J-Stage Released Online Advance Publication: April 1, 2024

Food Science and Technology Research, 30 (4), 491–499, 2024
Copyright©2024, Japanese Society for Food Science and Technology
doi: 10.3136/fstr.FSTR-D-23-00237

https://jsfst.smoosy.atlas.jp

Technical Paper
Trajectory plots that highlight statistically different periods among
multiple foods studied using the temporal dominance of sensations
methodHiroharu NATSUME1, Shogo OKAMOTO1*, and Hikaru NAGANO2
1 Tokyo Metropolitan University, 6-6, Asahigaoka, Hino, Tokyo 191-0065, Japan
2 Kobe University, 1-1, Rokkodai, Higashinada-ku, Kobe 657-8501, Japan

Received December 17, 2023 ; Accepted March 13, 2024
The temporal dominance of sensations (TDS) method allows us to record temporal changes in multiple
sensations during an experience, such as eating food. A trajectory plot is used to visualize the experimental
results of the TDS method and is useful for comprehending multidimensional sensory transitions in a lowdimensional principal component space. Castura et al. (2023) proposed an approach to statisticallycompare multiple food products in such spaces. We rectify the hypothesis-testing method of that approachand modify the visualization method, such that the periods in which the products are statistically different
from each other are easily recognized. The developed method helps us understand when and how a foodproduct feels different from others during the eating experience and can contribute to designing new
products by focusing on their time-series experience.
Keywords: temporal dominance of sensations (TDS), bootstrap resampling, t-squared test, data visualization

Introduction
The temporal dominance of sensations (TDS) method(Pineau et al., 2009; ISO 13299, 2016; Visalli et al., 2023)
allows us to record time-evolving changes in multiple
sensation types. Popular visualization methods for TDS task
results include TDS curves (Pineau et al., 2009) and trajectoryplots (Lenfant et al., 2009). The former method exhibits
temporal changes in the dominance of individual tastes,
flavors, and mouth sensations on a plane in which thehorizontal axis is time. Hence, TDS curves are suitable forinspecting changes in individual sensations over time. In
contrast, TDS trajectories present no time axis and exhibittransitions of multiple types of sensations in a
multidimensional space, which is typically a two-dimensional
plane owing to visual clarity. The TDS trajectory method
allows for the simultaneous comparison of multiple foodproducts in the same space (Merlo et al., 2019; Nguyen andWismer, 2022; Delompré et al., 2020).
Nonetheless, until recently, no statistical methods or
hypothesis tests were available to compare TDS trajectories

among food products. The biggest reason for this is that theuncertainty or potential variability of the trajectories cannot be
estimated based on the results of the standard protocols of the
TDS method. Castura et al. (2023) solved this problem using
a bootstrap resampling method. The bootstrap resampling
method for the TDS method was introduced by Okamoto et al.
(2020) and Okamoto (2021). and enables the estimation of
population parameters related to uncertainty and data
augmentation from a population with limited samples. Using
the resampling method, Castura et al. (2023) computed theconfidence intervals of trajectory plots and contrasted them;then, they used this method to discriminate three types ofwines. Their approach allowed a statistical comparison ofeating experiences between more than two distinct foods using
trajectory plots.
This study modifies the method of Castura et al. (2023)
in two ways. First, to compare two trajectories in a
multidimensional space, 𝑡 -tests involving multiple
dimensions were used, whereas, Castura et al. (2023)
conducted a hypothesis test for each dimension. Second, we––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––*To whom correspondence should be addressed.
E-mail address: okamotos@tmu.ac.jp

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introduced a visualization method that clarified the periods ofa trajectory that were significantly different from others,
which allowed us to easily recognize meaningful differencesamong multiple TDS trajectories. Castura et al. (2023)
effectively used animations to show the confidence intervalsof multiple trajectories at each moment, whereas our method
is more suitable for still images. The trajectory plot is aneffective method to facilitate the simultaneous comparison of
more than two food products. The method of visualizing
statistical differences between the trajectories of different
products provides analysts with valuable insights into
temporal experiences. Owing to the modifications made in thisstudy, TDS trajectories will become a more effective
visualization and analysis method. This study is an extensionof the research presented by Natsume et al. (2023b), where
only two food products were compared in a non-statisticalmanner, whereas, the method of the present study allows forthe comparison of two or more products using hypothesis
testing.

H. NATSUME et al.

Fig. 1.
The graphical interface used in thetemporal dominance of sensations (TDS) tasks.
Each button represents an attribute word. Adaptedfrom Natsume et al. (2023a).

