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Complexity
Volume 2018 (2018), Article ID 2173640, 9 pages
https://doi.org/10.1155/2018/2173640
Research Article

Invariant Measures Based on the U-Correlation Integral: An Application to the Study of Human Voice

1Laboratorio de Señales y Dinámicas no Lineales, Facultad de Ingeniería, Universidad Nacional de Entre Ríos, Ruta Prov. 11, Km 10, Oro Verde, Entre Ríos, Argentina
2Instituto de Investigación y Desarrollo en Bioingeniería y Bioinformática, Oro Verde, Entre Ríos, Argentina

Correspondence should be addressed to Juan F. Restrepo; ra.ude.aireinegnioib@opertserj

Received 27 September 2017; Revised 20 February 2018; Accepted 12 March 2018; Published 17 April 2018

Academic Editor: Roberto Tonelli

Copyright © 2018 Juan F. Restrepo and Gastón Schlotthauer. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Nonlinear measures such as the correlation dimension, the correlation entropy, and the noise level were used in this article to characterize normal and pathological voices. These invariants were estimated through an automated algorithm based on the recently proposed U-correlation integral. Our results show that the voice dynamics have a low dimension. The value of correlation dimension is greater for pathological voices than for normal ones. Furthermore, its value also increases along with the type of the voice. The low correlation entropy values obtained for normal and pathological type 1 and type 2 voices suggest that their dynamics are nearly periodic. Regarding the noise level, in the context of voice signals, it can be interpreted as the power of an additive stochastic perturbation intrinsic to the voice production system. Our estimations suggest that the noise level is greater for pathological voices than for normal ones. Moreover, it increases along with the type of voice, being the highest for type voices. From these results, we can conclude that the voice production dynamical system is more complex in the presence of a pathology. In addition, the presence of the inherent stochastic perturbation strengthens along with the voice type. Finally, based on our results, we propose that the noise level can be used to quantitatively differentiate between type and type voices.

1. Introduction

The human voice is the most important means of communication among individuals. Thanks to vocal communication, activities like asking for help are apparently trivial in our daily routine. Thus, a voice disorder can limit our ability to cover our most basic needs, producing a negative impact on our quality of life. For this reason, it is very important not only to increase our knowledge about the mechanism of voice production but also to characterize its dynamics in normal and pathological conditions.

In the literature, several methodologies to assess human voices can be found. However, their reliability depends on the nature of the studied voice. Consequently, Titze proposed a qualitative classification for voice signals [1]. The scheme proposed by Titze divides the signals into three types: type signals are nearly periodic voice signals, type signals have strong modulation or subharmonics, and type signals are characterized by a very irregular or even chaotic behavior. Sprecher et al. proposed a modification to this scheme. They redefined type voices as being deterministic chaotic signals, adding a fourth type that is characterized by a dominant random-like behavior [2].

Figure 1 shows the time series, the state space reconstruction, and the spectrogram of each type of voice. A normal voice (first column) is characterized for a quasiperiodic time representation and a smooth attractor in the reconstructed state space. Moreover, from its spectrogram, one can easily observe the fundamental frequency and its harmonics. A pathological type voice (second column) displays a more irregular time series than the normal voice. Notice that although their attractors are similar, pathological type attractor is not so smooth. Furthermore, both spectrograms are also similar in the sense that one can still distinguish a fundamental frequency and its harmonics. Nevertheless, the noise content blurs the harmonics at high frequencies. In the case of a pathological type voice (third column), it is possible to observe a less regular time series, compared to the pathological type voice. The volume occupied by its attractor has been reduced and it is more difficult to distinguish a smooth shape. In its spectrogram, one can find subharmonic frequencies. The pathological type voice (fourth column) is characterized by an irregular time series. The state space reconstruction is similar to the one of white Gaussian noise. Its spectrogram shows the fundamental frequency but its harmonics are rapidly blurred. Finally, a pathological type voice (fifth column) has an irregular time series representation and a regular state space reconstruction, like the type voice. In its spectrogram, one can find a fundamental frequency but not the harmonics.

Figure 1: Time series, state space reconstruction, and spectrogram of normal and pathological voices. First column: normal voice; second column: pathological type voice; third column: pathological type voice; fourth column: pathological type voice; fifth column: pathological type voice.

These representations are useful to differentiate between some kinds of voices, for example, between normal and pathological type voices. However, it is very difficult to differentiate between a normal and a pathological type voice or between a type and a type voice. The importance of this classification is based on the fact that traditional perturbation measures like jitter and shimmer give reliable results if they are applied to type and type signals. In contrast, nonlinear dynamics concepts should be used to characterize type voices, whereas they are unreliable when applied to type signals. Moreover, until now, the classification of type voices has been done subjectively by visual inspection [2].

