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Emotion as a Thermostat: Representing Emotion Regulation Using a Damped Oscillator Model.


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The authors present in this study a damped oscillator model that provides a direct mathematical basis for testing the notion of emotion as a self-regulatory thermostat. Parameters from this model reflect individual differences in emotional lability and the ability to regulate emotion. The authors discuss concepts such as intensity, rate of change, and acceleration in the context of emotion, and they illustrate the strengths of this approach in comparison with spectral analysis and growth curve models. The utility of this modeling approach is illustrated using daily emotion ratings from 179 college students over 52 consecutive days. Overall, the damped oscillator model provides a meaningful way of representing emotion regulation as a dynamic process and helps identify the dominant periodicities in individuals' emotions.
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Emotion as a Thermostat: Representing Emotion Regulation Using a
Damped Oscillator Model
Sy-Miin Chow and Nilam Ram
University of Virginia
Steven M. Boker
University of Notre Dame
Frank Fujita
Indiana University South Bend
Gerald Clore
University of Virginia
The authors present in this study a damped oscillator model that provides a direct mathematical basis for
testing the notion of emotion as a self-regulatory thermostat. Parameters from this model reflect
individual differences in emotional lability and the ability to regulate emotion. The authors discuss
concepts such as intensity, rate of change, and acceleration in the context of emotion, and they illustrate
the strengths of this approach in comparison with spectral analysis and growth curve models. The utility
of this modeling approach is illustrated using daily emotion ratings from 179 college students over 52
consecutive days. Overall, the damped oscillator model provides a meaningful way of representing
emotion regulation as a dynamic process and helps identify the dominant periodicities in individuals’
Keywords: differential equation, oscillator, dynamic, emotion regulation, spectral analysis
Building on popular belief in the “blue Monday” phenomenon,
Larsen and Kasimatis (1990; see also Huttenlocher, 1992; Reid,
2000) presented early evidence advocating the existence of a
weekly cycle in individuals’ daily mood fluctuations. Using spec-
tral analysis, these researchers found a strong weekly rhythm in a
group of college students’ average hedonic level. In broader con-
texts, rhythmicity has also been examined in relation to diurnal
changes in mood and circadian activity (e.g., Larsen, 1985a; Mur-
ray, Allen, Trinder, & Burgess, 2002; Rusting & Larsen, 1998) and
seasonal affective disorder (e.g., Johansson et al., 2003). As more
emerging evidence has shown clear associations between these
cyclicities and other key aspects of life (e.g., Brown, 2000; Pet-
tengill, 1993), a closer examination of the nature and determinants
of these dynamic processes is imperative.
Despite ample research suggesting the existence of different
physiological and affective cycles in everyday life (e.g., Brown,
2000; Stone, 1985), methodologies amenable to the modeling of
cyclic change are still lacking. Even though researchers have
begun to incorporate longitudinal designs into the study of affect
(e.g., Diener, Fujita, & Smith, 1995; Eid & Diener, 1999), the
focus on a linear notion of change is still prominent in these
studies. With the exceptions of spectral analysis (e.g., Larsen &
Kasimatis, 1990) and other related frequency-domain and time-
series analyses, most of the dominant longitudinal methodologies
(e.g., growth curves and hierarchical linear models; Bryk & Rau-
denbush, 1987; McArdle & Epstein, 1987; Meredith & Tisak,
1990) have been used to represent linear change, even though several
nonlinear extensions of these methodologies have been proposed
in the psychometric literature (e.g., Browne & du Toit, 1991).
In a recent theoretical model of mood regulation, Larsen (2000;
see also Carver & Scheier, 1982, 1990) proposed that mood
regulation is, by nature, a dynamic process. A weekly mood cycle,
for instance, imparts that emotions are dynamic—they exhibit
specific patterns of change over the course of the week. Following
Larsen’s (2000) lead, we present in this article a damped oscillator
model (Boker & Graham, 1998; Nesselroade & Boker, 1994) that
provides a direct mathematical basis for testing this mood regula-
tion model. In our presentation, we discuss the basic elements of
the model, including intensity, rate of change, and change in the
rate of change as they pertain to the study of emotion. We also
highlight some of the similarities and differences between this
differential equation modeling approach and other more widely
known approaches, such as growth curve model and spectral
analysis. Finally, we illustrate through an empirical example the
potential utility of the approach as a tool for understanding emo-
tion processes (e.g., interindividual differences in patterns of day-
to-day emotional variability; see Fleeson, Malanos, & Achille,
2002; Nesselroade & Baltes, 1979, for the rationales for such an
Sy-Miin Chow, Nilam Ram, and Gerald Clore, Department of Psychol-
ogy, University of Virginia; Steven M. Boker, Department of Psychology,
University of Notre Dame; Frank Fujita, Department of Psychology, Indi-
ana University South Bend.
Part of the writing of this article took place while while Sy-Miin Chow
was at the University of Notre Dame. Sy-Miin Chow and Nilam Ram
acknowledge the support provided by National Institute on Aging Grants
R01 AG18330 and T32 AG20500-01, both awarded to John R. Nessel-
roade. We thank John Nesselroade, Jack McArdle, other members of the
Institute for Developmental and Health Research Methodology at the
University of Virginia, and Anthony Ong for helpful comments on earlier
versions of this article.
Correspondence concerning this article should be addressed to Sy-Miin
Chow, who is now at the Department of Psychology, University of Notre
Dame, 108 Haggar Hall, Notre Dame, IN 46556. E-mail:
Emotion Copyright 2005 by the American Psychological Association
2005, Vol. 5, No. 2, 208–225 1528-3542/05/$12.00 DOI: 10.1037/1528-3542.5.2.208
Intensity, Rate of Change, and Acceleration in Emotion
The temporal aspect of emotion and its relatedness to individual
differences in emotion regulation have been discussed by several
researchers (e.g., Davidson, 1998; Larsen, 2000). The rate of
change and acceleration represent different temporal characteris-
tics of affective processes. Before presenting the mathematical
foundations of the damped oscillator model, we begin by clarifying
some basic terms and concepts pertinent to the differential equa-
tion model, including intensity (or level), rate of change, and
change in the rate of change (or acceleration).
Level or Intensity
Intensity (denoted as Yin the model equation to come), in the
context of the differential equation model presented here, repre-
sents the magnitude of displacement or deviation in an individual’s
emotion compared with some baseline affect level.
As certain
environmental cues or objects elicit a particular kind of emotion
(e.g., sadness) from an individual, the extent to which the individ-
ual’s level of emotion deviates from his or her baseline level
represents the intensity of that emotion. High intensities represent
large distances from baseline (regardless of direction), and low
intensities represent small distances from baseline. Here, intensity
is conceptualized and used explicitly to characterize states, rather
than as a personality trait. Stable individual differences in emotion
intensity observed across different situations and contexts (e.g.,
Moskowitz, 1982) may, in contrast, reflect a trait-like disposition
of affect intensity (Larsen, 1985b; Larsen, Diener, & Emmons, 1986).
Rate of Change
The rate of change (Y) represents the magnitude of change in
intensity over one unit of time (e.g., day to day, pretest vs. posttest,
year to year, etc.). In other words, rate of change describes the
change in emotion intensity in relation to time (i.e., first derivative
of emotion intensity with respect to time). In studies of affect,
individual differences in the rate of change have been found to
emerge independently of initial affect intensity (Hemenover, 2003)
and have been linked to attributes such as emotional clarity
(Salovey, Mayer, Goldman, Turvey, & Palfai, 1995) and trait
hostility (Fredrickson et al., 2000). Most of the findings with
regard to rate of change apply primarily to long-term developmen-
tal changes (e.g., Kim, Conger, Lorenz, & Elder, 2001; see
McArdle & Nesselroade, 2003, for a review). This concept of
change, however is applicable to short-term, within-person vari-
ability in emotion as well.
Acceleration or Change in Rate of Change
Whereas the rate of change describes how much an attribute
changes over time (i.e., change in intensity), acceleration indicates
how fast the attribute changes over the same amount of time—
thus, change in the rate of change (i.e., the second derivative of
intensity with respect to time). Upon hearing depressing news, for
example, an individual is likely to show a higher level (i.e.,
intensity) of sadness than his or her usual set point. The amount of
change in sadness (i.e., rate of change) may differ as a result of the
individual’s degree of neuroticism. The speed with which this
change takes place, by contrast, taps into the acceleration aspect of
the individual’s emotion. The nature of a process as conveyed by
the amount of acceleration is somewhat analogous to Davidson’s
(1998) concept of the “rise time” of an emotion, and he theorized
that abnormalities in the rise time (or other temporal characteris-
tics) of an individual’s emotion may be indicative of emotion
The concept of acceleration in emotion can be illustrated more
concretely using the following example. The speed shown on a car
speedometer (representing the rate of change) indicates the pro-
jected distance traveled by the car in one unit of time (e.g., 60
mph). However, the car may be speeding up or slowing down. This
change in the rate of change (i.e., acceleration or braking) is
indicated by how quickly the needle on the speedometer is moving.
In the context of emotion, suppose one must give an important
presentation the next day. It is possible, perhaps probable, that
one’s anxiety would increase. Assuming that this change in anxiety
level is assessed on an hourly basis, the amount of increase in
anxiety per hour would constitute the rate of change in anxiety.
