#  Multiple teleportation via the partially entangled states

Meiyu Wang, Fengli Yan<EMAIL_ADDRESS>
College of Physics Science and Information Engineering, Hebei Normal
University, Shijiazhuang 050016, China
Hebei Advanced Thin Films Laboratory, Shijiazhuang 050016, China

###### Abstract

We investigate the multiple teleportation with some nonmaximally entangled
channels. The efficiencies of two multiple teleportation protocols, the
separate multiple teleportation protocol (SMTP) and the global multiple
teleportation protocol (GMTP), are calculated. We show that GMTP is more
efficient than SMTP.

###### pacs:

03.67.Hk

Quantum teleportation s1  is one of the most significant components in quantum
information processing, which allows indirect transmission of quantum
information between distant parties by using previously shared entanglement
and classical communication between them. Indeed, it is considered as a basic
building block of quantum communication nowadays. Not only is it one of the
most intriguing phenomena in the quantum world, but also a very useful tool to
perform various tasks in quantum information processing and quantum computing
s14 ; s15 . For example, controlled quantum gates are implemented by means of
quantum teleportation, which is very important in linear optical quantum
computation s2 ; s3 . Recently, the original scheme for teleporting a qubit
has been widely generalized in many different ways s5 ; s6 ; s7 ; s9 ; s10 ;
s11 ; s12 ; s13 ; s16 . In the previous teleportation protocols and in many
other applications of teleportation, we want to construct an unknown input
state with unity fidelity at another location while destroying the original
copy, which is always achieved if two parties share a maximally entangled
state. However, it might happen that our parties do not share a maximally
entangled state. This limitation can be overcome by distilling out of an
ensemble of partially entangled states a maximally entangled one s4 . But this
approach requires a large amount of copies of partially entangled states to
succeed. Another way to achieve unity fidelity teleportation with limited
resources is based on the probabilistic quantum teleportation protocols of
Refs. s5 ; s6 ; s7 .

Recently, in an interesting work, Modławska and Grudka s8  showed that if the
qubit is teleported several times via some nonmaximally entangled states, then
the “errors” introduced in the previous teleportations can be corrected by the
“errors” introduced in the following teleportations. Their strategy was
developed in the framework of the scheme proposed in Ref.s3  for linear
optical teleportation. In this paper, we show that this feature of the
multiple teleportation of Ref.s8  is not restricted to the teleportation
scheme stated in Ref.s3 . Based on the general teleportation language of the
original proposal shown in Ref.s1 , we compare the efficiencies of two
multiple teleportation protocols, the separate multiple teleportation and the
global multiple teleportation. In the former protocol, a complete
teleportation including error correction is strictly executed by neighboring
parties. On the other hand, in the latter protocol, all errors introduced in
the teleportation are corrected by the final receiver. We find the global
multiple teleportation is more efficient than the separate multiple
teleportation.

To illustrate two protocols clearly, let us first begin with the multiple
teleportation in the case of three parties.

Alice wants to teleport an unknown quantum state

$|\psi\rangle=a|0\rangle+b|1\rangle$ (1)

to Bob, where $a,b\in C$ and $|a|^{2}+|b|^{2}=1$. There is no direct
entanglment resource between Alice and Bob, fortunately, Alice and the third
party Charlie have a partially entangled state

$|\Psi\rangle=\alpha|00\rangle+\beta|11\rangle,$ (2)

while Charlie and Bob share the same entanglment resource, where $\alpha$ and
$\beta$ are real numbers and satisfy $\alpha^{2}+\beta^{2}=1$. Without loss of
generality, we suppose $|\alpha|\leq|\beta|$.

The simplest and directest strategy is to perform two separate teleportations,
i.e., Alice teleports the quantum state $|\psi\rangle$ to Charlie via the
first teleportation. Then Charlie teleports it to Bob via the second
teleportation. Because this procotol consists of two separate teleportations,
we call it the separate multiple teleportation procotol (SMTP).

According to the standard probabilistic teleportation protocol, in the first
separate teleportation, Alice performs the Bell-basis measurement (BM) on the
teleported qubit and the entangled qubit in her side. Charlie can apply the
corresponding Pauli transformation conditioned on the result of BM, i.e., $I$
if the BM yields $|\Phi^{+}\rangle$, $\sigma_{z}$ for $|\Phi^{-}\rangle$,
$\sigma_{x}$ for $|\Psi^{+}\rangle$, and ${\rm i}\sigma_{y}$ for
$|\Psi^{-}\rangle$ , where $I$ is the identity,
$\sigma_{x},\sigma_{y},\sigma_{z}$ are standard Pauli matrices and

$|\Phi^{\pm}\rangle=\frac{1}{\sqrt{2}}\left(|00\rangle\pm|11\rangle\right),$
(3)
$|\Psi^{\pm}\rangle=\frac{1}{\sqrt{2}}\left(|01\rangle\pm|10\rangle\right).$
(4)

Finally, the state Charlie received becomes

$|\psi_{\rm 1}\rangle=\frac{1}{\sqrt{p_{1}}}(\alpha a|0\rangle+\beta
b|1\rangle)$ (5)

with the probability $p_{1}=|a\alpha|^{2}+|b\beta|^{2}$ or

$|\psi_{\rm 2}\rangle=\frac{1}{\sqrt{p_{2}}}(\beta a|0\rangle+\alpha
b|1\rangle)$ (6)

with the probability $p_{2}=|a\beta|^{2}+|b\alpha|^{2}$. These states are in
accordance with the original state $|\psi\rangle$ only if the quantum channel
is a maximally entangled state, i.e. $\alpha=\beta$. For the case of non-
maximally entangled channel, there exists the “error” in $|\psi_{1}\rangle$
and $|\psi_{2}\rangle$. These states can be returned to the original state
with certain probability by performing the generalized measurerment given by
Kraus operators:

$\displaystyle E_{S1}$ $\displaystyle=|0\rangle\langle
0|+\frac{\alpha}{\beta}|1\rangle\langle 1|,$ (7a) $\displaystyle E_{F1}$
$\displaystyle=\sqrt{1-\frac{\alpha^{2}}{\beta^{2}}}|1\rangle\langle 1|$ (7b)

for $|\psi_{1}\rangle$ and

$\displaystyle E_{S2}$ $\displaystyle=\frac{\alpha}{\beta}|0\rangle\langle
0|+|1\rangle\langle 1|,$ (8a) $\displaystyle E_{F2}$
$\displaystyle=\sqrt{1-\frac{\alpha^{2}}{\beta^{2}}}|0\rangle\langle 0|$ (8b)

for $|\psi_{2}\rangle$. When $E_{S}$ is obtained, the qubit ends in its
original state $|\psi\rangle=a|0\rangle+b|1\rangle$. The success probability
in the first teleportation is

$p=\sum_{i=1}^{2}p_{i}\langle\psi_{i}|E^{\dagger}_{Si}E_{Si}|\psi_{i}\rangle=2\alpha^{2}.$
(9)

Next, Charlie teleports the recovered quantum state to Bob by the similar
process. Combining these two teleportations, the total probability that Bob
receives the quantum state $|\psi\rangle$ is

$P_{S}=p^{2}=4\alpha^{4}.$ (10)

However, the above teleportation protocol is not the optimal strategy. In
fact, the third party Charlie does not need to recover the quantum state to be
teleported, but teleports the “error state” to Bob directly. Lastly, Bob
corrects all “errors” of the quantum state in the teleportation process.
Formally, either Alice and Charlie or Charlie and Bob do not complete an
intact separate teleportation, so we call it the global multiple teleportation
protocol (GMTP).

Let us, thus, assume that Charlie does not correct the “error” introduced in
the first teleportation, he only makes a Pauli transformation according to
Alice’s measurement outcome, then he also performs BM on his two qubits and
broadcasts the measurement outcome to Bob. After making the corresponding
Pauli transformation conditioned on Charlie’s measurement outcome, Bob’s qubit
will collapse into one of the following states

$\displaystyle|\phi_{1}\rangle=\frac{1}{\sqrt{p^{\prime}_{1}}}(\alpha^{2}a|0\rangle+\beta^{2}b|1\rangle),$
(11a)
$\displaystyle|\phi_{2}\rangle=\frac{1}{\sqrt{p^{\prime}_{2}}}(\beta^{2}a|0\rangle+\alpha^{2}b|1\rangle),$
(11b) $\displaystyle|\phi_{3}\rangle=a|0\rangle+b|1\rangle)$ (11c)

with the probabilities $p^{\prime}_{1}=\alpha^{4}|a|^{2}+\beta^{4}|b|^{2},$
$p^{\prime}_{2}=\beta^{4}|a|^{2}+\alpha^{4}|b|^{2},$
$p^{\prime}_{3}=2\alpha^{2}\beta^{2}$ respectively. When the state is in
$|\phi_{3}\rangle$, we do not have to perform the error correction. It is very
joyful to see that the second teleportation corrects the “error” introduced by
the first teleportation. This effect is called error self-correction. For
$|\phi_{1}\rangle$ and $|\phi_{2}\rangle$, one can recover the original state
by performing generalized measurement given by Kraus operators:

$\displaystyle E^{\prime}_{S1}$ $\displaystyle=|0\rangle\langle
0|+\frac{\alpha^{2}}{\beta^{2}}|1\rangle\langle 1|,$ (12a) $\displaystyle
E^{\prime}_{F1}$
$\displaystyle=\sqrt{1-\frac{\alpha^{4}}{\beta^{4}}}|1\rangle\langle 1|$ (12b)

and

$\displaystyle E^{\prime}_{S2}$
$\displaystyle=\frac{\alpha^{2}}{\beta^{2}}|0\rangle\langle
0|+|1\rangle\langle 1|,$ (13a) $\displaystyle E^{\prime}_{F2}$
$\displaystyle=\sqrt{1-\frac{\alpha^{4}}{\beta^{4}}}|0\rangle\langle 0|$ (13b)

respectively. The total probability of successfully recovering the original
state is

$P_{G}(3)=2\alpha^{2}\beta^{2}+\sum_{i=1}^{2}p^{\prime}_{i}\langle\phi_{i}|E^{\prime{\dagger}}_{Si}E^{\prime}_{Si}|\phi_{i}\rangle=2\alpha^{2}.$
(14)

The ratio of efficiency of GMTP to that of SMTP

$P_{G}(3)/P_{S}=\frac{1}{2\alpha^{2}}.$ (15)

We can easily see $P_{G}/P_{S}\geq 1$ because of
$\alpha\leq\frac{1}{\sqrt{2}}$ . It is obvious that for the maximally
entangled channel, the two protocols are equivalent, but for the partially
entangled channel, GMTP is more efficient than SMTP. Moreover, the less
$\alpha$ is, the more efficient the GMTP is.

