Sleep, 7(1):3-17
© 1984 Raven Press, New York
The Detection of Behavioral State Cycles and
Classification of Temporal Structure in
Behavioral States
*Helena Chmura Kraemer, tWilliam T. Hole, and *Thomas F. Anders
*Department of Psychiatry and Behavioral Sciences, and fDepartment of Pediatrics,
Stanford University School of Medicine, Stanford, California, U.S.A.
Summary: Previous methods for the analysis of temporal structure in sleep
and other state time series have described cycles, rhythms, and semi-Markov
chains. Methods, however, have been subjective and arbitrary. We propose an
objective system of classification for these series, based on definitions of temporal structure which are consistent with those long used in the analysis of
quantitative series. An ordered sequence of statistical tests is described which
classifies observed behavioral state time series into four primary categories.
The system is illustrated with examples from normal infant sleep. The results
show that some infant sleep series are cycles, as previously reported, some
are semi-Markov chains, and some are neither. The proposed objective
methods promise consistency, clarity, and a richer understanding of behaviors
such as sleep. Key Words: Cycles-Rhythms-Sleep-Behavioral statesClassification - Methodology.
I,
Temporal organization of behavioral states, especially sleep states, has been described in terms such as "cycle," "rhythm," or "period." When recurrent phenomena
are measured by quantitative (interval or ratio) scales, methods for defining, detecting,
and describing temporal organization are readily available (1,2). In contrast, the
methods for analyzing temporal organization of categorical (dichotomous or nominal)
data such as sleep states have been varied, subjective, and arbitrary. Results of state
time series analysis have been inconsistent and unclear. Stratton (3), discussing infant
state cycles, concluded that "the literature is in fact devoid of any analysis which
would indicate a rhythmic character of states .... "
Terms with precise definitions in the physical and biological sciences are used in
unique or less stringent ways when describing states in the behavioral sciences (4).
The term cyclic, for example, has been applied to state alternations with no more
organization than a random number series. Yet scientists assume that cycles, periods,
and rhythms are highly organized temporal sequences.
Historically, the concepts of cycles and periodicity evolved from study of the physics
of vibrating strings, sound waves, and electrical signals. These phenomena involve
Accepted for publication September, 1983.
Address correspondence and reprint requests to William T. Hole, M.D., Stanford University Department
of Pediatrics, 520 Willow Road, Palo Alto, CA 94304, U.S.A.
3
H. C. KRAEMER ET AL.
4
nearly exact repetitions of patterns at constant time intervals. Definitions and methods
used in quantitative time series analysis incorporate this central property of repetition
at a fixed interval. For example, one definition of a cyclic or periodic time series
requires
a component with the property that fit + T) = fit) where T is the period of
the cycle. The word ("cyclic") is also used in a less exact sense to denote up
and down movements which are not strictly periodic. This usage is to be
deplored. (5)
This definition is exacting and identifies time series which demonstrate a high degree
of organization. Such perfect cycles are seldom seen in nature. Consequently, the
detection of cycles in time series analysis involves setting specific statistical limits
within which recurring phenomena may be considered cyclic. At the very least, it must
be shown that a series is more cyclic than randomly generated observations.
Consistent definitions and rigorous methods for state time series should require that
only series with repetition at fixed intervals within objective statistical limits be called
cyclic, periodic, or rhythmic.
Behavioral scientists, however, have used time series terminology loosely to describe
any seemingly recurrent pattern of behavioral states. The term cycle, for example, has
been used to describe the time span from the onset of one state to its next onset in
two-state series. The terms period or cycle length have been applied to the means of
such time spans (4). Such a time span is more appropriately termed a recurrence time
(5). Furthermore, since every two-state series consists of bouts of the alternating states,
this use suggests that every two-state time series is cyclic with a period equal to the
mean recurrence time.
Idiosyncratic definitions of cyclicity have also been attempted. For example, a state
series has been defined as cyclic if "the lengths of adjacent cycles (sic) are not independent but negatively correlated" (6). A two-state series in which each state recurs
every 30 min would be judged cyclic by most. Yet if there is a small random error in
measurement, the series would not be cyclic by this definition, because the correlation
between the adjacent recurrence times is zero.