Over the last three decades, strong evidence about the nonlinear behavior of the voice production mechanism has been collected [36], leading the scientific community to develop concepts and methods based on nonlinear dynamics and chaos theories. There exist an extensive set of publications in which those concepts have been used to characterize healthy and pathological voices. The correlation dimension and the correlation entropy are two quantities that are used to characterize the complexity of a dynamical system. The former can be thought of as an estimation of the number of variables involved in the dynamics (degrees of freedom) [7]. The latter measures the rate at which information about the dynamics is lost over time. More complex systems are commonly characterized by having higher dimensions and entropy values [8].

The correlation integral is the quantity used to estimate and [9]. It has been widely used in the biomedical field since it was proposed by Grassberger and Procaccia in the early 19s [9].

According to what we know so far, this correlation integral has been used in all studies that have estimated and from voice signals. However, it is well known that the Grassberger and Procaccia approach is not robust in the presence of noise, and it needs special conditions to converge [10]. More robust variants of the correlation integrals have been proposed, like the Gaussian kernel correlation integral (GCI) [11] and the U-correlation integral [12].

The objective of this article is to characterize the dynamics of normal and pathological voices through the correlation dimension, the correlation entropy, and the noise level . Furthermore, we seek to statistically analyze these invariants for the four types of voices. As a novelty, we obtain these invariants using U-correlation integral through an automated algorithm [12] which allowed us to avoid subjective judgements. Finally, we propose a new quantitative methodology to differentiate between type and type voices.

2. Materials and Methods

The correlation dimension and the correlation entropy are invariants that characterize the natural measure of a dynamical system [13]. These invariants can be easily computed from the correlation integral which is obtained from indirect temporal measures of one of the variables of the system [9]. For an observed stationary time series of length , , the reconstructed -dimensional delay vectors , must be formed. The correlation integral , in its general form, is defined as the probability that the distance between two randomly selected -dimensional delay vectors and is smaller than a value [13]:where is a kernel function and is the probability density function (pdf) of the distance between pairs of delay vectors. In this article, we used the Euclidean distance.

The Grassberger and Procaccia correlation integral uses as a kernel function , where is the Heaviside step function [9]. On the other hand, the GCI proposed by Diks et al., , adopts [11]. The main advantage of the GCI over the Grassberger and Procaccia correlation integral is that the former allows us to model the influence of additive noise on the scaling law.

For a zero-mean time series of variance with additive white Gaussian noise of variance , the scaling rule is [11, 14, 15]where

It is important to say that all time series used in this work were rescaled to have unitary standard deviation. In this sense, is the noise level after rescaling; that is, corresponds to clean time series and implies a pure stochastic process. The signal-to-noise ratio (SNR) can be calculated as

In practical applications, it is very important to be able to quantify , the noise level in the time series. This is because it allows us to correct the estimations of and . Moreover, the conclusions driven by the interpretation of the invariants calculated in the presence of high levels of noise should be taken carefully. For this reason, an estimate of the noise level must be also reported, allowing the readers to be aware of the limitations of the estimates. On the other hand, when the data is taken under controlled conditions, like the voices in the database analyzed herein, a high noise level can be seen as an indicator of a dominant underlying additive stochastic process.

The main disadvantage of the GCI is that it requires high values of to obtain a convergent estimate of [10, 15]. This is because the pdf of the interpoint distance changes with the embedding dimension, whereas the GCI’s kernel does not (see (1)). As a result, the convergence of the GCI to is slowed down [12]. The last statement also applies to the Grassberger and Procaccia correlation integral. As a solution, we have recently proposed the U-correlation integral (UCI) [12]: where is the upper incomplete Gamma function, is the Gamma function, and is the pdf of the squared interpoint distance. There are two important aspects about this correlation integral that deserve to be mentioned. First, note that the UCI’s kernel function, given by has a parameter that is used to incorporate information about the embedding dimension. In other words, this kernel function is able to change according to . Second, we are now working with the square of the interpoint distance, that is, , to reduce the computational cost.

The scaling law for the UCI is [12] where is a normalization constant and is the Gauss hypergeometric function. The UCI is approximated with the U-correlation sum which, in this article, is calculated as follows: first, we obtain the squared distances between each pair of -dimensional delay vectors , where , and thenwhere . The evaluation of the function requires numerical integration which is computationally expensive. Instead, we performed the evaluation through spline interpolation of the function , where for each value of .

