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6Modeling of Speech Parameter Sequence
Considering Global Variance for HMM-Based
Speech Synthesis
Tomoki Toda
Nara Institute of Science and Technology
Japan

1. Introduction
Speech technologies such as speech recognition and speech synthesis have many potential
applications since speech is the main way in which most people communicate. Various
linguistic sounds are produced by controlling the conﬁguration of oral cavities to convey amessage in speech communication. The produced speech sounds temporally vary and are
signiﬁcantly affected by coarticulation effects. Thus, it is not straightforward to segmentspeech signals into corresponding linguistic symbols. Moreover, the acoustics of speech varyeven if the same words are uttered by the same speaker due to differences in the manner of
speaking and articulatory organs. Therefore, it is essential to stochastically model them inspeech processing.
The hidden Markov model (HMM) is an effective framework for modeling the acoustics ofspeech. Its introduction has enabled signiﬁcant progress in speech and language technologies.
In particular, there have been numerous efforts to develop HMM-based acoustic modelingtechniques in speech recognition, and continuous density HMMs have been widely used inmodern continuous speech recognition systems (Gales & Young (2008)). Moreover, several
approaches have been proposed for applying the HMM-based acoustic modeling techniquesto speech synthesis technologies (Donovan & Woodland (1995); Huang et al. (1996)) such
as Text-to-Speech (TTS), which is ... from a given text. Recently, HMM-based speechsynthesis has been proposed (Yoshimura et al. (1999)) and has generated interest owing to
its various attractive features such as completely data-driven voice building, ﬂexible voice
quality control, speaker adaptation, small footprint, and so forth (Zen et al. (2009)).
A basic framework of HMM-based speech synthesis consists of training and synthesis
processes. In the training process, speech parameters such as spectral envelope andfundamental frequency (F0) are extracted from speech waveforms and then their time
sequences are modeled by context-dependent phoneme HMMs. To model the dynamic
characteristics of speech acoustics with HMMs, which assume piecewise constant statistics
within an HMM state and conditional independence, a joint vector of static and dynamic
features is usually used as an observation vector. In the synthesis process, a smoothly
varying speech parameter trajectory is generated by maximizing the likelihood of a composite
sentence HMM subject to a constraint between static and dynamic features with respect to
not the observation vector sequence including both static and dynamic features but the staticfeature vector sequence (Tokuda et al. (2000)). Finally, a vocoding technique is employed

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Hidden Markov Models, Theory and Applications

to generate a speech waveform from the generated speech parameters. This frameworkfor directly generating speech parameters from the HMMs has the potential to be used fordeveloping very ﬂexible TTS systems. On the other hand, the quality of the synthetic speech
is noticeably degraded compared with the original spoken audio. It is known that the staticfeature vectors generated from the HMMs are often oversmoothed, which is one of the main
factors causing the mufﬂed effect in synthetic speech.
There have been some attempts to model new features of speech acoustics to ensure thatthe generated parameters exhibit similar properties to those of natural speech, thus reducing
the oversmoothing effect. As one of the most effective features for capturing properties notwell modeled by traditional HMMs, Toda and Tokuda (Toda & Tokuda (2007)) proposed
the global variance (GV), which is the variance of the static feature vectors calculated overa time sequence (e.g., over an utterance). It has been found that the GV is inversely correlatedwith the oversmoothing effect. Therefore, a metric on the GV of the generated parameters
effectively acts as a penalty term in the parameter generation process. Several papers (Toda& Tokuda (2007); Zen et al. (2007a; 2008)) have reported that the naturalness of synthetic
speech is signiﬁcantly improved by considering the GV in HMM-based speech synthesis.
Moreover, it has been reported that the use of the GV is also effective for improving otherstatistical parametric speech synthesis techniques such as voice conversion (Toda et al. (2007)).
It is not an exaggeration to say that GV modeling has signiﬁcantly contributed to the recent
improvements in those techniques.
This chapter presents an overview of the techniques for modeling a speech parametersequence considering the GV for HMM-based speech synthesis. First, the traditionalframework for HMM-based speech synthesis and its weaknesses are described. Then, theparameter generation algorithm considering the GV (Toda & Tokuda (2007)) is presented as
an effective approach to addressing the oversmoothing problem caused by the traditionalmodeling process. This algorithm is capable of generating the parameter trajectory, yielding
a signiﬁcant improvement in synthetic speech quality while maintaining its GV close to its
natural value. Furthermore, a training method considering the GV (Toda & Young (2009))
is presented, which is derived by introducing the GV-based parameter generation into the
HMM training process. It is shown that this method yields some additional advantages
such as the use of a consistent optimization criterion between the training and synthesis
processes and the use of a closed-form solution for parameter generation, whereas only aniterative solution is available if the GV is considered in only the parameter generation process.
Finally, several experimental results are presented to demonstrate that these methods yield asigniﬁcant improvement in the naturalness of synthetic speech.

2. Basic framework of HMM-based speech synthesis
Figure 1 shows a schematic image of the basic training and synthesis processes of HMM-based
speech synthesis. In the training process, the time sequence of the static feature vectors c islinearly transformed into a higher-dimensional space using the linear transformation function
f o (c). Then, the transformed feature vector sequence consisting of static and dynamic feature
vectors o is modeled with HMMs. In the synthesis process, the static feature vector sequence
is determined by maximizing the likelihood of the HMMs for the static and dynamic feature
vectors with respect to only the static feature vectors under the constraint given by the linear
transformation function. The following subsections give more details of these training andsynthesis processes.

