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Identifying the Recurrence of Sleep Apnea Using A Harmonic Hidden Markov Model

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We propose to model time-varying periodic and oscillatory processes by means of a hidden Markov model where the states are defined through the spectral properties of a periodic regime. The number of states is unknown along with the relevant periodicities, the role and number of which may vary across states. We address this inference problem by a Bayesian nonparametric hidden Markov model, assuming a sticky hierarchical Dirichlet process for the switching dynamics between different states while the periodicities characterizing each state are explored by means of a transdimensional Markov chain Monte Carlo sampling step. We develop the full Bayesian inference algorithm and illustrate the use of our proposed methodology for different simulation studies as well as an application related to respiratory research which focuses on the detection of apnea instances in human breathing traces.
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The Annals of Applied Statistics
2021, Vol. 15, No. 3, 1171–1193
https://doi.org/10.1214/21-AOAS1455
©Institute of Mathematical Statistics, 2021
IDENTIFYING THE RECURRENCE OF SLEEP APNEA USING A
HARMONIC HIDDEN MARKOV MODEL
BYBENIAMINO HADJ-AMAR1,*,BÄRBEL FINKENSTÄDT1,,MARK FIECAS2AND
ROBERT HUCKSTEPP3
1Department of Statistics, University of Warwick, *Beniamino.Hadj-Amar@rice.edu;B.F.Finkenstadt@warwick.ac.uk
2School of Public Health, Division of Biostatistics, University of Minnesota, mfiecas@umn.edu
3School of Life Sciences, University of Warwick, R.Huckstepp@warwick.ac.uk
We propose to model time-varying periodic and oscillatory processes by
means of a hidden Markov model where the states are defined through the
spectral properties of a periodic regime. The number of states is unknown
along with the relevant periodicities, the role and number of which may vary
across states. We address this inference problem by a Bayesian nonparamet-
ric hidden Markov model, assuming a sticky hierarchical Dirichlet process for
the switching dynamics between different states while the periodicities char-
acterizing each state are explored by means of a transdimensional Markov
chain Monte Carlo sampling step. We develop the full Bayesian inference
algorithm and illustrate the use of our proposed methodology for different
simulation studies as well as an application related to respiratory research
which focuses on the detection of apnea instances in human breathing traces.
1. Introduction. Statistical methodology for identifying periodicities in time series can
provide meaningful information about the underlying physical process. Nonstationary be-
havior seems to be the norm rather than the exception for physiological time series as time-
varying periodicities, and other forms of rich dynamical patterns are commonly observed in
response to external perturbations and pathological states. For example, body temperature
and rest activity might exhibit changes in their periodic patterns as an individual experiences
a disruption in its circadian timing system (Krauchi and Wirz-Justice (1994), Komarzynski
et al. (2018)). Heart rate variability and electroencephalography are other examples of data
that are often characterized by time-changing spectral properties, the quantification of which
can provide valuable information about the well being of a subject (Malik (1996), West, Prado
and Krystal (1999), Cohen (2014), Bruce et al. (2018)). This paper is motivated by model-
ing airflow trace data, obtained from a sleep apnea study, where our objective is to identify
and model the recurrence of different periodicities which are indicative of the apneic and
hyponeic events.
1.1. A case study on sleep apnea in humans. Our study focuses on sleep apnea (Heinzer
et al. (2015)), a chronic respiratory disorder characterized by recurrent episodes of temporary
(2 breaths) cessation of breathing during sleep (about 10 seconds in humans). Sleep apnea
negatively affects several organ systems, such as the heart and kidneys in the long term. It
is also associated with an increased likelihood of hypertension, stroke, several types of de-
mentia, cardiovascular diseases, daytime sleepiness, depression and a diminished quality of
life (Ancoli-Israel et al. (1991), Teran-Santos et al. (1999), Peker et al. (2002), Young, Pep-
pard and Gottlieb (2002), Yaggi et al. (2005), Cooke et al. (2009), Dewan, Nieto and Somers
(2015)). Instances of sleep apnea can be subclassified based on the degree of reduction in
Received March 2020; revised December 2020.
Key words and phrases. Sleep apnea, time-varying frequencies, reversible-jump MCMC, Bayesian nonpara-
metrics, hierarchical Dirichlet process.
1171
1172 HADJ-AMAR, FINKENSTÄDT, FIECAS AND HUCKSTEPP
FIG.1. Airflow trace collected over a period of five and half minutes of continuous breathing where instances
of simulated apnea and hypopnea (highlighted on the graph)were recurring over time.
airflow to the lungs, whereby apneas are classified as a reduction of airflow by 90% and
hypopneas require a reduction in airflow by at least 30% (with a reduction of blood oxygen
levels by at least 3%). For example, the airflow trace shown in Figure 1was collected from
a human over a time span of 5.5 minutes of continuous breathing. During this time, apneic
and hyponeic events were simulated; apneas appear in the first and second minute and around
the start of the fifth minute, where there are two instances of hypopneas in the first half of
the second minute and at the start of the fourth minute, as marked in Figure 1. Note these
events were classified by eye by an experienced experimental researcher. Detecting apneic
and hyponeic events during sleep is one of the primary interests of researchers and clinicians
working in the field of sleep medicine and relevant healthcare (Berry et al. (2017)). Man-
ual classification is a time-consuming process, and hence there is a need of a data-driven
approach for the automated classification of these types of events.
1.2. Hidden Markov models and spectral analysis. Approaches to spectral analysis of
nonstationary processes were first developed by Priestley (1965) who introduced the concept
of evolutionary spectra, namely, spectral density functions that are time dependent as well
as localized in the frequency domain. This modeling framework was formalized as a class
of nonstationary time series called locally stationary (Dahlhaus (1997)). Locally stationary
processes can be well approximated by piecewise stationary processes, and several authors
proposed to model the time-varying spectra of locally stationary time series through the piece-
wise constant spectra of the corresponding stationary segments (Adak (1998), Ombao et al.
(2001), Davis, Lee and Rodriguez-Yam (2006)). This framework was extended to a Bayesian
setting by Rosen, Stoffer and Wood (2009)andRosen, Wood and Stoffer (2012) who esti-
mated a time-varying spectral density using a fixed number of smoothing splines and approx-
imated the likelihood function via a product of local Whittle likelihoods (Whittle (1957)).
Their methodology is based on the assumption that the time series are piecewise station-
ary, and the underlying spectral density for each partition is smooth over frequencies. In
order to deal with changes in spectral densities with sharp peaks which can be observed for
some physiological data sets such as respiratory data, Hadj-Amar et al. (2020) proposed a
change-point analysis where they introduced a Bayesian methodology for inferring change
points along with the number and values of the periodicities affecting each segment. While
these approaches allow us to analyse the spectral changing properties of a process from a
IDENTIFYING THE RECURRENCE OF SLEEP APNEA 1173
retrospective and exploratory point of view, in order to develop a more comprehensive un-
derstanding of the process driving the data, further modeling assumptions are needed that
quantify the probabilistic rules governing the transitions as well as recurrence of different os-
cillatory dynamic patterns. For example, in the context of experimental sleep apnea research,
both, correctly classifying the states of apnea as well as quantifying their risk of recurrence,
possibly in the context of real-time monitoring of patients, are of major interest to the devel-
opment of treatments for breathing disorders.
Here, we address the switching dynamics between different oscillatory states in the frame-
work of a hidden Markov model (HMM) that assumes a discrete latent state sequence whose
transition probabilities follow a Markovian structure (see, e.g., Ephraim and Merhav (2002),
Rabiner (1989), Cappé, Moulines and Rydén (2005)). Conditioned on the state sequence,
the observations are assumed to be independent and generated from a family of probabil-
ity distributions, which hereafter we refer to as the emission distributions. HMMs are, ar-
guably, among the most popular statistical approaches used for modeling time series data
when the observations exhibit nonstationary characteristics that can be represented by an un-
derlying and unobserved hidden process. These modeling approaches, also known as hidden
Markov processes and Markov-switching models, became notable by the work of Baum and
Petrie (1966)andBaum and Eagon (1967), and HMMs have since been successfully used in
many different applications (Krogh et al. (1994), Yau et a l . (2011), Langrock et al. (2013),
Yaghouby and Sunderam (2015), Huang et al. (2018)).
As we are interested in modeling the recurrence of periodicities in the airflow trace data,
we propose a harmonic HMM where the discrete latent state sequence reflects the time-
varying changes as well as recurrence of periodic regimes, as defined by their spectral prop-
erties. Furthermore, we pursue a flexible nonparametric specification within a Bayesian ap-
proach by assuming the infinite-state hierarchical Dirichlet process (HDP) as a building block
(Teh et al. (2006)). The HDP-HMM approach places a Dirichlet process (DP) prior on the
Markovian transition probabilities of the system, while allowing the atoms associated with
the state-specific conditional DPs to be shared between each other, yielding an HMM with a
countably infinite number of states. The HDP-HMM, therefore, not only provides a nonpara-
metric specification of the transition distributions but also removes the need for specifying
a priori the number of states. In our case study, while it is true that by looking at Figure 1
there seems to be a substantial difference between the apnea states and normal breathing (i.e.,
neither apnea or hypopnea), it is also conceivable that normal breathing may exhibit many
distinct periodic patterns, both with respect to this subject and possibly across different in-
dividuals. Yet we are interested in classifying, in an unsupervised fashion, the occurrence of
apnea/hypopnea states; it might also be necessary to model states corresponding to normal
breathing and other aspects that characterize respiration, such as a sigh, for example. Paz
and West (2013) reported at least 13 forms of breathing patterns, including forms of apnea,
further justifying the need to not prespecify the number of hidden states and hence using a
more versatile HDP for this application. We focus on the sticky HDP-HMM by Fox et al.
(2011), where an additional parameter is introduced to promote selftransition with the effect
that the sticky HDP-HMM more realistically explains the switching dynamics between states
that exhibit some temporal mode persistence. We hence extend the Bayesian methodology
for the sticky HDP-HMM to a spectral representation of the states where inference for the
variable dimensionality regarding the number of periodicities that characterize the emission
distributions of the states is achieved by developing an appropriate form of transdimensional
MCMC sampling step (Green (1995)).
This article introduces a dynamic oscillatory model that is remarkably flexible while being
developed in a framework that is still computationally accessible. To the best of our knowl-
edge, it is the first statistical methodology that exploits an HMM for analyzing the spectral
1174 HADJ-AMAR, FINKENSTÄDT, FIECAS AND HUCKSTEPP
properties of a time series while quantifying the probabilistic mechanism governing the tran-
sitions and recurrence of distinct dynamic patterns. The rest of the paper is organized as
follows. Section 2presents the model and the general framework of our Bayesian approach.
