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Figure 3: Probability density functions: experimental (continuous line) and simulated (dashed line). Figure 3 shows the probability density function constructed from experimental data and the probability density func- ion constructed from simulations. The function ksdensity from MATLAB was used. 4.2 Description of the second part

Figure 3 Probability density functions: experimental (continuous line) and simulated (dashed line). Figure 3 shows the probability density function constructed from experimental data and the probability density func- ion constructed from simulations. The function ksdensity from MATLAB was used. 4.2 Description of the second part

Related Figures (6)

Figure 1: Two-mass model of the vocal folds. The dynamics of the system is given by Eqs. (1) and (2) (Cataldo et. al., 2008, 2009): Some authors have modeled the vocal folds dynamics, mainly in a deterministic way (Koizumi et al., 1976; Lous et al., 1998; Zhang et al., 2005). One of these models is the well-known model proposed by Ishizaka and Flanagan (1972) and it will be used here because it has provided a simple and effective representation of the system for studying the underlying dynamics of voice production.
Figure 1: Two-mass model of the vocal folds. The dynamics of the system is given by Eqs. (1) and (2) (Cataldo et. al., 2008, 2009): Some authors have modeled the vocal folds dynamics, mainly in a deterministic way (Koizumi et al., 1976; Lous et al., 1998; Zhang et al., 2005). One of these models is the well-known model proposed by Ishizaka and Flanagan (1972) and it will be used here because it has provided a simple and effective representation of the system for studying the underlying dynamics of voice production.
In order to validate the development presented here, voice signals produced by one person have been analyzed and their statistics have been compared with simulations. A voice signal corresponding to a sustained vowel /a/ has been recorded from one person and 1,800 frames were obtained from this signal, each one with 0.01s of length. For each frame, the corresponding fundamental frequency was calculated. So, a corresponding p.d-f., the so-called experimental, can be constructed. Figure 2 shows the p.d.f. of the fundamental frequency constructed from the experimental data. The problem to be solved to update the p.d.f. of Q, using Bayesian statistics, will be divided into two parts: at first. an inverse problem will be solved in order to obtain, from simulations, a p.d.f. of the fundamental frequency near to the experimental one. Then, at the second part, the p.d.f. obtained in the first part will be updated, using experimental date and the Bayesian method. Consequently, the updated p.d.f. of Q will be obtained. As explained above, the stochastic system is deduced from the deterministic one substituting ago, y, g by the random variables Ago, Y, Q. Consequently, the random variable associated to the fundamental frequency Fo is given by Fo = M (Ago, Y,Q). However, the nonlinear mapping -@ is not explicitly known and it is implicitly defined by Eqs. (1) and (2) substituting ago, y, g by random variables Ago, Y, Q. The fundamental frequency associated to each realization of the voice signal is calculated through the glottal signal, calculating the inverse of its period.
In order to validate the development presented here, voice signals produced by one person have been analyzed and their statistics have been compared with simulations. A voice signal corresponding to a sustained vowel /a/ has been recorded from one person and 1,800 frames were obtained from this signal, each one with 0.01s of length. For each frame, the corresponding fundamental frequency was calculated. So, a corresponding p.d-f., the so-called experimental, can be constructed. Figure 2 shows the p.d.f. of the fundamental frequency constructed from the experimental data. The problem to be solved to update the p.d.f. of Q, using Bayesian statistics, will be divided into two parts: at first. an inverse problem will be solved in order to obtain, from simulations, a p.d.f. of the fundamental frequency near to the experimental one. Then, at the second part, the p.d.f. obtained in the first part will be updated, using experimental date and the Bayesian method. Consequently, the updated p.d.f. of Q will be obtained. As explained above, the stochastic system is deduced from the deterministic one substituting ago, y, g by the random variables Ago, Y, Q. Consequently, the random variable associated to the fundamental frequency Fo is given by Fo = M (Ago, Y,Q). However, the nonlinear mapping -@ is not explicitly known and it is implicitly defined by Eqs. (1) and (2) substituting ago, y, g by random variables Ago, Y, Q. The fundamental frequency associated to each realization of the voice signal is calculated through the glottal signal, calculating the inverse of its period.
