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ORIGINAL RESEARCH
published: 19 April 2016
doi: 10.3389/fpsyg.2016.00522
Frontiers in Psychology | www.frontiersin.org 1April 2016 | Volume 7 | Article 522
Edited by:
Luiz Pessoa,
University of Maryland, USA
Reviewed by:
Belinda Jayne Liddell,
University of New South Wales,
Australia
H. Andac Demirtas-Madran,
Ba ¸skent University, Turkey
*Correspondence:
Radek Trnka
trnkar@volny.cz
Specialty section:
This article was submitted to
Emotion Science,
a section of the journal
Frontiers in Psychology
Received: 16 April 2015
Accepted: 29 March 2016
Published: 19 April 2016
Citation:
Trnka R, La ˇ
cev A, Balcar K, Kuška M
and Tavel P (2016) Modeling Semantic
Emotion Space Using a 3D
Hypercube-Projection: An Innovative
Analytical Approach for the
Psychology of Emotions.
Front. Psychol. 7:522.
doi: 10.3389/fpsyg.2016.00522
Modeling Semantic Emotion Space
Using a 3D Hypercube-Projection: An
Innovative Analytical Approach for
the Psychology of Emotions
Radek Trnka 1, 2*, Alek La ˇ
cev 3, Karel Balcar 1, Martin Kuška 1and Peter Tavel1, 3, 4
1Science and Research Department, Prague College of Psychosocial Studies (PVSPS), Prague, Czech Republic, 2Faculty of
Humanities, Charles University in Prague, Prague, Czech Republic, 3Olomouc University Social Health Institute (OUSHI),
Palacky University in Olomouc, Olomouc, Czech Republic, 4Health Psychology Unit – Institute of Public Health, Medical
Faculty, P. J. Safarik University, Kosice, Slovakia
The widely accepted two-dimensional circumplex model of emotions posits that most
instances of human emotional experience can be understood within the two general
dimensions of valence and activation. Currently, this model is facing some criticism,
because complex emotions in particular are hard to define within only these two
general dimensions. The present theory-driven study introduces an innovative analytical
approach working in a way other than the conventional, two-dimensional paradigm.
The main goal was to map and project semantic emotion space in terms of mutual
positions of various emotion prototypical categories. Participants (N=187; 54.5%
females) judged 16 discrete emotions in terms of valence, intensity, controllability and
utility. The results revealed that these four dimensional input measures were uncorrelated.
This implies that valence, intensity, controllability and utility represented clearly different
qualities of discrete emotions in the judgments of the participants. Based on this data, we
constructed a 3D hypercube-projection and compared it with various two-dimensional
projections. This contrasting enabled us to detect several sources of bias when working
with the traditional, two-dimensional analytical approach. Contrasting two-dimensional
and three-dimensional projections revealed that the 2D models provided biased insights
about how emotions are conceptually related to one another along multiple dimensions.
The results of the present study point out the reductionist nature of the two-dimensional
paradigm in the psychological theory of emotions and challenge the widely accepted
circumplex model.
Keywords: emotions, emotional experience, affect, semantic, dimensions, circumplex model
INTRODUCTION
Language is a primary tool of emotion research and the primary access to the affective experience
of the self and others (Storm and Storm, 2005). The key importance of research instruments
inspired in psycholinguistics is apparent when exploring the research designs of empirical studies
in this field. Aside from some experimental (e.g., Gerdes et al., 2013; Grol and De Raedt, 2014;
Yu et al., 2015) and observational research (e.g., Jensen, 2014; Rohlf and Krahé, 2015), a huge
number of current empirical studies used linguistic properties when exploring the terrain of
Trnka et al. Emotion Space in 3D Hypercube-Projection
human emotional experience (e.g., Crutch et al., 2013; Bayer and
Schacht, 2014; Gallant and Yang, 2014; Schindler et al., 2014).
People construct and understand their emotional experience
through the abstract representations of emotions in language. For
this reason, it is not surprising that psycholinguistics attract many
researchers who seek out the links between language, cognitive
processes and emotional experience (Aznar and Tenenbaum,
2013; Fisher et al., 2014; Verhees et al., 2015). The present study
follows just this line of research and investigates interrelations
between the experiential component of emotions and the
anchoring of emotional concepts in language.
First of all, it is necessary to say that human emotional life is
fascinating due to its very high degree of complexity (Grühn et al.,
2013). The terrain of human emotional experience seems to be a
little bit complicated and non-transparent, which, on the other
hand, attracts psychologists who are motivated to explore fields
that are not simply structured. In contrast, we can sometimes see
a tendency to shape research designs in a simplified manner in an
effort to provide readers with clear solutions. The effort to present
scientific results in a simple and structured form may even lead
to a superfluous reduction of the phenomena. As an example, it
is very difficult to approach the complexity of human emotions
within only two general dimensions (Roberts and Wedell, 1994;
Fontaine et al., 2007). The newest research findings on the global
meaning structure of the emotion domain pointed out that more
than two dimensions are needed to describe the nature of the
human emotional experience sufficiently (Fontaine et al., 2007;
Fontaine and Scherer, 2013). The present study was inspired by
this current progress in emotion research and continues in the
further development of a multidimensional approach to the study
of human emotions.
At the beginning we will focus our attention on the conceptual
embedding of psychological research of emotions. The general
organization of the semantic field, linguistic labels and the
conceptual structure actually attract the attention of researchers
(Boutonnet et al., 2014; Kuehnast et al., 2014; Troche et al., 2014).