A relatively small variance in recurrence times has been the basis of a claim of
cyclicity or rhythmicity (7,8). How small the variance must be, however, was not
defined. This results in a definition that is subjective and arbitrary.
Attempts have been made to distinguish cycles from renewal processes in state
series. A renewal process is a time series in which there is some recurring "renewal
event" such that all that occurs after each event is completely independent of all that
has occurred before it. The above example of alternating 30-min bouts with a small
measurement error describes both a cyclic process and a renewal process, since each
entry into each state is a renewal event. Attempting to classify the process as either
renewal or cyclic is problematic, since a state series may be one or the other, neither,
or both (6,9).
The difficulty in assessment of state time series has led some investigators to formulate new, objective classification and detection methods. These methods have generally applied statistical techniques developed for quantitative time series. They have
been only partially successful.
For example, a time series may be divided into successive time blocks (10). The
proportion of time spent in a state is computed. for each block. If the blocks are large
Sleep, Vol. 7, No.1, 1984
THE DETECTION OF BEHA VIORAL STATE CYCLES
5
compared with the state's recurrence time, the series of calculated proportions can be
treated as a quantitative time series. However, the smallest detectable period using
this method is twice the block length. Furthermore, if block length is large relative to
recurrence time, this method quickly loses resolution and power.
Another method -difficult to justify - uses weighted numerical state labels as scores
in a quantitative time-series analysis (11). Such labels are not necessarily interval, or
even ordinal, data. The sums and products of such "scores," which are the basis of
quantitative time-series analysis, may have little meaning.
More appropriate strategies for the analysis of state time series are those of Globus
(12) and Sackett (13), which were designed specifically for categorical data.
Globus developed an Index of Rhythmicity. He computed the percentage agreement
between time points at all possible intervals (2t,3t,4t, ... , nt/2 = TI2). The Index of
Rhythmicity is defined as the difference between agreement at the first peak and the
minimum agreement. If this index is large, the state series is said to be cyclic. The
period is defined as the interval at which the first peak of agreement occurs. Unfortunately, the magnitude of this index is influenced by the relative durations of bouts
of each state and by the length of observation relative to recurrence time. How large
the index must be to identify a significant cycle is not known. The method, therefore,
remains descriptive.
Lag Sequential Analysis (13) incorporates tests of statistical significance. Sackett
warns, though, that the analysis is based on too many nonindependent tests, making
false positive identification of cycles very likely.
We propose a new approach, expanding the methods, of Globus and Sackett, which
we believe is a conceptual advance in the study of state: time series. We (a) clarify and
specify the use of terms such as cycle or period in state time series; (b) provide an
initial classification schema; and (c) indicate objective methods for this classification,
based on identification of significant nonrandom temporal organization. All of our
methods employ standard statistical techniques or are adaptations of standard methods.
We have used the infant sleep records of normal-term babies, videotaped at home,
to illustrate our time series methods. The states Active Sleep, Quiet Sleep, and Awake
were scored from observation of the sleeping infant's behavior (14). Most examples
are from the longest continuous sleep of the night, as recently described by Anders
et al. (15).
DEFINITION OF TERMS
A state time series is most succinctly recorded as the durations of successive states.
Hypothetical examples illustrating this notation are shown in Table 1 (cf. 13). For
example, in the three-state series (Table 1, a) the subject spends 13 min in State A,
then 63 min in State C, 29 min in State A, 14 min in State B, etc.
In such a multi state series it is possible to study one state at a time by considering
only bouts (blocks of time in one state) and pauses (times not in that state). For State
B in Table 1, a, there is a pause of 105 min (13 + 63 + 29), a bout of 14 min, a pause
of 136 min (59 + 59 + 18), etc. Thus any multistate series can be studied as separate
two-state series, where the two states are defined as the presence or absence of one of
the original states. There is one such two-state series for each of the original states.
There is an advantage in studying multistate series in this way-essentially one state
at a time. If one of the states is poorly defined or poorly measured, the structure of
other well-defined and -measured states will not be concealed.
Sleep, Vol. 7, No. I, 1984
H. C. KRAEMER ET AL.
6
TABLE 1. Four hypothetical state series
a
States
Sequence no.