Sections 3and 4provide the inference scheme and simulation studies to show the perfor-
mance of the proposed method. In Section 5we illustrate the use of our approach to detect
instances of apnea in human breathing traces.
2. A sticky HDP-HMM with oscillatory emissions. We propose a Bayesian approach
relevant for analyzing observations of oscillatory dynamical systems based on an HMM.
The observational model follows Andrieu and Doucet (1999)andHadj-Amar et al. (2020),
where the state-dependent data generating process is expressed via a Gaussian harmonic re-
gression with an unknown number of periodicities. We integrate this methodology with the
nonparametric HDP-HMM model introduced by Teh et al. (2006), where the rows of the
infinite-dimensional transition matrix are framed to enable linkage between the probabilities
associated with each hidden state in a hierarchical manner. Temporal mode persistence of
the latent state sequence is achieved using the sticky HDP-HMM formulation of Fox et al.
(2011).
Let y=(y1,...,y
T)be a realization of a time series whose oscillatory behavior may
switch dynamically over time, and let z=(z1,...,z
T)denote the hidden discrete-valued
states of the Markov chain that characterize the different periodic regimes, where ztdenotes
the state of the Markov chain at time t. Any observation yt, given the state zt, is assumed
to be conditionally independent of the observations and states at other time steps (Rabiner
(1989)). Here, a highly flexible nonparametric approach is postulated by assuming that the
state space is unbounded, that is, has infinitely many states, as in Beal, Ghahramani and
Rasmussen (2002)andTeh et al. (2006). Thus, the Markovian structure on the state sequence
zis given by
(1) zt|zt1,(πj)
j=1πzt1,t=1,...,T,
where πj=j1
j2,...) represents the (countably infinite) state-specific vector of transi-
tion probabilities and, in particular, πjk =p(zt=k|zt1=j),wherep(·)is used as a generic
notation for probability density or mass function, whichever appropriate. We assume that the
initial state has distribution π0=01
02,...), namely, z0π0.
Next, assume that each state jrepresents a periodic regime that is characterized by dj
relevant periodicities whose frequencies are denoted by ωj=j1,...,ω
jdj), recalling that
periodicity is the inverse of frequency. Let βj=(β
j1,...,β
jdj)be the vector of linear coef-
ficients that can be associated with the amplitude and phase corresponding to each frequency
ωjl that is of relevance to state j,whereβjl = (1)
jl (2)
jl )and l=1,...,d
j.Furthermore,
let us define θj=(dj,ω
j,β
j2
j),whereσ2
jaccounts for a state-specific variance. Then,
each observation is assumed to be generated from the following emission distribution:
(2) yt|zt=j,(θj)
j=1Nftj 2
j,t=1,...,T,
where the mean function ftj for state jat time tis (Andrieu and Doucet (1999), Hadj-Amar
et al. (2020)) specified to be oscillatory, that is,
(3) ftj =xt(ωj)βj,
and the vector of basis functions xt(ωj)is defined as
(4) xt(ωj)=cos(2πωj1t),sin(2πωj1t),...,cos(2πωjdjt),sin(2πωjdjt).
IDENTIFYING THE RECURRENCE OF SLEEP APNEA 1175
The dimension of each oscillatory function depends on the unknown number djof peri-
odicities relevant to state j. Given a prefixed upper bound for the number of relevant peri-
odicities per state, dmax, the parameter space jfor the vector of emission parameters θj
can be written as j=dmax
dj=1{dj}×{IR2dj×dj×IR+},wheredj=(0,0.5)djdenotes
the sample space for the frequencies of the jth state. Hadj-Amar et al. (2020) introduced this
modeling approach for oscillatory data that show regime shifts in periodicity, amplitude and
phase. They assume that, conditional on an unknown number of change points at unknown
positions, the time series process can be approximated by a sequence of segments, each with
mean functions specified by Gaussian harmonic models of the form given in equation (3).
Here, this approach will be combined with a nonparametric sticky HDP-HMM model (Fox
et al. (2011)) which provides a structure for modeling switching dynamics and connectivity
between different states.
2.1. A Bayesian nonparametric framework for unbounded Markov states. Dirichlet pro-
cesses provide a simple description of clustering processes where the number of clusters is not
fixed a priori. Suitably extended to a hierarchical DP, this form of stochastic process provides
a foundation for the design of state-space models in which the number of modes is random
and inferred from the data. In contrast to classic methods that assume a parametric prior on
the number of states or use model selection techniques to determine the number of regimes in
an HMM, here we follow Beal, Ghahramani and Rasmussen (2002), Teh et al. (2006)andFox
et al. (2011) and assume the number of states to be unknown. We, therefore, do not need to
prespecify the number of hidden states which provides a more flexible modeling framework.
The DP may be used in frameworks where an element of the model is a discrete random vari-
able of unknown cardinality (Walker (2010)). The unbounded HMM (i.e., where the number
of possible states is unknown) can be seen as an infinite mixture model, where the mixing
proportions are modelled as DPs (Beal, Ghahramani and Rasmussen (2002), Rasmussen and
Ghahramani (2002), Teh et al. (2006)).
The current state ztindexes a specific transition distribution πztover the positive integers,
whose probabilities are the mixing proportions for the choice of the next state zt+1. To allow
the same set of next states to be reachable from each of the current states, we introduce a set
of state-specific DPs whose atoms are shared between each other (Teh et al. (2006)). As in
Fox et al. (2011), we implement the sticky version by increasing the expected probability of
selftransitions. In particular, the state-specific transition distribution πjfollows the HDP
(5) πj|η, κ, αDPη+κ, ηα+κδj
η+κ,
where
α|γGEM(γ ).
Here, the sequence α=k)
k=1can be seen as a global probability distribution over the
positive integers that ties together the transition distributions πjand guarantees that they
have the same support. We denote by GEM (γ)
1the stick-breaking construction (Sethuraman
(1994), Pitman (2002)) of αas
(6) αk=νk
k1
l=1
(1νl),
1GEM is an abbreviation for Griffiths, Engen and McCloskey; see Ignatov (1982), Perman, Pitman and Yor
(1992), Pitman (1996) for background.
1176 HADJ-AMAR, FINKENSTÄDT, FIECAS AND HUCKSTEPP
where
(7) νk|γBeta(1,γ),
for k=1,2,...,andγis a positive real number that controls the expected value of the
number of elements in αwith significant probability mass. Equations (6)and(7) can be mo-
tivated by the equivalent process where a stick of length one is split into lengths specified by
the weights αk,wherethekth proportion is a random fraction νkof the remaining stick af-
ter the preceding (k 1)proportions have been constructed. The stick-breaking construction
ensures that the sequence αsatisfies
k=1αk=1 with probability one.
Conditional on α, the hierarchical structure given in equation (5) indicates that the state-
specific transition distributions πjare distributed according to a DP with concentration pa-
rameter η+κand base distribution α+κδj)/(η +κ) that is itself a DP. Here, ηis a
positive real number that controls the variability of the πjs around α, while κis a positive
real number that inflates the expected probability of a selftransition (Fox et al. (2011)) and
δjdenotes a unit-mass measure concentrated at j. By setting κ=0 in equation (5), we ob-
tain the nonsticky HDP-HMM framework proposed by Teh et al. (2006). It was noted that
this specification could result in an unrealistically rapid alternation between different (and
often redundant) states. The sticky formulation of Fox et al. (2011) allows for more tempo-
ral state persistence by inflating the expected probabilities of selftransitions by an amount
proportional to κ,thatis,
E[πjk|η, κ, α]= η
η+καk+κ
η+κδ(j,k),
where δ(j,k) =1, if k=jand zero otherwise. Though the sticky parameter parameter in-
creases temporal persistence of the hidden state sequence, several simulations studies (see,
e.g., Fox et al. (2011)) suggest that the sticky HDP-HMM is still capable to capture and
identify states that are short in duration by inferring a small probability of selftransition.
3. Inference. Our inference scheme is formulated within a full Bayesian framework,
where our proposed sampler alternates between updating the emission and the HMM pa-
rameters. Section 3.1 presents a reversible jump MCMC based algorithm to obtain posterior
samples of the emission parameters θj, where a transdimensional MCMC sampler is devel-
oped to explore subspaces of variable dimensionality regarding the number of periodicities
that characterize state j. This part of the inference scheme follows a similar structure to the
one presented by Andrieu and Doucet (1999)andHadj-Amar et al. (2020). In Section 3.2
we address model search on the number of states by exploiting the Chinese restaurant fran-
chise with loyal customers (Fox et al. (2011)), a metaphor that provides the building blocks
to perform Bayesian nonparametric inference for updating the HMM parameters. The details
related to this part of the sampler follow the scheme presented in Fox et al. (2011). The re-
sulting Gibbs sampler for the full estimation algorithm is described in Section 3.2.1 while in
Section 3.2.2 we address the label switching problem related to our proposed approach.
3.1. Emission parameters. Conditional on the state sequence z, the observations yare
implicitly partitioned into a finite number of states, where each state refers to at least one seg-
ment of the time series. When a type of periodic behavior recurs over time, the corresponding
state is necessarily related to more than one segment. Let y
j=(y
j1,y
j2,...,y
jRj)be the
vector of (nonadjacent) segments that are assigned to state j,whereyjr denotes the rth seg-
ment of the time series for which zt=jand Rjis the total number of segments assigned to
that state. Then, the likelihood of the emission parameter θj, given the observations in y
j,is
(8) Lθj|y
j=2πσ2
jT
j/2exp1
2σ2
j
tI
jytxt(ωj)βj2,
IDENTIFYING THE RECURRENCE OF SLEEP APNEA 1177
where I
jand T
jdenote the set of time points and number of observations, respectively,
associated with y
j.
Following Hadj-Amar et al. (2020), we assume independent Poisson prior distributions for
the number of frequencies djfor each state j, constrained on 1 djdmax. Conditional
on dj, we choose a uniform prior for the frequencies ωj,l Uniform(0
ω), l =1,...,d
j,
where 0
ω<0.5. The value of φωcan be chosen to be informative in the sense that it may
reflect prior information about the significant frequencies that drive the overall variation in
the data, for example, φωmay be assumed to be in the low frequencies range 0
ω<0.1.
Analogous to a Bayesian regression (Bishop (2006)), a zero-mean isotropic Gaussian prior is
assumed for the coefficients of the jth regime, βjN2dj(02
βI), where the prior variance
σ2
βis fixed at a relatively large value (e.g., in our case 102). The prior on the residual variance
σ2
jof state jis specified as Inverse-Gamma(ξ0
2,τ0
2),whereξ0and τ0are fixed at small values,
noticing that, when ξ0=τ0=0, we obtain Jeffreys’ uninformative prior (Bernardo and Smith
(2009)).