Initially, there may be a steady increase in anxiety level that
persists for hours. However, as the presentation hour approaches,
one’s anxiety might really kick in, increasing more rapidly (accel-
erating) as the moment of truth arrives. Such changes in the rate of
change are explicitly included in the forthcoming model.
To further illustrate the relationships between acceleration, rate
of change, and intensity, as they might apply to emotion processes,
we ask the reader to consider the scenario depicted in Figure 1.
This figure, representing the progression of sadness, was generated
using the damped oscillator model (the corresponding mathemat-
ical equation will be presented shortly). Initially (i.e., at Time 0),
the individual’s level of sadness (represented with a solid line) is
perturbed and deviates momentarily from his or her usual set point
(represented using a dashed line).
Over time, this heightened level
of sadness dissipates, and the individual slowly returns to his or her
usual baseline (Y0), overshooting a few times before settling
into an equilibrium state. Instead of visualizing merely the rela-
tionship between level of sadness and time (as in Figure 1), we
can, alternatively, examine the relationship between level (denoted
as Y) and rate of change (denoted as Y; see Figure 2A), and between
level and acceleration in sadness (denoted as Y; see Figure 2B).
Examining Figure 2A, one can see that as the level of sadness
decreases from an intensity level of 5.0 to about 1.0 (see also the
corresponding change in level in Figure 1 with respect to time), the
rate of change (Y) drops from 0.6 to about 6.0 (signifying a
greater decrease in sadness). After going past an intensity level of
1.0, the magnitude of negative change slowly decreases and after
Individuals may differ from one another in their baseline affect level.
For instance, individuals high on neuroticism might have a higher dispo-
sition for unpleasant affect at baseline. However, such differences in
individuals’ trait-like baseline (also referred to as set point by Lykken &
Tellegen, 1996) are not the focus of the oscillator model presented here.
Rather, our focus is on how and when individuals deviate from and return
to their baseline affect level, and their ability to minimize the discrepancies
between their current emotional state and their equilibrium level.
A set point is always located at Y(t)0. The focus of the oscillator
model, as stated previously, is not on individual differences in trait-like
baseline. Therefore, individual differences in trait-level baseline will have
to be removed (thus putting everyone’s baseline at 0) before a state-based
model like the oscillator model is fitted.
hitting an uncharacteristically low sadness level (around 3.0), the
individual begins to show elevations in sadness again (rate of
change becomes positive). This dynamic interplay between level
and rate of change continues until the individual returns to his or
her set point, in which case level, rate of change, and acceleration
would all be zero (where the two dashed lines cross in Figure 2A).
A similar inward spiral pattern is seen in the relationship be-
tween level and acceleration (see Figure 2B). As the individual’s
level of sadness decreases from 5.0 to 1.0, there is a deceleration
(i.e., Yis negative) in sadness. In other words, the decrease in
sadness unfolds at a progressively slower pace. Then, acceleration
in sadness (i.e., Ybecomes positive) takes place as the individu-
al’s sadness shows a steeper descent to Y⫽⫺3, and this accel-
eration subsequently propels the individual’s sadness level up to
the set point again.
In short, intensity rate of change and acceleration all manifest
changes but maintain a lawful relationship with one another as the
individual experiences the ebb and flow of daily emotions. Fur-
thermore, the damped oscillator model we present in this study
allows researchers to effectively separate individual differences in
emotion intensity from individual differences in frequency. In the
context of the damped oscillator model, frequency describes how
rapidly individuals experience ups and downs in their emotions,
and it is similar to the concept of emotional lability (Harvey,
Greenberg, & Serper, 1989; Larsen & Diener, 1985). As noted by
many researchers, the magnitude of intraindividual variability is
often operationalized by using an individual’s within-person,
across-occasion standard deviation (Eaton & Funder, 2001; Eid &
Diener, 1999). However, this index does not distinguish linear
change from cyclic change (Larsen, 1985a), nor does it identify
individuals who manifest gradual change in high amplitudes from
those who exhibit rapid ups and downs in small magnitudes (Eaton
& Funder, 2001). The latter can be pinpointed using the damped
oscillator model presented here. Perhaps of most interest to affect
researchers is the concept of damping portrayed by this model.
This will be elaborated in greater detail next.
Emotion Regulation Based on Homeostatic Principles
Several researchers (Bisconti, Bergeman, & Boker, 2004;
Carver & Scheier, 1982; Gross, Sutton, & Ketelaar, 1998; Headey
& Wearing, 1989; Larsen, 2000; Lykken & Tellegen, 1996) have
compared mood regulation and other regulatory behaviors with the
characteristics of a thermostat—as discrepancies arise between
one’s ideal set point and the current (e.g., emotion) state, a kind of
natural homeostasis kicks in and these discrepancies are succes-
sively minimized until one returns to the ideal set point. The speed
with which individuals self-regulate corresponds directly to the
idea of damping represented by the damped oscillator model. For
example, highly neurotic individuals, because of their heightened
sensitivity to stimuli that generate negative affect (Larsen & Kete-
laar, 1991), may not show any damping in their negative emotions
in specific settings, for example, when they are constantly per-
turbed by external cues. Other possible sources of individual
differences in ability to minimize discrepancies, such as the effec-
tiveness of different mood regulation strategies and differential
sensitivity to affect-relevant stimuli, have also been outlined by
Larsen (2000). In response to Larsen’s (2000) mood regulation
model, several researchers have suggested that these homeostatic
principles can arguably be applied to the case of pleasant emotions
Figure 1. Trajectory of a construct as it returns to its set point (i.e., zero) over time. Y(t)level at time t.
as well—that is, the case of a “happy thermostat” (Erber & Erber,
2000; see also other commentaries in the same issue, e.g., Freitas
& Salovey, 2000; Isen, 2000; Watson, 2000). In the current study,
we illustrate the possibility of fitting the thermostat model to both
pleasant and unpleasant emotions. The differential equation that
gives rise to the damped oscillatory behavior presented earlier will
be reviewed next.
Damped Oscillator Model
Given a hypothetical construct, arbitrarily denoted here as Y
(representing, e.g., sadness), the independent oscillator model
specifies the relationship among the intensity, rate of change, and
acceleration of sadness as
Y⬘共t, (1)
where Y
(t) represents the acceleration in sadness at time tfor
person i,Y(t) represents the rate of change in sadness at time tfor
person i, and Y
(t) represents the intensity of sadness at time tfor
person i. Thus, person i’s sadness can be said to evolve continu-
ously over time as a self-regulatory thermostat. The parameter
Figure 2. Plots reflecting the fluctuations in intensity over time as depicted in Figure 1 but with rate of change
against intensity of emotion (A) and acceleration against intensity of emotion (B). Y(t)level at time t;Y(t)
rate of change at time t;Y(t)acceleration in sadness at time t.
describes the frequency of oscillation.
The parameter
how promptly person ireturns to his or her set point after
Higher absolute values of
indicate more frequent fluctuations
(i.e., oscillations), or in other words, more rapid ups and downs in
sadness. Note that the
parameter is analogous to the frequency
parameter used by Larsen and Diener (1987) in that both param-
eters capture how rapidly a construct changes, rather than how
frequently a certain emotion or behavior is observed in the abso-
lute sense (Diener, Larsen, Levine, & Emmons, 1985; Zelinski &
Larsen, 2000). Thus, this parameter provides a direct representa-
tion for the concept of emotional lability (Harvey et al., 1989;
Larsen & Diener, 1985).
The parameter
, by contrast, governs the speed with which
person i’s sadness returns to or moves away from an ideal set
point. When
is less than zero, there is evidence of damping, that
is, there is a decrease in oscillation magnitude over time. When
is greater than zero, there is evidence of amplification (i.e., move-
ment away from equilibrium). Finally, a
estimate of zero indi-
cates that the pattern of fluctuations (i.e., oscillations) is constant
over the entire duration that the phenomenon was observed. In
sum, the model is parameterized such that it can represent a
particular construct’s frequency of oscillations and rate of return to
equilibrium (or rate of divergence from equilibrium) over time.
Dynamic Interplay Between Frequency and Damping
The dynamic interplay between frequency and damping deter-
mines how fast a dynamic process (e.g., emotion) returns to a
target equilibrium level (see, e.g., Nesselroade & Boker, 1994). In
fact, this can help shed light on the distinctions between mood and
emotion. Figures 3A–F represent trajectories generated using the
damped oscillator model with different frequency and damping
parameters, and one of two initial emotion intensities (either 5.0 or
5.0). Compared with Figure 2, the process depicted in Figures
3A and 3B unfolds at a slightly slower frequency, coupled with a
greater magnitude of damping. Thus, this may represent individ-
uals who have developed a heightened sensitivity to physiological
cues or other affect-relevant cues and are thus able to start regu-
lating their emotion very promptly upon perturbation (see other
examples in Larsen, 2000).
As the damping parameter,
, is increased to 4.0 in Figures 3C
and 3D, the discrepancies between current emotion state and one’s
set point are minimized even more effectively, and no signs of
oversuppression are shown. The resultant shape may represent, for
instance, the case of mood—instead of showing rapid ups and
downs, there is now a gradual but stable decline (or increase) in the
construct under study. This scenario can be used to represent other
more gradual changes in stable individual traits. Note that the time
scales used in this illustration are completely arbitrary and can
reflect, contingent upon one’s research question, processes that
unfold over minutes, days, years, or decades.
