2014 IEEE International Conference on Acoustic, Speech and Signal Processing (ICASSP)MIXTURE OF GAUSSIAN PROCESS EXPERTS FOR PREDICTING SUNG MELODIC
CONTOUR WITH EXPRESSIVE DYNAMIC FLUCTUATIONS
Yasunori Ohishi† , Daichi Mochihashi‡ , Hirokazu Kameoka† , Kunio Kashino††NTT Communication Science Laboratories, NTT Corporation, ‡The Institute of Statistical Mathematics
ABSTRACT
We present a generative model for predicting the sung melodic contour, i.e., F0 contour, with expressive dynamic fluctuations, such as
vibrato and portamento, for a given musical score. Although several
studies have attempted to characterize such fluctuations, no systematic method has been developed for generating the F0 contour withthem in connection with musical notes. In our model, the relationship between a musical note sequence and F0 contour is directly
learned by a mixture of Gaussian process experts. This approach allows us to automatically characterize the fluctuations by utilizing thekernel function for each Gaussian process expert and predict the F0
contour for an arbitrary musical note sequence. Experimental results
show that our model can better predict the F0 contour than a baseline
method can. Additionally, we discuss the effective musical contexts
and the amount of training data for the prediction.
Index Terms— Singing voice, fundamental frequency (F0 ),mixture of Gaussian process experts, multiple kernel learning,
Markov chain Monte Carlo method
1. INTRODUCTION
The goal of this study is to build a generative model that can characterize the singing style of a singer in sung melodic contours, i.e.,
F0 contours, and predict the F0 contour that reflects the personal
singing style for an arbitrary musical score. Although no firm definition has yet been established for “singing style” in music information processing research, several studies have reported the relationship between singing styles and signal features, such as singing
formant [1,2] and singing fluctuations [3–6]. Specifically, various research efforts have attempted to characterize the fluctuations, such as
vibrato and portamento [5,7–9]. Vibrato is a quasi-periodic variationin the frequency of a sustained musical note and is described with thefollowing parameters: rate, extent, waveform, regularity, time delay
until vibrato onset, and the percentage of vibrato present for the duration [8–13]. Portamento is a gradual sliding of pitch smoothly andcontinuously from one note to another when well-separated musicalnotes are sung in the same breath [14,15]. Accordingly, modeling theexpressive fluctuations enables us to characterize the singing style ofa singer, and it is potentially very beneficial for any singing voice application, such as singing style conversion, singer identification, andthe synthesis of more varied singing voices [16–20].
Several application systems that characterize the fluctuations
have been reported. [21] proposed a method for calculating the vibrato rate and extent to evaluate singing skill automatically. [8] verified the vibrato features (rate, extent, and duration) for synthesizing
more varied singing voices. [9] modeled vibrato and portamento as
the sum of the periodic modulation plus a slow continuous variation,and used the estimated parameters for singer identification. In [7]and [22], a generative process for an F0 contour was proposed using978-1-4799-2893-4/14/$31.00 ©2014 IEEE