Materials and Methods
Procedures for the temporal dominance of sensations
method Here, general procedures for a TDS task are outlined
to help understand this paper. Further details are provided in
the literature (ISO, 2016; Pineau et al., 2009; Pineau andSchilch, 2015). The TDS task is performed using a computer.
As shown in Fig. 1, the attribute words are displayed on thescreen as buttons, including the start and stop buttons. The task
begins when the panelist places the food sample in its mouthand presses the start button. Then, the panelist selects an
attribute word that describes the dominant sensation. When thedominant sensation changes, the panelist selects anotherbutton, and the currently selected button is automaticallyunselected. In the standard method of ISO 13299, multiple
buttons cannot be selected simultaneously (ISO, 2016). Some
attribute words are selected multiple times or never selected.
When the food sample is swallowed, the panelist presses thestop button to end the task. The panelist repeats the task
several times for the same food. More than ten panelists arerecommended for each food product (ISO, 2016).
Temporal dominance of sensations curves The result ofa TDS task is a set of binary functions of a continuous time 𝑡.
The function for the 𝑖 -th attribute of the 𝑗 -th trial isrepresented as 𝑓 𝑡 . The parameter 𝑡 is the timenormalized between 0 and 1, which represent the moments
when the start and stop buttons are pressed, respectively. At an
arbitrary 𝑡 , 𝑓 𝑡
1 if word 𝑖 is selected, otherwise,𝑓 𝑡
0.
TDS curves (ISO, 2016; Pineau et al., 2009) are typicallyused to visualize the results. The TDS curve is the mean of alltrials at 𝑡. Hence, the curve 𝑝 𝑡 for the 𝑖-th attribute is

Sour
Sweet

BitterFig. 2. An example of the TDS curves of three
attributes, namely sweet, sour, and bitter.
given by:
𝑝 𝑡

∑

𝑓

𝑡 ,

･･････ Eq. 1

where 𝑛 is the total number of trials. The TDS curve foreach attribute word indicates the dominance proportion, whichis the proportion of trials in which the attribute word isselected at time 𝑡. Fig. 2 shows an example of the TDS curves.
Note that TDS tasks typically involve several attributes;
however, for visual clarity, this example includes only three
attributes. The sweet and bitter attributes are prominent at
approximately 𝑡 0.3 and 0.6 , respectively. The sour
attribute is the most prominent at the end of the task, with aproportion of approximately 0.6, indicating that it is selected
in 60 % of all the trials in the last phase.
Trajectory plots of the temporal dominance of sensations

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Trajectory plots highlighting statistically different periods of TDS trajectories
curves in the principal component space Before computing
the trajectory plots, the continuous TDS curves are discretized
into arbitrary 𝑅 intervals in the time domain using Eq. 2. In
this study, we computed trajectory plots at 1 000 discretizedintervals (𝑅 1 000).
𝑃 𝑘

𝑅

𝑝 𝑡 𝑑𝑡･･････ Eq. 2

In Eq. 2, 𝑃 𝑘 is the discretized curve value on the 𝑘-thinterval (𝑘 0, 1, … , 𝑅 1). This conversion uses the averageof 𝑝 𝑡 for each discretized interval.
A trajectory plot (Lenfant et al., 2009) shows the
temporal evolution of the dominant proportions of all
attributes for a certain food using a single curve. In otherwords, multiple TDS curves, that is, multidimensional sensorychanges, are compressed into a single curve. Trajectory plotsof TDS curves are typically drawn by plotting the first andsecond principal component scores of 𝑃 𝑘 on a twodimensional principal component plane. However, they can be
determined in a space with higher dimensions using the third
and remaining principal components. Scores are obtained by
performing a principal component analysis on vectors
with attribute words as variables,
𝑃 𝑘 ,…,𝑃 𝑘 ,…,𝑃 𝑘
where 𝑖′ is the number of attributes. There are 𝑅 scores per
set of TDS curves. Fig. 3 shows an example of a trajectory plot.
The vectors for the attributes are also presented. The
horizontal axis represents the score for the first principal
component and is mostly aligned with the sour vector,
indicating that higher scores along this axis are sourer.
Similarly, the vertical axis corresponds to the second principal

493

score and is characterized as sweet and bitter.
Bootstrap resampling of the temporal dominance ofsensations data In the TDS method, a food product produces
a set of TDS curves, from which one trajectory curve is drawn.
Bootstrap resampling can be used to compute the uncertaintyof the TDS curves and trajectory (Okamoto et al., 2020;
Okamoto, 2021; Castura et al., 2023). As shown in Fig. 4,
using this method, a new sample set is established by sampling

Sour
Bitter
Sweet

Fig. 3. An example of a trajectory plot, whichcorresponds to the trajectory of the TDS curves inFig. 2. Straight lines starting from the origin are thevectors of attributes. The blank circles and squareat the end of the trajectory split the whole tasting
period into ten equal sub-periods at 𝑡
0.1, 0.2, … , 1.0. The trajectory transits between thesweet and bitter attributes in the first half of theperiod and between the bitter and sour attributes inthe second half.

