ᚑくᄢቇℂᎿቇ⎇ⓥႎ๔
J. Fac. Sci. Tech., Seikei Univ.
Vol.46 No.2 (2009) pp.53-62㧔ᣂછ⠪ߩ⺰ᢥ㧕

Modeling of Sleep States and Estimation of Sleep Stages
Yosuke KURIHARA㧖1
ABSTRACT㧦The judgment standards of R-K method include ambiguities and are thus compensated bysubjective interpretations of sleep stage scorers. This paper presents a novel method to compensate
uncertainties in judgments by the subjective interpretations by the sleep model estimation approach andby describing the judgments in probabilistic terms. A full-order Luenberger observer based on the twosleep models with no body movement and with body movement was designed. Sleep stages judged bythree different scorers were re-judged by the observer. The two sleep models were stochasticallyestimated from bio-signals from 15 subjects and the re-judged scores by the observer were evaluated by
the data from 5 subjects. The average values of kappa statistics, which show the degree of agreement,
were 0.85, 0.89 and 0.81, respectively, for the original sleep stages. Because the new method provides
probabilities on how surely the sleep belongs to each sleep stage, we were able to determine the most,second most and third most probable sleep stage. The kappa statistics between the most probable sleep
stages were improved to 0.91, 0.92 and 0.85, respectively. Those of sleep stages determined from themost and second most probable were 0.94, 0.96 and 0.90 and those from the most, second most and third
most probable were 0.95, 0.97 and 0.92. The sleep stages estimated by the observer are expressed byprobabilistic manner which are more reasonable tin expression than those given by deterministic manner.
The expression could compensate the uncertainties in each judgments and thus were more accurate thanthe direct judgments.
Keywords㧦sleep stage, R-K method, sleep stage transition probability matrix, sleep stage state variable
equation
㧔Received July 30, 2009㧕㧝㧚㪠㫅㫋㫉㫆㪻㫌㪺㫋㫀㫆㫅

reliability of judgments [5].

Sleep plays a vital role in recovery from mental andtransition model that shows the all-night sleeping trend

Japan’s national health improvement projects involving

judgments given by scorers, would improve the reliability

We propose that having another reference, a sleep

characteristics and applying it to compensate thephysical fatigue [1, 2]. Its importance is reflected inof judgment. There have been various reports on using asleep [3] and many organizations are researching

sleep model [6, 7], but our model and method are different

non-invasive techniques for measuring sleep conditions.

in that we use a new sleep transition model and apply a

Sleep is categorized into six different stages using thefull-order Luenberger observer to the model. Here, we cite

R-K method, in which a nominal scale is applied to brain

the sleep stage transition equation estimated from clinicalwaves, eye movement and myoelectricity of the submental

data and use it as a state variable equation for designing a

muscles [4] at time intervals divided independently for

full-order Luenberger observer. The measurements used

all-night sleep. The stages are AWAKE, REM-Sleep andfor the observer are the temporal changes in sleep stages

Non-REM-sleep 1, 2, 3 and 4. In using the R-K method,

judged by a scorer or automatic sleep stage estimator. The

however, some of the rules include ambiguities and thestate variables estimated by the observer are probabilitiesscorers must subjectively interpret what happens during
the sleep from the bio-signals above. As a result, different

scorers might judge different sleep stages for the same data.
A method called kappa statistics is used to evaluate the*1Assistant professor, Department of Computer andInformation Science (yosuke-kurihara@st.seikei.ac.jp)

of how surely the sleep can be categorized into each sleep
stage. Here, we describe how to build the sleep stagetransition probability matrix, sleep stage transition equationand optimal observer.

㧙㧙㧞䋮㪦㪹㫁㪼㪺㫋㫀㫍㪼㩷This paper is aimed at describing a novel scheme tocompensate uncertainties of sleep stage judged by sleep
medical specialists. The uncertainties are due to the(F3) Body movement occurs just before and/or after REM.
(F4) AWAKE stage frequently occurs after a body
movement.

ambiguities of the standards of the R-K method andfluctuations of sleep characteristics in the judgment time

Let A be the probability transition matrix. The element ininterval. In spite of such the uncertainties, the conventional

matrix A can be determined statistically from clinical data.

not reasonable. Here we present a method to express theone sleep stages to another as described above; here, we

expression of the sleep stage was deterministic, which issleep stage by probabilities on how surely the sleep

belongs to each sleep stage. Further, we present a novel

method to compensate the fluctuations in judgments bycomparing the judged sleep state with the normal sleep

Body movement plays a key function in switching fromobtain two different probabilistic state transition matrices.

