foods
Communication

Bootstrap Resampling of Temporal Dominance of Sensations
Curves to Compute Uncertainties
Shogo Okamoto
Department of Computer Sciences, Tokyo Metropolitan Universities, Hino, Tokyo 191-0065, Japan;
okamotos@tmu.ac.jp

Abstract: In the last decade, temporal dominance of sensations (TDS) methods have proven to bepotent approaches in the field of food sciences. Accordingly, thus far, methods for analyzing TDS
curves, which are the major outputs of TDS methods, have been developed. This study proposesa method of bootstrap resampling for TDS tasks. The proposed method enables the production of
random TDS curves to estimate the uncertainties, that is, the 95% confidence interval and standard
error of the curves. Based on Monte Carlo simulation studies, the estimated uncertainties are
considered valid and match those estimated by approximated normal distributions with the numberof independent TDS tasks or samples being 50–100 or greater. The proposed resampling method
enables researchers to apply statistical analyses and machine-learning approaches that require a large
sample size of TDS curves.
Keywords: temporal dominance of sensations; confidence interval; standard error; Markov chain

Citation: Okamoto, S. Bootstrap
Resampling of Temporal Dominanceof Sensations Curves to Compute
Uncertainties. Foods 2021, 10, 2472.
https://doi.org/10.3390/foods
10102472
Academic Editor: Derek V. Byrne
Received: 25 August 2021
Accepted: 7 October 2021
Published: 15 October 2021

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1. Introduction
Over the last decade, temporal dominance of sensations (TDS) methods, in which
multiple types of temporally evolving subjective responses are recorded, have been provento be effective methods of sensory appraisal in the field of food sciences by many researchers [1–3]. Furthermore, in some recent studies, TDS methods have been applied
to tasks involving visual, auditory, and haptic cues [4–7]. TDS methods are becoming astandard of sensory appraisal, irrespective of the type of modality.
Typically, in the task of the TDS method, adjective descriptors (Except for adjective
descriptors, onomatopoeic words are used, for example [8]) listed on a computer screen aresequentially selected by assessors. These descriptors correspond to the dominant sensations
felt while experiencing food stimuli. The dominant sensation is defined as “the sensationthat catches his/her attention” [9] or constitutes the intensity of or change in the sensory
profile [10]. In most data analysis methods, the period that each descriptor is selected as
dominant is calculated; in other words, this is the dominance duration [8,11–13]. These
periods are treated as variables, and principal component analysis or canonical variate
analysis are then applied to evaluate the differences among the food stimuli. Furthermore,as the most typical analysis [9], by accumulating the results of independent TDS tasks,
a proportion that a certain descriptor is dominant in is computed at each instant. These
proportions are functions of continuous time and are displayed as “TDS curves.” TheseTDS curves are visually inspected, and their key values, such as maximum values andthe time when the maximum values are observed, are calculated and compared using
multivariate analysis techniques among different food products [2,14,15].
In the standard data analysis method established by earlier studies, a set of TDS
curves are computed using the results of all the independent tasks. This method of curve
production prevents the use of many analytical methods requiring a large sample size.
For instance, if we are interested in the peak curve value of a certain descriptor, there is
no standard method to estimate how it fluctuates, meaning we cannot judge whether the