Bayesian inference on θjis built upon the following factorization of the joint posterior
distribution:
(9) pθj|y
j=pdj|y
jpωj|dj,y
jpβj|ωj,d
j,y
jpσ2
j|βj,ωj,d
j,y
j.
Sampling from (9) gives rise to a model selection problem regarding the number of peri-
odicities, thus requiring an inference algorithm that is able to explore subspaces of variable
dimensionality. This will be addressed by the reversible-jump sampling step introduced in
the following section.
3.1.1. Reversible-jump sampler. Here, we provide the details for drawing θjfrom the
posterior distribution p(θj|y
j)given in equation (9). Our methodology follows Andrieu and
Doucet (1999)andHadj-Amar et al. (2020) and is based on the principles of reversible-jump
MCMC introduced in Green (1995). Notice that, conditional on the state sequence z,the
emission parameters θjcan be updated independently and in parallel for each of the current
states. Hence, for the rest of this subsection and for ease of notation, we drop the subscript
corresponding to the jth state.
At each iteration of the algorithm, a random choice with probabilities given in (10), based
on the current number of frequencies d, will dictate whether to add a frequency (birth step)
with probability bd, remove a frequency (death step) with probability rdor update the fre-
quencies (within step) with probability μd=1bdrd,where
(10) bd=cmin1,p(d +1)
p(d) ,r
d+1=cmin1,p(d)
p(d +1),
for some constant c∈[0,1
2]and p(d) is the prior probability. Here, as in Hadj-Amar et al.
(2020), we fixed c=0.4, but other values are admissible as long as cis not larger than
0.5 to guarantee that the sum of the probabilities does not exceed 1 for some values of c.
Naturally, bdmax =r1=0. An outline of these moves is as follows (further details are provided
in Supplementary Material A (Hadj-Amar et al. (2021))).
Within-model move. Conditional on the number of frequencies d, the vector of frequencies
ωis sampled, where we update the frequencies, one-at-a-time, using Metropolis–Hastings
(M–H) steps with target distribution
(11) pω|β2,d,yexp1
2σ2
tIytxt(ω)β21[ωd].
Specifically, the proposal distribution is a combination of a Normal random walk centred
around the current frequency and a draw from the periodogram of ˆ
y,where ˆ
ydenotes a
1178 HADJ-AMAR, FINKENSTÄDT, FIECAS AND HUCKSTEPP
segment of data randomly chosen from ywith probability proportional to the number of
observations belonging to that segment. Naturally, when a state does not recur over time, that
is, when a state refers to only one segment of the time series, that segment is chosen with
probability one. Next, updating the vector of linear coefficients βand the residual variance
σ2is carried out, as in the fashion of the usual normal Bayesian regression setting (Gelman
et al. (2014)). Hence, βis updated in a Gibbs step from
(12) β|ω2,d,yN2d(ˆ
β,Vβ),
where
Vβ=σ2
βI+σ2X(ω)X(ω)1,
ˆ
β=Vβσ2X(ω)y,
(13)
and we denote with X(ω)the design matrix with rows given by xt(ω) (equation (4)), for
tI. Finally, σ2is drawn in a Gibbs step directly from
(14) σ2|β,ω,d,yInverse-GammaT+ξ0
2,τ0+tI{ytxt(ω)β}2
2.
Transdimensional moves: For these types of move, the number of periodicities is either
proposed to increase by one (birth) or decrease by one (death) (Green (1995)). If a birth
move is attempted, we have that dp=dc+1, where we denote with superscripts cand p,
the current and proposed values, respectively. The proposed vector of frequencies is obtained
by drawing an additional frequency to be included in the current vector. On the other hand,
if a death move is chosen, we have that dp=dc1 and one of the current periodicities
is randomly selected to be deleted. Conditional on the proposed vector of frequencies, the
vector of linear coefficients and the residual variance are sampled, as in the within-model
move described above. For both birth and death moves, the updates are jointly accepted or
rejected in a M–H step.
3.2. HMM parameters. We explain how to perform posterior inference about the prob-
ability distribution α, the transition probabilities πjand the state sequence z.TheChinese
restaurant franchise with loyal customers presented by Fox et al. (2011), which extends the
Chinese restaurant franchise introduced by Teh et al. (2006), is a metaphor that can be used
to express the generative process behind the sticky version of the HDP and provides a general
framework for performing inference. A high-level summary of the metaphor is as follows: in
aChinese restaurant franchise the analogy of a Chinese restaurant process (Aldous (1985))
is extended to a set of restaurants, where an infinite global menu of dishes is shared across
these restaurants. The process of seating customers at tables happens in a similar way, as
for the Chinese restaurant process, but is restaurant-specific. The process of choosing dishes
at a specific table happens franchisewide; namely, the dishes are selected with probability
proportional to the number of tables (in the entire franchise) that have previously served that
dish. However, in the Chinese restaurant franchise with loyal customers, each restaurant in
the franchise has a speciality dish which may keep many generations of customers eating in
the same restaurant.
Let yj1,...,y
jNjdenote the set of customers in restaurant j,whereNjis the number of
customers in restaurant jand each customer is preallocated to a specific restaurant designated
by that customer’s group j. Let us also define indicator random variables tji and kjt such that
tji indicates the table assignment for customer iin restaurant j,andkjt the dish assignment
for table tin restaurant j. In the Chinese restaurant franchise with loyal customers, customer
iin restaurant jchooses a table via tji ˜
πj,where ˜
πjGEM+κ) and ηand κare
IDENTIFYING THE RECURRENCE OF SLEEP APNEA 1179
as in Section 2.1. Each table is assigned a dish via kjt α+κδj)/(η +κ) so that there
is more weight on the house speciality dish, namely, the dish that has the same index as
the restaurant. Here, αfollows a DP with concentration parameters γand can be seen as
a collection of ratings for the dishes served in the global menu. Note that, in the HMM
formulated in equation (1), the value of the hidden state ztcorresponds to the dish index, that
is, kjtji =zji =zt, where we suppose there exist a bijection f:tji of time indexes tto
restaurant-customer indexes ji. Furthermore, as suggested in Fox et al. (2011), we augment
the space and introduce considered dishes ¯
kjt and override variables ojt so that we have the
following generative process:
¯
kjt|αα,
ojt|η, κ Bernoulliκ
η+κ,
kjt|¯
kjt,o
jt =¯
kjt,o
jt =0,
j, ojt =1.
Thus, a table first considers a dish ¯
kjt without taking into account the dish of the house, that
is, ¯
kjt is chosen from the infinite buffet line, according to the ratings provided by α. Then,
the dish kjt that is actually being served can be the house-speciality dish j, with probability
ρ=κ/(η +κ), or the initially considered dish ¯
kjt, with probability 1 ρ.AsshowninFox
et al. (2011), table counts ¯mjk of considered dishes are sufficient statistics for updating the
collection of dish ratings α,where ¯mjk denotes how many of the tables in restaurant jcon-
sidered dish k. The sampling of ¯mjk is additionally simplified by introducing the table counts
mjk of served dishes and override variables ojt. In the next section we describe a Gibbs
sampler which alternates between updating the hidden states z, dish ratings α, transition
probabilities πj, newly introduced random variables mjk,ojt and ¯mjk, emission parameters
θjas well as the hyperparameters γ,ηand κ.
3.2.1. Gibbs sampler. We follow Kivinen, Sudderth and Jordan (2007)andFox et al.
(2011) and consider a Gibbs sampler which uses finite approximations to the DP to al-
low sampling in blocks of the state sequence z. In particular, conditioned on observations
y, transition probabilities πjand emission parameters θj, the hidden states zare sam-
pled using a variant of the well-known HMM forward-backward procedure (see Supple-
mentary Material B) presented in Rabiner (1989). In order to use this scheme, we must
truncate the countably infinite transition distributions πj(and global menu α), and this is
achieved using the Kmax-limit approximation to a DP (Ishwaran and Zarepour (2002)), that
is, GEMKmax (γ ) := Dir(γ /Kmax ,...,γ/K
max), where the truncation level Kmax is a number
that exceeds the total number of expected HMM states and Dir(·)denotes the Dirichlet dis-
tribution. Following Fox et al. (2011), conditioned on the state sequence zand collection of
dish ratings α, we sample the auxiliary variables mjk,ojt and ¯mjk, as described in Supple-
mentary Material C.2. Dish ratings αand transition distributions πjare then updated from
the following posterior distributions:
α|¯
mDir(γ /Kmax m·1,...,γ/K
max m·Kmax ),
πj|z,αDir(ηα1+nj1,...,ηα
j+κ+njj ,...,ηα
Kmax +njKmax ),
for each state j=1,...,K
max. Here, ¯
mis the vector of table counts of considered dishes for
the whole franchise, and marginal counts are described with dots so that ¯m·k=Kmax
j=1¯mjk is
the number of tables in the whole franchise considering dish k. We denote with njk the num-
ber of Markov chain transitions from state jto state kin the hidden sequence z.Next,given
1180 HADJ-AMAR, FINKENSTÄDT, FIECAS AND HUCKSTEPP
the state sequence zand transition probabilities πj, we draw the emission parameters θjfor
each of the currently instantiated state, as described in Section 3.1, where each reversible-
jump MCMC update is run for several iterations. We also need to update the emission pa-
rameters for states which are not instantiated (namely, those states among {1,...,K
max}that
are not represented during a particular iteration of the sampler), and hence we draw the cor-
responding emission parameters from their priors. For computational or modeling reasons,
the latter may be also performed for those instantiated states that do not contain a minimum
number of observations. Finally, we sample the hyperparameters γ,ηand κin a Gibbs step
(see Supplementary Material C.3).
For the HDP-HMM, different procedures have been applied for sampling the hidden state
sequence z.Teh et al. (2006) originally introduced an approach based on a Gibbs sam-
pler which has been shown to suffer from slow mixing behavior due to strong correlations,
that is, frequently observed in the data at nearby time points. Van Gael et al. (2008)pre-
sented a beam sampling algorithm that combines a slice sampler (Neal (2003)) with dynamic
programming. This allows to constrain the number of reachable states at each MCMC it-
eration to a finite number, where the entire hidden sequence zis drawn in block using a
form of forward-backward filtering scheme. However, Fox et al. (2011) showed that appli-
cations of the beam sampler to the HDP-HMM resulted in slower mixing rates, compared to
the forward-backward procedure that we use in our truncated model. Recently, Tripuraneni
et al. (2015) developed a particle Gibbs MCMC algorithm (Andrieu, Doucet and Holenstein
(2010)) which uses an efficient proposal and makes use of ancestor sampling to enhance the
mixing rate.