Finally, when damping is omitted in Figures 3E and 3F and the
frequency parameter is set to 0.8, the resultant trajectories cor-
respond to two 7-day cycles. As we will elaborate in further detail,
when there is no damping in the system and
is negative, the
integral solution to Equation 1 is identical to the sinusoidal
model—a cyclic model that serves as the core of spectral analysis
(for specific details on this integral solution see, e.g., Zill, 1993).
Under this specific constraint, the sinusoidal model can in fact be
viewed as a nested version of Equation 1. The respective strengths
of spectral analysis and the damped oscillator approach presented
in this study are discussed more thoroughly in Appendix A. Next,
we present an overview of how Equation 1 can be fitted as a
structural equation model using available software such as
LISREL (Jo¨reskog & So¨rbom, 1993), Mx (Neale, Boker, Xie, &
Maes, 1999) and Mplus (Muthe´n & Muthe´n, 2001).
Fitting the Damped Oscillator Model as a Structural
In this study, a fourth-order latent differential structural ap-
proach is used to fit the damped oscillator model as a structural
equation model (Boker, 2003; see similar but alternative ap-
proaches in Boker & Bisconti, in press; Boker & Graham, 1998;
Boker, Neale, & Rausch, 2004). This approach is functionally
similar to the approach used in growth curve modeling (e.g.,
McArdle & Epstein, 1987; Meredith & Tisak, 1990) and polyno-
mial regression (Cohen, 1968; Wishart, 1938). More specifically,
relationships among level, rate of change, and acceleration are
specified using a set of fixed factor loadings in similar fashion to
how fixed loadings are used to specify a construct’s patterns of
change (including, e.g., linear, quadratic, and other components of
change) in a growth curve model.
Given the novelty of this approach, we begin by first introducing
a simplified second-order estimation approach (Boker & Bisconti,
in press; Boker et al., 2004). We will then expand this to a more
complex fourth-order approach. A path diagram of the damped
oscillator model estimated using the second-order estimation ap-
proach is shown in Figure 4. We consider a hypothetical scenario
in which a particular emotion, sadness, is measured using three
indicators: unhappiness, depression, and loneliness. In the corre-
sponding path diagram, latent constructs and observed variables
are represented using circles and squares, respectively. Variances
and covariances among different variables are represented using
two-headed arrows, whereas factor loadings and regression paths
are represented using one-headed arrows. The latent factors SAD,
DSAD, and D2SAD represent the intensity of sadness and its
corresponding first and second derivatives (representing rate of
change and acceleration, respectively).
The second-order latent structural approach is essentially used
to specify the curve of a latent factor, SAD. Because the different
components of this model can have important implications for the
modeling of emotion, we will provide a general overview of the
elements depicted in Figure 4. The full model encompasses two
basic parts: a measurement model that specifies the relationship
between factors and their associated indicators (see Figure 4B) and
a dynamic model that imposes a certain functional curve on the
factors (see Figure 4A). A state-space embedding technique
(Boker & Bisconti, in press; Boker et al., 2004) is first used to lag
each individual’s time series against itself to create a matrix
containing the measurements at time t,t1, t2, and so on. This
technique is typically used in approaches wherein the data being
Formally, the frequency,
, is equal to 1
. Note that only
negative values of
are interpretable from a mathematical standpoint.
analyzed involve a large number of measurement occasions (e.g.,
P-technique model; Cattell, 1963; Cattell, Cattell, & Rhymer,
1947; and dynamic factor analysis model; Molenaar, 1985; Nes-
selroade, McArdle, Aggen, & Meyers, 2002).
Measurement model and the role of shocks. The measurement
model in Figure 4B is just a usual factor analytic model, in which
three indicators (unhappiness, depression, and loneliness) are used
to identify the latent factor SAD, and one of the factor loadings is
fixed at 1.0 for identification purposes. The terms S
and S
are shocks or state components associated with sadness
at each of the four particular time points. These shock terms have
potentially interesting meanings from a substantive perspective—
they capture a certain amount of common variance among the
three indicators at each particular time point and yet they do not
show systematic patterns of variation over time. Browne and
Nesselroade (in press) used the example of daily hassles to illus-
trate the role of these shock terms. More specifically, today’s
hassles can influence a person’s unhappiness, depression, and
For example, one may choose to reformat a time series (say, for the
indicator depression) with 100 measurement occasions into four blocks of data
points, dep
, dep
, dep
, and dep
. The vector of depression scores at
time t, dep
, would contain data from time t4 to 100, dep
would contain
data from time t3 to 99, dep
would contain data from t2 to 98, and
would contain data from time t1 to 97. Thus, the number of
manifest indicators included in the structural model in Figure 4 is significantly
reduced, and yet patterns of intraindividual variability and any covariations in
intraindividual variability among items are preserved.
Figure 3. Plots generated using the damped oscillator model with two initial emotion intensities (5.0 and 5.0,
respectively) and different frequency (i.e.,
) and damping (i.e.,
) parameters. A and B: The hypothesized
process shows gradual damping in magnitude over time under two initial conditions. C and D: The process
approaches its equilibrium state very quickly without overshooting. E and F: There is no damping in the system.
Y(t)level at time t.
loneliness all at the same time (thus contributing to some amount
of shared variance among these three items). However, the impact
of these daily hassles does not persist in a systematic manner
beyond today and hence they do not covary over time. Therefore,
these state components can be conceived as shocks to one’s
emotion status at a particular time point.
Dynamic model: The curve of a factor. The dynamic model in
Figure 4A is used to specify the trajectory of sadness over time. By
using the specific loadings in Figure 4, the latent components
SAD, DSAD, and D2SAD represent the intensity, rate of change,
and acceleration in sadness, respectively. The element tis a user-
specified scaling value that determines the time interval between
two successive measurement occasions. Therefore, if two indica-
tors (e.g., items or tests) are measured over different intervals,
different values of tcan be specified for each of these indicators to
incorporate unequal measurement intervals.
The parameters
are estimated as regressions of accel-
eration (D2SAD) on intensity (SAD) and rate of change (DSAD).
If the model depicted in Figure 4 is fitted to data from a single
individual over many measurement occasions, variances of the
components S_SAD, S_DSAD, and D2SAD (where S_ represents
shock to sadness) capture the magnitudes of systematic within-
person variability in level and rate of change of sadness over time.
The covariance between these two components represents the
amount of covariations between these two sources of intraindi-
vidual variability. If this model is fitted to data from multiple
individuals, variances of S_SAD and S_DSAD encompass both
intraindividual variability and interindividual differences in level
and rate of change. If that is the case, these two sources of variance
are, to some extent, confounded.
Finally, the component U_D2SAD is the residual (or uncer-
tainty) in D2SAD not accounted for by the damped oscillator
model. This modeling uncertainty is not attributable to measure-
ment errors in the indicators and essentially reflects the discrep-
ancy between one’s hypothesized model and the true mechanism
that underlies the dynamics of sadness. In general, all the variance
and covariance components in the dynamic model (i.e., Figure 4A)
capture systematic patterns of variation (or covariation) over time
that are quite distinct from the instantaneous shock components in
the measurement model. Researchers may choose to estimate or
omit some of these components to test their specific hypotheses of
interest. In this study, we omit the shock components and focus
Figure 4. A path diagram showing the relationships among level of sadness (SAD) and its corresponding rate
of change (DSAD) and acceleration (D2SAD) as hypothesized in the damped oscillator model for the dynamic
model (A) and the measurement model (B). Unhp unhappiness; Dep depression; Lone loneliness; t
time; afactor loading of SAD on depression constrained to be invariant over time; bfactor loading of SAD
on loneliness constrained to be invariant over time. S_SAD and S_DSAD shocks to an individual’s sadness
and to its corresponding first derivative, respectively; C
covariance between the shock terms
associated with SAD and DSAD; U_D2SAD residuals in D2SAD not accounted for by the model. S
, and S
are shocks to an individual’s sadness unique to a particular measurement occasion. This is in
contrast to S_SAD, which represents systematic shocks to an individual’s sadness that persist over all
measurement occasions.
instead on a more parsimonious model that captures only the
systematic variability over time. This will be elaborated further in
a moment.
In short, combining the measurement and dynamic models in
Figure 4 yields a growth curve model for the factor SAD that
conforms to the damped oscillator model. However, the current
approach is different from conventional growth curve models in
some subtle ways. These differences are detailed in Appendix B.
Similarities between the current approach and time series models
(or more specifically, autoregressive moving average models) will
also be highlighted briefly.
A fourth-order approach with no shock components. The
damped oscillator model defined in Equation 1 is formulated on
the basis of information up to the second derivative (i.e., acceler-
ation). Recently, Boker (2003) demonstrated that when one incor-
porates the third and fourth derivatives (i.e., D3SAD and D4SAD)
into the estimation process, the redundancy in the relationships
between successive derivatives can help yield more accurate pa-
rameter estimates. The idea is simply to capitalize on the fact that
the regression estimates of
linking SAD to D2SAD, DSAD to
D3SAD, and D2SAD to D4SAD are all mathematically equivalent
and can thus be constrained to be equal to one another. The same
procedures are used to constrain the
estimates linking DSAD to
D2SAD, D2SAD to D3SAD, and D3SAD to D4SAD to be equal
to one another. Thus, one estimate for
and one estimate for
obtained on the basis of information from the first to fourth
In this alternative approach, the measurement model and dy-
namic model are also expressed as one single model. This is
accomplished by defining a matrix of factor loadings, L, that
combines the relationships between factors and indicators (see
Figure 4B) with the growth curve loadings (see Figure 4A). The
resultant model is shown in Figure 5. By doing this, the instanta-
neous shock components (S
, and S
) are not
estimated. Because these shock components do not show any
continuity over time, they are pushed down and are estimated
instead as part of the measurement errors. We emphasize, how-
ever, that these shocks may represent an integral and important
part of emotion processes—we opted for a more constrained model
because we wish to focus on the dynamics of emotion over time.