It is straightforward to generalize the above two protocols to arbitrary
parties. Let us first discuss the GMTP. Since error self-correction only
appears in the even times Bell-basis measurements, so here we discuss the
$(2N+1)$-party teleportation. Suppose that Alice 1 wants to teleport a quantum
state $|\psi\rangle=a|0\rangle+b|1\rangle$ to Alice $2N+1$. There is no direct
entanglement resource between them, but they can link through $2N-1$
intermediaries called Alice 2, Alice 3, $\cdots$, Alice $2N$, respectively.
Two neighboring parties share the partially entangled state described by
Eq.(2). They can complete the task through cooperative teleportation. After
$2N$ Bell-basis measurements and corresponding Pauli transformations
conditioned on previous parties, the final receiver’s qubit will be in one of
the states

$|\phi_{i}\rangle=\frac{1}{\sqrt{p^{G}_{i}}}(\alpha^{2N-i}\beta^{i}a|0\rangle+\alpha^{i}\beta^{2N-i}b|1\rangle),\\\
$ (16)

with the probability $C_{2N}^{i}p_{i}^{G}\equiv
C_{2N}^{i}(\alpha^{2(2N-i)}\beta^{2i}|a|^{2}+\alpha^{2i}\beta^{2(2N-i)}|b|^{2})$,
$i=0,1,2,\cdots,2N$. By correcting the error, the total success probability is

$P_{G}(2N+1)=C_{2N}^{N}\alpha^{2N}\beta^{2N}+2\sum_{i=0}^{N-1}C_{2N}^{i}\alpha^{2(2N-i)}\beta^{2i}.$
(17)

On the other hand, in the case of SMTP, we must perform $2N$ separate
teleportations, then the total success probability equals

$P_{S}(2N+1)=p^{2N}=2^{2N}\alpha^{4N}.$ (18)

It is easy to verify that $P_{G}(2N+1)\geq P_{S}(2N+1)$.

In order to show how the total success probabilities of two protocols depend
on the entanglement of channels for different $N$, we will choose concurrence
$C$ defined by Wootters as a convenient measure of entanglement Wootters . The
concurrence varies from $C=0$ of a separable state to $C=1$ of a maximally
entangled state. For a pure partially entangled state described by Eq. (2),
the concurrence may be expressed explicitly by $C=2|\alpha\beta|$.

In Fig.1, we plot $P_{S}$ and $P_{G}$ as the function of concurrence $C$ for
different $N$. We can see that both the total success probabilities of two
protocols declines with the decrease of the entanglement of channels.
Moreover, the greater $N$ is, the more sharper the success probabilities
declines. It shows that the quantum channel with small entanglement will
become unpractical with the increase of $N$. Fig.1 also indicates explicitly
that the GMTP is more efficient than SMTP. For example, for the case of
$N=10$, the total success probability of GMTP $P_{G}\approx 21\%$ while the
total success probability of SMTP $P_{S}$ only attains $0.14\%$ when the
concurrence of channels is $C=0.96$.

Figure 1: The total success probability $P_{G}$ and $P_{S}$ versus concurrence
$C$ for different $N$ (Solid line: $P_{S}$, dashed line: $P_{G}$). From top to
bottom, $N$ corresponding takes $1,5,10$.

The ratio of $P_{G}$ to $P_{S}$ as a function of $C$ for different $N$ is
illustrated in Fig.2. Here we only take the concurrence from $0.9$ to $1$
because the small entanglement channels are unpractical for large $N$. From
Fig.2, we can see that the greater $N$ is, the larger $P_{G}/P_{S}$ is. In
other words, the efficiency of GMTP is far higher than that of SMTP when the
steps of teleportation increase.

Figure 2: The ratio $P_{G}$ to $P_{S}$versus concurrence $C$ for different $N$
. From bottom to top, $N$ corresponding takes $1,5,10$.

When the entanglement of the quantum channel is different between neighboring
parties, the circumstance becomes complicated. Here we only consider the case
of three parties .

Alice wants to teleport an unknown quantum state
$|\psi\rangle=a|0\rangle+b|1\rangle$ to Bob. There is no direct entanglement
resource between Alice and Bob, fortunately, Alice and the third party Charlie
share a partially entangled state

$|\Psi_{1}\rangle=\alpha_{1}|00\rangle+\beta_{1}|11\rangle,$ (19)

while Charlie and Bob share another entanglement resource

$|\Psi_{2}\rangle=\alpha_{2}|00\rangle+\beta_{2}|11\rangle,$ (20)

where $\alpha_{i}$ and $\beta_{i}$ are real numbers and satisfy
$|\alpha_{i}|\leq|\beta_{i}|$ and $\alpha_{i}^{2}+\beta_{i}^{2}=1$. After two
Bell-basis measurements and Pauli operations, the qubit of Bob will be in one
of the following states

$|\psi_{ij}\rangle=\frac{1}{\sqrt{p_{ij}}}(\alpha_{1}^{1-i}\alpha_{2}^{1-j}\beta_{1}^{i}\beta_{2}^{j}a|0\rangle+\alpha_{1}^{i}\alpha_{2}^{j}\beta_{1}^{1-i}\beta_{2}^{1-j}b|1\rangle)$
(21)

with the probabilities

$p_{ij}=|\alpha_{1}^{1-i}\alpha_{2}^{1-j}\beta_{1}^{i}\beta_{2}^{j}a|^{2}+|\alpha_{1}^{i}\alpha_{2}^{j}\beta_{1}^{1-i}\beta_{2}^{1-j}b|^{2}(i,j=0,1)$

respectively. The qubit can be returned to its original state by performing
the generalized measurement given by Kraus operators:

$\displaystyle E_{S}$ $\displaystyle=|0\rangle\langle
0|+\frac{\alpha_{1}^{1-i}\alpha_{2}^{1-j}\beta_{1}^{i}\beta_{2}^{j}}{\alpha_{1}^{i}\alpha_{2}^{j}\beta_{1}^{1-i}\beta_{2}^{1-j}}|1\rangle\langle
1|,$ (22a) $\displaystyle E_{F}$
$\displaystyle=\sqrt{1-\frac{|\alpha_{1}^{1-i}\alpha_{2}^{1-j}\beta_{1}^{i}\beta_{2}^{j}|^{2}}{|\alpha_{1}^{i}\alpha_{2}^{j}\beta_{1}^{1-i}\beta_{2}^{1-j}|^{2}}}|1\rangle\langle
1|$ (22b)

for
$|\alpha_{1}^{1-i}\alpha_{2}^{1-j}\beta_{1}^{i}\beta_{2}^{j}|\leq|\alpha_{1}^{i}\alpha_{2}^{j}\beta_{1}^{1-i}\beta_{2}^{1-j}|$.
A similar measurement exists if
$|\alpha_{1}^{1-i}\alpha_{2}^{1-j}\beta_{1}^{i}\beta_{2}^{j}|\geq|\alpha_{1}^{i}\alpha_{2}^{j}\beta_{1}^{1-i}\beta_{2}^{1-j}|$.

By tedious but standard calculation we can obtain the success probability of
teleportation

$P=\min\\{2\alpha_{1}^{2},2\alpha_{2}^{2}\\}.$ (23)

It is an interesting result, the success probability of teleportation is
completely determined by the channel of less entanglement. For another channel
of more entanglement, its entanglement does not affect the success probability
at all. In other words, the channel of more entanglement is equivalent to the
maximally entangled channel in the total teleportation process.

In summary, we have presented two multiple teleportation protocols via some
partially entangled state, the separate multiple teleportation and the globe
multiple teleportation. In the former protocol, a complete teleportation
including error correction is strictly executed by neighboring parties.
However, in the latter protocol, all errors introduced in the teleportation
are corrected by the final receiver. It has been shown that the property of
self error-correction is a general feature of multiple teleportations, not
being restricted to the scheme proposed in Ref.s3 . We also have compared the
efficiencies of the two multiple teleportation protocols and found the globe
multiple teleportation is more efficient than the separate multiple
teleportation due to the property of self error-correction.

## References

  * (1) C.H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W.K. Wootters, Phys. Rev. Lett. 70, 1895 (1993).
  * (2) X.B. Wang, T. Hiroshima, A. Tomita, and M. Hayashi, Phys. Rep. 448, 1 (2007).
  * (3) G.L. Long, F.G. Deng, C. Wang, X.H. Li, K. Wen, and W.Y. Wang, Front. Phys. China 2, 251 (2007).
  * (4) D. Gottesman and I.L. Chuang, Nature (London) 402, 390 (1999).
  * (5) E. Knill, R. Laflamme, and G.J. Milburn, Nature (London) 409, 46 (2001).
  * (6) P. Agrawal and A.K. Pati, Phys. Lett. A 305, 12 (2002).
  * (7) G. Gordon and G. Rigolin, Phys. Rev. A 73, 042309 (2006).
  * (8) W.L. Li, C.F. Li, and G.C. Guo, Phys. Rev. A 61, 034301 (2000).
  * (9) F.G. Deng, C.Y. Li, Y.S. Li, H.Y. Zhou, and Y. Wang, Phys. Rev. A 72, 022338 (2005).
  * (10) F.L. Yan and D. Wang, Phys. Lett. A 316, 297 (2003).
  * (11) T. Gao, F.L. Yan, and Y.C. Li, Europhys. Lett. 84, 50001 (2008).
  * (12) M.Y. Wang and F.L. Yan, Phys. Lett. A 355, 94 (2006).
  * (13) M.Y. Wang, F.L. Yan, T. Gao, and Y.C. Li, International Journal of Quantum Information 6, 201 (2008).
  * (14) T. Gao, F.L. Yan, and Z.X. Wang, Quantum Information and Computation 4, 186 (2004).
  * (15) C.H. Bennett1, G. Brassard, S. Popescu, B. Schumacher, J.A. Smolin, and W.K. Wootters, Phys. Rev. Lett. 76, 722 (1996).
  * (16) J. Modławska and A. Grudka, Phys. Rev. Lett. 100, 110503 (2008).
  * (17) W.K. Wootters, Phys. Rev. Lett. 80, 2245 (1998)


# Automated Sex Classification of Children’s Voices and Changes in
Differentiating Factors with Age

Fuling Chen<EMAIL_ADDRESS>Roberto Togneri  Dept. of Electrical,
Electronic and Computer Engineering, University of Western Australia, Perth,
Western Australia, 6155, Australia  Murray Maybery  Diana Weiting Tan  Also
at: Macquarie School of Education, Macquarie University, Sydney, New South
Wales, 2109, Australia.  School of Psychological Science, University of
Western Australia, Perth, Western Australia, 6155, Australia

###### Abstract

Sex classification of children’s voices allows for an investigation of the
development of secondary sex characteristics which has been a key interest in
the field of speech analysis. This research investigated a broad range of
acoustic features from scripted and spontaneous speech and applied a
hierarchical clustering-based machine learning model to distinguish the sex of
children aged between 5 and 15 years. We proposed an optimal feature set and
our modelling achieved an average F1 score (the harmonic mean of the precision
and recall) of 0.84 across all ages. Our results suggest that the sex
classification is generally more accurate when a model is developed for each
year group rather than for children in 4-year age bands, with classification
accuracy being better for older age groups. We found that spontaneous speech
could provide more helpful cues in sex classification than scripted speech,
especially for children younger than 7 years. For younger age groups, a broad
range of acoustic factors contributed evenly to sex classification, while for
older age groups, F0-related acoustic factors were found to be the most
critical predictors generally. Other important acoustic factors for older age
groups include vocal tract length estimators, spectral flux, loudness and
unvoiced features.