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Considering Global Variance for HMM-Based Speech Synthesis

133

Fig. 1. Schematic image of basic training and synthesis processes of HMM-based speechsynthesis.
2.1 Speech parameter sequence modeling

Let us assume a D-dimensional static feature vector of a speech parameter ct = [ct (1) , ct (2),· · · , ct (d), · · · , ct ( D )] ⊤ at frame t. To suitably model the dynamic properties of the speechparameter with HMMs, the ﬁrst dynamic feature vector Δ (1) ct and the second dynamic feature
vector Δ (2) ct are calculated frame by frame as follows:
(n)

Δ

( n)

L+

w( n) (τ )ct+τ ,

∑

ct =

n = 1, 2,(1)(n)τ =− L−

where w( n) (t) is a regression coefﬁcient used to calculate the dynamic features from the
( n)

( n)

current frame, L − previous frames, and L + succeeding frames,1 and then a joint static and⊤( 1) ⊤ ( 2) ⊤dynamic feature vector ot = c⊤is used as the observation vector. The
t , Δ ct , Δ ct
time sequence vectors of the joint static and dynamic feature vector ot and the static feature
⊤
⊤


⊤
⊤
,
and c = c1⊤ , c2⊤ , · · · , c⊤vector ct are written as o = o1⊤ , o2⊤ , · · · , o⊤t , · · · , cTt , · · · , oT
respectively. The relationship between these two time sequence vectors is represented as alinear transformation as follows:o = Wc,(2)where W is the 3DT-by-DT matrix written as
W = [W 1 , W 2 , · · · , W t , · · · , W T ] ⊤ ⊗ I D × D ,


( 2)
( 1)
( 0)
W t = wt , wt , wt ,

⊤
( n)
( n)
( n)
wt = 
0 , · · · , 0, w( n) (− L − ), · · · , w( n) (0), · · · , w( n) ( L + ), 0, · · · , ,0
  
  


1st



(n)

t− L−

-th

( t)-th



(n)

t+ L+

-th

(3)
(4)
(5)

T-th

1 For example, when the number of frames to be used for calculating dynamic features is set to L (1) =
−
(1)
(2)
(2)
L + = 1 and L − = L + = 1, the regression coefﬁcients can be set to w (1) ( t − 1) = −0.5, w (1) ( t) = 0,
w (1) ( t + 1) = 0.5, w (2) ( t − 1) = 1, w (2) ( t) = −2, and w (2) ( t + 1) = 1.

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( 0)

( 0)

where L − = L + = 0, and w(0) (0) = 1. The operator ⊗ denotes the Kronecker product. The
matrix I D× D denotes the D-by-D identity matrix. It is important to note that the joint staticand dynamic feature vector sequence o is generated from the static feature vector sequence cusing the linear transformation function given by Eq. (2).
The observation sequence vector o is modeled by context-dependent phoneme HMMs, whose
the parameter set is denoted by λ. The state output probability density function (p.d.f.) ofot at an HMM state q is usually modeled by a single multivariate Gaussian distribution as
follows:
⊤1
1
ot − µq(6)
U q−1 ot − µq ,exp −N ot ; µq , U q =2(2π )3D |U q |where µq and U q are the mean vector and the covariance matrix, respectively. The p.d.f. of
the observation sequence vector o given an HMM state sequence q = {q1 , q2 , · · · , q t , · · · , q T }is written asTP ( o| q, λ) = ∏ N ot ; µ qt , U qt
t =1

= N o; µq , U q ,(7)

where µq and U q are the 3D-by-1 mean vector and the 3D-by-3D covariance matrix,
respectively, which are given by

⊤
⊤
⊤
⊤
µq = µ⊤,q1 , µ q2 , · · · , µ q t , · · · , µ q T

U q = diag U q1 , U q2 , · · · , U qt , · · · , U q T
⎤
⎡
U q1
⎥
⎢
U q2
⎥
⎢
=⎢
⎥.
.
..
⎦
⎣
U qT

(8)

(9)

The operator diag[·] denotes the transformation from a rectangular matrix (or a vector) to a
block diagonal matrix (or a diagonal matrix). The likelihood function given the HMM set forthe observation sequence vector o is given by
P (o| λ) =∑ P (o, q|λ)all q

= ∑ P (o| q, λ) P (q| λ) ,
all q

(10)

where P (q| λ) is usually modeled by state transition probabilities in speech recognition.
However, in speech synthesis explicit duration models are often incorporated into the HMMs
to model the temporal structure of a speech parameter sequence appropriately (Yoshimura etal. (1999); Zen et al. (2007b)). In training, the HMM parameter set λ is optimized for the given
observation sequence vectors o(1) , o(2) , · · · , o( k) , · · · , o( K ) in the sense of maximum likelihood

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Modeling of Speech Parameter Sequence
Considering Global Variance for HMM-Based Speech Synthesis

135(ML) as follows:Kλ̂ = arg max ∏ P o( k) | λ ,(11)

λ k =1

where K is the total number of observation sequence vectors.
2.2 Parameter generation based on maximum likelihood criterion

In synthesis, a time sequence of static feature vectors is determined by maximizing the HMM
likelihood under the condition given by Eq. (2) as follows:ĉ = argmax P (o| λ)subject to

co = Wc.

(12)

To reduce the computational cost, the HMM likelihood is usually approximated with a single
HMM state sequence as follows:
P (o| λ) ≃ P (o| q, λ) P (q | λ),(13)

and then the HMM state sequence and the static feature vector sequence are sequentiallydetermined. First a suboptimum HMM state sequence is determined byq̂ = argmax P (q| λ).