One is when body movement is included and the other iswhen no body movement is included. Let Ej,i be an eventin which the sleep stage switches from i stage to j stage fortransition model. The state of sleep and/or awake wheni, j = 6, 5, 4, 3, 2, 1, let Ej be an event in which the sleep

body movement occurs are different, and so we prepare

body movement occurs and no body movement occurs,

state estimation theory is newly introduced to realize theprobability of these events occurring, respectively.

methodology of scoring the sleep stages based onstage at k discrete time switches to j stage at k+1 discrete

body movement occurs and the state of sleep when no
two different transition models. An approach based on thealgorithm for the compensation. Here we describe a newstage switches toj stage, let Eb and En be events in which

respectively, and let P(Ej,i), P(Ej), and P(Eb) be the
The element a7-j, 7-i, which is the probability that sleep in i

probabilistic terms instead of scoring in a deterministic

time, is given by conditional probability as follows:

validity of the method is examined using only youngP( E j ,i | E j , Eb ) (body movement occurs)a7− j ,7−i = 
P( E j ,i | E j , En ) (no body movement occurs)manner to compensate uncertainties in judgments. The
subjects as examples.

㧟㧚㪤㪼㫋㪿㫆㪻

where

∑a
6

i =1

7 − j ,7 −i

=1

(1)

must be satisfied.

㪝㫆㫉㫄㫌㫃㪸㫋㫀㫆㫅㩷㫇㫉㫆㪺㪼㫊㫊First, in building the sleep stage transition model,㪪㫃㪼㪼㫇㩷㫊㫋㪸㪾㪼㩷㫊㫋㪸㫋㪼㩷㫍㪸㫉㫀㪸㪹㫃㪼㩷㪼㫈㫌㪸㫋㫀㫆㫅

WAKE, 2 for REM, 3 for Non-REM1, 4 for Non-REM2, 5

equation. Let Tib be the total time that the subject is in

numbers are assigned to the sleep stages as follows: 1 for

Here, we describe the sleep stage state variable

for Non-REM3 and 6 for Non-REM4. The number isbed, let k㧔= 1, 2, 3, … Tib㧕be a discrete time of every

mathematical representations.

and let x6(k), x5(k), x4(k), x3(k), x2(k) and x1(k) be thenominal and is used instead of the sleep category inIn the sleep stage transition, body movement plays the

key functions [2, 8] as follows:(F1) Non-REM1 sleep stage frequently occurs after REM,

Non-REM3 and Non-REM4, triggered by a body
movement.

(F2) The sleep stage frequently switches just after body
movement when the stage is Non-REM3 or

one minute with the sleep stage judged within that time,

probabilities of how surely the sleep stages belong to 6,
5, 4, 3, 2, and 1, respectively.

Furthermore, let the vector x(k)=[x6(k), x5(k), x4(k),x3(k), x2(k), x1(k)]T be the state vector and the vectory(k)=[y6(k), y5(k), y4(k), y3(k), y2(k), y1(k)]T be the sleep

stage judged by a certain method such as the R-K
method. y(k) is deterministically given corresponding to

the judgment of sleep stage as follows:

Non-REM4.

㧙㧙

(Wake)[1 0 0 0 0 0]T[0 1 0 0 0 0]T(REM)
( Non - REM1)
[0 0 1 0 0 0]Ty(k) = T( Non - REM2)
[
0
0
0
1
0
0
]

T
(Non - REM3)
[0 0 0 0 1 0][0 0 0 0 0 1]Twhen              ( Non - REM4)

⁓1,1
Wake

⁓

1,2

⁓3,1
⁓1,3

2 ,1

⁓4,1

⁓5,1

⁓2,3

⁓2,5

⁓6,3
⁓1,5

⁓ 1,4

4

⁓1,6

(3)

3

NonREM2

The elements of state transition vector directly correspond

⁓4,4

⁓6
⁓4,6 ,4⁓4,5x(Tib)=[1 0 0 0 0 0]Ttransition equation and the coefficient a7-j, 7-i is the (7-j, 7-i)

⁓6,6
probability
transition matrixz

Figure 1 shows a Shannon diagram of the sleep statex(k+1) = A x(k)element in matrix A. The element a7-j, 7-i has a different

⁓5,5

Zz

A

the subject must be in the AWAKE stage; thus, x(0)=⁓5,6 5⁓ 6,

NonREM4

to the sleep stage judgments (measurements). Thus,y(k)=x(k) =Ix(k). Just after bed-in and just before bed-out,

NonREM3⁓5,4⁓2,6

I is an identity matrix with dimensions (6×6) in Eq. (3).

⁓5,3
⁓3,5

⁓3,

⁓4,

⁓3,6

⁓6,2

⁓5,2

⁓4,2
⁓2,4

y(k)=Ix(k)⁓6,1Then, the sleep stage state variable equation andmeasurement equation are given as follows:(2)

NonREM1⁓3,2

REM

x(k+1)=Ax(k)

⁓3,3

⁓

⁓2,2

sleep stage statevariable equation

value for the body movement condition, as described in Eq.
(1).