3.2.2. Label switching. The proposed approach may suffer from label switching (see,
e.g., Redner and Walker (1984), Stephens (2000), Jasra, Holmes and Stephens (2005)) since
the likelihood is invariant under permutations of labelling of the mixture components for both
hidden state labels {1,...,K
max}and frequency labels {1,...,d
max}in each state. The label
switching problem occurs when using Bayesian mixture models and needs to be addressed
in order to draw meaningful inference about the posterior model parameters. In our multi-
ple model search the frequencies (and their corresponding linear coefficients) are identified
by keeping them in ascending order for every iteration of the sampler. Posterior samples of
the model parameters, corresponding to different hidden states, are postprocessed (after the
full estimation run) using the relabelling algorithm developed by Stephens (2000). The ba-
sic idea behind this algorithm is to find permutations of the MCMC samples in such a way
that the Kullback–Leibler (KL) divergence (Kullback and Leibler (1951)) between the “true”
distribution on clusterings, say P(θ), and a matrix of classification probabilities, say Q,
is minimized. The KL distance is given by d(Q,P(θ))KL =tjptj (θ)log ptj (θ)
qtj ,where
ptj (θ)=p(zt=j|zt1,y,π,θ)is part of the MCMC output, obtained as in Supplementary
Material C.1, and qtj is the probability that observation tis assigned to class j. The algo-
rithm iterates between estimating Qand the most likely permutation of the hidden labels for
each MCMC iteration. We chose the strategy of Stephens (2000) since it has been shown to
perform very efficiently in terms of finding the correct relabelling (see, e.g., Rodríguez and
Walker (2014)). However, it may be computationally quite intensive in memory since it re-
quires the storage of a matrix of probabilities of dimension N×T×Kmax,whereNis the
number of MCMC samples. Furthermore, at each iterative step the algorithm requires to go
over Kmax!permutations of the labels for each MCMC iteration which might significantly
slow down the computation when using large values of Kmax . Related approaches to the
label switching issue include pivotal reordering algorithms (Marin, Mengersen and Robert
(2005)), label invariant loss functions (Celeux, Hurn and Robert (2000), Hurn, Justel and
Robert (2003)) and equivalent classes representative methods (Papastamoulis and Iliopoulos
(2010)), where an overview of these strategies can be found in Rodríguez and Walker (2014).
IDENTIFYING THE RECURRENCE OF SLEEP APNEA 1181
FIG.2. Illustrative Example.Dots represent the simulated time series,where the different colors corresponds to
(true)different regimes.The state-specific estimated oscillatory mean function is displayed as a solid curve,and
the estimated state sequence as a piecewise horizontal line at the top part of the graph.
4. Simulation studies. This section presents results of simulation studies to explore the
performance of our proposed methodology in two different settings. In the first scenario the
data are generated from the model described in Section 2, and, thus, this simulation study
provides a “sanity” check that the algorithm is indeed retrieving the correct prefixed param-
eters. We also investigate signal extraction for the case that the innovations come from a
heavy-tailed t-distribution instead of a Gaussian. Our second study deals with artificial data
from an HMM whose emission distributions are characterized by oscillatory dynamics gen-
erated by state-specific autoregressive (AR) time series models. Julia code that implements
our procedure is available at https://github.com/Beniamino92/HHMM.
4.1. Illustrative example. We generated a time series consisting of T=1450 data points
from a three-state HMM with the following transition probability matrix showing high proba-
bilities of selftransition along the diagonal and Gaussian oscillatory emissions, as specified in
equation (2), where the parameters of each of the three regimes and the transition probability
matrix are given in Supplementary Material A. A realization from this model is displayed in
Figure 2. The prior mean on the number of frequencies djis set equal to 1, and we place a
Gamma (1,0.01)prior on the concentration parameters γand +κ) and a Beta (100,1)
prior on the selftransition proportion ρ,asinFox et al. (2011). The maximum number of
periodicities per regime dmax is set to 5, while the truncation level Kmax for the DP approx-
imation is set equal to 7. Also, we set φω=0.25 as a threshold for the uniform prior. The
proposed estimation algorithm is run for 15,000 iterations, 3000 of which were discarded as
burn-in. At each iteration for each instantiated set of emission parameters, two reversible-
jump MCMC updates were performed. The full estimation algorithm took 31 minutes with a
program written in Julia 0.62 on an Intel®CoreTM i7 2.2 GHz Processor 8 GB RAM. For our
experiments we used the R package label.switching of Papastamoulis (2016) to postprocess
the MCMC output with the relabelling algorithm of Stephens (2000).
Table 1(left panel) shows that our estimation algorithm successfully detects the correct
number of states in the sense that a model with k=3 regimes has the highest posterior prob-
ability. In addition, our approach correctly identifies the right number of frequencies in each
regime, as shown in Table 1(right panel). Table 2displays the estimated posterior mean
and standard deviation of the frequencies along with the square root of the power of the
corresponding frequencies, where the results are conditional on three estimated states and
the modal number of frequencies within each state. Here, the power of each frequency ωjl
1182 HADJ-AMAR, FINKENSTÄDT, FIECAS AND HUCKSTEPP
TABLE 1
Illustrative example.(left panel)posterior probabilities for number of distinct states k;(right panel)posterior
probabilities for number of frequencies in each state,conditioned on k=3
kˆπ(k|y)mˆπ(d1|k=3,y)ˆπ(d2|k=3,y)ˆπ(d3|k=3,y)
1 0.00 1 0.99 1.00 0.01
2 0.00 2 0.01 0.00 0.99
3 0.99 3 0.00 0.00 0.00
4 0.01 4 0.00 0.00 0.00
5 0.00 5 0.00 0.00 0.00
60.00
70.00
is summarized by the amplitude Ajl =β(1)2
jl +β(2)2
jl , namely, the square root of the sum
of squares of the corresponding linear coefficients (see, e.g., Shumway and Stoffer (2017)).
Our proposed method seems to provide a good match between true and estimated values
for both frequencies and their power for this example. We also show in Figure 2the state-
specific estimated signal (equation (3)), and the estimated state sequence using the method
of Stephens (2000) (as a piecewise horizontal line). The rows of the estimated transition
probability matrix were ˆ
π1=(0.9921,0.0073,0.0006), ˆ
π2=(0.0005,0.9956,0.0040)and
ˆ
π3=(0.0051,0.0006,0.9942). The high probabilities along the diagonal reflect the esti-
mated posterior mean of the selftransition parameter ˆρ=0.9860, which is indeed centered
around the true probability of selftransition.
Diagnostics for verifying convergence were performed in several ways. For example, we
observed that the MCMC samples of the likelihood of the HMM reached a stable regime
while initializing the Markov chains from overdispersed starting values (see Figure 3(b)).
This diagnostic might be very useful, for example, in determining the burn-in period. How-
ever, we note that it does not guarantee convergence since steady values of the log likelihood
might be the result of a Markov chain being stuck in some local mode of the target posterior
distribution. The likelihood of an HMM with Gaussian emissions can be expressed as
(15) L(z,π,θ|y)=p(z1|y,π,θ)Ny1;f1z12
z1
T
t=2
p(zt|zt1,y,π,θ)Nyt;ftzt2
zt,
where N(yt;fjt2
j)denotes the density of a Gaussian distribution with mean fjt =
xt(ωj)βj(as in equation (3)) and variance σ2
j, evaluated at yt. Conditioned on the modal
TABLE 2
Illustrative example.Estimated posterior mean (and standard deviation)of frequencies and square root of the
power of the corresponding frequencies
ω11 ω21 ω31 ω32
True 0.0400 0.0526 0.0833 0.1250
Estimated 0.0399 0.0526 0.0833 0.1249
(8.8 ·106)(6.3·106)(9.6·106)(9.4·106)
A11 A21 A31 A32
True 1.131 0.283 1.414 1.414
Estimated 1.069 0.281 1.380 1.367
(0.029)(0.004)(0.022)(0.022)
IDENTIFYING THE RECURRENCE OF SLEEP APNEA 1183
FIG.3. Illustrative example.(a)Trace plots (after burn-in of 3000 updates)for posterior sample of frequencies,
conditional on modal number of states and number of frequencies in each state;red lines correspond to true
values of the frequencies.(b)Trace plots (including burn-in)of the likelihood for three Markov chains initialized
at different starting values (where the initial 100 updates are omitted from the graph).
number of states, we also validated convergence for the state-specific emission parameters by
analyzing trace plots and running averages of the corresponding MCMC samples, with ac-
ceptable results as each trace reached a stable regime. As an example, we show in Figure 3(a)
trace plots (after burn-in) for the posterior values of the frequencies.
Finally, while we notice that we have not set Kmax to a very large value, this choice has
been made a posteriori after we found that the estimation algorithm assigned negligible prob-
abilities to a large number of components. For example, in this simulation study we initially
set Kmax =20 and ran the full estimation algorithm; after observing that the posterior proba-
bilities for the number of distinct states were equal to zero for all the models with more than
four hidden states, we reran the estimation algorithm with a smaller value for the HDP trun-
cation, that is, Kmax =7, obtaining the same correct results. This yielded some benefits from
a computational perspective, in particular, in terms of facilitating storage and memory access
of the posterior sample and also speeding up the relabelling algorithm developed by Stephens
(2000). Furthermore, we notice that, even in the case where the maximum complexity of the
model is set to Kmax =7, we are still dealing with a framework that assumes a relatively high
number of regimes (within a fairly complicated setting of oscillations in each state). Conven-
tional models might not even be able to achieve satisfactory estimation performances when
specifying such a large number of Markov modes.
Signal extraction with non-Gaussian innovations: In many scientific experiments it may
be of interest to extract the underlying signal that generates the observed time series and
HMMs can be used to this end. Here, we study the performance of our proposed approach
in estimating the time-varying oscillatory signal fjt (equation (3)) when the Gaussian as-
sumption of εtin equation (2) is violated. In particular, we generated 20 time series, each
consisting of 1024 observations from the same simulation setting introduced above, where
the innovations were simulated from heavy-tailed t-distributions with 2, 3, 2 degrees of
freedom for states 1, 2, 3, respectively. The linear basis coefficients were chosen to be
β11 =(3,2),β21 =(1.2,4.0),β31 =(1.0,5.0),β32 =(4.0,3.0). As a measure of perfor-
mance, we computed the (in-sample) mean squared error MSE =1
1024 1024
t=1(ftztˆ
ftzt)2
between true and estimated signal and compared the proposed approach with the method
of Hadj-Amar et al. (2020), referred to as AutoNOM (Automatic Nonstationary Oscillatory
Modelling), which we believe is the state-of-the-art in extracting the signal of nonstationary
periodic processes. Our proposed estimation algorithm was run with the same parameteri-
zation as above, while AutoNOM was performed for 15,000 updates, 3000 of which were
1184 HADJ-AMAR, FINKENSTÄDT, FIECAS AND HUCKSTEPP
FIG.4. Signal extraction with non-Gaussian innovations.Boxplots of the MSE values for AutoNOM and our
proposed HHMM when (a)the data exhibit recurrent patterns,(b)the data do not exhibit recurrent patterns and
(c)the innovations are skewed.
discarded as burn-in, where we fixed 15 maximum number of change points and five maxi-
mum number of frequencies per segment. The prior means for the number of change points
and frequencies per segment are fixed at two and one, respectively, and the minimum distance
between change-points is set at 10. For both methodologies the estimated signal was obtained
by averaging across MCMC iterations. AutoNOM was run using the Julia software provided
by the authors at https://github.com/Beniamino92/AutoNOM.