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2 of 11peak value is different from a hypothesized value. Further, in general, machine-learningtechniques to predict or utilize TDS curves require a large set of TDS curves. To acquire
multiple sets of TDS curves, for example, TDS tasks need to be repeated by the same panels
for the same food stimuli [15]. These multiple sets of TDS curves are necessary to consider
the curves’ uncertainties for some statistical analyses, including hypothesis testing. Such
repetitions are acceptable for acquiring a few sets of TDS curves; however, more curve
sets are required for some analysis techniques. To circumvent this problem, bootstrap
resampling methods can be used [16–18]. In these resampling methods, as described in
Section 2.2, tasks are sampled with replacement from a set of independent TDS tasks,
and many sets of TDS curves are acquired. For instance, in [17], to apply the principal
motion analysis on TDS curves, dozens of TDS curve samples were produced by bootstrap
resampling to avoid overfitting. However, thus far, the validity of bootstrap resampling of
TDS curves has remained unclear.
Bootstrap resampling for TDS curves yields some benefits. First, it allows us to adopt
analytical methods that require many curve samples. Second, bootstrap resampling enables
the estimation of curve uncertainties. More specifically, we can test whether a curve value
at a certain instance is significantly greater than a hypothesized value. Further, a hypothetical test of curve values between two or multiple different descriptors can be conducted.
Nonetheless, Pineau et al. [2] suggested that the uncertainties of TDS curves can be estimated based on approximated normal distributions because TDS curve values correspondto proportions. Provided that the uncertainties estimated by bootstrap resampling andnormal distributions match, the validity of bootstrap resampling is supported.
However, the validity of the bootstrap resampling method for TDS curves has yet to bestudied. As such, the present study proposes a bootstrap resampling method for estimating
the uncertainties of TDS curves, demonstrates the method’s validity, and calculates thenecessary sample size. In particular, our interest is how many samples are necessary toaccurately estimate the uncertainties of TDS curves and to create random TDS curves, ofwhich fluctuations represent the uncertainty of the population. As the most popular indicesof uncertainties, 95% confidence intervals and standard errors are computed using the
distribution of TDS curve values that are generated by resampled data. These uncertainties
are validated via comparison with the population produced via Monte Carlo simulation,
where Markov chain models that simulate TDS tasks [19–21] are used to produce a large
number of random samples. The present study is the first attempt to utilize Markov chain
models and Monte Carlo simulation to yield a large dataset of TDS curves.
2. Methods
2.1. TDS Method
Herein, the typical procedures of TDS tasks and computation of TDS curves are
introduced to help understand the remaining part of the paper. Regarding further details,
earlier publications [1–3,9] are referred to.
In TDS tasks, a graphical user interface is used as shown in Figure 1a. The assessor
pushes the start button at the moment of putting the food in his/her mouth. S/he thenpushes the button corresponding to the sensation that comes to mind. Different buttons
are sequentially pushed as the dominant sensations change. Once the button is pushed,
it remains selected until the next button is pushed. The stop button is pushed when thefood vanishes in the mouth or prominent sensations fade after swallowing the food. These
procedures comprise a single TDS task. As shown in Figure 1b, from a single task, thatis, the i-th task, binary functions of continuous time that take either 0 (unselected) or 1
(selected) are yielded for individual p descriptors:bij (t) ∈ {0, 1}
where j is the index of descriptors.

( j = 1, ..., p),(1)

Foods 2021, 10, 2472

3 of 11(a) Graphical user interface(b) Binary function from a single task
1
0
1
Sweet
0
1
Juicy
0
Sour

0
(Start)1(Stop)

Normalized time: t’…

Sour fromassessor 1

Dominance
proportion(c) Temporal dominance curve (time function of dominanceproportion)Sour fromassessor n

TD curveof sour01
Normalized time: t’Figure 1. Temporal dominance task. (a) Graphical user interface. (b) Binary function of selected/nonselected from a single TDS task. (c) Temporal dominance curves as the accumulation of binary
functions by descriptors. Partly adapted from [18].

This task is repeated by n assessors. TDS tasks do not necessarily require well-trainedor professional assessors [3,15,22]. Typically, only familiarization with the TDS tasks andcomprehension of the descriptors used in the tasks are required [9,23]. Based on the
results of n tasks, the proportion at which each descriptor is dominant at moment t iscomputed. For this computation, typically, the period of each task is normalized such thatthe time elapsed between pushing the start and stop buttons is equal to 1. The normalizedbinary functions are accumulated for individual descriptors. As shown in Figure 1c, thedominance proportions are computed by dividing the accumulated values by n.
bij (t0 )
,
n
i =1
n

d j (t0 ) = ∑(2)

where t0 is the normalized time (t0 ∈ [0, 1]). The time function d j (t0 ) is the TDS curve fordescriptor j.
2.2. Bootstrap Resampling of TDS Tasks
Herein, the bootstrap resampling method for TDS tasks is introduced, which follows
a standard bootstrap resampling method [24]. Note that this method is very different frombootstrap resampling methods for time-series data, such as block bootstrap [25,26]. It is
unique to the temporal dominance tasks and has been adopted in some earlier studies [16–18].

