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# Categorical Functional Data Analysis applied to Temporal Dominance of Sensations data

Caroline Peltier, Michel Visalli, Pascal Schlich, Hervé Cardot

caroline.peltier@inrae.fr
* Presentation of Temporal Dominance of Sensations (TDS) and Categorical Functional Data Analysis (CFDA)

* Illustration of the application of CFDA on a simulated dataset

* Illustration on real TDS dataset
# Center of Taste and Feeding Behavior

**A problematic:**

What happens during a tasting?

<figure><img src="image_9.png" /></figure>

Two approaches to understand

<figure><img src="image_10.png" /></figure>

**Chemical methods:** measuring
directly in the mouth

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>

**Sensory methods:** asking the
subjects what they perceive

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>

Static or
temporal
data...
# Temporal Dominance of Sensations

**Temporal Dominance of Sensations (TDS):** protocol of sensory analysis that consists in asking the subject to declare dynamically the dominant sensation (one among a list of descriptors)

Sour

Bitter

Sweet

START

STOP
# Temporal Dominance of Sensations

**Temporal Dominance of Sensations (TDS):** protocol of sensory analysis that consists in asking the subject to declare dynamically the dominant sensation (one among a list of descriptors)

<figure><img src="image_16.png" /></figure>
# Temporal Dominance of Sensations

**Temporal Dominance of Sensations (TDS):** protocol of sensory analysis that consists in asking the subject to declare dynamically the dominant sensation (one among a list of descriptors)

<figure><img src="image_17.png" /></figure>
# Temporal Dominance of Sensations

**Temporal Dominance of Sensations (TDS):** protocol of sensory analysis that consists in asking the subject to declare dynamically the dominant sensation (one among a list of descriptors)

<figure><img src="image_18.png" /></figure>
# Temporal Dominance of Sensations

**Temporal Dominance of Sensations (TDS):** protocol of sensory analysis that consists in asking the subject to declare dynamically the dominant sensation (one among a list of descriptors)

Sour

Bitter

Sweet

START

STOP
# Temporal Dominance of Sensations

**Temporal Dominance of Sensations (TDS):** protocol of sensory analysis that consists in asking the subject to declare dynamically the dominant sensation (one among a list of descriptors)

<figure><img src="image_19.png" /></figure>

<figure><img src="image_20.png" /></figure>

These are categorical functional data... a method adapted for this data is categorical functional data analysis (CFDA)
# History of Categorical Functional Data Analysis

## CFDA as « analyse harmonique qualitative »

DEVILLE J.C. et SAPORTA G. (1979) Analyse harmonique qualitative. In Data Analysis and Informatics (E. DIDAY et al. Edito rs), 375-389, North-Holland, Amsterdam.

1982

1979

1981

GILBERT SAPORTA

Méthodes exploratoires d'analyse de données temporelles

Cahiers du Bureau universitaire de recherche opérationnelle. Série Recherche, tome 37-38 (1981), p. 7-194

<http://www.numdam.org/item?id=BURO_1981__37-38__7_0>

<figure><img src="image_21.png" /><figcaption>G. Saporta</figcaption></figure>

Analyses de données chronologiques qualitatives : comment analyser des calendriers?

par Jean-Claude DEVILLE*

La statistique décroîtive des processus alla- bours est un domaine à peur très vaste. Cet

1996

L'analyse harmonique qualitative, une synthèse de la théorie
Gilbert Saporta

Didactic synthesis of CFDA

<figure><img src="image_22.png" /><figcaption>J.-C. Deville</figcaption></figure>

<figure><img src="image_23.png" /></figure>

mathematics

MDPI

Article

Categorical Functional Data Analysis. The cfda R Package

Cristian Preda L**A**, Quentin Grimonprez 4 and Vincent Vandewalle M**

<figure><img src="image_24.png" /></figure>

<figure><img src="image_25.png" /></figure>

C. Preda
2021
# CFDA: an extension of Correspondance Analysis

**Correspondence Analysis:** a usual statistical analysis allowing two qualitative variables

X: $\Omega \to S_1$ and Y: $\Omega \to S_2$ to be conjointly analyzed.

It can also be seen as optimal encoding finding numerical values associated to each modality by finding $\varphi_1: S_1 \to \mathbb{R}$ and $\varphi_2: S_2 \to \mathbb{R}$ minimizing:

$$E((\varphi_1(X) - \varphi_2(Y))^2)$$

Under constraints $E(\varphi_1(X)) = E(\varphi_2(Y)) = 0$ and $E(\varphi_1(X)^2) = E(\varphi_2(Y)^2) = 1$

## Example of encoding with previous data considering

X: state chosen during period 1 and Y: state chosen during period 2

Period. 1

<table><thead><tr><th>Modality</th><th>Numerical Encoding</th></tr></thead><tbody><tr><td>●</td><td>NA</td></tr><tr><td>●</td><td>-0.35</td></tr><tr><td>●</td><td>0.7</td></tr></tbody></table>