Figure 4(a) presents boxplots of the MSE values for AutoNOM and our proposed ap-
proach which will be referred to as HHMM (Harmonic Hidden Markov Model). It becomes
clear that the estimates of the signal obtained using HHMM are superior to those obtained
using AutoNOM. However, this result is not surprising, as the two approaches make differ-
ent assumptions. In particular, AutoNOM does not assume recurrence of a periodic behavior
and hence needs to estimate the regime-specific modeling parameters each time it detects
a new segment, while our HHMM has the advantage of using the same set of parameters
whenever a particular periodic pattern recurs in the time series. Hence, we also compared the
performance of the two approaches in extracting the signal (under non-Gaussian innovations)
in a scenario where the time series do not exhibit recurrence. Specifically, we generated 15
time series manifesting two change points (where the oscillatory behavior corresponding to
the three different partitions are parameterized as above) and computed the MSE between
true and estimated signal, as we did in the previous scenario. The corresponding boxplots
displayed in Figure 4(b) show that the two approaches seem to perform in similar way with
AutoNOM being slightly more accurate than our HHMM. Moreover, we have further exam-
ined a scenario where the noise is generated from exponentially distributed random variables
in order to introduce a large skew. Here, we simulated 15 time series from a three-state HMM,
as above, where now the innovations corresponding to the three different states are generated
from exponentially distributed random variables with rates 0.5, 1, and 0.2, respectively. The
draws from the exponential distribution are centered in such a way that they have mean zero
to avoid noise that takes on strictly positive values. Figure 4(c) presents boxplots of the MSE
values for AutoNOM and our proposed HHMM, showing that our methodology seems to be
superior to AutoNOM, in terms of extracting the signal when the innovations are skewed.
We conclude that both approaches have their own strengths. Our proposed procedure is supe-
rior to AutoNOM in the sense that the additional HMM provides a framework for modeling
and explicitly quantifying the switching dynamics and connectivity between different states.
On the other hand, AutoNOM is better suited to scenarios where there are nonstationarities
IDENTIFYING THE RECURRENCE OF SLEEP APNEA 1185
arising from singular change points, and the observed oscillatory processes evolve without
recurrent patterns.
4.2. Markov switching autoregressive process. We now investigate the performance of
our approach in detecting time-changing periodicities in a scenario where the data generat-
ing process shows large departures from our modeling assumptions. The HMM hypothesis,
which assumes conditionally independent observations given the hidden state sequence, such
as the one formulated in equation (2), may sometimes be inadequate in expressing the tempo-
ral dependencies occurring in some phenomena. A different class of HMMs that relaxes this
assumption is given by the Markov switching autoregressive process,alsoreferredtoasthe
AR-HMM (Juang and Rabiner (1985), Albert and Chib (1993), Frühwirth-Schnatter (2006)),
where an AR process is associated with each state. This model is used to design autoregres-
sive dynamics for the emission distributions while allowing the state transition mechanisms
to follow a discrete state Markov chain.
We generated T=900 observations from an AR-HMM with two hidden states and autore-
gressive order fixed at p=2, that is,
ztπzt1,
yt=
p
l=1
ψ(zt)
lytl+ε(zt),
(16)
where π1=(0.99,0.01)and π2=(0.01,0.99). The AR parameterization ψ(1)=(1.91,
0.991)and ψ(2)=(1.71,0.995)is chosen in such a way that the state-specific spec-
tral density functions display a pronounced peakedness. Furthermore, ε(1)
t
i.i.d.
N(0,0.12)
and ε(2)
t
i.i.d.
N(0,0.052). A realization from this model is shown in Figure 5(top) as a blue
solid line. Our proposed estimation algorithm was run for 15,000 iterations 5000 of which
are used as burn-in. At each iteration, we performed two reversible-jump MCMC updates for
each instantiated set of emission parameters. The rate of the Poisson prior for the number of
periodicities is fixed at 101, and the corresponding truncation level dmax was fixed to 3. The
maximum number of states Kmax was set equal to 10, whereas the rest of the hyperparameters
are specified as in Section 4.1. Our procedure seems to overestimate the number of states, as a
model with eight regimes had the highest posterior probability ˆπ(k =8|y)=97%. However,
this is not entirely unexpected, as a visual inspection of the realization displayed in Figure 5
(top) suggest more than two distinct spectral patterns in the sense that the phases, amplitudes
and frequencies appear to vary stochastically within a regime. Figure 5(bottom) shows the
estimated time-varying frequency peak along with a 95% credible interval obtained from the
posterior sample. The estimate was determined by first selecting the dominant frequency (i.e.,
the frequency with the highest posterior power) corresponding to each observation and then
averaging the frequency estimates over MCMC iterations. While our approach identifies a
larger number of states when the data were generated from an AR-HMM, we note that the
data generating process are very different from the assumptions of our model, and the pro-
posed procedure still provides a reasonable summary of the underlying time changing spec-
tral properties observed in the data. Furthermore, by setting the truncation level Kmax equal
to 2, we retrieve the true transition probability matrix that generates the switching dynamics
between the two different autoregressive patterns, as the vectors of transition probabilities
obtained using our estimation algorithm are ˆ
π1=(0.99,0.01)and ˆ
π2=(0.98,0.02).
In addition, we simulated 10 time series from model (16) and computed the mean squared
error MSE =1
900 900
t=1t−ˆωt)between the true time-varying frequency peak ωtand its esti-
mate ˆωtfor the proposed approach, AutoNOM and the procedure of Rosen, Wood and Stoffer
1186 HADJ-AMAR, FINKENSTÄDT, FIECAS AND HUCKSTEPP
FIG.5. (To p)A realization from model (16), where the piecewise horizontal line represents the true state se-
quence.(Bottom)True time varying frequency peak (dotted red line)and the estimate provided by our proposed
approach (solid blue line)where we highlight a 95% credible interval obtained from the posterior sample.(Right)
Boxplots of the MSE values for AutoNOM,our proposed HHMM and AdaptSPEC.
(2012), referred to as AdaptSPEC (Adaptive Spectral Estimation). For both AutoNOM and
AdaptSPEC, we ran the algorithm for 15,000 MCMC iterations (5000 of which were used as
burn-in), fixed the maximum number of change-points at 15 and set the minimum distance be-
tween change-points to 30. The number of spline basis functions for AdaptSPEC is set to 10.
AutoNOM is performed using a Poisson prior with rate 101for both number of frequencies
and number of change points. AdaptSPEC was performed using the R package BayesSpec
provided by the authors. Boxplots of the MSE values for the three different methodologies
are displayed in Figure 5(right), showing that our proposed HHMM seems to outperform
the other two approaches in detecting the time-varying frequency peak, for this example.
However, our procedure finds some very short sequences (such as in Figure 5(bottom) for
t200,500,700), demonstrating that the sticky parameter might not always be adequate
enough in capturing the correct temporal mode persistence of the latent state sequence. Au-
toNOM and AdaptSPEC are less prone to this problem, as both methodologies are able to
specify a minimum time distance between change-points; though, we acknowledge that this
constraint might not be optimal when the observed data exhibit relatively rapid changes. We
also notice that, not surprisingly, the estimates of the time-varying frequency peak obtained
using AutoNOM and our HHMM, which are based on a line-spectrum model, are both su-
perior than the ones obtained via the smoothing spline basis of AdaptSPEC which is built
upon a continuous-spectrum setting; this is consistent with the findings in Hadj-Amar et al.
(2020). However, it is important to keep in mind that, while AutoNOM and AdaptSPEC allow
to retrospectively analyse the spectral changing properties of a process from an exploratory
angle, unlike our proposed HHMM, they do not quantify a probabilistic mechanism for the
recurrence of periodic dynamic patterns.
5. Analysis of the airflow trace data. TheairowtraceshowninFigure1was collected
from a human over a time span of 5.5 minutes of continuous breathing and measured via a
facemask attached to a pressure transducer. Airflow pressure signals were amplified using
IDENTIFYING THE RECURRENCE OF SLEEP APNEA 1187
the NeuroLog system connected to a 1401 interface and acquired on a computer using Spike2
software (Cambrdige Electronic Design). The data are sampled at rate of four Hertz, that is,
four observations per second, for a total of 1314 data points. The airflow data was captured
and amplified only, and no preprocessing was carried out afterward. We notice that the signal
is clean since the pressure transducer is sensitive enough to pick up breathing, even from a
mouse in a plethysmograph, and it was attached to a medical face-mask directly in front of
the mouth. Therefore, the signal to noise ratio is extraordinarily high.
We fitted our HHMM to the time series displayed in Figure 1for 200,000 iterations,
125,000 of which were discarded as burn-in, where, at each iteration, we carried out 10
reversible-jump MCMC updates for each instantiated set of emission parameters. The trun-
cation level Kmax was set to 10, whereas the maximum number of frequencies per state dmax
was fixed to 3. With respect to the harmonic regression part of the model, we specified the
prior for the innovations σ2
jas Inverse-Gamma (3.11,3.17)so that it is centered at the empir-
ical variance 1.5 and has a standard deviation of 2.5. The Poisson prior for the number of
frequencies djis chosen equal to 102to favour models with a small number of components,
and the prior on the frequencies ωj,l is set informative as Uniform(0, φω),whereφω=0.3
was selected by looking at the raw periodogram of the data and noting that the power of
the frequencies was approximately zero for any values larger than 0.3. Finally, we specified
weakly informative prior on the linear basis coefficients βjas N2dj(02
βI), where the prior
variance σ2
β=4 is now chosen in such a way that the mass of the prior is concentrated in
reasonable regions based on the data. Regarding the part of the model relative to the HDP,
we specified for both concentration parameters η+κand γa weakly informative hyperprior
Gamma (1,0.01)so that the corresponding priors for the base measures favour a DP model
with a small number of components (see stick breaking construction, equations (6)and(7)).
The prior on the selftransition proportions ρis specified informative as Beta (103,1)so that
we force high probability of selftransitions.
