Abstract
Abstract Introduction
This research investigates the applicability of a relatively new estimator of mutual information, KSG estimator, to find the reconstruction delay of phase space in dynamical systems. There are evidences that the KSG estimator is more accurate than the naive method commonly used.
Methods
In this paper we estimated mutual information between the voice signals and their delayed versions, with KSG method. The voice signals were obtained from a disordered voice database. Then, we found the reconstruction delay where mutual information reached its first minimum. We applied the encountered value of reconstruction delay in linear discriminant analysis, in order to discriminate between healthy and pathological voices or to discriminate between pathologies. Discrimination between voice pathologies using nonlinear measurements is still not much explored. Moreover, in this paper we used a single nonlinear measurement: reconstruction delay.
Results
The results show that the reconstruction delay obtained with KSG method has increased classification rates in most cases, in terms of accuracy, sensitivity and specificity, when compared to the naive estimator usually adopted.
Conclusion
The KSG estimator is a promising technique to improve the diagnosis of voice related pathologies.
Keywords
KSG estimator; Reconstruction delay; Vocal disorders
Introduction
The mechanisms of voice production are of great complexity ( Pontes et al., 2005 Pontes P, Brasolotto A, Behlau M. Glottic characteristics and voice complaint in the elderly. J Voice. 2005; 19(1):8494. http://dx.doi.org/10.1016/j.jvoice.2004.09.002. PMid:15766853.
http://dx.doi.org/10.1016/j.jvoice.2004...
). There are many laryngeal diseases that cause changes in the voice. These pathologies may be of organic origin, such as nodules, cysts or edemas, or of neurological origin, such as paralysis in the vocal folds ( Davis, 1979 Davis SB. Acoustic characteristics of normal and pathological voices. Speech and Language. 1979; 1:271335. http://dx.doi.org/10.1016/B9780126086010.500103.
http://dx.doi.org/10.1016/B978012608...
; Quek et al., 2002 Quek F, Harper M, Haciahmetoglou Y, Chen L, Ramig LO. Speech pauses and gestural holds in parkinsons disease. In: ICSLP2002  INTERSPEECH 2002: Proceedings of the 7th International Conference on Spoken Language Processing; 2002 Sep 1620; Denver, Colorado, USA. Denver: Causal Productions Pty; 2002. p. 24852488. ). The laryngeal pathologies nodule, Reinke's edema and paralysis in the vocal folds are widely used in studies involving the classification of laryngeal pathologies in adults, both male and female ( BarbosaBranco and Romariz, 2006 BarbosaBranco A, Romariz MS. Doenças das cordas vocais e sua relação com o trabalho. Comum Ciênc Saúde. 2006; 17(1):3745. ; Costa et al., 2013 Costa WCA, Costa SLNC, Assis FM, Aguiar BG No. Classificação de sinais de vozes saudáveis e patológicas por meio da combinação entre medidas da análise dinâmica não linear e codificação preditiva linear. Res Biomed Eng. 2013; 29(1):314. http://dx.doi.org/10.4322/rbeb.2013.010.
http://dx.doi.org/10.4322/rbeb.2013.010...
; Cummings, 2008 Cummings L. Clinical linguistics. Edimburgo: Edinburgh University Press; 2008.
Edimburgo...
; Pinho et al., 2016 Pinho PHU, Couras MFKB, Dantas ECS, Costa SLC, Correia SEN. Classificação de patologias laríngeas por meio de características do espaço de fase reconstruído. In: SBrT2016: Anais do 34° Simpósio Brasileiro de Telecomunicações; 2016 ago 30 set 2; Santarém, PA. Rio de Janeiro: SBrT; 2016. p. 5837. ).
The methods of acoustic evaluation have raised interest in research for the development of tools to support the diagnosis of laryngeal pathologies, because it is a less traumatic method, not causing discomfort to the patient, when compared to traditional examinations for the detection of laryngeal pathologies. The methods of acoustic evaluation can be used both to evaluate the quality of voice, to perform prediagnosis of laryngeal pathologies, as well as the evolution of a medical or postsurgical treatment ( Rabiner and Schafer, 1978 Rabiner LR, Schafer RW. Digital processing of speech signals. USA: Prentice Hall; 1978. ).