(14)

Then, the static feature vector sequence is determined by maximizing the HMM likelihood
given the HMM state sequence q as follows:ĉ = argmax P (o| q, λ)subject too = Wc.

(15)

The objective function L q to be maximized with respect to the static feature vector sequenceis given by

L q = log P (o| q, λ)1
1⊤ −1∝ − o⊤ U −q o + o U q µq211
⊤ ⊤ −1= − c⊤ W ⊤ U −q Wc + c W U q µq21
= − c⊤ Rq c + c⊤ r q ,
2

(16)

1
Rq = W ⊤ U −
q W,

(18)

⊤

(19)

(17)

where

−1

r q = W U q µq .
The ML estimate of the static feature vector sequence cq is given bycq = Pq r q ,−1

Pq = Rq .

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(21)

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Hidden Markov Models, Theory and Applications

2
4
6
Frequency [kHz]a) Natural

8600

Power [dB]

700
650

Time [ms]Power [dB]

20
0

750

80
60
40
20
0

750
700
650
2
4
6
Frequency [kHz]8Time [ms]

800

800
80
60
40

600

b) Generated

Fig. 2. An example of natural and generated spectral segments.
Since the matrix Pq is generally full owing to the inverse of the band matrix Rq , the state
output p.d.f. at each HMM state affects the ML estimates of the static feature vectors
at all frames over a time sequence. This parameter generation algorithm is capable ofgenerating speech parameter trajectories that vary frame by frame from the p.d.f. sequencecorresponding to discrete state sequences so that the generated trajectories exhibit suitablestatic and dynamic properties.
The calculation of the ML estimate is efﬁciently performed by the Cholesky decompositionof Rq and the forward/backward substitution operation. It is also possible to implementa recursive estimation process so as to generate the static feature vectors frame by frame(Tokuda et al. (1995)). Furthermore, although only the case of using a single multivariate
Gaussian distribution as the state output p.d.f. is described in this section, Gaussian mixture
models can also be employed in this framework by using the Expectation-Maximization (EM)algorithm (Tokuda et al. (2000)).
2.3 Oversmoothing effect

In HMM-based speech synthesis, speech samples synthesized with the generated speechparameters often sound mufﬂed. One of the factors causing the mufﬂed sound is theoversmoothing of the generated speech parameters. Figure 2 shows an example of naturaland generated spectral segments. It can be observed from this ﬁgure that the generated
spectra are often excessively smoothed compared with the natural spectra. The statisticalmodeling process with HMMs tends to remove the details of spectral structures. Although thissmoothing results in reduced error in the generation of spectra, it also causes the degradationof naturalness of synthetic speech because the removed structures are still necessary forsynthesizing high-quality speech.

3. Parameter generation considering global variance
To reduce oversmoothing, a parameter generation algorithm considering the GV has been
proposed (Toda & Tokuda (2007)). A schematic image of the training and synthesis processes
is shown in Figure 3. In the training process, the linearly transformed feature vector sequence
consisting of static and dynamic feature vectors o is modeled with HMMs in the same manneras discussed in Section 2. Also, the time sequence of the static feature vectors c is nonlinearlytransformed into the GV v using the nonlinear transformation function f v (c), and then its
p.d.f. is modeled with a continuous density distribution. In the synthesis process, the staticfeature vector sequence is determined by maximizing the product of the HMM likelihood andwww.intechopen.com

Modeling of Speech Parameter Sequence
Considering Global Variance for HMM-Based Speech Synthesis

137

Fig. 3. Schematic image of training and synthesis processes of HMM-based speech synthesis
with GV-based parameter generation algorithm.
the GV likelihood. The following subsections give more details of these training and synthesis
processes.
3.1 Global Variance (GV)The GV vector v(c) = [ v(1), · · · , v(d), · · · , v( D )] ⊤ of a static feature vector sequence c is
calculated byv(d) =1 T(ct (d) − c(d))2 ,
T t∑
=1

(22)

1 T
c τ ( d ).
T τ∑
=1

(23)

where

c(d) =The GV is often calculated utterance by utterance.
Figure 4 shows a time sequence of the 2nd mel-cepstral coefﬁcients extracted from natural
speech and that generated from the HMM, where mel-cepstrum is one of the most effective
spectral parameters (Tokuda et al. (1994)). It can be observed that the GV of the generated
mel-cepstra (given by Eq. (20)) is smaller than that of the natural ones. The ML criterion
usually makes the generated trajectory close to the mean vector sequence of the HMM.
Consequently, the reduction of GV is often observed.
3.2 Parameter generation algorithm considering GV

To consider the GV in the parameter generation process, the p.d.f. of the GV is modeled by asingle multivariate Gaussian distribution, which is given by

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138

1

0
v(d)

2nd mel-cepstral coefficient

Hidden Markov Models, Theory and Applications

NaturalGenerated

-1

0

1

2
Time [sec]3

Fig. 4. Natural and generated mel-cepstrum sequences. The square root of the GV of eachsequence is shown as a bidirectional arrow.