Period. 2

<table><thead><tr><th>Modality</th><th>Numerical Encoding</th></tr></thead><tbody><tr><td>●</td><td>-0.35</td></tr><tr><td>●</td><td>0.7</td></tr><tr><td>●</td><td>0</td></tr></tbody></table>

<figure><img src="image_26.png" /></figure>
$X_t: \Omega \to S = \{x_1, ..., x_S\} : (X_t)_{t \in T}$ is a random qualitative process = qualitative functional data

**_Idea of CFDA:_** Finding optimal encodings of the states evolving over time

$$ \varphi : S \times T \to \mathbb{R} \\ (x, t) \to \varphi(x, t), $$

such as the encoded process $\varphi(X_t, t)$ minimizes

$$ \iint_{T \times T} E((\varphi(X_t, t) - \varphi(X_s, s))^2) dt ds $$

$$ \text{u. c. } \forall t, E(\varphi(X_t, t)) = 0; \text{Var}(\varphi(X_t, t)) = 1 $$

$\varphi(X_t, \cdot)$ can be approximated by projection on a basis of functions
<figure><img src="image_27.png" /><figcaption>Coordinates of individuals</figcaption></figure>

<figure><img src="image_28.png" /></figure>

<figure><img src="image_29.png" /></figure>

Basis of functions

order=3

n_bases=6

<figure><img src="image_30.png" /></figure>

<figure><img src="image_31.png" /></figure>

Optimal encodings of states
Simulation

<figure><img src="image_32.png" /></figure>
<figure><img src="image_33.png" /></figure>

Individual map

<figure><img src="image_34.png" /></figure>

Cumulative eigenvalues

<figure><img src="image_35.png" /></figure>

Harm. 1

<figure><img src="image_36.png" /></figure>

Harm. 2

<figure><img src="image_37.png" /></figure>
<figure><img src="image_38.png" /></figure>

Individual map

<figure><img src="image_39.png" /></figure>

Cumulative eigenvalues

<figure><img src="image_40.png" /></figure>

Harm. 1

<figure><img src="image_41.png" /></figure>

Harm. 2

<figure><img src="image_42.png" /></figure>
<figure><img src="image_43.png" /></figure>

Individual map

<figure><img src="image_44.png" /></figure>

Cumulative eigenvalues

<figure><img src="image_45.png" /></figure>

Harm. 1

<figure><img src="image_46.png" /></figure>

Harm. 2

<figure><img src="image_47.png" /></figure>
<figure><img src="image_48.png" /></figure>

<figure><img src="image_49.png" /></figure>

<figure><img src="image_50.png" /></figure>

<figure><img src="image_51.png" /></figure>

<figure><img src="image_52.png" /></figure>
<figure><img src="image_53.png" /></figure>

<figure><img src="image_54.png" /></figure>

<figure><img src="image_55.png" /></figure>

<figure><img src="image_56.png" /></figure>

<figure><img src="image_57.png" /></figure>
<figure><img src="image_58.png" /></figure>

<figure><img src="image_59.png" /></figure>

<figure><img src="image_60.png" /></figure>

<figure><img src="image_61.png" /></figure>

<figure><img src="image_62.png" /></figure>
<figure><img src="image_63.png" /></figure>

Individual map

<figure><img src="image_64.png" /></figure>

Cumulative eigenvalues

<figure><img src="image_65.png" /></figure>

Harm. 1

<figure><img src="image_66.png" /></figure>

Harm. 2

<figure><img src="image_67.png" /></figure>
# A real dataset with gustometer

A gustometer delivers controlled solutions in the mouth

<figure><img src="image_68.png" /><figcaption>## TDS screen</figcaption></figure>

<figure><img src="image_69.png" /></figure>

<figure><img src="image_70.png" /></figure>

Study on a dataset with three signals evaluated by 50 subjects on 8 descriptors<fnref n="1" />,<fnref n="2" />

<figure><img src="image_71.png" /></figure>

<footnote n="1">¹Beno, Nicolle, Visalli (2023) A dataset of consumer perceptions of gustometer-controlled stimuli measured with three temporal sensory evaluation methods. Data in Brief</footnote>

<footnote n="2">²Visalli, M., Beno, N., Nicolle, L., Schlich, P. (2023) Assessment of the validity and reliability of temporal sensory evaluation methods used with consumers on controlled stimuli delivered by a gustometer. Food Quality and Preference 110</footnote>
* Linewidth of the harmonic proportional to the number of citations

