2014 IEEE International Conference on Acoustic, Speech and Signal Processing (ICASSP)MONDRIAN HIDDEN MARKOV MODEL FOR MUSIC SIGNAL PROCESSING
Masahiro Nakano, Yasunori Ohishi, Hirokazu Kameoka, Ryo Mukai, Kunio Kashino
NTT Communication Science Laboratories, NTT Corporation
3-1 Morinosato Wakamiya, Atsugi, Kanagawa 243-0198, Japannakano.masahiro@lab.ntt.co.jp

ABSTRACTThis paper discusses a new extension of hidden Markov models that can capture clusters embedded in transitions between
the hidden states. In our model, the state-transition matrices
are viewed as representations of relational data reflecting a
network structure between the hidden states. We specificallypresent a nonparametric Bayesian approach to the proposed
state-space model whose network structure is represented by aMondrian Process-based relational model. We show an application of the proposed model to music signal analysis throughsome experimental results.
Index Terms— Bayesian nonparametrics, hidden Markov
model, Mondrian process
1. INTRODUCTION
Hidden Markov models (HMMs) have been a major success story in time series modeling. In particulars, the recent
nonparametric Bayesian approaches constitute one of themost successful directions. [1] described the infinite hidden Markov model (IHMM) and [2] further formalized thehierarchical Dirichlet process hidden Markov model (HDPHMM), that develops an HMM with a (potentially) infinite
state space. However, after ten years of work on Bayesiannonparametric HMMs [3, 4], a number of significant improvements are still required. This paper addresses an important issue, namely whether we can capture “clusters” intransitions between the hidden states.
When we apply HMMs (which share hidden states) tomultiple time series, a network structure between the hidden
states is considered to be embedded in the state-transition
matrices, since they represent relational data. Actually, many
sequences (whose state-transition matrices are analyzed by
HMMs) often have square and rectangular clusters. Chord
progressions in music are a telling example. In the context
of HMMs, for the observed audio signals, each hidden stateis expected to correspond to an individual chord , such as C,
Dm7 and Edim. That is, the sequences of hidden states are
regarded as chord progressions. Popular songs are usuallyin a particular key. For example, the chords used within
the key “C major” are generally drawn from the “C major