##  I  Introduction

The past two decades have witnessed great success in automated sex
classification 111The term “sex classification” refers to the binary
classification of males and females. We acknowledge while there are different
sex identities, including males, females, and other common identities such as
LGBTIQA+ (lesbian, gay, bisexual, transgender, intersex, queer/questioning,
asexual and other identities), previous works have focused on differences
between males and females based on their self-identification with reference to
sex as assigned at birth. We adopt this approach given the nature of the
available substantial dataset, and also because sex as assigned at birth is a
dominant factor in the development of the vocal system. using speech which
aims to identify the binary sex (male or female) of speakers by using
computational modelling [2, 3, 4, 5]. Automated sex classification assists in
the study of the development of secondary sex characteristics which may be
associated with health status [6, 7] and the presence of certain developmental
disorders in children[8, 9, 10]. Sex classification has been found to be
highly accurate for adult speakers. By contrast, the sex classification of
young prepubescent child speakers is much less accurate [11, 12]. Automated
sex classification from speech signals for children remains a challenge
because most acoustic features are not highly distinguishable between boys and
girls. Additionally, another impediment to research in this space is that
collecting children’s voices is time consuming and resource intensive. In the
present study, we aim to develop a machine learning model for sex
classification of children's voices and characterise the differentiating
factors with age.

Numerous studies mainly used statistical analysis to investigate sex
differences for individual acoustic features, and reported several features to
be statistically significantly different between groups of boys and girls. The
most typically studied acoustic features are fundamental frequency (F0) [13,
14, 15, 16], the first three formant frequencies (F1, F2 and F3, which will be
referred to as Fn from here on) [17, 18, 16], acoustic perturbation features,
including jitter, shimmer and Harmonic-to-Noise Ratio (HNR) [19, 20, 15], and
the vocal tract length (VTL) estimator [21]. However, we have limited
knowledge of whether these acoustic features are sufficient for sex
classification for children, and whether the utility of the different acoustic
features in classification varies with age.

Some studies have developed machine learning models to address the sex
classification problem for children and achieved promising results. Such
studies deployed several low-level descriptors, such as the commonly used 57
mel-frequency cepstral coefficients (19 static MFCCs, 19 $\Delta$MFCCs and 19
$\Delta^{2}$MFCCs) [22, 23], the Computational Paralinguistics Challenge
baseline set ComParE (6373 features) [24], the extended Geneva Minimalistic
Acoustic Parameter Set (eGeMAPS, with 88 features) [25, 26] and the linear
predictive cepstral coefficients (LPCCs, with 13 features) [27]. However, such
large numbers of low-level descriptors with severe multicollinearity and
undefined physical meanings are hard to interpret, especially the cepstral-
based features. Moreover, it is not clear which descriptors are discriminant
in sex classification. The reason could be that the models were designed for
sex classification only, but not considering the characterisation of the
contributing factors. Additionally, these feature sets do not cover all
acoustic features that could discriminate boys and girls as mentioned earlier,
such as the VTL estimators.

Furthermore, it is known that children’s voice attributes are age dependent
[28]. Therefore, any successful models of sex classification in children are
likely to be sensitive to the speaker’s age. What is of specific interest is
the age at which a specific voice attribute shows a sex difference. The most
important factor that influences children’s voice attributes is puberty [29,
30], which usually starts between ages 8 and 13 years in girls and between
ages 9 and 15 years in boys [31, 32]. Despite such wide age ranges for puberty
onset, each child experiences his/her puberty in a short period of time.
Children tend to present with different voice quality before, during and after
their puberty. Some studies investigated the variation of acoustic features
with age in children of individual years [18]; while other studies grouped
children by their age, but with different aggregations (e.g., 5-8 years, 9-12
years and 13-15 years in [22]; 4-7 years, 8-11 years, 12-16 years in [17]; 8-9
years, 10-11 years, 12-13 years, 14-15 years in [11]). It is not clear whether
sex classification in children is any better when based on aggregation of year
groups rather than based simply on each age year.

There has been limited investigation of sex classification for children that
has compared their scripted speech and spontaneous speech. Most studies of sex
differences have used scripted productions [19, 13, 20, 21, 17, 22, 14, 15,
18, 16], spontaneous speech [24, 22, 14], or both scripted and spontaneous
speech combined into one cohort [22, 14]. However, it was shown in several
studies that significant differences existed between scripted and spontaneous
speech in terms of acoustic and linguistic parameters [33, 34, 35].
Accordingly, it is of interest to know more about the utility of scripted and
spontaneous speech in sex classification of children and in mapping variation
in distinguishing features across age.

Additionally, as mentioned above, broad sets of features were largely
popularly used in sex classification by means of machine learning models [24,
27, 22, 36]. However, severe multicollinearity could occur in these large
feature sets. For example the mean value and the percentile values of the same
acoustic source are likely to be strongly correlated. Few studies have taken
multicollinearity into account and by not doing so limit the capability to
properly characterise the important contributing factors in sex
classification.

Lastly, several studies have shown that some acoustic features could be used
in differentiating the two sexes in children, but the particular features
identified are not consistent across studies. For example, some studies
reported girls had significantly higher F0 than boys [13, 16], but other
studies reported no difference between boys and girls in F0 [14, 15]. Glaze et
al. [13] investigated F0 on the sustained neutral /a/ vowel; Whiteside and
Hodgeson [16] analysed F0 for the entire duration of the phrase-final vowel
/a:/ following /b/, /k/ and /d$\mathfrak{z}$/; and Sussman and Sapienza [15]
analysed F0 on the sustained vowels /a/ /i/ and /u/. Whiteside and Hodgeson
[16] also reported that girls generally had higher Fn than boys across all
ages. More specifically, Perry et al. [17] established that Fn differentiated
sexes for children aged 4 years and above, while F0 differentiated sexes after
12 years of age. A systematic review [18] showed age- and sex- related changes
in F0 and Fn. Vorperian and Kent found a gradual reduction in Fn and the
standard deviation of Fn (Fn std) with age, the emergence of male-female
differences in Fn by age 4 years with more obvious differences by 8 years, and
jumps in Fn at ages corresponding to growth spurts of the VTL. The acoustic
perturbation features, including jitter, shimmer and HNR, have also been
investigated for children in previous studies. For instance, it was found that
significant sex differences in jitter were evident for adults, but not for
children [15]. Ferrand [19] reported that girls presented significantly higher
HNRs than boys. But another study [20] investigating F0, Fn and perturbation
features reported no significant sex differences for most of the acoustic
features, except for shimmer and F3 for specific vowels. Thus, as alluded to
above, although previous research has identified some acoustic cues that may
discriminate boys’ and girls’ voices, outcomes are not consistent.

The contributions of the present study are three-fold. Firstly, we identify
the optimal feature set for voice sex classification in children, by comparing
classification performance using different feature sets. These features
include the acoustic features (abbreviated as AF from here on, including F0,
Fn, VTL estimators and voice perturbation features), the extended Geneva
Minimalistic Acoustic Parameter Set (eGeMAPS) [37] which covers most features
in AF except for VTL estimators, and the combined feature set of eGeMAPS and
VTL. Secondly, we establish the optimal age grouping strategy for sex
classification by building models for children of each year and children of
larger age bands (5-8 years, 9-12 years, and 13-15 years) and comparing the
models’ performances. Thirdly, we use a correlation-matrix-based hierarchical
clustering method applied on an optimal set of features to address
multicollinearity while retaining the critical information. Lastly, we
identify the critical factors in sex classification as they change across the
childhood age range.

##  II  Dataset

The OGI Kids’ Speech Corpus [38] is a collection of spontaneous and scripted
speech recorded in the Northwest Regional School District near Portland,
Oregon. A sex-balanced group of approximately 100 children per grade from
Kindergarten (5 years) through to grade 10 (15 years), approximately 1100
children in total, participated in the collection. The protocol consisted of
scripted speech and spontaneous speech. The scripted speech is composed of
three types of stimulus; 205 isolated words, 100 prompted sentences, and 10
numeric strings. In the present study, each stimulus was regarded as one
sample of scripted speech. All scripted speech utterances were rated on a
three-point scale consisting of “good” utterances (the word is clearly
intelligible with no significant background noise or extraneous speech),
“questionable” (intelligible but accompanied by other sounds) or “bad”
utterances (unintelligible or wrong word spoken). We only included good
utterances in the present study. After the collection of scripted speech was
completed, the experimenter asked the participants a series of questions
intended to elicit spontaneous speech (e.g., “Tell me about your favourite
movie”). In the present study, each recording of the spontaneous speech was
segmented into 5 second timeframe samples. The total amount of speech recorded
per subject was approximately 8-10 minutes. A summary of the number of
participants and number of samples of scripted and spontaneous speech is
shown, per year, in TABLE 1.

Table 1: Number of subjects and samples per year. age | subjects | scripted samples | spontaneous samples
---|---|---|---
|  girls |  boys |  girls |  boys |  girls |  boys
5 |  49 |  39 |  1393 |  1092 |  1133 |  925
6 |  31 |  58 |  1585 |  3131 |  668 |  1190
7 |  61 |  53 |  3237 |  2685 |  1367 |  1184
8 |  52 |  63 |  3262 |  3726 |  1223 |  1410
9 |  44 |  47 |  2508 |  2642 |  844 |  877
10 |  50 |  49 |  2972 |  2902 |  957 |  952
11 |  55 |  57 |  2001 |  1992 |  1004 |  1097
12 |  51 |  46 |  1872 |  1754 |  958 |  917
13 |  50 |  49 |  1923 |  1808 |  930 |  910
14 |  39 |  69 |  1586 |  2773 |  596 |  1171
15 |  29 |  75 |  1132 |  2898 |  486 |  1388

##  III  Methodology

###  III.1  Feature sets

####  III.1.1  Acoustic features

A set of 23 widely known AF were used to describe the vocal characteristics of
each sample. Among these features, mean value of F0 (F0), standard deviation
of F0 (F0 std), HNR, all jitter features (local Jitter, local absolute Jitter,
rap Jitter, ppq5 Jitter and ddp Jitter) and all shimmer features (local
Shimmer, apq3/5/11 Shimmer and dda Shimmer) were obtained from Parselmouth
0.3.3 which is a Python package for the Praat Software [39]. The mean values
of F1, F2, F3 and F4 measure the corresponding formants at each glottal pulse
using the formant position formula [40]. The apparent VTL was estimated using
six features based on F1, F2, F3 and F4, namely formant dispersion (fdisp)
[41], average formant frequency (avgF) [42], geometric mean formant frequency
(mff) [43], Fitch formant estimate (fitch_vtl) [41], formant spacing
($\Delta$f) [44] and formant position (pF) [40].