㪦㪹㫊㪼㫉㫍㪼㫉㩷 㫋㫆㩷 㪼㫊㫋㫀㫄㪸㫋㪼㩷 㫊㫃㪼㪼㫇㩷 㫊㫋㪸㪾㪼㩷㪸㫇㫇㪼㪸㫉㪸㫅㪺㪼㩷(KI㩷 㪪㪿㪸㫅㫅㫆㫅㩷㪻㫀㪸㪾㫉㪸㫄㩷㫆㪽㩷㫊㫃㪼㪼㫇㩷㫊㫋㪸㪾㪼㩷㫋㫉㪸㫅㫊㫀㫋㫀㫆㫅㩷㫇㫉㫆㪹㪸㪹㫀㫃㫀㫋㫐㩷
Here, we consider an observer to estimate the probabilityof how surely the sleep belongs to each sleep stage basedon the state variable equation. The observer is a state

estimator for an object whose characteristics are given by
an a priori mathematical model described as the statexˆ (k + 1) = ( A − KI ) xˆ (k ) + Ky(k )(4)

If one of the absolute values of eigenvalues of matrix

A-KI is greater than 1, the observer equation Eq. (4)variable equations [9] Eq. (2) and Eq. (3). It estimates all

diverges, whereas if the absolute values of all eigenvalues

state variable in Eq. (2) is the probability, thus we estimate

the theory of observer [9].

which may include errors.

determined by genetic algorithm [10]. Genetic algorithms

matrix of the observer so that all absolute eigenvalues ofexact or approximate solutions to optimization and searchstate variables from measurements corrupted by noise. The
the probability from the sleep judgments (measurements)
Let xˆ (k ) be the estimate of x(k) and let K be the gain

matrix A-KI i.e., the characteristic roots of the observer areless than 1, then the following state variable equation is afull-order Luenberger observer for Eqs. (2) and (3):are less than 1, it converges and xˆ (k ) tends to x(k) from
Eigenvalues ǳ 1- ǳ 6 of matrix A-KI are optimallyare one of the search techniques used in computing to findproblems. A degree of adaptation index shows the degreeof how the process approaches the exact or approximate
solutions. As an index, we selected the coincidence rate of
the measurements y(k) and the most probable state inxˆ (k )

estimatedbytheobservermeasurement time for the test subjects.

㧙㧙

for

the

entire㧠㧚㪩㪼㫊㫌㫃㫋㫊㩷㪋㪅㩷㪈㩷 㪜㫊㫋㫀㫄㪸㫋㫀㫆㫅㩷㫆㪽㩷㫊㫃㪼㪼㫇㩷㫊㫋㪸㪾㪼㩷㫋㫉㪸㫅㫊㫀㫋㫀㫆㫅㩷㪼㫈㫌㪸㫋㫀㫆㫅In order to estimate a normal sleep stage transition

20 nights was randomly divided into two groups: G1

age 22.2 years old, and we conducted clinical tests over aremaining 5 nights’ data. The G1 data was used to estimate

First, we estimated the transition matrix A. Data from theconsisting of 15 nights’ data, and G2 consisting of theequation, we employed 10 normal sleep subjects, average

the transition matrices and the G2 data was used toperiod of 20 nights. Figure 2 shows the sleep conditions.

evaluate the models. The occurrence of body movement

We obtained informed consent from each subject. Sleep

stage was evaluated by the international 10/20 method [8]and so EEG at points C4-A1 and C3-A2 of the head, eye

was detected from artifacts in the brainwave measurements.

movement and EMG at submental muscles were measured.
Electrocardiograms were measured using the I-induction
correction method [8]. The sampling interval of the dataacquisitions was 0.01 s. For the measurements we
employed a polygraph (SANEI FIT 2500). To follow theR-K procedure, we let scorers judge two times for every
one minute i.e., every 30 s, and let them make a final

single judgment from two judgments for every minute.

Body movements can be detected by EMG artifacts
occurring in the measurements of brain wave and eye
movements. The body movements can be classified into

The transition matrix was calculated from the stages
judged by Scorer 1 using the R-K method. Transition

matrices when body movement occurred and when no
body movement occurred were obtained as follows:
When body movement occurred:0.333
0.103
0.154
Ab = 
0.4100.0000.000

small body movements which continue for less than 0.5 s,

and large body movements which continue for more than
0.5s [8]. We detected body movements using this detection

procedure. The method was verified by comparison withthe pneumatic body movement detection method [11].
Sleep stages were judged by three different scorers.
Polygraph

Electrodes forR-K method

0.034 0.000 0.008 0.030 0.044
0.729 0.318 0.023 0.000 0.011
0.106 0.273 0.002 0.067 0.066

0.131 0.409 0.952 0.127 0.066
0.000 0.000 0.015 0.732 0.198

0.000 0.000 0.000 0.044 0.615 (5)When no body movement occurred:0.333
0.167
0.000An = 
0.5000.0000.000

0.014 0.000 0.006 0.003 0.000
0.866 0.167 0.014 0.000 0.000
0.014 0.333 0.001 0.000 0.000

0.106 0.500 0.951 0.059 0.000
0.000 0.000 0.028 0.879 0.145

0.000 0.000 0.000 0.059 0.855 (6)Figure 3 shows the Shannon diagram of the sleep

transition equation.