Fig. 4. Bootstrap resampling, in which a dataset, namely a set of TDS trials, of the same size as the original dataset isgenerated by randomly resampling elements with duplicates from the original dataset. For each resampled dataset, a
trajectory curve is calculated. This operation is repeated many times to acquire the distribution of trajectory curves.

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H. NATSUME et al.

Sour

Product B

Bitter
Sweet

Product A

Fig. 5. The trajectory plot of Fig. 3 with overlaid 95 %confidence ellipsoids.

Fig. 6. An example of the visualization of statisticallydifferent periods. During the periods represented bybold lines, the two products are significantly different
from each other on the principal component space.

with replacement. When an experiment originally includes 𝑛trials for a food product, a new sample set includes 𝑛 trials
drawn from the original sample set with a replacement. The
TDS curves and corresponding trajectories are computed forthe new sample set. This process is repeated hundreds of times
to compute the uncertainty, that is, the confidence interval of
the trajectory. In our study, the number of repetitions for eachfood product was 200.
Confidence intervals for the trajectory plots
The
uncertainty of the locus of each discrete point in the trajectory
plot can be estimated using a dataset generated by thebootstrap resampling method (Castura et al., 2023). Here, as
an index of uncertainty, we used a 95 % confidence interval(Okamoto et al., 2021; Castura et al., 2023). For each time
point, based on the distribution of the samples, the 95 %confidence interval ellipsoid is given by:component space at a particular time 𝑘 (𝑘 0, 1, … , 𝑅 1).
To compare the trajectories of A and B, a hypothesis test
was used to determine the significant differences in theprincipal component space at each discretized moment. We
used the 𝑡 -test to determine whether the centroids of the twogroups differed significantly. As described in the Introduction,
Castura et al. (2023) judged the difference between two
products by performing a 𝑡 -test on each dimension of theprincipal component space. However, in a multidimensional
space, the judgment should be based on the distance between
the centroids of the two groups, considering the covariancestructure among multiple variables.
The 𝑡 statistic is computed as

𝒙

𝒙 𝚺

𝒙

𝒙

𝜒 , .

,

･･････ Eq. 3

where 𝒙𝑥 ,…,𝑥is the coordinate on the qdimensional space, and 𝒙 and 𝚺 ∈ ℝ
are the centroid
vector and variance-covariance matrix for the resampled
points, respectively. The parameter 𝜒 , . is the chi-squared
value of the degree of freedom 𝑞 at a significance probabilityof 0.05. In the case of the two-dimensional principal
component plane, 𝑞 2 and 𝜒 , .
5.99. Fig. 5 shows the
trajectory plot of Fig. 3 with 95 % confidence intervals. The
gray areas represent the overlaid confidence ellipses for all
discrete time points.
Judgment of the significant differences between multiple
trajectory plots First, we describe a method for comparing
two food products, namely A and B, where 𝒙 𝑘
and 𝒙 𝑘,𝑥̅
𝑘 , … , 𝑥̅𝑘𝑥̅ 𝑘 , … , 𝑥̅𝑘respectively, are the mean coordinate vectors of the resampled
trajectories of A and B in a 𝑞 -dimensional principal

𝑡

𝒙

𝒙

𝑺

𝒙

𝒙 ,

･･････ Eq. 4

where 𝑛 and 𝑛 are the sizes of the resampled datasetsfor products A and B, respectively. In addition, 𝑺 ∈ ℝ
isthe pooled variance-covariance matrix for the two products.
The 𝑡 value approximately follows an F distribution:𝑡 ∼ 𝐹 𝑞, 𝑛

𝐹

𝑛

1

𝑞 . ････ Eq. 5

If the 𝐹 value is greater than the critical value of𝑛
1 𝑞 at a significant probability α, then the
𝐹 𝑞, 𝑛trajectory points of products A and B are consideredsignificantly different.
In the interval immediately after the start button ispressed, the two products are not compared because the 𝑺
cannot be computed. In such an interval, we treat thecoordinates of the trajectory as fixed at the origin, and themean coordinate vector of one product in Eq. 4 is set to 𝟎, thatis, 𝒙𝟎, and 𝑡 is computed using Eq. 6.
𝑡

𝑛 𝒙

𝟎 𝑺

𝒙

𝟎 .