The structural model shown in Figure 4 can be fitted to empir-
ical data using conventional structural equation modeling pro-
grams (e.g., LISREL, Mplus, and Mx). Currently, the condensed
version depicted in Figure 5 can only be fitted using Mx because
of the constraints one needs to place on the factor loadings matrix.
However, if the fourth-order model in Figure 5 is parameterized as
an expanded model depicting the curve of a factor (i.e., analogous
to the model in Figure 4 but with the third and fourth derivatives
included as well), then it can be fitted using any structural equation
modeling software. In the present context, we fitted first the
condensed model using Mx, and later, fitted the expanded version
using Mplus to utilize the program’s specific option for handling
incomplete data (i.e., the option for performing full information
maximum likelihood estimation in the presence of incomplete
Figure 5. A fourth-order differential structural approach used in this study to fit the damped oscillator model.
The dynamic and measurement portions of the model are not defined jointly by a factor loading matrix Lshown
in Table 1. ttime; Unhp unhappiness; Dep depression; Lone loneliness; SAD intensity of sadness;
DSAD, D2SAD, D3SAD, and D4SAD first, second, third, and fourth derivatives of sadness, respectively;
S_SAD, S_DSAD, S_D2SAD, and S_D3SAD shocks to an individual’s sadness and to its corresponding first,
second, and third derivatives, respectively; C
covariance between the shock terms associated
with SAD and DSAD; U_D4SAD residuals in D4SAD not accounted for by the model.
data). The matrices of parameters to be estimated include the
matrices L,A, and Sshown in Table 1. The matrix Lcarries the
associated factor loadings that define the relationships among level
and its higher-order derivatives, the matrix Acontains the regres-
sions among latent derivatives, and the covariance matrix Scarries
the variances and covariances among sources of intraindividual
variability in level and rate of change and residuals in acceleration.
An Empirical Example
We used the damped oscillator approach presented herein to
replicate Larsen and Kasimatis’s (1990) finding of a weekly cycle
in average hedonic level using data that have been published
elsewhere (e.g., Diener et al., 1995; Eid & Diener, 1999). The
sample consisted of 179 college students (98 men and 81 women,
average age 20.24, SD 1.81) at the University of Illinois at
Urbana–Champaign. Participants completed a set of affect ratings
on 52 consecutive days. In addition to testing the generalizability
of Larsen and Kasimatis’s (1990) earlier findings, we also present
some interemotion differences in oscillation frequency and inter-
individual differences in frequency and damping.
Because emotion was measured in days and the dominant cycle
in most participants’ data was a weekly cycle, damping is not very
meaningful in this particular context. As a result, this data set does
not demonstrate fully the strengths of the damped oscillator ap-
proach as a homeostatic emotion regulation model. However, this
data set is ideal for illustration purposes because most participants
in the study are characterized by a clear 7-day affect cycle.
Affect Measures
Participants completed daily self-reports of 24 emotions. They
were asked to rate how often they felt each of the emotions on a
7-point scale, ranging from 1 (none)to7(always). Using the
damped oscillator approach, we examined patterns of intraindi-
vidual variability in six emotion factors that have been identified
Table 1
Matrices Involved in Estimating the Damped Oscillator Model Using a Latent Differential
Structural Approach
Lmatrix of factor loadings
/2 (1.5t)
/6 (1.5t)
/2 (0.5t)
/6 (0.5t)
1 0.5t(0.5t)
/2 (0.5t)
/6 (0.5t)
1 1.5t(1.5t)
/2 (1.5t)
/6 (1.5t)
/2 a(1.5t)
/6 a(1.5t)
/2 a(0.5t)
/6 a(0.5t)
/2 a(0.5t)
/6 a(0.5t)
/2 a(1.5t)
/6 a(1.5t)
/2 b(1.5t)
/6 b(1.5t)
/2 b(0.5t)
/6 b(0.5t)
/2 b(0.5t)
/6 b(0.5t)
/2 b(1.5t)
/6 b(1.5t)
Amatrix with regressions among latent derivatives
SAD 0 0 0 0 0
DSAD 0 0 0 0 0
D4SAD 0 0
Scovariance matrix of intraindividual variability and residuals
—— — —
S_D2SAD 0 0 V
S_D3SAD 0 0 0 V
U_D4SAD 0 0 0 0 V
Note. SAD intensity of sadness; DSAD, D2SAD, D3SAD, and D4SAD the first, second, third, and fourth
derivatives of sadness, respectively; afactor loading of SAD on depression; bfactor loading of SAD on
loneliness; Unhp unhappiness; Dep depression; Lone loneliness; S_SAD, S_DSAD, S_D2SAD, and
S_D3SAD shocks to an individual’s sadness and to its corresponding first, second, and third derivatives,
respectively; V
, and V
the variances of the shock terms; C
covariance between the shock terms associated with SAD and DSAD; U_D4SAD residuals in D4SAD not
accounted for by the model, and V
is its associated variance.
elsewhere (Diener et al., 1995). These factors are love, joy, sad-
ness, fear, anger, and shame. Each factor was measured using four
items (see Table 2).
Data Analysis
Each participant’s time series for each of the six factors was
detrended prior to model fitting to eliminate spurious correlations
among the six emotions due to any common linear trend (Mc-
Cleary & Hay, 1980). In addition, this procedure removes individ-
ual differences in equilibrium level. In other words, all partici-
pants’ equilibrium level on each variable is shifted to a zero point.
We organize our results into three sections to (a) replicate Larsen
and Kasimatis’s (1990) finding on weekly cycle in individuals’
aggregate hedonic level, (b) present interemotion differences in
periodicity, and (c) demonstrate individual differences in fre-
quency and damping.
First, to illustrate the damped oscillator approach’s utility in
recovering systematic oscillation frequency, we computed each
participant’s daily hedonic level based on the definition used in
Larsen and Kasimatis (1990),
and we aggregated these daily
measures across all participants to yield a single time series of
hedonic level. We then fitted the damped oscillator model to this
single time series with six estimation occasions to obtain a fre-
quency estimate and a damping estimate for the aggregate hedonic
level. On the basis of preliminary analysis, we were best able to
recover the 7-day cycle evident in the participants’ aggregate
hedonic level by using six estimation occasions. We therefore
chose six estimation occasions for all subsequent analyses. Note
that because the damped oscillator model involves an oscillatory
(i.e., nonlinear) function, the estimates yielded from this aggregate
curve will not be the same as the estimates obtained from averag-
ing across different individual curves. In other words, this aggre-
gate curve may, in fact, characterize no one individual’s curve.
However, this step was performed primarily for illustration pur-
poses because there is a clear 7-day affect cycle in the aggregate
Second, we examined interemotion differences in frequency and
damping. Daily emotion scores were aggregated across partici-
pants to yield a 52 4 data matrix for each emotion (4 manifest
indicators measured over 52 days). This matrix was lagged against
itself to yield a 24 24 covariance matrix (with 4 manifest
indicators 6 estimation occasions) for model fitting. This pro-
cedure was performed separately for each of the six emotions. Six
frequency estimates and six damping estimates were obtained by
fitting the damped oscillator model separately to each of the six
emotion measures. Finally, individual differences in affective pe-
riodicity and damping were examined by fitting the independent
oscillator model to each individual’s data separately. We only
focused in this case on three emotions: love, joy, and sadness.
Replicating a 7-Day Cycle in Aggregate Hedonic Level
Consistent with Larsen and Kasimatis’s (1990) finding, a 7-day
cycle is evident in these participants’ aggregate hedonic level (see
Figure 6). Examination of the plot in Figure 6 indicates that the
aggregate average hedonic level peaks on the 7th day of each
week, in this case a Saturday. Results from fitting the damped
oscillator model to the aggregate average hedonic level data
yielded an
estimate of .77. In day units, this equals an oscil-
lation period of 7.16 days, thus showing close correspondence to a
weekly cycle. Discrepancy from a precise 7-day estimate was
evaluated by fitting a second model wherein
was fixed at .80
(a 7-day period of oscillation). The change in fit was very small,
(1) 1.9, p.05, indicating that the 7.16-day estimate was
not significantly different from a 7-day estimate. Thus, the damped
oscillator model was able to recover the 7-day frequency in ag-
gregate hedonic level accurately.
Interemotion Differences in Periodicity
The damped oscillator model was used to estimate the period-
icity present in the six aggregate emotions using Mplus (Muthe´n &
Muthe´n, 2001) with full information maximum likelihood estima-
tion. In all cases, incomplete data were treated as missing at
random (Little & Rubin, 1987). The corresponding parameter
estimates are summarized in Table 3.
The frequency estimates indicated strong weekly cycles in the
pleasant emotions, joy and love, and in the unpleasant emotions,
sadness, fear, and shame. Only anger seemed to diverge slightly
from a weekly cycle, as it was characterized by a higher frequency
compared with other emotions. However, despite all six emotions
exhibiting weekly cycles, an inspection of the plots associated with
each emotion revealed that emotions of different valences tended
to peak on different days of the week. Love and joy were observed
to peak over weekends, whereas unpleasant emotions (sadness,
fear, anger, and shame) typically manifested surges in magnitude
in the middle of the week.