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The results of m independent temporal dominance tasks for the same food are collected.
We call this sample set P0 . The set of TDS curves computed using all the samples in P0
is called C0 . From P0 , m tasks are randomly sampled with replacement and form a new
sample set P1 . Note that some tasks in P1 may be the same. A set of TDS curves, C1 , iscomputed based on P1 . This process of resampling is repeated q times, and q curve sets
C1 , ..., Cq are acquired. The expected values of q curve sets match the values of curves in C0 .
2.3. Monte Carlo Simulation of Temporal Dominance Tasks Based on Markov Chains
The combination of Monte Carlo simulation and Markov chains is used to simulatea large number of TDS tasks. The Monte Carlo method is a simulation based on many
random samples, while Markov chains express the probabilistic transitions of dominantsensations while eating food [19–21,27] and are used to simulate TDS tasks.
In the present study, two cases are simulated. In Cases 1 and 2, four and six descriptors
are used, respectively. As shown in Figure 2, descriptors are expressed as states, and theinitial state at t0 = 0 is determined by the initial distribution of (D1, D2, D3, D4) = (0.5,
0.25, 0.25, 0) for Case 1 and (D1, D2, D3, D4, D5, D6) = (1/6, 1/3, 1/3, 1/6, 0, 0) for Case 2.
Here, Di (i ∈ {1, 2, 3, 4} for Case 1 and i ∈ {1, 2, 3, 4, 5, 6} for Case 2) indicates descriptor
i. For example, in Case 1, D1 is selected at the probability of 0.5 at t0 = 0. The state at
t0 = ∆t after one transition is probabilistically determined by the transition probabilities inTables 1 and 2. For simplification, each task is assumed to consist of 20 transitions; that is,∆t = 1/20. The state-transition table varies according to t0 . The entire period is split into
three phases: initial (t0 = 0–6/20), middle (t0 = 7/20–13/20), and last (t0 = 14/20–1.0). The
transition probabilities in the tables were arbitrarily determined by the author. For example,
when D1 is selected at t0 = 0, D1 is likely to be kept selected at t0 = ∆t at the probabilityof 0.5, according to Table 1. D2 is also likely to be selected after D1 at the probability of
0.3. By continuing this process along the timeline, the temporal evolution of descriptors is
acquired. Consequently, the process to sequentially select descriptors referring to dominantsensations is simulated.
Table 1. Tables of transition probabilities for Case 1 involving four descriptors. Di indicates descriptor
i. Initial, middle, and last phases range t0 = 0–6/20, t0 = 7/20–13/20, and t0 = 14/20–1, respectively.

Initial
Phase

From

D1
D2
D3
D4

Middle
Phase

From

D1
D2
D3
D4

Last
Phase

From

D1
D2
D3
D4

D1

Destination
D2

D3

D4

0.5
0.1
0.1
0

0.3
0.5
0.2
0

0.2
0.3
0.5
0.4

0
0.1
0.2
0.6

D1

Destination
D2

D3

D4

0.4
0.1
0.1
0.1

0.3
0.4
0.1
0.1

0.2
0.3
0.6
0.2

0.1
0.2
0.2
0.6

D1

Destination
D2

D3

D4

0.2
0.1
0.1
0.1

0.2
0.3
0.2
0

0.3
0.3
0.3
0.2

0.3
0.3
0.4
0.7

Foods 2021, 10, 2472

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ଵଵ
D1
ଵଶ

ଵଷ
ଶଵ

ସଵ

ଶଶ
D2

ଷଵ

ଷଷ

ଷଶ

ଶଷ
ଶସ
ଵସ
ସଶ

D3

ସଷ
ଷସ

D4
ସସ

Figure 2. Markov chain model for Case 1 involving four descriptors. D1–D4 indicate descriptors 1–4.
p ab indicates the probability of transitioning to Db from Da.
Table 2. Transition tables for Case 2 involving six descriptors.

