Higher-Order Correlation Analysis of Pitch Fluctuations in Sustained Normal Vowels by the Method of Surrogate Data

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1 a) Higher-Order Correlation Analysis of Pitch Fluctuations in Sustained Normal Vowels by the Method of Surrogate Data Isao TOKUDA a), Takaya MIYANO, and Kazuyuki AIHARA [1] [3] Department of Computer Science and Systems Engineering, Muroran Institute of Technology, Muroran-shi, Japan COE Center for Promotion of the COE Program, Ritsumeikan University, Kusatsu-shi, Japan Department of Mathematical Informatics, The University of Tokyo, Bunkyo-ku, Tokyo, Japan a) tokuda@agnld.uni-potsdam.de AR [2] [4] [5] [6] [10] [11] [12] [14] [15] [3] [2] [21], [22] 3 [18], [19] Schreiber [23] A Vol. J87 A No. 3 pp

2 2004/3 Vol. J87 A No. 3 Schreiber oo, ao, ka, yo /a/ Table 1 Data length, number of extracted pitches, mean pitch period, and coefficient of variation of a sustained vowel /a/ recorded from 4 subjects (oo, ao, ka, yo). oo 800 msec msec % ao 450 msec msec % ka 900 msec msec % yo 900 msec msec 1.10 % oo, ao 2 ka, yo /a/ 4 10 khz khz [20] {T i : i =1, 2,.., N} (1) 1 4 oo, ao, ka, yo N = 137, 71, 223, (a) oo coefficient of variation [16] α = 100 T 1 N N (T i T ) 2 [%] (2) i=1 1 {T 1,T 2,...} Fig.1 Pitch extraction via the zero-crossing method and generation of the pitch period sequence {T 1,T 2,...}. T = 1 N Ti 4 N i=1 oo, ao, ka, yo 1 [17] 1.05±0.40% [18], [19] method of surrogate [21], [22] statistical hypothesis testing 356

3 (a) (b) (c) (d) (e) 2 (a) oo {T 1,T 2,...,T 137} (b) (a) (c) 2 (d) 3 (e) 4 Fig.2 (a) Original pitch period sequence {T 1,T 2,..., T 137} extracted from the subject oo.(b) Surrogate data generated by random shuffling of the original data (a).(c) Surrogate data that preserves the 2nd order correlation function.(d) Surrogate data that preserves the 3rd order correlation function.(e) Surrogate data that preserves the 4th order correlation function. FT [21] Schreiber [23] {T i} 357

4 2004/3 Vol. J87 A No {T i} T σ τ µ } rst (τ ) {{} m times 1 = N (m 1)τ N (m 1)τ t=1 1 σ r+s+t+ (T i T ) r (T i+τ T ) s (T i+2τ T ) t (3) 2 κ 2(τ ) κ 2(τ )=µ 11(τ ) = 1 σ 1 2 N τ N τ i=1 (T i T ) (T i+τ T ) (4) 2 κ 2(τ )=µ 11(τ ) (5) 3 4 [18], [24] 3 d κ d d (C1) [21] {T i} {T i} (C2) {T i} {T i} d κ k, κ k (k =2,...,d) E({T })= d k=2 τ {κ k (τ ) κ k(τ )} 2 (8) SA (8) {T } SA (T i,t j)(i j) {...,T i,...,t j,...} {..., T j,...,t i,...} SA l t =1/log(l) E(T i T j) 1/{1 + exp( E/t)} oo {T i} 4 {T i} 3 l SA κ 3(τ )=µ 111(τ ) (6) 4 κ 4(τ )=µ 1111(τ ) µ 0011(τ ) µ 1100(τ ) µ 1010(τ ) µ 0101(τ ) µ 1001(τ ) µ 0110(τ ) (7) 4 [18], [24] 3. 2 {T 1,T 2,...} {T 1,T 2,...} 3 (8) Fig.3 Process of minimizing the error function (8) by the simulated annealing method. 358

5 2 (e) 2 (a) (a) (b) 4. 1 LPC [20] {T i} 5 (S1) (S2) N (c) 4 oo 4 (a) 2 κ 2(τ ) (b) 3 κ 3(τ ) (c) 4 κ 4(τ ) Fig.4 Higher order correlation functions of the subject oo s original pitch sequence (bold line) and the surrogate data that preserves the 4th order correlation function (crosses).(a) The 2nd order correlation functions κ 2(τ ), (b) The 3rd order correlation functions κ 3(τ ), (c) The 4th order correlation functions κ 4(τ ). Fig.5 5 Speech synthesis based on waveform editting. 359

6 2004/3 Vol. J87 A No. 3 {T 1,T 2,...,T N} (S3) [4] (S3) (S2) [25] (a) (b) (c) 2 (d) 3 (e) 4 5 [26] 20 4 oo, ao, yo, ka [3] [2] 3 AR ao 4 ao 1 {T 1,...,T 71} {T 1,...,T 71} 2 {T 1,...,T 71,T 1,...,T 71}, {T 1,...,T 71,T 1,...,T 71}

(a) (b) (c) (d) 6 2 3 4 (a) oo (b) ao (c) yo (d) ka Fig.

