# Efficient Characterization of the Human Balance Using HMMs

A J Tsagmo Denkeng, A. Mohammad Mourad, Sylvain Iloga, R M Mballa

Mba, Hajer Baazaoui, T. Djotio Ndié, Olivier Romain

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A J Tsagmo Denkeng, A. Mohammad Mourad, Sylvain Iloga, R M Mballa Mba, Hajer Baazaoui, et al. Efficient Characterization of the Human Balance Using HMMs. IEEE Access, 2025, 13, pp.183456-183479. 10.1109/ACCESS.2025.3622375. hal-05346618

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Distributed under a Creative Commons Attribution 4.0 International License
Received 18 September 2025, accepted 10 October 2025, date of publication 16 October 2025, date of current version 30 October 2025.

Digital Object Identifier 10.1109/ACCESS.2025.3622375

RESEARCH ARTICLE

# Efficient Characterization of the Human Balance Using HMMs

A. J. TSAGMO DENKENG<sup>1</sup>, A. MOHAMMAD MOURAD<sup>2</sup>, SYLVAIN ILOGA<sup>1,3,4</sup>, R. M. MBALLA MBA<sup>5</sup>, HAJER BAAZAOUI<sup>1</sup>, T. DJOTIO NDIÉ<sup>6</sup>, AND OLIVIER ROMAIN<sup>1</sup>

Corresponding author: Sylvain Iloga (sylvain.iloga@gmail.com)

This work was supported by the French National Research Agency (ANR) under the Labcom 2024 call for projects, grant number ANR-24-LCV2-0020-01.

**ABSTRACT** Human balance evaluation has become crucial over the past decades for several clinical applications in stabilometry. Consequently, various reliable clinical tests related to the human balance and many variables characterizing the displacements of the human Center Of Pressure (COP) recorded in a statokinesigram have been proposed. Unfortunately, objectively selecting the accurate subset of COP variables to use for a specific task is actually a serious challenge due to the high number of existing variables. Additionally, when taken separately, each variable aggregates all the statokinesigram data in one single value. Furthermore, the computation schemes of the existing COP variables do not involve the automatic learning of the overall behavior of the COP during the traversal of its trajectory. To overcome these limitations, the current work proposes a new technique that trains Hidden Markov Models (HMMs) for learning the sequential variations of some geometric features that occur in the statokinesigrams of each subject, irrespective of the test conditions. Meta-data derived from the resulting HMMs are then used for deriving one unique user descriptor vector. Classification experiments conducted on two open datasets with the vectors generated by the proposed approach exhibited best accuracies of 80.36% for age-group prediction, of 77.91% for fall risk prediction and of 98.87% for the prediction of Parkison's disease. This performance is +15.36% and +3.91% better than existing techniques for age-group and fall risk prediction.

**INDEX TERMS** Human balance, hidden Markov models, statokinesigram, stabilometry, COP variables.

## I. INTRODUCTION

Age-related disabilities and some illnesses negatively affect balance in humans and impact their health and quality of life [1].<fnref n="1" /> Human balance evaluation has therefore become crucial over the past decades for many clinical applications in stabilometry, which is the scientific study of the human postural stability under various balance conditions. These clinical applications include age-group [2], [3], [4], fall risk [4], [5], [6], [7] and illness [8], [9] prediction. Clinicians

The associate editor coordinating the review of this manuscript and approving it for publication was Ines Domingues.

have consequently proposed various reliable clinical tests related to the human balance. Among the most widespread clinical tests one can list: the *Mini Balance Evaluation Systems Tests* (Mini-BEST) [10], the *Falls Efficacy Scale International* (FES-I) [11], the *International Physical Activity Questionnaire* (IPAQ) [12] or the *Trail Making Test* (TMT) [13]. The results of these clinical tests are generally used by researchers to enrich the description of each subject in dedicated public datasets [1], [8]. These results can further be used for data labeling purposes. This is the case in fall risk prediction where the authors relied on the Mini-BEST score [9] and the FES-I score [6], [7] to label the data.

<footnote n="1">¹ETIS UMR 8051, CNRS, ENSEA, CY Cergy Paris Université, 95000 Cergy, France</footnote>

<footnote n="2">²Department of Mathematics and Computer Science, Faculty of Sciences, University of Maroua, Maroua, Cameroon</footnote>

<footnote n="3">³Department of Computer Science, Higher Teachers' Training College, University of Maroua, Maroua, Cameroon</footnote>

<footnote n="4">⁴CIRD, UMMISCO, University of Sorbonne, 93143 Bondy, France</footnote>

<footnote n="5">⁵Central Hospital of Yaoundé, Geriatric Unit, Yaoundé, Cameroon</footnote>

<footnote n="6">⁶Department of Computer Engineering, National Advanced School of Engineering, University of Yaoundé I, Yaoundé, Cameroon</footnote>

<footnote n="1">¹See the Introduction of [1]</footnote>
Beside these clinical tests, researchers have proposed several variables that characterize the displacements of the human Center Of Pressure (COP) recorded as time-series numeric data in a *statokinesigram*. A recent review of the existing COP variables is available in [14]. These variables can be organized into two main groups: time-domain variables and frequency 2 variables. Time-domain variables focus on the temporal information related to the displacements of the COP and are divided into three main subgroups: positional, 3 dynamic 4 and stochastic 5 variables. Positional variables are dedicated to the characterization of the dispersion of the COP trajectory, but they do not consider the dynamic of the local displacements of the COP. Dynamic variables enable to overcome this limitation through the quantification of the velocity of the COP trajectory, by correlating each position of the COP with the next position. But they are unable to correlate pairs of non consecutive positions of the COP. In order to overcome this drawback, existing stochastic variables can correlate each position of the COP with any previous position through the analysis of the mean square displacements of the COP. Their main challenge is the selection of the appropriate previous positions with which the correlation should effectively be made to obtain the best human balance evaluation. Unlike time-domain variables, frequency variables provide spectral information related to the displacements of the COP. But they induce a loss of temporal localization and a high computational complexity.

An important drawback shared by the existing COP variables is related to the fact that, objectively selecting the accurate subset of COP variables to use for a specific purpose is actually a serious challenge due to the high number of existing variables. Indeed, researchers must always combine various existing COP variables into a vector to describe a subject because when taken separately, each variable aggregates in one single value all the high volume of data contained in a statokinesigram. This induces a high level of data compression and a significant risk of loss of information. Additionally, the computation schemes of the existing COP variables do not involve the automatic learning of the overall behavior of the COP during the traversal of its trajectory. Yet, this may produce finer descriptors of the COP displacements that can lead to better performances in dedicated clinical applications.

The motivation of the current work is to make a finer characterization of the human balance capable to attenuate the aforementioned drawbacks of the existing COP variables. In order to achieve this goal, the current work proposes a new stochastic technique that trains a Hidden Markov Model (HMM) for automatically learning the sequential variations of some geometric features that occur in the

<table><caption>TABLE 1. Main parameters used in the current work.</caption><thead><tr><th>Parameter</th><th>Description</th></tr></thead><tbody><tr><td>u, v</td><td>Subjects</td></tr><tr><td>κ</td><td>The time-shift</td></tr><tr><td>u&#775;κ</td><td>Final vector associated with u</td></tr><tr><td>ξ&#775;</td><td>Time required to compute u&#775;κ</td></tr><tr><td>D<sub>κ</sub>(u, v)</td><td>Distance between the subjects u and v</td></tr><tr><td>n</td><td>Number of test conditions</td></tr><tr><td>m</td><td>Number of trials</td></tr><tr><td>(p<sub>l</sub>, p&#775;<sub>l</sub>)</td><td>Pair of points of a statokinesigram</td></tr><tr><td>S<sub>i</sub></td><td>Set of statokinesigrams recorded under the i<sup>th</sup> test condition</td></tr><tr><td>S&#775;<sub>i,κ</sub></td><td>Mean vector derived from S<sub>i</sub></td></tr><tr><td>S<sub>i,j</sub></td><td>j<sup>th</sup> statokinesigram recorded under the i<sup>th</sup> test condition</td></tr><tr><td>δ<sub>κ</sub>(S<sub>i,j</sub>)</td><td>Long MC derived from S<sub>i,j</sub></td></tr><tr><td>Δ<sub>κ</sub>(S<sub>i,j</sub>)</td><td>Set of short MCs derived from δ<sub>κ</sub>(S<sub>i,j</sub>)</td></tr><tr><td>λ<sub>κ</sub>(S<sub>i,j</sub>)</td><td>HMM derived from Δ<sub>κ</sub>(S<sub>i,j</sub>)</td></tr><tr><td>S&#775;<sub>i,j,κ</sub></td><td>Single vector derived from λ<sub>κ</sub>(S<sub>i,j</sub>)</td></tr><tr><td>χ<sub>κ</sub><sup>l</sup></td><td>l<sup>th</sup> component of S&#775;<sub>i,j,κ</sub></td></tr><tr><td>λ</td><td>A HMM</td></tr><tr><td>(A, B&#775;, π)</td><td>The parameters of λ</td></tr><tr><td>S</td><td>Set of states of λ</td></tr><tr><td>ϑ</td><td>Set of symbols of λ</td></tr><tr><td>φ</td><td>Stationary distribution of λ</td></tr><tr><td>γ</td><td>Maximum number of iterations of the Baum-Welch algorithm</td></tr><tr><td>r</td><td>Number of matrix products during the computation of φ</td></tr><tr><td>σ</td><td>Down-sampling factor</td></tr><tr><td>N</td><td>Number of slices of the interval [0, 100]</td></tr><tr><td>ε</td><td>Enables to avoid division by zero during the HMM initialization</td></tr></tbody></table>

statokinesigram of a given subject. Meta-data derived from this HMM are then used for generating one descriptor vector associated with this statokinesigram. Given that several statokinesigrams are generally recorded for the same subject depending on the experimental conditions (vision, surface, etc.), the descriptor vectors corresponding to these statokinesigrams are then concatenated to derive one single descriptor vector associated with the subject, irrespective of the experimental conditions. The resulting descriptor vectors are finally experimented for age-group, fall risk and illness prediction using data from two open datasets containing statokinesigrams. Table 1 summarizes the main parameters used throughout the current work with their corresponding descriptions.