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scale”, and are C, Dm, Em, F, G, Am and Bm-5. As a result,transitions between these chords tend to be preferred. On theother hand, when we want to add some “colors” to a piece,
various chords can be used, including borrowed chords,
altered chords, secondary dominants. A brief example isthe progression: “F→G→{Em, E7, Gdim}→Am” (often
adopted by Eurobeat music), i.e., E7 or G are sometimes
used in place of Em. Thus, “{E7, Gdim}→Am” may forma rectangular cluster on the state-transition matrices. In fact,
the concept of chord substitution in music theory has been auseful technique for assigning one chord in place of another.
Therefore, we believe that a technique for automatically finding the clusters in chord transitions is essential when building
machines that can understand music.
To capture such cluster structures, we have to deal withthe unknown numbers of hidden states and the unknown numbers of clusters in the transitions. This paper addresses thisissue using a Bayesian nonparametric fusion of HMMs andrelational models. Bayesian nonparametric approaches havebeen developed in both time-series modeling and relational
data analysis.
2. RELATED WORK AND PRELIMINARIES
2.1. HMM and hierarchical Dirichlet processes
Briefly, an HMM consists of a hidden state sequence z =(z1 , . . . , zT ) and a corresponding observation sequencey = (y1 , . . . , yT ). Each state zt denotes the index of thestate at time t. Transitions between states are governed byMarkov dynamics parameterized by the transition probabilities π, where πi,j = p(zt = j | zi = i) and zt ∼ π zt−1 . The
current state zt indexes the parameter θzt which parametrizes
the observation likelihood for that state: yt | zt ∼ f (θzt ).
Generally, it is not easy to determine the model complexity, i.e., the number of states K. To address this problem,
we can typically employ the hierarchical Dirichlet process,which can be used to develop an HMM with a potentially infinite state space. A Dirichlet process (DP) provides a discreterandom probability measure. It is very useful for constructing an (countably) infinite number of states. Given a positive concentration parameter γ and a base measure H on aparameter space Θ, the measure drawn from a Dirichlet process, G0 ∼ DP(γ, H) can be expressed as follows: G0 =∞
k=1 βk δθk , θk ∼ H, where δ· denotes the Dirac measures.
By using a well-known method, the weights β can be sampledk−1
via a stick-breaking construction [5]: βk = βk l=1 (1 −βl ), βk ∼ Beta(1, γ). This is often expressed as β ∼GEM(γ). Intuitively, the weight drawn from DP can be regarded as the mixture proportion of an infinite-dimensional
multinomial distribution. We might thus use DP priors over
the transition probabilities π i (i = 1, 2, . . . ) to obtain an
infinite HMM. However, there may be no coupling acrosstransitions out of different states if the transition probabilities are given independent priors. To avoid such a problem, we can use the following hierarchical Dirichlet process.
A hierarchical Dirichlet process (HDP) is a set of Dirichlet
processes coupled through a shared base measure that is itself drawn from a DP: Gi ∼ DP (α, G0 ) (i = 1, 2, . . . ).
Each
∞ random measure can be expressed as follows: Gi =k=1 πi,k δθk (i = 1, 2, . . . ). Identifying πi,j and θk as describing the transition probabilities from state i to state j andthe emission parameters, we can define the HDP-HMM asβ ∼ GEM(γ), π i ∼ DP(α, β),zt | zt−1 ∼ π zt−1 , θk ∼ H, yt | zt , θ∼f (θzt ). (1)A number of extensions to the HDP-HMM have been proposed. We briefly review the two extensions that are mostclosely related to our work. [6] described a sticky HDPHMM, which captures smoothly varying dynamics. Whenmodeling systems with state persistence, we would like to beable to express slow dynamics. They proposed sampling trani
sition distributions π i as follows: π i ∼ DP(α + κ, αβ+κδα+κ ).
An amount κ > 0 exhibits a bias for self-transitions.
[7] proposed a block diagonal infinite HMM (BD-IHMM),
which biases the transition matrix so that it is block diagonal.
When modeling sequences consisting of sub-process, there
is a higher tendency of transitions occurring between stateswith the same sub-process, i.e. a transition matrix will exhibit
nearly block diagonal. They introduced ρ ∼ GEM(ζ) to partition the countably infinite number of states into blocks. Eachstate k belongs to the block bk ∼ ρ. Then, they propose sampling transition distributions π i as follows: π i ∼ DP(α, β ∗i ),
∗
=
βi,j

ξ
1
βj (ξi∗ )δ(bi =bj ) , ξi∗ = 1 + ,
1+ξ
k βk δ(bi = bk )where a nonnegative parameter ξ control the amount of priorbias for within-block transitions.
2.2. Relational models and the Mondrian Process
Stochastic block models have been developed to model relational data, which are observations of relationships between
sets of entities. While we only focus on binary relations here,