A
1
2
3
13
C
A
B
20
59
7
18
8
9
10
77
30
40
30
10
20
20
93
10
30
20
81
10
30
10
30
10
20
20
59
B
10
10
20
A
30
20
14
5
6
B
10
30
29
A
d
States
20
63
4
Classification:
B
c
States
b
States
10
50
30
10
10
etc.
etc.
etc.'
etc.
Simple
Structure,
Non-Cyclic
Organized
Transitions
(ACAB ,ACAB ,AC)
Complex
Structure,
Complex
Cycle
Simple
Structure,
Simple
Cycle
Complex
Structure
(trend)
In addition, there is a pattern of state-to-state transitions (e.g., ACAB, ACAB, AC
in Table 1, a) in state time series that provides' an opportunity to use the methods
developed for Markov chains (16-18).
Our system of classification is based on an ordered sequence of tests for statistically
significant organization in a state time series. In order to test for significance, it is
essential to know the random case. For state time series there are two relatively similar
random cases that must be considered: a Random States Series and a Random Bouts
Series.
A Random States Series is defined as one that completely mimics coin tossing. Both
the bout durations and the pattern of transitions are completely random, and the bout
durations have exponential distributions. A Random Bouts Series is one produced by
sampling several different distributions (not all exponential) in random order, once
again producing random bout durations and random transitions. The difference is
minor, but not trivial. When a Random Bouts Series is used as the null hypothesis,
statistical methods become more complex. Frequently the nature ofthe bout duration
distributions must be known before a test can be proposed. For these reasons, we have
chosen to use a Random States Series whenever we require a model for a random time
series.!
In contrast to these random cases, maximum organization is seen in a Pure States
Cycle. A state series is defined as a Pure States Cycle when there is exact repetition
at a fixed time interval.
For an understanding of this definition and those that follow, it would be helpful to
1 A more complete discussion of the choice of null hypotheses and of the distribution of kappa under
Random States and Random Bouts null hypotheses, including simulation results, is available from the authors.
Sleep. Vol. 7. No.1. 1984
THE DETECTION OF BEHA VIORAL STATE CYCLES
7
study carefully the examples given in the tables, as references to them occur in the
text.
In a Pure States Cycle:
(a) The time series of bout durations for each state is composed of either constants
(Example c, Table 1) or is a cyclic time series (Example b, Table 1);
(b) There is a fixed pattern of state transitions (ABABAB ... in both examples);
(c) The set of bout durations for each state has only a few discrete values; i.e., it is
not continuous (10, 20, 30 in both examples).
In reality, Random States (or Bouts) Series and Pure States Cycles are hypothetical
extremes. They serve only to clarify the three characteristics that distinguish state time
series, namely (a) the nature of the bout durations, (b) the structure of the transitions,
and (c) the distribution of the bout durations.
A systematic schema for analyzing state time series, based on these distinguishing
characteristics, is summarized in Table 2. Terms used in Table 2 are defined in order
of their appearance. Methods of analysis are discussed in the section Empirical Classification of State Series.
A state series is defined as a Complex Structure if one or more of the time series of
bout durations is nonrandom. Such a series may be a Complex Cycle if the time series
of bout durations is itself cyclic (Table 1, b). Alternatively, such a time series may
demonstrate a trend (Table 1, d) or a combination of trends and cycles. If a state series
is not a Complex Structure (i.e., if all the time series of state bout durations are
random), the state series is termed a Simple Structure (Table 1, a and c).
A Simple Structure is defined as a Simple Cycle if one or more of its states is cyclic.
TABLE 2. Classification of state time series
All series
/
Test
'\
Adequate for
Testing
Inadequate for
Testing
/ \
Complex Structure
Simple Structure
(Complex Cycles, /
\
1tends)
/
Simple Cycle
Non-Cyclic
(M>2) /
Organized Transitions
(Semi-Markov Chains)
\M>= 2)
Random Transitions
/
\
Random
Bouts
Organ~zed -
state senes
Random
States
!orgamzatlOn
- N? e~idence of-I
*Test each state for at
least 4 bouts of
duration > = 2t
[Empirical Classification of
State Series (a)]
*Test for Random Bout
Durations, e.g., Turning
Points Test
[Empirical Classification of
State Series (b)]
*Kappa Test for Simple
Cyclicity
[Empirical Classification of
State Series (c)]
*Test for Random Transition
matrix
[Empirical Classification of
State Series (d)]
*Test for exponential
distribution
[Empirical Classification of
State Series (e)]
Sleep, Vol. 7, No. I, 1984
H. C. KRAEMER ET AL.
8
A state is cyclic if it tends io repeat itself at a fixed interval. Note that a Simpte Cycle
differs from a Pure States Cycle, which must repeat exactly at a fixed interval.