There are several nonlinearities involved in vocal fold vibration and glottal wave generation. Due to such factors, classical methods of data analysis based on a linear model have been enriched with methods that are derived from the theory of nonlinear dynamical systems ( Jiang et al., 2006 Jiang JJ, Zhang Y, McGilligan C. Chaos in voice, from modeling to measurement. J Voice. 2006; 20(1):217. http://dx.doi.org/10.1016/j.jvoice.2005.01.001. PMid:15964740.
http://dx.doi.org/10.1016/j.jvoice.2005...
). Over the last two decades, researches considering the techniques of nonlinear dynamical systems and chaos theory include: phoneme classification ( Kokkinos and Maragos, 2005 Kokkinos I, Maragos P. Nonlinear speech analysis using models for chaotic systems. IEEE Trans Speech Audio Process. 2005; 13(6):1098109. http://dx.doi.org/10.1109/TSA.2005.852982.
http://dx.doi.org/10.1109/TSA.2005.8529...
), automatic speaker recognition ( Reynolds and Heck, 2000 Reynolds DA, Heck LP. Automatic speaker recognition. In: Proceedings of the AAAS 2000 Meeting Humans, Computers and Speech Symposium; 2000; Washington. Washington, DC: American Association for the Advancement of Science; 2000. p. 101104. ), discrimination between healthy and pathological voices ( Henriquez et al., 2009 Henriquez P, Alonso JÚB, Ferrer MA, Travieso CM, GodinoLlorente JI, DiazdeMaria F. Characterization of healthy and pathological voice through measures based on nonlinear dynamics. IEEE Trans Audio Speech Lang Process. 2009; 17(6):118695. http://dx.doi.org/10.1109/TASL.2009.2016734.
http://dx.doi.org/10.1109/TASL.2009.201...
), diagnosis of laryngeal pathologies and effects of clinical treatments ( Awan et al., 2010 Awan SN, Roy N, Jiang JJ. Nonlinear dynamic analysis of disordered voice: the relationship between the correlation dimension (D 2) and Pre/Posttreatment change in perceived dysphonia severity. J Voice. 2010; 24(3):28593. http://dx.doi.org/10.1016/j.jvoice.2008.11.003. PMid:19502002.
http://dx.doi.org/10.1016/j.jvoice.2008...
; Chai et al., 2011 Chai L, Sprecher AJ, Zhang Y, Liang Y, Chen H, Jiang JJ. Perturbation and nonlinear dynamic analysis of adult male smokers. J Voice. 2011; 25(3):3427. http://dx.doi.org/10.1016/j.jvoice.2010.01.006. PMid:20472394.
http://dx.doi.org/10.1016/j.jvoice.2010...
; Vaziri et al., 2010 Vaziri G, Almasganj F, Behroozmand R. Pathological assessment of patients’ speech signals using nonlinear dynamical analysis. Comput Biol Med. 2010; 40(1):5463. http://dx.doi.org/10.1016/j.compbiomed.2009.10.011. PMid:19962694.
http://dx.doi.org/10.1016/j.compbiomed....
).
This paper has as main objective to increase the classification rates between healthy or pathological voices, as well as to discriminate between pathologies (discrimination among pathologies is still little explored in the literature) by using a more efficient method to estimate one parameter used in nonlinear dynamical analysis (NDA). This parameter is the reconstruction delay. Fraser and Swinney (1986) Fraser AM, Swinney HL. Independent coordinates for strange attractors from mutual information. Phys Rev A Gen Phys. 1986; 33(2):113440. http://dx.doi.org/10.1103/PhysRevA.33.1134. PMid:9896728.
http://dx.doi.org/10.1103/PhysRevA.33.1...
introduced a method for estimating the reconstruction delay as the first minimum of the mutual information in place of the interval in which autocorrelation first crosses zero. Mutual information is an established concept from information theory, which measures dependency between random variables and was firstly introduced to measure channel capacity ( Cover and Thomas, 2006 Cover TM, Thomas JA. Elements of information theory. 2nd ed. USA: John Wiley & Sons; 2006. ; Shannon and Weaver, 1949 Shannon CE, Weaver W. The mathematical theory of information. Urbana: Illinois Books Edition; 1949. ). By estimating mutual information with a more efficient mutual information estimator, which was proposed by Kraskov et al. (2004) Kraskov A, Stögbauer H, Grassberger P. Estimating mutual information. Phys Rev E Stat Nonlin Soft Matter Phys. 2004; 69(6):066138. http://dx.doi.org/10.1103/PhysRevE.69.066138. PMid:15244698.
http://dx.doi.org/10.1103/PhysRevE.69.0...
and is called here KSG estimator, we could observe improved classification rates. These results were obtained in contrast with the naive estimator commonly used (which is based on adaptive partitioning).