A proper defining of various theoretical concepts is necessary at
the beginning of our work.
Emotional meaning systems are culturally-specific systems
that shape the ways in which people experience, express, organize,
and modulate their emotions (Parkinson et al., 2005). Emotional
meanings specifically are abstract devices that shape general
emotion knowledge; they may be subjectively chosen by the
cognitive processing of an individual from a collective emotional
meaning system shared in a given culture.
Emotion concepts are abstract representations of experiences
of various emotions in one’s mind (Oosterwijk et al., 2009).
They are components of the general emotion knowledge of
an individual (Oosterwijk et al., 2009) that is formed by
storing previously experienced sensory, motor, physiological and
introspective states. Faucher and Tappolet (2008) distinguished
three forms of knowledge about emotions: conceptual knowledge
about emotions, personal knowledge about emotions and
knowledge about others’ emotions.
Some emotion concepts represent a group of feelings that is
qualitatively different from other emotional experiences. They
are called emotion categories (Russell and Lemay, 2000), emotion
prototype categories (Reilly and Seibert, 2003) or prototypes
(Parkinson et al., 2005). Specific emotion terms, emotion words
or emotion names mean linguistic labels for emotion categories
in language (Hupka et al., 1999). Mutual relations between the
semantic fields of all emotion prototypes in one’s mind define
the overall structure of the semantic space for emotions (Scherer,
2005), also called the subjective emotion space (Sokolov and
Boucsein, 2000; Trnka, 2013). In a similar vein, Reisenzein
and Schimmack (1999) used the term “affect structure” as
the constitutional makeup and interrelations between emotion
prototypes. The subjective emotion space is not equally limited
in each person, and its size is given by the maximum extremes
and minimal minimums of the range of each of the possible
dimensions of experience (Trnka, 2013). The extent of subjective
emotion space may change throughout the life course, due to the
process of evolution of emotion concepts and emotion prototypes
over time (Scherer, 2005).
The fascinating question is what qualities does subjective
emotion space have? The discussion about the dimensionality
of human emotional experience is dynamic and long-lasting
(see Scherer, 2013, for an overview). Currently, most empirical
studies have utilized the two-dimensional circumplex model
(Russell and Lemay, 2000) that includes only two general
dimensions of valence and activation (Kuppens et al., 2013).
Even as the most frequently used, the two-dimensional model
faces several problems, especially, when one thinks about the
positions of various discrete emotions within the dimensions
of valence and activation (Roberts and Wedell, 1994; Trnka,
2013). To understand the complex structure and varieties of
human emotional experience on two quite general dimensions
is reductionist. The question is if it is possible to investigate
complex emotions like shame, guilt, envy, or compassion within
a simple two-dimensional paradigm working with dimensions
of valence and activation? We offer to disagree. It can be said
that the above-mentioned emotions are somehow pleasant or
unpleasant and that they are somehow intense, but, in doing
so many slight variations of such complex feelings remain
hidden. Also, assessing the semantic similarity between complex
emotions in the semantic space of individuals is problematic
when using only the above-mentioned basic dimensions of
valence and activation. We argue that using a simple two-
dimensional model for an in-depth analysis of the human
emotional experience may lead to a risk of reduction of the
complexity of the phenomena.
Further, the critique of utilizing only a limited number of basic
dimensions was not focused only on complex emotions, but on
basic emotions, as well. Roberts and Wedell (1994) pointed out
the high degree of reductionism when utilizing the dimension
of valence and activation within the framework of simple multi-
dimensional scaling (MDS) techniques. For instance, anger and
fear are usually placed very near to each other in two-dimensional
space, since they mean a high amount of arousal with a strongly
negative valence (Russell, 1980; Watson and Tellegen, 1985;
Larsen and Diener, 1992). This might lead to the impression
that these two emotions are very similar to each other (Roberts
and Wedell, 1994), yet in reality they both have their specific
experiential character.
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Trnka et al. Emotion Space in 3D Hypercube-Projection
Given the above-mentioned arguments, it seems
indispensable to think about further conceptual and
methodological development in this field. One way of
overcoming the limitations of the two-dimensional model
for assessment of the complex structure of emotional experience
is to explore some possibilities of a multidimensional conceptual
embedding of emotion research, for example, as proposed by
the theory of multi-dimensional emotional experience (Trnka,
2013).
Sokolov and Boucsein (2000) pointed out that the general
preference of researchers for a reduction of emotion space
to a small number of dimensions is probably influenced by
attempts to visualize such a dimensional system within familiar
Euclidian geometrical space. However, the current inductive
study of Fontaine and Scherer (2013) showed that the global
meaning structure of emotions can be optimally described by
four dimensions: valence, arousal, power/control, and novelty.
In a similar vein, Sokolov and Boucsein (2000) introduced an
alternative, four-dimensional theoretical model for approaching
subjective emotion space. They showed that discrete emotions
can be analyzed on a hypersphere in four-dimensional space. The
concept of a hypersphere probably provides a less-reductionist
paradigm for the investigation of human emotional experience.
Actually, it is not clear how many and what kinds of
dimensions could be optimal for the construction of a
hyperspace that would fit well for the analysis of human
emotional experience. Empirical studies employing more than
two-dimensional solutions of emotional experience are almost
lacking, with some exceptions (Fontaine et al., 2007; Fontaine
and Scherer, 2013). Therefore, the field is now open for
new, experimental work exploring various multidimensional
approaches in the psychological research of emotions. This was
also the challenge and the starting point for the present study.