More precisely, a Simple Cycle is a Simple Structure with a period (d) only if the
probability that the subject is in the same state at times separated by multiples of that
period exceeds the probability for any other time separation. (All symbols, such as d,
used in the text are listed with their definitions in Appendix 1.)
A Simple Structure which does not meet the criteria for a Simple Cycle is called
Non-Cyclic.
A Non-Cyclic multiple state series with nonrandom transitions is defined as an Organized Transitions Series (Table 1, a). A Semi-Markov chain is one such series (9).
A Non-Cyclic multiple state series in which there are random state transitions, or any
non-cyclic two-state series, is called a Random Transitions Series.
A Random Transitions Series, in which bout durations are exponentially distributed,
defines a Random States Series; otherwise, the series is a Random Bouts Series. Both
such temporal structures may be regarded as "noise" or totally random.
EMPIRICAL CLASSIFICATION OF STATE SERIES
To classify any state series according to the schema outlined in Table 2, one performs
an hierarchical analysis that searches serially for evidence of a Complex Structure, a
Simple Cycle, an Organized Transitions Series, a Random Bouts Series, or a Random
States Series. During the search, the probability of misclassification of the state series
depends on the length of the observation, the sampling interval, the number and choice
of statistical tests, the reliability of the observations, and the nature and clarity of the
underlying temporal structures. Potential errors are discussed in detail later (Consideration of Detection Errors). Criteria and methods are discussed here in the order in
which they are applied, shown in Table 2.
(a) Criteria for adequacy of series
If there are too few bouts of each state, no statistical test can distinguish a Complex
from a Simple Structure. The number of bouts required will depend on the types of
Complex Structure being sought and which specific test is chosen. Four bouts are
required to detect the simplest Complex Structure, a monotonic trend, with a onetailed 5% significance level. Only one of the 24 (four factorial) possible cases contradicts the null hypothesis, and 1/24 is less than 5%. Four bouts are not always adequate
for every test, but are the minimum required for any test. Therefore, if fewer than four
bouts of each state are observed in a series, it is classified as "Inadequate for Testing."
If a bout duration of any state is less than the sampling interval, missing bouts in
the observed state series are likely and the durations of all states may be biased.
Analysis of such time series is highly questionable. For this reason no observed bout
duration should be less than twice the sampling interval. In difficult cases, this criterion
may be met by choosing a very small sampling interval, by imposing appropriate
smoothing criteria (7,8), by reducing the total number of states, or by combining the
rarer states. Series failing to meet this criterion are also classified as Inadequate for
Testing.
(b) Complex Structure versus Simple Structure
By definition, a Complex Structure requires that one or more of the bout duration
Sleep, Vol. 7, No.1, 1984
THE DETECTION OF BEHA VIORAL STATE CYCLES
9
TABLE 3. A Complex Structure sleep record
Active
Quiet
17
59
28(peak)
25(trough)
16(trough)
29(peak)
34(peak)
General statistics:
Duration = 444 min
Sampling Interval = I min
Proportion Active Sleep (P) = 0.57
Median Active Bout = 28 min
Median Quiet Bout = 23.5 min
Recurrence Time = 51.5 min
22
Complex Structure:
Z (active) = 4.13 (p < 0.05)
Z (quiet) = 0.00 (ns)
26(trough)
22
50(peak)
28(peak)
32(trough)
9(trough)
33(peak)
22
15
series be nonrandom. There are many tests of randomness appropriate for this task.
One such test is the nonparametric Turning Points Test (19).
