Trajectory Biplot Highlighting Statistically Different
Periods for Temporal Sensory Evaluation
Hiroharu Natsume

Shogo Okamoto

Hikaru Nagano

Tokyo Metropolitan UniversityTokyo, JapanTokyo Metropolitan UniversityTokyo, Japan

Kobe University
Kobe, Japan

Abstract—The temporal dominance of sensations (TDS)
method is used to record time-series sensory changes during
daily experiences such as eating food, listening to music, andwatching videos. Trajectory biplots are typically used to visualize
the data recorded by this method. This trajectory shows how thesubjective sensory experience temporarily changes on a principal
component plane computed from the records of TDS method.
Despite its popularity, there is no method for determining
whether two biplots differ statistically. We propose a method forillustrating the periods in which two biplots differ signiﬁcantly
based on their conﬁdence intervals, which are computed using abootstrap resampling method. This method achieves an effective
visualization of the results of the TDS method that can be utilized
to evaluate user experiences in consumer electronics.
Index Terms—dynamic experience, sensory evaluation, datavisualization

I. I NTRODUCTIONThe temporal dominance of sensations (TDS) method isused to record multiple sensory changes over time during anexperience such as eating food and watching a movie [1]–[3].
Researchers in consumer electronics will beneﬁt from TDS
methods because they allow us to investigate how subjective
feelings evolve when consumers use products. The experimental results of the TDS method are often visualized using trajectory biplots [1], [4], which present multidimensional timeseries data on a two-dimensional principal component plane.
However, there is no method for visualizing the periods when
the two biplots or experiences from two different products
differ statistically. In this study, we propose a method for
determining and visualizing such periods using the conﬁdence
intervals of trajectories [5]. Thus far, TDS methods have been
widely used in food science. We used the TDS data of two
ham brands, which were part of the TDS data acquired in aprevious study [6].
II. T EMPORAL DOMINANCE OF SENSATIONS METHOD
A. Task of TDS method
In an experiment using the TDS method, an assessor (panel)reports their experience using a computer application overtime. The computer screen displays buttons, each of which
has a label describing an attribute word such as sweet andsour. When the panel places a food piece in their mouth,
they press the start button. The panel then presses a button
This study was in part supported by Lotte Foundation (#2023-B-1).

979-8-3503-4017-4/23/$31.00 ©2023 IEEE

that corresponds to the most dominant feeling at each momentand selects another button every time the dominant sensation
changes. However, multiple buttons cannot be selected simultaneously. As soon as the food vanishes in the mouth, the stop
button is pressed, and the task ends.
B. TDS curves
The result of the TDS task is multidimensional time-series
data. For each attribute word, information on whether it is selected (1) or not (0) at an arbitrary time t is recorded. Here, we
formulate the results of TDS following our earlier study [7].
The time series for ith attribute of jth trial (j = 1, ..., n) is a(j)function of time t, and is represented by fi (t) ∈ {0, 1}. The
number of TDS tasks to be analyzed is n. t is the standardized
time ranging from 0 to 1. The moment when the stop button
is pressed corresponds to t = 1.
The TDS curves are the averages of multiple TDS tasks.
The TDS curve of a certain attribute word indicates thetemporal evolution of the proportion of dominance, which isthe proportion of trials in which an attribute is selected at anarbitrary time. The function pi (t) corresponding to the TDS
curve of ith attribute isn1 ∑ (j)pi (t) =f (t).
(1)
n j=1 iIII. C OMPUTATION OF TRAJECTORY PLOTS WITHCONFIDENCE INTERVALS

A. Computation of trajectory plot
We followed the method proposed by Lenfant et al. [4]to compute the trajectory plots of the TDS curves. First, acontinuous TDS curve pi (t) was discretized into 1000 intervals
in the time domain over 0 ≤ t ≤ 1. The discretized curve Pi [k](k = 0, 1, ..., 999) was obtained using the following formula:∫ (k+1)/1000
Pi [k] = 1000pi (t) dt.
(2)
k/1000We performed principal component analysis on Pi [k] witheach attribute i (i = 1, ..., q) as a variable and obtained1000 × s principal component scores. Here, s was the numberof stimuli evaluated in the TDS tasks. In this study, s = 2
because two types of products or experiences were compared.
We acquired a trajectory biplot by plotting these scores intime order on a plane deﬁned by the ﬁrst and second principal