The rest of this paper is organized as follows: the background is presented in Section II, followed by a synthetic description of HMMs in Section III. The proposed approach is described in Section IV, while relevant clinical applications of the proposed approach are described in Section V. Experimental results are presented in Section VI and the last section is dedicated to the conclusion.

<footnote n="2">2 See the Table 5 and the Section III-D of [14]</footnote>

<footnote n="3">3 See the Table 3 and the Section III-B of [14]</footnote>

<footnote n="4">4 See the Table 4 and the Section III-C of [14]</footnote>

<footnote n="5">5 See the Table 6 and the Section 3.5 of [14]</footnote>
## II. BACKGROUND

Researches in stabilometry have been conducted by both clinicians and researchers. To evaluate the balance of a subject, clinicians have developed clinical tests, while researchers proposed COP variables. These researches rely on experiments conducted using dedicated recording devices according to various data acquisition protocols. The resulting recorded statokinesigrams are most often saved in open datasets that are later used for several applications including age-group, fall-risk and illness prediction. The current section provides details on all these relevant components of the background in human balance evaluation.

### A. CLINICAL TESTS

The following clinical tests have been encountered in the related work reviewed in the current paper:

1) The Mini-BEST [10]: It evaluates the fall risk of the participants through multiple dynamic balance tasks related to the four following sub-components of the human balance <fnref n="6" />: anticipatory, reactive postural control, sensory orientation and dynamic gait. It evaluates 14 items scored on a 3-point scale (0-2), for a maximum score of 28 points.

2) The FES-I [11]: It assesses the fear of fall of the participants who are asked to rank their level of concern that they might fall if doing a hypothetical activity. Its short version includes 7 items scored on a 4-point scale (1-4), for a maximum score of 28 points.<fnref n="7" />

3) The IPAQ [12]: In order to measure their practice of health-related physical activities on a 3-level scale (low, moderate, high), the participants of the IPAQ must answer to questions about the time they spend being physically active.

4) The TMT [13]: It is a neuropsychological assessment that aims at characterizing the cognitive functions (attention, processing speed, mental flexibility, executive function) of participants by asking them to connect a set of labeled circles as quickly as possible. It produces scores that are interpreted based on the completion times and the error rates.

### B. RECORDING DEVICES

The most common practice used in research laboratories and clinical environments to obtain a statokinesigram of a subject *u* consists in recording various information related to the trajectory followed by the COP of *u*, while *u* is standing still in the upright position, barefooted and arms laying at the side, on a dedicated device for a relatively short duration. The trajectory followed by the COP is generally sampled at a user-defined sampling frequency, the resulting signal is then smoothed with a selected low-pass filter. Two types of devices have yet been used for this purpose: the Force Platform (FPF) [1], [8] and the Wii Balance Board (WBB) [5], [14].

FPFs can be considered as the gold standard in human balance assessment due to their reliability and their extensive validation in most of the existing work. The first FPF was designed in the early 1950s by the constructor Kistler Instrumente AG,<fnref n="8" /> followed by several other manufacturers including AMTI,<fnref n="9" /> Bertec<fnref n="10" /> and Pasco.<fnref n="11" /> FPFs are highly accurate and highly precise with typical sampling frequencies ranging from 100Hz to 2000Hz or more. They record high-resolution measurements of the COP coordinates (COPₓ, COPᵧ) in the Medio-Lateral (ML) x-axis and in the Antero-Posterior (AP) y-axis,<fnref n="12" /> as well as many other parameters like the ground forces (Fₓ, Fᵧ, Fz) in the 3-dimensional space, alongside with the moments (Mₓ, Mᵧ, Mz) of these forces.<fnref n="13" /> Examples of graphics that present a statokinesigram and the curves of all the aforementioned parameters are available in [1].<fnref n="14" /> The main barriers that most often discourage institutions and individuals to use FPFs are: their elevated prices between €2000 and €50,000, the fact that they are less portable because they require a stable setup environment, and their high-level technical requirements for software manipulation and for data interpretation.

<table><caption>TABLE 2. Relevant characteristics of the main recording devices in stabilometry.</caption><thead><tr><td></td><td>FPF</td><td>WBB</td></tr></thead><tbody><tr><td>Domain</td><td>Medical</td><td>Gaming</td></tr><tr><td>Accuracy</td><td>High ✓</td><td>Low ✗</td></tr><tr><td>Precision</td><td>High ✓</td><td>Low ✗</td></tr><tr><td>Resolution</td><td>High ✓</td><td>Low ✗</td></tr><tr><td>Price</td><td>High ✗</td><td>Low ✓</td></tr><tr><td>Portability</td><td>Low ✗</td><td>High ✓</td></tr><tr><td>Technicality</td><td>High ✗</td><td>Low ✓</td></tr></tbody></table>

Exclusively constructed by Nintendo,<fnref n="15" /> the WBB was initially released in 2007 as a game accessory. It is in 2015 that Yoshikawa et al. examined the possibility of using the WBB in stabilometry [15] and it was finally validated as a device for stabilometry measurements by Nagymate et al. in 2016 [16]. Generally, the WBB is adopted when the budget is very small because of its significantly low prices (between €50 and €100) compared to FPFs, thus being accessible to a broader audience. Its manipulation does not require extensive technical knowledge and it can be used in various less restricted environments, including homes due to its high portability. But unlike FPFs, the WBB is less accurate and less precise with an unstable sampling frequency that can reach approximately 100Hz. Its sensors only provide low-resolution measurements of the COP coordinates and

<footnote n="6">⁶See the Table 1 of [9]</footnote>

<footnote n="7">⁷See the first Box of [11]</footnote>

<footnote n="8">⁸https://www.kistler.com/US/en/force-plate/C00000113</footnote>

<footnote n="9">⁹https://www.amti.biz/product-line/force-plates/</footnote>

<footnote n="10">¹⁰https://www.bertec.com/products/force-plates/</footnote>

<footnote n="11">¹¹https://www.pasco.com/products/sensors/pasport/pasport-force-platform</footnote>

<footnote n="12">¹²See the Equations 3 and 4 of [7]</footnote>

<footnote n="13">¹³See the Figure 1 of [7]</footnote>

<footnote n="14">¹⁴See the Figures 4.A to 4.H of [1]</footnote>

<footnote n="15">¹⁵https://www.nintendo.com/region-selector/</footnote>
they do not provide all the other parameters that are available with FPFs. This can negatively affect the quality of balance assessment, particularly for detailed biomechanical analysis. Table 2 summarizes the main characteristics of each device.

## C. DATA ACQUISITION PROTOCOLS

In stabilometry, the data acquisition phase must always be approved by an ethical committee. A statokinesigram is generally recorded according to a step-by-step protocol defined by clinicians which must strictly be followed by each subject during each recording session. Although there might be little differences regarding the test conditions from one protocol to the other, existing protocols most often rely on the four following main parameters of test conditions that can significantly impact the human balance according to the clinicians:

1) The **vision**: the subject can either open his eyes or keep them closed during the experiment.

2) The **surface**: the subject can either stand still in the upright position on a firm (rigid) surface or on a foam (smooth) surface.

3) The **feet placement**: the subject must position his feet on the recording device according to several specific measurements as the space between heels, the angle between the two feet, etc.<fnref n="16" /> Although there are actually no standard fixed measurements related to the feet placement, a detailed analysis of the impact of 16 feet placements during human balance evaluation tests is available in [17].

4) The **recording duration**: the subject must try to remain in a balanced position for a fixed duration. The recording duration of each session is strongly related to the volume of recorded data and must be long enough to enable the recording of sufficient data for an adequate description of the subject balance. Generally, a couple of seconds is adopted (e.g., 60s in [1], 30s in [8] and 25s in [5]).

In practice, several recording sessions (trials) under different test conditions are usually necessary to hope to obtain sufficiently informative data related to the subject's balance. Once the feet placements and the recording duration have been fixed, the vision and the surface enable to define the four following most widespread test conditions [1], [8]: *eyes open* on a *firm surface*, *eyes open* on a *foam surface*, *eyes closed* on a *firm surface* and *eyes closed* on a *foam surface*. In some cases, the vision exclusively determines the two possible test conditions [5]: *eyes open* or *eyes closed*.

## D. OPEN DATASETS

In order to collect data during human balance evaluation projects, several subjects of varied physiologic (age, height, weight, etc.) and demographic (nationality, skin color, etc.) characteristics are generally invited to voluntarily participate to the experiments. The participants can be asked to provide

information related to their health conditions (illnesses, disabilities, etc.) and to declare their number of non-intentional falls in the last few months. The results of some clinical tests can further be added to the subjects data and used as class labels to organize the subjects into classes. Sometimes, the resulting datasets are made open so that they can be used in the future by other researchers and clinicians. This section reviews the main characteristics of three open datasets in stabilometry.

The first dataset is the most popular open dataset used for performing the qualitative and quantitative evaluation of the human balance proposed in 2016 by Santos and Duarte [1].<fnref n="17" /> This dataset is generally used for age-group and fall risk prediction. A total of 163 subjects including 116 females and 47 males participated to this project. Their ages ranged from 18 to 85 years old and they were organized into the two following age-groups: 87 young (*age*<60) and 76 old (*age*≥60). The results of the following clinical tests were attached to the data of each subject: the Mini-BEST, the short FES-I, the IPAQ and the TMT. The subjects were asked to declare their number of non-intentional falls in the last 12 months. Various physiologic and demographic characteristics were added to data of each subject. During each recording session realized on a 40cm ×60cm FPF (OPT400600-1000; AMTI, Watertown, MA, USA) at a sampling frequency of 100Hz, the subject's feet were placed with an angle of 20 degrees between them, and their heels were kept 10 cm apart. Each recording session took 60s (i.e 6000 samples per statokinesigram) and was realized under one of the four possible test conditions defined by the vision and the surface. Three trials were performed by each subject under each test condition, for a total of 12 statokinesigrams and 720s recorded per subject.

The second dataset is the open dataset proposed by de Oliveira et al. in 2022 [8]<fnref n="18" /> and designed for illness prediction. That dataset was recorded in quite similar conditions as those of [1] (FPF, frequency, vision, surface, feet placement, clinical tests). A total of 32 participants including 8 females and 24 males, aged from 44 to 81 years old participated to this project. The peculiarity of this dataset is that all the participants suffered from the *idiopathic Parkinson's disease* which is characterized by clinical motor manifestations and postural instability. Each recording session took 30s and six trials were performed by each subject under each test condition (three trials ON medications and three trials OFF medications), for a total of 24 statokinesigrams and 720s recorded per subject.