Once the correlation sum is calculated, we estimated , , and using coarse-grained estimators. These are explicit expressions for , , and as functions of and [13]. Such functions are useful because they allow us to estimate the invariants and to visually confirm a scaling regime. In [16], we have presented three coarse-grained functions: , , and which are based on the UCI. As an advantage, they can be calculated in an automated manner from two U-correlation integrals. Moreover, we have proposed an automated algorithm to estimate , , and from these coarse-grained estimators [16].

2.1. Coarse-Grained Estimators Based on UCI

Here, we present the coarse-grained estimators used in this study. A more detailed description can be found in [12, 16].

We define the noise level functional as [12] where

This functional is the difference between the logarithmic derivatives of two U-correlation integrals: , which is the UCI calculated with squared distances coming from -dimensional delay vectors and a kernel with the parameter , and . The last one is the correlation integral obtained from distances between pairs of -dimensional delay vectors and a kernel with the parameter .

We have shown in [12] that is a function that decreases monotonically from to on a scale proportional to the value of . From this noise functional, we can define the noise level coarse-grained estimator as [12]

The coarse-grained estimators for and can be read as [16] respectively.

The calculation of requires the estimation of and two correlation integrals: and . On the other hand, depends on the correlation integrals and , , the logarithmic derivative , and the coarse-grained estimator .

2.2. Database

This study was conducted using the Massachusetts Eye & Ear Infirmary (MEEI) Voice Disorders Database, distributed by Kay Elemetrics [17]. We have chosen a subset of records ( normal and pathological) of sustained phonation of vowel /a/ [18]. Each record was downsampled to  kHz and the subjects’ descriptive statistics are presented in Table 1. From this set of voices, we excluded records because they contained less than sample points, ending with a total of records to be analyzed.

Table 1: Analyzed subjects: number and age.

For each record, a centered window of data points was selected and normalized to have zero mean and unitary standard deviation. Then, the U-correlation sums were computed for , , and . Moreover, the nearest temporal neighbors of each delay vector were discarded [7]. Finally, Algorithm 1 was applied. We must clarify that the range of values for the embedding dimension was selected taking into account the studies conducted in [6, 1921], where a low-dimensional vocal system is suggested ( for pathological voices). On the other hand, the embedding lag was selected as the average lag (over all voices) where the first minimum of the mutual information function occurs [7, 10].

Algorithm 1: Automated estimation of attractors’ invariants.

In a preliminary study, the pathological records were classified into Titze’s scheme based on the visual observation of the time series and its spectrograms. The classification resulted in type records and type records. The remainder voices where taken apart since these techniques cannot differentiate between type and type voices. Following the definition of type voices given by Sprecher et al. in [2], we calculated the coarse-grained estimators and for these voices. Then, we grouped together the voices that did show a scaling regime (finite correlation dimension and entropy) as type and those that did not as type .

3. Results

The first result of the simulations can be observed in Figure 2. It shows the coarse-grained estimators , , and for a normal voice (Figures 2(a), 2(f), and 2(k)), a pathological type voice (Figures 2(b), 2(g), and 2(l)), a pathological type voice (Figures 2(c), 2(h), and 2(m)), a pathological type voice (Figures 2(d), 2(i), and 2(n)), and a pathological type voice (Figures 2(e), 2(j), and 2(o)).

Figure 2: Coarse-grained estimators , , and for normal and pathological voices. First column: normal voice; second column: pathological type voice; third column: pathological type voice; fourth column: pathological type voice; fifth column: pathological type voice. The curves for are color-coded in grayscale where the lightest gray corresponds to .

It is important to observe that all coarse-grained estimators present a scaling region, except the ones for correlation dimension and entropy of type voices. This highlights the suitability of these estimators to analyze normal and pathological (type , type , and type ) voices.

As it can be observed in the first row of Figure 2, the noise level is greater for the pathological voices than for the normal voice and, in the pathological case, it increases along with the type. This means that, for both normal and pathological voices, there is an underlying stochastic component and its level increases in the presence of pathology.

The behavior of the estimator is presented in the second row of Figure 2. As it can be seen, this estimator suggests a value of for normal voices and slightly greater values for pathological type and type voices. However, it is difficult to say whether there is any difference between the analyzed normal and pathological voice. Nevertheless, these results suggest that the voice production system has a relatively low dimension.

Regarding the estimator , it can be observed from the third row of Figure 2 that, for the voices analyzed herein, the estimator converges to values close to zero. This suggests the presence of a strong harmonic component. Furthermore, there exists a small increase of entropy from normal voice to pathological type voice and from the latter to pathological type voice. This reflects an increasing degree of irregularity which is often associated with an increase in complexity.