Initial
Phase

From

D1
D2
D3
D4
D5
D6

Middle
Phase

From

D1
D2
D3
D4
D5
D6

Last
Phase

From

D1
D2
D3
D4
D5
D6

D1

Destination
D2
D3

D4

D5

D6

0.4
0.1
0.1
0
0
0

0.3
0.4
0.2
0.1
0
0

0.2
0.2
0.4
0.4
0.3
0.2

0.1
0.2
0.2
0.3
0.2
0.2

0
0.1
0
0.1
0.3
0.3

0
0
0.1
0.1
0.2
0.3

D1

Destination
D2
D3

D4

D5

D6

0.3
0.0
0.1
0
0
0

0.3
0.3
0.1
0.1
0
0

0.2
0.3
0.4
0.2
0.1
0.1

0.1
0.2
0.3
0.4
0.2
0.2

0.1
0.1
0.1
0.2
0.3
0.3

0
0.1
0
0.1
0.4
0.4

D1

Destination
D2
D3

D4

D5

D6

0.1
0.1
0.1
0.1
0
0

0.2
0.3
0.2
0
0
0

0.3
0.3
0.4
0.5
0.1
0.1

0.1
0
0.1
0.2
0.6
0.3

0
0
0
0
0.3
0.5

0.3
0.3
0.3
0.2
0
0.1

Following the abovementioned methods, for both Cases 1 and 2, the results of n tasks
are produced and treated as the population. Figure 3 shows the TDS curves computed by
all the tasks in the population when n = 10000.

Foods 2021, 10, 2472

6 of 11(b) Case 2 with six descriptors
1.0
Descriptor 1
Descriptor 2
Descriptor 3(a) Case 1 with four descriptors
Descriptor 1
Descriptor 2

Descriptor 3
Descriptor 4

Dominance proportion (-)

Dominance proportion (-)

1.0

0.5

Descriptor 4
Descriptor 5
Descriptor 6

0.5

0

0
0

0

1.0

Normalized time (-)

Normalized time (-)

1.0

Figure 3. Temporal dominance curves, that is, dominance proportions, computed from the populations of which n = 10,000.
(a) Case 1 with four descriptors. (b) Case 2 with six descriptors.

2.4. Estimation of 95 % Confidence Intervals and Standard Errors
As mentioned above, as representative indices of uncertainties, 95% confidence intervals and standard errors are adopted. First, these values are estimated using resampled
TDS curves. Second, these values are estimated using approximated normal distributions.
The values estimated by the two approaches are then compared.
As in Figure 4, m tasks are randomly sampled from the population, including 10,000
tasks: n = 10,000. Hence, m is the sample size and is 15, 30, 50, 75, 100, 150, 200, 250, 300,
350, 400, 450, or 500. Based on m samples, the bootstrap resampling procedure is repeated1000 times (q = 1000), and 1000 sets of TDS curves are computed (C1 , . . . , C1000 ). At each
instant t0 , for each descriptor, the 95% confidence interval is estimated as the range between
the 25th largest curve value and 25th smallest curve value among the 1000 curve values.
Then, it is checked whether the dominance proportion value in the population is includedin this range. These procedures are repeated 1000 times, and the proportion for which
the population value was included in the estimated range is calculated. Note that this
proportion is expected to be close to 0.95.
Population P (size n = 10,000)

Population curves CʞʞNormalized time
Random sampling without replacement
Sample set ܲ଴ (size m)
1

3

2

4

Sample curves C0ʞ m

Random sampling with replacement
Resampled set ܲଵ (size m)
3

7

3

5

Resampled curves C1

ʞ 5

ʞResampled set ܲଵ଴଴଴ (size m)
8

1

1

6

Resampled curves C1000

ʞ 4

Figure 4. Method of simulation. Individual circles indicate independent TDS tasks.