The last open dataset was proposed by Audiffren et al. in 2016 [5]<fnref n="19" /> and was designed for fall risk prediction. Each statokinesigram was recorded with a WBB during around 25s at the frequency of 25Hz. Only the vision (*eyes open* or *closed*) was considered during each recording session.

<footnote n="16">¹⁶See the Figure 1.A of [1]</footnote>

<footnote n="17">¹⁷https://doi.org/10.6084/m9.fishshare.3394432.v2</footnote>

<footnote n="18">¹⁸https://doi.org/10.6084/m9.fishshare.13530587.v6</footnote>

<footnote n="19">¹⁹https://doi.org/10.6084/m9.fishshare.4244456.v1</footnote>
<table><caption>TABLE 3. Main characteristics of three open datasets in stabilometry.</caption><thead><tr><td>Authors [Ref]</td><td>Year</td><td>Designed to predict</td><td>Number of subjects</td><td>Age range</td><td>Device</td><td>Vision</td><td>Surface</td><td>Session time</td><td>Trials</td><td>Total time</td></tr></thead><tbody><tr><td>Audiffren et al. [5] <sup>19</sup></td><td>2016</td><td>Fall risk</td><td>84</td><td>65-87</td><td>WBB</td><td>Yes</td><td>No</td><td>25s</td><td>1</td><td>50s</td></tr><tr><td>Santos & Duarte [1] <sup>17</sup></td><td>2016</td><td>-Age-group -Fall risk</td><td>163</td><td>18-85</td><td>FPF</td><td>Yes</td><td>Yes</td><td>60s</td><td>3</td><td>720s</td></tr><tr><td>Oliveira et al. [8] <sup>18</sup></td><td>2022</td><td>Illness (Parkinson)</td><td>32</td><td>44-81</td><td>FPF</td><td>Yes</td><td>Yes</td><td>30s</td><td>6</td><td>720s</td></tr></tbody></table>

A total of 84 elderly subjects aged between 65 and 87 years old participated to this project. Their feet were placed in the most comfortable position for the subject without exceeding the shoulder width. The participants were organized into the two following classes based on their personal declarations: 24 fallers (at least one fall in the six last months) and 60 non fallers. Only one single trial was performed by each subject under each test condition, for a total of 2 statokinesigrams and around 50s recorded per subject. Table 3 summarizes the main characteristics of the three open datasets reviewed in the current work.

## E. COP VARIABLES

This section provides a summarized presentation of the most popular existing time-domain (positional, dynamic, stochastic) and frequency variables that enable to characterize the displacements of the COP using time-series numeric data extracted from a statokinesigram of a subject. Each existing COP variable is one single numeric value calculated using a specific computation scheme and researchers must always combine several variables as the components of one vector describing the balance of the subject. The resulting vectors serve as inputs for clinical applications. The technical details (goal, computation scheme, mathematical formula, etc.) related to each variable are not presented here due to space constraints, but they are available in the survey realized by Quijoux et al. in 2021 [14]. Table 4 presents the most popular members of each group of the existing COP variables selected from 27 articles according to the objective criteria listed in [14].<sup>20</sup>

### 1) POSITIONAL VARIABLES

Positional variables<sup>1</sup> describe characteristics of the dispersion of the COP trajectory, but they do not consider the dynamic of the local displacements of the COP. Positional variables consider each COP coordinates ($COP_x$, $COP_y$) found in the statokinesigram as a distinct point in the 2-dimensional space $\mathbb{R}^2$, regardless of the recording time of this point.

### 2) DYNAMIC VARIABLES

Dynamic variables<sup>1</sup> quantify the velocity of the COP trajectory. They also consider each COP coordinates found

in the statokinesigram as a distinct point of $\mathbb{R}^2$. But unlike positional variables that do not consider the recording time of each point, they correlate each point recorded at the time $t_i$, with the next point recorded at a time $t_{i+1} = (t_i + \frac{1}{f})$,<sup>21</sup> where f is the sampling frequency used for recording the statokinesigram. Dynamic variables are mainly limited by their inability to correlate pairs of non-consecutive points of the statokinesigram.

### 3) STOCHASTIC VARIABLES

Stochastic variables<sup>1</sup> analyze the mean square displacements of the COP. As it was the case for positional and dynamic variables, they also consider each COP coordinates found in the statokinesigram as a distinct point of $\mathbb{R}^2$. But unlike dynamic variables that are unable to correlate pairs of non-consecutive points of the statokinesigram, stochastic variables can correlate each point recorded at a time $t_i$, with any point previously recorded at the time $t_{i-\kappa} = (t_i - \frac{\kappa}{f})$ with $(1 \le \kappa \le i)$, where f is the sampling frequency used for recording the statokinesigram. Their main challenge is the selection of the appropriate time-shift (value of $\kappa$) with which the correlation should effectively be made in order to obtain the best evaluation of the subject's balance.

### 4) FREQUENCY VARIABLES

Frequency variables<sup>1</sup> analyze the power spectral density of the COP trajectory by providing relevant spectral information related to the overall displacements of the COP. But they suffer from the loss of temporal data localization and from high computational time complexity due to the conversion of the raw data from the time domain to the spectral one.

## F. RELATED WORK

### 1) AGE-GROUP PREDICTION

Age-group prediction refers to the estimation of a subject's age range based on stabilometric measurements. In 2019, Cetin and Bilgin [2] organized the subjects of the dataset proposed in [1] into eight separated groups (BC1 to BC8), each group containing all the 163 subjects of the dataset. These groups were created in such a way that the members of each group differ by only one of the three following binary criteria whose values are also considered as class labels: the age-group (young or old), the vision (eyes open or closed)

<footnote n="20"><sup>20</sup>See the Section II-B of [14]</footnote>

<footnote n="21"><sup>21</sup>See the Figure 5 of [14]</footnote>
<table><caption>TABLE 4. The four main groups of existing COP variables reviewed in [14].</caption><thead><tr><th>Groups</th><th>Variables</th><th>Descriptions</th><th>Drawbacks</th></tr></thead><tbody><tr><td rowspan="10">Positional <sup>3</sup></td><td>Mean values</td><td rowspan="10">Characterize the dispersion of the COP</td><td rowspan="10">Do not consider the dynamic of the local displacements of the COP</td></tr><tr><td>Mean distance</td></tr><tr><td>Maximal distance</td></tr><tr><td>Root mean square</td></tr><tr><td>Amplitude</td></tr><tr><td>Ratio of amplitudes</td></tr><tr><td>Planar deviation</td></tr><tr><td>Coefficient of sway direction</td></tr><tr><td>95% confidence ellipse area</td></tr><tr><td>Principal sway direction</td></tr><tr><td rowspan="10">Dynamic <sup>4</sup></td><td>Mean velocity</td><td rowspan="10">Correlate each point of the COP trajectory with the next point</td><td rowspan="10">Do not correlate non-consecutive points of the COP trajectory</td></tr><tr><td>Sway area per second</td></tr><tr><td>Phase plane parameter</td></tr><tr><td>Length over area</td></tr><tr><td>Fractal dimension</td></tr><tr><td>Zero-crossing velocity</td></tr><tr><td>Mean velocity peak</td></tr><tr><td>Mean sway density peak</td></tr><tr><td>Mean spatial distance</td></tr><tr><td>Mean frequency</td></tr><tr><td rowspan="6">Stochastic <sup>5</sup></td><td>Short-term diffusion coefficient</td><td rowspan="6">Correlate each point of the COP trajectory with any previous point</td><td rowspan="6">Selecting the appropriate time-shift is a challenge</td></tr><tr><td>Long-term diffusion coefficient</td></tr><tr><td>Short-term scaling coefficient</td></tr><tr><td>Long-term scaling coefficient</td></tr><tr><td>Critical time</td></tr><tr><td>Critical mean square displacement</td></tr><tr><td rowspan="5">Frequency <sup>2</sup></td><td>Total power</td><td rowspan="5">Provide spectral information related to the displacements of the COP</td><td rowspan="5">Loss of temporal localization and high computational complexity</td></tr><tr><td>Power spectrum density mode</td></tr><tr><td>Centroidal frequency</td></tr><tr><td>Frequency dispersion</td></tr><tr><td>Energy content of frequencies intervals</td></tr></tbody></table>

and the surface (firm or foam). Each subject was initially represented as a 2-dimensional vector whose coordinates were calculated as the standard deviations of the FPF forces ($F_x, F_y$), then as the standard deviations of the COP coordinates ($COP_x, COP_y$).<fnref n="22" /> Finally, binary classification experiments were performed with several classifiers on each group initially considering the points generated by the FPF forces, then the points generated by the COP coordinates, for a total of 16 classification experiments.<fnref n="23" /> The Support Vector Machine (SVM) [18] classifier showed the best accuracy of 81.67% with the points generated by the FPF forces when the subjects were labeled by their age-group, standing on a foam surface with their eyes open (group BC2). Unfortunately, one separate prediction is made per test conditions. Consequently, the best performance only considers one single test condition (eyes open on a foam surface).

More recently in 2024, Alcan [3] also performed age-group prediction on 152 subjects from the dataset proposed in [1]. He computed and saved several dynamic COP variables<fnref n="24" /> as the components of a vector describing the balance of each subject, in each test condition when the vision and the surface are both considered. The vectors derived for

each test condition were later processed by the Partial least squares-discriminant analysis (PLS-DA) algorithm which is a supervised learning algorithm used for classification specifically designed for datasets containing high-dimensional vectors. The best age-group classification accuracy, 89%, was observed when the subjects were standing on a foam surface with open eyes. As it was the case in [2], this performance also only considers one single test condition because one separate prediction is made per test condition.

The same year, Pennone et al. [4] computed a 53-dimensional vector from the statokinesigrams of each of the 163 subjects of the dataset proposed in [1]. This vector was computed considering all the three trials of each subject in all the test conditions.<fnref n="25" /> The components of these vectors included the values of several COP variables, the results of clinical tests and the subject's physiologic characteristics. Age-group classification experiments realized with the Naive Bayes (NB) [19] classifier showed a best accuracy of 65%<fnref n="26" /> after a 10-folds cross-validation. Unlike [2], [3], this is the only work where separate predictions are not made for each test condition, because each prediction simultaneously considers all the four test conditions instead.

