarian cancer
significantly increases survival rates, any improvement in detection methods
is highly significant. The proposed approach captures new mass-to-charge ratio
ranges with minimal data preprocessing, while also encompassing mass-to-charge
ratio ranges that encompass the majority of previously identified biomarkers.

However, it is essential to acknowledge certain limitations of the proposed
method, which could be addressed in future research. Specifically, accounting
for machine effects and measurement noise in modeling efforts is crucial, as
these factors can adversely impact classification performance. This would
involve exploring models with careful fine-tuning to mitigate the influence of
such nuisance information. Notably, [40] discusses potential machine learning
methods to address these challenges, providing a valuable avenue for future
research.

In the spirit of reproducible research, the software used in this paper is
posted on https://github.com/jihyun-byun/ovarian-cancer-classification.

## Acknowledgements

This research was supported in part by the H.O. Hartley endowment at Texas A&M
University. The study sponsor had no involvement in the study design,
collection, analysis and interpretation of the data; in the writing of the
manuscript; and in the decision to submit the manuscript for publication.

##  Appendix A Discrete Wavelet Transform

Suppose a data signal ($Y$) is a vector of size $N\times 1$. The DWT of $Y$,
denoted as $\mathbf{d}$, is represented as

$\mathbf{d}=WY,$ (3)

where $W$ is an orthogonal matrix of size ${N\times N}$. The elements in $W$
are determined by selecting a particular wavelet basis, such as Haar,
Daubechies, and Symmlet. Although $N$ can be arbitrary, it is usually selected
to be a power of two, i.e., $N=2^{J}$, $J\in\mathbb{Z}^{+}$ for the ease of
calculating the DWT.

When $N$ is large, and calculating $\mathbf{d}$ as in (3) becomes
computationally expensive, a fast algorithm based on filtering proposed by
Mallat is used for computational efficiency. With this algorithm the DWT is
obtained by performing a series of successive convolutions that involve a
wavelet-specific low-pass filter $\mathbf{h}$ and its quadrature-mirror
counterpart, high-pass filter $\mathbf{g}$. These repeated convolutions using
the two filters accompanied with the operation of decimation (keeping every
second coefficient of the convolution) generate a multiresolution
representation of a signal, consisting of a smooth approximation coefficients,
$c_{jk}$, and a hierarchy of detail coefficients $d_{jk}$ at different
resolutions (indexed by a scale index $j$) and different locations within the
same resolution, indexed by $k$. The convolutions with filters $\mathbf{h}$
and $\mathbf{g}$ are repeated until a desired decomposition level $j=J_{0}$ is
reached ($1\leq J_{0}\leq J-1$ and $J=\log_{2}N$).

Thus, the vector $\mathbf{d}{}$ in (3) has a structure,

$\mathbf{d}=(\mathbf{c}_{J_{0}},\mathbf{d}_{J_{0}},\dots,\mathbf{d}_{J-2},\mathbf{d}_{J-1}),$
(4)

where $\mathbf{c}_{J_{0}}$ is a vector of coefficients corresponding to a
smooth trend in signal, and $\mathbf{d}_{j}$ are detail coefficients at
different resolutions $j$ where $J_{0}\leq j\leq J-1.$ It is the logarithm of
the variance of coefficients in $\mathbf{d}_{j}$ that is used to define
wavelet spectra.

##  Appendix B Wavelet Packet Decomposition

The wavelet packet decomposition (WPD) extends the wavelet transform by
analyzing the detail coefficients in addition to the approximation
coefficients, thereby generating a tree of coefficients that may better
characterize certain qualities of the signal.

The WPD of a $N\times 1$ signal $Y$ can be calculated as $\mathbf{d}=WY$.
Here, $W$ is a $NJ\times NJ$ matrix that is determined by the choice of
wavelet and number of levels of decomposition, $J$.

The wavelet packet coefficients may be calculated as follows. Let $\mathbf{h}$
and $\mathbf{g}$ be the low- and high-pass filters, respectively, determined
by the choice of wavelet, and let $\mathcal{W}_{0}=\psi(x)$ and
$\mathcal{W}_{1}=\phi(x)$ be the corresponding scaling and wavelet functions.
Next, define a library of wavelet packet functions,
$\\{\mathcal{W}_{n}(x)\\}$, as follows, which generates the entire wavelet
packet tree or “table”:

$\displaystyle\mathcal{W}_{2n}(x)$
$\displaystyle=\sum_{k}h_{k}\sqrt{2}\mathcal{W}_{n}(2x-k),$
$\displaystyle\mathcal{W}_{2n+1}(x)$
$\displaystyle=\sum_{k}g_{k}\sqrt{2}\mathcal{W}_{n}(2x-k),n=0,1,2,\ldots\ .$