AIMS AND SIGNIFICANCE
The present study brings further knowledge to a new, developing
field of multidimensional research in the psycholinguistics
of emotions. We challenge the use of the widely accepted
theoretical two-dimensional circumplex model working only
with the dimensions of valence and activation. The goal of
the present study was not to solve the question of how many
dimensional qualities subjective emotion space actually has but
to examine the use of a new model working with more than
two dimensions in the analysis of the overall structure of the
semantic space for emotions. More specifically, we used four
different dimensional measures as input data for this model. We
hypothesized that all four input dimensional measures will be
uncorrelated and, therefore, can be considered as clearly different
qualities of subjective understanding of emotion prototypes.
We follow current progress in psycholinguistics (Fontaine and
Scherer, 2013) and introduce an innovative analytical approach
for emotion research working in other than the conventional,
two-dimensional paradigm.
We focused the study on methodological improvements
in data gathering that fit well for the assessment of slight
differences in the mutual positions between various emotion
prototypes in the participants’ judgments (see the Materials and
procedure subsection for more details). The main goal was to
construct a three-dimensional model based on this data. The
main output of the present study is, therefore, the construction
of a 3D hypercube-projection including the positions of the main
emotion prototypes as the basic constitutive elements of semantic
emotion space. Various aspects of the constructed 3D hypercube-
projection and limitations of the proposed methodological
approach are discussed in the final part of the study.
METHODS
Subjects
Participants were 187 university students with an age ranging
from 19 to 38 years (M=22.6; SD =3.2). Sex distribution was
54.5% female. All participants were Czech native speakers and
participated voluntarily in the study. The research design was
approved by the institutional ethics committee and also by the
principal governmental research institution, the Czech Science
Foundation. All participants signed the informed written consent
with their participation in the study.
Materials and Procedure
At the beginning, participants filled in the basic demographical
characteristics. Following a discrete emotions paradigm
(Kunzmann et al., 2014), 16 emotion words were judged by
participants: anger, fear, sadness, happiness, disgust, hope, love,
hate, contempt, guilt, compassion, shame, gratefulness, envy,
disappointment, jealousy. These emotion prototypes cover
five basic emotions (anger, fear, sadness, happiness, disgust)
as well as complex emotions like hope, love, hate, contempt,
guilt, compassion, shame, gratefulness, envy, disappointment
and jealousy. The whole judging procedure was conducted
in the Czech language. All of the emotion words used were
non-synonymous.
Participants judged all 16 emotion words four-times. A
separate list including the 16 above-mentioned emotion words
was provided for each judgment. Each of four judgments
measured subjective understanding of the provided emotion
words on another dimension. First, participants judged all
16 emotion words on the dimension of valence. A 10 cm
horizontal line was provided next to each of the 16 emotion
words. Participants were asked to rate the degree to which
they experienced this emotion as pleasant/unpleasant using the
instruction: “Please, mark on the following lines how much you
experience the particular emotion as pleasant or unpleasant.” The
participants then marked the position of each emotion word on
the 10 cm lines provided next to each of the 16 emotion words.
The same tool was also used in the following three judgments.
Second, the participants judged the same 16 emotions on the
dimension of intensity introduced by the instruction: “Please,
mark on each line how much you experience a particular emotion
as calm or aroused.” Third, the same 16 emotion words were
rated on the dimension of control. Participants were provided
with the instruction: “Please, mark on each line how much you
are able to control a particular emotion in the sense that it does
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Trnka et al. Emotion Space in 3D Hypercube-Projection
not influence your thinking or behavior”. Fourth, the same 16
emotion words were rated on the dimension of utility using the
instruction: “Please, mark on each line how much you perceive
that the following emotions are harmful or beneficial for you?” A
separate sheet was used for each of the four judgments.
The used measurement was dimensional, but the data
obtained from the participants’ judgments will be called “aspects”
or “input dimensional aspects” throughout the remaining part
of the paper. This change was made to provide a clear
differentiation between dimensional data entering the analysis
and the dimensions that were constructed in the course of data
analysis.
Data Analysis
The above-described methodological instrument for data
gathering enabled a fine-grained assessment of slight differences
in the mutual positions between the semantic fields of various
emotion prototypes in the semantic spaces of participants. By
measuring the position of the marked points on the line segment,
the millimeter positions from the left end were obtained, thus
providing a scale from 0 to 100 (mm). These positions varying
between 0 and 100 were then entered into the data analysis and
were used for the construction of a 3D hypercube-projection.
This assessment of semantic fields on a line segment is more
fine-grained in comparison with standard Likert-type scales
using five of seven non-continuous options. Therefore, the 3D
hypercube-projection constructed based on this methodology
captured the mutual positions of participants’ judgments more
accurately than projections based on data from a Likert-type
scale.
To obtain the multidimensional emotional space and assess
the necessary number of dimensions needed to properly explain
the relations between the individual emotion prototypes,
multidimensional scaling based on correlations between
the individual emotions in each of the aspects was used.