####  III.1.2  Extended Geneva Minimalistic Acoustic Parameter Set

The extended Geneva Minimalistic Acoustic Parameter Set (eGeMAPS) [37] is
composed of 88 low-level descriptors which cover frequency related parameters
(F0, Jitter, Fn frequencies and bandwidths), energy and amplitude related
parameters (shimmer, loudness and HNR), spectral parameters (Alpha Ratio,
Hammarberg Index, Spectral Slope, Fn energy, Harmonic difference H1-H2, H1-A3)
and temporal features (rate of loudness peaks, voiced and unvoiced regions,
and the number of continuous voiced regions per second). The parameters were
derived using the default configuration in the openSMILE Python package [45].

####  III.1.3  eGeMAPS+VTL

Considering that eGeMAPS covers most features in the AF feature set except for
the estimates of vocal tract length which could be important features in
discriminating the two sexes, we proposed an extended feature set called
eGeMAPS+VTL. eGeMAPS+VTL comprises 94 features that include the 88 low-level
descriptors in eGeMAPS and the 6 estimators of VTL from the AF feature set,
namely pF, fdisp, avgF, mff, fitch_vtl and $\Delta$f.

###  III.2  Acoustic characterisation (correlation-matrix-based hierarchical
clustering)

The correlation matrices of the feature sets for the children aged 5 to 15
years showed that some features were highly and consistently correlated
throughout all ages (e.g., all jitter features in AF, all shimmer features in
AF, and Fn bandwidths in eGeMAPS), and some features were highly correlated
for younger children but were independent for older children (e.g., F0 related
features were strongly correlated with Fn bandwidths in children aged 5 to 7
years, but became independent for children aged above 8 years). Such
complexity of inter-correlations and severe multicollinearity introduces
challenges in interpreting the contributing factors in sex classification.

Feature selection and variable clustering are the most popular methods for
tackling the multicollinearity problem. Feature selection aims to select the
most related feature among a group of ``similar” features. It is suitable for
features that are known to be replaceable by other ``similar” features in the
same group. In the present work, due to the complex inter-relationships among
multiple features and the uniqueness of each feature, feature selection is not
suited. Therefore, we used a correlation-matrix-based hierarchical clustering
method to group the strongly correlated features and represent them with newly
generated variables.

The method is one of the agglomerative clustering methods, which are popular
in multiple fields [46, 47]. The idea of such methods is to group variables
according to their ``similarity”. In the present work, we defined
``similarity” as the absolute value of the Pearson’s correlation coefficient
of the two objects, which was also used for the multicollinearity diagnostic.
The most similar clusters are merged at each step to produce larger clusters.
The outcome of this clustering method is the distribution of the factors and
their inclusive features. It should be noticed that some factors could be
composed of multiple features, and some factors are just the features
themselves, depending on the independency of the features. In the following
sections, we will use the term ``factor” to refer to the independent cluster
generated by the clustering algorithm, and the term ``feature” to refer to the
original feature that was extracted as described in the Section III.1. The
workflow of the clustering method and the details of the algorithm we
implemented can be found in the Appendix.

###  III.3  Extreme Random Forest Model

The Extreme Random Forest (ERF) model is one of the most popular machine
learning algorithms used for classification. It is an extension of the Random
Forest (RF) model, and is an ensemble machine learning algorithm [48]. The ERF
algorithm works by creating a large number of unpruned decision trees from the
training dataset. In the case of classification, predictions are made by using
majority voting. The ERF provides good predictive performance and low over-
fitting by selecting a split point at random and fitting each decision tree on
the whole training set.

Furthermore, the ERF provides factor importance by computing the impurity of
each node. The more a factor decreases the impurity, the more important the
factor is. The impurity decreases provided by factors can be averaged across
trees to determine the factor importance. Factors that are selected at the top
of trees are in general more important than factors that are selected at the
end nodes of trees, as top splits lead to bigger information gains.

###  III.4  Experiments and evaluation

We conducted two steps of experimentation. In the first step, three sets of
ERF models were trained by using the three sets of features (AF, eGeMAPS and
eGeMAPS+VTL) with no application of the clustering method. This modelling was
initially on all speech samples including scripted speech and spontaneous
speech. In each set of models, we built 11 models by using samples of children
per year, and 3 models by using samples of children per age band (5-8 years,
9-12 years, and 13-15 years), giving a total of 14 models per feature set.
Depending on the performance of these three sets of models, two decisions were
made to proceed to the next step: determining the optimal feature set, and
whether one model per year or one model per age band is more appropriate for
sex classification in children. In the second step, two sets of ERF models
were trained by using the optimal feature set and the optimal age division
with the application of the clustering method on scripted speech and
spontaneous speech. Based on the outcomes of the second step, we then obtained
the weights of the factors in distinguishing the male and female voices in
each age division.

Because the samples of the two classes (boys and girls) are not strictly
balanced for size, we applied Borderline-SMOTE [49] from the Python imblearn
package to oversample the minor class to balance with the major class.

The most suitable hyper-parameters of each ERF model were obtained by
exhaustive search over specified parameter values and using a cross-validation
splitting strategy of five folds with five repetitions, evaluated by the
weighted F1 score. The F1 score is a measure of a test’s accuracy in
statistical analysis of binary classification, which is calculated as in Eqs
(1), (2), and (3). The final sex prediction of each subject was decided by
majority voting of the predictions across all boy/girl samples from the test
sets.

$\displaystyle F1$ $\displaystyle=2\ddagger\frac{precision\ddagger
recall}{precision+recall}$ (1) $\displaystyle precision$
$\displaystyle=\frac{truepositive}{truepositive+falsepositive}$ (2)
$\displaystyle recall$
$\displaystyle=\frac{truepositive}{truepositive+falsenegative}$ (3)

To evaluate the results of sex classification performance, we evaluated the F1
scores for both classes and the mean of the F1 scores of the two classes for
the overall performance of each model. We also compared our results with the
previous studies of sex classification using the same cohort [22, 23].

##  IV  Results and discussion

###  IV.1  Sex classification by using AF, eGeMAPS and eGeMAPS+VTL

TABLE 2 presents the F1 scores for girls and boys and the mean F1 scores for
both classes per age year and per age band, when using each of the three
feature sets described in Section III.1 (AF, eGeMAPS, and eGeMAPS+VTL).

Table 2: F1 scores for girls (G) and boys (B) and the mean F1 scores for both
classes (Mean). The feature set sizes are presented in parentheses.

age AF(23) eGeMAPS(88) eGeMAPS +VTL(94) G B Mean G B Mean G B Mean 5 0.68 0.63
0.66 0.87 0.82 0.85 0.86 0.82 0.84 6 0.57 0.84 0.7 0.72 0.89 0.81 0.69 0.88
0.79 7 0.72 0.57 0.64 0.73 0.69 0.71 0.75 0.7 0.73 8 0.58 0.62 0.6 0.71 0.77
0.74 0.75 0.8 0.77 9 0.79 0.81 0.8 0.79 0.81 0.8 0.8 0.8 0.8 10 0.67 0.67 0.67
0.78 0.77 0.78 0.8 0.78 0.79 11 0.78 0.75 0.77 0.82 0.82 0.82 0.83 0.83 0.83
12 0.81 0.77 0.79 0.85 0.82 0.83 0.89 0.86 0.87 13 0.91 0.9 0.91 0.94 0.93
0.94 0.94 0.93 0.94 14 0.82 0.9 0.86 0.83 0.91 0.87 0.85 0.92 0.89 15 0.74
0.88 0.81 0.95 0.98 0.96 0.95 0.98 0.96 5-8 0.61 0.66 0.64 0.67 0.73 0.7 0.68
0.73 0.71 9-12 0.71 0.7 0.7 0.77 0.76 0.77 0.77 0.75 0.76 13-15 0.76 0.81 0.79
0.81 0.85 0.83 0.82 0.86 0.84

In comparing the performance of using AF, eGeMAPS, and eGeMAPS+VTL,
eGeMAPS+VTL is generally more accurate in differentiating boys from girls than
the other two feature sets. This result indicates the importance of the VTL
features in sex classification for children, which is consistent with the
findings in [21]. Our results also show that the eGeMAPS feature set provided
more helpful information in sex classification than the AF set across all
ages.

As expected, sex classification is more successful in older children than in
younger children. Interestingly, we found that the F1 scores were higher
(i.e., sex classification was more successful) for the analyses conducted for
each year of age rather than for the broader age bands. The likely reason is
that some acoustic factors vary with children’s growth, but at different rates
for boys and girls. So the two sexes could experience similar changes in
voice, but over different age ranges. For instance, Vorperian and Kent [18]
showed that the average F1-F2 acoustic space for boys at 4-6 years old largely
overlapped with the acoustic space for girls at 5-8 years, with a similar
pattern also observed for boys at 7-11 years and girls at 11-16 years. By
mixing children of different ages, younger boys or older girls could be
misclassified because of their similar acoustic characteristics. Therefore, we
strongly suggest, given plenty of data across the age range, that children’s
voice sex classification should be conducted based on each year, instead of
broader age bands.

Sex classification has been investigated in the recent studies [22, 23], which
used the same stimulus cohort but different features and models. In [22], the
researchers used cepstral features (57 MFCCs), and fed them into a Gaussian
Mixture Model with a Universal Background Model (GMM-UBM) which achieved an
accuracy of 0.77 for 5-8 year olds, 0.84 for 9-12 year olds, and 0.76 for
13-15 year olds. sex. And in [23], a Deep Neural Network (DNN) model was
applied on 60 MFCCs to the whole samples of children which resulted in an
overall accuracy of 0.82 . It is worth pointing out two limitations of the
studies [22, 23]. Firstly, the MFCC features are difficult to interpret in
terms of human voice quality , therefore there is no evident acoustic analysis
of how MFCC features could be different in sex for children. In our present
work, we used the more explainable feature set eGeMAPS which could be used to
describe voice quality in multiple ways. Secondly, the GMM-UBM model and the
DNN model showed great capability in classification tasks, but these models do
not provide evidence on how they make the classification decision with regard
to the input data (e.g., factor importance). While the models in these two
studies provide rates of classification accuracy comparable to the rates
reported for the model used in the present study, output from the models in
[22, 23] is not as amenable to interpretation as the output from the present
model.

As discussed above, due to the best performance being given by the combination
feature set of eGeMAPS+VTL based on each year, in the following sections, we
will only consider using eGeMAPS+VTL features for children of each year.

###  IV.2  Sex classification of different types of speech

TABLE 3 presents the F1 scores for girls and boys and the average F1 scores
across the two classes, when conducting analyses separately for scripted and
spontaneous speech.