In this procedure one counts "peaks" and "troughs" in one state's observed duration series. A peak is defined as a bout duration greater than the bout durations
immediately before and after it, a trough, as one less than those on either side. The
number of turning points is the sum of the number of peaks and the number of troughs.
If there are N bouts in a series with TP turning points, then according to the null
hypothesis of random time series as an approximation:
Z
=
(TP - TP)/s
~
N(O,I)
where
TP = 2(N - 2)/3 and s2 = (16N - 29)/90
This test identifies both trends (too few Turning Points) and Complex Cycles (too
many Turning Points). It requires at least four bouts to detect trends and at least eight
bouts to detect Complex Cycles with a 5% significance level.
The test for Complex Structure is applied to each bout duration time series. If one
or more are significant, the state series is classified as a Complex Structure. Further
analysis, using established quantitative time series methods, may differentiate complex
cycles from trends.
Table 3 presents the record of a 6-month-old baby's longest sustained nighttime sleep,
consisting of Quiet and Active Sleep bouts. Peaks and troughs are indicated. The
number of Turning Points in the Active Sleep duration series is seven. Since the number
of Active Sleep bouts is nine,
TP
S2
z
= 2(9 - 2)/3 = 14/3 = 4.67
=
[16(9) - 29]/90 = 1.28; s = 1.13
4.67/1.13 = 4.13, p < 0.05
Sleep, Vol. 7, No. I, 1984
H. C. KRAEMER ET AL.
10
The Active Sleep bouts are therefore nonrandom. Results for Quiet Sleep (Table 3) are
not significant (z = 0.0). Since one duration series is nonrandom, this state series is
classified as a Complex Structure.
On examination, the record shows alternating long (median = 34 min) and short
(median = 17 min) Active Sleep bouts with relatively similar intervening (=24 min)
bouts of Quiet Sleep. The complex cycle's median length is 114 min, with each cycle
consisting of two Active (34 and 17 min) and two Quiet Sleep (24 min) bouts. There is
also a suggestion of a trend toward longer Active Sleep bouts later in the series (9). In
this example there is a significant Complex Structure which, with earlier methods, is
misrepresented as a cycle with a period (i.e., mean recurrence time) of 5l.5 min.
(e) Simple Cycle versus Non-Cyclic
A Simple Cycle is a Simple Structure with at least one state that cycles with a period
equal to its recurrence time.
For two-state series the period is estimated by the sum of the median bout and the
median pause duration. Medians are chosen rather than means, because observations
in the rather long tails of these distributions tend to distort the mean. For example, in
Table 4 the mean Active Sleep bout is 33.0 min, whereas the median is 42.5 min, largely
because of the two extremely short 8- and lO-min bouts. The sum of the medians is
used instead of the median recurrence time, since there is frequently one more bout
or pause than there are recurrence times. In Table 4, for example, there are five recurrence times, but six Active Sleep bouts. When sample sizes are so small, it is
important to use all available data.
The estimate of period (a) is rounded to the nearest multiple of the sampling interval
(in Table 4, 69.5 is rounded to 70 min, so d = 70) and the expected number of full
cycles (C) is calculated. This number is the largest integer less than or equal to the
number of observations, divided by the estimated period (in Table 4, 325170 = 4.6, so
C = 4). As mentioned above, at least four full recurrences are required for analysis.
The observations in the series are then arranged as a Raster Plot with the width
equal to the estimated period. This is a matrix, the first row containing the first d
observations, the second row containing the next d observations, and so forth. This
format graphically presents the state cycles and facilitates their analysis.
Figure 1 shows a Raster Plot of the sleep state series presented in Table 4. If this
series were a Pure Cycle of period 70 min, the observations in each column would be
identical; i.e., the proportion of pairwise agreements for the presence of the state within
a column would be 1.0. On the other hand, if the state series were random, the agreement between pairs of observations within the same column would be the same as the
agreement between any randomly selected observations.
The kappa coefficient (K) is a statistic that measures such agreement, although it has
not been used in this context before (20-22). It is expressed as: K = (P - Q)/(l - Q)
where P is the proportion of pairs in the same column which are in agreement and Q
is the overall proportion of pairs in agreement. The kappa coefficient is zero for a
Random States Series and increases to 1.0 for a Pure Cycle.
