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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’

emotions.

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

examination).

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: schow@nd.edu

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

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.

1

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

dysfunctions.

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).

2

Over time, this heightened level

of sadness dissipates, and the individual slowly returns to his or her

usual baseline (Y⫽0), 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

1

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.

2

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.

209

EMOTION AS A THERMOSTAT

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., Y⬙is negative) in sadness. In other words, the decrease in

sadness unfolds at a progressively slower pace. Then, acceleration

in sadness (i.e., Y⬙becomes 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.

210 CHOW, RAM, BOKER, FUJITA, AND CLORE

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⬙

i共t兲⫽

Yi共t兲⫹

Y⬘共t兲, (1)

where Y⬙

i

(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

i

(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.

211

EMOTION AS A THERMOSTAT

describes the frequency of oscillation.

3

The parameter

describes

how promptly person ireturns to his or her set point after

perturbation.

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

Model

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,t⫺1, t⫺2, and so on. This

technique is typically used in approaches wherein the data being

3

Formally, the frequency,

, is equal to 1

2

冑

⫺

. Note that only

negative values of

are interpretable from a mathematical standpoint.

212 CHOW, RAM, BOKER, FUJITA, AND CLORE

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).

4

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

t

,S

t–1

,S

t–2

,

and S

t–3

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

4

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

t

, dep

t⫺1

, dep

t⫺2

, and dep

t⫺3

. The vector of depression scores at

time t, dep

t

, would contain data from time t⫽4 to 100, dep

t⫺1

would contain

data from time t⫽3 to 99, dep

t⫺2

would contain data from t⫽2 to 98, and

dep

t⫺3

would contain data from time t⫽1 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.

213

EMOTION AS A THERMOSTAT

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

and

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; a⫽factor loading of SAD on depression constrained to be invariant over time; b⫽factor 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

S_SAD

,

S_DSAD

⫽covariance between the shock terms

associated with SAD and DSAD; U_D2SAD ⫽residuals in D2SAD not accounted for by the model. S

t

,S

t⫺1

,

S

t⫺2

, and S

t⫺3

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.

214 CHOW, RAM, BOKER, FUJITA, AND CLORE

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

are

obtained on the basis of information from the first to fourth

derivatives.

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

t

,S

t–1

,S

t–2

, and S

t–3

) 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. t⫽time; 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

S_SAD

,

S_DSAD

⫽covariance between the shock terms associated

with SAD and DSAD; U_D4SAD ⫽residuals in D4SAD not accounted for by the model.

215

EMOTION AS A THERMOSTAT

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

Variable SAD DSAD D2SAD D3SAD D4SAD

L⫽matrix of factor loadings

Unhp

t⫺3

1⫺1.5t(⫺1.5t)

2

/2 (⫺1.5t)

3

/6 (⫺1.5t)

4

/24

Unhp

t⫺2

1⫺0.5t(⫺0.5t)

2

/2 (⫺0.5t)

3

/6 (⫺0.5t)

4

/24

Unhp

t⫺1

1 0.5t(0.5t)

2

/2 (0.5t)

3

/6 (0.5t)

4

/24

Unhp

t

1 1.5t(1.5t)

2

/2 (1.5t)

3

/6 (1.5t)

4

/24

Dep

t⫺3

aa(⫺1.5t)a(⫺1.5t)

2

/2 a(⫺1.5t)

3

/6 a(⫺1.5t)

4

/24

Dep

t⫺2

aa(⫺0.5t)a(⫺0.5t)

2

/2 a(⫺0.5t)

3

/6 a(⫺0.5t)

4

/24

Dep

t⫺1

aa(0.5t)a(0.5t)

2

/2 a(0.5t)

3

/6 a(0.5t)

4

/24

Dep

t

aa(1.5t)a(1.5t)

2

/2 a(1.5t)

3

/6 a(1.5t)

4

/24

Lone

t⫺3

bb(⫺1.5t)b(⫺1.5t)

2

/2 b(⫺1.5t)