Multidimensional scaling is a useful tool to help understand
people’s judgments considering the similarity of members or
objects and thus to produce inductive, but empirically derived
“maps of elements.” In multidimensional scaling we try to find a
configuration of points in space in which the distance between
these points match as close as possible the original proximities
between the objects (Busing, 1998). Thus, the MDS technique
enables us to construct a 3D hypercube-projection based on
the perceptions of a diverse set of individuals who are blind to
the exact purpose of the give study. PROXSCAL with multiple
matrices as the source (as provided by the statistical software
SPSS 20) was used to examine dimensions within the data. This
algorithm minimizes raw normalized stress, and thus the result
is a far more “honest” Euclidean space. Compared with other
existing scaling options PROXSCAL has a number of important
advantages (Busing et al., 1997). As a source matrices for MDS
Spearman correlation matrices—one for each aspect was created
and transformed to proximities so that a high positive correlation
meant high proximity, while a high negative correlation meant
low proximity (i.e., r=1 was transformed to distance 0, r=0
was transformed to distance 1 and r= − 1 was transformed to a
distance of 2, etc.).
RESULTS
Descriptive Data
The four originally measured aspects—i.e., valence, intensity,
control and utility—were not significantly correlated (α=
0.05) when using Bonferroni correction for multiplicity. Table 1
includes the arithmetic mean (M) scores for each emotion in
each aspect as well as its standard deviation (SD). Each emotion
was transformed to values from 0 to 100 according to a mark
in the answer sheet—where each 1 point corresponds with
1 mm distance from the left-side beginning of the line. Thus,
for valence, the more the numbers drop below 50, the more
unpleasantness they express and vice versa for numbers where
a number reaching 100 means maximum pleasantness. The
numbers for arousal (ranging from 0—calm to 100—aroused),
control (ranging from 0—uncontrolled to 100—controlled) and
finally utility (ranging from 0—harmful to 100—beneficial)
function in a similar way.
Multidimensional Scaling
To simply analyze proximities between emotions in these original
four aspects we can construct a matrix using arithmetic means as
coordinates in 4D space (using w-, x-, y-, and z-axis). To analyze
proximities we assume the same weight of all aspects and thus use
the simple distance (s) obtained by the following formula (where
w, x, y, and z are the differences between the values of arithmetic
means of two emotions in each of the four dimensions):
s=qw2+x2+y2+z2
The lower the number in the resulting proximity matrix, the
closer the two emotions are in theoretical 4D space. The range
of possible distances is (0–200) where 200 is the length of the
hypotenuse of a theoretical hypercube with each side of length
of 100. The distances are listed in Table 2.
Following the descriptive analysis, we used the individual
correlation matrices (as described above) for each of the original
aspects and obtained PROXSCAL solutions for one through
twelve dimensions. Based on these results (see Figure 1), a three-
dimensional solution with a total fit of 0.96 was chosen, because
the increase in the total fit by adding a fourth dimension was very
small.
The values of stress obtained with a simplex start and
the amount of variance accounted for by a three-dimensional
solution are shown in Table 3. The iterations were stopped at 34
because the S-stress improvement was less than 0.001. However,
there is no rule of thumb to interpret the quality based on the
normalized raw stress results. Busing et al. (1997),Busing (1998)
calculates the total fit by subtracting the total stress from 1. We
can than borrow a rule of thumb from Kruskal (1964): 0.2 =poor;
0.1 =fair; 0.05 =good; 0.025 =excellent and 0.0 =perfect.
A two dimensional (2D) representation of each of three
resulting dimensions shows clear groups of emotions on three
different planes. Figure 2 of the Dimensions 1 and 2 plane
indicates that there are several clusters of emotions close
together—happiness and love accompanied also by slightly more
distant hope and gratefulness on one side of the plane, and anger
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Trnka et al. Emotion Space in 3D Hypercube-Projection
TABLE 1 | Descriptive results of assessment of emotions (whole sample, n=187).
Emotion Valence Arousal Control Utility
M SD M SD M SD M SD
Anger 25.3 19.4 79.4 16.9 47.7 27.4 32.4 23.1
Fear 18.1 17.8 75.6 18.7 50.0 24.9 39.9 24.4
Sadness 17.4 17.3 47.1 26.6 49.9 25.6 36.1 21.9
Happiness 91.8 12.0 69.1 30.4 73.9 62.2 89.5 11.0
Disgust 26.1 17.5 53.8 21.9 60.4 22.6 35.9 16.8
Hope 77.1 17.4 53.6 28.1 71.7 20.4 79.5 16.9
Love 91.2 14.9 67.9 33.7 55.2 31.9 90.2 14.3
Hate 18.7 18.2 65.9 21.7 48.8 24.3 24.7 20.7
Contempt 22.1 17.8 51.6 43.8 57.2 23.7 27.4 17.8
Guilt 15.6 13.9 55.4 24.4 45.2 22.9 39.0 23.5
Compassion 51.4 20.6 39.1 21.0 63.4 20.8 64.5 19.7
Shame 24.3 15.9 60.6 20.9 45.5 23.7 40.9 20.8
Gratefulness 68.3 18.3 40.4 23.9 69.2 20.6 73.0 16.2
Envy 22.9 16.2 56.5 49.3 65.8 24.6 22.9 18.5
Disappointment 17.3 14.1 53.5 25.6 45.3 24.7 41.0 20.3
Jealousy 23.9 17.7 71.7 19.7 48.0 26.6 26.5 21.7
TABLE 2 | Original emotional proximities in theoretical 4D space based on originally measured aspects.