In more mathematical terms,
d
K
= 1
2:
j=1
Sleep, Vol. 7, No.1, 1984
Pi! - P)/(dp[1 - p])
11
THE DETECTION OF BEHA VIORAL STATE CYCLES
t
1
2
(minutes)
3
4
567
.... I.... o.... I.... o.... I.... o.... 1.... o.... I.... o.... I.... o.... I.... 0
--
........................... .
............................ .
*************************
FIG.1. A raster plot of the sleep state series presented in Table 4 . •
not observed. Period (d) = 70 min. Number of cycles (C) = 4.
*=
=
Active Sleep; •
=
Quiet Sleep;
where Pj is the proportion of the observations in columnj in which the state is present,
and p, as before, is the overall proportion of observations in which the state is present.
Since ~ P/l - P)ld is a measure of the within-column variance of the dichotomous
obserVations and p(l - p) is a measure of overall variance, K measures the proportion
of total variance accounted for by a Simple Cycle of length l1. The statistic K functions
in a manner similar to spectral power in spectral analysis.
The kappa coefficient is also an intraclass correlation coefficient (23), and so is
comparable to the maximum autocorrelation coefficient in quantitative time series analysis. Both the Globus and Sackett procedures are based on measures similar to these.
To demonstrate that a two-state series meets the criteria for a Simple Cycle, kappa
must be significantly greater than zero. For a Random States null hypothesis, approximately
-
K
SK
~
n(O 1)
'
where
~[l
SK2
T
- 4p(l - P)(l _ td) + 2td(l _ td)]
p(l - p)
T
T
T'
TABLE 4. A Simple Cycle sleep record
Active
Quiet
8
25
46
27
41
28
49
General statistics:
Duration (D = 325 min
Sampling Interval = 1 min
Proportion Active Sleep (P) = 0.61
Median Active Bout = 42.5 min
Median Quiet Bout = 27.0 min
Recurrence Time = 70 min
17
44
30
10
Complex Structure:
Z (active) = 0.39 (ns, 6 bouts)
Z (quiet) = 0.00 (ns, 5 bouts)
Simple Cycle:
K = 0.78
Z = 20.0 (p < 0.01)
Sleep, Vol. 7, No.1, 1984
H. C. KRAEMER ET AL.
12
Thble 4 illustrates the manner in which these procedures are applied serially to the
sleep of a normal 8-week-old infant.
There is no evidence for Complex Structure (Z = 0.39 and Z = 0.00, but there are
fewer then eight bouts for each series). The kappa coefficient, however, is 0.78 (ZK =
20.0, p < 0.00. This significant kappa suggests a Simple Cycle of 70 min, comprising
an Active Sleep duration of about 43 min and a Quiet Sleep duration of 27 min. Figure
1 is the Raster Plot of this child's data, and visually confirms the presence of a strong
Simple Cycle.
(d) Organized Transitions versus Random Transitions Series
Non-cyclic two-state series are always alternating bouts of the two states. If there
are more than two states, a non-cyclic multiple-state series mayor may not display
organized transitions between the states (cf. Table 1, a).
To demonstrate organization at this level, one estimates the expected proportions of
state-to-state transitions when transitions are random. These proportions are
%=
(no. transitions to J)
+ (no. transitions from j)
2(total no. of transitions)
where j is any of the possible states. The expected number for each possible transition
in the random case is then
E· =
lJ
o·/.
1 - qi
i#-j
where 0i. is the number of transitions out of state i. We use Oij to represent the number
of observed transitions from state i to state j, and Eij to represent the expected number
of transitions from state ito statej. We define Eii = 0 for all i. Under the null hypothesis
of random transitions the test statistic [sum of (observed - expected)2/expected] is
x2
=
2: (Oi)
- Ei)?/Ei}
i""J
This statistic, X2 , has a chi-squared distribution with (M - 1)2/2 degrees of freedom.
If this test is to be used, no more than 20% of the expected values (Eij) may be less
than 5 and none may be less than 1 (24). If the test proves significant, the distinction
between one-step versus multiple-step semi-Markov chains may be pursued (cf. 9).