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Fig. 1. Musical note sequence and singing voice F0 contour
a second-order differential equation for singing style conversion.
Although these studies attempted to characterize the fluctuations as
a cue for singing skill and style, no systematic method has beendeveloped for generating the various fluctuations in connection withthe musical contexts as shown in Fig. 1. The fluctuations singers use
differ depending on the musical score, and so it is important to learn
this relationship. In singing voice synthesis based on hidden Markov
models (HMMs), F0 contours are characterized by Gaussian distributions for context-dependent states [18, 19, 23, 24]. However, an
HMM is not always an appropriate model, because its hidden-state
space is discrete in spite of the continuous and rapid changes in thefluctuations. Furthermore, a context-dependent decision-tree clustering has been employed to make robust models for new contexts,
but averaging the F0 contours of a leaf node of the tree causes anover-smoothing effect.
In this paper, we propose a generative model that uses a mixtureof Gaussian process experts (MoGPEs) [25] to predict F0 contours
with expressive fluctuations for a given musical score. Each GPE
directly learns the relationship between the musical contexts and an
expressive fluctuation utilizing the kernel function. The MoGPEs
represents the continuous transition of the fluctuations as a mixture
model, and then the F0 contour for an arbitrary musical score is generated by the predictive distribution of the MoGPEs. One merit ofthis representation is that since the F0 contour is not deterministic,i.e., it varies across singing styles, the fluctuations are characterized
stochastically. Thus we design the kernel functions for characterizing the fluctuations elaborately. Experimental results show the effectiveness of using the MoGPEs in terms of predicting the F0 contour
for a new musical score. We also discuss the effective musical contexts and the amount of training data for learning our model.
2. GENERATIVE MODELING OF F0 CONTOURFirst, we briefly explain the input and output features for the MoGPEs. As shown in Fig. 1, we use singing voice signals synchronizedFig. 2. An example allotment of input space in the MoGPEs. Each
elliptical region represents a two-dimensional full covariance Gaussian distribution. The shaded regions are owned by the same GPE.
with melodies, which are represented in standard MIDI file format.
Fig. 1 (a) shows the musical note sequence of a melody. Input vector
xt consists of the position and duration of the current musical noteon the second time scale, the relative pitch between the current noteand the preceding note, and the relative pitch between the current
note and the succeeding note, all of which are extracted from thissequence every 10 ms. It is certainly possible to add phonemes labels and dynamic marks such as crescendo to the input vector. On
the other hand, we use the backward-differential values of the F0
contour as the outputs {yt }Tt=1 to remove musical note informationfrom the F0 contour [Fig. 1 (c)]. The F0 value is estimated every 10
ms using YIN [26]. Then, the F0 frequency in hertz is converted tocents byyHz,
(1)
ycent = 1200 log2
3
440 × 2 12 −5
so that one equal-tempered semitone corresponds to 100 cents.
Next, we explain the standard GP regression [27]. Suppose that
we have T observations as a training set D = {xt , yt }Tt=1 . When
output vector y = [y1 , . . . , yT ]T follows a GP, the probabilistic density function is represented as a multivariate Gaussian distributionp(y) = GP(y; 0, K + η 2 I) = N (y; 0, K + η 2 I),(2)where K is a Gram matrix whose element is given by Ki,j =k(xi , xj ). k(xi , xj ) is calculated by the kernel function, which
defines “similarity” between xi and xj . η 2 and I denote the noise
variance and identity matrix, respectively. The goal of GP regression
is to infer the predictive distribution of y∗ given a new input vectorx∗ . The predictive distribution is given byp(y∗ |y, X, x∗ ) = N (y∗ ; μ∗ , σ∗2 ),

(3)

2
−1
y,
μ∗ = k T∗ (K + η I)

Fig. 3. Graphical representation of our model
specific separate experts make predictions. Each GPE learns different characteristics of the fluctuation. Of course, the learning of theexperts and that of the gating network are intimately coupled. Finally, the outputs are represented as the mixture of these experts.
Two types of the MoGPEs have been proposed [25, 29], and the difference between these formulations is whether or not a full generative model over inputs and outputs is defined. Since [25] defines afull generative model of the MoGPEs that has a number of potential
advantages, such as the ability to deal with partial specified data andinfer inverse functional mappings, we utilize this formulation. The
generative process is as follows:
1. Construct a partition of T observations into R groups using
the Dirichlet-multinomial distribution. This assignment is denoted by using a set of the indicator variables {zt }Tt=1 .
2. For each grouping of indicators {zt : zt = r}, sample theinput space parameters θrx ≡ {μr , Σr }. θrx defines a fullcovariance Gaussian.
3. Given the parameters θrx for each group, sample the input
vectors Xr ≡ {xt : zt = r}.
4. For each group, estimate parameters θrGP for the GP expert.
5. Using the input vectors Xr and parameters θrGP for the individual groups, formulate the GP output covariance matrix
and sample the set of output values, y r ≡ {yt : zt = r},from the joint Gaussian distribution.
The graphical representation of this process is shown in Fig. 3. The
full joint distribution is given byx R
p({xt , yt }Tt=1 , {zt }Tt=1 , {θrGP }R
r=1 , {θr }r=1 |Ω)