Table 3: F1 scores for girls (G) and boys (B) and the macro average F1 scores for both classes (Mean), using scripted and spontaneous speech. age | scripted | spontaneous
---|---|---
|  G |  B |  Mean |  G |  B |  Mean
5 |  0.82 |  0.76 |  0.79 |  0.86 |  0.81 |  0.84
6 |  0.69 |  0.88 |  0.79 |  0.74 |  0.86 |  0.8
7 |  0.73 |  0.69 |  0.71 |  0.82 |  0.8 |  0.81
8 |  0.73 |  0.79 |  0.76 |  0.73 |  0.79 |  0.76
9 |  0.83 |  0.84 |  0.84 |  0.83 |  0.84 |  0.83
10 |  0.8 |  0.79 |  0.8 |  0.79 |  0.8 |  0.8
11 |  0.74 |  0.73 |  0.73 |  0.87 |  0.88 |  0.87
12 |  0.87 |  0.83 |  0.85 |  0.87 |  0.84 |  0.85
13 |  0.94 |  0.93 |  0.94 |  0.91 |  0.89 |  0.9
14 |  0.85 |  0.92 |  0.89 |  0.9 |  0.94 |  0.92
15 |  0.95 |  0.98 |  0.96 |  0.96 |  0.99 |  0.98

Generally, the model using spontaneous speech generated better performances
than the models using the scripted speech. Few studies have investigated the
acoustic differences between scripted and spontaneous speech, which means
there is a lack of knowledge about the salient acoustic factors that could be
used to classify children by sex when using spontaneous speech rather than
scripted speech. This could be due to several reasons, such as small samples,
inconsistent data size between scripted speech and spontaneous speech, and
weak methods in analysing the discriminant acoustic factors.

###  IV.3  Sex classification using individual features and clustered features

TABLE 4 presents the results of F1 scores for classification before clustering
(BC-Mean) and after clustering (AC-Mean), by using scripted and spontaneous
speech samples. The results in the “BC-Mean” columns in TABLE 4 are the same
as the results in the “Mean” columns in TABLE 3, which were obtained by using
the individual 94 eGeMAPS+VTL features, while the results in the “AC-Mean”
columns in TABLE 4 were obtained by applying the clustering algorithm.
Generally, after clustering, the F1 scores were slightly lower than the F1
scores before clustering, which is within expectation. This is because, after
clustering, data dimensionality was reduced considerably from the original 94
dimensions to between 41 and 63 dimensions (shown as column “n” in TABLE 4),
with n varying with the age of the children and the type of speech. After
clustering, the redundant information carried by highly correlated features
was eliminated, and the essential information was condensed within each
independent cluster.

Table 4: The F1 scores before clustering (BC-Mean) and after clustering (AC-Mean), by using scripted and spontaneous speech samples. age | scripted | spontaneous
---|---|---
|  BC-Mean |  AC-Mean |  n111n is the number factors generated after clustering. |  BC-Mean |  AC-Mean |  n
5 |  0.79 |  0.73 |  54 |  0.84 |  0.74 |  46
6 |  0.79 |  0.77 |  56 |  0.8 |  0.78 |  50
7 |  0.71 |  0.66 |  55 |  0.81 |  0.76 |  44
8 |  0.76 |  0.71 |  60 |  0.76 |  0.68 |  47
9 |  0.84 |  0.75 |  59 |  0.83 |  0.8 |  41
10 |  0.8 |  0.79 |  58 |  0.8 |  0.73 |  48
11 |  0.73 |  0.77 |  59 |  0.87 |  0.86 |  50
12 |  0.85 |  0.8 |  59 |  0.85 |  0.85 |  45
13 |  0.94 |  0.94 |  58 |  0.9 |  0.92 |  49
14 |  0.89 |  0.86 |  63 |  0.92 |  0.88 |  51
15 |  0.96 |  0.94 |  59 |  0.98 |  0.96 |  49

As the F1 scores after clustering are not significantly less than the F1
scores before clustering, we regarded these independent clusters suitable for
further investigation of the salient acoustic factors in sex classification.

###  IV.4  Critical factors

In summary of Sections IV.1, IV.2 and IV.3, the sex classification model
benefits more from using spontaneous speech for children of each individual
year, and the sex classification performance is acceptable after clustering.
In this section, we present and discuss the important factors derived from the
modelling outlined in Section IV.3 for spontaneous speech.

We took the clusters generated for the results in the column ``AC-Mean” in
TABLE 4, and referring to the clusters as acoustic factors, then extracted the
factor importance of these clusters from the ERF models. The factor importance
was presented in the form of the weight which was the percentage of the
contribution of the factor.

TABLE LABEL:tab:clustertop10 presents the clustering pattern and the weights
of the top 10 important factors across all ages. The features column lists the
eGeMAPS+VTL features which constitute the clustering factors. The remaining
columns provide the weights for each year across the listed features. For each
age year, the 11 weights shown (the top 10 shown in percentage and the last
row) sum up to 100%. The features which are grouped in the same cluster are
marked with the same symbol and their cells merged where possible (e.g., the
cluster marked with ``$\ddagger$" includes all features from F0 mean to UVL
for most age years, with the exception of ages 13 and 14 where only the first
5 features are included). The weight associated with the cluster is provided
in the merged cells. For completeness, the acoustic factors not in the top 10
for some age years are also included but without their weights (e.g., the
cluster marked with ``$\ddagger$" is not in the top 10 for ages 8 and 10). We
excluded features that were not among the top factors of any age year and
included them in the last row labeled "not in top 10" with their aggregated
weight. More detailed information about TABLE LABEL:tab:clustertop10 can be
found in the supplementary material222See supplementary material at
https://github.com/FulingChen/Automated-Sex-Classification-of-Children-Voices-
and-Changes-in-Differentiating-Factors-with-Age/blob/main/SuppPub1.xlsx for
the clustering pattern and the weights of the factors across all ages, which
includes all the 94 original features of eGeMAPS+VTL.

Table 5: The clustering pattern and the weights of the top 10 important factors across all ages.  | Weights of each year
---|---
features | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15
F0 mean | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$
F0 p20 | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$
F0 p50 | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | 4% | 10% | 31% | 14% | 6%
F0 p80 | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$
HNR | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$
H1-A3 | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $||$ | $||$ | $||$ | $||$ | $||$ | $||$
mfcc1V | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | 7% | 5% | 5% | 6% | 6% | 3%
F1 | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\bullet$ | $\bullet$ | $\ddagger$
F2 | 6% | 4% | 3% | $\ddagger$ | 4% | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\bullet$ | $\bullet$ | $\ddagger$
F3 | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\bullet$ | $\bullet$ | $\ddagger$
F1 BD | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\bullet$ | $\bullet$ | $\ddagger$
F2 BD | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\bullet$ | 3% | $\ddagger$
F3 BD | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\bullet$ | $\bullet$ | $\ddagger$
aR V | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\bullet$ | $\bullet$ | $\ddagger$
hI V | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\bullet$ | $\bullet$ | $\ddagger$
F1 std | $\ddagger$ | $\diamond$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\diamond$ | $\diamond$ | $\diamond$ | $\diamond$ | $\ddagger$
F2 std | $\ddagger$ | 3% | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\diamond$ | 3% | $\diamond$ | $\diamond$ | $\ddagger$
F3 std | $\ddagger$ | $\diamond$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\diamond$ | $\diamond$ | $\diamond$ | $\diamond$ | $\ddagger$
UVL | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ | $\ddagger$ |  |  |  |  |  | $\ddagger$
slpV0-500 |  | 3% | 3% | 3% |  |  |  | $\ddagger$ |  | 4% |
F1 BD std |  |  | $\ddagger$ |  |  |  |  |  |  |  |
avgF |  |  |  |  |  |  |  |  |  |  |
$\Delta$f |  |  |  |  |  |  |  |  |  |  |
pF | 5% | 4% | 3% |  |  | 7% |  | 4% | 7% | 4% | 7%
mff |  |  |  |  |  |  |  |  |  |  |
fitch_vtl |  |  |  |  |  |  |  |  |  |  |
aR UV |  |  |  |  |  |  |  |  |  |  |
hI UV | 3% |  |  | 3% | 4% |  | 3% | 3% |  | 5% | 5%
slpUV0-500 |  |  |  |  |  |  |  |  |  |  |
SF mean | // | // | // | // | // | // | // | // | // | // | //
SF UV | // | // | // | 3% | 4% | // | // | // | // | // | //
SF V | // | // | // | // | // | // | // | // | // | // | //
LD mean | $\star$ | $\star$ | $\star$ | $\star$ | $\star$ | // | $\star$ | // | // | $\star$ | //
LD p20 | $\star$ | $\star$ | $\star$ | $\star$ | $\star$ | // | $\star$ | // | // | $\star$ | //
LD p50 | $\star$ | $\star$ | $\star$ | $\star$ | $\star$ | // | $\star$ | // | // | $\star$ | //
LD p80 | 3% | 3% | 4% | 3% | 6% | 8% | $\star$ | 4% | 3% | $\star$ | 8%
SL | $\star$ | $\star$ | $\star$ | $\star$ | $\star$ | // | $\star$ | // | // | $\star$ | //
slpUV500-1500 | $\star$ |  |  | $\star$ | $\star$ | // |  | // | // |  |
LD p02 |  |  | 3% | // | // | // |  | // | // |  | //
LD rslp | // |  |  | // | // | // |  | // | // |  | //
LD rslp std | // |  |  | // | // | // |  | // | // |  | //
LD fslp |  |  |  | // | // | // |  | // | // |  | //
LD fslp std |  |  |  | // | // | // |  | // | // |  | //
mfcc1 |  | 3% |  |  | 4% | // |  | // |  |  | 5%
F0 std |  |  |  | 3% |  |  | 3% | 4% |  |  |
LD std | + |  |  | 3% | + | 7% |  |  |  |  |
mfcc1 std |  |  |  |  | + |  |  |  |  |  |
F2 BD std | + | 3% |  |  | + |  |  |  |  |  |
F3 BD std | 4% | 4% |  |  | 4% |  |  |  |  |  |
aR V std | + |  |  |  | + | 5% |  |  |  |  |
hI V std | + |  |  |  | + | + |  |  |  |  |
mfcc1V std | + |  |  |  | + |  |  |  |  |  |
F1amp |  |  |  |  |  |  | $\sim$ |  |  |  |
F2amp |  |  |  |  |  |  | $\sim$ |  |  |  |
F3amp |  |  |  |  |  |  | $\sim$ |  |  |  |
F1amp std |  |  |  |  |  |  | 3% |  |  |  |
F2amp std |  |  |  |  |  |  | $\sim$ |  |  |  |
F3amp std |  |  |  |  |  |  | $\sim$ |  |  |  |
VPS |  |  |  |  |  |  | $\sim$ |  |  |  |
mfcc2 | 3% |  |  |  | 4% |  | 3% |  |  | 3% |
mfcc2V | 3% | 3% | 5% | 3% | $\triangleright$ |  | 3% |  |  | 3% |
mfcc2 std |  |  |  |  |  |  | 3% |  |  |  |
mfcc3 |  |  |  | 3% | 3% |  |  |  | 2% |  |
mfcc3V | 3% |  | 3% | 3% | $\triangleleft$ |  |  |  | $\triangleleft$ |  |
mfcc3 std |  |  |  |  |  | 5% |  |  |  |  |
jitter std |  |  |  |  |  |  |  |  | 6% |  |
shimmer std |  |  |  |  | 4% |  |  |  |  |  |
HNR std |  |  |  |  | 3% | 4% |  |  | 2% |  |
H1-H2 | 3% |  | 3% |  |  |  |  | 4% | 2% |  | 3%
H1-A3 std |  |  |  |  |  |  | 4% | 3% | 5% | 2% |
slpV500-1500 |  |  | 3% |  |  | 7% |  | 4% | 3% | 3% | 3%
mfcc4V |  | 2% |  |  |  | 7% |  |  |  |  | 3%
LDPPS | 3% |  |  |  |  | 4% |  |  |  |  |
fdisp |  |  | 3% | 3% |  |  | 3% |  |  |  | 5%
not in top 10 | 66% | 67% | 64% | 70% | 60% | 39% | 67% | 56% | 34% | 54% | 51%

We found that, for children aged below 12 years old, there were no obvious
acoustic factors that could significantly differentiate boys from girls, with
all weights lower than 10%. Each acoustic factor counts evenly in sex
classification for young children, and the final judgements were made based on
considering all the involved factors. As children’s age increased, some
acoustic factors started to play important roles in sex classification.