Thble 5 presents the nighttime state series of a 20-week-old infant, including Awake
as well as Active and Quiet Sleep. The calculations described above are detailed in
Table 5. Here, X2 = 9.26 (p < 0.05); but despite the relatively large number of transitions (38), the expected values do not meet the minimum criteria for valid testing;
33% of expected values (2/6) are less than 5. The transitions do show a pattern typical
of younger infants: there are no direct transitions between Quiet Sleep and Awake.
The bout series for Active Sleep in Table 5 has a significant Turning Points test (Z
= - 2.47, p < 0.05). Thus this series is classified as a Complex Structure. If the
Sleep, Vol. 7, No.1, 1984
r
TABLE 5. An Organized Transitions sleep record
Awake
Active
11
39
16
24
25
6
3
11
43
46
12
II
,
I
From QS
From AS
From AW
5
=
38
To
AS
To
AW
x
11
0
11
x
8
0
8
x
To
QS
To
AS
To
AW
Expected:
7
18
7
From QS
From AS
From AW
26
20
x
11.0
2.9
7.7
x
5.1
3.3
8.0
x
35
31
8
21
13
X2 = 9.26, df = 2
But two cells (33%) are <5, and
data are thus not adequate for
testing.
35
6
f'
Random Transitions:
Number of transitions
Observed:
To
QS
9
I
I
General statistics:
Duration (D = 579 min
Sampling Interval = I min
Proportion Active Sleep (P) = 0.51
Median Active Bout = 12 min
Median Quiet Bout = 8 min
Recurrence Time = 20 min
33
23
10
I
Quiet
14
I
I
13
THE DETECTION OF REHA VIORAL STATE CYCLES
4
transitions were significantly nonrandom, they would provide an example of subclasses
within our schema.
(e) Random Bouts versus Random States Series
As noted earlier, Random Bouts and Random States series display little organization.
Both might be regarded as examples of "pure noise." If one is interested in distinguishing between the two, the analysis is easily done.
According to the Random States null hypothesis, the durations of each state are
independent and have an exponential distribution. Goodness of Fit tests such as the
chi-square test, the Kolmogorov-Smirnov'test (e.g., 24), or a (-test comparing the
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H. C. KRAEMER ET AL.
14
observed and expected mean durations can be used to make the distinction. None of
our sleep records have Random States structure.
Within each of the categories discussed above, finer subclassifications are possible.
For example, in a Complex Structure each single state having a nonrandom bout series
may be cyclic with or without harmonics and may include trends as well. There also
may be structure to the pattern of transitions among states within a Complex Structure
or a Simple Cycle. Our four proposed categories describe the trunk and major branches
of a classification tree. Smaller branches will define these subclasses clearly.
CONSIDERATION OF DETECTION ERRORS
We have proposed a new nosology for the study of state time series: Complex
Structure, Simple Cycle, Organized Bouts, and Random Bouts/States. The four classes
can be objectively identified, using clearly defined criteria, in a manner consistent with
the methods used in time series analysis in other fields.
There are certain risks in such testing that cannot be ignored. To classify a two-state
series one may use up to three tests; to classify a series of M states, up to (2M + 1)
tests. Each test has a probability of a false positive result (Type I error), reflected in
its significance level. Traditionally, a significance level of 5% is chosen for each test.
To limit the overall probability of this error to 5%, each test should use a level of 0.05/
3 for two states or of 0.05/(2M + 1) for M states. The more stringent criteria would
reduce the chance of detection of true temporal organization (Type II e r r o r ) . .
The power (absence of Type II error) of tests to distinguish Complex from Simple
Structure depends on the number of bouts of each state. As we have shown in our
examples of sleep state recordings, the number of observed bouts is often small. The
chance of missing true Complex Structure with small numbers of bouts is large.
The power of the kappa test to distinguish a Simple Cycle from a Non-Cyclic series,
if there are at least four bouts and pauses of the state, depends more strongly on the
number of sampling intervals and is therefore greater.
The power of the test to distinguish an Organized Transitions Series from a Random
Transitions Series depends on the number of states and the number of observed transitions. It will be limited, particularly when the number of states is large, or if some
of the states are rare. In infant sleep data, it is unusual for a single night to contain an
adequate number of transitions to allow testing. Tests on pooled data for successive
nights or different babies are, however, possible. For example, state-to-state transition
matrices may be compiled for several consecutive nights and added together to produce
one composite matrix. The test for Organized Transitions may be applied to this composite matrix.