This library of functions is overcomplete, or redundant in a sense, and an
arbitrary subset chosen from it may not have desirable properties (i.e. being
an orthonormal and/or complete set). In order to choose an appropriate subset
(an orthonormal basis), instead define the wavelet packet functions in terms
of three parameters: $j$ \- the scaling parameter (corresponding to the level
of the tree); $n$ \- the oscillation parameter (corresponding to the node
within a level); and $k$ \- the translation parameter (corresponding to the
coefficient index within a node). Then, the wavelet packet functions can be
written as follows:

$\mathcal{W}_{j,n,k}=2^{j/2}\mathcal{W}_{n}(2^{j}x-k),(j,n,k)\in\mathbb{Z}\times\mathbb{N}\times\mathbb{Z},$
(5)

and any suitable (i.e. $\mathbb{L}_{2}$) function $f$ can now be represented
as:

$f(x)=\sum_{(j,n)\in\mathbb{P}}{\sum_{k\in\mathbb{Z}}{w_{j,n,k}\mathcal{W}_{j,n,k}(x),}}$
(6)

where $\mathbf{w}_{J,0}$ is conventionally the discrete dataset.

Here, $(j,n)\in\mathbb{P}$ means that we must select nodes from the wavelet
packet table that meet special criteria in order for them to form a valid
orthonormal basis. Formally, as described in, e.g., [41], the dyadic intervals
$\mathcal{I}_{j,n}=\\{[2^{j}n,2^{j(n+1)},\\{j,n\\}\in\mathbb{P}\in\mathbb{Z}\times\mathbb{N})\\}$
must form a disjoint, countable covering of the interval $[0,\infty)$.
Informally, if we draw an arbitrary vertical line down a wavelet packet table,
then the line must intersect a selected node exactly once. More than one
intersection results in redundancy, while no intersection results in non-
completeness.

##  Appendix C Wang et al. method calculations

For a signal of length $N=2^{J}$, level $j=J,J-1,\ldots,J-j_{0}$ of the
wavelet packet tree contains $2^{J-j-1}$ detail nodes (indexed by $k$) with
$N/2^{J-j}$ coefficients (indexed by $i$) in each node. The energy for a node
is calculated similarly as in the DWT method,
$E_{jk}=\sum_{i=1}^{N/(2^{J-j})}{c_{ijk}^{2}}$. Then, the energy for the level
is the average of the detail node energies,
$E_{j}=(1/2^{J-j-1})\sum_{k=1}^{2^{J-j-1}}{E_{jk}}$. This results in an energy
$E_{j}$ for each level. The wavelet spectra is plotted and the spectral slope
is estimated in the same way as the DWT method (described in Section 3.3). As
described in [22], the Hurst exponent is estimated as $H=-(slope/2)$.

The following levels were used when estimating the slope with the DWT method:
Ovarian 4-3-02 - windows 1-11, levels 7-10; windows 12-29, levels 6-9; Ovarian
8-7-02 - windows 1-11, levels 8-10; windows 12-29, levels 7-10. And with the
Wang et al. method, the following levels were used: Ovarian 4-3-02 - windows
1-10, levels 7-10; windows 11-29, levels 6-9; Ovarian 8-7-02 - windows 1-15,
levels 8-10; windows 16-29, levels 6-10.

##  Appendix D Best Basis Selection

A wavelet packet decomposition can be seen as a library of orthonormal bases.
The best basis for a signal, which minimizes an information cost functional,
can be found efficiently by exploiting the natural binary tree structure of
the decomposition. In this study, the non-normalized Shannon entropy is used
as the information cost functional, which for a vector
$\mathbf{x}=(x_{1},\ldots,x_{n})$ can be calculated as

$\mathcal{C}(\mathbf{x})=-\sum_{i=1}^{n}x_{i}^{2}\log x_{i}^{2}\text{.}$ (7)

The method from [28] finds the best basis in a way that only needs to examine
each node in a tree twice. The algorithm can be carried out as follows,
adapted from [24].

  1. 1.

The cost function is calculated for each node in the tree, then all nodes in
the final (bottom) level, $j=0$, are “marked”.

  2. 2.

Next, visit each node in the level above, $j=1$. If the cost of a parent node
from level 1 is less than the sum of the costs of its child nodes in level 0,
then mark the parent.

  3. 3.

Otherwise, assign the sum of the child nodes’ costs to the parent node.

  4. 4.

Proceed upwards in the tree, applying steps 2-3 accordingly.

  5. 5.

After examining all nodes in the tree, the topmost marked nodes are kept and
all nodes under a marked node are pruned from the tree. The remaining nodes
constitute the best basis.

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