Damping was not significant for most of the emotions. Even
when damping was statistically different from zero (i.e., for love
and sadness), the parameters were still small (–.01 and .02,
respectively). This indicates that the weekly cycle continues un-
abated through the entire 52 days of study. In sum, interemotion
differences in weekly cycle indicate that the well-known specula-
tion of a blue Monday phenomenon is, for the most part, attribut-
able to a decline in pleasant emotions on Mondays. Furthermore,
because of the lack of damping, the blue Monday decline in
pleasant emotions persisted throughout the duration of this study.
We computed each participant’s average hedonic level on a particular
day as the difference between his or her pleasant emotion (averaged across
two items: love and joy) and unpleasant emotion (averaged across four
items: sadness, fear, anger, and shame).
Table 2
List of Manifest Variables Used to Identify Six Latent Emotions
Factor Manifest indicators
Love Love, affection, caring, and fondness.
Joy Joy, happiness, contentment, and satisfaction.
Sadness Sadness, unhappiness, depression, and loneliness.
Fear Fear, worry, anxiety, and nervousness.
Anger Anger, irritation, disgust, and rage.
Shame Shame, guilt, regret, and embarrassment.
Interindividual Differences in Frequency and Damping
The damped oscillator model was fitted separately to each
individual’s data on love, joy, and sadness. We obtained a total of
three frequency estimates and three damping estimates for each
person. The ranges and averages of these parameters are shown in
Table 4. Individual differences were quite apparent in both the
frequency of oscillation and the rate of damping. A subset of
individuals’ estimation results did not meet the statistical criteria
for convergence (possibly because of high degrees of incomplete-
ness in their data). Their estimates were therefore excluded from
the interindividual analyses.
We then examined the individual differences in frequency and
damping estimates by using gender, affect intensity, extraversion,
and neuroticism as predictors in a series of multiple regression
analyses. Significant gender differences were found only in the
periodicity of sadness F(1, 125) 4.27, p.041. On average,
women exhibited a higher frequency of fluctuation in sadness
(M⫽⫺1.02, SD 0.41; i.e., average period of 6.22 days)
compared with men (M⫽⫺0.87, SD 0.42; i.e., average period
of 6.74 days; p.036). This indicates that men, on average, were
slightly more entrained to a weekly cycle (i.e., their average
estimate was closer to the ideal
estimate of .80 corresponding
to a period of 7 days) than were women. In addition, there was also
a marginally significant effect of affect intensity (as measured by
the Affect Intensity Measure; Larsen, 1985b) on the frequency of
love, F(1, 137) 3.76, p.055. In particular, participants who
were higher on affect intensity also experienced fluctuations in
love at a higher frequency. Individual differences in other fre-
quency and damping estimates were not significantly related to
differences in gender, affect intensity, neuroticism, and extraver-
sion ( p.05).
The purpose of this study was to present a damped oscillator
model and to demonstrate how it can be fitted as a structural
equation model to empirical data. The damped oscillator model
provides a direct representation of the concept of emotion as a
thermostat. The specific parameters of this model, including fre-
quency and damping, offer a practical way for modeling individ-
uals’ emotional lability and the effectiveness of their regulatory
behaviors within a process-oriented framework. Furthermore, the
particular estimation approach used to fit the damped oscillator
model in this study is highly flexible and can be used to fit other
dynamic models in the form of differential or difference equation
Figure 6. A plot of the participants’ changes in average hedonic level as a group. A weekly trend was apparent
in the group’s surges in hedonic level over weekends (on Days 7, 14, 21, etc., which are all marked with dotted
Table 3
Estimated Values of
Based on All Participants’ Data
SE Period in days
Love .83 .005*** 6.90 .01 .005*
Joy .83 .007*** 6.90 .01 .006
Sadness .84 .006*** 6.86 .02 .005*
Fear .84 .006*** 6.86 .01 .005
Shame .84 .008*** 6.86 .01 .007
Anger .93 .007*** 6.52 .01 .006
*p.05. *** p.001.
Using the damped oscillator approach, we replicated Larsen and
Kasimatis’s (1990) earlier finding on a 7-day cycle in college
students’ aggregate hedonic level. Consistent with Stone’s (1985)
earlier findings, results from model fitting also reveal that the blue
Monday phenomenon is more attributable to postweekend declines
in pleasant emotions rather than increases in unpleasant emotions.
Generally, even though all six emotions manifested similar peri-
odicity (i.e., close to a period of 7 days), emotions of opposite
valence (i.e., pleasant or unpleasant) tended not to occur together
in high intensity. This difference in affective dynamics is consis-
tent with previous findings regarding the independence of pleasant
and unpleasant emotions (e.g., Diener & Emmons, 1984; Diener &
Iran-Nejad, 1986).
In addition to using the damped oscillator model to examine
interemotion differences, we also used it to examine interindi-
vidual differences in emotion oscillations. Using each individual’s
frequency estimates for love, joy, and sadness as indicators of the
individual’s periodicities in these three emotions, we found sig-
nificant gender differences in the periodicity of sadness. Male
participants’ sadness was slightly more entrained to a weekly
cycle, whereas female participants exhibited changes in sadness at
a slightly higher frequency. In addition, there was also a margin-
ally significant relationship between affect intensity and the fre-
quency of love—participants who were higher on affect intensity
also manifested fluctuations in love at a higher frequency. How-
ever, we did not find other personality differences in the period-
icity of average hedonic level, love, joy, or sadness. It is possible
that interindividual differences in personality may relate, in the
context of a weekly cycle, more to damping than frequency.
Given the individual differences in class and/or work schedules
among the college students in this study, the blue Monday phe-
nomenon as reflected in the aggregate data might be potentially
stronger if data from individuals with a more homogeneous work
schedule (e.g., individuals working regularly on a Monday–Friday
schedule) are analyzed. Because different individuals’ rhythms are
likely to be slightly off phased (i.e., each individual’s pleasant and
unpleasant emotions are likely to peak on different days of the
week, even though they may all conform to a weekly cycle), these
individual-level dynamics can only be extracted if model fitting or
analysis is done at the individual level. Although we did fit the
oscillator model to each participant’s data, we did not examine the
individuals’ idiosyncratic reactions to Mondays. In other words,
we did not investigate whether the postweekend declines in pleas-
ant emotions observed at the aggregate level reflect similar reac-
tions at the individual level. Spectral analysis may be a particularly
useful descriptive tool in this case as it provides a quick and
convenient way to extract peaks in each individual’s emotions. If
researchers are interested in extracting the different cycles embed-
ded in an individual’s data, spectral analysis again offers a con-
venient way of answering this research question.
We have demonstrated the potential utility of assessing the
frequency of emotion fluctuations in addition to the intensity (i.e.,
amplitude) of mood. We have also pinpointed the correspondence
between the damped oscillator model and a homeostatic emotion
regulation model suggested by various researchers (e.g., Gross,
Sutton, & Ketelaar, 1998; Headey & Wearing, 1989; Larsen, 2000;
Lykken & Tellegen, 1996). As we have stated previously, the
current data set does not illustrate fully the potentials of the
damped oscillator model as an emotion regulation model. To do
this, we would need data sampled at much closer intervals. For
example, we would need to start measuring an individual’s emo-
tion status upon being exposed to an affect-relevant cue and record
his or her fluctuations in emotion closely as the individual’s
emotion returns to its equilibrium set point (e.g., by using ap-
proaches such as the experience sampling method; Csikszentmi-
halyi & Larson, 1987). Interindividual differences in this damping
rate can provide interesting insights into sources of individual
difference in emotion regulation.
Another possible extension to the approach presented herein is
to expand the damped oscillator model to examine how different
processes might be dynamically coupled to one another. One
example of such models is the coupled oscillators model presented
by Boker and Graham (1998). When one uses this approach, two
or more processes can be modeled as multiple oscillators that are
coupled to one another. For instance, by coupling an individual’s
unpleasant emotion to his or her pleasant emotion, the two pro-
cesses may fluctuate in perfect synchrony with one another. This
may occur under specific environmental influences (e.g., under
stress; Zautra, Potter, & Reich, 1998). By incorporating this cou-
pling term, an individual’s failure to suppress (i.e., to damp) his or
Table 4
Summary Statistics of Parameter Values Estimated Individually for Each Participant
Factor MSD Range N
Average hedonic level 0.96 (6.41) .63 2.91 to 0.11 (3.68 to 18.94) 150
Love 0.92 (6.55) .42 2.70 to 0.11 (3.82 to 18.94) 142
Joy 0.89 (6.66) .41 2.09 to 0.07 (4.34 to 23.74) 135
Sadness 0.94 (6.48) .42 2.04 to 0.13 (4.40 to 17.43) 130
Average hedonic level 0.01 .19 0.96 to 0.86 150
Love 0.02 .07 0.43 to 0.20 142
Joy 0.01 .09 0.76 to 0.27 135
Sadness 0.02 .09 0.61 to 0.21 130
Note. Values in parentheses represent period in days.
her unpleasant emotion can also lead to more rapid oscillations in
pleasant emotion. These dynamic models thus offer ample oppor-
tunities for researchers to examine the linkages or divergences
between emotions of different valences or activation poles (Feld-
man Barrett, 1998; Green & Citrin, 1994; Trierweiler, Eid, &
Lischetzke, 2002; Watson, Wiese, Vaidya, & Tellegen, 1999).