<footnote n="22">²²See the Figures 3.a and 3.b of [2]</footnote>

<footnote n="23">²³See the Table 2 and the Figure 4 of [2]</footnote>

<footnote n="24">²⁴See the Equations 1 to 13 of [3]</footnote>

<footnote n="25">²⁵See the Section 'Analyzed parameters' of [4]</footnote>

<footnote n="26">²⁶See the Table 3 of [4]</footnote>
## 2) FALL RISK PREDICTION

Fall risk prediction is the assessment of a subject's likelihood of falling based on stabilometric analysis. In 2016, the dataset constructed by Audiffren et al. with a WBB [5] was used by these same authors to perform fall risk prediction. The authors combined five positional and dynamic COP variables as the components of a vector describing the balance of each of the 84 subjects, in each test condition (eyes open or closed).<fnref n="27" /> The resulting 168 vectors (2 vectors per subject) were labeled as fallers (48 vectors) and non-fallers (120 vectors) irrespective of the test conditions, then the dataset was split into a training set (70%) and a testing set (30%). Classification experiments were conducted using the Ranking Forest (RF) [20] classifier, but no classification accuracy was stated by the authors. They rather constructed the Receiver Operating Characteristic (ROC) curve of their classifier and they obtained an Area Under the Curve (AUC) of 0.75.

In 2019, Reilly [6] used the dataset proposed in [1] to perform fall risk prediction. He selected a subset of 73 subjects and organized them into two classes according to the following criteria: 45 low-risk subjects (age: 72±6.5, FES score: 9.2±1.7) and 28 high-risk subjects (age: 71±6.3, falls history: 2.5±9.4, FES score: 12±4.2). He then extracted several positional, dynamic and frequency variables from the statokinesigrams of each subject considering all the four possible test conditions. This produced a 528-dimensional vector for each subject. Binary classification experiments were later conducted using several classifiers after performing feature selection and a 10-folds cross-validation. The Multi-layer perceptron (MLP) [21] classifier exhibited the best classification accuracy of 80%.<fnref n="28" /> The major drawback of this work is the fact that young subjects are not experimentally considered because only a limited subset of 73 subjects over the 163 subjects of the dataset was considered.

In 2021, Savadkoohi et al. [7] also used the dataset proposed in [1] to perform fall risk prediction. They initially merged the statokinesigrams of all the 163 subjects in one unique file containing a total of 11.580.000 rows of COP data.<fnref n="29" /> A 23-dimensional vector was then extracted for each row (not for each subject) of the unique file. The components of this vector were raw values taken from 23 selected columns of the unique file.<fnref n="30" /> The resulting 11.580.000 vectors are later organized into three classes according to the their fear of falling<fnref n="31" /> given by the FES score as follows: Low fear (FES score in 2 – 9), Moderate fear (FES score in 9 – 13) and High fear (FES score > 14). These vectors are finally classified using several types of Neural Networks (NN) [22] that exhibited accuracies always greater than 99%.<fnref n="32" /> Unfortunately, this very high classification performance does not enable to predict the fear of falling of

a subject, but it only determines the fear of falling of each of the 6000 samples of his statokinesigram taken separately. Indeed, each vector does not represent a subject, but it rather refers to only one sample in his statokinesigram sampled at 100Hz.

In 2024, Pennone et al. [4] whose work has already been reviewed in Section II-F1 also conducted experiments related to fall risk prediction in the dataset proposed in [1]. During these experiments, a participant was labeled as a non-faller if he declared no intentional fall during the past 12 months, otherwise he was labeled as a faller. These experiments showed a best accuracy of 74% II-F1 with the SVM classifier. The major limitation of this study is the fact that the falls history is declared by the subject himself, which is biased (the subject may tell lies or forget some falls) and less reliable than the result of a clinical test like the Mini-BEST that effectively evaluates the human balance through diverse dynamic physical tasks.

## 3) ILLNESS PREDICTION

In stabilometry, illness prediction is the process of identifying potential health conditions or disorders of a subject based on the analysis of his postural stability. Despite the existence of open datasets designed for illness prediction in the field of stabilometry (e.g., the dataset proposed in 2022 by de Oliveira et al. [8] for the Parkinson disease), no study that strictly performs illness prediction through classification experiments was found.

Nevertheless in 2023, Zhu et al. [9] used the Mini-BEST to assess the balance of subjects suffering from Bilateral Vestibulopathy (BVP) which is a pathology that negatively affects the human balance (unsteadiness when walking and falls). No statokinesigram was recorded in this project, but a total of 50 subjects with BVP rather completed the Mini-BEST. Their Mini-BEST scores were then compared to the Mini-BEST scores of 190 healthy subjects from the literature. This study demonstrated that subjects with BVP had significantly lower Mini-BEST total scores than healthy subjects. More precisely, anticipatory, reactive postural control and sensory orientation sub-scores of the Mini-BEST were significantly lower for the subjects with BVP, while dynamic gait sub-scores were not significantly different. Table 5 gives details about the relevant existing work related to age-group, fall risk and illness prediction reviewed in the current paper.

## G. PROBLEM STATEMENT

As it was demonstrated in Section II-E, researchers have yet proposed a high number of COP variables. As a consequence, objectively selecting the accurate subset of COP variables to use for a specific task is now a serious challenge. Additionally, each existing COP variable aggregates in one single numeric value all the high volume of time-series data contained in a statokinesigram. This induces a high level of data compression and a significant risk of loss of information. Researchers must consequently always combine

<footnote n="27">²⁷See the Table 1 of [5]</footnote>

<footnote n="28">²⁸See the Tables 3 to 6 of [6]</footnote>

<footnote n="29">²⁹See the Section III-B of [7]</footnote>

<footnote n="30">³⁰See the second paragraph of the Section III-C of [7]</footnote>

<footnote n="31">³¹See the end of the Section III-C of [7]</footnote>

<footnote n="32">³²See the Table 4 of [7]</footnote>
<table><caption>TABLE 5. Relevant existing work related to age-group, fall risk and illness prediction.</caption><thead><tr><th>Prediction</th><th>Authors[Ref]</th><th>Years</th><th>Datasets</th><th>Models</th><th>Accuracies</th><th>Drawbacks</th></tr></thead><tbody><tr><td rowspan="3">Age-group</td><td>Cetin &amp; Bilgin [2]</td><td>2019</td><td>[1]</td><td>SVM</td><td>81.67%</td><td>Separate predictions per test condition</td></tr><tr><td>Alcan [3]</td><td>2024</td><td>[1]</td><td>PLS-DA</td><td>89%</td><td>Separate predictions per test condition</td></tr><tr><td>Pennone et al. [4]</td><td>2024</td><td>[1]</td><td>NB</td><td>65%</td><td>Low accuracy</td></tr><tr><td rowspan="4">Fall risk</td><td>Auiffren et al. [5]</td><td>2016</td><td>[5]</td><td>RF</td><td>AUC:0.75</td><td>No cross-validation and falls history declared by subjects</td></tr><tr><td>Reilly [6]</td><td>2019</td><td>[1]</td><td>MLP</td><td>80%</td><td>Only considers old subjects</td></tr><tr><td>Savadkoohi et al. [7]</td><td>2024</td><td>[1]</td><td>NN</td><td>&gt;99%</td><td>Predictions are made for samples</td></tr><tr><td>Pennone et al. [4]</td><td>2024</td><td>[1]</td><td>SVM</td><td>74%</td><td>Falls history declared by subjects</td></tr><tr><td>Illness</td><td>Zhu et al. [9]</td><td>2023</td><td>Private (BVP)</td><td>—</td><td>—</td><td>No statokinesigram recorded</td></tr></tbody></table>

various existing COP variables as the components of one descriptor vector. Furthermore, the computation schemes of the existing COP variables do not involve the automatic learning of the overall behavior of the COP during the traversal of its trajectory. However, this may lead to a finer description of the COP displacements.

The current paper attempts to attenuate these limitations by proposing a new stochastic technique that trains a HMM for automatically learning the sequential variations of some geometric features that occur in the statokinesigram of a given subject. Meta-data derived from this HMM are then used for generating one descriptor vector associated with this statokinesigram. HMMs are preferred here because they are suitable for managing time-series data and also because they are manipulated by robust well-known algorithms whose efficiency no longer needs to be demonstrated.

### III. PRESENTATION OF HMMs

#### A. DEFINITION OF A HMM

A HMM $\lambda = (A, B, \pi)$ is fully characterized by [23]$^{33}$:

1) Its number $N$ of states, the set of states being $S = \{s_1, \dots, s_N\}$. The state of the model at time $t$ is generally noted $q_t \in S$.

2) Its number $M$ of symbols, the set of symbols being $\theta = \{z_1, \dots, z_M\}$. The symbol observed at time $t$ is generally noted $o_t \in \theta$.

3) Its state transition probability matrix $A$ verifying $A[s_x, s_y] = Prob(q_{t+1} = s_y|q_t = s_x)$ with $1 \le x, y \le N$.

4) Its symbols observation emission probability matrix $B$ verifying $B[s_x, z_t] = Prob(z_t$ at time $t|q_t = s_x)$ with $1 \le x \le N$ and $1 \le z_t \le M$.

5) Its initial state probability vector $\pi$ verifying $\pi[s_x] = Prob(q_1 = s_x)$ with $1 \le x \le N$.

#### B. GENERATION OF SEQUENCE

A HMM $\lambda = (A, B, \pi)$ can be used for generating a sequence $O = o_1 \dots o_T$ composed of $T$ symbols observed following the sequence of states $q = q_1 \dots q_T$ as described in the *Markov chain* (MC) shown in Figure 1.

<figure><img src="image_2.png" /><figcaption>FIGURE 1. A Markov chain.</figcaption></figure>

In order to obtain this MC, the following algorithm is executed:

1) Select the initial state $s=(s_x \in S)$ according to $\pi$ and set $t = 0$.

2) Set $t = t + 1$ and change the current state to $q_t = s$

3) Select the symbol $o_t \in \theta$ to be observed at state $q_t$ according to $B$.

4) If $(t < T)$ **go to** step 5, else **terminate**.

5) Select the state transition to be realized from the current state $q_t$ to another state $s=(s_y \in S)$ according to $A$, then **go to** step 2.