Foods 2021, 10, 2472

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Similar processes are conducted for standard errors. The range between the 159thlargest and 159th smallest values is used as the estimated standard error. The proportion atwhich the population value is included in the estimated range of standard error is expected
to be close to 0.682.
The values of TDS curves are thought to be subject to normal distributions, because
the values are proportions [2]. The uncertainties can then be computed using normal
distributions. When the curve value for descriptor i at time t0 is denoted by di (t0 ), the 95%
confidence interval of di (t0 ) is estimated byr0

di (t ) ± 1.96

di (t0 )(1 − di (t0 ))
.
m

(3)

Similarly, the standard error is estimated byr0

di ( t ) ±di (t0 )(1 − di (t0 ))
.
m

(4)

3. Results
Figure 5 shows the examples of sample TDS curves and confidence intervals that were
estimated by resampling. The three figures show those simulated when the sample sizes
are m = 50, 100, and 200, respectively. Following the principles of statistical estimation, agreater sample size leads to smaller confidence intervals. Although they are not shown,
this trend is the same for the standard errors.
(a) Case 1

Dominance proportion (-)

1

Descriptor 1
1

m = 501

m = 100

0.5

Lower limits of 95% CI

0.5

0.5

0
0

Normalized time (-)

1.0

0
0

(b) Case 2
1.0

1.0

1.0

Lower limits of 95% CI
0.5

0.5

0

0

0
Normalized time (-)

1.0

1.0

m = 200

TD curves from m samples(Sample curves)

0.5

Normalized time (-)

Descriptor 4

m = 100

Upper limits of 95% CI0

1.0

Normalized time (-)
Descriptor 1

m = 500m = 200

TD curves from m samples(Sample curves)

Upper limits of 95% CI

0Dominance proportion (-)

Descriptor 40Normalized time (-)

1.00Normalized time (-)

1.0

Figure 5. Examples of temporal dominance curves and 95% confidence intervals for Cases 1 (a) and 2 (b). For visual clarity,
only Descriptors 1 and 4 are shown.

Figure 6 shows the inclusion probabilities that the dominance proportions of the
population are included in the estimated confidence intervals and standard errors at eachdiscrete moment t0 for different m values. Note that the uncertainties were computed by

Foods 2021, 10, 2472

8 of 11the two methods. One was based on resampling, and the other was based on normal
distributions following (3) and (4). When m was relatively small, the inclusion probabilities
were smaller than the expected values, that is, 0.95 and 0.682 for confidence intervals
and standard errors, respectively. The confidence intervals and standard errors were
underestimated with small m values. Note that it is empirically known that uncertainties
based on approximated normal distributions are inaccurate when mp < 5 or m(1 − p) < 5,where p is the sample proportion, suggesting that estimated uncertainties are not reliable
for small sample sizes. As in the figures, when m was large, the uncertainties estimated byresampling and normal distributions were close to each other.
Comparing Case 1 (four descriptors) and Case 2 (six descriptors), the inclusion probabilities of Case 1 seem larger than those of Case 2 when m = 15 and 30. The present study
does not further investigate this point. Note that the differences between the two cases
include the number of descriptors and profiles of TDS curves. Hence, the root causes of thedifferences in the inclusion probabilities between the two cases remain unclear.
Uncertainties estimated by approximated normal distribution

1.0

95% confidence interval

0.95 line

0.9

0.8Standard error

0.682 line

0.7

0.6
0

100

200

300
m: Sample size

400

500

Probability at which the TD curves in the
population are included in the intervals(b) Case 2(a) Case 1
Probability at which the TD curves in the
population are included in the intervals

Uncertainties estimated by bootstrap resampling

1.0

95% confidence interval

0.95 line

Standard error

0.682 line

0.9

0.8

0.7

0.6
0

100

200
300
m: Sample size

400

500

Figure 6. Probabilities that the dominance proportions in the population are included in the estimated uncertainties, thatis, confidence intervals and standard errors. (a) Case 1. (b) Case 2. The uncertainties are computed using resampling andnormal distributions. Probabilities are computed at m = 15, 30, 50, 75, 100, 150, 200, 250, 300, 350, 400, 450, and 500.