For the type voice, it can be observed in the fourth column of Figure 2 that the noise level coarse-grained estimator converges to a higher value than the one corresponding to the type voice. However, it is still possible to observe a scaling range in both and . Moreover, these estimators converge to higher values than the ones for type voice. On the other hand, for the type voice (fifth column of Figure 2), it is not possible to find a scaling range either in dimension or in the entropy estimator. This behavior is expected since, by definition, these types of voices have an infinite value of dimension and entropy.

As we mentioned before, this kind of nonlinear techniques should not be used with type voices. Note that, from the curves of and (Figures 2(j) and 2(o), resp.), it is not easy to decide whether there is a scaling range. In this sense, an untrained person could erroneously determine its existence, resulting in misleading estimations.

On the other hand, a scaling range can be found for (Figure 2(e)), and it converges to a very high noise level value (SNR  dB). This suggests that the dynamics are mostly ruled by the underlying stochastic component. It is important to mention that this behavior is consistent in all voices that did not have a scaling region on and (type voices). Based on this result, we suggest that a high value of could be a quantitative indicator of type voices.

In order to find a noise level value that allows us to differentiate between type and type voices, we select by visual inspection all voices that did not show a scaling region on the estimators and . Then, for each signal, we estimate the noise level value from and select the minimum of these values () as a threshold.

3.1. Automated Estimation of Invariants

In order to obtain reliable estimations of , , and , it is essential to verify the existence of a scaling behavior over the coarse-grained estimators. Once it is found, one has to choose a range of values from which to estimate the invariants. This choice is critical since, in practical application, the coarse-grained estimators strongly vary as a function of the scale. To avoid subjective judgements, we have proposed an algorithm (see Algorithm 1) for the automated estimation of , , and based on the coarse-grained estimators , , and , respectively [16].

This algorithm selects the scaling range where those invariants should be estimated based on the next criteria: (i) the coarse-grained estimator must be constant for a range of scale values and (ii) the value of the invariant should converge as the embedding dimension increases.

The algorithm begins by approximating the U-correlation integrals and for different values () using (8). Note that must be calculated for since the shape parameter of the incomplete Gamma and Gamma functions must be greater than zero. Next, the logarithmic derivatives and must be computed in order to obtain (see (9)). Observe that the correlation integral is equal to the correlation integral evaluated at . In this article, the logarithmic derivatives were obtained using a wavelet transform approach [22]. This produces a smooth version of the derivatives allowing us to better estimate the invariants.

The noise level must be calculated from a range of where the coarse-grained estimator is nearly constant and its variation across the values is the smallest. In this sense, for the noise level, we define the functions [16]where and is the average of across . is the average over of the derivative of with respect to , gives the variation of across , and is the product of the two aforementioned functions. We propose to estimate within a range of centered at the value at which is minimum. This way, is estimated in a range of centered in a plateaued region (scaling region) of , and its value is consistent through the parameter .

The correlation dimension and the correlation entropy can be determined using the coarse-grained estimators (see (12)) and (see (13)), respectively. To find a range of values to estimate and , we use a similar approach, but from functions and , respectively. The functions and can be computed similarly to but using the coarse-grained estimators and , respectively.

The estimation of each invariant for the whole group of signals is presented in Figure 3. Figure 3(a) shows a box plot of the noise level estimation for normal (N), pathological type (PT), pathological type (PT), pathological type (PT), and pathological type (PT) voices. As it can be observed, is greater for pathological voices than for normal ones. However, it is not possible to establish a statistical difference between normal and pathological type and type voices. On the other hand, we can differentiate normal from pathological type and type voices. It is very important to say that the group labeled as TP was selected because they have a noise level value . Note that, using this threshold, we can separate type from type voices.

Figure 3: Invariant estimation box plot: (a) noise level, (b) correlation dimension, and (c) correlation entropy. Normal (N), pathological type (PT), pathological type (PT), pathological type (PT), and pathological type (PT) voices.

The box plot of the correlation dimension is presented in Figure 3(b). There is an increase of from normal to pathological voices. For normal voices, the median is ; for pathological type voices, ; for type voices, ; and for type voices, . From these results, we can establish that the vocal system has a low dimension, even in the presence of a pathology. The group TP is not shown in this plot since, by definition, its dimension is infinite. Regarding the correlation entropy, it is shown in Figure 3(c) that there is a small increase of from normal to pathological voices. However, it is not possible to find a statistical difference between normal, pathological type , and pathological type voices. The median correlation entropy estimated for normal voices is , for pathological type voices is , for pathological type voices is , and for pathological type voices is .