P (v(c )| λv ) = N (v (c); µv , U v )

= 

1
1
1
exp − (v(c ) − µv )⊤ U −v (v (c) − µ v ) ,2(2π ) D |U v |(24)

where λ v denotes the GV parameter set consisting of the mean vector µ v and the covariancematrix U v . The GV parameter set λv and the HMM parameter set λ are independently trained
from speech samples in the training data.
In the synthesis process, given a suboptimum HMM state sequence q, the static feature vector
sequence is determined by maximizing a new likelihood function given by a product of the
HMM likelihood for the static and dynamic feature vectors and the GV likelihood as follows:ĉ = argmax P (o| q, λ ) P (v(c)| λv )ωTsubject too = Wc,(25)

where the constant ω denotes the GV weight, used for controlling the balance between the
two likelihoods, which is usually set to the ratio between the number of dimensions of vectors
v (c) and o (i.e., ω = 3). Note that this likelihood function with an additional constraint on the
GV of the generated trajectory is still a function of the static feature vector sequence c. The GV
likelihood P (v (c)| λv ) can be viewed as a penalty term for a reduction of the GV. The objective( GV )

function L q

to be maximized with respect to the static feature vector sequence is written as( GV )Lq= log P (o| q, λ) + ωT log P (v (c)| λv )

(26)

1
1
∝ − c⊤ Rq c + c⊤ r q + ωT − v(c)⊤ U v−1 v(c) + v(c)⊤ U v−1 µv .
2
2

(27)

This objective function is equivalent to the traditional function given by Eq. (16) when the GV
weight ω is set to 0. To determine the static feature vector sequence maximizing the objectivefunction, an iterative process for updating ĉ employing the gradient method is necessary as

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Modeling of Speech Parameter Sequence
Considering Global Variance for HMM-Based Speech Synthesis

139

follows:ĉ( i+1)-th = ĉ( i)-th + α · δĉ( i)-th,(28)

where α is the step size parameter. The following two gradient methods are basically
employed to calculate the vector δĉ ( i)-th.
Steepest descent algorithm: Using the steepest descent algorithm, δĉ( i)-th is written as( GV ) ∂Lqδĉ( i)-th =
(29)
∂c 
c=ĉ(i)-th .

The ﬁrst derivative is calculated by( GV )∂L q∂c= − Rq c + r q + ωx,⊤⊤,x = x1⊤ , x2⊤ , · · · , x⊤t , · · · , xT

(30)
(31)
⊤

x t = [ xt (1), xt (2), · · · , xt (d), · · · , xt ( D )] ,

(32)

(d)
xt (d) = −2 (ct (d) − c(d)) (v (c) − µ v )⊤ p v ,

(33)

(d)

where pv is the d-th column vector of the precision matrix Pv (= U v−1 ).
Newton-Raphson method: If the initial value of the static feature vector sequence ĉ(0)-th isclose to the optimum value, the Newton-Raphson method using not only the ﬁrst derivative
but also the second derivative, i.e., the Hessian matrix, may also be used. The vector δĉ( i)-th iswritten as⎛⎞( GV ) −1( GV ) 
2
∂ Lq
∂L q

⎠
δĉ( i)-th = − ⎝
(34)

∂c 
∂c∂c ⊤
c=ĉ(i)-th .
Because a Hessian matrix is not always a positive-deﬁnite matrix, the following second
derivative, approximated using only diagonal elements, is used:( GV )∂2 L q




−1
≃
diag
−
diag
R
+
ωy
,
q
∂c∂c⊤

⊤
⊤
,
y = y1⊤ , y2⊤ , · · · , y⊤t , · · · , yT
δxt (d)δxt ( D ) ⊤δxt (1) δxt (2),,··· ,,··· ,,yt =δct (1) δct (2)δct (d)δct ( D )
δxt (d)2(d)(d)( T − 1) (v (c) − µv )⊤ p v + 2 ( ct (d) − c(d))2 pv (d) ,=−δct (d)
T

(35)
(36)
(37)
(38)

where the operator diag−1 [·] denotes the inverse operation of diag[·]; i.e., the extraction of
only the diagonal elements (or block diagonal matrices) from a square matrix.

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Hidden Markov Models, Theory and Applications

There are two main methods for setting the initial value of the static feature vector sequenceĉ(0)-th . One is to use the ML estimate cq given by Eq. (20) in the traditional parameter
generation process. The other is to use the static feature vector sequence c linearly converted
from the ML estimate so that its GV is equivalent to the mean vector of the GV p.d.f., µv , asfollows:μ v (d)ct (d) =(39)(ct (d) −  c(d)) + c(d) ,v(d)where μ v (d) is the d-th element of the mean vector µv . The former sequence maximizes the
HMM likelihood P (o| q, λ ), while the latter sequence maximizes the GV likelihood P (v(c)| λv ).
It is reasonable to start the iterative update from the static feature vector sequence yielding ahigher value of the objective function given by Eq. (26).
3.3 Effectiveness of considering GV

It is well known that postﬁltering to increase the sharpness of spectral peaks is effective
for improving synthetic speech quality (Koishida et al. (1995)), in which it is necessary toempirically adjust a parameter to control the degree of emphasis. Moreover, this parameter isusually kept constant over different frames. On the other hand, when considering the GV, at a
certain dimension the trajectory movements are greatly emphasized, but at another dimension
they remain almost the same. The degree of emphasis varies between individual dimensions
and frames, and it is automatically determined from the objective function given by Eq. (26).
This process may be regarded as statistical postﬁltering.
Using more mixture components to model the probability density also reduces
oversmoothing. However, it also causes another problem of overtraining due to an increasein the number of model parameters, which often causes performance degradation for data
samples not included in the training data. One of the advantages of considering the GV isthat the number of parameters is kept almost equal to that when not considering the GV.
In addition, since the proposed framework is based on a statistical process, it retains many
advantages of statistical parametric speech synthesis, such as allowing model adaptation(Yamagishi et al. (2009)) in a manner supported mathematically.
3.4 Weakness