Air cushion sensor to detect body movements(KI㩷 㪤㪼㪸㫊㫌㫉㪼㫄㪼㫅㫋㩷㫊㫐㫊㫋㪼㫄㩷㧙㧙

0.333

0.333

Wake

NonREM1

0.318

0.0
66

NonREM4

98
0.1 4
4
0.0

00

06

08
0.0

30

A

11
0.0

0. 0
66

0.952

NonREM2

NonREM3

0.127

0. 0

0.732

03

0 .0

0.044

0.015

0 .0

02

00

0.106
0.014

0.067

0.131

0.023

NonREM2

NonREM1

0.167
0.5

09
0.4
0 .0

0.333

0.014

REM

0 .0

0.106

REM

14
0.0 67
0.1

0.866

0.273
0.4
10

34
0 .0 3
0
0.1

0.729

0.15
4

0 .5

Wake

01

0.028

0.951

NonREM4

A

0.615

45
0.1 59
0.0

0.879

0.855

No body movement
occursnZz

b Body movement
occursNonREM3

0.059

z

(a)

Body movement occurs(b) No body movement occurs㩷㪝㫀㪾㪅㩷㪊 㪪㪿㪸㫅㫅㫆㫅㩷㪻㫀㪸㪾㫉㪸㫄㩷㫆㪽㩷㫋㪿㪼㩷㫄㪸㫋㫉㫀㫏㩷㪘㩷

When no body movement occurred, the probability oftransition

was

zero

for

“Non-REM3→Non-REM1”,

㪫㪸㪹㫃㪼㩷㪈㩷 㩷 㪙㫆㪻㫐㩷㫄㫆㫍㪼㫄㪼㫅㫋㩷㩿㪝㪈㪀㩷“WAKE→Non-REM1”,

Matrix element“Non-REM4→AWAKE”, “Non-REM4→REM”,

a3,2 (REM→Non-REM1)“Non-REM4→Non-REM1”and

a3,5 (Non-REM3→NonREM1)“Non-REM4→Non-REM2”.

a3,6 (Non-REM4→Non-REM1)

When body movement occurred, the probability increased

and thus sleep stage transition occurred frequently.

0.014

0.067

0.066

0.000

0.000

“Non-REM1 sleep stage frequently occurs after REM,

conjunction with body movement as described in Section 3.
and An.

0.106

occurred, with a 5% significance level. Thus, (F1)

Shannon diagram in Figure 3 or the transition matrices in

a3,6 in AbAn

are greater than those in Eq. (6) when no body movement

to a shallower stage is high. Here, we analyze whether the

Table 1 shows the value of elements a3,2, a3,5,Ab

The elements in Eq. (5) when body movement occurred

Especially, the probability of switching from Non-REM4

Eqs. (5) and (6) satisfy the sleep feature (F1) – (F4) inTransition matrix

Non-REM3

and

Non-REM4,

triggered

by

body

movement” is shown. Table 2 shows the values of
elements a5,5, a6,6.

㪫㪸㪹㫃㪼㩷㪉㩷 㩷 㪙㫆㪻㫐㩷㫄㫆㫍㪼㫄㪼㫅㫋㩷㩿㪝㪉㪀㩷
Matrix element
a5,5 (Non-REM3→Non-REM3)
a6,6 (Non-REM4→Non-REM4)㧙㧙

Transition matrix
Ab

An

0.732

0.879

0.615

0.855

These are the probabilities of maintaining the same sleep

The elements a1,2, a1,5, a1,6 in Eq. (5) are greater thanstage. Those in Eq. (5) are less than those in Eq. (6), with athose in Eq. (6), with a 5% significance level. This means

5% significance level. Thus, (F2) “Sleep stage transition

that when body movement occurs in REM, Non-REM3,stage is Non-REM3 or Non-REM4” is shown.

Thus, (F4) “AWAKE frequently occurs after bodyfrequently switches just after body movement when theand Non-REM4 sleep, sleepers tend to switch to AWAKE.

Table 3 shows the values of elements a2,1– a2,6, except a2,2,

and a1,2– a6,2.

movement” is shown. Figure 4 shows the transition of

sleep to a steady state from the WAKE stage for theequation when body movement occurs and no body
movement occurs.

㪫㪸㪹㫃㪼㩷 㩷 㪙㫆㪻㫐㩷㫄㫆㫍㪼㫄㪼㫅㫋㩷㩿㪝㪊㪀㩷a2,1 (Wake →REM)

a2,3 (Non-REM1→REM)
a2,4 (Non-R2→REM)
a2,5 (Non-R3→REM)
a2,6 (Non-R4→REM)
a1,2 (REM →Wake)

Ab

An

0.103

0.167

0.318

0.023
0.000
0.011

0.034

0.000

0.5
0.4
0.3

0.014

0.000
0.000

0.6

0.000

a5,2 (REM→Non-REM3)
a6,2 (REM→Non-REM4)

0.7

0.000

0.014

Non-REM2

0.8

0.014

0.106
0.131

0.9

0.167

a3,2 (REM→Non-REM1)
a4,2 (REM→Non-REM2)

1

Transition matrix

Probability

Matrix element

0.2
0.1

0.106

Non-REM4

0

50

100

150

200

REM

Non-REM3

Non-REM1

Wake

250

Time [min]

300

350

400

450

(a) Transition of sleep stage probability when body

0.000

movement occurs

The elements a2,3, a2,4 in Eq. (5) are greater than those in1

Eq. (6), with a 5% significance level. This means that just

0.9

after body movement in Non-REM1 and Non-REM2 sleep,

0.8

there is a tendency to switch to REM. The elements a1,2,

0.7
Probability

a3,2, a4,2 in Eq. (5) are greater than those in Eq. (6), with a

5% significance level. This means that just after body
movement in REM sleep, there is a tendency to switch to

Non-REM1 or Non-REM2. Thus, (F3) “Body movement

Non-REM2

0.6
0.5
0.4

Non-REM3

0.3

occurs just before and/or after REM” is shown. Table 4

0.2

shows the values of elements a1,2, a1,5, a1,6.