#### C. MANIPULATION OF A HMM

Consider a sequence of symbols $O = o_1 \dots o_T$ and a HMM $\lambda = (A, B, \pi)$. The probability $Prob(O|\lambda)$ to observe $O$ given $\lambda$ is efficiently calculated by the *Forward-Backward* algorithm [23]$^{34}$ that runs in $\theta(T \cdot N^2)$. Given a sequence of symbols $O = o_1 \dots o_T$, it is possible to iteratively re-estimate the parameters of a HMM $\lambda = (A, B, \pi)$ in order to maximize the value of $Prob(O|\bar{\lambda})$, where $\bar{\lambda} = (\bar{A}, \bar{B}, \bar{\pi})$ is the re-estimated model. The *Baum-Welch* algorithm [23]$^{35}$ is used for this purpose. This algorithm runs in $\theta(\gamma \cdot T \cdot N^2)$ where $\gamma$ is the user-defined maximum number of iterations.

<footnote>$^{33}$See the Section II-B of [23]</footnote>

<footnote>$^{35}$See the Section III-C of [23]</footnote>
<figure><img src="image_3.png" /><figcaption>FIGURE 2. Sequential variations of the distances and the angles between $p_l$ and $\bar{p}_l$ observed from $\mathcal{O}(0, 0)$ when the time-shift is ($\kappa=3$) for $l = 3, 4, 5, 6$.</figcaption></figure>

The Baum-Welch algorithm can also be used to train a HMM for learning multiple sequences.<fnref n="36" /> In this case, the algorithm maximizes the value of $Prob(O|\lambda) = \sum_{k=1}^{K} Prob(O^{(k)}|\lambda)$ where $O = \{O^{(1)}, \dots, O^{(K)}\}$ is a set of K sequences and $O^{(k)} = O_{1}^{(k)} \dots O_{T_k}^{(k)}$ is the $k^{th}$ sequence of O. In this context, only the distributions A and B are re-estimated by the Baum-Welch algorithm because $\pi$ can be statistically determined from the initial states of the K observed sequences. The time cost of the Baum-Welch algorithm for multiple sequences is approximated by $\theta \left( \gamma \cdot \left( \sum_{k=1}^{K} T_k \right) \cdot N^2 \right)$.

### D. STATIONARY DISTRIBUTION

A vector $\varphi = (\varphi[s_1], \dots, \varphi[s_N])$ is a stationary distribution of a HMM $\lambda = (A, B, \pi)$ if [24]<fnref n="37" />:

1) $\forall y, \varphi[s_y] \ge 0$ and $\sum_y \varphi[s_y] = 1$

2) $\varphi = \varphi.A \Leftrightarrow (\varphi[s_y] = \sum_{x} \varphi[s_x] \times A[s_x, s_y], \forall y)$

$\varphi[s_y]$ estimates the overall proportion of time spent by $\lambda$ in state $s_y$ after a sufficiently long time. $\varphi$ can be extracted from any line of the matrix $A^r = A \times A \times \dots \times A$ ($r$ times) when $r \to +\infty$. Its computation requires $\theta(r \cdot N^3)$ operations.

## IV. THE PROPOSED APPROACH

### A. MAIN IDEA

The originality of the current work relies on the idea that the balance of a subject recorded into a statokinesigram in one specific test condition can be well characterized by the overall behavior of the COP throughout its trajectory in $\mathbb{R}^2$. This behavior can be learned from the sequential variations of

distances and angles between regularly separated points of his statokinesigram $S$ recorded at a sampling frequency of $f$ Hz. More precisely, suppose that the statokinesigram contains $|S|$ points and let $(\kappa \ge 1)$ be a fixed user-defined time-shift. If $p_l$ and $\bar{p}_l$ respectively refer to the COP points of rank $l$ and rank $(l - \kappa)$ with $(\kappa \le l \le |S| - 1)$, then the pairs of points $(p_l, \bar{p}_l)$ are chronologically browsed from the first pair when $(l=\kappa)$ to the last pair when $(l=|S| - 1)$ and at each step, the distance $d$ between $p_l$ and $\bar{p}_l$ is observed from the point $\mathcal{O}(0, 0)$. The angle of observation is therefore $(\hat{p}_l, \hat{\bar{p}}_l)$. Figures 2a to 2b depict the sequential variations of distances and angles between the points $p_l$ and $\bar{p}_l$ observed from the point $\mathcal{O}(0, 0)$ when the time-shift is ($\kappa=3$) for $l = 3, 4, 5, 6$. In these figures, the COP points are black bullets connected with blue lines to construct the statokinesigram and the red triangles materialize the angle of observation. When the Euclidean distance is for example selected, Table 6 shows the resulting distances and angles calculated according to the coordinates of the COP points appearing in these figures.

<table><caption>TABLE 6. Euclidean distances and angles between the pairs of points $(p_l, \bar{p}_l)$ of the Figures 2a to 2b for $l = 3, 4, 5, 6$.</caption><thead><tr><td>Pairs</td><td>Distances</td><td>Angles</td></tr></thead><tbody><tr><td>(p<sub>3</sub>, p&#773;<sub>3</sub>)</td><td>0.412</td><td>32.90°</td></tr><tr><td>(p<sub>4</sub>, p&#773;<sub>4</sub>)</td><td>0.403</td><td>39.58°</td></tr><tr><td>(p<sub>5</sub>, p&#773;<sub>5</sub>)</td><td>0.390</td><td>36.57°</td></tr><tr><td>(p<sub>6</sub>, p&#773;<sub>6</sub>)</td><td>0.212</td><td>17.62°</td></tr></tbody></table>

The distances and the angles derived from the aforementioned process enable to transform $S$ into one very long MC that will then be split into several short MCs at the rate of one short MC per second. The resulting MCs are used to initialize

<footnote n="36">³⁶See the Section V-B of [23]</footnote>

<footnote n="37">³⁷See the Definition 9.1 of [24]</footnote>
<figure><img src="image_4.png" /><figcaption>FIGURE 3. Workflow used for generating the single vector $\vec{\tilde{S}}_{i,j,\kappa}$ associated with the $j^{th}$ statokinesigram of a subject recorded under the $i^{th}$ test condition given a fixed time-shift $\kappa$.</figcaption></figure>

then to train one HMM. Meta-data derived from this HMM will be saved as the components of one descriptor vector associated with $S$. When this process is repeated for each test condition, the resulting descriptor vectors are concatenated to obtain one single descriptor vector associated with $S$, irrespective of the test conditions.

## B. METHODOLOGY

Consider a subject $u$ whose balance is evaluated under $n$ different test conditions (i.e vision, surface) and suppose that $m$ recording sessions (trials) are realized under each test condition. Let also assume that $S_{i,j}$ is the $j^{th}$ statokinesigram recorded under the $i^{th}$ test condition such that the set $S_i = \{S_{i,1}, \dots, S_{i,m}\}$ contains all the statokinesigrams recorded under the $i^{th}$ test condition. According to these notations, the subject can be represented by the collection $u = (\cup_{i=1}^n S_i)$ containing all his statokinesigrams, irrespective of the test conditions. Given a fixed user-defined time-shift ($\kappa \ge 1$), the methodology proposed in the current work for deriving the vector $\vec{\tilde{S}}_{i,j,\kappa}$ associated with $S_{i,j}$ is composed of the five following steps as described in the Figure 3:

1) **Down-sampling:** In the case where the statokinesigram was recorded at a relatively high frequency $f$ which may lead to extremely long MCs, it is recommended to down-sample the COP data with a user-defined factor $\sigma$ as it was the case in [25].$^{38}$ The sampling frequency thus becomes ($f = \frac{1}{\sigma}$). The down-sampling step also enables to avoid overfitting during the learning process, but it is optional since it can be skipped (i.e., $\sigma=1$) if the original sampling frequency is estimated suitable.

2) **Transformation into a MC:** The distances and the angles between the pairs of points of the statokinesigram are captured and used to construct one long MC $\delta_{\kappa}(S_{i,j})$ of length ($|S_{i,j}| - \kappa$).

3) **MC split:** $\delta_{\kappa}(S_{i,j})$ is later split into several short MCs at the rate of one short MC per second. The resulting short MCs are saved in the set $\Delta_{\kappa}(S_{i,j})$.

4) **HMM initialization and training:** The MCs contained in $\Delta_{\kappa}(S_{i,j})$ are used to initialize then to train one HMM $\lambda_{\kappa}(S_{i,j})$ for learning the sequential variations of distances and angles that occur in the statokinesigram.

5) **Meta-data extraction:** Meta-data extracted from $\lambda_{\kappa}(S_{i,j})$ are saved as the components of one single vector $\vec{\tilde{S}}_{i,j,\kappa}$ associated with the statokinesigram.

When the process described in the Figure 3 is applied to the $m$ statokinesigrams contained in the set $S_i$, the arithmetic mean of the $m$ resulting vectors is considered as the descriptor vector $\vec{\tilde{S}}_{i,\kappa}$ associated with the statokinesigrams recorded under the $i^{th}$ test condition as shown in the Figure 4. Finally, the descriptor vectors associated with the $n$ test conditions are concatenated to obtain the unique final descriptor vector $\vec{u}_{\kappa}$ associated with the subject $u$, irrespective of the test conditions as depicted in the Figure 5.

### 1) TRANSFORMATION INTO A MC

Consider a fixed user-defined time-shift ($\kappa \ge 1$) and a valid distance measure $d$ between two points of $\mathbb{R}^2$. In the current work, the pairs of points $(p_l, \bar{p}_l)$ of the statokinesigram $S_{i,j}$ recorded for evaluating the balance of a user $u$ during the $j^{th}$ trial under the $i^{th}$ test condition are chronologically browsed (i.e., $l = \kappa, \kappa + 1, \dots, |S_{i,j}| - 1$) and at each step, the angle of observation $(\bar{p}_l, \bar{p}_l)$ is considered as the current state, while the observed distance $d(p_l, \bar{p}_l)$ is considered as the current symbol. When this process is applied, the pseudo MC $\tilde{\delta}_{\kappa}(S_{i,j})$ of length ($|S_{i,j}| - \kappa$) is obtained as depicted in the Figure 6 where $\alpha = (|S_{i,j}| - 1)$.