The critical factors could vary with age. For instance, the 19 features (F0
mean, F0 p20, F0 p50, F0 p80, HNR, H1-A3, mfcc1V, F1, F2, F3, F1 BD, F2 BD, F3
BD, aR V, hI V, F1 std, F2 std, F3 std and UVL) formed a large cluster (marked
with ``$\ddagger$") and became an important factor for young children aged
between 6 to 9 years, as shown in TABLE LABEL:tab:clustertop10. However, with
the increasing age, this big cluster started to break down into several small
clusters, and some of these small clusters were identified to be important
factors in sex classification.

For those features that were consistently highly correlated and commonly
appeared in the top 10 important factors across the age range, we identified
them as critical acoustic factors and discussed them in the following
sections. These important factors are 1) F0 and HNR (including F0 mean, F0
p20, F0 p50, F0 p80 and HNR), 2) Fn and Fn BD (including F1, F2, F3, F1 BD, F2
BD, F3 BD), 3) H1-A3 and mfcc1V, 4) VTL (including avgF, $\Delta$f, pF, mff
and fitch_vtl), 5) Spectral flux (including SF mean, SF UV and SF V), 6)
unvoiced features (including aR UV, hI UV and slpUV0-500), and 7) loudness
(including LD mean, LD p20, LD p50, LD p80 and SL). Considering the inclusive
features are highly correlated (r$>$.75), in the following sections, we
selected one or a few commonly discussed features to represent the cluster
without loss of generality. We did not analyse the representatives generated
by the clustering algorithm, because the application of PCA involved in the
clustering algorithm on multiple features would affect the interpretation. We
report the t-test statistic (t), p-value (p) and Cohen’s d (d) for each
comparison. Due to the large data size, we relied on Cohen’s d (d) in drawing
conclusions about the statistical analyses. Our interpretation is to refer to
effect sizes as small ($\mid$d$\mid<$0.2), small to medium
(0.2$\leq\mid$d$\mid<$0.5), medium to large (0.5$\leq\mid$d$\mid<$0.8, marked
in bold), and large ($\mid$d$\mid\geq$0.8, marked in bold), based on
benchmarks suggested by Cohen [51].

Table 6: Summary based on t tests of differences on the features between boys and girls. (#$>$0.05, †$<$0.05, ††$<$0.01, †††$<$0.001; negative values of d and t indicate that the mean value of the feature is larger in boys than in girls) feature | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15
---|---|---|---|---|---|---|---|---|---|---|---
F0 mean | t | 6.3 | 1.2 | -3.0 | 1.2 | 7.8 | 6.6 | 11.9 | 19.9 | 45.9 | 23.7 | 22.1
p | ††† | # | †† | # | ††† | ††† | ††† | ††† | ††† | ††† | †††
d | 0.3 | 0.1 | -0.1 | 0.1 | 0.4 | 0.3 | 0.5 | 1.0 | 2.2 | 1.2 | 1.2
HNR | t | 5.4 | 0.3 | -2.1 | -0.2 | 6.6 | 12.1 | 7.9 | 19.6 | 42.4 | 27.6 | 24.8
p | ††† | # | † | # | ††† | ††† | ††† | ††† | ††† | ††† | †††
d | 0.3 | 0.0 | -0.1 | 0.0 | 0.3 | 0.6 | 0.4 | 0.9 | 2.0 | 1.4 | 1.3
F1 | t | 4.8 | 2.4 | 2.3 | -1.5 | 3.7 | 11.6 | 8.9 | 14.0 | 15.4 | 14.4 | 13.9
p | ††† | † | † | # | ††† | ††† | ††† | ††† | ††† | ††† | †††
d | 0.2 | 0.1 | 0.1 | -0.1 | 0.2 | 0.6 | 0.4 | 0.7 | 0.7 | 0.8 | 0.8
F2 | t | 3.9 | 4.3 | 3.6 | 0.4 | 6.1 | 11.5 | 9.3 | 15.0 | 16.9 | 11.7 | 12.5
p | ††† | ††† | ††† | # | ††† | ††† | ††† | ††† | ††† | ††† | †††
d | 0.2 | 0.2 | 0.2 | 0.0 | 0.3 | 0.5 | 0.4 | 0.7 | 0.8 | 0.6 | 0.7
F3 | t | 3.9 | 4.0 | 1.4 | 0.7 | 5.2 | 9.4 | 9.6 | 14.3 | 15.8 | 10.9 | 13.1
p | ††† | ††† | # | # | ††† | ††† | ††† | ††† | ††† | ††† | †††
d | 0.2 | 0.2 | 0.1 | 0.0 | 0.3 | 0.4 | 0.4 | 0.7 | 0.8 | 0.6 | 0.7
H1- A3 | t | 2.3 | 5.7 | 2.1 | 4.3 | -2.8 | -10.6 | -11.8 | -13.0 | -25.7 | -22.8 | -7.6
p | † | ††† | † | ††† | †† | ††† | ††† | ††† | ††† | ††† | †††
d | 0.1 | 0.3 | 0.1 | 0.2 | -0.1 | -0.5 | -0.5 | -0.6 | -1.2 | -1.2 | -0.4
fitch _vtl | t | 0.5 | 7.0 | 2.2 | 5.3 | 0.0 | -13.4 | -2.6 | -8.5 | -18.0 | -15.0 | -13.3
p | # | ††† | † | ††† | # | ††† | †† | ††† | ††† | ††† | †††
d | 0.0 | 0.3 | 0.1 | 0.2 | 0.0 | -0.6 | -0.1 | -0.4 | -0.8 | -0.8 | -0.7
SF mean | t | 2.5 | 3.8 | -7.8 | -4.8 | -8.6 | -13.1 | -1.4 | -8.6 | -13.5 | -0.1 | 7.3
p | † | ††† | ††† | ††† | ††† | ††† | # | ††† | ††† | # | †††
d | 0.1 | 0.2 | -0.3 | -0.2 | -0.4 | -0.6 | -0.1 | -0.4 | -0.6 | 0.0 | 0.4
aR_ UV | t | -3.1 | 2.4 | 7.0 | 4.8 | -2.9 | 0.0 | 3.7 | -5.3 | -0.4 | -15.5 | -15.5
p | †† | †† | ††† | ††† | †† | # | ††† | ††† | # | ††† | †††
d | -0.1 | 0.1 | 0.3 | 0.2 | -0.1 | 0.0 | 0.2 | -0.2 | 0.0 | -0.8 | -0.8
LD mean | t | 2.0 | 0.9 | -9.4 | -8.1 | -7.5 | -11.2 | 0.2 | -7.3 | -9.3 | 5.1 | 16.5
p | † | # | ††† | ††† | ††† | ††† | # | ††† | ††† | ††† | †††
d | 0.1 | 0.1 | -0.4 | -0.3 | -0.4 | -0.5 | 0.0 | -0.3 | -0.4 | 0.3 | 0.9

Figure1a.jpg.5(a) Figure1b.jpg.5(b)

Figure1c.jpg.5(c) Figure1d.jpg.5(d)

Figure1e.jpg.5(e) Figure1f.jpg.5(f)

Figure1g.jpg.5(g) Figure1h.jpg.5(h)

Figure1i.jpg.5(i) Figure1j.jpg.5(j)

Figure 1: Box plots of the features. (white - girls, grey - boys)

####  IV.4.1  F0 and HNR

F0 mean, F0 p20, F0 p50 and F0 p80 (these 4 features are named F0 from here
on) and HNR consistently appeared in the same factor across all ages. From
TABLE LABEL:tab:clustertop10, it is observed that F0 and HNR did not
contribute substantially to sex classification in younger children (weights
less than 6% for 5 to 11 year-olds) but started to be more discriminant for
older children (weights 10%-31% in 12-, 13-, and 14-year-olds). As described
earlier, for younger children, F0 and HNR, together with other acoustic
features, formed a large cluster which consists of 19 features in total, which
gradually broke down into several small clusters and individual features with
the increase in age. At ages 13 and 14, the cluster composed of F0 and HNR
became independent of other features and was also the most important factor in
sex classification.

Because the F0 and HNR related features are strongly correlated, we selected
the F0 mean and HNR to represent the cluster and conducted the following
analysis. FIG.1, 1 and TABLE 6 presents the F0 mean and HNR distributions and
the statistical summary for both sexes across the age range from 5 to 15 years
old. Specifically, the effect sizes for F0 and HNR are large in children older
than 12, and are medium to large in 10- and 11-year-olds. F0 and HNR have
small effects for children aged between 6 and 8 years.

####  IV.4.2  Fn and Fn BD

Fn and Fn BD consistently appeared in the same cluster throughout the age
range as they were strongly correlated, and they were grouped with F0 and HNR
for young children ($<$13 years old). As Fn and Fn BD were highly correlated,
we selected Fn to represent their cluster. FIG.1, 1, 1 and TABLE 6 show the Fn
distributions and the statistical summary.

The sex differences in Fn and Fn BD and the variation with age have been
rarely reported in developmental studies. In the present study, the results
reveal that: (1) Fn reduces with advancing age in both sexes; (2) Fn is
generally lower in boys than in girls across all ages, an outcome consistent
with results reported by [52, 53, 18], and (3) the differences in Fn between
boys and girls become more consistent and more pronounced with increasing age,
which is consistent with the findings in [53, 18].

It is known that Fn is related to the wave resonance of the pharynx [54],
which, together with changes associated with puberty [55], explains why the
sex differences in Fn increase with increasing age. Another possible reason
for the sex differences in Fn are sex-specific articulatory behaviours. It was
found that most vowels are articulated with considerable variation in the size
of the mouth opening, tongue dorsum position and jaw movements, and Fn is
sensitive to such changes [56, 57, 58]. For instance, males tend to lengthen
their vocal tract by using a more pronounced degree of lip protrusion than
females which could effectively decrease Fn [58]. It was also reported that
the sex differences in F1 could be derived from the sex differences in the
width and height of the lip opening and amount of jaw opening during the
production of /æ/ [59, 58].

####  IV.4.3  H1-A3 and mfcc1V

Because mfcc1V is difficult to interpret, we selected H1-A3 and conducted the
analysis on it. H1-A3 represents the harmonic difference which is defined as
the ratio of energy of the first harmonic (H1) to the energy of the highest
harmonic in the third formant range (A3). FIG.1 and TABLE 6 demonstrate the
H1-A3 distributions and statistical summary across the age range for both
sexes. It is noteworthy that the emergence of a substantial sex difference
happens at the age of 10, with girls showing decreasing values in H1-A3 with
advancing age while there is no clear age trend in boys. It is known that
H1-A3 is correlated with the source spectral tilt [60], where lower spectral
tilt values indicate more abrupt glottal closures and higher spectral tilt
values reflect less non-simultaneous closure. The results of our study
indicate that the source spectral tilt is sex dependent in children older than
10 years.