Methods
This section presents the methods and data used in this paper in order to compare and to improve the classification rates in NDA.
Naive estimation of reconstruction delay
As mentioned above, to find the reconstruction delay it is often used the average mutual information method ( Fraser and Swinney, 1986 Fraser AM, Swinney HL. Independent coordinates for strange attractors from mutual information. Phys Rev A Gen Phys. 1986; 33(2):113440. http://dx.doi.org/10.1103/PhysRevA.33.1134. PMid:9896728.
http://dx.doi.org/10.1103/PhysRevA.33.1...
). According to this method, one can ensure reconstruction vectors with the lowest level of redundant information, yet still correlated. Information theory aims to identify how much information a measurement made at a given time
The average mutual information between
where
http://dx.doi.org/10.1103/PhysRevA.33.1...
).
However, there is evidence that the naive estimator for mutual information, obtained as above from histograms, is severely biased ( Assis et al., 2016 Assis JM, Santos MO, Assis FM. Auditory stimuli coding by postsynaptic potential and local field potential features. PLoS One. 2016; 11(8):e0160089. http://dx.doi.org/10.1371/journal.pone.0160089. PMid:27513950.
http://dx.doi.org/10.1371/journal.pone....
; Darbellay and Vajda, 1999 Darbellay GA, Vajda I. Estimation of the information by na adaptive partitioning of the observation space. IEEE Trans Inf Theory. 1999; 45(4):131521. http://dx.doi.org/10.1109/18.761290.
http://dx.doi.org/10.1109/18.761290 ...
). That is, the estimated mutual information value may not depict a true mutual information value nor a value close to the true mutual information value. This happens because this estimate strongly depends on the number
KSG estimation of reconstruction delay
Kraskov et al. (2004) Kraskov A, Stögbauer H, Grassberger P. Estimating mutual information. Phys Rev E Stat Nonlin Soft Matter Phys. 2004; 69(6):066138. http://dx.doi.org/10.1103/PhysRevE.69.066138. PMid:15244698.
http://dx.doi.org/10.1103/PhysRevE.69.0...
developed a mutual information estimator (KSG estimator). KSG estimator is based on the work of Kozachenko and Leonenko (1987) Kozachenko L, Leonenko NN. Sample estimate of the entropy of a random vector. Probl. Peredachi Inf. 1987; 23(2):916. , which estimates entropy based on the
The basic idea of KSG estimator is to use different neighbours to estimate the marginal entropies
http://dx.doi.org/10.1103/PhysRevE.69.0...
with similar performance, one of these is adopted here. This estimator considers the distance of each point to its
Determination of
The estimate is performed as:
where _{<}⬚_{>} denote the arithmetic mean,
If the distributions are very skewed and/or uneven, the authors of the method suggest to transform them so as to become more uniform (or at least singlehumped and more or less symmetric). In this case the KSG estimator gave excellent results after transforming the variables to
Figure 2 illustrates reconstructed phase space examples with the optimum KSG estimated reconstruction delay τ to all speech classes considered in this paper.
Reconstructed phase space examples of voice signals: (A) healthy, (B) with Reinke´s edema, (C) with nodule on vocal folds and (D) with paralysis on vocal folds.
Experimental data
The Disordered Voice Database (Model 4337, of the Kay Elemetrics, recorded by Massachusetts Eye and Ear Infirmary (MEEI) Voice and Speech Lab ( Kay Elemetrics Corp., 1994 Kay Elemetrics Corp. Disordered voice database. USA: Kay Elemetrics Corp.; 1994. ) was used in the experiments. There were 53 talkers voice signals with healthy larynges and 118 talkers voice signals affected by laryngeal pathologies (55 voice signals of larynges affected by paralysis in the vocal folds, 45 voice signals of larynges affected by Reinke's edema, and 18 voice signals of larynges affected by vocal nodules). The voice signals are from sustained vowel /a/. The voice signals of healthy larynges, originally sampled at 50000 samples/s, were subsampled to 25000 samples/s to match the sampling rate of the voice signals of pathological larynges.
Five classes of signals were considered in this study: healthy voice (SDL), voice signal with paralysis on vocal folds (PRL), voice signal with Reinke's edema (EDM), voice signal with nodules on vocal folds (NDL) and all pathologies grouped (PTL). The linear discriminant analysis was used to investigate seven cases of discrimination: SDL vs. PTL, SDL vs. PRL, SDL vs. EDM, SDL vs. NDL, PRL vs. EDM, PRL vs. NDL, and EDM vs. NDL.