The main problem that arises from the Figure 6 is the fact that the states and the symbols do not belong to finite sets. Indeed, the angles belong to the continuous interval $[0, 180]$ and the distances belong to the continuous interval $[0, +\infty[$. However, as it can be observed in the Figure 1, a MC is composed of states and symbols, both from finite sets. Analogically to the solution proposed in [26]$^{39}$ to solve a similar problem, the three following steps are adopted here to obtain states and symbols that belong to finite sets:

1) Replace the angles and the distances appearing in $\tilde{\delta}_{\kappa}(S_{i,j})$ by their normalized values belonging to the continuous interval $[0, 100]$. Equation 1 is used to calculate each normalized angle $\beta(p_l, \bar{p}_l)$ and Equation 2 is used to calculate each normalized distance $\tilde{d}(p_l, \bar{p}_l)$. In these two equations, $\alpha = (|S_{i,j}| - 1)$. These

<footnote>$^{38}$See the Section II-B of [25]</footnote>

<footnote>$^{39}$See the Equations 11 to 13 of [26]</footnote>
<figure><img src="image_5.png" /><figcaption>FIGURE 4. Generation of the mean vector $\tilde{S}_{i,\kappa}$ associated with the statokinesigrams recorded under the $i^{th}$ test condition given a fixed time-shift $\kappa$.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>FIGURE 5. Generation of the final vector $\tilde{u}_{\kappa}$ associated with the subject $u$, irrespective of test condition given a time-shift $\kappa$.</figcaption></figure>

replacements enable to obtain a modified pseudo MC $\tilde{\delta}_{\kappa}(S_{i,j})$.

$$ \left\{ \begin{array}{l} \beta(\widehat{p_l}, \widehat{p_l}) = 100 \times \frac{\widehat{p_l}, \widehat{p_l}}{\beta_{max}} \quad \text{where} \\ \beta_{max} = \max\{\widehat{p_{\omega}}, \widehat{p_{\omega}}) | (\kappa \le \omega \le \alpha)\} \end{array} \right. \quad (1) $$

$$ \left\{ \begin{array}{l} \tilde{d}(p_l, \bar{p}_l) = 100 \times \frac{d(p_l, \bar{p}_l)}{d_{max}} \quad \text{where} \\ d_{max} = \max\{d(p_{\omega}, \bar{p}_{\omega}) | (\kappa \le \omega \le \alpha)\} \end{array} \right. \quad (2) $$

2) Given a user-defined positive integer $N$, use Equation 3 to split the interval $[0, 100]$ into $(N + 1)$ sub-intervals denoted $I_0, I_1, \dots, I_N$.

$$ \left\{ \begin{array}{l} I_0 = \{0\} \\ I_k = \left[ 100 \times \frac{(k-1)}{N}, 100 \times \frac{k}{N} \right] \quad \text{with } (1 \le k \le N) \end{array} \right. \quad (3) $$

$$ \begin{array}{ccccc} d(p_{\kappa}, \bar{p}_{\kappa}) & d(p_{\kappa+1}, \bar{p}_{\kappa+1}) & \dots & d(p_{\alpha}, \bar{p}_{\alpha}) & d(p_{\kappa+1}, \bar{p}_{\kappa+1}) \end{array} $$

<caption>FIGURE 6. The pseudo MC $\tilde{\delta}_{\kappa}(S_{i,j})$.</caption>

3) Replace each normalized angle and each normalized distance appearing in $\tilde{\delta}_{\kappa}(S_{i,j})$ by the index of the sub-interval to which it belongs. This replacement is done using the function $\eta:[0, 100] \to \{0, 1, \dots, N\}$ defined in Equation 4 and enables to obtain a long MC $\delta_{\kappa}(S_{i,j})$. As a consequence, the new finite set of states and set of symbols are $S=\vartheta=\{0, 1, \dots, N\}$.

$$ (\eta(x) = k) \Leftrightarrow (x \in I_k) \text{ with } (0 \le k \le N) \quad (4) $$

To illustrate this transformation, consider for example a subject $u=\{S_{1,1}, S_{1,2}, S_{2,1}, S_{2,2}\}$ whose balance is evaluated
<table><caption>TABLE 7. Examples of COP points contained in the statokinesigrams of a subject $u=\{S_{1,1}, S_{1,2}, S_{2,1}, S_{2,2}\}$ recorded during 4 seconds at 3Hz under two test conditions ($S_1$=eyes open and $S_2$=eyes closed).</caption><thead><tr><th>Ranks</th><th>S<sub>1,1</sub></th><th>S<sub>1,2</sub></th><th>S<sub>2,1</sub></th><th>S<sub>2,2</sub></th></tr></thead><tbody><tr><td>0</td><td>(0.1054, -0.3928)</td><td>(0.0775, 0.4762)</td><td>(-0.0168, -0.2637)</td><td>(-0.0654, 0.7709)</td></tr><tr><td>1</td><td>(0.1054, -0.3853)</td><td>(0.0727, 0.4843)</td><td>(-0.0091, -0.2783)</td><td>(-0.0589, 0.7631)</td></tr><tr><td>2</td><td>(0.1056, -0.3743)</td><td>(0.0679, 0.4933)</td><td>(-0.0081, -0.2931)</td><td>(-0.0530, 0.7580)</td></tr><tr><td>3</td><td>(0.1060, -0.3600)</td><td>(0.0624, 0.5031)</td><td>(0.0108, -0.3079)</td><td>(-0.0468, 0.7553)</td></tr><tr><td>4</td><td>(0.1063, -0.3428)</td><td>(0.0561, 0.5134)</td><td>(0.0231, -0.3222)</td><td>(-0.0395, 0.7546)</td></tr><tr><td>5</td><td>(0.1066, -0.3234)</td><td>(0.0486, 0.5239)</td><td>(0.0366, -0.3356)</td><td>(-0.0307, 0.7553)</td></tr><tr><td>6</td><td>(0.1067, -0.3024)</td><td>(0.0402, 0.5337)</td><td>(0.0507, -0.3476)</td><td>(-0.0205, 0.7570)</td></tr><tr><td>7</td><td>(0.1066, -0.2806)</td><td>(0.0310, 0.5423)</td><td>(0.0645, -0.3582)</td><td>(-0.0093, 0.7586)</td></tr><tr><td>8</td><td>(0.1064, -0.2584)</td><td>(0.0215, 0.5488)</td><td>(0.0771, -0.3673)</td><td>(0.0020, 0.7593)</td></tr><tr><td>9</td><td>(0.1062, -0.2364)</td><td>(0.0120, 0.5528)</td><td>(0.0878, -0.3751)</td><td>(0.0124, 0.7581)</td></tr><tr><td>10</td><td>(0.1063, -0.2150)</td><td>(0.0028, 0.5539)</td><td>(0.0959, -0.3819)</td><td>(0.0212, 0.7544)</td></tr><tr><td>11</td><td>(0.1069, -0.1947)</td><td>(-0.0057, 0.5519)</td><td>(0.1014, -0.3878)</td><td>(0.0281, 0.7477)</td></tr></tbody></table>

<figure><img src="image_7.png" /><figcaption>FIGURE 7. Step-by-step computation of the long MC $\delta_3(S_{1,1})$ when ($\mathcal{N}=10$).</figcaption></figure>

through the recording of the four statokinesigrams presented in the Table 7. Assume that these statokinesigrams are recorded on a firm surface under two test conditions (i.e $S_1$=eyes open and $S_2$=eyes closed), at the rate of two trials (statokinesigrams) per test condition. Suppose also that each statokinesigram is recorded during 4 seconds at the sampling frequency $f=3$Hz, for a total of 12 sequential COP points per statokinesigram. When the overall procedure described in the current section is applied for ($\sigma=1$), ($\mathcal{N}=10$) and ($\kappa=3$) to the statokinesigram $S_{1,1}$, the following step-by-step outputs are obtained:

1) The pairs of COP points ($p_l, \bar{p}_l$) are browsed chronologically from ($l=3$) to ($l=11$) and at each step, the Euclidean distance $d(p_l, \bar{p}_l)$ between $p_l$ and $\bar{p}_l$ is observed from the point $O(0, 0)$ such that the angle of observation is $(\hat{p}_l, \bar{p}_l)$. The output of this step is the pseudo MC $\tilde{\delta}_3(S_{1,1})$ depicted in the Figure 7a.

2) Equations 1 and 2 are used to replace the angles and the distances appearing in $\tilde{\delta}_3(S_{1,1})$ by their normalized values such that ($\beta_{max}=6.38$) and ($d_{max}=0.066$). This enables to obtain the modified MC $\tilde{\delta}_3(S_{1,1})$ shown in the Figure 7b.

3) Equation 3 is used to split the interval $[0, 100]$ into the 11 following sub-intervals: $I_0 = \{0\}$, $I_1 = \{0, 10\}$, $I_2 = \{10, 20\}$, ..., $I_{10} = \{90, 100\}$.

4) Equation 4 is finally used to replace each normalized angle and each normalized distance appearing in $\tilde{\delta}_3(S_{1,1})$ by the index of the sub-interval to which it belongs. This enables to obtain the long MC $\tilde{\delta}_3(S_{1,1})$ as depicted in the Figure 7c.

## 2) MC SPLIT

The length of $\delta_\kappa(S_{i,j})$ is ($|S_{i,j}| - \kappa$) and this value is very high in practice. For instance, when ($\kappa=3$) and $S_{i,j}$ is recorded at 20Hz during 60s, then $|S_{i,j}|=20 \times 60=1200$ and $|\tilde{\delta}_3(S_{i,j})|=1197$. It is not reasonable to train a HMM for efficiently learning a sequence of symbols from such a long MC due to high training time it will require. However, it is experimentally less time consuming to learn many short sequences with the Baum-Welch algorithm, than to learn one very long sequence with the same algorithm. Furthermore, it is easier for a human being to analyze the content of many short sequences than that of one very long sequence.

For these reasons, the long MC $\delta_\kappa(S_{i,j})$ is split to obtain ($|S_{i,j}| - \kappa) \times \frac{1}{f}$ short MCs, each having a length equal to the
<figure><img src="image_8.png" /><figcaption>FIGURE 8. Construction of $\Delta_3(S_{1,1})$ by splitting the long MC $\delta_3(S_{1,1})$ of the Figure 7c.</figcaption></figure>

number $f$ of COP points recorded per second, except the last one that may be shorter. This splitting is initially done by deleting the suitable state transitions (right arrows) in $\delta_\kappa(S_{i,j})$ as shown in the Figure 8a where the long MC $\delta_3(S_{1,1})$ is split into $(12-3) \times \frac{1}{3}=3$ short MCs of length 3.