=

R 


(4)



p(θrx |Ω)p(Xr |θrx )p(y r |Xr , θrGP , Ω) × p({zt }Tt=1 |Ω),

r=1

2
−1
k∗ ,
σ∗2 = k(x∗ , x∗ ) − kT
∗ (K + η I)

where k∗ is the column Gram vector, which consists of the elements
k(xt , x∗ ) (t = 1, . . . , T ). The exemplars in the training set contribute to the prediction based on their correlation to the new input,
as measured in input space.
The requirement for the kernel function is that the Gram matrix
should be positive semi-definite and symmetric. On the assumptionthat the output signal is stationary, the squared exponential (SE) covariance function and the rational quadratic covariance function are
generally used. However, as shown in Fig. 1, the output signal is
not always stationary because a singer sings while using different
fluctuations depending on the musical note sequence. We employ
the MoGPEs to model the expressive fluctuations which have nonstationary dynamics.
In the MoGPEs, as shown in Fig. 2, the input space is probabilistically divided by a gating network [28] into regions within which

3743

where R and Ω are the number of experts and hyper-parameters,
respectively, and we directly represent the indicators {zt }Tt=1 andsample them to capture their dependence using the Gibbs sampler[25]. The individual distributions in Eq. (4) are defined as follows:p({zt }Tt=1 |Ω) =RΓ(α)  Γ(Tr + α/R),Γ(α + T ) r=1 Γ(α/R)

p(θrx |Ω) = N (μr ; m0 , Σr /β0 )W(Σ−1r ; W 0 , ν0 ),p(Xr |θrx ) = N (Xr ; μr , Σr ),2p(y r |Xr , θ GP
r , Ω) = GP y r ; 0, K r + ηr I r ,

(5)
(6)
(7)
(8)

where α is the hyper-parameter of the Dirichlet-multinomial distribution. Tr , I r , ηr2 , and W denote the number of elements in Xr , theTr × Tr identity matrix, noise variance, and Wishart distribution,
respectively.

where the first term and the second term can be written as2),p(y∗ |y r , Xr , x∗ , z∗ = r, θrGP ) = N (y∗ ; μr,∗ , σr,∗

(12)

2
−1
yr ,
μr,∗ = kTr,∗ (K r + ηr I r )2
2−1= kr (x∗ , x∗ ) − kTkr,∗ ,σr,∗r,∗ (K r + ηr I r )p(z∗ = r|x∗ , θrx ) =p(x∗ |z∗ = r, θrx )p(z∗ = r),p(x∗ )(13)p(x∗ |z∗ = r, θrx ) = N (x∗ ; μr , Σr ), p(z∗ = r) = 1/Tr .
Therefore, the predictive distribution of Eq. (11) can be rewritten asp(y∗ |{yt , xt }Tt=1 , x∗ , Θ, Ω) = N (y∗ ; μ∗ , σ∗2 ),Fig. 4. Gram matrix based on multiple kernel learning
The Gram matrix K r is calculated using the input vectors Xr
and the parameters θrGP . Using multiple kernel leaning [30, 31], we
represent the function as the linear combination of kernel functions:kr (xi , xj ) = wr2

M

ψr,m kr,m (xi , xj ),

(9)

m=1

where θrGP represents the amplitude wr2 and the weight ψr,m of eachMkernel, and then xi , xj ∈ Xr ,m=1 ψr,m = 1. M is the number of kernel functions. For the input vector consisting of various
features, we define kr,m (xi , xj ) as the product of two kernels
(p)

(p)

(c)

(c)

(p)
(c)
kr,m (xi , xj ) = kr,m(xi , xj )kr,m(xi , xj ),
(p)

(p)

(p)

(c)

(c)

(10)

(c)

where kr,m (xi , xj ) and kr,m (xi , xj ) denote the position
kernel and musical context kernel, respectively, as shown in Fig. 4.
The position kernel represents the similarity between the positionsin the musical notes, whereas the musical context kernel represents
the similarity between the musical contexts. Here vector xi is di(p)(c)
vided into xi for the position and xi for the musical context, andthen the kernel functions are calculated for each variable. To capture
the continuity and periodicity of the expressive fluctuations, the SE
covariance function and the periodic covariance function are used
for the position kernel:
(p)

(p)

(p)
(xi , xj ) = exp −
kr,m

(p)

(p)

(xi

(p)

(p)

− xj )T (xi

(p)

(p)
kr,m
(xi , xj ) = exp −2 sin2

(p) 2
2lm

(p)

− xj )

(p)

lm
(p)
(p)
(x − xj )
2π i

,
.