From left side, random shuffed surrogate data, 2nd order surrogate data, 3rd order

(a) subject oo, (b) subject ao, (c) subject yo, (d) subject ka. 7 2 ao Fig.

7 (a) (b) (c) (d) (a) oo (b) ao (c) yo (d) ka Fig.6 Measurement of the naturalness of the synthesized sounds. From left side, random shuffed surrogate data, 2nd order surrogate data, 3rd order surrogate data, 4th order surrogate data, and the original data.(a) subject oo, (b) subject ao, (c) subject yo, (d) subject ka. 7 2 ao Fig.7 Measurement of the naturalness of the synthesized sounds from the subject ao in the case when the pitch sequence was repeated twice. 2 2 [3] [2] 3 3 AR 2 [6] [10] [27] 361

8 2004/3 Vol. J87 A No. 3 3 I II LPC III AR [1] L.Dolansky and P.Tjernlund, On certain irregularities of voiced-speech waveforms, IEEE Trans.Audio Electroacoust., vol.16, pp.51 56, [2] vol.47, no.8, pp , [3] vol.47, no.12, pp , [4] vol.47, no.12, pp , [5] N.Aoki and T.Ifukube, Analysis and perception of spectral 1/f characteristics of amplitude and period fluctuations in normal sustained vowels, J.Acoust. Soc. Am., vol.106, no.1, pp , [6] H.Herzel, D.Berry, I.R.Titze, and M.Saleh, Analysis of vocal disorders with method from nonlinear dynamics, J. Speech Hear. Res., vol.37, pp , [7] W.Mende, H.Herzel, and I.R.Titze, Bifurcations and chaos in newborn cries, Phys.Lett.A, vol.145, pp , [8] S.S. Narayanan and A.A. Alwan, A nonlinear dynamical systems analysis of fricative consonants, J. Acoust. Soc. Am., vol.97, no.4, pp , [9] A.Behrman, Global and local dimensions of vocal dynamics, J. Acoust. Soc. Am., vol.105, no.1, pp , [10] M.Banbrook, S.McLaughlin, and I.Mann, Speech characterization and synthesis by nonlinear methods, IEEE Trans.Speech Audio.Process.vol.7, no.1, pp.1 17, [11] M.Sato, K.Joe, and T.Hirahara, APOLONN brings us to the real world, Proc.Int.Joint Conf. Neural Networks, vol.1, pp , [12] I.Tokuda, R.Tokunaga, and K.Aihara, A simple geometrical structure underlying speech signals of the Japanese vowel /a/, Int. J. Bif. Chaos, vol.6, no.1, pp , [13] I.Tokuda, T.Miyano, and K.Aihara, Surrogate analysis for detecting nonlinear dynamics in normal vowels, J. Acoust. Soc. Am., vol.110, no.6, pp , [14] NLP99-152, [15] T.Ikeguchi and K.Aihara, Estimating correlation dimensions of biological time series with a reliable method, J. Int. Fuzzy Sys., vol.5, no.1, pp.33 52, [16] N.B. Pinto and I.R. Titze, Unification of perturbation measures in speech signals, J.Acoust.Soc.Am., vol.87, pp , [17] R.C. Scherer, V.J. Vail, and C.G. Guo, Required number of tokens to determine representative voice perturbation values, J. Speech Hear. Res., vol.38, pp , [18] [19] [20] [21] J.Theiler, S.Eubank, A.Longtin, B.Galdrikian, and J.D. Farmer, Testing for nonlinearity in time series: The method of surrogate data, Physica D, vol.58, pp.77 94, [22] J.Theiler and D.Prichard, Constrained-realization Monte-Carlo method for hypothesis testing, Physica D, vol.94, pp , [23] T.Schreiber, Constrained randomization of time series data, Phys. Rev. Lett., vol.80, no.10, pp , [24] A.Stuart and J.K.Ord, Kendall s Advanced Theory of Statistics, Fifth Ed., Charles Griffin, [25] CAS , [26] 362

9 1991. [27] I.Tokuda, T.Riede, J.Neubauer, M.J.Owren, and H.Herzel, Nonlinear analysis of irregular animal vocalizations, J. Acoust. Soc. Am., vol.111, no.6, pp , COE IEEE ME INNS SMB 363

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