Distance (s)
Fear
Sadness
Happiness
Disgust
Hope
Love
Hate
Contempt
Guilt
Compassion
Shame
Gratefulness
Envy
Disappointment
Jealousy
Anger 11.3 33.6 92.1 28.8 78.4 88.7 16.9 30.0 26.9 59.9 20.8 74.1 30.8 28.6 9.8
Fear 28.8 92.3 25.8 77.5 89.3 18.0 28.3 21.0 56.9 16.9 72.4 25.3 22.7 15.2
Sadness 97.2 15.2 77.2 93.9 22.1 13.1 10.1 47.0 16.5 66.1 21.5 9.2 27.2
Happiness 87.2 23.8 18.8 100.9 96.5 96.8 57.2 88.4 40.9 114.0 94.8 96.2
Disgust 67.9 86.0 21.5 10.3 18.8 40.9 17.2 58.4 36.6 18.2 23.8
Hope 28.2 84.1 77.1 78.2 34.1 70.8 17.2 96.4 75.9 80.7
Love 97.9 94.8 92.6 56.0 84.0 42.0 114.2 90.5 93.0
Hate 17.2 18.4 59.9 18.2 76.6 31.7 20.8 8.0
Contempt 18.3 49.3 20.1 67.0 29.1 18.8 22.2
Guilt 50.2 10.3 68.8 44.7 3.2 22.3
Compassion 45.6 19.9 72.6 47.4 59.1
Shame 62.8 45.8 10.0 18.3
Gratefulness 87.6 66.1 74.6
Envy 28.0 23.6
Disappointment 24.3
and jealousy accompanied with slightly more distant fear on the
other side of the plane; the third vertex of the imaginary triangle
consists of a larger cluster of envy, disappointment, sadness and
contempt accompanied also by the more distant hate, disgust and
possibly also shame and guilt. Other emotions are somewhere in
between these clusters (e.g., fear) or somewhat outside creating
its own category, such as compassion.
A different two-dimensional representation of Dimensions 2
and 3 (Figure 3) provides a different image. The emotions are
further apart and clusters are possibly less intuitive, e.g., anger,
fear and love clustered together, while jealousy, happiness, disgust
and possibly hate create another cluster; a third is formed by
sadness and shame; a fourth by hope, disappointment and guilt;
and a fifth by contempt and gratefulness. Compassion seems to
again stand apart, as does envy.
The final available 2D representation provides a look at
Dimensions 1 and 3 (Figure 4). In this case there is a plausible
cluster containing envy, disgust, contempt, hate and jealousy
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Trnka et al. Emotion Space in 3D Hypercube-Projection
FIGURE 1 | Scree plot. Total fit calculated according to Busing et al. (1997).
on one hand; another created by guilt, disappointment, fear
and anger; a third one consisting of compassion, happiness and
gratefulness’ and finally one of love and hope. Sadness and shame
seem to be somewhat separated from the others, or possibly they
create some cluster of their own.
A more significant separation of the emotions across all
three dimensions in a 3D representation and aggregation of
the original four emotional aspects can be seen in Figure 5.
It seems that Dimension 1 corresponds somewhat with the
original aspect of valence, where highly pleasant items (e.g.,
happiness, hope, love) are on the opposite side from unpleasant
items (e.g., hate, disgust, sadness or fear), but it mixes with
some of the original utility dimension as well. Dimension 2
seems to correspond mostly with arousal, although in reversed
numbering dividing on one side highly arousing items (e.g.,
anger, fear, happiness, love) and on the other those which
tend to be rather calm (e.g., compassion, guilt, gratefulness).
Finally, Dimension 3 seems to correspond somewhat with
control and partially with utility, as well. It lists on one
side items that are controllable, such as envy or disgust,
and on the rather uncontrollable side emotions of shame
and sadness. This interpretation should not be overestimated
because certain ratios of original four emotional aspects in the
aggregated three-dimensional hypercube-model have not been
determined.
Table 4 lists the final positions of the emotions in this
model, while Table 4 lists the distances of the emotions in this
MDS-constructed model.
Similarly to the construction of the proximity matrix from
the original data, we can obtain such a matrix from the MDS-
results using the final positions as coordinates on x-, y-, and z-
axes. The same rule applies, i.e., the lower the number in the
resulting proximity matrix, the closer the two emotions are in
MDS-aggregated 3D-space. The distances are listed in Table 5.
It is also possible to assess the correlation between
the original 4D-space proximities matrix and the resulting
MDS-aggregated 3D-space proximities matrix. The Pearson
correlation is r=0.758 (sig <0.001, N=120);
while statistical procedures obviously reduced some amount of
information, there is still very strong correlation in the resulting
model.
TABLE 3 | S-stress improvement (for a three-dimensional solution).
Iteration S-Stress Improvement
0 0.42697
1 0.09666 0.33030
2 0.07608 0.02059
3 0.06733 0.00875
4 0.06154 0.00579
5 0.05736 0.00418
6 0.05421 0.00315
7 0.05170 0.00251
8 0.04964 0.00206
9 0.04792 0.00172
10 0.04648 0.00144
11 0.04528 0.00121
12 0.04427 0.00101
13 0.04342 0.00084
14 0.04272 0.00071
15 0.04212 0.00060
16 0.04161 0.00051
17 0.04118 0.00044
18 0.04080 0.00038
19 0.04047 0.00033
20 0.04017 0.00030
21 0.03991 0.00026
22 0.03967 0.00024
23 0.03946 0.00021
24 0.03926 0.00020
25 0.03908 0.00018
26 0.03891 0.00017
27 0.03876 0.00015
28 0.03862 0.00014
29 0.03848 0.00013
30 0.03836 0.00013
31 0.03824 0.00012
32 0.03813 0.00011
33 0.03802 0.00011
34 0.03792 0.00010
DISCUSSION
The present study introduces an innovative analytical tool
for approaching emotions in other than the conventional,
two-dimensional paradigm. The widely accepted theoretical
two-dimensional circumplex model implies that the character
of most human emotions is possible to define within two
general dimensions of valence and activation. The present
study challenges this theoretical paradigm and provides
new inspiration for further development of emotion
theory.