This notion of dynamic linkages between different emotions also
adds interesting perspectives to other prevalent models of affect
(e.g., circumplex models; Browne, 1992; Fabrigar, Visser, &
Browne, 1997).
In sum, this study examined weekly periodicity in a sample of
college students using a damped oscillator approach. We replicated
previous findings on the existence of a weekly cycle in average
hedonic level (e.g., Larsen & Kasimatis, 1990) and found that the
blue Monday phenomenon seems primarily to be a result of
declines in pleasant emotion. Furthermore, although aggregated
data showed clear 7-day cycles, analysis at the individual level
revealed substantial individual differences in entrainment to this
cycle. Implications are that dynamics of emotional experience are
much more complex than what snapshots at a particular time point
could convey. However, as we hope we have illustrated here, the
damped oscillator approach provides important methodological
and theoretical advantages as a tool for representing emotion
regulation as a dynamic process— or in the present context, as a
self-regulatory thermostat.
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Appendix A
Spectral Analysis and the Damped Oscillator Approach
Spectral analysis is a descriptive approach that decomposes a time series
into a set of sine and cosine functions (i.e., oscillatory functions) in the
frequency domain (see Chatfield, 1996; Gottman, 1979; Warner, 1998). In
brief, this data-analytic technique identifies the weights or densities of all
possible frequencies of sine and cosine waves that exist within a time
series. By examining the magnitudes of these weights, one can identify
hidden cycles that may not be apparent by inspection. For example, if a set
of time-series data shows high spectral density–weight for a 7-day fre-
quency and low weights for other frequencies, one might conclude that the
time series is characterized by a strong weekly rhythm (e.g., Larsen &
Kasimatis, 1990). That is, much of the variance in the time series is
accounted for by a particular sinusoidal cycle.
Currently, spectral analysis is one of the most popular tools for detecting
the existence of cycles in time-series data (Gottman, 1979; Warner, 1998).
The sinusoidal model that forms the basis of spectral analysis (see Warner,
1998) is identical to the integral solution of the damped oscillator model
(see Zill, 1993) when there is no damping and
is negative.
In this
section, we highlight some of the similarities and differences between the
spectral analysis approach adopted by Larsen and Kasimatis (1990) and the
damped oscillator approach used in this study. We demonstrate that al-
though both of these approaches yield similar information concerning the
periodicity of a construct, these approaches have their own strengths and
weaknesses in helping to address different research questions.
Distinctions Between Spectral Analysis and the
Latent Structural Approach
Typically, spectral analysis is used when researchers often do not have
preconceived notions or expectations of the periodicity of a construct or
hypotheses on how this periodicity is related to other constructs. Most
often, spectral analysis is used as a tool to identify and describe the
periodicity or seasonality in the data, rather than to model this periodicity
in relation to other constructs (Warner, 1998; for an exception, see Larsen
& Kasimatis, 1990).
Because the sinusoidal model is identical to the damped oscillator model
with no damping, all the estimates available from spectral analysis can also
be obtained from estimates of the damped oscillator model. However,
because spectral analysis is available in most statistical packages (e.g.,
SAS, S-plus, R, and SPSS), it is conveniently equipped with options and
estimates that these programs execute and output automatically. These
estimates will, however, have to be computed in some additional steps
when the oscillator approach is used. For instance, in bivariate spectral
analysis, wherein two time series are subject to spectral analysis simulta-
neously, most software packages output the cross-phase between the two
series automatically. This parameter represents the difference between the
first peaks of the two series in radian units and provides an indication of the
amount of time one construct is lagging the other by (e.g., pleasant emotion
may precede unpleasant emotion by one day). To obtain the same infor-
mation using the damped oscillator approach, one will have to fit the
oscillator model separately to the two time series and subsequently com-
pute the cross-phase in an additional step.
The damped oscillator model does have an important feature that spec-
tral analysis does not offer—it incorporates a damping parameter that is not
part of the sinusoidal model assumed in spectral analysis. This gives the
former some added flexibility in shaping the corresponding trajectory of
change into different functional forms (see Figure 3) and conveys impor-
tant meanings in the context of the homeostatic emotion regulation model
discussed earlier. As an illustration, we analyzed the time series in Figures
3A and 3C using spectral analysis. The resultant periodograms are shown
in Figure A1. Even though the true frequency (
–1, corresponding to a
period of 6.823 marked with a dashed line)
can still be recovered in the
first case, the damping shown in Figure 3A is manifested as a nonstationary
trend in the periodogram (see Panel A in Figure A1).
When the magni-
tude of damping is increased further to
4, in which case the resultant
trajectory no longer appears cyclic, spectral analysis fails to recover the
true frequency (see Panel B in Figure A1). This, however, is not a problem
if the damped oscillator approach is used.
Under a special condition where [(|
|] is less than zero, the
integral solution for the damped oscillator model in Equation 1 (Zill, 1993)
is expressed as
, (A1)
where c
and c
are arbitrary constants derived from one’s initial level and
rate of change. In the case of
0, it simplifies to
t. (A2)
Equation A2 is identical to the sinusoidal model used in harmonic analysis
and is reexpressed in an alternative but equivalent form in spectral analysis
(see Warner, 1998).
Given a known oscillation period of
, the theoretical value of
be computed as
, where
represents the period of
oscillation. For example, a 7-day cycle would yield an
estimate of
When spectral analysis is used to fit a series of different frequencies
to data, the particular frequencies fitted are a function of the number of
occasions in the data. Panel A in Figure A1 is an example of commonly
observed phenomenon often termed the leakage effect (Warner, 1998). In
this case, the dominant frequency leaks into the nearest fitted frequency.
Thus, the estimated period is close but does not coincide perfectly with the
true period.
Although spectral analysis is a well-known analytic tool that is available
in most software packages, the associated functions for spectral analysis do
not handle incomplete data. The damped oscillator approach, however, can
be implemented within a structural equation modeling framework and thus
offers several options that have not been implemented in spectral analysis
(e.g., full information maximum likelihood and multiple imputation). This
is due, however, to limitations imposed by the software packages, rather
than spectral analysis itself. In addition, multiple indicators can be used as
markers of a latent construct, and this multivariate measurement model can
be combined with the cyclic dynamic model in one single step, rather than
in a two-step procedure (as in spectral analysis). If a researcher chooses,
the oscillator model can also be fitted simultaneously to multivariate data
from multiple individuals. In spectral analysis, however, one must first
derive composite or factor scores and then conduct the spectral analyses
separately for each participant.
In addition to the benefits of convenience and accessibility outlined
earlier, spectral analysis does have another important strength in help-
ing to answer a specific type of research question. When a data set is
characterized by multiple cycles (e.g., daily cycles, weekly cycles, and
menstrual cycles all embedded in the same data set), and a researcher is
interested in extracting all of these cycles, spectral analysis offers a
quick and easy way to accomplish this task. The damped oscillator
approach, however, only extracts the most dominant cycle in a data set.
To accomplish the same purpose, one will have to fit the oscillator
model repeatedly— each time extracting the most dominant cycle and
then reanalyzing the residuals.
(Appendixes continue)
Figure A1. Periodograms of simulated data with the same parameters as in Figure 3A, with
⫽⫺1.0 and
0.7 (A) and of simulated data with the same parameters as in Figure 3C, with
⫽⫺1.0 and
⫽⫺4.0 (B).
A: The peak in spectral density signifies the dominant period recovered by spectral analysis, and the true period
is marked with a dashed line. B: No single period was identified by spectral analysis as the dominant period, and
the true period is marked with a dashed line.
Appendix B
A Comparison Between the Fourth-Order Differential Structural Approach and Growth Curve
Contemporary growth curve or hierarchical linear models (Bryk &
Raudenbush, 1987; McArdle & Epstein, 1987; Meredith & Tisak, 1990)
allow researchers to form testable hypotheses regarding level (or intensity),
rate of change, and associated interindividual differences. Although
second-order change (or acceleration) can readily be incorporated into
growth curve models, few researchers have focused on capturing second-
order changes.
In a typical growth curve analysis, the latent process of interest is usually
identified using a single indicator. A path diagram depicting how linear and
quadratic slopes are typically defined in growth curve models with one
indicator is shown in Figure B1. This quadratic growth curve model only
captures part of the dynamic model defined in Figure 4A. Furthermore, the
multivariate measurement model in Figure 4B is not part of the hypotheses
tested. Because of this, the shock components shown in Figure 4 are not
explicitly modeled in Figure B1, unless multivariate information is incor-
porated. Therefore, the model in Figure 4 can be interpreted as a curve of
the factor, sadness, whereas the model in Figure B1 is used to represent the
curve of only one indicator, unhappiness.
Four other important distinctions exist between the differential structural
approach and the growth curve approach. First, the state-space embedding
technique used in the differential structural approach to capture systematic
patterns of covariation for long time series is not usually used in growth
curve modeling. The typical use of growth curve models to represent
long-term developmental changes also precludes the need for using this
technique. Secondly, the scaling value for time tused in the matrix of
factor loadings L(see Table 1) to define measurement intervals is unique
to this approach and can be used to accommodate unequal measurement
intervals among tests or items.