For the musical context kernel, we use the SE covariance function
1 (c)
(c)
(c)
(c)
(c)
(c)
(c)
kr,m
(xi , xj ) = exp − (xi − xj )T Λ(xi − xj ) ,
2
(c) 2

(c) 2

(c)

2

Λ−1 = diag(lm,1 , lm,2 , . . . , lm,Dc ),
(c)

=

R


r=1

(14)

R

2
cr μr,∗ , σ∗2 =c2r σr,∗, cr ≡ p(z∗ = r|x∗ , θrx ).
r=1

m
Based on this predictive distribution, the F0 contour, {ft }Tt=1, isreproduced by m
ft = ot + c − ō, ot = tn=1 μ∗,n , ō = Tt=1
ot /Tm , (15)where c and Tm are the pitch and duration of a target musical note,
respectively.
The GP regression has been applied to speech synthesis [32,33],
F0 contour prediction of speech [34], voice conversion [35], and music performance rendering [36]. In [33], the relationship between the
input features, which consist of linguistic information, and the output, which consists of the spectral feature calculated at each frame,
was learned. This method outperformed conventional HMM-based
speech synthesis. Our model is related to these approaches but differs from them in that it capitalizes on characterizing the dynamics of the fluctuations elaborately utilizing the MoGPEs and various
kinds of kernel functions, which was not considered in these earlier
approaches. Furthermore, using GP regressions as experts, we gain
the advantage that computation for each expert is cubic only in thenumber of data points in its region, rather than in the entire dataset.
3. INFERENCE
All the necessary parameter updates are straightforwardly carried out
using a Markov Chain Monte Carlo (MCMC) and Expectation Maximization (EM) scheme [37]. Specifically, Gibbs sampling is used forthe inference of {zt }Tt=1 and {θrx }R
r=1 [25]. Then we consider themultivariate Gaussian distribution of Eq. (8) as the sum of independent Gaussians, and then employ the EM algorithm for the inferenceof {θrGP }R
r=1 [31, 38].
4. EVALUATION

where Dc is the number of dimensions of xi . The parameters andx
the hyper-parameters are Θ = {z1 , . . . , zT , θ1x , . . . , θR
, θ1GP , . . . ,(p)(p) (c)(c)GP
θR } and Ω = {α, m0 , W 0 , β0 , ν0 , l1 , . . . , lM , l1,1 , . . . , lM,Dc },respectively. For information on how to set up the hyper-parameters,
see the footnote of Section 4.
Finally, we derive the predictive distribution of y∗ for a new
input vector x∗ . The predictive distribution is given byp(y∗ |{yt , xt }Tt=1 , x∗ , Θ, Ω)

μ∗ =

R


(11)

p(y∗ |y r , Xr , x∗ , z∗ = r, θrGP )p(z∗ = r|x∗ , θrx ),

r=1

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We tested our model in terms of the predictive performance of theoutput for a new input vector. In this experiment, we used the F0
contours and the MIDI signals of melodies annotated manually in
the AIST annotation [39] (RWC-MDB-P-2001 [40], Song No. 38,
39, 42, 44, 45, 46, 64, 72, 74, and 76). These songs are sung by
an identical singer. Although the F0 contour should essentially be
estimated from the acoustic signal, we used these data to evaluate
the upper limit of the performance of our model.
Input vector xt and the backward-differential value yt are calculated every 10 ms from the musical note sequence and the F0 contour, respectively. Since the F0 values are not obtainable in the unvoiced segments, we removed those segments. We used song No.