Figure 4: Probability density functions pp,jg (100 plots) and the probability density function of the fundamental frequency obtained from experimental data (thick line). pitas i a 2 01 8 To construct the function pyjg, 100 deterministic values of g (from 0.6153 up to 0.6442) were considered and, for each value of q, a p.d.f. of the fundamental frequency was obtained by simulation. The corresponding conditional p.d.f.’s are shown in fig. 4. So, corresponding values of Prial fo|q) can now be calculated, for giving values of fo and g. Here, values of fo wer considered from 109 Hz up to 125 Hz, with 0.1 Hz of spacing.
Figure 4: Probability density functions pp,jg (100 plots) and the probability density function of the fundamental frequency obtained from experimental data (thick line). pitas i a 2 01 8 To construct the function pyjg, 100 deterministic values of g (from 0.6153 up to 0.6442) were considered and, for each value of q, a p.d.f. of the fundamental frequency was obtained by simulation. The corresponding conditional p.d.f.’s are shown in fig. 4. So, corresponding values of Prial fo|q) can now be calculated, for giving values of fo and g. Here, values of fo wer considered from 109 Hz up to 125 Hz, with 0.1 Hz of spacing.
Figure 5: Updated p.d.f.’s of the fundamental frequency for different values of Vexp. Let Fy XP be the random variable associated to the fundamental frequencies obtained experimentally and pom the random variable associated to the simulated fundamental frequencies. The aim is to update the probability density function of Fsim, using experimental values of the fundamental frequency, applying the Eq. 15. Fig. 5 shows the updated p.d_f. (pee d, of the fundamental frequency considering Vexp = 1, 10, 100, 800and 1,800. Starting from Vexp = 1000, the p.d.f. does not change anymore. It means that the same p.d.f. is obtained considering Vv = 1,000 or more. It should be observed that the p.d.f. obtained for Vexp = 1,800 is almost the same of the p.d.f. constructed with experimental values.
Figure 5: Updated p.d.f.’s of the fundamental frequency for different values of Vexp. Let Fy XP be the random variable associated to the fundamental frequencies obtained experimentally and pom the random variable associated to the simulated fundamental frequencies. The aim is to update the probability density function of Fsim, using experimental values of the fundamental frequency, applying the Eq. 15. Fig. 5 shows the updated p.d_f. (pee d, of the fundamental frequency considering Vexp = 1, 10, 100, 800and 1,800. Starting from Vexp = 1000, the p.d.f. does not change anymore. It means that the same p.d.f. is obtained considering Vv = 1,000 or more. It should be observed that the p.d.f. obtained for Vexp = 1,800 is almost the same of the p.d.f. constructed with experimental values.
Figure 6: Functions | pre — pam | (dashed line) and | Pre — pipe | (continuous line). Calculating the area under the plots of the Fig. 6, the values found were:
Figure 6: Functions | pre — pam | (dashed line) and | Pre — pipe | (continuous line). Calculating the area under the plots of the Fig. 6, the values found were:
Figure 7: Prior p.d.f. of Q and the corresponding updated p.d.f’s. Now, it is possible to obtain the p.d.f. of Q, using Eq. (11). Figure 7 shows the plots of the prior probability density function of Q and the posterior probability density function of Q, obtained with Vexp = 1, 100, and1, 800. The p.d.f. of Q is near a delta function located in Q = 0.633. With this value of the parameter g, and considering the random variables Ago and Y, it is possible, using the model, to obtain the p.d.f. of the fundamental frequency constructed with experimental values.
Figure 7: Prior p.d.f. of Q and the corresponding updated p.d.f’s. Now, it is possible to obtain the p.d.f. of Q, using Eq. (11). Figure 7 shows the plots of the prior probability density function of Q and the posterior probability density function of Q, obtained with Vexp = 1, 100, and1, 800. The p.d.f. of Q is near a delta function located in Q = 0.633. With this value of the parameter g, and considering the random variables Ago and Y, it is possible, using the model, to obtain the p.d.f. of the fundamental frequency constructed with experimental values.
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MathematicsComputer ScienceUncertainty QuantificationVoice Production
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