4. Discussion
Two types of uncertainties, standard error and 95% confidence interval, of TDS curve
values were repeatedly estimated through a combination of Monte Carlo simulation andbootstrap resampling. Approximately 68.2% of the estimated standard errors included
the population values. Similarly, 95% of the estimated confidence intervals included
the population values. These results indicate that resampling of TDS tasks is valid for
estimating uncertainties with large sample sizes m. Thus, the resampling produces random
TDS curves centering around the sample curves. In terms of the acceptable m values, basedon Figure 6, when m ≥ 100, the uncertainties were fairly estimated. However, practically,
the sample size of 100 is unlikely for most settings of sensory appraisal. Most of the previous
studies on temporal dominance methods do not satisfy this sample size. Considering thissituation, the present study proposes an acceptable value range of m = 50–100 while notingthat the uncertainty indices tend to be underestimated with small m values.
As mentioned above, bootstrap resampling is beneficial for analysis techniques requiring many TDS curves. Because of the bootstrap method, temporal dominance methods
and a variety of statistical and machine-learning techniques will be closely tied. Typically, the concern of overfitting can be circumvented by using resampled curves. Anothersuggestion of the present study is that hypothesis testing of dominance proportions orFoods 2021, 10, 2472

9 of 11values of TDS curves can be conducted via approximated normal distributions. A standard
parametric method for proportions is available when the sample size is large. For example,
a dominance proportion at t0 for a certain brand of food can be statistically comparedwith that for another brand. Further, a dominance proportion at t0 is tested with regard towhether it is significantly greater than zero or the chance level. Note that these hypothesis
tests using approximated normal distributions should involve a sample size above 50–100,
although such restrictions on the sample size were not mentioned in the original study by
Pineau et al. [2] nor in the international standards [9].
Some limitations of this study should be mentioned. First, the acceptable m values
may depend on populations, but only two types of population models were investigatedin this study. Particularly, when the TDS curves of the population include small valuesof nearly zero, it is difficult to accurately estimate uncertainties. However, such smallvalues below the chance level are typically regarded as trivial and ignored for analyses [2].
Second, this study is limited to the TDS method, where only one descriptor can be selected
at each moment, and the dominance proportions can be computed. As a variation of the
TDS method, the temporal check-all-that-apply (TCATA) method [28,29] is also popular,
as multiple descriptors can be selected simultaneously. Although TDS and TCATA methods [30,31] are often compared, the curves yielded by the TCATA method do not indicate
dominance proportions. Moreover, thus far, mathematical models that can be used for aMonte Carlo simulation of the TCATA method have not been determined, whereas Markov
chains are used for TDS methods [19–21].
5. Conclusions
The TDS method is a time-series sensory appraisal method that has been increasingly
used in the field of food science. In general, a set of dominance-proportion curves is computed based on all the samples, whereas some statistical and machine-learning techniques
require several or many sets of curves. Furthermore, it is thought that the uncertainties ofdominance proportions are computed using approximated normal distributions; however,their validity is yet to be investigated. Hence, thus far, the estimation of uncertainties has
rarely been utilized. Resampling is expected to resolve these issues, and this study proposesa resampling method for TDS curves. Through Monte Carlo simulation, the 95% confidence
intervals and standard errors could be adequately estimated when the sample sizes weregreater than 50–100. Furthermore, these uncertainties were close to those estimated using
approximated normal distributions. Note that these results are based on the simulation
study, and mathematical generalization remains unstudied. The resampling method for
TDS curves can be used to expand the analyses of TDS tasks. In the future, resampling
methods for a modified version of the TDS method, such as the TCATA method, need to beestablished, and successful analysis methods using the bootstrap resampling method are
expected. For example, machine-learning methods using TDS curves to distinguish foodproducts or judge food preferences are expected applications.
Funding: This research was funded by the Tojuro Iijima Foundation for Food Science and Technology(2020-40).
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Conflicts of Interest: The author declares no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript:
TCATA
TDS

temporal check-all-that-apply
temporal dominance of sensations

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