####  IV.4.4  VTL

The factor of VTL is composed of 5 features across the age range, including
avgF, $\Delta$f, pF, mff and fitch_vtl. As these features were all highly
correlated, we selected the Fitch formant estimate (fitch_vtl) as the
representative feature of this factor. FIG.1 and TABLE 6 show the fitch_vtl
distribution and effect size across the age range for both sexes. As expected,
fitch_vtl, as an estimator of VTL, shows a substantial sex difference, but
restricted to older children (aged 13-15), likely a consequence of puberty.
Specifically, boys show a significant increase in fitch_vtl after 13 years,
while fitch_vtl does not vary significantly with age in girls. This detailed
pattern of results supports adding the VTL estimators to the eGeMAPS feature
set to improve sex classification for children, as we demonstrated in Section
IV.1.

####  IV.4.5  Spectral flux

Spectral flux (SF) represents a quadratic, normalised version of the simple
spectral difference between the spectra of two consecutive speech frames. This
factor is composed of three features, including the SF of all regions (SF
mean), voiced regions (SF V), and unvoiced regions (SF UV). SF mean was
selected as representative of these three features, as they are highly
correlated. FIG.1 and TABLE 6 show the distribution and the statistical
summary of SF mean values for both sexes across the age range.

We found that boys and girls showed different age trends for SF mean, with SF
mean being consistently low for boys up until age 9, then increasing from age
10 to a consistently higher level, whereas for girls this transition occurred
late, with SF mean levels remaining relatively low across ages 5-13 before
increasing substantially for ages 14 and 15. These trends contributed to
substantial sex differences favouring boys in the age range 9-13 followed by a
reversal of the sex difference at age 15. SF mean is one indicator of the
timbre of an audio signal, which means the timbre of children’s voices
experience a significant change for boys at 9 years and for girls at 14 years.
The sex difference favouring girls at age 15 is consistent with outcomes of a
study investigating SF in discriminating adult male and female voices [61].

####  IV.4.6  Unvoiced features

The unvoiced features (UV) include the features of aR, hI and spectral slopes
0-500Hz of the unvoiced segments (aR UV, hI UV and slpUV0-500). Unvoiced
speech is a type of speech without any periodic nature, but it is different to
silence. During the production of unvoiced speech, the air exhaling out of the
lungs through the trachea is not interrupted by the vibrating vocal folds. The
air flow could be completely or narrowly obstructed by total or partial
closure of somewhere along the length of the vocal tract, which results in
stop or frication excitation and excites the vocal tract system to produce
unvoiced speech. The typical examples of voiced signals are /a/, /e/, /i/,
/u/, /o/; while the typical examples of unvoiced signals include stop
consonants /p/, /t/, /k/ and some unvoiced fricative consonants /f/, /$\phi$/,
//.

As the four inclusive features are highly correlated, we selected aR_UV as a
representative of this cluster. FIG.1 and TABLE 6 present the distributions
and the statistical summary of aR_UV. In the present work, we found that the
sex difference appears at ages 14 and 15, with the aR_UV decreasing in girls
while increasing in boys.

####  IV.4.7  Loudness

The factor of loudness (LD) includes LD mean, LD p20, LD p50, LD p80 and SL.
Loudness is an estimate of perceived signal intensity from an auditory
spectrum. Due to the high correlations among the inclusive features, we
analysed the LD mean as a representative of the factor. FIG.1 and TABLE 6
shows the loudness distribution and effect size across the age range for both
sexes. It provides evidence of girls sounding softer than boys at age 10 but
louder than boys at age 15. Most previous studies used the F0 to estimate the
highness or lowness of a tone perceived by the ear or used decibel (dB) to
measure vocal intensity, and assess sex differences. The loudness related
features used in the current study, different from F0 and vocal intensity, are
a set of features that approximate humans’ non-linear perception of sound by
applying an auditory spectrum in the Perceptual Linear Prediction[62]. Limited
studies were found to compare with our findings regarding the sex differences
in loudness varying with age, given that the loudness features are relatively
new and not well understood.

##  V  Conclusions

This study investigated a broad range of acoustic features and proposed a
hierarchical clustering-based machine learning model for the sex
classification of children and characterisation of the salient acoustic
factors. The set of models provided promising results in predicting sexes in
children aged from 5 to 15 years, by using the combination feature set of
eGeMAPS and VTL estimators. We reported greater success in predicting sexes by
using spontaneous speech rather than scripted speech, especially for younger
children. Smaller sets of independent acoustic factors were then obtained
after the application of hierarchical clustering for children of each year,
which showed equivalent performance in sex classification to performance when
using the full set of features. The independent acoustic factors were used for
salient acoustic factors characterisation.

We then analysed and discussed the important acoustic factors regarding the
sex differences and age variation. We found that, for young children, all
acoustic factors contributed to the sex classification equivalently. In other
words, there are no noticeable acoustic factors that contribute substantially
to distinguishing the voices of young girls and boys. For older children, sex
differences become more prominent. Among the important factors, we found F0,
HNR, Fn and H1-A3 showed significant sex differences for children aged above
10 years; VTL estimators are critical contributors in sex classification in
children aged above 13 years; SF has medium to large effect size in
differentiating sexes for children at 10 and 13 years old; unvoiced features
differentiate the two sexes for children aged above 14 years old; and girls
and boys have different loudness in their speech when they are at 10 and 15
years old.

The limitations of this study are that: (1) the data were not strictly
balanced between sexes for each year. For instance, in terms of the
spontaneous samples, the number of boys was almost twice the number of girls
at the age of 6 (1190:668) and 14 (1171:596), and the ratio was 2.86:1 between
boys and girls at age 15 (1388:486). This imbalance could cause bias in sex
classification, where the samples of the minority class could be misclassified
to be the majority class; and (2) though we found the classification models
were more accurate for spontaneous speech than for scripted speech, the sample
size of spontaneous speech is considerably smaller than the sample size of
scripted speech, especially for younger children (see TABLE 1). Future studies
could consider deploying a more sex-balanced dataset and a larger dataset with
more spontaneous speech.

###### Acknowledgements.

The authors would like to thank Khaldoun Shobaki, John-Paul Hosom, and Ronald
A. Cole for making the OGI Kids’ Speech Corpus available.

*

##  Appendix A Procedures of Clustering

Figure 2: Scheme of correlation-matrix-based hierarchical clustering

Step 1. Each feature of the original feature set was assigned to be the
simplest object of size 1. Assuming the original feature set was composed of m
features, a correlation matrix of size m x m was first obtained. During the
following steps, m would be reduced by 1 per iteration.

Step 2. If the maximum absolute value of the correlation coefficients between
two objects A and B, max(abs(r(A,B)), was larger than the cut-off value
(0.75), the decision was made that the objects A and B were highly correlated
and should be merged in the subsequent steps. Otherwise, all the objects in
the current pool were regarded to be independent and there was no severe
multicollinearity. The objects A or B could be clusters of any size larger
than 1. There is no universal agreement on the cut-off value of the
correlation coefficient in multicollinearity diagnostic. Investigators have
used a cut-off value of 0.5 or greater [63]. In the present study, we set the
cut-off value to be 0.75.

Step 3. Once the objects A and B were determined to be highly correlated, a
one-component principal component analysis (PCA) was applied on all features
wrapped in A and B to generate a new object X to represent A and B. For
instance, if A was composed of 2 features, and B was composed of 3 features, a
PCA would be applied on these 5 inclusive features. We regarded the newly
generated object X as representative of the 5 features.

Step 4. The two objects A and B were then removed from the pool that generated
the correlation matrix in step 1 and replaced with X. The current pool would
then be composed of m-1 objects, including the one newly generated object. The
correlation matrix was then updated to be of size (m-1)$\times$(m-1).

Step 5. Steps 2 to 4 were repeated until all the absolute values of the
pairwise correlation coefficients were lower than 0.75.