Table 1. Average RMSEs for MoPRs and our model (MoGPEs)
Num. of experts
5
10
20
30
40
50
MoPRs
65.6 71.8 59.7 73.1 79.6 56.5
MoGPEs
26.4 25.0 24.0 23.9 23.1 22.3

Fig. 5. Comparison of the prediction results: blue, green, and redlines correspond to the actual outputs, the result of the MoPRs, andthe result of the MoGPEs, respectively.
38, 39, 42, 44, and 45 for training and song No. 46, 64, 72, 74, and76 for testing. The training and test data consist of 649.3 sec. and625.6 sec. of the singing voices, respectively. Using the training
data, parameters Θ are estimated, and then the outputs are predicted
for the input vectors of the test data. We initialize the indicators
as the assignments obtained by k-means clustering of all the input
vectors of the training data. Using the input vectors assigned to eachexpert, the input space parameters θrx are initialized. The number ofkernel functions is set at M = 216, and the initial values of θrGP are
set at wr2 = 100, ψr,1 = 1/M, . . . , ψr,M = 1/M, ηr2 = 10 (r =1, . . . , R). The settings of the hyper-parameters are shown in thefootnote 1 . The parameter inference was run for 50 iterations.
As the evaluation measure, we use the root mean square error(RMSE) between the actual outputs and the predictive means μ∗ forthe input vectors in the test data as follows:
RMSE = Tt

2
t=1 (yt − μ∗,t ) /Tt ,(16)

where Tt is the signal length of the test data. As a conventional
method, we defined a mixture of polynomial regressions (MoPRs) inwhich the relationship between the input and the output is modeled
as a mixture of nth order polynomials. Specifically, we replace the
GP regression of Eq. (8) with the polynomial regression [41].
Tab. 1 shows the average RMSEs when the number of experts
is changed. Here we only used song No. 35 for the training. In
the MoPRs we set n at 8, because it was difficult sometimes to stably estimate the parameters of the higher order polynomials. Ourmodel outperformed the simple method consisting of the polynomial regressions, and the performance of our model improved as thenumber of experts increased. Fig. 5 compares the prediction resultsof our model and the MoPRs. We confirmed the effectiveness of
1 The hyper-parameters are fixed to α = 1, β = 0.1, ν = D + 1,00where D is the number of dimensions of the input vector. m0 is set
to the mean value over all the input vectors for training data. W 0 is
set to the inverse matrix of the covariance matrix over all the input vectors divided by ν0 . The hyper-parameters of the position kernel are set to(p){lm , m = 1, . . . , 108|0.05, 0.11, 0.23, 0.5} for the SE covariance func(p)tions and to {lm , m = 109, . . . , 216|0.13, 0.15, 0.17, 0.2} for the periodic covariance functions. The hyper-parameters of the musical context ker(c)(c)(c)
nel are set to {lm,1 |1, 2.2, 5}, {lm,2 |1, 2.2, 5}, and {lm,3 |0.1, 0.55, 3}.

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Table 2. Average RMSEs for the number of musical contexts
Num. of musical contexts
1
2
3
MoGPEs23.8 23.1 22.3
Table 3. Average RMSEs for the number of songs in training data
Num. of songs
1
3
5
MoGPEs22.3 20.7 20.5