The posterior distribution over the number of states had a mode at 6, with posterior proba-
bilities ˆπ(k =6|y)=94%, ˆπ(k =7|y)=5% and ˆπ(k =8|y)=1%. Indeed, it is conceivable
that the state corresponding to normal breathing (i.e., neither apnea or hypopnea) may exhibit
more than one distinct periodic pattern, which further justifies the need to use a nonparamet-
ric HMM. Paz and West (2013) reported at least 13 forms of breathing patterns including
forms of apnea. Figure 6shows the estimated hidden state sequence (piecewise horizontal
line), where we highlight our model estimate for apnea state (red) and hypopnea state (blue)
while reporting the ground truth at the top of the plot. We have also included a posterior pre-
dictive graphical check consisting of the airflow trace alongside 20 draws from the estimated
posterior predictive (Gelman et al. (2014)), where the latter is obtained by first drawing a
sample path and then, conditioned on the hidden state sequence, the predicted values are sim-
ulated from the appropriate emission distributions. Our model seems to be able to capture
the underlying signal that characterizes this time series. Conditional on the modal number of
regimes, the number of periodicities belonging to apnea and hypopnea had a posterior mode
at 3 and 2, respectively. Conditional on the modal number of frequencies, Table 3displays
the posterior mean and standard deviation of periodicities (in seconds) and powers that char-
acterize the two states classified as apnea and hypopnea, showing that apnea instances seem
to be characterized by larger periods and lower amplitude than hypopnea.
Our estimation algorithm detected all known apnea and hypopnea instances. In order for
them to qualify as a clinically relevant obstructive event they must have a minimum length
of 10 seconds (Berry et al. (2017)). Thus, we only highlight the clinically relevant instances
in Figure 6, discarding sequences of duration less than 10 seconds. We also detected a post-
sigh apnea (after the third minute) which is a normal phenomenon to observe in a breathing
trace and hence should not count as a disordered breathing event. Again, such an event after
1188 HADJ-AMAR, FINKENSTÄDT, FIECAS AND HUCKSTEPP
FIG.6. Case study.Dots represent the airflow trace collected over a period of five and half minutes of continuous
breathing.The grey lines represent draws from the estimated posterior predictive.The piecewise horizontal line
corresponds to the estimated state sequence where we highlight the states corresponding to estimated apnea (red)
and hypopnea (blue)while reporting the ground truth at the top of the plot.
a sigh can be identified as a sigh is characterized by an amplitude which is always higher
than any other respiratory event and hence can be easily detected. Subtracting the number
of sighs from the total number of apneas/hypopneas results in a measure of all apneas of
interest without the confounding data from post-sigh apneas. A common score to indicate the
severity of sleep apnea is given by the Apnea-Hypopnea Index (AHI) which consists of the
number of apneas and hypopneas per hour of sleep (Ruehland et al. (2009)). Our proposed
approach provides a realistic estimate of the total number of apnea and hypopnea instances
recurring in this case study. While an essential aim of this paper is to detect apnea instances
retrospectively, which is currently a time consuming and demanding task as it is performed
by eye, we have also investigated the out-of-sample predictive performance of our proposed
approach in Supplementary Material E.
6. Summary and discussion. In this paper we developed a novel HMM approach that
can address the challenges of modeling periodic phenomena whose behavior switches and
recurs dynamically over time. The number of states is assumed unknown as well as their
TABLE 3
Case study.Posterior mean and standard deviation of frequencies and corresponding powers that characterize
the two states classified as apnea and hypopnea
Apnea Hypopnea
Freq Power Freq Power
0.0159 0.1376 0.0455 0.261
(6.66·105) (1.81·102) (1.29·104) (3.37·102)
0.0353 0.2153 0.0542 0.620
(3.73·105) (1.87·102) (4.72·105)(3.4·102)
0.0379 0.2065
(3.89·105) (1.97·102)
IDENTIFYING THE RECURRENCE OF SLEEP APNEA 1189
relevant periodicities which may differ over the different regimes since each regime is rep-
resented by a different periodic pattern. To address flexibility in the number of states, we
assumed a sticky HDP-HMM that penalises rapid changing dynamics of the process and pro-
vides effective control over the switching rate. The variable dimensionality with respect to
the number of frequencies that specifies the different states is tackled by a reversible-jump
MCMC algorithm.
While being noticeably flexible, the model proposed in this article is developed in a frame-
work that is still computationally accessible. Naturally, an alternative strategy would involve
fitting several finite-state HMMs and then performing in a second stage model selection by
means of the marginal likelihood (Kass and Raftery (1995)). Nevertheless, reliably approx-
imating this quantity from the posterior sample is not straightforward, though several tech-
niques have been proposed in the literature to overcome this burden, such as sequential Monte
Carlo (SMC, Jasra et al. (2008)), population MCMC (PMCMC, Liang and Wong (2001),
Jasra, Stephens and Holmes (2007)) or bridge sampling (Meng and Wong (1996), Meng and
Schilling (2002)), where we refer the reader to Zhou, Johansen and Aston (2016)foranex-
cellent summary of these developments. However, these methods involve algorithms that are
often computationally challenging and difficult to implement efficiently, especially within
our modeling framework.
We illustrated the use of our approach in a case study relevant to respiratory research,
where our methodology was able to identify recurring instances of sleep apnea in human
breathing traces. Despite the fact that here we have focused on the detection of apnea in-
stances, our proposed methodology provides a very flexible and general framework to analyze
different breathing patterns. A question of interest is whether similar dynamical patterns can
be identified across a heterogeneous patient cohort and be used for the prognosis of patients’
health and progress. The growth of information and communication technologies permits new
advancements in the health care system to facilitate support in the homes of patients in order
to proactively enhance their health and well being. We believe that our proposed HMM ap-
proach has the potential to aid the iterative feedback between clinical investigations in sleep
apnea research and practice with computational, statistical and mathematical analysis.
As pointed out by a referee, apnea states have certain features, such as low amplitude and
low frequency behaviour, that may suggest that assuming symmetry among the parameters in
their prior distribution might not be an ideal modeling approach. However, as we discussed in
this article, it is plausible that normal breathing is exhibiting more than one distinct periodic
pattern. While it would be of interest to integrate prior knowledge into the model to fully
remove permutations of labelling of the HDP-HMM mixture components, we believe it would
not be trivial to fully characterize in an identifiable way the different states corresponding to
normal breathing. These are early days for such data to be analyzed in this way, and we are
at the beginning of being able to construct a catalogue of more informative priors that might
help this type of analysis in future.
Although both parametric and nonparametric HMMs have been shown to be good mod-
els in addressing learning challenges in time-series data, they have the drawback of limiting
the state duration distribution, that is, the distribution for the number of consecutive time
points that the Markov chain spends in a given state, to a geometric form (Ephraim and Mer-
hav (2002)). In addition, the selftransition bias of the sticky HDP-HMM used to increase
temporal state persistence is shared among all states and thus does not allow for inferring
state-specific duration features. In our application, learning the duration structure of a spe-
cific state may be of interest to health care providers, for example, in assessing the severity
of sleep apnea. Future work will address extending our approach to a hidden semi-Markov
model (HSMM) setting (Guédon (2003), Yu (2010), Johnson and Willsky (2013)), where the
generative process of an HMM is augmented by introducing a random state duration time
1190 HADJ-AMAR, FINKENSTÄDT, FIECAS AND HUCKSTEPP
which is drawn from some state-specific distribution when the state is entered. However, this
increased flexibility in modeling the state duration has the cost of increasing substantially the
computational effort to compute the likelihood: the message-passing procedure for HSMMs
requires O(T 2K+TK2)basic computations for a time series of length Tand number of
states K, whereas the corresponding forward-backward algorithm for HMMs requires only
O(T K2).
Acknowledgments. We wish to thank Maxwell Renna, Paul Jenkins and Jim Griffin for
their insightful and valuable comments. The work presented in this article was developed
as part of the first author’s Ph.D. thesis at the University of Warwick, and he is currently
affiliated with the Department of Statistics at Rice University.
Funding. B. Hadj-Amar was supported by the Oxford-Warwick Statistics Programme
(OxWaSP) and the Engineering and Physical Sciences Research Council (EPSRC), Grant
Number EP/L016710/1. R. Huckstepp was supported by the Medical Research Council
(MRC), Grant Number MC/PC/15070.
SUPPLEMENTARY MATERIAL
Supplement to “Identifying the recurrence of sleep apnea using a harmonic hidden
Markov model” (DOI: 10.1214/21-AOAS1455SUPP; .pdf). We provide supplemental ma-
terial to the manuscript. Section A contains additional details about the sampling scheme for
updating the emission parameters via reversible-jump MCMC steps, and in Section B we
present the sampling scheme for drawing the HMM parameters within the Chinese restaurant
franchise framework. Section C gives the parameterization of the simulation setting presented
in Section 4.1. Section D provides further diagnostics about our MCMC sampler and in Sec-
tion E we investigate the out-of-sample predictive performance of the proposed approach in
the air flow case study. Julia code that implements our proposed approach is also available at
https://github.com/Beniamino92/HHMM.
REFERENCES
ADAK, S. (1998). Time-dependent spectral analysis of nonstationary time series. J.Amer.Statist.Assoc.93 1488–
1501. MR1666643 https://doi.org/10.2307/2670062
ALBERT,J.H.andCHIB, S. (1993). Bayes inference via Gibbs sampling of autoregressive time series subject to
Markov mean and variance shifts. J.Bus.Econom.Statist.11 1–15.
ALDOUS, D. J. (1985). Exchangeability and related topics. In École D’été de Probabilités de Saint-Flour,XIII—
1983. Lecture Notes in Math.1117 1–198. Springer, Berlin. MR0883646 https://doi.org/10.1007/BFb0099421
ANCOLI-ISRAEL,S.,KLAUBER,M.R.,BUTTERS,N.,PARKER,L.andKRIPKE, D. F. (1991). Dementia in
institutionalized elderly: Relation to sleep apnea. J.Amer.Geriatr.Soc.39 258–263.
ANDRIEU,C.andDOUCET, A. (1999). Joint Bayesian model selection and estimation of noisy sinusoids via
reversible jump MCMC. IEEE Trans.Signal Process.47 2667–2676.
ANDRIEU,C.,DOUCET,A.andHOLENSTEIN, R. (2010). Particle Markov chain Monte Carlo methods. J.R.
Stat.Soc.Ser.B.Stat.Methodol.72 269–342. MR2758115 https://doi.org/10.1111/j.1467-9868.2009.00736.x
BAUM,L.E.andEAGON, J. A. (1967). An inequality with applications to statistical estimation for probabilistic
functions of Markov processes and to a model for ecology. Bull.Amer.Math.Soc.73 360–363. MR0210217
https://doi.org/10.1090/S0002-9904-1967-11751-8
BAUM,L.E.andPETRIE, T. (1966). Statistical inference for probabilistic functions of finite state Markov chains.