Third, the loadings of the acceleration factor on manifest indicators
differ slightly from typical loadings used to define a quadratic factor (see,
e.g., Biesanz, Deeb-Sossa, Papadakis, Bollen, & Curran, 2004; Willett &
Sayer, 1994). Basically, the loadings for rate of change are identical to the
loadings for slope in a growth curve model (except for an additional scaling
value tin the former). However, instead of simply squaring these linear
loadings to define the acceleration factor, the squared loadings are divided
by two to indicate the derivative relationships among level, rate of change,
and acceleration. More specifically, one will have to take the derivative of
these acceleration loadings with respect to time to obtain the loadings for
the rate of change. Therefore, acceleration is represented as the change in
the rate of change with respect to time. This approach was used by Wishart
(1938) in the context of polynomial regression to define average growth
rate and rate of change of growth rate in bacon pigs.
Finally, because the specific loadings used to define the latent deriva-
tives establish their roles as level, rate of change, acceleration, and other
higher-order changes, the fourth-order (or any higher order) differential
structural approach can be modified slightly to fit other kinds of differential
equation models. More important, the differential structural approach
opens the opportunity for formulating specific testable hypotheses of the
dynamics of factors (e.g., as conformed to the damped oscillator model). In
fact, the damped oscillator model imposes an alternative autoregressive
Figure B1. Path diagram of a typical univariate growth curve model with linear and quadratic slopes. The
triangle k represents a constant term with its variance fixed at 1.0. Regression estimates from k to SAD, DSAD,
and D2SAD (i.e.,
, and
, respectively) correspond to the means of the three latent
components. SAD intensity of sadness; DSAD and D2SAD first and second derivatives of sadness,
respectively; Unhp1, Unhp2, Unhp3, and Unhp4 unhappiness measured at Times 1, 2, 3, and 4, respectively.
moving average model process (more specifically an AMRA[2,1] model)
process on the latent derivatives. Essentially, the
weights represent lag-2 and lag-1 autoregressive weights in a second-order
autogressive process (i.e., AR[2]), respectively. Sources of intraindividual
variability in level and rate of change (i.e., S_SAD and S_DSAD in Figure
4) replace the random shocks in a moving average process, and the lag-1
moving average weight is just manifested as covariance between these two
sources of intraindividual variability (for details, see Browne & Nessel-
roade, in press). Yule (1927) pointed out that a stationary AR(2) process
with a positive lag-1 autoregressive weight (denoted as
) between 0 and
2 and a negative lag-2 autoregressive weight (denoted as
) between –1
and –
/4 yields an autocorrelation function that follows a damped sine
wave, which can be used to model a pendulum that is subjected to random
shocks (for details, see Box & Jenkins, 1976; Browne & Nesselroade, in
press; Wei, 1990). Therefore, the dynamic process characterizing the
damped oscillator model is analogous to an autoregressive moving average
model (2,1) process. Using the differential structural approach, however,
one can effectively capitalize on information from high-order derivatives to
yield more accurate estimates for
(or the lag-1 and lag-2 autore-
gressive weights). This approach also provides a more general way of
fitting other kinds of differential or difference equation models.
That is, the hypothesized model can be decomposed into a lag-2
autoregressive process and a lag-1 moving average structure.
Received February 17, 2004
Revision received September 20, 2004
Accepted November 15, 2004
... For example, suppose members of a couple rate their daily levels of positive and negative affect for one month. These ratings are expected to fluctuate over time (Chow et al., 2005), and may follow a specific oscillatory pattern such that affect is higher on some days (e.g., weekends) compared to others (e.g., weekdays). In standard regression, these affect ratings would be treated as a dependent variable and modeled as a function of some set of independent variables. ...
... The second derivative of affect at time represents the acceleration in affect, or the change in momentary velocity, at a given time point. When plotted, this equation yields an oscillatory functional form that maps on to fluctuations in affect over time (Chow et al., 2005) and can be used to model dyadic processes using the CLO (see Randall et al., 2021). While the mathematical principles underlying this model are beyond the scope of this article, the CLO exemplifies a shift toward modeling relationships as dynamic change processes as opposed to a series of discrete, independent events. ...
Increasing the representation of diverse voices in relationship science requires statistical methodologies that are inclusive of individuals in relationships who identify as a sexual minority (i.e., lesbian, gay, or bisexual) or gender diverse (i.e., transgender, nonbinary, genderqueer, etc.) individuals. Research questions related to the initiation, development, and maintenance of romantic relationships for these individuals should be explored using quantitative methods that are sensitive to diversity and individual differences within a population. Analytical tools relevant to the study of interdependent, yet indistinguishable dyads, including references to extended technical guides for those wishing to conduct this work are presented.
... Mean net change provides a general measure of how much scores varied from week-to-week but does not indicate direction of change or consider how fluctuation varies over time. Methods designed specifically to assess stability, such as P(Δ) or damped oscillator models (Chow et al. 2005) may be informative. ...
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Objectives Prior studies indicate risk for recidivism declines with time spent in the community post-incarceration. The current study tested whether declines in risk scores occurred uniformly for all individuals in a community corrections sample or whether distinct groups could be identified on the basis of similar trajectories of change in acute risk and time to recidivism. We additionally tested whether accounting for group heterogeneity improved prospective prediction of recidivism. Methods This study used longitudinal, multiple-reassessment data gathered from 3,421 individuals supervised on parole in New Zealand ( N = 92,104 assessments of theoretically dynamic risk factors conducted by community corrections supervision officers). We applied joint latent class modelling (JLCM) to model group trajectories of change in acute risk following re-entry while accounting for data missing due to recidivism (i.e., missing not at random). We compared accuracy of dynamic predictions based on the selected joint latent class model to an equivalent joint model with no latent class structure. Results We identified four trajectory groups of acute dynamic risk. Groups were consistently estimated across a split sample. Trajectories differed in direction and degree of change but using the latent class structure did not improve discrimination when predicting recidivism. Conclusions There may be significant heterogeneity in how individuals’ assessed level of acute risk changes following re-entry, but determining risk for recidivism should not be based on probable group membership. JLCM revealed heterogeneity in early re-entry unlikely to be observed using traditional analytic approaches.
Longitudinal research demonstrates that child ADHD symptoms and behaviors exhibit reciprocal associations with parenting behaviors over time. However, minimal research has investigated these associations and their dynamic links at the daily level. Intensive longitudinal data can disentangle stable between-person differences from within-person fluctuations and reveal nuanced short-term family dynamics on a micro timescale. Using 30-day daily diary data from a community sample of 86 adolescents (Mage = 14.5, 55% female, 56% White, 22% Asian) and latent differential equation modeling, this study examined the links between perceived daily parental warmth and ADHD symptoms as coupled dynamical systems. The results show that the magnitude of fluctuations in perceived daily parental warmth generally remains stable, while elevated ADHD symptoms return to their normal level over time. Perceived parental warmth is sensitive to change in ADHD symptoms such that adolescents feel that their parents will fine-tune their warmth with gradual changes when adolescents demonstrate heightened symptoms. There are substantial between-family differences in these regulating system dynamics. Among families with more baseline parental non-harsh discipline, both perceived parental warmth and ADHD symptoms tend to be more stable and fluctuate less often. Intensive longitudinal data and dynamical systems approaches offer a new lens to uncover short-term family dynamics and adolescent adjustment at a refined micro level. Future research should explore antecedents and consequences of between-family differences in these short-term family dynamics on multiple timescales.
Recent advances in technology contribute to a fast-growing number of studies utilizing intensive longitudinal data, and call for more flexible methods to address the demands that come with them. One issue that arises from collecting longitudinal data from multiple units in time is nested data, where the variability observed in such data is a mixture of within-unit changes and between-unit differences. This article aims to provide a model-fitting approach that simultaneously models the within-unit changes with differential equation models and accounts for between-unit differences with mixed effects. This approach combines a variant of the Kalman filter, the continuous-discrete extended Kalman filter (CDEKF), and the Markov chain Monte Carlo method often employed in the Bayesian framework through the platform Stan. At the same time, it utilizes Stan's functionality of numerical solvers for the implementation of CDEKF. For an empirical illustration, we applied this method in the context of differential equation models to an empirical dataset to explore the physiological dynamics and co-regulation between couples.
The dynamics of parent-infant physiology are essential for understanding how biological substrates of emotion regulation are organized during infancy. Although parent-infant physiological processes are dyadic in nature, research is limited in understanding how one person's physiological responses predict one's own and as well as the other person's responses in the subsequent moment. In this study, we examined mother-infant respiratory sinus arrhythmia (RSA) dynamics during the Still-Face Paradigm (SFP) among 106 mothers (Mage = 29.54) and their 7-month-old infants (55 males). Given mothers' role in shaping dyadic interactions with their infant, we also tested how mothers' self-reported emotion dysregulation (measured via the Difficulties in Emotion Regulation Scale) associated with these dynamics. Results showed that both mothers' and infants' RSA tended to return to their respective homeostatic points (i.e., exhibited return strength) during each SFP episode, indicating stability in RSA for mother-infant dyads. Significant shifts in mother and infant RSA return strength were observed across SFP episodes, highlighting the role of contextual demands on each individual's physiological dynamics. Mother-infant RSA dynamics varied as a function of maternal self-reported emotion dysregulation. Specifically, RSA levels of infants with more dysregulated mothers had a weaker tendency to return to homeostasis during the Reunion episode and were less affected by their mothers' RSA during the Still-Face and Reunion episodes of the SFP, suggesting a less effective coregulatory influence. Our findings have implications for the intergenerational transmission of emotion dysregulation via mother-infant physiological dynamics.