Non-REM4

REM

Wake

0.1

Non-REM1
0

50

100

150

200

250

Time [min]

300

350

400

450

㪫㪸㪹㫃㪼㪙㫆㪻㫐㩷㫄㫆㫍㪼㫄㪼㫅㫋㩷㩿㪝㪋㪀㩷
Matrix element
a1,2 (REM →Wake)

a1,3 (Non-REM1→Wake)
a1,4 (Non-REM2→Wake)
a1,5 (Non-REM3→Wake)
a1,6 (Non-REM4→Wake)(b) Transition of sleep stage probability when no body

Transition matrix
Ab

An

0.034

0.014

0.000
0.008
0.030
0.044

movement occurs(KI㩷 㪫㫉㪸㫅㫊㫀㫋㫀㫆㫅㩷㫆㪽㩷㫊㫃㪼㪼㫇㩷㫊㫋㪸㪾㪼㩷㫇㫉㫆㪹㪸㪹㫀㫃㫀㫋㫐㩷

0.000

When body movement occurs, the most probable stage is

0.006

Non-REM2 with a probability of 0.80 and the second most

0.000

when no body movement occurs, the most probable stage

0.003

probable stage is REM with a probability of 0.11, whereas
is Non-REM2 with a probability of 0.56, the second mostprobable is Non-REM3 with a probability of 0.25, the㧙㧙third most probable is Non-REM4 with a probability of

ǳ1= -1.0910-2,

ǳ2= -3.5210-2,

These tendencies show that the sleep transition equation

ǳ5= -9.0310-3,

ǳ6= -6.7910-3

0.10 and the fourth is REM with a probability of 0.07.

ǳ3= -3.4410-4,

when no body movement occurs demonstrates deeper

sleep, whereas that when body movement occurs

ǳ4= -6.9310-3,

and we select the genes as characteristic roots of thedemonstrates shallower sleep.

observer or the eigenvalues of A-KI. Thus, the gain matrix

K is determined as follows:㪦㫇㫋㫀㫄㪸㫃㩷 㪼㫊㫋㫀㫄㪸㫋㫀㫆㫅㩷 㫆㪽㩷 㫆㪹㫊㪼㫉㫍㪼㫉㩷 㪸㫅㪻㩷 㪺㪿㪸㫉㪸㪺㫋㪼㫉㫀㫊㫋㫀㪺㩷㫉㫆㫆㫋㫊㩷

When body movement occurred:calculated if the eigenvalues ǳ1-ǳ6 of A-KI are given.
Among the various values ofǳ1-ǳ6, we select those sothat the coincidence rate of measurements y(k) and themost probable state in xˆ (k ) estimated by the observer
for the entire measurement time for the test subjects inGroup 1 is maximum, using the genetic algorithm.

When no body movement occurred:0.344
0.167
0.000Kn = 
0.5000.0000.000

Assuming 30 individuals that have ǳ 1- ǳ 6 in eachgeneration, digenesis of 200 generations is continuous.

Figure 5 shows the highest and the average coincidence
rates between the measurements y(k) and the mostprobable state in xˆ (k ) estimated by the observer.
䋺Maximum agreement ofeach generations

0.034 0.000 0.008 0.030 0.044
0.764 0.318 0.023 0.000 0.011
0.106 0.273 0.002 0.067 0.066

0.131 0.409 0.959 0.127 0.066
0.000 0.000 0.015 0.741 0.198

0.000 0.000 0.000 0.044 0.622 (7)0.344
0.103
0.154
Kb = 
0.4100.0000.000

Here, we determine the gain matrix K, which is uniquely

62.3%

0.014 0.000 0.006 0.003 0.000
0.901 0.167 0.014 0.000 0.000
0.014 0.333 0.001 0.000 0.000

0.106 0.500 0.958 0.059 0.000
0.000 0.000 0.028 0.888 0.145

0.000 0.000 0.000 0.059 0.862 (8)Scorer 1

Body movement

62

Wake
REM
NR1
NR2
NR3

60

Agreement[㧑]

]
[
e
e
r
A

NR4

100

150

200
250
time [min]

300

200

250
time [min]

300

200
250
time [min]