The main weakness of considering the GV is the inconsistency between the training andsynthesis criteria. In the training, the likelihood for the joint static and dynamic feature vectors
is used to optimize the HMM parameters and the likelihood for the GV is used to optimize theGV p.d.f. parameters. These two models are independently optimized. On the other hand, inthe synthesis, the product of these two likelihoods is used to optimize only the static feature
vectors. Consequently, the trained model parameters are not optimum for this parameter
generation process.
The GV p.d.f. given by Eq. (24) is context-independent. Hence, it does not capture variationsof the GV caused by different contextual factors. A context-dependent model may be used as
the GV p.d.f. to capture them. However, if the GV is calculated utterance by utterance, thenumber of GV samples used to train the context-dependent model is relatively small; i.e., only
the number of utterances in the training data. Therefore, the number of context-dependent
GV p.d.f.s is often limited to avoid the overtraining problem.
The parameter generation process with the GV requires the gradient method. Thus, it has agreater computational cost than the traditional parameter generation process, for which theclosed-form solution can be used.

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Modeling of Speech Parameter Sequence
Considering Global Variance for HMM-Based Speech Synthesis

141

Fig. 5. Schematic image of training and synthesis processes of HMM-based speech synthesis
with GV-constrained trajectory training algorithm.

4. HMM training considering global variance
The traditional framework described in Section 2 also suffers from the inconsistency between
the training and synthesis criteria. To address this issue, Zen et al. (Zen et al. (2007c))
proposed the trajectory HMM, which is derived by imposing an explicit relationship between
static and dynamic features on the traditional HMM. This method allows the utilization of a
uniﬁed criterion, i.e., the trajectory likelihood, in both training and synthesis processes. In asimilar spirit, Wu and Wang (Wu & Wang (2006)) proposed minimum generation error (MGE)training. This method optimizes the HMM parameters so that the error between the generated
and natural parameters is minimized.
Inspired by these approaches, the idea of considering the GV has been introduced to themodel training process to make it possible to use the same objective function consisting of
the GV metric in both the training process and the synthesis process (Toda & Young (2009)). A
schematic image of the training and synthesis processes is shown in Figure 5. The basic HMMs
that model the p.d.f. of the joint static and dynamic feature vector sequence are reformulated
as the trajectory HMM, which models the p.d.f. of the static feature vector sequence by
imposing the constraint given by the linear transformation function f o (c). Furthermore, theGV p.d.f. is generated from the trajectory HMM based on the constraint given by the nonlinear
transformation function f v (c). Then, the basic HMM parameters are optimized so that theobjective function deﬁned as the product of the likelihood of the trajectory HMM and thelikelihood of the generated GV p.d.f. is maximized. In the synthesis process, the static feature
vector sequence is determined by maximizing the same objective function. The following
subsections give more details of these training and synthesis processes.
4.1 Trajectory HMM

The traditional HMM is reformulated as a trajectory HMM by imposing an explicit
relationship between static and dynamic features, which is given by Eq. (2) (Zen et al. (2007c)).

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The p.d.f. of c in the trajectory HMM is given by∑ P (c|q, λ) P (q|λ) ,all q
1
P (c| q, λ ) =P (o| q, λ )
Zq

(40)

= N (c; cq , Pq ),(41)

P (c| λ) =where the normalization term Zq is given by
P (o| q, λ)dc

Zq =(2π ) DT | Pq |=(2π )3DT |Uq1
exp − (µ⊤U −1 µ − r ⊤q Pq r q ) .
2 q q q|

(42)

In Eq. (41), the mean vector cq (given by Eq. (20)) varies within the states, and the interframe
correlation is modeled by the temporal covariance matrix Pq (given by Eq. (21)) even whenusing the same number of model parameters as in the traditional HMM. Note that the mean
vector of the trajectory HMM is equivalent to the ML estimate of the generated static feature
sequence in the traditional parameter generation process. Namely, the traditional parameter
generation process is equivalent to the maximization process of the likelihood function of
the trajectory HMM with respect to the static feature vector sequence. The utilization of the
trajectory likelihood as a uniﬁed criterion in both training and synthesis processes makes it
possible to optimize the HMM parameters for parameter generation.
4.2 GV-constrained trajectory training

The parameter generation process considering the GV given by Eq. (25) has been modiﬁed
and integrated into a training framework as follows:λ̂ = arg max P (c| q, λ) P (v(c)| q, λ)ωT ,λ(43)

where P (c| q, λ) is given by Eq. (41) and P (v(c)| q, λ) is the modiﬁed GV p.d.f., which is given
by
"
!
(44)
P (v (c)| q, λ ) = N v(c); v (cq ), U v .
Note that the mean vector of the GV p.d.f. is deﬁned as the GV of the mean vector of the
trajectory HMM, which is equivalent to the GV of the generated parameters from the HMMs
given by Eq. (20). Hence, the GV likelihood P (v(c)| q, λ) acts as a penalty term to make theGV of the generated parameters close to that of the natural ones. The balance between the
two likelihoods P (c| q, λ) and P (v (c)| q, λ) is controlled by the GV weight ω. The objective( GV ) function L q

used in both training and synthesis processes is given by( GV ) 

Lq

www.intechopen.com= log P (c| q, λ) + ωT log P (v (c)| q, λ ).