･･････ Eq. 6

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Table 1. Definitions of eleven attributes used for TDS tasks.
Attributes

Description

Umami

One of the basic tastes. No definition was provided.

Juicy

Lots of juice extracted in the mouth.

FragileEasily broken down into pieces.

Sweet

One of the basic tastes. No definition was provided.

Elastic

After being bitten, the ham is likely to return to its original shape.

Salty

One of the basic tastes. No definition was provided.

Dry

Ham includes little juice.

Fatty

The ham includes fat that melts in the mouth.

Soft

Small biting force is required.

Smoky

Taste and smell of smoked food.

FibrousHam includes lots of fibers.

The trajectory of product B is fixed to the origin, and the
𝑺 is computed using only product A. The corresponding 𝐹
statistic is determined as follows:𝐹𝑡 ∼ 𝐹 𝑞, 𝑛𝑞 .

･･････ Eq. 7

Note that such intervals exist near the origin in theprincipal component space.
When more than two products are compared, the 𝑡 -test
is performed multiple times. The Bonferroni correction of thecritical probability is applied when comparing 𝑚 (𝑚 2)
products (Castura et al., 2023). For example, when we want toknow whether a product is different from others, we repeat
the 𝑡 -tests to make comparisons between the products. Hence,
the hypothesis tests are repeated 𝑚 1 times for each time
point. For each test, the significant probability α is adjusted
by a factor of 𝑚 1.
Visualization of multiple trajectories with significantly
different periods
We draw trajectory plots of multiple
products for comparison in the same principal component
space. Fig. 6 shows an example of the trajectories for two
products. The intervals in which the two products are
significantly different are represented by bold lines, and theintervals with no significant differences are represented bythin lines. This helps us visually understand when theseproducts are experienced distinctly during eating. When threeor more products are being addressed simultaneously,
regarding a certain product, bold lines are drawn along periods
when that product is significantly different from any of theothers.
Example using Five Processed Hams
Temporal dominance of sensations data for processedhams We used data collected by Hariu et al. (2023), in which
TDS tasks were performed on commercially available
processed hams. We used the data for five types of ham priced

at approximately 30–50 JPY per slice. The five products werelabeled as Hams 1, 2, 3, 4, and 5. For each product, 55 trials
were conducted, that is, two to four trials were conducted forindividual assessors. Eighteen university students in their
twenties (15 males and 3 females) joined as assessors and ate
a piece of ham cut into 3
5 cm pieces, which were kept atroom temperature.
The attributes used in the assessment were umami, juicy,
fragile, sweet, elastic, salty, dry, fatty, soft, smoky, and fibrous.
These attributes were selected from the pool of 85 attributewords established in the literature, such as Lorido et al. (2016).
For the selection, six panelists including the authors and theircolleagues rated all the attributes applicable to express thetastes of ham in a check-all-that-apply manner. Fifteen highly
voted for attributes were then examined via preliminary TDS
trials, in which four attributes were rarely used by the panels.
Table 1 lists the definitions of 11 attributes provided to thepanelists.
Fig. 7 shows the TDS curves for the five products. For
Ham 1, the elastic and salty attributes are prominent in theearly and later phases, respectively. For Ham 2, the soft, elastic,
and sweet attributes are dominant in the early phase, whereas
the fragile attribute is prominent in the last phase. For Ham 3,
the elastic and salty attributes are dominant in the early andremaining periods, respectively. For Ham 4, the dry and elastic
attributes are prominent in the early phase, whereas the salty,
fragile, and fibrous attributes are prominent in a later phase.
For Ham 5, the elastic, fatty, and umami attributes areprominent in the early phase, and the salty attribute isprominent in a later phase.
Trajectory plots with confidence intervals andsignificantly different periods For each ham product, we
obtained 200 sets of resampled TDS curves with 𝑛 55 and𝑛
200 to compute the 95 % confidence intervals.
𝑛
The overall significance level for the multiple comparison was