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the ideas can be straightforwardly extended to other likelihood models. Suppose we are given D relational data as arrays of random binary variables Rd,i,j , where i and j index
entities xi ∈ X and yj ∈ Y . The goal of the stochastic blockmodels is to assign each pair (i, j) to a cluster.
[8] presented a Bayesian nonparametric approach to relational models called the infinite relational model (IRM),
which is expressed as the following generative model. Bothsets x1 , x2 , . . . and y1 , y2 , . . . are first clustered using a Chinese restaurant process prior. Then, each relation Rd,i,j isgenerated as follows: Rd,i,j ∼ Bernoulli(φBi,j ), where Si,jshows the cluster to which the pair (xi and yj ) belong, andφB denotes a cluster-specific Beta variable. The partitions
obtained by the IRM are not usually parsimonious, since suchpartitions look like regular grids. When one area of the array
requires a split, it causes the other parts to be divided, even if
the data suggest there is no such structure. This issue motivated the proposal of the Mondrian process.
Mondrian processes generate random partitions on product spaces not constrained to be regular grids. [9] and [10]presented a Markov process on hierarchical partitioning.
As shown in [9] and [10], we can construct a Mondrian
process based on the following famous recursive procedure. For simplicity, we focus only on a Mondrian process
M ∼ MP(λ, (a, A), (b, B)) on a 2-dimensional rectangle(a, A) × (b, B) [11]. Starting with an initial budget λ, theMondrian process makes a sequence of cuts that split rectangles into sub-rectangles. Each cut incurs a random cost
E. If E exceeds the budget, the process halts and returns thecurrent partitions. Otherwise, the Mondrian process makes anaxis-aligned cut uniformly at random along (a, A) and (b, B).
The cost E to cut the rectangle (a, A) × (b, B) is distributed
exponentially: E ∼ Exp(A − a + B − b). After a cut ismade (for example, suppose a cut split (a, A) into (a, x) and(x, A)), the cost E is subtracted from the budget λ and a new
budget λ = λ − E is calculated. Both sub-rectangles are
drawn independently from M< ∼ MP(λ , (a, x) × (b, B))
and M> ∼ MP(λ , (x, A) × (b, B)). The partition on(a, A) × (b, B) can be expressed as: M< ∪ M> .
As shown in [9], we obtain the MP-based relational model
as follows: We first generate a hierarchical partitioning of[0, 1]2 from the Mondrian process, and then the column/row
location of each entry is generated based on a geometrical interpretation of the Uniform[0, 1] random variable.
3. MONDRIAN HIDDEN MARKOV MODEL
The goal of a Mondrian HMM is not only to analyze thetemporal dynamics of time series but also to capture clusters embedded in transitions between the hidden states. Suppose we are given D time series {Yd,1 , Yd,2 , . . . , Yd,Td } (d =1, . . . , D) (Fig. 1 (left) illustrates a case of two time series).
Although the lengths of the time series are not necessarily thesame, subscripts are omitted for simplicity of notation, i.e.,

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Fig. 1. Illustration of Mondrian HMM.

Fig. 2. Matrices of transition counts from ground truth (left), andwe express T1 , . . . , TD as T . We assume that D time series
share the hidden states and their corresponding parameters foremission distributions. A network structure between the hidden states is considered embedded in state-transition matrices.
As relational models explain the relationships between sets ofentities, the clusters on state-transition matrices provide an insight into the network structure of the hidden states. We thus
consider that state-transition matrices should be partitioned
into square and rectangular clusters.
Using a Mondrian process with the Lebesgue measure,
we first sample a random partition M of the unit square into
blocks: M ∼ MP(λ, [0, 1], [0, 1]). We next sample hypothetical locations Sk (related to transitions “out”) and Sk (related
to transitions “in”) of the hidden state k on the unit [0, 1]:
Sk ∼ Uniform[0, 1], Sk ∼ Uniform[0, 1]. By using the partition M, and the hypothetical locations Sk (k = 1, 2, . . . )and Sk (k = 1, 2, . . . ), we can express “clusters” in the transitions between the hidden states.
Binary variables Rd,i,j show whether the transition
from state i to state j is preferable. That is, whether thetransition i → j will be more likely, when Rd,i,j = 1.
They are controlled independently according to the clusters:
Rd,i,j ∼ Bernoulli(φd,Bi,j ), φd,B ∼ Beta(a0 , a1 ), where
Bi,j denotes the block B ∈ M such that (Si , Sj ) ∈ B. The
transition probabilities are formulated as: β ∼ GEM(γ),∗∗
dπ d,i ∼ DP(α, β ∗d,i ), ξd,i
= 1 +  Rξd,i,lβl , βd,i,j =Mondrian partitions and means of binary relations on transition matrices inferred by the Mondrian HMM (right). Larger black squares
have larger E[Rd,i,j ] values. Mondrian HMM can capture an active
network structure between the hidden states.

l1∗ Rd,i,j. These modified weights β ∗d,i are reminis1+ξd βj (ξd,i )cent of the BD-IHMM [7]. β ∗d,i are scaled to favor relatively

higher probabilities πd,i,j for transition from state i to state jsuch that Rd,i,j = 1. As a result, the Markov chains are controlled by the nearly Mondrian-textured transition matrices.
Additionally, we would like to avoid rare transitions causing
unreasonable relational structures in the transition between
hidden states. We thus regard rare transitions as the spiky
noise of sequences. We introduce binary indicators, as usedin image processing [12]:cd,t ∼ Bernoulli(cd ), cd ∼ Beta(a2 , a3 ).