(45)

Modeling of Speech Parameter Sequence
Considering Global Variance for HMM-Based Speech Synthesis

143

Given the HMM state sequence q,2 the GV weight ω, and the GV covariance matrix U v , the( GV ) 

HMM parameter set is optimized by maximizing the proposed objective function L q. Themean vectors and diagonal precision matrices at all HMM states (from 1 to N), which are
given by⊤
m = µ1⊤ , µ2⊤ , · · · , µ⊤
,
N
⊤

1
,
Σ−1 = U 1−1 , U 2−1 , · · · , U −
N

(46)
(47)

are simultaneously updated since they depend on each other. The mean vectors m areiteratively updated using the following gradient:( GV ) 

∂L q

∂m
where

!
"
−1
= A⊤
q U q W c − cq + ωPq xq ,

⊤

⊤
⊤
⊤
,
xq = x ⊤
,
x
,
·
·
·
,
x
,
·
·
·
,
x
q
,t
q,1 q,2
q,T

(48)

(49)
⊤

xq,t = x q,t (1), x q,t (2), · · · , x q,t (d), · · · , x q,t ( D ) ,!
#$" !
"⊤ ( d)x q,t (d) = −2 c q,t (d) − c q (d)v (cq ) − v(c ) pv ,

(50)
(51)

and Aq is a 3DT-by-3DN matrix whose elements are 0 or 1 depending on the state sequenceq. The precision matrices Σ−1 are iteratively updated using the following gradient:( GV ) 

∂L q

∂Σ−1

=


1 ⊤
⊤
⊤
⊤ ⊤
Aq diag−1 W (Pq + cq c⊤
q − cc )W − 2µq (cq − cq ) W
2+2ωWPq xq (µq − Wcq )⊤ .

(52)

This update is performed on a logarithmic domain to ensure that the updated covariancematrices are positive deﬁnite. Although only the case of using a single observation sequence
for training data is described here, it is straightforward to extend this training algorithm tomultiple observation sequences. This training algorithm enables the HMM parameters to be
optimized so that both the generated parameter trajectory and its GV are close to the naturalones.
The proposed objective function L q is also used in parameter generation. The static feature
vector sequence is determined by( GV ) ĉ = arg max L qc= arg max N (c; cq , Pq )N (v(c ); v(cq ), U v )c= cq .

2(53)

A suboptimum HMM state sequence may be determined by the Viterbi algorithm with the traditional
likelihood, i.e., P ( q|λ) P ( o|q, λ).

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Note that this estimate is equivalent to the ML estimate (given by Eq. (20)) in the traditionalparameter generation algorithm because the GV likelihood is always maximized by setting
the generated parameters to the mean vector of the trajectory HMM. Therefore, it is no longer
necessary to practically consider the GV term in parameter generation, and the traditionalparameter generation algorithm can be directly employed. Note that the traditional algorithm
is computationally much more efﬁcient than the parameter generation algorithm considering
the GV, which requires an iterative process as mentioned in Section 3.2.
4.3 Discussion

GV-constrained trajectory training addresses some of the issues of parameter generationconsidering the GV described in Section 3.4. It provides a uniﬁed framework using the same
objective function in both training and synthesis processes. It also allows the closed-form
solution to be used in parameter generation. This makes it possible to implement the recursive
estimation process, which generates the static feature vectors frame by frame (Tokuda etal. (1995)), which is very effective for achieving a low-delay synthesis process. Moreover,context-dependent GV modeling can be easily implemented without increasing the numberof model parameters. It has been found that the variations of the GV tend to be larger
with decreasing number of frames (i.e., the total duration of an utterance). Therefore, theparameter generation algorithm considering the GV sometimes causes highly artiﬁcial sounds
in synthetic speech when synthesizing very short utterances such as one word. This problem
is effectively addressed by GV-constrained trajectory training.
In an attempt to apply the idea of considering the GV in the HMM training process, Wu
et al. (Wu et al. (2008)) proposed MGE training that considers the error in the GV between
natural and generated parameters as well as the generation error mentioned above. The HMM
parameters are optimized so that both the generated parameters and their GV are similar to
the natural ones. This method is similar to GV-constrained trajectory training. One of thedifferences between these two methods is that not only frame-by-frame (weighted) generation
errors but also the correlation between the errors over a time sequence is considered in
GV-constrained trajectory training because of the use of the temporal covariance matrix ofthe trajectory HMM, Pq .

5. Experimental evaluation
The effectiveness of parameter generation considering the GV and that of GV-constrained
trajectory training were evaluated separately.
5.1 Experimental conditions

The 0th through 24th mel-cepstral coefﬁcients were used as spectral parameters and log-scaled
F0 was used as the excitation parameter. A high-quality speech analysis-synthesis method
called Speech Transformation and Representation using Adaptive Interpolation of weiGHTed
spectrum (STRAIGHT) (Kawahara et al. (1999)) was employed for the analysis-synthesis
method. Each speech parameter vector included the static features and their delta anddelta-deltas. The frame shift was set to 5 ms.
Context-dependent phoneme HMMs were trained for each of the spectral and F0 components
using a decision-tree-based context-clustering technique based on the minimum description
length (MDL) criterion (Shinoda & Watanabe (2000)). The spectral component was modeledby the continuous density HMM, of which each state output p.d.f. was modeled by asingle Gaussian with a diagonal covariance matrix. The F0 component was modeled by themultispace probability distribution HMM (MSD-HMM) (Tokuda et al. (2002)) to model a time

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Modeling of Speech Parameter Sequence
Considering Global Variance for HMM-Based Speech Synthesis

