R Rosipal

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MEASUREMENT 2013, Proceedings of the 9th International Conference, Smolenice, Slovakia 
101 
Clustering Probabilistic Sleep Microstate Curves: a Functional Data 
Analysis Approach   
R. Rosipal 
Department of Theoretical Methods, Institute of Measurement Science 
Slovak Academy of Sciences, Bratislava, Slovak Republic 
 Email: roman.rosipal@savba.sk 
Abstract. We introduced and validated an EEG data-based model of the sleep process with an 
arbitrary number of different sleep states and a high time resolution allowing modelling of 
sleep microstructure. The proposed probabilistic sleep model describes sleep via posterior 
probabilities of a finite number of microstates. Using the model, we extracted objective sleep 
parameters describing quantitative and qualitative characteristics of the probabilistic sleep 
microstate curves and proved their usefulness when assessing selected aspects of sleep 
quality. In the current work we are investigating functional data clustering methods applied 
to sleep microstate posterior curves. The hierarchical structure of the data given by the 
repeated visits of subjects in the sleep lab motivates our focus on recently proposed multilevel 
functional clustering analysis approaches. We are applying the multilevel functional principal 
component analysis to the sleep posterior curves. Preliminary results show promising 
potential of the approach to separate age-related sleep profiles and extracting subjects' 
specific night deviations from the mean sleep profiles. 
Keywords: Probabilistic Sleep Model, Functional Data Clustering, Multi-level Functional 
Principal Component Analysis  
1. Introduction 
In spite of the fact that sleep accounts for about one-third of human live, the purposes and 
mechanisms of sleep are only partially clear and are the subject of substantial research. One of 
the long-term open questions in sleep research is how to define and objectively measure 
‘normal’ sleep and its quality. A different but not less important question is how sleep 
structure relates to selected daytime quality of life measures, including cognitive, emotional, 
psychometric or physiological tests and measures. The conventional description of sleep 
architecture from polysomnographic (PSG) recordings is carried out through applying the 
standardized Rechtschaffen and Kales (R&K) scoring manual [1] or the recently published 
update of the rules [2]. However, some aspects of the R&K sleep staging rules were criticized 
in the past and new ways of analysing sleep have been discussed [3]. With the aim of 
avoiding known drawbacks of R&K, an alternative computerized sleep model was introduced 
[4]. The model based on solid probabilistic principles allows describing sleep on an arbitrarily 
fine time scale and allows considering sleep as a continuous process of transitions between a 
larger number of sleep sub-states (microstates). Microstates can be combined into subsets. 
This feature allows defining new sleep states whose physiological interpretation and specific 
task-related performances can be studied. In the current study we considered posterior values 
of combined microstates as continuous time curves and we used advanced functional data 
analysis tools [5,6] to cluster curves into a meaningful structure. The validity of the approach 
is demonstrated on tasks of clustering age-dependent sleep profiles and clustering specific 
night deviations observed on subjects spending two consecutive nights in the sleep lab.    
2. Subject and Methods 
Data of 175 subjects (94 males and 81 females), age between 20 and 95 (mean 51 years and 
MEASUREMENT 2013, Proceedings of the 9th International Conference, Smolenice, Slovakia 
102 
standard deviation 20 years), from the sleep database created during the EU Siesta project 
were used [7]. One aim of the Siesta project was to create a normative database of healthy 
sleepers and sleep-disturbed patients. According to the Siesta recording protocol all subjects 
had to document their sleep habits over 14 nights. Subjects spent two consecutive nights (7th 
and 8th night) in the sleep laboratory during which PSG recordings were obtained. Subjects 
used in this study were classified as healthy sleepers without sleep related disorders [7].  
The PSG recording protocol specified 16 channels of biosignals: 6 EEG channels with 
mastoid as reference (Fp1-M2, C3-M2, O1-M2, Fp2-M1, C4-M1, O2-M1), an additional EEG 
channel (M1-M2) for re-referencing, 2 EOG channels, submental EMG and EMG recorded 
from electrodes placed at the musculus anterior tibialis of the left and right leg (electrodes 
were linked), electrocardiogram and respiratory signals (airflow; movements of the chest wall 
and abdomen and O2 saturation of arterial blood). 
First, using the PSG recording the sleep structure was analysed in 30 sec epochs according to 
the standard R&K scoring rules for sleep staging [1]. To this aim, the computerized system 
Somnolyzer 24x7 was used [8]. Next, a new probabilistic sleep model (PSM) was used to 
represent sleep as a continuum [4].  
Probabilistic Sleep Model (PSM)  
We use the C3-M2 (or C4-M2 as a substitute) EEG channel. For each 3 sec segment a AR(10) 
parameter vector = ( ,… , ) is fitted: = { } + { } + ⋯+ { } +  
To each 3 sec interval a value 	 ∈ 	 {0,1,2,3} is assigned (0: no spindle, 1,2,3: spindle with 
increasing certainty). The R&K labels c are assigned by the automatic sleep scoring system to 
each 30 sec long data segment. The PSM assumes the existence of a latent variable Z with K 
possible states (Fig. 1, left) ( , , ) = 	 ( ) ( | ) ( | ) ( | ) 
 