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components. The trajectory exhibited a multidimensional TDS
curve on a two-dimensional principal component plane.
B. Computation of conﬁdence intervals
To obtain conﬁdence intervals for the trajectory plots, weresampled the TDS curves using the bootstrap method [3], [5],[8]. In this method, new samples were extracted by sampling
the recorded TDS data set with replacement. The size of theresulting sample was the same (n) as the original dataset. The
TDS curves and trajectories were computed from a resampled
data set.
We repeated the bootstrap method 200 times to generate
200 trajectories for one type of sensory stimulus, i.e., abrand of ham. The standard deviations of the ﬁrst and second
principal component scores of the trajectories were obtainedfor each ham at each discretized moment. We regarded 1.96
times the standard deviation as the conﬁdence interval. A
conﬁdence interval ellipse was then drawn for each moment ona trajectory plot. The conﬁdence intervals for all the discrete
moments were overlaid to visualize the uncertainty of thetrajectory.
C. Visualization of periods signiﬁcantly different between two
trajectory plots
When a trajectory is inside the conﬁdence interval ofanother trajectory, the two are assumed not to be signiﬁcantlydifferent. To visualize these intervals, we drew these parts ina different line style.
IV. R ESULT: E XAMPLES OF TRAJECTORY PLOTS WITHCONFIDENCE INTERVALS

We applied the above method to the TDS task results
recorded in a previous study [6]. The trajectories of thetwo ham products are presented in Fig. 1. The solid curves
represent the mean trajectories of the two ham brands. Vectors
starting from the origin indicate the direction of individual
attributes. The trajectory of Ham A (red) starts from theorigin and transitions in directions of juicy (dotted red line),fragile (orange dotted line), and umami (blue dotted line).
In contrast, the trajectory of Ham B directs to sweet (khakiline) in its early phase. The trajectories of the two hams wereclose to each other before t = 0.2. The blue curve nearly
ended at the extension of the dotted orange line, indicating astrongly fragile texture. The red curve ended approximately at
the center of the extended orange dotted (fragile) and brown(salty) lines, indicating that it was fragile and salty at the last
moment.
The grey areas were the overlays of the 95% conﬁdence
intervals for t = 0–1. Light-colored parts of the trajectory
showed the intervals in which the trajectory of one productoverlapped with another’s conﬁdence interval. For example,
in the very early phase, these trajectories were not consideredsigniﬁcantly different (t = 0–0.1). During the periods when
the trajectories were drawn in dark red or blue, the sensory
experiences of the two brands of hams were signiﬁcantlydifferent．

Fig. 1. Trajectory biplots of two brands of hams. The red and blue curves
were the trajectories of Hams A and B, respectively. The light grey areas
were the overlaid ellipsoids of 95% conﬁdence interval of individual brandsof hams. The dark grey areas were the overlaid of the conﬁdence intervalsof two brands of hams. At periods during which the curves were dark redor blue, the two trajectories were signiﬁcantly different from each other. Incontrast, during the periods shown by light red or blue trajectories, the twoham products were not signiﬁcantly different. White circles are plotted every
0.1 normalized time.

V. C ONCLUSION
The trajectory plots are a popular method to visualize theexperimental results of the TDS method, which is a timeseries sensory analysis method. Thus far, there has been noway to visually indicate whether two trajectory plots drawn
on the same trajectory plane differ statistically. We proposed amethod to visualize periods in which the trajectories of the two
products were signiﬁcantly different based on the conﬁdence
intervals of the trajectory plots. With this visualization method,
it is easy to recognize when two stimuli differ signiﬁcantly
in terms of their temporal sensory experiences. In the future,some extensions are needed such that more than two products
or trajectories can be visually and statistically compared on
the same plane.
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