Classification
The selected voice signals database are analyzed by estimating reconstruction delay. Then, linear discriminant analysis and the statistical model using CrossValidation kfold stratified were performed to detect the presence of voice disorders caused by Reinke's edema, paralysis on vocal folds and nodules on vocal folds pathologies and to compare with the results obtained using the naive estimator. In this work, the value of k is equal to 10 in the crossvalidation process.
Initially, the classification performance is analyzed considering only two groups signals: one with all grouped pathologies and other with healthy voices (SDL vs. PTL). Posteriorly, the classification performance is analyzed among the other six classification cases: SDL vs. PRL, SDL vs. EDM, SDL vs. NDL, PRL vs. EDM, PRL vs. NDL, and EDM vs. NDL.
Evaluation and interpretation
In order to measure the accuracy of classifiers in each study case, three measures are commonly used: accuracy, sensitivity and specificity. These measures are related to the ability of a classifier in diagnosing a disease in a sick patient (True Positive  TP) or healthy patient (False Positive  FP), or, still, diagnosing a healthy state in a healthy patient (True Negative  TN) or sick patient (False Negative  FN) ( Costa et al., 2012 Costa WCA, Assis FM, Aguiar BG No, Costa SLC, Vieira VJD. Pathological voice assessment by recurrence quantification analysis. In: Proceedings of the 2012 ISSNIP Biosignals and Biorobotics Conference: Biosignals and Robotics for Better and Safer Living (BRC); 2012 Jan 911; Manaus, Brazil. USA: IEEE; 2012. p. 16. ).
The accuracy of classification measures the global correct classification rate, reflecting the ability of the classifier to identify correctly when there is a disorder. The accuracy is defined as the ratio between the number of correctly classified cases and all presented cases to classifier ( Costa et al., 2012 Costa WCA, Assis FM, Aguiar BG No, Costa SLC, Vieira VJD. Pathological voice assessment by recurrence quantification analysis. In: Proceedings of the 2012 ISSNIP Biosignals and Biorobotics Conference: Biosignals and Robotics for Better and Safer Living (BRC); 2012 Jan 911; Manaus, Brazil. USA: IEEE; 2012. p. 16. ):
The sensitivity of classification measures the classifier ability to identify the disorder when it actually exists. Sensitivity is defined as the ratio between the number of correctly classified cases with the disorder and the total number of cases of the disorder:
The specificity of classification measures the ability of the classifier to identify the absence of the disorder when it actually does not exist. Specificity is defined as the ratio between the number of correctly classified healthy cases and the total number of healthy cases:
The classifier presents high performance if it is able to obtain high values for accuracy, sensitivity and specificity. The representation of sensitivity and specificity is clearer when it comes to discrimination between healthy and pathological classes. When there is discrimination between pathologies, it must be defined in the classifier which group will have its correct classification measured by sensitivity and which group will have its correct classification measured by specificity.
Results
Before presenting the classification results, in this section we evaluate both naive and KSG estimators using synthetic signals. In the simulations, we generated bivariate Gaussian datasets, with
For these cases, there is an analytical value for mutual information, which can be used to compare the estimates:
Figure 4 illustrates the estimation with both methods.
Mutual information estimates using the naive estimator (A) and the KSG estimator (B). The naive estimates are plotted as a function of the number of bins used in the procedure. The KSG estimates are plotted as a function of the number of neighbours used in the estimation. The data had Gaussian distributions with zero mean, unit variance and correlation coefficients
It is visible from the simulations that KSG estimator is minimally biased, that is, the mean of the estimator is close to the analytical mutual information value, for all five values of tested
Now we present the results obtained from the classification process of healthy and pathological voices in order to investigate the discriminatory potential of a more precisely estimated reconstruction delay. The objective is to compare the detection and the discrimination of voice disorders with the results obtained with the naive estimator. The voice disorders analyzed here are Reinke's edema, paralysis on the vocal folds and nodules on the vocal folds.
Reconstruction delay estimation
Initially we estimate the value of reconstruction delay using the naive estimator and the KSG estimator, in order to compare which method was able to estimate lower values. This was an interesting investigatory step, since it is desirable to estimate the lowest value of
http://dx.doi.org/10.1103/PhysRevE.69.0...
).
Distribution of mean values of reconstruction delay τ for signals of healthy voices (SDL) and voices with Reinke’s edema (EDM), nodule (NDL) and paralysys (PRL), estimated with the naive and KSG methods.