(2)Thus, the Markov chains are generated from the mixture of
the general transition probabilities π d,i and the special proportions π d : π d ∼ DP(α, β),Zd,t | Zd,t−1 ∼ I[cd,t = 1]π d,Zd,t−1 + I[cd,t = 0]π d .
Finally, emission parameters for θ are drawn from H and theobservation sequences Yd,t are generated from f (θZd,t ).

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4. EXPERIMENTS“Chord progression analysis” in music is an interesting andattractive application for machine learning. The goal of theexperiments was to capture clusters in the transitions between
chords from music signals. Note that chord recognition tasks
themselves are not our main focus although chord detection
implicitly becomes part of the procedure. We are interested in
not only chord recognition but also chord progression analysis
from audio signals, i.e., we want to find clusters in transitions
between the hidden states. We used a standard dataset consisting of 180 Beatles songs collected from 12 albums. The
chord annotations are provided for these songs [13]. We employed the beat-synchronous chroma features [14] as input
data. Each frame of the features represents the intensity associated with each of the 12 semitones (e.g. piano keys) of themusical octave onto which the entire spectrum is projected(i.e., all octaves are folded together). The beat-synchronous
chroma features consist of one (12-dimensional) feature vector per beat, which identifies the beat segmentation times in
the music audio. We excerpted major scale sections and transposed them to C major, because chords depend on the musicalkey. That is, chroma features were circularly shifted so that
all the songs were transposed to C major. The pre-measured
sequences had 64703 frames.
Synthetic data: Our first experiment was intended toillustrate the behavior of the Mondrian HMM. We generated
12 sequences from a 12-state HMM with single-Gaussian
emissions. Each state randomly chose one label from thechord annotation data, and learned means and variances fromthe corresponding chroma features. Transition probabilities
favored active blocks at a rate of 20/21. Each sequence had
5000 steps generated from the synthetic HMMs. Fig. 2(left) shows the matrices of transition counts in the true state
sequences (ground truth). As shown in Fig. 2 (right), theMondrian HMM can capture nearly the active network structure even though there are little-used states. To evaluate thepredictive power of learned models, we compare the test-set
log probabilities of the Mondrian HMM and the HDP-HMM
on two types of synthetic data: (1) 100 randomly-generated 1dimensional Poisson emission dataset, and (2) 10 randomlygenerated 12-dimensional Poisson emission dataset. Each

−1.1

x 10

−1

−1.05
−1.14
−1.1
−1.16

−1.2

HDP−HMM

−1.2

G#m
F#
D#m
E
Fm
C#m
B
D#
Bm
Gm
Cm
A#
F#m
C#
G#
A
Em
Am
D
Dm
F
G
C
Dm
Em
Am
D
G
C
F
C#m
Fm
F#m
G#
B
A#
Cm
Bm
F#
D#m
G#m
D#
C#
E
A
Gm

−1.15

Mondrian HMM

Little Child

Honey Pie
G#m
F#
D#m
E
Fm
C#m
B
D#
Bm
Gm
Cm
A#
F#m
C#
G#
A
Em
Am
D
Dm
F
G
C

−1.12

−1.18

With A Little Help
From My Friends

6

Mondrian HMM

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F#
D#m
E
Fm
C#m
B
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Bm
Gm
Cm
A#
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C#
G#
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Em
Am
D
Dm
F
G
C
Dm
Em
Am
D
G
C
F
C#m
Fm
F#m
G#
B
A#
Cm
Bm
F#
D#m
G#m
D#
C#
E
A
Gm

−0.95

HDP−HMMDig ItLet It Be

consists of 12 training sequences (each has 5000 steps) and
12 test sequences (each has 5000 frames) generated from12-state HMMs. We performed the Mondrian HMM and the
HDP-HMM once per each data. The result of the test-set log
probabilities for 100 synthetic data is shown in Fig. 3.
Beatles chord progression analysis: Our second experiment applies the Mondrian HMM to real-world data, 180
Beatles songs. It is very challenging to identify “chords”
of hidden states in a fully unsupervised manner because themodel does not know the correspondence relationship between “chord names” and chroma features. We therefore
employed a semi-supervised learning approach. Of the 180
pre-measured songs, we divided into test songs and training
songs. The songs for training provided pairs of the chromafeatures and the “known” chord labels, but did not tell us
anything about the state transitions. The state-transition probabilities were inferred only from the test data. Chords werereduced to 25 labels (12 major + 12 minor + no chord) byanalogy with the classical chord recognition tasks. We used
an infinite Dirichlet process mixture of Gaussians as emission
distributions similar to [6], since we found single-Gaussian
emission distributions to be less effective.
We compared the performance of the HDP-HMM, thesticky HMM, the Mondrian HMM. For the Mondrian HMM,
we evaluate two variants: one is the original version; the otheris the IRM for the Mondrian process (called IRM-HMM). We
evaluated the chord recognition performance based on thebest of the five runs. We ran each algorithm for 2000 iterations, and then the last-sampled hidden sequences werecompared with the ground truth to calculate the accuracy ofthe chord detection. Table 1 shows results for 3-fold and6-fold cross validations. This experimental result is strongly
in favor of the sticky HMM. The reason is that, for exam-