Ann.Math.Stat.37 1554–1563. MR0202264 https://doi.org/10.1214/aoms/1177699147
BEAL,M.J.,GHAHRAMANI,Z.andRASMUSSEN, C. E. (2002). The infinite hidden Markov model. In Advances
in Neural Information Processing Systems 577–584.
BERNARDO,J.-M.andSMITH, A. F. M. (2009). Bayesian Theory. Wiley, Chichester. MR1274699
https://doi.org/10.1002/9780470316870
BERRY,R.B.,BROOKS,R.,GAMALDO,C.,HARDING,S.M.,LLOYD,R.M.,QUAN,S.F.,TROESTER,M.T.
and VAUGHN, B. V. (2017). AASM scoring manual updates for 2017 (version 2.4). J.Clin.Sleep.Med.13
665–666. https://doi.org/10.5664/jcsm.6576
IDENTIFYING THE RECURRENCE OF SLEEP APNEA 1191
BISHOP, C. M. (2006). Pattern Recognition and Machine Learning.Information Science and Statistics. Springer,
New York. MR2247587 https://doi.org/10.1007/978-0-387-45528-0
BRUCE,S.A.,HALL,M.H.,BUYSSE,D.J.andKRAFTY, R. T. (2018). Conditional adaptive Bayesian spectral
analysis of nonstationary biomedical time series. Biometrics 74 260–269. MR3777946 https://doi.org/10.1111/
biom.12719
CAPPÉ,O.,MOULINES,E.andRYDÉN, T. (2005). Inference in Hidden Markov Models.Springer Series in
Statistics. Springer, New York. MR2159833
CELEUX,G.,HURN,M.andROBERT, C. P. (2000). Computational and inferential difficulties with mixture
posterior distributions. J.Amer.Statist.Assoc.95 957–970. MR1804450 https://doi.org/10.2307/2669477
COHEN, M. X. (2014). Analyzing Neural Time Series Data:Theory and Practice. MIT Press, Cambridge.
COOKE,J.R.,AYA LO N ,L.,PALMER,B.W.,LOREDO,J.S.,COREY-BLOOM,J.,NATARAJAN,L.,LIU,L.
and ANCOLI-ISRAEL, S. (2009). Sustained use of CPAP slows deterioration of cognition, sleep, and mood
in patients with Alzheimer’s disease and obstructive sleep apnea: A preliminary study. J.Clin.Sleep.Med.5
305–309.
DAHLHAUS, R. (1997). Fitting time series models to nonstationary processes. Ann.Statist.25 1–37. MR1429916
https://doi.org/10.1214/aos/1034276620
DAVIS ,R.A.,LEE,T.C.M.andRODRIGUEZ-YAM, G. A. (2006). Structural break estimation for non-
stationary time series models. J.Amer.Statist.Assoc.101 223–239. MR2268041 https://doi.org/10.1198/
016214505000000745
DEWAN,N.A.,NIETO,F.J.andSOMERS, V. K. (2015). Intermittent hypoxemia and OSA: Implications for
comorbidities. Chest 147 266–274. https://doi.org/10.1378/chest.14-0500
EPHRAIM,Y.andMERHAV, N. (2002). Hidden Markov processes. IEEE Trans.Inf.Theory 48 1518–1569.
MR1909472 https://doi.org/10.1109/TIT.2002.1003838
FOX,E.B.,SUDDERTH,E.B.,JORDAN,M.I.andWILLSKY, A. S. (2011). A sticky HDP-HMM with applica-
tion to speaker diarization. Ann.Appl.Stat.51020–1056. MR2840185 https://doi.org/10.1214/10-AOAS395
FRÜHWIRTH-SCHNATTER, S. (2006). Finite Mixture and Markov Switching Models.Springer Series in Statistics.
Springer, New York. MR2265601
GELMAN,A.,CARLIN,J.B.,STERN,H.S.,DUNSON,D.B.,VEHTARI,A.andRUBIN, D . B. (2014). Bayesian
Data Analysis,3rded.Texts in Statistical Science Series. CRC Press, Boca Raton, FL. MR3235677
GREEN, P. J. (1995). Reversible jump Markov chain Monte Carlo computation and Bayesian model determina-
tion. Biometrika 82 711–732. MR1380810 https://doi.org/10.1093/biomet/82.4.711
GUÉDON, Y. (2003). Estimating hidden semi-Markov chains from discrete sequences. J.Comput.Graph.Statist.
12 604–639. MR2002638 https://doi.org/10.1198/1061860032030
HADJ-AMAR,B.,RAND,B.F.,FIECAS,M.,LÉVI,F.andHUCKSTEPP, R. (2020). Bayesian model search
for nonstationary periodic time series. J.Amer.Statist.Assoc.115 1320–1335. MR4143468 https://doi.org/10.
1080/01621459.2019.1623043
HADJ-AMAR,B.,FINKENSTÄDT,B.,FIECAS,M.andHUCKSTEPP, R. (2021). Supplement to “Identi-
fying the recurrence of sleep apnea using a harmonic hidden Markov model.” https://doi.org/10.1214/
21-AOAS1455SUPP
HEINZER,R.,VAT,S.,MARQUES-VIDAL,P.,MARTI-SOLER,H.,ANDRIES,D.,TOBBACK,N.,MOOSER,V.,
PREISIG,M.,MALHOTRA, A. et al. (2015). Prevalence of sleep-disordered breathing in the general popula-
tion: The HypnoLaus study. The Lancet Respiratory Medicine 3310–318.
HUANG,Q.,COHEN,D.,KOMARZYNSKI,S.,LI,X.-M.,INNOMINATO,P.,LÉVI,F.andFINKENSTÄDT,B.
(2018). Hidden Markov models for monitoring circadian rhythmicity in telemetric activity data. J.R.Soc.
Interface 15 20170885.
HURN,M.,JUSTEL,A.andROBERT, C. P. (2003). Estimating mixtures of regressions. J.Comput.Graph.Statist.
12 55–79. MR1977206 https://doi.org/10.1198/1061860031329
IGNATOV, T. (1982). A constant arising in the asymptotic theory of symmetric groups, and Poisson–Dirichlet
measures. Theory Probab.Appl.27 136–147.
ISHWARAN,H.andZAREPOUR, M. (2002). Exact and approximate sum representations for the Dirichlet process.
Canad.J.Statist.30 269–283. MR1926065 https://doi.org/10.2307/3315951
JASRA,A.,HOLMES,C.C.andSTEPHENS, D. A. (2005). Markov chain Monte Carlo methods and the label
switching problem in Bayesian mixture modeling. Statist.Sci.20 50–67. MR2182987 https://doi.org/10.1214/
088342305000000016
JASRA,A.,STEPHENS,D.A.andHOLMES, C. C. (2007). On population-based simulation for static inference.
Stat.Comput.17 263–279. MR2405807 https://doi.org/10.1007/s11222-007-9028-9
JASRA,A.,DOUCET,A.,STEPHENS,D.A.andHOLMES, C. C. (2008). Interacting sequential Monte
Carlo samplers for trans-dimensional simulation. Comput.Statist.Data Anal.52 1765–1791. MR2418470
https://doi.org/10.1016/j.csda.2007.09.009
1192 HADJ-AMAR, FINKENSTÄDT, FIECAS AND HUCKSTEPP
JOHNSON,M.J.andWILLSKY, A. S. (2013). Bayesian nonparametric hidden semi-Markov models. J.Mach.
Learn.Res.14 673–701. MR3033344
JUANG,B.-H.andRABINER, L. (1985). Mixture autoregressive hidden Markov models for speech signals. IEEE
Trans.Acoust.Speech Signal Process.33 1404–1413.
KASS,R.E.andRAFTERY, A. E. (1995). Bayes factors. J.Amer.Statist.Assoc.90 773–795. MR3363402
https://doi.org/10.1080/01621459.1995.10476572
KIVINEN,J.J.,SUDDERTH,E.B.andJORDAN, M. I. (2007). Learning multiscale representations of natural
scenes using Dirichlet processes. In 2007 IEEE 11th International Conference on Computer Vision 1–8. IEEE,
New York.
KOMARZYNSKI,S.,HUANG,Q.,INNOMINATO,P.F.,MAURICE,M.,ARBAUD,A.,BEAU,J.,
BOUCHAHDA,M.,ULUSAKARYA,A.,BEAUMATIN, N. et al. (2018). Relevance of a mobile Internet plat-
form for capturing inter- and intrasubject variabilities in circadian coordination during daily routine: Pilot
study. J.Med.Internet Res.20 e204. https://doi.org/10.2196/jmir.9779
KRAUCHI,K.andWIRZ-JUSTICE, A. (1994). Circadian rhythm of heat production, heart rate, and skin and core
temperature under unmasking conditions in men. Am.J.Physiol., Regul.Integr.Comp.Physiol.267 R819–
R829.
KROGH,A.,BROWN,M.,MIAN,I.S.,SJÖLANDER,K.andHAUSSLER, D. (1994). Hidden Markov models in
computational biology: Applications to protein modeling. J.Mol.Biol.235 1501–1531.
KULLBACK,S.andLEIBLER, R. A. (1951). On information and sufficiency. Ann.Math.Stat.22 79–86.
MR0039968 https://doi.org/10.1214/aoms/1177729694
LANGROCK,R.,SWIHART,B.J.,CAFFO,B.S.,PUNJABI,N.M.andCRAINICEANU, C . M. (2013). Combining
hidden Markov models for comparing the dynamics of multiple sleep electroencephalograms. Stat.Med.32
3342–3356. MR3074361 https://doi.org/10.1002/sim.5747
LIANG,F.andWONG, W. H. (2001). Real-parameter evolutionary Monte Carlo with applications to
Bayesian mixture models. J.Amer.Statist.Assoc.96 653–666. MR1946432 https://doi.org/10.1198/
016214501753168325
MALIK, M. (1996). Heart rate variability: Standards of measurement, physiological interpretation, and clinical
use. Annals of Noninvasive Electrocardiology 1151–181.
MARIN,J.-M.,MENGERSEN,K.andROBERT, C. P. (2005). Bayesian modelling and inference on mix-
tures of distributions. In Bayesian Thinking:Modeling and Computation.Handbook of Statist.25 459–507.
Elsevier/North-Holland, Amsterdam. MR2490536 https://doi.org/10.1016/S0169-7161(05)25016-2
MENG,X.-L.andSCHILLING, S. (2002). Warp bridge sampling. J.Comput.Graph.Statist.11 552–586.