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Constrained fourth-order latent differential equation (FOLDE) models have been proposed (e.g., Boker et al. 2020) as alternative to second-order latent differential equation (SOLDE) models to estimate second-order linear differential equation systems such as the damped linear oscillator model. When, however, only a relatively small number of measurement occasions T are available (i.e., T=50), the recommendation of which model to use is not clear (Boker et al. 2020). Based on a data set, which consists of T=56 observations of daily stress for N=44 individuals, we illustrate that FOLDE can help to choose an embedding dimension, even in the case of a small T. This is of great importance, as parameter estimates depend on the embedding dimension as well as on the latent differential equations model. Consequently, the wavelength as quantity of potential substantive interest may vary considerably. We extend the modeling approaches used in past research by including multiple subjects, by accounting for individual differences in equilibrium, and by including multiple instead of one single observed indicator.
This study investigated the associations between momentary emotion dynamics and posttraumatic stress disorder (PTSD) symptoms. Using a sample of 61 couples (N = 122 individuals) in which all individuals were trauma-exposed and at least one partner screened positive for PTSD, we examined the intra- and interpersonal regulation of vocally encoded emotional arousal (fundamental frequency, f0) and how these momentary emotion regulatory patterns relate to specific PTSD symptoms during two couple conversations, one designed to elicit conflict and one to elicit intimacy. PTSD symptoms were assessed using gold standard clinical interview. In both conversations, higher reexperiencing symptoms were associated with greater emotional inertia (i.e., more resistance to change in emotional state following deviation from one’s emotional equilibrium), and higher avoidance symptoms were associated with less emotional inertia (i.e., quicker return to emotional equilibrium). In the intimacy conversations, individuals also responded to their partners’ arousal. Furthermore, individuals whose partners exhibited higher emotional numbing symptoms exhibited more emotional inertia, suggesting that emotion regulation may be a function of both one’s own and one’s partner’s PTSD symptoms. Attending to the interpersonal context of emotion dynamics during PTSD treatment may enhance outcomes.
Psychology has witnessed a dramatic increase in the use of intensive longitudinal data (ILD) to study within-person processes, accompanied by a growing number of indices used to capture individual differences in within-person dynamics (WPD). The reliability of WPD indices is rarely investigated and reported in empirical studies. Unreliability in these indices can bias parameter estimates and yield erroneous conclusions. We propose an approach to (a) estimate the reliability and (b) correct for sampling error of WPD indices using “Level-1 variance-known” (V-known) multilevel models (Raudenbush & Bryk, 2002). When WPD indices are calculated for each individual, the sampling variance of the observed WPD scores is typically falsely assumed to be zero. V-known models replace this “zero” with an approximate sampling variance fixed at Level 1 to estimate the true variance of the index at Level 2, following random effects meta-analysis principles. We demonstrate how V-known models can be applied to a broad range of emotion dynamics commonly derived from ILD, including indices of the average level (mean), variability (intraindividual standard deviation), instability (probability of acute change), bipolarity (correlation), differentiation (intraclass correlation), inertia (autocorrelation), and relative variability (relative standard deviation) of emotions. A simulation study shows the usefulness of V-known models to recover the true reliability of these indices. Using a 21-day diary study, we illustrate the implementation of the proposed approach to obtain reliability estimates and to correct for unreliability of WPD indices in real data. The techniques may facilitate psychometrically sound inferences from WPD indices in this burgeoning research area.
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Objectives The present research examines genomics and in vivo dynamics of family context and experienced affect following discharge from psychiatric hospitalisation for suicidal thoughts and behaviours (STBs). The purpose of this paper is to provide an overview of a new model, description of model-guided integration of multiple methods, documentation of feasibility of recruitment and retention and a description of baseline sample characteristics. Design The research involved a longitudinal, multimethod observational investigation. Setting Participants were recruited from an inpatient child and adolescent psychiatric hospital. 194 participants ages 13–18 were recruited following hospitalisation for STB. Primary and secondary outcome measures Participants underwent a battery of clinical interviews, self-report assessments and venipuncture. On discharge, participants were provided with a phone with (1) the electronically activated recorder (EAR), permitting acoustic capture later coded for social context, and (2) ecological momentary assessment, permitting assessment of in vivo experienced affect and STB. Participants agreed to follow-ups at 3 weeks and 6 months. Results A total of 71.1% of approached patients consented to participation. Participants reported diversity in gender identity (11.6% reported transgender or other gender identity) and sexual orientation (47.6% reported heterosexual or straight sexual orientation). Clinical interviews supported a range of diagnoses with the largest proportion of participants meeting criteria for major depressive disorder (76.9%). History of trauma/maltreatment was prevalent. Enrolment rates and participant characteristics were similar to other observational studies. Conclusions The research protocol characterises in vivo, real-world experienced affect and observed family context as associated with STB in adolescents during the high-risk weeks post discharge, merging multiple fields of study.
Daily emotion dynamics provide valuable information about individuals’ emotion processes as they go about their lives. Emotion dynamics such as emotion levels (mean), emotion variability (degree of fluctuation), and emotion network density (strength of temporal connections among emotions) are associated with risks for various psychopathology in youth and adults. Prior work has shown that caregivers and friends play crucial socializing roles in adolescent emotional well‐being, but less is known about their roles in daily emotion dynamics. This study examined whether caregiver emotion coaching, caregiver‐adolescent closeness, and friendship quality were associated with adolescents’ emotion levels, emotion variability, and emotion network density. Further, we examined whether caregiver‐adolescent closeness moderated the associations between coaching and emotion dynamics. Participants were 150 adolescents (61% girls; Mage = 14.75) and one of their caregivers (95% female; Mage = 43.35) who completed a baseline survey and 21 daily surveys. Results showed that caregiver emotion coaching interacted with caregiver‐adolescent closeness in predicting emotion levels and variability. Specifically, when closeness was higher, emotion coaching was significantly associated with lower sadness and anger levels, higher happiness levels, and lower happiness variability. Caregiver emotion coaching, independent of closeness, was also associated with lower anxiety levels, lower sadness variability, and lower emotion network density. Friendship quality was significantly associated with lower levels of sadness, anxiety, and anger, higher levels of happiness, and lower variability in anxiety and anger. These findings suggest that caregivers and friends are central to everyday emotion levels and variability and a more flexible emotion system in adolescents.
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A self‐report measure of changeable affect was developed, with a goal of identification of patterns of instability in mood. Scales measuring lability in anxiety, depression, anger, and hypomania, and labile shifts between anxiety and depression and hypomania and depression were constructed. These scales were then evaluated for internal consistency, retest reliability, score stability across samples, and for discriminant validity through assessment of association with measures of dysphoria and intensity of affect. The final versions of the scales are short scales that yield highly stable estimates of affect lability. It was noted that these scales are highly correlated in unselected students and it is believed that ongoing research with clinical populations will better allow for determination of the independence of these scales.
Several models of the dimensionality of emotional expressivity were examined in a multitraitmultimethod study. Targets and peer raters completed measures of the target's emotional expressivity (Berkeley Expressivity Questionnaire, BEQ; J. J. Gross & O. P. John, 1995; and a measure of emotion-specific expression) and the Big 5 personality dimensions. The results of structural equation modeling and analysis of variance revealed that an emotion-specific model was superior to models of valence-specific or unidimensional expressivity. The distinct emotions differed in their relations with the dimensions of the 5-factor model. These results were corroborated by self- and other reports. Finally, the degree of convergence between self- and other ratings differed between emotions, demonstrating the multidimensional character of emotional expressivity.
The literature on symptoms of depression has included diurnal changes in mood. The morning-worse pattern is commonly mentioned. This pattern is often associated with endogenous or vegetative symptoms (e.g., weight and appetite loss, loss of pleasure, psychomotor retardation). However, depression researchers have also identified an evening-worse pattern of mood. This pattern is sometimes thought to be associated with milder depressive symptoms, and may characterize chronic dysthymia rather than clinical depression. The present study examines a nonclinical sample to test the hypothesis that an evening-worse diurnal pattern of mood would be associated with trait neuroticism, anxiety, and subclinical depressive symptoms. An experience sampling methodology was employed to assess mood three times a day for 60 consecutive days. This allowed us to calculate a reliable aggregate score for diurnal mood patterns. The evening-worse pattern was associated with many neurotic features, with scores on depression and anxiety measures, and with a cognitive style indicative of hopelessness. Discussion focuses on how an evening-worse diurnal pattern of mood may be indicative of mild subclinical depression, chronic dysthymia, or personality traits associated with negative affectivity.
The question of how affect arises and what affect indicates is examined from a feedback-based viewpoint on self-regulation. Using the analogy of action control as the attempt to diminish distance to a goal, a second feedback system is postulated that senses and regulates the rate at which the action-guiding system is functioning. This second system is seen as responsible for affect. Implications of these assertions and issues that arise from them are addressed in the remainder of the article. Several issues relate to the emotion model itself; others concern the relation between negative emotion and disengagement from goals. Relations to 3 other emotion theories are also addressed. The authors conclude that this view on affect is a useful supplement to other theories and that the concept of emotion is easily assimilated to feedback models of self-regulation.