The researcher must carefully weigh the balance between Type I and Type II errors
in applying these tests, as well as in interpreting results. We have elected, in our
examples, to use a 5% significance level for each of the three tests. Others might
appropriately choose a more conservative approach.
CONCLUSIONS
The definitions and methods proposed in this paper offer conceptual advances for
the analysis of state time series. First, the definitions of terms are consistent with the
Sleep, Vol. 7, No.1, 1984
THE DETECTION OF REHA VIORAL STATE CYCLES
i'
15
accepted definitions of like terms in quantitative time series analysis. Second, the
standard statistical methods of quantitative time series analysis and Markov chain
methods are employed in conjunction wtih new methods. Third, classification is
achieved by objective testing rather than by subjective evaluation. If our proposed
classification and definitions are adopted, a term such as cycle could be used in the
behavioral sciences just as precisely, consistently, and meaningfully as it is in the
biological and physical sciences.
Furthermore, there is an added richness in the proposed approach. Since organization of states may exist in terms of either durations or transitions, or both, the use of
methods identifying both types of organization promises increased understanding of
state phenomena.
There is a price to be paid for these methodologic and substantive advantages. The
meanings of some traditional state time series terms would require modification. We
suggest that many studies reporting the p'resence of sleep "cycles," rest-activity "cycles," or other state "rhythms" would need reexamination.
By our definitions, for example, the data that Stern et al. (7,8) present are not sleep
cycles, but are, instead, sleep recurrence times. No evidence of cyclicity was presented, although cyclicity (in particular, a Simple Cycle) was assumed to be present.
Bowe and Anders (9) used a Semi-Markov (Organized Bouts Series) model to describe infant sleep. They had tried the Globus method but decided, on subjective
grounds alone, that "it was difficult to ascribe significance to these seemingly irregular
perturbations." In effect, they assumed that the state series were non-cyclic.
In reanalyzing the original Anders data using the methods outlined here, we found
evidence of Complex Structure for some infants, contradicting the assumptions of both
Stern et al. and Bowe and Anders (for example, see Table 3). For other infants, we
have found Simple Cycles with periods similar to those found by Stern et al. (for
example, see Table 4). For still other infants, even with long sleep periods and many
bouts, we find no evidence supporting anything other than Random Bouts, as reported
by Bowe and Anders.
As infants mature, there may be developmental changes, not only in the proportions
of Quiet and Active Sleep (7,8,25) but also in temporal organization (9,15,26). The
nature and degree of organization will be clarified by objective classification, and the
patterns of developmental change may lead to further understanding of sleep phenomena.
The sleep of premature infants, brain-injured infants, or other infants at risk may
differ from the sleep of normal children in temporal organization (27). Differences may
be found in the ability to sleep in developmentally appropriate ways. Similar differences
in organization may be found in the sleep of the elderly, of psychiatrically disturbed
patients, of sleep-disturbed patients, or of subjects under biological or emotional stress.
The methods presented here provide tools to address these and other questions raised
in basic research and the clinical study of any state time series.
I
Acknowledgments: We appreciate the support of the following organizations: The National
Institute of Mental Health, Grant MH 31845 to Dr. Dement, for Dr. Kraemer; the Robert Wood
Johnson Foundation, to Dr. Hole; and the William T. Grant Foundation, to Dr. Anders.
Sleep, Vol. 7, No. /, 1984
H. C. KRAEMER ET AL.
16
Appendix!
Number of cycles
d Period estimate
True period
d
Eij Expected number of transitions
from state i to state j
iJ One of possible states
kappa
K
M Number of states
N Number of bouts
Oij Observed number of transitions
from state i to state j
Pj Percent agreement in column j
C
Proportion of time in the state
Qj Percent agreement by chance
Transition probabilities
q
Standard deviation
S
S State indicator
Length of observation
T
Sampling interval (epoch length)
t
TP Number of turning points
X2 chi-square
ZK Z-score for kappa
Z Z score
p
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