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Long Long Long

Dm
Em
Am
D
G
C
F
C#m
Fm
F#m
G#
B
A#
Cm
Bm
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D#m
G#m
D#
C#
E
A
Gm

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Gm
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C

Please Please Me

Sexy Sadie

Two of Us
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Gm
Cm
A#
F#m
C#
G#
A
Em
Am
D
Dm
F
G
C

Dm
Em
Am
D
G
C
F
C#m
Fm
F#m
G#
B
A#
Cm
Bm
F#
D#m
G#m
D#
C#
E
A
Gm

G#m
F#
D#m
E
Fm
C#m
B
D#
Bm
Gm
Cm
A#
F#m
C#
G#
A
Em
Am
D
Dm
F
G
C

G#m
F#
D#m
E
Fm
C#m
B
D#
Bm
Gm
Cm
A#
F#m
C#
G#
A
Em
Am
D
Dm
F
G
C
Dm
Em
Am
D
G
C
F
C#m
Fm
F#m
G#
B
A#
Cm
Bm
F#
D#m
G#m
D#
C#
E
A
Gm

Table 1. Chord recognition performance
HDP
stickyIRM
Mondrian
30 songs
57.2
62.8
61.6
61.1
60 songs
58.6
63.8
64.2
64.5

Dm
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Fm
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G#
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Gm

Fig. 3. Left: Test-set log probabilities for the 100 synthetic datasets (1-dimensional Poisson emissions) for models
learned by the Mondrian HMM and the HDP-HMM. Right:
Test-set log probabilities for the 10 synthetic datasets (eachdata was tested 10 times) (12-dimensional Poisson) for models learned by the Mondrian HMM and the HDP-HMM. We
can verify that the flexibility of the Mondrian HMM does not
significantly degrade the predictive power.

G#m
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5

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x 10

Dm
Em
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G
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F
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−1.08

Fig. 4. Learned Mondrian partitions and E[Rd,i,j ] of 9 songs.

Four blocks in the bottom left show typical transitions, nearly
Diatonic chords {C, Dm, Em, F, G, A, Bm} generated from C
major scale. What is very interesting for us is that these blocks include “D”. The Beatles often used “D” as secondary dominants,
while the transition “Dm”→G is generally an extremely typical pattern (called two-ﬁve). Moreover, the bottom-left block captures theset {C, F, G}, which is called three chords. As some songs are
actually built around three chords, these chords have strong connections. The model could find such a structure from the songs.

ple, 4 beat songs typically stay in the same states threetimes. However, the Mondrian HMM performs similarly tothe sticky HMM. We thus confirm that the flexibility of theMondrian HMM does not degrade the performance of hidden
state assignments.
Finally, Fig. 4 shows the E[Rd,i,j ] values of 9 songs (randomly chosen from 30 test songs) are illustrated. “Am” (noassignments) and “no chord” are omitted for visibility. We
used a similar procedure to that used for the synthetic data toevaluate E[Rd,i,j ]. Larger black squares shows the “active”
state transitions.
5. CONCLUSION
This paper deals with a new extension of HMMs, which cancapture clusters in transitions between the hidden states. Our
method is based on the nonparametric Bayesian fusion of the
HDP-HMM and a Mondrian process-based relational model.
While we confirmed that the Mondrian HMM sees the possibility of capturing clusters in transitions between the hidden
states through experiments on both synthetic and real-world
data, we anticipate many directions of improvement. In the
future, we are interested in embedding relational models in
the n-gram models.

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