The results presented in Figure 5 revealed some differences in values of reconstruction delay between naive estimator and KSG estimator for all classes considered. In some cases the values of reconstruction delay fall more than a half. For example, for edema class, the naive method estimates a value of reconstruction delay
We performed statistical tests to confirm the hypothesis that the distribution for each class and method were significantly different. We chose nonparametric statistical tests because they are more broadly acceptable. To evaluate differences in the medians of the reconstructed τ’s with both methods within a single class, we used Wilcoxon signedrank test. On the other hand, to evaluate differences between classes using KSG method, we used Wilcoxon rank sum test. The obtained results were: SDL(Naive) x SDL(KSG) – p= 0.0042 and h=1, EDM(Naive) x EDM(KSG) – p= 0,0001and h=1, NDL(Naive) x NDL(KSG) – p= 0.0005 and h= 1, PRL(Naive) x PRL(KSG) – p=
Thus, there are significant differences between KSG estimated
Classification between healthy voices versus affected voices by pathologies
This section presents the results obtained between healthy voice signals (SDL) and voice signals affected by diseases, Reiken's Edema (EDM), Paralysis (PRL) and Nodule (NDL), individually: SDL x EDM, SDL x PRL and SDL x NDL, and grouping all the pathologies in a single class: SDL x PTL.
Table 1 presents the obtained values of accuracy, sensitivity and specificity in signal healthy voices (SDL) and pathological (PTL) using KSG and naive estimators. The pathological voice signals comprise, in this case, the signals of all the pathologies (Reinke's edema, paralysis and nodule) grouped in the same class.
Table 2 presents the obtained values of accuracy, sensitivity and specificity in healthy voice signals (SDL) and voices signals affected by (EDM), healthy voice signals (SDL) and voices signals affected by (PRL), and in healthy voice signals (SDL) and voices signals affected by (NDL) using KSG estimator.
As seen from the previous tables, results obtained using KSG estimation of the reconstruction delay, in all classification cases, revealed significant differences from the results obtained using the naive estimator. In the classification between healthy voices and pathological voices, the accuracy obtained value was 68,86%, from 60,66% the naive estimator. In classification Healthy x Reinke's Edema, the accuracy obtained value was 69,22%, from 63,78% of the naive estimator. In the classification Healthy x Paralysis on vocal folds, the accuracy obtained value was 70,18%, from 68,64% of the naive estimator and classification Healthy x Nodule on vocal folds, the accuracy obtained value was 80,00%, from 73,32% of the naive estimator.
Classification of pathologically affected voices
This section presents the results obtained in the classification of voice signals affected by Reiken's edema (EDM), Nodules on the vocal folds (NDL) and Paralysis on the vocal folds (PRL) pathologies: EDM x NDL, PRL x NDL and PRL x EDM.
Table 3 presents the values obtained of accuracy, sensitivity and specificity in voice signals affected by EDM and voice signals affected by NDL. There are also the results obtained in the classification of voice signals affected by PRL and voice signals affected by NDL, and voice signals affected by PRL and voice signals affected by EDM.
As it can be seen, the results obtained using the reconstruction delay estimated with the KSG estimator, for classification case EDM x NDL, presented higher average accuracy of the classifier when compared with results from naive estimator. Moreover, the obtained results using the reconstruction delay estimated with the KSG estimator also offered greater mean sensitivity.
For the classification case PRL x NDL, the results obtained using the reconstruction delay estimated with the KSG estimator presented higher average accuracy of the classifier when compared with results from naive estimator. Sensitivity and specificity also presented increased values.
For the classification case PRL x EDM, the results obtained using KSG estimated reconstruction delay presented lower average accuracy of the classifier when compared with results from the naive estimator, the accuracy obtained value was 43,44% from 55,56% of the naive estimator. However, the variance in this case was lower for all criteria with the KSG estimator.