First of all, the results of the present study indicate that
various discrete emotions have more qualities perceived by
individuals than only valence and activation. Participants in our
study judged 16 discrete emotions in terms of valence, intensity,
controllability and utility. All of these four kinds of judgments
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Trnka et al. Emotion Space in 3D Hypercube-Projection
FIGURE 2 | Two-dimensional projection of discrete emotions on Dimension 1 and Dimension 2.
FIGURE 3 | Two-dimensional projection of discrete emotions on Dimension 2 and Dimension 3.
were not significantly correlated and represent independent
qualities within the participants’ subjective knowledge about
emotions. This finding diverges from the assumption that two
basic qualities, valence and activation, are sufficient for describing
the prototypical character of various discrete emotions. Neither
controllability nor utility were significantly correlated with
valence or with intensity, and they therefore represent clearly
different qualities of discrete emotions.
The independence of four kinds of dimensional measurement
in our study justified the later construction of a four-dimensional
model of semantic emotion space, which was then transformed
into a 3D hypercube-model (Figure 5). The 3D hypercube-
projection (Figure 5) was constructed based on the mutual
positions of emotion prototypes that were extracted from
participants’ judgments of four different qualities of emotion
words. This model helps to analyze and interpret the structure
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Trnka et al. Emotion Space in 3D Hypercube-Projection
FIGURE 4 | Two-dimensional projection of discrete emotions on Dimension 1 and Dimension 3.
FIGURE 5 | Aggregated three-dimensional hypercube-model of emotional space based on the original four measured emotional aspects.
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Trnka et al. Emotion Space in 3D Hypercube-Projection
of semantic emotion space in a more complex manner than in
cases of standard, two-dimensional analytical projections.
The present study revealed the following very important
insights: (1) sources of bias when working in a two-dimensional
paradigm were identified; (2) emotions that represent limits
or frontiers of semantic emotion space were found; (3) no
emotion prototype was settled in the central area of 3D
emotion space; (4) the mutual multidimensional positions of
emotional prototypes in the 3D hypercube-projection enable
the multidimensional semantic similarity of used emotional
TABLE 4 | Final positions in MDS generated mean-centered 3D model.
Emotion Dimension 1 Dimension 2 Dimension 3
Anger −0.133 −0.588 −0.296
Fear −0.386 −0.389 −0.264
Sadness −0.412 0.095 −0.481
Happiness 0.794 −0.347 0.273
Disgust −0.478 −0.165 0.409
Hope 0.795 0.163 −0.182
Love 0.753 −0.335 −0.159
Hate −0.548 −0.046 0.193
Contempt −0.483 0.256 0.280
Guilt −0.190 0.484 −0.199
Compassion 0.315 0.537 0.162
Shame 0.043 0.192 −0.559
Gratefulness 0.652 0.312 0.256
Envy −0.277 0.183 0.496
Disappointment −0.345 0.183 −0.237
Jealousy −0.101 −0.535 0.308
prototypes to be defined more clearly than when working in
a two-dimensional paradigm, (5) the mutual multidimensional
positions of emotional prototypes in the 3D hypercube-model
enabled the pairs of emotions that are opposite in terms of their
multidimensional semantic similarity to be identified.
Most importantly, the results of the present study pointed out
the risk of confusion when interpreting data based on the two-
dimensional theoretical paradigm. All of the above-mentioned
key insights will be discussed below.
Sources of Biases in the Two-Dimensional
Model of Emotion
In the following text, our two-dimensional projections
(Figures 2–4) will be used as hypothetical examples of
projections from studies working with measures of qualities of
discrete emotions only within two dimensions. The reader may
compare the two-dimensional projections (Figures 2–4) with
the 3D hypercube-projection (Figure 5) in the results section of
this study. All of these projections are based on the same data
set. Such comparisons revealed some potential sources of bias
when working only within the two-dimensional paradigm. For
example, when looking at the positions of emotion prototypes
using two-dimensional projections, one may get the impression
that some of the emotion prototypes are located in the central
area of the space that is determined by the emotion prototypes
with peripheral positions. For example, love and hate appear to
be located close to the central area when using a standard two-
dimensional projection of emotion prototypes on Dimension
2 and Dimension 3 (Figure 3). Similarly, compassion, guilt,
disappointment, fear and anger all seem to be located close
to the central area when using a two-dimensional projection
on Dimension 1 and Dimension 3 (Figure 4). However, these
TABLE 5 | Emotional proximities in MDS-aggregated 3D space.