Fig. 6. Reproduction of F0 contour: blue and red lines correspondto an actual F0 contour and the reproduction, respectively.
using our model to capture the periodic dynamics of the expressive
fluctuations.
Next, we discussed the effective musical contexts and theamount of training data. To find the effective musical contexts,
we trained our model by selecting some of musical contexts described in Section 2, and then we predicted the outputs. The numberof experts is set at R = 50. As shown in Tabs. 2 and 3, as thenumber of contexts and the number of songs in the training data
increased, the performance of our model improved.
Fig. 6 shows the reproduction result for an F0 contour obtainedby Eq. (15). The reproduced F0 contour is exactly the same as theactual F0 contour, but it is still inadequate for predicting the individual fluctuations. To further improve the predictive performance,
it is necessary to adjust the number of experts and the amount oftraining data. We can sidestep the model selection problem by using
an infinite number of experts and employing the gating network related to the Dirichlet process to specify a spatially varying Dirichlet
process [25, 29, 42]. Adding the long-term musical note sequence,the dynamic marks, and the articulation in the musical score to the
input vector as binary variables is also future work. Finally, the F0
contour reproduced by our model should be compared with the F0
contour generated by HMM-based singing voice synthesis.
5. CONCLUSIONSWe proposed a generative model for predicting the sung melodic
contour with the expressive dynamic fluctuations for an arbitrary
musical note sequence. Specifically, each GPE directly learns therelationship between the musical contexts and an expressive fluctuation utilizing the kernel function, and then the MoGPEs represents
the continuous transition of the fluctuations as a mixture model. The
experimental results showed that our model is promising for predicting the F0 contour for a given musical score.
To further improve the predictive performance, we plan to adjust
the number of experts and the hyper-parameters, and simultaneously
characterize the dynamics of the F0 contour and the voice volume
contour utilizing the MoGPEs. Although our model has great potential for recognizing and converting singing styles, we have not tested
it yet. In future work, we will evaluate its ability to automaticallyclassify singing styles and identify singers using Eq. (14), and apply
our model to singing style conversion using a speech manipulation
system such as STRAIGHT [43].

6. REFERENCES[1] J. Sundberg, “Singing and timbre,” Music room acoustics, vol.
17, pp. 57–81, 1977.
[2] J. Sundberg, The Science of the Singing Voice, Northern Illinois University Press, 1987.
[3] K. Kojima et al., “Variability of vibrato -a comparative study
between Japanese traditional singing and bel canto-,” in Proc.
Speech Prosody 2004, pp. 151–154.
[4] I. Nakayama, “Comparative studies on vocal expressions inJapanese traditional and western classical- style singing, using
a common verse,” in Proc. ICA 2004, pp. 1295–1296.
[5] T. Saitou et al., “Development of an F0 control model basedon F0 dynamic characteristics for singing-voice synthesis,”
Speech Communication, vol. 45, no. 3-4, pp. 405–417, 2005.
[6] V. Verfaille et al., “Perceptual evaluation of vibrato models,”in Proc. CIM 2005.
[7] T. Saitou et al., “Speech-To-Singing synthesis: Converting
speaking voices to singing voices by controlling acoustic features unique to singing voices,” in Proc. WASPAA 2007, pp.
215–218.
[8] N. Migita et al., “A study of vibrato features to control singing
voices,” in Proc. ICA 2010, pp. 23–27.
[9] L. Regnier, Localization, Characterization and Recognition ofSinging Voices, Ph.D. thesis, IRCAM / UPMC in Paris, France,
2013.
[10] M. Metfessel, The Vibrato in Artistic Voices, University of
Iowa Press.
[11] C. E. Seashore, “A musical ornament, the vibrato,” in Psychology of Music. pp. 33–52, McGraw-Hill Book Company.
[12] D. Myers et al., “Vibrato and pitch transitions,” J. Voice, vol.
1, no. 2, pp. 157–161, 1987.
[13] Y. Horii, “Acoustic analysis of vocal vibrato: A theoreticalinterpretation of data,” J. Voice, vol. 3, no. 1, pp. 36–43, 1989.
[14] E. Pollastri, “Some considerations about processing singing
voice for music retrieval,” in Proc. ISMIR 2002.
[15] J. Potter, “Beggar at the door: the rise and fall of portamento in
singing,” Music and Letters, vol. 87, no. 4, pp. 523–550, 2006.
[16] T. Nakano et al., “VOCALISTENER2: A singing synthesis
system able to mimic a user’s singing in terms of voice timbrechanges as well as pitch and dynamics,” in Proc. ICASSP 2011,
pp. 453–456.
[17] H. Kenmochi et al., “Singing synthesis as a new musical instrument,” in In Proc. ICASSP 2012.
[18] K. Oura et al., “Pitch adaptive training for HMM-based singing
voice synthesis,” in Proc. ICASSP 2012, pp. 5377–5380.
[19] T. Nose et al., “A style control technique for singing voicesynthesis based on multiple-regression HSMM,” in Proc. INTERSPEECH 2013, pp. 378–382.
[20] H. Doi et al., “Evaluation of a singing voice conversion method
based on many-to-many eigenvoice conversion,” in Proc. INTERSPEECH 2013, pp. 1067–1071.
[21] T. Nakano et al., “An automatic singing skill evaluation method
for unknown melodies using pitch interval accuracy and vibrato features,” in Proc. Interspeech 2006, pp. 1706–1709.