MR1938446 https://doi.org/10.1198/106186002457
MENG,X.-L.andWONG, W. H. (1996). Simulating ratios of normalizing constants via a simple identity: A the-
oretical exploration. Statist.Sinica 6831–860. MR1422406
NEAL, R. M. (2003). Slice sampling. Ann.Statist.31 705–767. MR1994729 https://doi.org/10.1214/aos/
1056562461
OMBAO,H.C.,RAZ,J.A.,VON SACHS,R.andMALOW, B. A. (2001). Automatic statistical analysis of
bivariate nonstationary time series. J.Amer.Statist.Assoc.96 543–560. MR1946424 https://doi.org/10.1198/
016214501753168244
PAPASTAMOULIS, P. (2016). Label.switching: An R package for dealing with the label switching problem in
MCMC outputs. Journal of Statistical Software,Code Snippets 69 1–24.
PAPASTAMOULIS,P.andILIOPOULOS, G. (2010). An artificial allocations based solution to the label switching
problem in Bayesian analysis of mixtures of distributions. J.Comput.Graph.Statist.19 313–331. MR2758306
https://doi.org/10.1198/jcgs.2010.09008
PAZ,J.C.andWEST, M. P. (2013). Acute Care Handbook for Physical Therapists. Elsevier Health Sciences,
Elsevier.
PEKER,Y.,HEDNER,J.,NORUM,J.,KRAICZI,H.andCARLSON, J. (2002). Increased incidence of cardiovas-
cular disease in middle-aged men with obstructive sleep apnea: A 7-year follow-up. Am.J.Respir.Crit.Care
Med.166 159–165.
PERMAN,M.,PITMAN,J.andYOR, M. (1992). Size-biased sampling of Poisson point processes and excursions.
Probab.Theory Related Fields 92 21–39. MR1156448 https://doi.org/10.1007/BF01205234
PITMAN, J. (1996). Blackwell–Macqueen urn scheme. Statistics,Probability,and Game Theory:Papers in Honor
of David Blackwell 30 245.
PITMAN, J. (2002). Poisson–Dirichlet and GEM invariant distributions for split-and-merge transformation
of an interval partition. Combin.Probab.Comput.11 501–514. MR1930355 https://doi.org/10.1017/
S0963548302005163
PRIESTLEY, M. B. (1965). Evolutionary spectra and non-stationary processes.(With discussion). J.Roy.Statist.
Soc.Ser.B27 204–237. MR0199886
IDENTIFYING THE RECURRENCE OF SLEEP APNEA 1193
RABINER, L. R. (1989). A tutorial on hidden Markov models and selected applications in speech recognition.
Proc.IEEE 77 257–286.
RASMUSSEN,C.E.andGHAHRAMANI, Z. (2002). Infinite mixtures of Gaussian process experts. In Advances
in Neural Information Processing Systems 881–888.
REDNER,R.A.andWALKER, H. F. (1984). Mixture densities, maximum likelihood and the EM algorithm.
SIAM Rev.26 195–239. MR0738930 https://doi.org/10.1137/1026034
RODRÍGUEZ,C.E.andWALKER, S. G. (2014). Label switching in Bayesian mixture models: Deterministic re-
labeling strategies. J.Comput.Graph.Statist.23 25–45. MR3173759 https://doi.org/10.1080/10618600.2012.
735624
ROSEN,O.,STOFFER,D.S.andWOOD, S. (2009). Local spectral analysis via a Bayesian mixture of smoothing
splines. J.Amer.Statist.Assoc.104 249–262. MR2504376 https://doi.org/10.1198/jasa.2009.0118
ROSEN,O.,WOOD,S.andSTOFFER, D. S. (2012). AdaptSPEC: Adaptive spectral estimation for nonstation-
ary time series. J.Amer.Statist.Assoc.107 1575–1589. MR3036417 https://doi.org/10.1080/01621459.2012.
716340
RUEHLAND,W.R.,ROCHFORD,P.D.,ODONOGHUE,F.J.,PIERCE,R.J.,SINGH,P.andTHORNTON,A.T.
(2009). The new AASM criteria for scoring hypopneas: Impact on the apnea hypopnea index. Sleep 32 150–
157.
SETHURAMAN, J. (1994). A constructive definition of Dirichlet priors. Statist.Sinica 4639–650. MR1309433
SHUMWAY,R.H.andSTOFFER, D. S. (2017). Time Series Analysis and Its Applications:With R Examples,4th
ed. Springer Texts in Statistics. Springer, Cham. MR3642322 https://doi.org/10.1007/978-3-319-52452-8
STEPHENS, M. (2000). Dealing with label switching in mixture models. J.R.Stat.Soc.Ser.B.Stat.Methodol.62
795–809. MR1796293 https://doi.org/10.1111/1467-9868.00265
TEH,Y.W.,JORDAN,M.I.,BEAL,M.J.andBLEI, D. M. (2006). Hierarchical Dirichlet processes. J.Amer.
Statist.Assoc.101 1566–1581. MR2279480 https://doi.org/10.1198/016214506000000302
TERAN-SANTOS,J.,JIMENEZ-GOMEZ,A.,CORDERO-GUEVARA,J.andBURGOS-SANTANDER,C.G.
(1999). The association between sleep apnea and the risk of traffic accidents. N.Engl.J.Med.340 847–851.
TRIPURANENI,N.,GU,S.S.,GE,H.andGHAHRAMANI, Z. (2015). Particle Gibbs for infinite hidden Markov
models. In Advances in Neural Information Processing Systems 2395–2403.
VAN GAEL,J.,SAATCI,Y.,TEH,Y.W.andGHAHRAMANI, Z. (2008). Beam sampling for the infinite hidden
Markov model. In Proceedings of the 25th International Conference on Machine Learning 1088–1095. ACM,
New York.
WALKER, S. G. (2010). Bayesian nonparametric methods: Motivation and ideas. In Bayesian Nonparametrics.
Camb.Ser.Stat.Probab.Math.28 22–34. Cambridge Univ. Press, Cambridge. MR2722988 https://doi.org/10.
1017/CBO9780511802478.002
WEST,M.,PRADO,R.andKRYSTAL, A. D. (1999). Evaluation and comparison of EEG traces: Latent structure
in nonstationary time series. J.Amer.Statist.Assoc.94 375–387.
WHITTLE, P. (1957). Curve and periodogram smoothing. J.Roy.Statist.Soc.Ser.B19 38–47 (discussion 47–63).
MR0092331
YAGGI,H.K.,CONCATO,J.,KERNAN,W.N.,LICHTMAN,J.H.,BRASS,L.M.andMOHSENIN, V. (2005).
Obstructive sleep apnea as a risk factor for stroke and death. N.Engl.J.Med.353 2034–2041.
YAGHOUBY,F.andSUNDERAM, S. (2015). Quasi-supervised scoring of human sleep in polysomnograms using
augmented input variables. Comput.Biol.Med.59 54–63. https://doi.org/10.1016/j.compbiomed.2015.01.012
YAU,C.,PAPASPILIOPOULOS,O.,ROBERTS,G.O.andHOLMES, C. (2011). Bayesian non-parametric hidden
Markov models with applications in genomics. J.R.Stat.Soc.Ser.B.Stat.Methodol.73 37–57. MR2797735
https://doi.org/10.1111/j.1467-9868.2010.00756.x
YOUNG,T.,PEPPARD,P.E.andGOTTLIEB, D . J. (2002). Epidemiology of obstructive sleep apnea: A population
health perspective. Am.J.Respir.Crit.Care Med.165 1217–1239.
YU, S.-Z. (2010). Hidden semi-Markov models. Artificial Intelligence 174 215–243. MR2724430
https://doi.org/10.1016/j.artint.2009.11.011
ZHOU,Y.,JOHANSEN,A.M.andASTON, J. A. D. (2016). Toward automatic model comparison: An adaptive
sequential Monte Carlo approach. J.Comput.Graph.Statist.25 701–726. MR3533634 https://doi.org/10.1080/
10618600.2015.1060885
... We now provide a brief introduction to the standard HMM and HSMM approaches before considering the special structure of the transition matrix presented by Zucchini et al. (2017), which allows the state dwell distribution to be generalized with arbitrary accuracy. HMMs, or Markov switching processes, have been shown to be appealing models in addressing learning challenges in time series data and have been successfully applied in fields such as speech recognition (Rabiner, 1989;Jelinek, 1997), digit recognition (Raviv, 1967;Rabiner et al., 1989) as well as biological and physiological data (Langrock et al., 2013;Huang et al., 2018;Hadj-Amar et al., 2021). An HMM is a stochastic process model based on an unobserved (hidden) state sequence s = (s 1 , . . . ...
... Full details of the sampling scheme and our prior choice are provided in the Supplementary Material. This algorithm is similar to the within-model move of the "segment model" presented in Hadj-Amar et al. (2019, 2021, but with the number of frequencies fixed at one. ...
Article
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We propose a Bayesian hidden Markov model for analyzing time series and sequential data where a special structure of the transition probability matrix is embedded to model explicit-duration semi-Markovian dynamics. Our formulation allows for the development of highly flexible and interpretable models that can integrate available prior information on state durations while keeping a moderate computational cost to perform efficient posterior inference. We show the benefits of choosing a Bayesian approach for HSMM estimation over its frequentist counterpart, in terms of model selection and out-of-sample forecasting, also highlighting the computational feasibility of our inference procedure whilst incur- ring negligible statistical error. The use of our methodology is illustrated in an application relevant to e-Health, where we investigate rest-activity rhythms using telemetric activity data collected via a wearable sensing device. This analysis considers for the first time Bayesian model selection for the form of the explicit state dwell distribution. We further investigate the inclusion of a circadian covariate into the emission density and estimate this in a data-driven manner.
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... We now provide a brief introduction on the standard HMM and HSMM approaches before considering the special structure of the transition matrix presented by Zucchini et al. (2017), which allows the state dwell distribution to be generalized with arbitrary accuracy. HMMs, or Markov switching processes, have been shown to be appealing models in addressing learning challenges in time series data and have been successfully applied in fields such as speech recognition (Rabiner 1989, Jelinek 1997, digit recognition (Raviv 1967) as well as biological and physiological data (Langrock et al. 2013, Huang et al. 2018, Hadj-Amar et al. 2020). An HMM is a stochastic process model based on a unobserved (hidden) state sequence s = (s 1 , . . . ...
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We propose a Bayesian hidden Markov model for analyzing time series and sequential data where a special structure of the transition probability matrix is embedded to model explicit-duration semi-Markovian dynamics. Our formulation allows for the development of highly flexible and interpretable models that can integrate available prior information on state durations while keeping a moderate computational cost to perform efficient posterior inference. We show the benefits of choosing a Bayesian approach over its frequentist counterpart, in terms of incorporation of prior information, quantification of uncertainty, model selection, and out-of-sample forecasting. The use of our methodology is illustrated in an application relevant to e-Health, where we investigate rest-activity rhythms using telemetric activity data collected via a wearable sensing device.
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