Distance (s)
Fear
Sadness
Happiness
Disgust
Hope
Love
Hate
Contempt
Guilt
Compassion
Shame
Gratefulness
Envy
Disappointment
Jealousy
Anger 0.324 0.761 1.114 0.891 1.199 0.932 0.840 1.081 1.078 1.294 0.842 1.315 1.115 0.802 0.607
Fear 0.532 1.298 0.715 1.307 1.146 0.594 0.850 0.898 1.237 0.781 1.356 0.958 0.575 0.656
Sadness 1.490 0.929 1.246 1.284 0.702 0.781 0.529 1.066 0.472 1.312 0.990 0.267 1.057
Happiness 1.292 0.683 0.434 1.378 1.413 1.372 1.011 1.244 0.674 1.216 1.356 0.915
Disgust 1.441 1.366 0.256 0.440 0.935 1.087 1.156 1.236 0.411 0.746 0.538
Hope 0.501 1.411 1.363 1.037 0.699 0.842 0.484 1.269 1.142 1.237
Love 1.379 1.439 1.251 1.028 0.971 0.776 1.326 1.217 0.994
Hate 0.321 0.751 1.042 0.986 1.254 0.467 0.528 0.673
Contempt 0.607 0.854 0.993 1.137 0.308 0.541 0.880
Guilt 0.623 0.519 0.972 0.763 0.341 1.143
Compassion 0.844 0.416 0.767 0.848 1.159
Shame 1.024 1.103 0.504 1.141
Gratefulness 0.968 1.119 1.135
Envy 0.737 0.763
Disappointment 0.934
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Trnka et al. Emotion Space in 3D Hypercube-Projection
impressions are misleading, because they are given by the
limitations of two-dimensional projections, mostly by the
flattening the emotion space.
Frontiers of Semantic Emotion Space
The positions of frontiers of semantic emotion space are exactly
defined by coordinates on the x-, y-, and z- axes (Table 4).
However, the question is how to interpret the positions of
emotion prototypes toward the originally measured aspects, like
valence, intensity, control and utility? When comparing the
descriptive results (Table 1), the two-dimensional projections
(Figures 3–5) and the final positions of the emotion prototypes
after the MDS three-dimensional solution (Table 4), it seems
that the final Dimension 1 is saturated mostly by valence and
possibly partially by utility, Dimension 2 mostly by intensity of
arousal, and Dimension 3 is mixed, saturated by some proportion
of control and utility. However, this is only a rough estimate
and not sufficiently exact. It is obvious that transforming the
original four input dimensional measures (aspects) into the final
three dimensions in the 3D hypercube-model (Figure 5) changed
the character of the resulting three dimensions. In other words,
the original judgments of participants on the dimensions of
valence, intensity, control and utility saturated the resulting three
dimensions (Dimension 1, Dimension 2, Dimension 3) by certain
ratios.
When projecting the same data using the 3D hypercube-
model, no emotion prototype is settled in central area of
the three-dimensional emotion space (Figure 5). Thus, two-
dimensional projections visibly flatten the space and elicit
biased impressions about the positions of individual emotions.
Emotions that seem to be located in the central area of the
space in two-dimensional projections are indeed located far from
the central area. This effect is caused by the combination of
the four input dimensional judgments of participants, which set
up the position in the final 3D hypercube-model. The thing
is that the positions of some emotions are based on such a
specific combination of judged valence, intensity, controllability
and utility that their positions appear to be located close to the
central area when using two-dimensional projection. Therefore,
the use of two-dimensional designs in the empirical investigation
of emotions is confronted with the risk of reductionist bias and
oversimplification of highly complex phenomena. On the other
hand, multidimensional designs and spatial, three-dimensional
data projections may enrich future studies with more complex
insights on the positions of semantic fields of emotional
prototypes in the whole semantic emotion space.
Our results indicated that no emotions are located in the
central area in the identified semantic emotion space. When
interpreting this finding, we should turn our attention to the
input measures, i.e., to the initial judgments of participants.
Participants judged various discrete emotions according to the
perceived valence, intensity, controllability and utility. Emotions
generally represent states of mind that are different from
emotionally-neutral or non-emotional states of mind. People
usually name such emotionally-neutral states by words like
“calm” or “serenity.” In the present study, the central area of the
semantic emotion space did not include a prototypical semantic
category for any kind of emotion. Indeed, this is not surprising.
The emotional prototypes that were judged by our participants
are different discrete emotions, and we would suppose that they
should not yield zero or close to zero scores for their valence,
intensity, controllability or utility.
Semantic Similarity of Emotional
Prototypes
The above-mentioned source of bias was revealed when
we contrasted the positions of emotion prototypes toward
dimensions in the two- and three-dimensional models. Now, we
turn our attention to another source of bias. It emerges when
interpreting the mutual positions of discrete emotions, i.e., the
semantic similarity of emotional prototypes. In the present study,
anger and jealousy look to be very similar in the projection
using Dimension 1 and Dimension 2 (Figure 2). However,
when looking at the 3D hypercube-projection (Figure 5) or the
projection using Dimension 2 and Dimension 3 (Figure 3), a very
different look is available for an observer. Here it is necessary to
point out that the two-dimensional projections are only another
view of the same semantic emotion space.
The same bias occurs when we compare the mutual positions
of, for example, sadness and contempt, or love and happiness.
Some emotions look to be very similar in the two-dimensional
projection, but they are indeed very different (a reader may
also compare mutual proximities in the MDS-aggregated 3D
space in Table 5) when a third dimension is taken into account.
The two above-mentioned sources of bias indicate that research
designs using traditional, two-dimensional paradigm, face a
significant risk of confusion. Therefore, the combination of
standard, two-dimensional projections and the 3D hypercube-
projection proposed by the present study may provide a more
accurate starting point for the interpretation of the results yielded
in this type of research.
Further, discrete emotions that represent limits or frontiers
of semantic emotion space were identified in the present study.