The deletion of some state transitions induces information loss which may negatively impact the HMM learning process. Indeed, the deleted state transitions significantly contribute to the description of the subject's balance because they denote angle variations between pairs of COP points. Therefore, if these state transitions are not learned, this may introduce an important bias in the parameters of resulting trained HMM. In order to capture the information contained in the deleted state transitions, the short MCs are sequentially browsed and at each step, the first observation of the next short MC is added at the end of the current short MC. This enables to construct the finite set $\Delta_\kappa(S_{i,j})$ of short MCs. Figures 8b to 8d show the short MCs contained in the set $\Delta_3(S_{1,1})$.

### 3) HMM INITIALIZATION AND TRAINING

Given a user-defined positive constant $\varepsilon$, we propose here to parameter the initial HMM $\bar{\lambda}_\kappa(S_{i,j})=(\bar{\lambda}_\kappa, \bar{B}_\kappa, \bar{\pi}_\kappa)$ associated with the statokinesigram $S_{i,j}$ as described below for statistically capturing the state transitions and the symbol probability distributions from the content of $\Delta_\kappa(S_{i,j})$:

1) The set of states is $S=\{0, 1, \dots, N\}$ where $N$ is the user-defined positive integer used in Equation 3 to split the interval $[0, 100]$.

2) The set of symbols is $\vartheta=\{0, 1, \dots, N\}$.

3) The probability of transiting from state $s_x$ to state $s_y$ is calculated in Equation 5 where $\text{transit}(s_x, s_y, \Delta_\kappa(S_{i,j}))$ is the number of transitions from state $s_x$ to state $s_y$ in $\Delta_\kappa(S_{i,j})$ and $\text{transit}(s_x, -, \Delta_\kappa(S_{i,j}))$ is the number of transitions from state $s_x$ to any destination in $\Delta_\kappa(S_{i,j})$.

$$ \bar{A}_\kappa[s_x, s_y] = \frac{\text{transit}(s_x, s_y, \Delta_\kappa(S_{i,j}))}{\text{transit}(s_x, -, \Delta_\kappa(S_{i,j})) + \varepsilon} \quad (5) $$

4) The probability to observe symbol $z_l$ from state $s_x$ is calculated in Equation 6 where $\text{observe}(z_l, s_x, \Delta_\kappa(S_{i,j}))$ is the number of times symbol $z_l$ is observed from state $s_x$ in $\Delta_\kappa(S_{i,j})$, and $\text{observe}(-, s_x, \Delta_\kappa(S_{i,j}))$ is the

number of occurrences of state $s_x$ in $\Delta_\kappa(S_{i,j})$.

$$ \bar{B}_\kappa[s_x, z_l] = \frac{\text{observe}(z_l, s_x, \Delta_\kappa(S_{i,j}))}{\text{observe}(-, s_x, \Delta_\kappa(S_{i,j})) + \varepsilon} \quad (6) $$

5) The probability of starting with the state $s_x$ is calculated in Equation 7, where $\text{start}(s_x, \Delta_\kappa(S_{i,j}))$ is the number of MCs in $\Delta_\kappa(S_{i,j})$ where the initial state is $s_x$.

$$ \bar{\pi}_\kappa[s_x] = \frac{\text{start}(s_x, \Delta_\kappa(S_{i,j}))}{|\Delta_\kappa(S_{i,j})| + \varepsilon} \quad (7) $$

The parameters of $\bar{\lambda}_\kappa(S_{i,j})$ are not probability distributions due to the $\varepsilon$ intentionally added to the denominators of its components for avoiding eventual divisions by zero as well as zero-probabilities in the initial HMM. This problem is solved by performing an equitable redistribution of the missing quantity to each line of each element in $\bar{\lambda}_\kappa(S_{i,j})$ as it was the case in [26].$^{40}$ This enables to derive the readjusted initial HMM $\tilde{\lambda}_\kappa(S_{i,j})=(\tilde{\lambda}_\kappa, \tilde{B}_\kappa, \tilde{\pi}_\kappa)$ whose parameters are calculated in Equation 8.

$$ \left\{ \begin{aligned} \tilde{A}_\kappa[s_x, s_y] &= \bar{A}_\kappa[s_x, s_y] + \frac{1}{|S|} \left( 1 - \sum_{h=1}^{|S|} \bar{A}_\kappa[s_x, s_h] \right) \\ \tilde{B}_\kappa[s_x, z_l] &= \bar{B}_\kappa[s_x, z_l] + \frac{1}{|\vartheta|} \left( 1 - \sum_{h=1}^{|\vartheta|} \bar{B}_\kappa[s_x, z_h] \right) \\ \tilde{\pi}_\kappa[s_x] &= \bar{\pi}_\kappa[s_x] + \frac{1}{|S|} \left( 1 - \sum_{h=1}^{|S|} \bar{\pi}_\kappa[s_h] \right) \end{aligned} \right. \quad (8) $$

Finally, the readjusted initial HMM $\tilde{\lambda}_\kappa(S_{i,j})$ is trained with the Baum-Welch algorithm for learning the sequences of symbols appearing in $\Delta_\kappa(S_{i,j})$. This enables to derive the final HMM $\lambda_\kappa(S_{i,j})=(A_\kappa, B_\kappa, \pi_\kappa)$ associated with $S_{i,j}$.

### 4) META-DATA EXTRACTION

Meta-data related to the behavior of $\lambda_\kappa(S_{i,j})$ after a sufficiently long time are extracted in order to generate the descriptor vector $\vec{S}_{i,j,\kappa}$ associated with $S_{i,j}$. The $l^{th}$

<footnote>$^{40}$See the Equation 17 of [26]</footnote>
<table><caption>TABLE 8. Vectors associated with the statokinesigrams of the subject $u=\{S_{1,1}, S_{1,2}, S_{2,1}, S_{2,2}\}$ of the Table 7 calculated for $\kappa=1, 3$. The vector components are truncated here to two decimal places.</caption><thead><tr><th></th><th>κ=1</th><th>κ=3</th></tr></thead><tbody><tr><td>S&#770;<sub>1,1,κ</sub></td><td>[0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2.46, 0.02]</td><td>[0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6.38, 0.06]</td></tr><tr><td>S&#770;<sub>1,2,κ</sub></td><td>[0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1.04, 0.01]</td><td>[0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 3.06, 0.04]</td></tr><tr><td>S&#770;<sub>2,1,κ</sub></td><td>[0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 3.59, 0.02]</td><td>[0, 0, 0, 0, 0, 0, 0, 0.10, 0.09, 0.12, 0.40, 0.29, 7.81, 0.06]</td></tr><tr><td>S&#770;<sub>2,2,κ</sub></td><td>[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.66, 0.33, 0, 0, 0, 0.85, 0.01]</td><td>[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.45, 0.55, 2.49, 0.03]</td></tr></tbody></table>

component $\chi_k^l$ of this vector with $(1 \le l \le |\vartheta|)$ is the overall proportion of time spent by $\lambda_\kappa(S_{i,j})$ observing the symbol $z_l$ after a sufficiently long time, irrespective of the state from which this observation is realized. In order to compute the value of $\chi_k^l$, the following principle is first adopted to evaluate the overall proportion of time spent by $\lambda_\kappa(S_{i,j})$ observing the symbol $z_l$ from state $s_x$ after a sufficiently long time:

1) The overall proportion of time spent by $\lambda_\kappa(S_{i,j})$ in state $s_x$ after a sufficiently long time is initially computed. This is the $x^{th}$ component $\varphi_\kappa[s_x]$ of the stationary distribution of $\lambda_\kappa(S_{i,j})$.

2) The value obtained at step 1 is multiplied by the probability of observing the symbol $z_l$ from state $s_x$ which is given by $B_\kappa[s_x, z_l]$.

The value of $\chi_k^l$ is finally obtained by repeating this process for every state $s_x$ and by summing the resulting proportions. In order to remember that the angles and the distances were normalized, the maximum angle $\beta_{max}$ and the maximum distance $d_{max}$ used to realize this normalization are inserted as the two last components of $\vec{S}_{i,j,\kappa}$ as described in Equation 9. As a consequence, $\vec{S}_{i,j,\kappa}$ has ($\mathcal{N} + 3$) components and we conventionally decided to associate the null vector $\vec{0}$ of $\mathbb{R}(\mathcal{N}+3)$ with any empty or missing statokinesigram.

$$ \left\{ \begin{array}{l} \vec{S}_{i,j,\kappa} = [\chi_{\kappa}^1, \chi_{\kappa}^2, \dots, \chi_{\kappa}^{|\vartheta|}, \beta_{max}, d_{max}] \quad \text{where} \\ \chi_{\kappa}^l = \sum_{x=1}^{|\vartheta|} (\varphi_{\kappa}[s_x] \times B_{\kappa}[s_x, z_l]) \quad \text{with } (1 \le l \le |\vartheta|) \end{array} \right. \qquad (9) $$

When Equation 9 is applied to the statokinesigrams of the subject $u=\{S_{1,1}, S_{1,2}, S_{2,1}, S_{2,2}\}$ of the Table 7 with different values of $\kappa$, the 13-dimensional descriptor vectors presented in the Table 8 are obtained. The components of the vectors presented in that table are truncated here to two decimal places due to space constraints. The following experimental parameters were used to calculate these vectors: $\sigma=1, \mathcal{N}=10, \varepsilon=1, r=100, \gamma=100$ and $\kappa=1, 3$. The content of Table 8 demonstrates the impact of the time-shift $\kappa$ on the vector components which vary from one value of $\kappa$ to the other. All the components of the vectors presented in the Table 8 are easily interpretable. As an example, the components of the vector $\vec{S}_{2,1,3}$ are interpreted as follows:

1) The last component is the maximum distance ($d_{max}=0.06$) observed during the transformation of the statokinesigram $S_{2,1,3}$ into a MC.

2) The component of rank 12 is the maximum angle of observation ($\beta_{max}=7.81^{\circ}$) encountered during the same transformation.

3) The component of rank 11 is the overall proportion (0.29) of time spent by the model $\lambda_3(S_{2,1})$ observing the interval $I_{10}]=90, 100]$ after a sufficiently long time, irrespective of the angles from which this observation is realized. In the context of the current work, it means that $\lambda_3(S_{2,1})$ relies on the current content of $S_{2,1,3}$ recorded during only four seconds, to predict that if the recording time was extended to a sufficiently long duration (e.g., several hours or even a full day), then distances belonging to the interval $]0.9 \times d_{max}, 1 \times d_{max}] =]0.054, 0.06]$ will be observed during 29% of this very long recording time.