Discussion
Discrimination between voices pathologies using nonlinear measurements is still not much explored. Some papers use several nonlinear combined measures such as Costa et al. (2013) Costa WCA, Costa SLNC, Assis FM, Aguiar BG No. Classificação de sinais de vozes saudáveis e patológicas por meio da combinação entre medidas da análise dinâmica não linear e codificação preditiva linear. Res Biomed Eng. 2013; 29(1):314. http://dx.doi.org/10.4322/rbeb.2013.010.
http://dx.doi.org/10.4322/rbeb.2013.010...
who combine 8 nonlinear measurements in healthy x pathological classification. They use the same database presented in this paper. In the classification between pathological voices the authors combine the 8 nonlinear measurements with LPC (Linear Predictive Coding). Pinho et al. (2016) Pinho PHU, Couras MFKB, Dantas ECS, Costa SLC, Correia SEN. Classificação de patologias laríngeas por meio de características do espaço de fase reconstruído. In: SBrT2016: Anais do 34° Simpósio Brasileiro de Telecomunicações; 2016 ago 30 set 2; Santarém, PA. Rio de Janeiro: SBrT; 2016. p. 5837. use great, medium, minimum and maximum values of τ to resconstruct phase space and use image measurements for discrimination between healthy x pathological voices. Henriquez et al. (2009) Henriquez P, Alonso JÚB, Ferrer MA, Travieso CM, GodinoLlorente JI, DiazdeMaria F. Characterization of healthy and pathological voice through measures based on nonlinear dynamics. IEEE Trans Audio Speech Lang Process. 2009; 17(6):118695. http://dx.doi.org/10.1109/TASL.2009.2016734.
http://dx.doi.org/10.1109/TASL.2009.201...
use 4 nonlinear measurements combined to discriminate between healthy x pathological voices.
In this paper we used a single nonlinear measurement both in healthy x pathological voices and classification between pathological voices. We presented the reconstruction delay as a parameter applied in order to detect the presence of voice pathologies or in order to discriminate pathologies. This reconstruction delay was estimated by a new method of estimation named KSG, which has presented less biased estimates when compared with the naive estimator usually adopted.
The choice of a more accurate estimator makes the classifier more efficient, since KSG estimation approaches a value that better fits the characteristics of a true reconstruction delay. It is a well known fact that a too short reconstruction delay will not capture the dynamics of the data, while a too large reconstruction delay will make it completely independent in a statistical sense ( Abarbanel, 1996 Abarbanel HD. Analysis of observed chaotic data. USA: Springer; 1996. http://dx.doi.org/10.1007/9781461207634.
http://dx.doi.org/10.1007/9781461207...
). The reconstruction delay estimated with KSG method captures the dynamics of voice data more reliably, which in turn allows improved performance of classification. This was observed in the results of this paper. The reconstruction delay estimated with KSG method diagnosed more accurately in most cases.
For the first analyzed case, discrimination between healthy voices versus pathological voices, the classification with KSG estimator presented improved performance when compared to classification with the naive estimator, for the criteria accuracy and sensitivity.
The comparison between heathy voices and voices with some particular pathology (the second classification case, Healthy x Edema, Healthy x Paralysis and Healthy x Nodule) showed that KSG estimator once again presented improved performance over the naive method.
In classification between pathology voices, the results obtained with KSG estimator, for classification cases Edema x Nodule and Paralysis x Nodule, confirm the superiority of this method. Only for the classification case Paralysis x Edema the results with naive estimator were better than using KSG estimator. This fact may have occurred by some particular characteristic of the signals affected by paralysis and edema. As a matter of fact, the statistical test showed that the KSG estimated τ for both classes (paralysis and edema) were not significantly different. This explains why the discriminatory potential of the reconstruction delay is reduced in this particular case.
We stress that the results obtained in this paper are interesting for the case of detection of a disease as also for the case of classification between pathologies. The classification between pathologies is still an unexplored subject in the literature.
Finally, from the present work, we conclude that the use of KSG estimation for the reconstruction delay on phase space is a promising technique, which can be considered to improve the diagnosis of voice related pathologies.
Acknowledgements
The authors wish to thank the CAPES and CNPq by partial financing this research and for the scientific support.

How to cite this article: Santos MO, Assis JM, Vieira VJD, Assis FM. KSG estimation of reconstruction delay to detect vocal disorders in nonlinear dynamical analysis. Res Biomed Eng. 2018; 34(3):xxxx. DOI: 10.1590/24464740.001018
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» http://dx.doi.org/10.1371/journal.pone.0160089  Awan SN, Roy N, Jiang JJ. Nonlinear dynamic analysis of disordered voice: the relationship between the correlation dimension (D 2) and Pre/Posttreatment change in perceived dysphonia severity. J Voice. 2010; 24(3):28593. http://dx.doi.org/10.1016/j.jvoice.2008.11.003. PMid:19502002.
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Publication Dates

Publication in this collection
23 Aug 2018 
Date of issue
JulSep 2018
History

Received
06 Feb 2018 
Accepted
25 July 2018