Fig. 1. Left: The schematic structure of the probabilistic sleep model  (PSM). Right: An example of the PSM 
posterior values (light grey) and the B-spline smoothed curve (black). SWS – slow wave sleep. 
The PSM with = 20	sleep microstates was fitted using the EM algorithm [4]. Applying a 
model means to present an  ( | , ) = ( ) ( | ) ( | )	∑ ( ) ( | ) ( | ) 
or the R&K posteriors  
                                    				 ( | , ) = 	∑ ( | , ) ( | )	                                  (1) 
   
MEASUREMENT 2013, Proceedings of the 9th International Conference, Smolenice, Slovakia 
103 
Two-way Functional ANOVA & Multilevel Functional Clustering 
We consider the two-way functional ANOVA model for our sleep profiles of subjects = 1,… , 175 and nights ∈ {1,2} ( ) = ( ) + ( ) +	 ( ) +	 ( ) + ( ) 
where ( ) and ( ) are fixed functional means specifying the global and night specific 
functional trends;  ( ) is the subject-specific deviation from the night-specific mean and ( ) is the subject- and night-specific deviation from the subject-specific mean. Posteriors 
curves were positive smoothed by using B-splines with 20 basis functions and down-sampled 
to 30 sec intervals (Fig. 1, right). Nights were aligned to 5.3 hours starting by sleep latency. 
We use multilevel functional PCA [5] for extracting intra- and inter- subject specific 
component scores ,  and eigenfunctions ( ) (t) = 	 ( ) ( ),						 ( ) = ( ) ( ) 
where  ( )( ) ∼ ( , ), 		 ( )( ) ∼ ( , )  
and ( , ), ( , )	and ( , ) ∶= 	 ( , ) − _ ( , )	are the total, between, and within 
subjects covariance functions.  The number of used eigenfunctions ( )( ) and ( )( )	was 
determined by the explained variance of 90%. Finally, we adapted clustering scheme 
proposed in [6]: 
Level-1 Clustering: 
Clustering of subject-specific means: subjects 	and  will be in the same cluster if their 
subject-specific deviations ( ) and ( ) are similar in shape.   
Level-2 Clustering: 
Clustering of night-specific deviations: subjects 	and  will be in the same cluster if their 
deviations from the subject-specific means , 	( ) and , ( ), ∈ 	 {1,2} are similar.  
3. Results 
First, 20 sleep microstates of the PSM were combined into the R&K representation (wake, S1, 
S2, SWS – slow wave sleep, and REM – rapid-eye movement) using eq. (1). In addition, 
microstates representing SWS ( (SWS| ) > 0.5) were combined into a sleep state, which we 
denote mixMicro-SWS. Using the Level-1 clustering and considering the existence of three 
clusters we observed for all sleep states a structure with two dominant clusters. The 
percentage of samples falling into these two clusters was for wake – 99%, S1 – 94%, S2 – 
95%, SWS  - 94%, REM – 92% and mixMicro-SWS – 94%.  
 