300

350

400

450

500

450

500

450

500

Scorer 2

55.7%

Body movement

㧦Mean agreement ofeach generations

56

50(a) Sleep stage judged by Scorer 1

maximum ofmean agreement

58

0

Wake

REM
NR1
NR2
NR3

54

NR4

0

50

0

20

40

60

80

100 120 140 160 180 200

Generation

150

350

400

(b) Sleep stage judged by Scorer 2

151st generation

52

100

Scorer 3

Body movement

Wake

REM
NR1

㪝㫀㪾㪅㪅㩷㪌㩷 㩷 㪩㪼㫊㫌㫃㫋㩷㫆㪽㩷㪾㪼㫅㪼㫋㫀㪺㩷㪸㫃㪾㫆㫉㫀㫋㪿㫄㩷

NR2
NR3

NR4

In Figure 5, increase in coincidence rate is saturated at

0

50

100

150

350

400

around 40 generations. Among 200 generations, the genes(c) Sleep stage judged by Scorer 2

maximum value of 55.7% and the highest coincidence rate

(KI㩷 㪪㫃㪼㪼㫇㩷㫊㫋㪸㪾㪼㫊㩷㫁㫌㪻㪾㪼㪻㩷㪹㫐㩷㫊㪺㫆㫉㪼㫉㫊㩷of ǳ 1- ǳ 6 in the 151

stgeneration have an averageis 62.3%. The genes ofǳ1-ǳ6 of this generation are:㧙㧙Table 5 shows the kappa statistics between scorers for㪫㪸㪹㫃㪼㩷㩷 㪢㪸㫇㫇㪸㩷㫊㫋㪸㫋㫀㫊㫋㫀㪺㫊㩷㪼㫍㪸㫃㫌㪸㫋㫀㫅㪾㩷㫉㪼㫃㫀㪸㪹㫀㫃㫀㫋㫐㩷㫆㪽㩷㫁㫌㪻㪾㫄㪼㫅㫋㩷uScorer 1

Scorer 1

Scorer 2

Scorer 2

Scorer 3

Scorer 3

vs

vs

vs

The sleep stages judged by the three scorers are

A1

22

0.85

0.88

0.76

B1

21

0.83

0.89

0.75

A2

r

o
u
p
1

B2
C1
C2
C3

23
23

0.81
0.85
0.76

0.89
0.92
0.89
0.78

0.81

0.87
0.79
0.76
0.81

0.86

0.82

D4

23

0.81

0.84

0.78

D2

22

0.80

0.88

0.75

E1

18

0.86

0.81

0.82

F2

23

0.74

0.85

0.82

Mean
S.D.

G

D3

o

G1

2

23

0.86

0.86

0.81

J1

p

22

0.78

22

H1

u

22

D1

F1

r

and the reliability of judgment is high.

㩷 㩷 㪜㫊㫋㫀㫄㪸㫋㫀㫆㫅㩷㫆㪽㩷㪪㫃㪼㪼㫇㩷㫊㫋㪸㪾㪼㩷㪸㫇㫇㪼㪸㫉㪸㫅㪺㪼㩷㫇㫉㫆㪹㪸㪹㫀㫃㫀㫋㫐㩷pG

sleep data in Groups 1 and 2. The statistics all exceed 0.74

22
23

25

22.3

0.85
0.77

0.81

0.81

0.91
0.83

0.88

0.87

0.84
0.85

0.85

0.81

1.5

0.04

0.04

0.04

23

0.81

0.94

0.85

E2

18

0.86

0.92

0.79

I1

22

0.88

0.90

0.79

J2

Mean
S.D.

23
25

22.2
2.6

0.79
0.90

0.85

0.05

0.81
0.90

0.89

0.05

employed as the measurements y(k) of the observer given

by Eq. (4) using the optimal gain matrix K. The estimatexˆ (k ) of the state vector x(k) provides the probability of

appearance of each sleep stage for each discrete time.

Figure 7 shows the markers for which body movement
occurred and the output from the observer for the subject.

The probabilities are shown by gray scale; the darker thelightness, the higher the probability. When we compare (a),(b) and (c) in Figure 7, the differences in deterministic

judgment in the confusin time intervals 10–50, 70–80,
170–180, 220–240, 350–380, and 450–470 min in Figure 6

are statistically compensated. In a deterministic expression,these differ, but in a statistic expression, the probabilities
are shown.
Scorer 1
Body movement

REM
NR1
NR2
NR3
NR4

0.81

0.03

and Scorer 3 for the third night’s data for a subject. The
kappa statistics between Scorer 1 and Scorer 2 is 0.81, thatbetween Scorer 1 and Scorer 3 is 0.87, and that between
Scorer 2 and Scorer 3 is 0.81 for G1 and 0.85, 0.89 and

100

150

200

250
300
time [min]

350

400

450

0

500

Body movement

1

Wake

REM
NR1
NR2
NR3

0

50

100

150

200

250
300
time [min]

350

400

450

0.5

0

500

(b) Sleep stage appearance probability judged by Scorer 2㪪㫃㪼㪼㫇㩷㫊㫋㪸㪾㪼㩷㫁㫌㪻㪾㫄㪼㫅㫋㩷㪹㫐㩷㪩㪄㪢㩷㪤㪼㫋㪿㫆㪻㩷occurred and the sleep stages judged by Scorer 1, Scorer 2

50

Scorer 2

NR4

Figure 6 shows the markers for which body movement

00.5(a) Sleep stage appearance probability judged by Scorer 1

0.79
0.81

1

Wake

Probability

o

ject

Age

Probability

r

Sub-

Scorer 3

Body movement

1

Wake

REM

Probability

G

NR1
NR2
NR3
NR4

0.81 for G2. These are high and the trend characteristics

0

50

100

150

200

250
300
time [min]