3

/6 b(⫺1.5t)

4

/24

Lone

t⫺2

bb(⫺0.5t)b(⫺0.5t)

2

/2 b(⫺0.5t)

3

/6 b(⫺0.5t)

4

/24

Lone

t⫺1

bb(0.5t)b(0.5t)

2

/2 b(0.5t)

3

/6 b(0.5t)

4

/24

Lone

t

bb(1.5t)b(1.5t)

2

/2 b(1.5t)

3

/6 b(1.5t)

4

/24

A⫽matrix with regressions among latent derivatives

SAD 0 0 0 0 0

DSAD 0 0 0 0 0

D2SAD

000

D3SAD 0

00

D4SAD 0 0

0

S⫽covariance matrix of intraindividual variability and residuals

S_SAD S_DSAD S_D2SAD S_D3SAD U_D4SAD

S_SAD V

S_SAD

—— — —

S_DSAD C

SAD, DSAD

V

S_DSAD

———

S_D2SAD 0 0 V

S_D2SAD

——

S_D3SAD 0 0 0 V

S_D3SAD

—

U_D4SAD 0 0 0 0 V

U_D4SAD

Note. SAD ⫽intensity of sadness; DSAD, D2SAD, D3SAD, and D4SAD ⫽the first, second, third, and fourth

derivatives of sadness, respectively; a⫽factor loading of SAD on depression; b⫽factor 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

S_SAD

,V

S_DSAD

,V

S_D2SAD

, and V

S_D3SAD

⫽the variances of the shock terms; C

SAD, DSAD

⫽

covariance between the shock terms associated with SAD and DSAD; U_D4SAD ⫽residuals in D4SAD not

accounted for by the model, and V

U_D4SAD

is its associated variance.

216 CHOW, RAM, BOKER, FUJITA, AND CLORE

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),

5

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

data.

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.

Results

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,

⌬

2

(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.

5

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.

217

EMOTION AS A THERMOSTAT

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).

Discussion

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

models.

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

lines).

Table 3

Estimated Values of

and

Based on All Participants’ Data

Emotion

SE Period in days

SE

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.

218 CHOW, RAM, BOKER, FUJITA, AND CLORE

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.

219

EMOTION AS A THERMOSTAT

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.

A1

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)

A2

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).

A3

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.

A1

Under a special condition where [(⫺|

|/2)

2

⫺|

|] is less than zero, the

integral solution for the damped oscillator model in Equation 1 (Zill, 1993)

is expressed as

Y共t兲⫽e⫺兩

兩

2t

冋

c1cos

冑

兩

兩⫺

冉

兩

兩

2

冊

2

t⫹c2sin

冑

兩

兩⫺

冉

兩

兩

2

冊

2

t

册

, (A1)

where c

1

and c

2

are arbitrary constants derived from one’s initial level and

rate of change. In the case of

⫽0, it simplifies to

Y共t兲⫽c1cos

冑

兩

兩t⫹c2sin

冑

兩

兩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).

A2

Given a known oscillation period of

, the theoretical value of

can

be computed as

⫽⫺

冉

2

冊

2

, where

represents the period of

oscillation. For example, a 7-day cycle would yield an

estimate of

⫽⫺

冉

2

7

冊

2

⫽⫺.80.

A3

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.

222 CHOW, RAM, BOKER, FUJITA, AND CLORE

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.

223

EMOTION AS A THERMOSTAT

Appendix B

A Comparison Between the Fourth-Order Differential Structural Approach and Growth Curve

Models

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.,

SAD

,

DSAD

, and

D2SAD

, 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.

224 CHOW, RAM, BOKER, FUJITA, AND CLORE

moving average model process (more specifically an AMRA[2,1] model)

B1

process on the latent derivatives. Essentially, the

and

regression

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

␣

1

) between 0 and

2 and a negative lag-2 autoregressive weight (denoted as

␣

2

) between –1

and –

␣

1

2

/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

and

(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.

B1

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 䡲

225

EMOTION AS A THERMOSTAT