## References

  * [1] The term “sex classification” refers to the binary classification of males and females. We acknowledge while there are different sex identities, including males, females, and other common identities such as LGBTIQA+ (lesbian, gay, bisexual, transgender, intersex, queer/questioning, asexual and other identities), previous works have focused on differences between males and females based on their self-identification with reference to sex as assigned at birth. We adopt this approach given the nature of the available substantial dataset, and also because sex as assigned at birth is a dominant factor in the development of the vocal system.
  * [2] G. Archana and M. Malleswari, ``Gender identification and performance analysis of speech signals,'' in _2015 Global conference on communication technologies (GCCT)_ , IEEE (2015), pp. 483–489.
  * [3] D. Childers, K. Wu, K. Bae, and D. Hicks, ``Automatic recognition of gender by voice,'' in _ICASSP-88., International Conference on Acoustics, Speech, and Signal Processing_ , IEEE Computer Society (1988), pp. 603–604.
  * [4] E. Ramdinmawii and V. Mittal, ``Gender identification from speech signal by examining the speech production characteristics,'' in _2016 International conference on signal processing and communication (ICSC)_ , IEEE (2016), pp. 244–249.
  * [5] Y.-M. Zeng, Z.-Y. Wu, T. Falk, and W.-Y. Chan, ``Robust gmm based gender classification using pitch and rasta-plp parameters of speech,'' in _2006 International conference on machine learning and cybernetics_ , IEEE (2006), pp. 3376–3379.
  * [6] W. Daniel Jr, W. R. Harlan, E. A. Harland, and G. P. Grillo, ``Secondary sex characteristics of girls 12 to 17 years of age: the us health examination survey,'' The Journal of pediatrics 96(6), 1074–1078 (1980).
  * [7] W. R. Harlan, G. P. Grillo, J. Cornoni-Huntley, and P. E. Leaverton, ``Secondary sex characteristics of boys 12 to 17 years of age: the us health examination survey,'' The Journal of pediatrics 95(2), 287–289 (1979).
  * [8] S. Tordjman, P. Ferrari, V. Sulmont, M. Duyme, and P. Roubertoux, ``Androgenic activity in autism,'' American Journal of Psychiatry 154(11), 1626–a (1997).
  * [9] S. Baron-Cohen, A. Tsompanidis, B. Auyeung, B. Nørgaard-Pedersen, D. M. Hougaard, M. Abdallah, A. Cohen, and A. Pohl, ``Foetal oestrogens and autism,'' Molecular Psychiatry 25(11), 2970–2978 (2020).
  * [10] G. Worley, C. M. Houlihan, M. E. Herman-Giddens, M. E. O’Donnell, M. Conaway, V. A. Stallings, W. C. Chumlea, R. C. Henderson, E. B. Fung, P. L. Rosenbaum, _et al._ , ``Secondary sexual characteristics in children with cerebral palsy and moderate to severe motor impairment: a cross-sectional survey,'' Pediatrics 110(5), 897–902 (2002).
  * [11] Y.-L. Shue and M. Iseli, ``The role of voice source measures on automatic gender classification,'' in _2008 IEEE International Conference on Acoustics, Speech and Signal Processing_ , IEEE (2008), pp. 4493–4496.
  * [12] G. Chen, X. Feng, Y.-L. Shue, and A. Alwan, ``On using voice source measures in automatic gender classification of children's speech,'' in _Eleventh Annual Conference of the International Speech Communication Association_ (2010).
  * [13] L. E. Glaze, D. M. Bless, P. Milenkovic, and R. D. Susser, ``Acoustic characteristics of children's voice,'' Journal of Voice 2(4), 312–319 (1988).
  * [14] D. N. Sorenson, ``A fundamental frequency investigation of children ages 6–10 years old,'' Journal of Communication Disorders 22(2), 115–123 (1989).
  * [15] J. E. Sussman and C. Sapienza, ``Articulatory, developmental, and gender effects on measures of fundamental frequency and jitter,'' Journal of Voice 8(2), 145–156 (1994).
  * [16] S. P. Whiteside and C. Hodgson, ``Some acoustic characteristics in the voices of 6-to 10-year-old children and adults: a comparative sex and developmental perspective,'' Logopedics Phoniatrics Vocology 25(3), 122–132 (2000).
  * [17] T. L. Perry, R. N. Ohde, and D. H. Ashmead, ``The acoustic bases for gender identification from children’s voices,'' The Journal of the Acoustical Society of America 109(6), 2988–2998 (2001).
  * [18] H. K. Vorperian and R. D. Kent, ``Vowel acoustic space development in children: A synthesis of acoustic and anatomic data,'' (2007).
  * [19] C. T. Ferrand, ``Harmonics-to-noise ratios in normally speaking prepubescent girls and boys,'' Journal of Voice 14(1), 17–21 (2000).
  * [20] M. Guzman, D. Muñoz, M. Vivero, N. Marín, M. Ramírez, M. T. Rivera, C. Vidal, J. Gerhard, and C. González, ``Acoustic markers to differentiate gender in prepubescent children's speaking and singing voice,'' International journal of pediatric otorhinolaryngology 78(10), 1592–1598 (2014).
  * [21] J. F. Lehman and R. Singh, ``Estimation of children's physical characteristics from their voices.,'' in _INTERSPEECH_ (2016), pp. 1417–1421.
  * [22] S. Safavi, M. Russell, and P. Jančovič, ``Automatic speaker, age-group and gender identification from children’s speech,'' Computer Speech & Language 50, 141–156 (2018).
  * [23] M. Sarma, K. K. Sarma, and N. K. Goel, ``Multi-task learning dnn to improve gender identification from speech leveraging age information of the speaker,'' International Journal of Speech Technology 23(1), 223–240 (2020).
  * [24] H. Kaya, A. A. Salah, A. Karpov, O. Frolova, A. Grigorev, and E. Lyakso, ``Emotion, age, and gender classification in children’s speech by humans and machines,'' Computer Speech & Language 46, 268–283 (2017).
  * [25] N. Cummins, B. Vlasenko, H. Sagha, and B. Schuller, ``Enhancing speech-based depression detection through gender dependent vowel-level formant features,'' in _Conference on artificial intelligence in medicine in Europe_ , Springer (2017), pp. 209–214.
  * [26] F. A. Shaqra, R. Duwairi, and M. Al-Ayyoub, ``Recognizing emotion from speech based on age and gender using hierarchical models,'' Procedia Computer Science 151, 37–44 (2019).
  * [27] P. B. Ramteke, A. A. Dixit, S. Supanekar, N. V. Dharwadkar, and S. G. Koolagudi, ``Gender identification from children's speech,'' in _2018 Eleventh International Conference on Contemporary Computing (IC3)_ , IEEE (2018), pp. 1–6.
  * [28] E. L. M. Tavares, R. B. d. Labio, and R. H. G. Martins, ``Normative study of vocal acoustic parameters from children from 4 to 12 years of age without vocal symptoms: a pilot study,'' Brazilian Journal of otorhinolaryngology 76, 485–490 (2010).
  * [29] J. Abitbol, P. Abitbol, and B. Abitbol, ``Sex hormones and the female voice,'' Journal of voice 13(3), 424–446 (1999).
  * [30] M. L. Harries, J. M. Walker, D. M. Williams, S. Hawkins, and I. Hughes, ``Changes in the male voice at puberty,'' Archives of disease in childhood 77(5), 445–447 (1997).
  * [31] W. A. Marshall and J. M. Tanner, ``Variations in the pattern of pubertal changes in boys,'' Archives of disease in childhood 45(239), 13–23 (1970).
  * [32] M. D. Wheeler, ``Physical changes of puberty.,'' Endocrinology and metabolism clinics of North America 20(1), 1–14 (1991).
  * [33] V. Farantouri, A. Potamianos, and S. Narayanan, ``Linguistic analysis of spontaneous children speech,'' in _First Workshop on Child, Computer and Interaction_ (2008).
  * [34] P. Howell and K. Kadi-Hanifi, ``Comparison of prosodic properties between read and spontaneous speech material,'' Speech communication 10(2), 163–169 (1991).
  * [35] M. Nakamura, K. Iwano, and S. Furui, ``Differences between acoustic characteristics of spontaneous and read speech and their effects on speech recognition performance,'' Computer Speech & Language 22(2), 171–184 (2008).
  * [36] A. Zourmand, H.-N. Ting, and S. M. Mirhassani, ``Gender classification in children based on speech characteristics: using fundamental and formant frequencies of malay vowels,'' Journal of Voice 27(2), 201–209 (2013).
  * [37] F. Eyben, K. R. Scherer, B. W. Schuller, J. Sundberg, E. André, C. Busso, L. Y. Devillers, J. Epps, P. Laukka, S. S. Narayanan, _et al._ , ``The geneva minimalistic acoustic parameter set (gemaps) for voice research and affective computing,'' IEEE transactions on affective computing 7(2), 190–202 (2015).
  * [38] K. Shobaki, J.-P. Hosom, and R. Cole, ``The ogi kids’ speech corpus and recognizers,'' in _Proc. of ICSLP_ (2000), pp. 564–567.
  * [39] Y. Jadoul, B. Thompson, and B. De Boer, ``Introducing parselmouth: A python interface to praat,'' Journal of Phonetics 71, 1–15 (2018).
  * [40] D. A. Puts, C. L. Apicella, and R. A. Cárdenas, ``Masculine voices signal men's threat potential in forager and industrial societies,'' Proceedings of the Royal Society B: Biological Sciences 279(1728), 601–609 (2012).
  * [41] W. T. Fitch, ``Vocal tract length and formant frequency dispersion correlate with body size in rhesus macaques,'' The Journal of the Acoustical Society of America 102(2), 1213–1222 (1997).
  * [42] K. Pisanski and D. Rendall, ``The prioritization of voice fundamental frequency or formants in listeners’ assessments of speaker size, masculinity, and attractiveness,'' The Journal of the Acoustical Society of America 129(4), 2201–2212 (2011).
  * [43] D. R. Smith and R. D. Patterson, ``The interaction of glottal-pulse rate and vocal-tract length in judgements of speaker size, sex, and age,'' The Journal of the Acoustical Society of America 118(5), 3177–3186 (2005).
  * [44] D. Reby and K. McComb, ``Anatomical constraints generate honesty: acoustic cues to age and weight in the roars of red deer stags,'' Animal behaviour 65(3), 519–530 (2003).
  * [45] F. Eyben, M. Wöllmer, and B. Schuller, ``Opensmile: the munich versatile and fast open-source audio feature extractor,'' in _Proceedings of the 18th ACM international conference on Multimedia_ (2010), pp. 1459–1462.
  * [46] Y. Gu and C. Wang, ``A study of hierarchical correlation clustering for scientific volume data,'' in _International Symposium on Visual Computing_ , Springer (2010), pp. 437–446.
  * [47] X. Liu, X.-H. Zhu, P. Qiu, and W. Chen, ``A correlation-matrix-based hierarchical clustering method for functional connectivity analysis,'' Journal of neuroscience methods 211(1), 94–102 (2012).
  * [48] P. Geurts, D. Ernst, and L. Wehenkel, ``Extremely randomized trees,'' Machine learning 63(1), 3–42 (2006).
  * [49] H. Han, W.-Y. Wang, and B.-H. Mao, ``Borderline-smote: a new over-sampling method in imbalanced data sets learning,'' in _International conference on intelligent computing_ , Springer (2005), pp. 878–887.
  * [50] See supplementary material at https://github.com/FulingChen/Automated-Sex-Classification-of-Children-Voices-and-Changes-in-Differentiating-Factors-with-Age/blob/main/SuppPub1.xlsx for the clustering pattern and the weights of the factors across all ages.
  * [51] J. Cohen, _Statistical power analysis for the behavioral sciences_ (Routledge, 2013).
  * [52] S. Bennett, ``Vowel formant frequency characteristics of preadolescent males and females,'' The Journal of the Acoustical Society of America 69(1), 231–238 (1981).
  * [53] R. D. Kent and H. K. Vorperian, ``Static measurements of vowel formant frequencies and bandwidths: A review,'' Journal of communication disorders 74, 74–97 (2018).
  * [54] G. Fant, ``Speech sounds and features.,'' (1973).
  * [55] W. T. Fitch and J. Giedd, ``Morphology and development of the human vocal tract: A study using magnetic resonance imaging,'' The Journal of the Acoustical Society of America 106(3), 1511–1522 (1999).
  * [56] G. Fant, _Acoustic theory of speech production_ , 2 (Walter de Gruyter, 1970).
  * [57] P. Ladefoged and R. Harshman, ``Formant frequencies and movements of the tongue,'' in _Proc. UCLA Work. Papers Phonetics_ (1979), Vol. 45, pp. 39–52.
  * [58] K. N. Stevens and A. S. House, ``Development of a quantitative description of vowel articulation,'' The Journal of the Acoustical Society of America 27(3), 484–493 (1955).
  * [59] B. E. Lindblom and J. E. Sundberg, ``Acoustical consequences of lip, tongue, jaw, and larynx movement,'' The Journal of the Acoustical Society of America 50(4B), 1166–1179 (1971).
  * [60] H. M. Hanson and E. S. Chuang, ``Glottal characteristics of male speakers: Acoustic correlates and comparison with female data,'' The Journal of the Acoustical Society of America 106(2), 1064–1077 (1999).
  * [61] A. Ghosal and S. Dutta, ``Automatic male-female voice discrimination,'' in _2014 International conference on issues and challenges in intelligent computing techniques (ICICT)_ , IEEE (2014), pp. 731–735.
  * [62] H. Hermansky, ``Perceptual linear predictive (plp) analysis of speech,'' the Journal of the Acoustical Society of America 87(4), 1738–1752 (1990).
  * [63] C. Donath, E. Gräßel, D. Baier, C. Pfeiffer, S. Bleich, and T. Hillemacher, ``Predictors of binge drinking in adolescents: ultimate and distal factors-a representative study,'' BMC public health 12(1), 1–15 (2012).