350

400

450

500

0.5

0

(c) Sleep stage appearance probability judged by Scorer 3

are similar. However, Scorer 2 and Scorer 3 tend to judge

AWAKE more frequently than Scorer 1, and Scorer 2
frequently judged REM sleep. Thus, details of the㪝㫀㪾㩷 㪪㫃㪼㪼㫇㩷㫊㫋㪸㪾㪼㩷㪸㫇㫇㪼㪸㫉㪸㫅㪺㪼㩷㫇㫉㫆㪹㪸㪹㫀㫃㫀㫋㫐㩷

judgments are different.

㪸㫅㪻㩷㪹㫆㪻㫐㩷㫄㫆㫍㪼㫄㪼㫅㫋㩷㧙㧙As shown in Table 6, when we use the three most㧡㧚㪛㫀㫊㪺㫌㫊㫊㫀㫆㫅㩷The results appear to show that the proposed algorithm isa simple smoother or low-pass filter, but it is not. The

Luenberger observer is a state estimator for an object
whose dynamics are given by an a priori mathematicalmodel, and it is designed by using the a priori informationof the object as described in Section 3.3. Thus, state

estimation by the observer is different from simple

smoothers and/or low-pass filters which do not use the apriori information of the object. The observer has asmoother (low-pass filter) function, and the cut-off

probable judgments, the average value of the kappa

statistics among the three scorers is over 0.92 and is highly
reliable statistical judgment. Furthermore, the confusing
judgment by scorers as shown in Figure 7 can be shownstatistically, which is a more accurate description of thesleep stage. If the probability of the most probable sleep

stage is low, we can understand that the sleep is hardly
judged as one of the six categories.

㪫㪸㪹㫃㪼㩷㩷 㩷 㪢㪸㫇㫇㪸㩷㫊㫋㪸㫋㫀㫊㫋㫀㪺㫊㩷㫆㪽㩷㪼㪸㪺㪿㩷㫊㫃㪼㪼㫇㩷㫊㫋㪸㪾㪼㩷㪸㫇㫇㪼㪸㫉㪸㫅㪺㪼㩷㫇㫉㫆㪹㪸㪹㫀㫃㫀㫋㫐㩷frequency which shows the degree of smoothness can beGmatrix i.e., the characteristic roots of the observer itself.

omeasurement noise is filtered out. In this paper, thepcontrolled by the eigenvalues of the observer coefficient
These

valuesshouldbedetermined

so

that

the

eigenvalues are automatically determined by the genetic

r

A1
A2

of three scorers. The cut-off frequency is automaticallyB1and optimally determined to compensate the fluctuations
maximizing the coincidence rates of their judgment
results.

Table 6 compares the sleep stages by the three scorers forthe most probable stage; by the mean value of the kappa
statistic, that between Scorer 1 and Scorer 2 is 0.89, thatbetween Scorer 1 and Scorer 3 is 0.92, and that between

G
r

o
u
p
1

B2
C1
C2
C3

D1
D2
D4

Scorer 2 and Scorer 3 is 0.85 for G1 and 0.91, 0.92 andE1mean values of 0.85, 0.89 and 0.81 in Table 5.

F2

Scorer 1

Scorer 1

Scorer 2

Scorer 2

Scorer 3

Scorer 3

vs

ject

u

algorithm from the measurements i.e., the judgment resultsin judgment by the three scorers in the sense of

Sub-

vs

vs

1st

2nd

3rd

1st

2nd

3rd

1st

2nd

3rd

0.88

0.92

0.94

0.89

0.94

0.96

0.77

0.87

0.91

0.87

0.94

0.95

0.90

0.96

0.97

0.82

0.90

0.92

0.91

0.95

0.95

0.95

0.96

0.97

0.90

0.92

0.93

0.91

0.93

0.95

0.91

0.96

0.97

0.87

0.91

0.94

0.88

0.92

0.95

0.95

0.97

0.98

0.89

0.90

0.93

0.91

0.93

0.94

0.95

0.96

0.97

0.88

0.91

0.92

0.82

0.91

0.92

0.79

0.93

0.94

0.82

0.85

0.95

0.86

0.89

0.96

0.91

0.94

0.96

0.83

0.92

0.93

0.89

0.90

0.93

0.96

0.96

0.97

0.87

0.88

0.91

0.88

0.92

0.94

0.94

0.96

0.97

0.82

0.89

0.94

0.94

0.91

0.94

0.91

0.95

0.96

0.86

0.87

0.91

0.89

0.92

0.94

0.93

0.95

0.97

0.85

0.88

0.94

0.91

0.93

0.94

0.93

0.96

0.98

0.87

0.90

0.92

0.90

0.93

0.95

0.91

0.96

0.97

0.88

0.89

0.93

0.85 for G2. These are all higher than the corresponding

F1

Furthermore, when we select the sleep stage from the most

H1
J1

0.85

0.94

0.95

0.92

0.96

0.97

0.86

0.91

0.92

between Scorer 1 and Scorer 2 is 0.92, that between Scorer

Mean

0.89

0.92

0.94

0.92

0.95

0.97

0.85

0.89

0.93

S.D.