<figure><img src="image_72.png" /></figure>

<figure><img src="image_73.png" /></figure>

<figure><img src="image_74.png" /></figure>

<figure><img src="image_75.png" /></figure>

<figure><img src="image_76.png" /></figure>

<figure><img src="image_77.png" /></figure>
CFDA results

* Positive scores on the first component: related to signal S06

* Sweet at the beginning, then Acid. Basilic all the time

* => coherent with the signal S06-S07-S04

<figure><img src="image_78.png" /></figure>

<figure><img src="image_79.png" /></figure>

<figure><img src="image_80.png" /></figure>

<figure><img src="image_81.png" /></figure>

<figure><img src="image_82.png" /></figure>

<figure><img src="image_83.png" /></figure>
CFDA results

* Negative scores on the first component: related to signal S07 and S04

* Acid and Lemon at the beginning then Sweet --> coherent with the signal

<figure><img src="image_84.png" /></figure>

<figure><img src="image_85.png" /></figure>

<figure><img src="image_86.png" /></figure>

<figure><img src="image_87.png" /></figure>

<figure><img src="image_88.png" /></figure>

<figure><img src="image_89.png" /></figure>

<figure><img src="image_90.png" /></figure>

<figure><img src="image_91.png" /></figure>

<figure><img src="image_92.png" /></figure>
* Positive scores on the second component: **Mint** then **Licorice** and finally **Lemon**
= DISTRACTORS: outlier detection

<figure><img src="image_93.png" /></figure>

<figure><img src="image_94.png" /></figure>

<figure><img src="image_95.png" /></figure>

<figure><img src="image_96.png" /></figure>

<figure><img src="image_97.png" /></figure>

<figure><img src="image_98.png" /></figure>
# How to characterize the three signals (if they were unknown)

**Objective:** attempt to explain the
differences between the signals

<figure><img src="image_99.png" /></figure>

Apply sPLSDA on the CFDA
components
signal
$$
\begin{pmatrix}
A \\
A \\
B \\
B \\
C \\
C
\end{pmatrix}
$$

11 CFDA components were
selected (cross-validation of
the model)

<figure><img src="image_100.png" /></figure>

**<u>Partial Least Square regression (PLS)</u>:**

a multivariate statistical method applied to two
matrices $X_1(n, p_1)$, and $X_2 (n, p_2)$, that finds the
weights $w_1, w_2$ maximizing

$$
\operatorname{cov}(X_1w_1, X_2w_2)
$$

under constraints of unit variance of the

$$
\text{weights } ||w_1||_2 = ||w_1||_2 = 1
$$

**<u>Sparse Partial Least Square (sPLS)</u>:**

Adding constraints of sparsity: ||wᵢ||₁ < σᵢ with
0 < σᵢ < 1 for selecting variables (more
variables than individuals)

**<u>sPLS - Discriminant Analysis (sPLS-DA)</u>:** consists
in choosing $X_2$ as a **dummy matrix**
corresponding to a qualitative variable

CFDA components

<figure><img src="image_101.png" /></figure>

?
sPLSDA

<figure><img src="image_102.png" /></figure>
# sPLS-DA results

The three signals are well discriminated.
First axis similar to the CFDA's one.

<figure><img src="image_103.png" /></figure>

<figure><img src="image_104.png" /></figure>

<figure><img src="image_105.png" /></figure>

<figure><img src="image_106.png" /></figure>

<figure><img src="image_107.png" /></figure>

<figure><img src="image_108.png" /></figure>
# sPLS-DA results

The second axis discriminated S07 and S04. S07 is **Acid** then **Sweet** in the middle of the tasting and **Salty** at the end

<figure><img src="image_109.png" /></figure>

<figure><img src="image_110.png" /></figure>

<figure><img src="image_111.png" /></figure>

<figure><img src="image_112.png" /></figure>

<figure><img src="image_113.png" /></figure>

<figure><img src="image_114.png" /></figure>

<figure><img src="image_115.png" /></figure>
# sPLS-DA results

The second axis discriminated S07 and S04. S04 is **Sweet** at the end of the tasting.

<figure><img src="image_116.png" /></figure>

<figure><img src="image_117.png" /></figure>

<figure><img src="image_118.png" /></figure>

## sPLS results

<figure><img src="image_119.png" /></figure>

## Harm. 1

<figure><img src="image_120.png" /></figure>

## Harm. 2

<figure><img src="image_121.png" /></figure>
<figure><img src="image_122.png" /></figure>

CFDA works well for these TDS data and returns results that are coherent with the original signals.

<figure><img src="image_123.png" /></figure>

**Some important technical parameters:**

Selection of the basis of functions (number, order, ...) ?)

Parameters of sPLSDA to be discussed
# Use of FDA techniques to other temporal sensory analysis protocols

* Multivariate CFDA with Temporal Check All That Apply

<figure><img src="image_124.png" /></figure>

* Multivariate FDA for temporal intensity values

<figure><img src="image_125.png" /></figure>

* Conjoint analysis of chemico-physical data (quantitative, analyzed with FDA) and sensory TDS data (qualitative, analyzed with CFDA)

<figure><img src="image_126.png" /></figure>
Thanks for your attention!

<figure><img src="image_127.png" /></figure>

ELSEVIER

Food Quality and Preference

Volume 109, July 2023, 104893

<figure><img src="image_128.png" /></figure>

Analyzing temporal dominance of
sensations data with categorical
functional data analysis

Caroline Peltier<sup>a, b</sup> 🌐, Michel Visalli<sup>a, b</sup>, Pascal Schlich<sup>a, b</sup>, Hervé Cardot<sup>c</sup>

Questions?

<figure><img src="image_129.png" /></figure>