Table 4 and Figures 2–4enable the most extreme positions of
emotion prototypes to be determined on all three dimensions
of the 3D hypercube-model. These extreme positions indeed
represent the extent of the three-dimensional overall semantic
space constructed on the basis of participants’ judgments.
Emotions that border the semantic space in Dimension 1 in
the positive direction are happiness, hope and love, and in the
negative direction hate. The emotion that borders the semantic
space in Dimension 2 in the positive direction is compassion,
and in the negative direction anger. The emotion that borders
the semantic space in Dimension 3 in the positive direction
is envy, and in the negative direction shame. These frontiers
of semantic emotion space have a very important function for
personal knowledge about emotions, i.e., they constitute the
personal awareness about maximal, as well as minimal, possible
general qualities of emotional experience, like valence, intensity,
controllability and utility. It borders the general emotionality
of a person, but it does not mean that the person may not
experience some emotions that would be out of this averaged
semantic emotion space. First, our participants differed in
their judgments of emotional qualities (see SDs in Table 1).
Second, the judgments of participants were based on the most
frequent experiences with particular emotions in their personal
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Trnka et al. Emotion Space in 3D Hypercube-Projection
histories. Some extreme and non-frequent emotional episodes
may sometimes occur, but it is unlikely that such experiences
would influence the judgment of an emotion prototype radically,
because personal knowledge about emotions is shaped in the
long-term course of emotional development and maturation. On
the other hand, merely different levels of emotional maturation
as well as differences in emotional personal traits may be some of
the factors influencing the participants’ judgments, and therefore,
also the interpersonal differences in the extent of semantic
emotion spaces (see SDs in Table 1).
Limitations
The innovative analytical tool introduced here has some
limitations. One of them is the problematic graphical depiction
of four-dimensional emotion space. The standard Euclidian
geometric space does not enable us to project all four originally
measured aspects proportionally. The graphical depiction is a
result of the MDS three-dimensional solution used and the
inspiration for future methodological shifts is a hypothetical
projecting tool that would be more suitable for working
with more than three-input dimensional measurements (e.g.,
an animated tesseract). Given the complexity of the human
emotional experience, it is the way to move the field forward and
not to permit some superfluous reduction of the phenomena.
Another limitation of the present study is that emotionally-
neutral semantic categories were not included in the research
design. Participants did not judge emotion-neutral semantic
categories such as calm or serenity. This limitation meant that
the core of the semantic emotion space could not be clearly
identified from the data. Also, the general orientation of semantic
emotion space toward other non-emotional semantic fields is
therefore hardly accessible. Future research in this field should
consider such emotionally-neutral semantic categories to avoid
the above-mentioned shortcomings.
CONCLUSION
The results of the present study pointed out to some limitations
of the two-dimensional paradigm in the psychological theory of
emotions. Contrasting two-dimensional and three-dimensional
projections helped us to identify some sources of bias that
may increase the risk of reductionism when approaching such
complex phenomena as human emotions. The discussion of the
results indicated that the 3D hypercube-projection may provide
researchers with more in-depth insights into the overall structure
of semantic emotion space than the traditional two-dimensional
projections widely used in this field.
As mentioned throughout the Discussion section, the findings
presented here signify very relevant implications for the
contemporary theory of emotions. Of course, we do not want
to reject the widely used circumplex model (Russell and Lemay,
2000). This model has been verified by many other empirical
studies in the past, and we believe that our study contributes
a new piece of knowledge to current theoretical discourse. The
results of the present study provide a slightly different look at
the phenomena and may inspire future researchers to continue
in the discussion about the dimensionality of human emotional
experience. Some alternative approaches, such as the theory of
multi-dimensional emotional experience (Trnka, 2013), may be
taken into account.
The current application of Sokolov and Boucsein’s (2000)
theoretical concept of the hypersphere in emotion research also
opens up new and exciting questions for future studies. For
example, what is the shape of such a hyperspace? May we
even think about visualization of intangible data in emotion
research (Murín, 2014)? Is there some kind of symmetry in
multidimensional semantic emotion space? Or, is it possible
to speak about a shape when exploring a phenomenon that is
probably even more multi-dimensional than we are currently
able to depict? These are rather philosophical questions and
answering them is far beyond the scope of the present study. But,
according to a famous bon mot of quantum physicist Niels Bohr
we conclude that “The question may not be, whether a theory is
too crazy, but whether it is crazy enough”!
AUTHOR CONTRIBUTIONS
All authors revised the manuscript critically, approved the final
version of the manuscript and agreed with all aspects of the work.
RT conceptualized the research design and drafted Introduction
and Discussion sections. AL conducted the data analysis and
drafted Methods and Results sections. MK gathered the data. KB
and PT worked on the conception of the work, the experimental
design, and the interpretation of data.
ACKNOWLEDGMENTS
The writing of this article was supported by the GA CR
project Spirituality and Health among Adolescents and Adults
in the Czech Republic [grant number 15-19968S] and by the
Technology Agency of the Czech Republic [grant number
TD020339]. Many thanks to Karel Hnilica and Daniel ˇ
Ríha for
their kind help during preliminary data analysis.
SUPPLEMENTARY MATERIAL
The Supplementary Material for this article can be found
online at: http://journal.frontiersin.org/article/10.3389/fpsyg.
2016.00522
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Conflict of Interest Statement: The authors declare that the research was
conducted in the absence of any commercial or financial relationships that could
be construed as a potential conflict of interest.
Copyright © 2016 Trnka, Laˇ
cev, Balcar, Kuška and Tavel. This is an open-access
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