0.03

0.01

0.01

0.04

0.01

0.01

0.03

0.02

0.01

D3

0.92

0.96

0.97

0.95

0.97

0.98

0.92

0.94

0.95

0.87

0.90

0.93

0.93

0.95

0.96

0.79

0.87

0.89

0.90

0.92

0.96

0.92

0.96

0.97

0.85

0.89

0.92

0.91

0.94

0.95

0.90

0.95

0.96

0.86

0.90

0.91

J2

0.93

0.94

0.96

0.91

0.95

0.96

0.81

0.92

0.93

Mean

0.91

0.94

0.95

0.92

0.96

0.97

0.85

0.90

0.92

S.D.

0.02

0.02

0.02

0.02

0.01

0.01

0.05

0.02

0.02

probable and the second most probable, the kappa statistic
1 and Scorer 3 is 0.95 and that between Scorer 2 andScorer 3 is 0.89 for G1 and 0.94, 0.96 and 0.90 for G2.
Similarly, when we select the sleep stage from the mostprobable, the second most probable and third mostprobable, the kappa statistic between Scorer 1 and Scorer 2
is 0.94, that between Scorer 1 and Scorer 3 is 0.97 and thatbetween Scorer 2 and Scorer 3 is 0.93 for G1 and 0.95,

0.97 and 0.92 for G2. The kappa statistics are increased.

G
r

o
u
p
2

For the judgment of the sleep stage for G2, which we did

not use for estimating the sleep stage transition matrix, thereliability is improved.

㧙㧙

E2

G1
I1

㧢㧚㪚㫆㫅㪺㫃㫌㫊㫀㫆㫅㩷㪩㪼㪽㪼㫉㪼㫅㪺㪼㫊㩷This paper describes a novel sleep stage transition[1] R. Kawahara, H. Maeda and S. Yoshioka “Sleep

equation in which the transition matrix is switched

disorder as present age disease,” Nippon-Hyoron-Sha

depending on whether or not body movement occurs

Co., Ltd., 2000.

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rhythm sleep disorder,” Shinkoh Igaku Shuppan Co.,An observer was built to estimate the probability of howLtd., 2003.

surely the sleep belongs to the six different stages. Sleep[3] Ministry of Economy, Trade and Industry of Japan.

stages judged by three different scorers were compared

Health service creation support project report, Marchusing kappa statistics.

2005. Available from:If we select the sleep stage from the most probable three

http://www.meti.go.jp/policy/servicepolicy/contents/hea

judgments on the group of data by which the transition

lthcare-service.pdf

matrix was estimated, the kappa statistic between Scorer 1[4] A. Rechtschaffen and A. Kales, Eds. “A Manual of

0.92, and that between Scorer 2 and Scorer 3 is 0.85.

System for Sleep Stage of Human Subjects,”

and Scorer 2 is 0.89, that between Scorer 1 and Scorer 3 isStandardized Terminology, Techniques and Scoring

Furthermore, for sleep not used to estimate the transition

Washington, D.C.; Public Health Service, U.S.

matrix, the kappa statistic between Scorer 1 and Scorer 2 isGovernment Printing Office, 1968.

0.91, that between Scorer 1 and Scorer 3 is 0.92 and that[5] C.W. Whitney, et al. “Reliability of Scoring

between Scorer 2 and Scorer 3 is 0.85. The reliability of

Respiratory Disturbance Indices and Sleep Staging,”

judgment is high and accurate in the sense that the results

Sleep, 21, 7, 749-57, 1998.

are statistically described.

[6] B. Kemp “A Model-Based Monitor of Human Sleep

sleep mathematical model using the sleep data of young[7] M. Schwaibold “Artificial intelligence in sleep analysis

The sleep modes change as people get older. For example,

classification,” Biomed Tech 46, 129-32, July 2001.

In this paper, we built the sleep transition equation, a

healthy subjects.

the occurrence rate of Non-REM4 sleep decreases almost
linearly in proportion to age. Thus, sleep transition mightalso change as people get older. Therefore, in order toStages,” Biol. Cybern. 57: 365-378, 1987.

(ARTISANA)-modeling visual processes in sleep[8] Handbook of Sleep Science and Sleep Medicine:

Asakura (The Japanese Society of Sleep Research),
1998.

apply the proposed algorithm, the sleep transition matrix[9] O’Reilly J. “Observer for linear system,” Academic

Furthermore, the sleep stages vary much more for subjects[10] D. E. Goldberg “Genetic Algorithms in Search,must be defined and estimated for different generations.
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must be prepared for each sleep disorder.

Press, 1983.

Optimization

andMachineLearning,”

Academic Publishers, Boston, MA, 1989.

Kluwer[11] Watanabe K. et al. “Noninvasive Measurement ofHeartbeat, Respiration, Snoring and Body Movementof a Subject in Bed via Pneumatic Method,” IEEE
Trans. BME, Vol. 52, No. 12, pp. 2100-2107, 2005.

㧙㧙