4. Discussion

Bearing in mind that the used records were taken in a controlled environment, we could associate the noise level with the strength of an additive stochastic component that coexists with the dynamic producing the voice. There is a tendency of to increase its value from normal voices to pathological ones. This can be seen as an increase of the power of the stochastic component caused by the presence of a pathology. Moreover, in pathological voices, increases its value along with the voice type.

The definition of type voices given by Sprecher et al. [2] states that these signals are characterized by pure stochastic oscillations; therefore, their dimension is infinite [10]. In practice, it is not possible to measure an infinite correlation dimension value since it is bounded by the embedding dimension used to calculate the correlation integral. Instead, an infinite correlation dimension is inferred if there is not a scaling regime in the coarse-grained estimator of . In previous studies, this was done through visual inspection, which is always a subjective judgement [2, 19].

One interesting aspect of the coarse-grained estimator is that, in the type signals analyzed here, it always presents a scaling range from which to estimate , although and have no scaling regions. These results led us to think that the noise level can be used as an objective measure to discriminate between type and type voices.

The threshold proposed here () was set using the voice signals that did not present a scaling regime in and . With this threshold, we were able to well separate type and type voices. Nevertheless, we are aware that this threshold was selected based on observations from this database. This finding must be validated with a more extensive study involving a larger number of records and voice care professionals.

Our estimations of correlation dimension are in concordance with other studies [6, 1921]. In [23], Choi et al. conducted a very similar simulation over the Kay Elemetrics database. They reported a mean value of for normal voices and an increased dimension value for pathological ones. Moreover, they obtained a decreasing SNR (it was calculated according to [24]) with the type of the voice, being the lowest for type voices. However, they did not analyze type voices. Another study by Zhang and Jiang conducted over the Kay Elemetrics database reported a mean correlation dimension of for normal voices and for a group with vocal tremor [25]. As far as we know, all researches conducted on normal and pathological voices have used the Grassberger and Procaccia correlation integral to estimate and . This methodology has the disadvantage that its estimations are sensitive to noise presence, requires large values to converge, and does not give an estimation of the noise level [10, 13]. However, other variants like the Gaussian correlation integral have not been used.

Regarding the correlation entropy, our results suggest that pathological type and type voices have a slightly greater value than normal ones. Furthermore, these three types of voices have a value close to zero, suggesting nearly periodic dynamics. For type voices, the values of are the greatest, meaning more irregular and unpredictable dynamics. These values are comparable with those presented by Yan et al. [26]. They reported an estimation of for normal (mean value ) and esophageal phonation (mean value ) subjects. In [27], Calawerts et al. calculated the largest Lyapunov exponent for type , type , and type voices using the Kay Elemetrics database. Their results suggest an increase of the value of the largest Lyapunov exponent along with the type of the voice. Our results are in concordance with this study since we obtain an estimation of correlation entropy that not only is greater for pathological voices than for normal ones but also increases with the type of the voice.

5. Conclusions

In this article, we have studied normal and pathological voices through the correlation dimension, the correlation entropy, and the noise level. These invariants were estimated using an automated algorithm based on coarse-grained estimators derived from the U-correlation integral. The results suggest that the voice production dynamical system has a low dimension. The value of is greater for pathological voices than for normal ones. Moreover, its value also increases along with the type of the voice. Regarding the correlation entropy, its value is very low for normal and for type and type pathological voices. Although a more extensive study is still needed, this finding suggests that the system dynamics have a harmonic oscillatory behavior. On the other hand, pathological type voices present higher values of , implying a more complex behavior which is reflected in a more irregular dynamic. The noise level can be interpreted as the power of a stochastic perturbation intrinsic to the voice production system. Our results show that is greater for pathological voices than for normal ones. Furthermore, it increases along with the type of voice, being the highest for type voices. This means that the presence of the stochastic component is stronger in pathological voices. Based on these results, in this work, we have proposed a quantitative criterion that can be used to differentiate between type and type voices. We are aware of the limitation of the reduced samples of type and type voices. In this sense, these preliminary results will be validated with a more extensive study involving a larger number of records and voice care professionals.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Scientific and Technical Research Council (CONICET) of Argentina, the National University of Entre Ríos (UNER), IBB, and Grants PID-6136 (UNER) and PIO-14620140100014CO (CITER-CONICET). The authors would like to thank Professor Cristina Jackson-Menaldi, Ph.D., and Juliana Codino, M.S., CCC-SLP, for the fruitful discussions.

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