 
 
 
 
Fig. 2. The averaged sleep 
profiles of the two dominant 
clusters (thicker-solid lines) 
and the averaged sleep 
profiles of the young and 
elderly subjects. Curves 
represent 5.3 hours of sleep 
aligned by sleep latency. 
SWS - slow wave sleep, 
REM - rapid-eye movement. 
0 1 2 3 4 5
0.
0
0.
2
0.
4
Wake
time [h]
clust1.mean
clust2.mean
mean.young
mean.elderly
0 1 2 3 4 5
0.
00
0.
05
0.
15
S1
time [h]
clust1.mean
clust2.mean
mean.young
mean.elderly
0 1 2 3 4 5
0.
2
0.
4
0.
6
S2
time [h]
clust1.mean
clust2.mean
mean.young
mean.elderly
0 1 2 3 4 5
0.
0
0.
2
0.
4
SWS
time [h]
clust1.mean
clust2.mean
mean.young
mean.elderly
0 1 2 3 4 5
0.
00
0.
10
0.
25
REM
time [h]
clust1.mean
clust2.mean
mean.young
mean.elderly
0 1 2 3 4 5
0.
00
0.
05
0.
13
mixMicro−SWS
time [h]
clust1.mean
clust2.mean
mean.young
mean.elderly
MEASUREMENT 2013, Proceedings of the 9th International Conference, Smolenice, Slovakia 
104 
Following our previous studies [9], we considered three age-related groups of subjects: young 
(< 40 years old), middle-age (40 to 60) and elderly (≥ 60). We compared averaged posterior 
curves computed for each age group with averages obtained by averaging posteriors curves 
falling into each of the two dominant clusters (Fig. 2). Close match between a mean curve of 
each cluster and averaged sleep profile either for the young or elderly subjects can be 
observed for every sleep state. For clarity, the averaged sleep profile of the middle-age 
subjects is not plotted, however, for all six sleep states this profile lies between the averaged 
sleep profiles of the young and elderly subjects. Finally, by considering the Level-2 clustering 
we observed strong night effects for wake state reflecting known phenomenon of the first-
night effect associated with a higher level of wakefulness and longer sleep latency. Nigh 
effects were also observed for other sleep states and these are subject of further studies and 
validation.  
4. Discussion 
Multilevel functional data clustering was applied to smoothed posterior sleep profiles of the 
PSM. Preliminary results show promising potential of the approach to separate age-related 
sleep profiles and extracting subjects' specific night deviations from the mean sleep profiles. 
Further studies will focus on correlating extracted cluster information and subjects' daytime 
performance. 
Acknowledgements 
This work was supported by the Slovak Research and Development Agency under the 
contract No. APVV-0096-10 and by the Scientific Grant Agency of the Ministry of Education 
of the Slovakia Republic and the Slovak Academy of Sciences VEGA 1/0503/13 and VEGA 
2/0043/13 grants. 
References 
[1] Rechtschaffen A, Kales A. A Manual of Standardized Terminology, Techniques and 
Scoring System for Sleep Stages of Human Subject. U.S. Dept. of Health, Education, and 
Welfare, Bethesda, MD, 1968. 
[2] Iber C, Ancoli-Israel S, Chesson A, Quan S. The AASM Manual for the Scoring of Sleep 
and Associated Events: Rules, Terminology and Technical Specifications, 2007 . 
[3] Himanen S, Hasan J. Limitations of Rechtschaffen and Kales. Sleep Medicine Reviews, 4, 
149-167, 2000. 
[4] Lewandowski A, Rosipal R, Dorffner G. Extracting more information from EEG 
recordings for a better description of sleep. Computer Methods and Programs in 
Biomedicine, 108, 961-972, 2012.  
[5] Di Ch-Z, Crainiceanu CM, Caffo BS, Punjabi NM. Multilevel Functional Principal 
Component Analysis, Annals of Applied Statistics,  3, 458-488, 2009.  
[6] Serban N, Jiang H. Multilevel functional clustering analysis. Biometrics, 68: 805-814, 
2012. 
[7] Klösch et al. The SIESTA project polygraphic and clinical database. IEEE Engineering in 
Medicine and Biology Magazine, 20, 51-57, 2001. 
[8] Anderer et al. An E-Health Solution for Automatic Sleep Classification according to 
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Database. Neuropsychobiology, 5, 115-133, 2005. 
[9] Rosipal R, Lewandowski A, Dorffner G. In search of objective components for sleep 
quality indexing in normal sleep. Biological Psychology, under review.  


Clustering Probabilistic Sleep Microstate Curves: a Functional Data Analysis Approach

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Clustering Probabilistic Sleep Microstate Curves: a Functional Data Analysis Approach

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