3746[22] Y. Ohishi et al., “A stochastic model of singing voice F0 contours for characterizing expressive dynamic components,” inProc. INTERSPEECH 2012.
[23] T. Yoshimura et al., “Simultaneous modeling of spectrum,
pitch and duration in HMM-based speech synthesis,” in Proc.
EUROSPEECH 1999.
[24] H. Zen et al., “Statistical parametric speech synthesis,” SpeechCommunication, vol. 51, no. 11, pp. 1039–1064, 2009.
[25] E. Meeds et al., “An alternative infinite mixture of Gaussian
process experts,” in Proc. NIPS 2006.
[26] A. de Cheveigné et al., “YIN, a fundamental frequency estimator for speech and music,” Journal of the Acoustical Society ofAmerica, vol. 111, no. 4, pp. 1917–1930, 2002.
[27] C. E. Rasmussen and C. K. I. Williams, Gaussian Processes
for Machine Learning, MIT Press, Cambridge, Mass, USA,2006.
[28] R. A. Jacobs et al., “Adaptive mixture of local experts,” NeuralComputation, vol. 3, pp. 79–87, 1991.
[29] C. E. Rasmussen et al., “Infinite mixtures of Gaussian processexperts,” in Proc. NIPS 2002.
[30] F. Bach et al., “Multiple kernel learning, conic duality, and the
SMO algorithm,” in Proc. ICML 2004, pp. 6–13.
[31] K. Yoshii et al., “Infinite kernel linear prediction for joint estimation of spectral envelope and fundamental frequency,” inProc. ICASSP 2013, pp. 463–467.
[32] G. E. Henter et al., “Gaussian process dynamical models for
nonparametric speech representation and synthesis,” in Proc.
ICASSP 2012, pp. 4505–4508.
[33] T. Koriyama et al., “Statistical nonparametric speech synthesis based on Gaussian process regression,” in Proc. INTERSPEECH 2013, pp. 912–916.
[34] R. Fernandez et al., “F0 contour prediction with a deep belief
network-Gaussian process hybrid model,” in Proc. ICASSP
2013, pp. 6885–6889.
[35] N. C. V. Pilkington et al., “Gaussian process experts for voice
conversion,” in Proc. INTERSPEECH 2011, pp. 2761–2764.
[36] K. Teramura et al., “Gaussian process regression for renderingmusic performance,” in Proc. ICMPC 2008.
[37] C. Andrieu et al., “An introduction to MCMC for machinelearning,” Machine Learning, vol. 50, no. 1-2, pp. 5–43, 2003.
[38] M. Feder et al., “Parameter estimation of superimposed signals
using the EM algorithm,” IEEE Transactions on Acoustics,
Speech, and Signal Processing, vol. 36, no. 4, pp. 477–489,
1988.
[39] M. Goto, “AIST annotation for the RWC music database,” inProc. ISMIR 2006.
[40] M. Goto et al., “RWC music database: Popular, classical, andjazz music databases,” in Proc. ISMIR 2002.
[41] C. M. Bishop, Pattern Recognition and Machine Learning (Information Science and Statistics), Springer-Verlag New York,
Inc., Secaucus, NJ, USA, 2006.
[42] A. Tayal et al., “Hierarchical double Dirichlet process mixtureof Gaussian processes,” in Proc. AAAI 2012.
[43] H. Kawahara et al., “TANDEM-STRAIGHT: A temporally
stable power spectral representation for periodic signals andapplications to interference-free spectrum, F0, and aperiodicity
estimation,” in Proc. ICASSP 2008, pp. 3933–3936.