145

Global variance

10
Generated w/o GV
using postfilter
Generated with GV
Natural

1

0.1

0.01
β = 0.8
β = 0.4
0.001
0

5

10
15
20
Order of mel-cepstrum

25

Fig. 6. GVs of several mel-cepstrum sequences. The values shown are GV mean values over
all test sentences and speakers.
sequence consisting of continuous values, i.e., log-scaled values of F0 , and discrete symbols
that represent unvoiced frames. Static, delta, and delta-delta values of F0 were modeled indifferent streams (Yoshimura et al. (1999)). Context-dependent duration models for modelingthe state duration probabilities were also trained.
In the synthesis, a sentence HMM for given input contexts was constructed by concatenating
the context-dependent phoneme HMMs, and then a suboptimum state sequence wasdetermined from the state duration model. Mel-cepstrum and F0 sequences were directly
generated from p.d.f. sequences corresponding to the determined suboptimum state
sequence. A speech waveform was synthesized by ﬁltering the excitation signal, which was
designed using the generated excitation parameters, based on the generated mel-cepstra with
the Mel Log Spectrum Approximation (MLSA) ﬁlter (Imai (1983)).
5.2 Evaluations of parameter generation considering GV

To evaluate the parameter generation algorithm considering the GV, voices were built for fourJapanese speakers (two males, MHT and MYI, and two females, FTK and FYM) in the ATR
Japanese speech database B-set (Sagisaka et al. (1990)), which consists of 503 phoneticallybalanced sentences. For each speaker, 450 sentences were used as training data and theother 53 sentences were used for evaluation. Context-dependent labels were prepared fromphoneme and linguistic labels included in the ATR database. A Gaussian distribution of theGV p.d.f. for each of the spectral and F0 components was trained using the GVs calculated
from individual utterances in the training data. The GV weight ω was set to 1.0.
Figure 6 shows the GVs of mel-cepstra generated with the traditional generation algorithm
and those with the generation algorithm considering the GV. For the traditional algorithm,
the GVs of the generated mel-cepstra when using the postﬁlter to emphasize the mel-cepstra(Koishida et al. (1995)) are also shown. The ﬁltering coefﬁcient β was set to 0.4 or 0.8. The GVof the natural mel-cepstra is also shown in the ﬁgure as a reference. It can be seen that the GVof the mel-cepstra generated with the traditional algorithm is small. Although postﬁltering
increases the GV, the GV characteristics of the emphasized mel-cepstra are obviously different
from those of the natural ones. On the other hand, mel-cepstra whose GV is almost equal tothat of the natural ones are generated when considering the GV.

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Generation method
MOS ± 95% conﬁdence interval
Generated w/o GV
2.53 ± 0.12
3.46 ± 0.15
Generated with GV
4.35 ± 0.12
Natural (analysis-synthesized)
Table 1. Mean opinion score (MOS) on naturalness given in opinion test to evaluate
parameter generation with GV.

100

100

90
80
Generated w/o GV
using postfilter
Generated with GV
Natural

70
60
50

0

0.2

0.4
β

0.6

0.8

GV likelihood

150

HMM likelihood

110

50
0
-50

-100
-150

0

0.2

0.4
β

0.6

0.8

Fig. 7. Log-scaled HMM and GV likelihoods on mel-cepstrum sequences as a function of
postﬁlter coefﬁcient β. The HMM likelihoods are normalized by the number of frames.
Figure 7 shows the logarithmic HMM and GV likelihoods on mel-cepstrum sequences, which
are normalized by the number of frames. It is reasonable that the largest HMM likelihood
is yielded by the traditional algorithm (β = 0.0) and that it decreases when the postﬁlter isapplied or the GV is considered. An interesting point is that the HMM likelihood for thenatural sequence is smaller than those for the generated sequences. This implies that we
do not necessarily generate the speech parameter sequence that maximizes only the HMM
likelihood, although it seems reasonable to keep the likelihood larger than that for the naturalsequence. The GV likelihoods are very small when using the traditional algorithm becauseof the GV reduction shown in Figure 6. Although it is observed that they are recovered by
postﬁltering, the resulting likelihoods are still much smaller than that for the natural sequence.
On the other hand, the algorithm considering the GV generates a sequence for which the GV
likelihood is sufﬁciently large. Consequently, it makes both HMM and GV likelihoods exceed
those for the natural sequence. These results demonstrate that the algorithm considering the
GV is capable of generating more similar speech parameter sequences to those of natural
speech from the viewpoint of satisfying a greater variety of characteristics than the traditionalalgorithm.
Table 1 shows the result of a subjective evaluation based on an opinion test on the naturalnessof the synthetic speech. The opinion score was set to a 5-point scale (5: excellent, 4: good, 3:fair, 2: poor, 1: bad) and ten Japanese listeners participated in the test. It is observed that thegeneration algorithm considering the GV yields signiﬁcant quality improvements comparedwith the traditional generation algorithm. It effectively reduces mufﬂed sounds of synthetic

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Modeling of Speech Parameter Sequence
Considering Global Variance for HMM-Based Speech Synthesis
0 2 4 6 8
Frequency (kHz)a) Generated
w/o GV147b) Generated
with GV

0 2 4 6 8
Frequency (kHz)c) Natural

0 2 4 6 8
Frequency (kHz)Fig. 8. Examples of spectrum sequences of generated speech using traditional algorithm,
generated speech using generation algorithm considering GV, and natural speech. Note thatthe phoneme duration of the natural sequence is different from those of the generated
sequences.
voices caused by the oversmoothing effect.3 An example of spectrum sequences is shownin Figure 8. The algorithm considering the GV generates much sharper spectral peaks thanthose generated by the conventional algorithm. Note that increasing the GV usually causes anincrease in mel-cepstral distortion between the generated sequence and the natural sequence,
which is strongly correlated with the decrease in the HMM likelihood shown in Figure 7.
It is possible that simply increasing the GV will cause the quality degradation of synthetic
speech because it does not always make the generated sequence close to the natural one.
The GV-based generation algorithm increases the GV by considering the GV likelihood while
also considering the HMM likelihood to reduce the quality degradation due to the excessive
increase in the GV.
5.3 Evaluation of GV-constrained trajectory training