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Fig. 7. TDS curves of five ham products. TDS curves of Hams (A) 1, (B) 2, (C) 3, (D) 4, and (E) 5.
set at α 0.01. The TDS trajectories were then drawn on a
two-dimensional principal component plane, that is, 𝑞 2.
Figs. 8(B)–(D) show examples of the comparisons between
two of the five ham products. Fig. 8(B) compares Hams 2 and5. There exists a short period in 0.1 𝑡 0.2 where thetrajectories are not significantly different. During the otherperiods, the two hams exhibited different tastes and textures.
Similarly, Fig. 8(C) compares Hams 3 and 4, whose
trajectories were not significantly different, particularly in theearly phase ( 𝑡 0.1). Fig. 8(D) shows the trajectories for
Hams 1 and 4, which were largely distinct over the entire
period. Fig. 8(E) shows the trajectories for the five hams in thesame plane at intervals where all the trajectories differ fromeach other. The tastes of all hams differed at certain momentsof the eating experience. In this case, the Bonferroni correction
factor was 10 (5C2).
A statistical comparison of the trajectories provides theanalyst with greater insights into the examined products. For

example, the trajectories for all the hams (Fig. 8(E)) show thatthe products can be apparently divided into two groups: Hams
3 and 4 comprise a group of fibrous hams with low moisture
content, whereas Hams 1, 2, and 5 belongs to a group of softand sweet hams. The meaningfulness of the differences
between these two groups cannot be asserted without a cloudof uncertainty. It is also unclear if there are differences
between hams within the same group. Hams 2 and 5 appear tobe similar, but as shown in Fig. 8(B), they provide
significantly different experiences for most of the periods.
Thus, the trajectory plots provide a visual representation of thesimilarity of multiple products, and the uncertainty-based tests
provide a statistical basis for judging their resemblances.

Discussion
Castura et al. (2023) computed the uncertainties of the
trajectory of TDS curves by using the bootstrap resampling
method to statistically compare multiple trajectories; in thisJ-Stage Released Online Advance Publication: April 1, 2024

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Fig. 8. Trajectory plots of five ham products. Labels near the blank circles indicate 𝑡 at that time. (A) The legend for
all attributes. Comparisons between hams (B) 2 and 5, (C) 3 and 4, and (D) 1 and 4. (E) Trajectory plots of all fivehams.
study, the method was modified regarding two aspects. First,
we rectified their statistical method. They repeated hypothesis
tests along different dimensions in the multidimensional space,which was alternated by the 𝑡 test. Second, we plotted theuncertainty of the trajectory using a cloud overlaid around the
trajectory, whereas Castura et al. (2023) visualized theevolution of the uncertainties using animations. Our method ismore suitable for still images of trajectories. Finally, we
underscored the periods of trajectories that are significantly
different from each other. Compared with the method
proposed by Castura et al. (2023), a clear flaw in our method

is that it does not visually provide the confidence intervals fora trajectory at arbitrary moments, since the confidence
intervals are overlaid throughout the entire period.
Furthermore, when comparing more than two products on thesame plane, the temporal evolution of the confidence intervals
for each product is not effectively visualized. Animation(Castura et al., 2023) can still be effective in solving these
problems.
As a problem commonly observed in the analyses of TDS
curves and trajectories, many researchers applied hypothesis
tests to the data at each instance of the tasks. As a result,

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several hypothesis tests are repeated for analyzing the same
TDS task, which remains a limitation of multiple tests. One
approach to avoid such excessive repetitions of hypothesis
tests was proposed by Okamoto et al. (2020), where all TDS
curves for a certain food product were projected into a single
point in principal component space. Using that method, TDS
curves for two different food products can be compared using
only a single hypothesis test. Nonetheless, the test cannot
determine the instance during which the TDS curves for thetwo products differ. Hence, the method proposed in this study
and that by Okamoto et al. (2020) can be used in acomplementary manner. Another concern is the number of
products being compared. When the number is large, multiple
tests may underestimate the significant differences. Thus, thenumber of products compared simultaneously should beseveral at most.
An important future research topic is determining how
statistically significant differences in the TDS curves andtrajectories relate to the value perceived by consumers. A
significant difference does not directly lead to distinctions in
the values perceived by consumers. To investigate this, theTDS can be combined with a temporal-liking method(Meyners, 2015; Thomas et al., 2015) or a questionnaire on
product preferences.

Conclusion
We modified the method of Castura et al. (2023) in two
aspects: visualization of significantly different periods andconfidence intervals for trajectory plots and hypothesis tests in
multidimensional spaces. Our visualization method clarified
the periods in the trajectories during which the products weresignificantly different. That is, it is possible to understand how
and when products provide different experiences. This method
will be used to differentiate strategies in a target market byproviding time-series experiences that differ from those forcompeting products. Furthermore, hypothesis tests using 𝑡
statistics enabled us to compare multiple curves in a more validmanner. Although we used one type of food product for
demonstration, the practical value of the method needs to be
confirmed using other types of food in the future.
Acknowledgements This study was supported in part by the
Lotte Foundation (2023-B-1).
Conflict of interest
declare.

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