To evaluate the GV-constrained trajectory training method, voices were built for four English
speakers (2 males: bdl and rms, and 2 females: clb and slt) in the CMU ARCTIC database(Kominek & Black (2003)). For each speaker, we used subset A, which consists of about 600
sentences as training data and the remaining subset B, which consists of about 500 sentences
for evaluation. Context-dependent labels were automatically generated from texts using a text
analyzer. After initializing the HMM parameters in the traditional training process, trajectory
training was performed for the spectral and F0 components. Finally, GV-constrained trajectory
training was performed for both components. The covariance matrix of the GV p.d.f. for eachcomponent was previously trained using the GVs calculated from individual utterances in the
training data. The GV weight ω was empirically set to 0.125.
Table 2 shows the log-scaled trajectory likelihood, GV likelihood, and total likelihood for the
training data and evaluation data, where the total likelihood is calculated as the product of
the trajectory and GV likelihoods. All values are normalized by the number of frames. The
trajectory training causes signiﬁcant improvements in the trajectory likelihoods because theHMM parameters are optimized so as to directly maximize the trajectory likelihoods. It isinteresting to note that the trajectory training also causes improvements in the GV likelihood,
although the improvements are not large. These results suggest that the trajectory training
3 Several samples are available from http://isw3.naist.jp/˜tomoki/INTECH/ModelGV/index.html

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Training method
Traditional
Trajectory
GV-constrained

Training data
Evaluation data
Trajectory GV Total Trajectory GV Total
19.06 -67.63 -48.57
16.26 -67.86 -51.60
30.78 -32.35 -1.57
29.30 -33.33 -4.03
30.36
94.98 125.34
28.89
82.51 111.40

Table 2. Log-scaled trajectory likelihood given by Eq. (41), GV likelihood given by Eq. (44),and total likelihood given by Eq. (43) (ω = 1.0) for each training method.
Training method
MOS ± 95% conﬁdence interval
Traditional
2.35 ± 0.14
2.79 ± 0.12
Trajectory
3.46 ± 0.15
GV-constrained
Natural (analysis-synthesized)
4.38 ± 0.13
Table 3. Mean opinion score (MOS) on naturalness given in opinion test to evaluate
GV-constrained trajectory training.
generates better parameter trajectories than the traditional training. The GV likelihoods aredramatically improved by GV-constrained trajectory training. Note that this training method
does not cause signiﬁcant reductions to the trajectory likelihoods. This can be observed inboth the training and evaluation data. Overall, these results suggest that the GV-constrained
trajectory training method leads to parameter trajectories that more closely resemble thevarious characteristic features of real speech.
Table 3 shows the result of a subjective evaluation based on an opinion test on the naturalnessof the synthetic speech. The opinion score was set to the same 5-point scale as before andten listeners participated in the test. GV-constrained trajectory training yields signiﬁcantquality improvements compared with the traditional training. This tendency is similar to thatobserved when considering the GV in the parameter generation process, as shown in Table 1.
The trajectory training also yields signiﬁcant quality improvements, but these improvements
are much smaller than those yielded by GV-constrained trajectory training. 4

6. Summary
This chapter has described the two main techniques for modeling a speech parametersequence considering the global variance (GV) for HMM-based speech synthesis: theparameter generation algorithm considering the GV and GV-constrained trajectory training.
The traditional framework of HMM-based speech synthesis suffers from the oversmoothing
effect in speech parameters generated from the HMM, which makes the synthetic speechsound mufﬂed. Since the GV is inversely correlated with the oversmoothing effect, a metricon the GV of the generated parameters is effectively used to reduce mufﬂed sounds. The
parameter generation algorithm considering the GV uses not only an HMM likelihood but alsoa GV likelihood to determine the generated parameters. GV-constrained trajectory training
has been proposed by integrating this idea into a training framework. Consequently, it
provides a uniﬁed framework for training and synthesizing speech using a common criterion,
context-dependent GV modeling, and a more efﬁcient parameter generation process with
the GV based on a closed-form solution. Experimental results have demonstrated that both
4 Several samples are available from http://isw3.naist.jp/˜tomoki/INTECH/ModelGV/index.html

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Considering Global Variance for HMM-Based Speech Synthesis

149

methods yield very signiﬁcant improvements in the naturalness of synthetic speech and that
GV modeling is very effective in HMM-based speech synthesis.
Acknowledgements: This research was supported in part by MEXT Grant-in-Aid for Young
Scientists (A).

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Hidden Markov Models, Theory and Applications
Edited by Dr. Przemyslaw DymarskiISBN 978-953-307-208-1
Hard cover, 314 pages
Publisher InTech

Published online 19, April, 2011

Published in print edition April, 2011
Hidden Markov Models (HMMs), although known for decades, have made a big career nowadays and are stillin state of development. This book presents theoretical issues and a variety of HMMs applications in speech
recognition and synthesis, medicine, neurosciences, computational biology, bioinformatics, seismology,
environment protection and engineering. I hope that the reader will find this book useful and helpful for theirown research.

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Speech Synthesis, Hidden Markov Models, Theory and Applications, Dr. Przemyslaw Dymarski (Ed.), ISBN:
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