4) Similarly, the component of rank 10 is the overall proportion (0.40) of time spent by $\lambda_3(S_{2,1})$ observing the interval $I_9=]80, 90]$ after a sufficiently long time, irrespective of the angles from which this observation is realized. This means that $\lambda_3(S_{2,1})$ predicts that if the recording time was extended to a sufficiently long duration, then distances belonging to the interval $]0.8 \times d_{max}, 0.9 \times d_{max}] =]0.048, 0.054]$ will be observed during 40% of this very long recording time.

5) Each remaining component of rank $l$ with $(1 \le l \le 9)$ is interpreted analogically.

The components of all the vectors of the Table 8 can be interpreted according to the same reasoning. Thus, the fact that the 11th component of the vector $\vec{S}_{1,1,1}$ is 1 means that $\lambda_1(S_{1,1})$ predicts that if the recording time of the statokinesigram $S_{1,1,1}$ was extended to a sufficiently long duration, then only distances belonging to the interval $]0.9 \times 0.02, 1 \times 0.02] =]0.018, 0.02]$ will be exclusively observed during the total (i.e., 100%) recording time.

## 5) FINAL VECTOR COMPUTATION

In order to derive an average characterization of the balance of a subject $u=\{\cup_{i=1}^n S_i\}$ under a specific test condition, the arithmetic mean of the descriptor vectors associated with the statokinesigrams corresponding to the $m$ trials recorded under the $i^{th}$ test condition is saved in the vector $\vec{S}_{i,\kappa}$ as described
<table><caption>TABLE 9. Vectors characterizing the balance of the subject *u* of the Table 7 under each test condition calculated for $κ=1, 3$.</caption><thead><tr><th></th><th>κ=1</th><th>κ=3</th></tr></thead><tbody><tr><th>$$\vec{S}_{1,κ}$$</th><td>[0, 0, 0, 0, 0, 0, 0, 0, 1, 1.75, 0.015]</td><td>[0, 0, 0, 0, 0, 0, 0, 0.5, 0, 0.5, 4.72, 0.05]</td></tr><tr><th>$$\vec{S}_{2,κ}$$</th><td>[0, 0, 0, 0, 0, 0.33, 0.165, 0.50, 0, 0, 2.22, 0.015]</td><td>[0, 0, 0, 0, 0, 0.05, 0.045, 0.06, 0.425, 0.42, 5.15, 0.045]</td></tr></tbody></table>

<table><caption>TABLE 10. Vectors characterizing the balance of the subject *u* of the Table 7 calculated for $κ=1, 3$.</caption><thead><tr><th></th><th>u̇<sub>κ</sub></th></tr></thead><tbody><tr><th>κ=1</th><td>[0, 0, 0, 0, 0, 0, 0, 0, 1, 1.75, 0.015, 0, 0, 0, 0, 0, 0.33, 0.165, 0.50, 0, 0, 2.22, 0.015]</td></tr><tr><th>κ=3</th><td>[0, 0, 0, 0, 0, 0, 0.5, 0, 0.5, 4.72, 0.05, 0, 0, 0, 0, 0, 0.05, 0.045, 0.06, 0.425, 0.42, 5.15, 0.045]</td></tr></tbody></table>

in Equation 10.

$$ \vec{S}_{i,κ} = \frac{1}{m} \left( \sum_{j=1}^{m} \vec{S}_{i,j,κ} \right) \quad (10) $$

Finally, the vector $\vec{u}_κ$ that characterizes the balance of the subject *u* while considering all the test conditions is obtained by concatenating the descriptor vectors associated with the *n* test conditions as described in Equation 11. Therefore, $\vec{u}_κ$ has $n \times (N + 3)$ components.

$$ \vec{u}_{\kappa} = \underbrace{\mathbf{concat}}_{i=1,...,n} (\vec{S}_{i,\kappa}) \quad (11) $$

Table 9 contains the mean descriptor vectors characterizing the balance of the subject *u* of the Table 7 under each test condition, while Table 10 shows the final descriptor vectors characterizing the balance of the same subject, both calculated for $κ=1, 3$.

### C. THEORETICAL TIME COST

Consider a subject $u = (\cup_{i=1}^n S_i)$ where $S_i = \{S_{i,1}, \dots, S_{i,m}\}$ contains the statokinesigrams recorded under the $i^{th}$ test condition. Given a user-defined time-shift $κ$, the goal of this section is to evaluate the theoretical time cost of the three main steps that constitute the major contributions of the current work which are:

1) The computation of the single vector $\vec{S}_{i,j,κ}$ associated with the $j^{th}$ statokinesigram recorded under the $i^{th}$ test condition.

2) The computation of the mean vector $\vec{S}_{i,κ}$ associated with all the statokinesigrams recorded under the $i^{th}$ test condition.

3) The computation of the final vector $\vec{u}$ associated with the balance of *u*, irrespective of the test conditions.

Suppose that all the statokinesigrams are saved into (text or binary) files and assume that all these files contain the same number of COP points since the recording sessions generally have the same duration in a given project. The time required for the computation of the single vector $\vec{S}_{i,j,κ}$ can be step-by-step estimated according to the content of the Figure 3 as follows:

1) **Down-sampling:** This step is dominated by the access to the statokinesigram data. More formally, given a user-defined down-sampling factor $σ$, the frequency becomes $f = \frac{f}{σ}$ and the $\frac{1}{σ} \times |S_{i,j,κ}| = |S|$ COP points that compose the statokinesigram are read from the file during this step. Hence, this step roughly runs in $θ(2c \cdot |S|)$ where $c$ is the time required to read a floating-point number (i.e., one COP coordinate) from a file.

2) **Transformation into a MC:** This step is dominated by the computation of the distances and the angles between the $(|S|-κ)$ pairs of points $(p_l, \bar{p}_l)$ of the statokinesigram $S_{i,j}$ with $(κ \le l \le |S|-1)$. Therefore, this step runs in $θ((a+b) \cdot (|S|-κ))$ where $a$ and $b$ respectively refer to the time required to compute the distance and the angle between each pair of points $(p_l, \bar{p}_l)$.

3) **MC split:** The time cost of this step can be neglected because it involves no particular computation.

4) **HMM initialization and training:** The time required for initializing a HMM was experimentally very low compared to the model training time. This step is consequently dominated by the model training phase, which runs in around $θ(γ \cdot (|S| - κ) \cdot (N + 1)^2)$ according to Section III-C, where $γ$ is the user-defined maximum number of iterations of the Baum-Welch algorithm.

5) **Meta-data extraction:** The most time consuming operation realized during this step is the computation of the stationary distribution, which runs in around $θ(r \cdot (N + 1)^3)$ as stated in Section III-D, where $r$ is the user-defined number of matrix products.

The overall time $ξ$ required for the computation of the single vector $\vec{S}_{i,j,κ}$ is calculated by Equation 12.

$$ \left\{ \begin{array}{ll} \xi = \theta \left( \sum_{i=1}^{4} \xi_i \right) & \text{where} \\ \xi_1 = 2c \cdot |S| & \text{and} \\ \xi_2 = (a+b) \cdot (|S| - \kappa) & \text{and} \\ \xi_3 = \gamma \cdot (|S| - \kappa) \cdot (N + 1)^2 & \text{and} \\ \xi_4 = r \cdot (N + 1)^3 & \end{array} \right. \quad (12) $$
As it can be observed in Equation 10, the computation of the mean vector $\vec{S}_{i,\kappa}$ runs $\theta(m.\xi + (m-1).(N+3)+1)$ because it involves:

1) The computation of $m$ single vectors which takes $m.\xi$ time units.

2) The sum of these vectors which requires to make $(m-1).(N+3)$ additions because each each vector has $(N+3)$ components.

3) The last division (by $m$) which is one single operation.

In order to obtain the final vector $\vec{u}$, the $n$ mean vectors need to be calculated the concatenated as it is described in Equation 11. This last step enables to use Equation 13 for estimating the overall time $\xi$ required to compute $\vec{u}$.

$$ \vec{\xi} = \theta[n.(m.\xi + (m-1).(N+3)+1)] \quad (13) $$

Since the beginning of the Section IV, it has been assumed that the same number $m$ of statokinesigrams was recorded under each test condition. But it may happen that for particular reasons, a specific number $m_i$ of statokinesigrams is recorded under the $i^{th}$ test condition, with $(1\le i\le n)$. The methodology proposed in the current paper for deriving the descriptor vector $\vec{u}_\kappa$ that characterizes the balance of a subject $u$, while considering all the test conditions, remains valid in that case if $m$ is replaced by $m_i$ wherever it is required (Figure 4, Equation 10, etc.).

# V. CLINICAL APPLICATIONS

## A. DISTANCE BETWEEN TWO SUBJECTS

Consider two subjects $u$ and $v$ whose balance was evaluated through statokinesigrams recorded under $n$ identical test conditions at the rate of $m$ trials per test condition. Given a user-defined time-shift $\kappa$ and a valid distance measure $d$ between two vectors of $\mathbb{R}^\rho$ where $\rho = n \times (N + 3)$, the distance $D_\kappa(u, v)$ that evaluates the gap separating the balance of $u$ from the balance of $v$ irrespective of the test conditions, is calculated through the computation of the distance between their respective descriptor vectors as shown in Equation 14. The formulas of more than 60 existing distance measures between two vectors of $\mathbb{R}^\rho$ that can be used for this purpose are available in [27].$^{41}$

$$ D_{\kappa}(u, v) = d(\vec{u}_{\kappa}, \vec{v}_{\kappa}) \quad (14) $$

In practice, that equation can be for example used for evaluating the gap separating the balance of a healthy subject, from that of a subject suffering from a specific postural disability or disorder.

## B. IMPACT OF A MEDICATION

The proposed approach can also be used for quantifying the impact of a particular medication on the balance of a given subject $u$. In that case, if $u_s$ refers to the statokinesigrams of the subject recorded when starting the medication, and if $u_e$ refers to the statokinesigrams of the same subject recorded at

<figure><img src="image_9.png" /><figcaption>FIGURE 9. Prediction of the class associated with a subject $u$ among the two classes $V$ and $\bar{V}$ using the proposed approach with a time-shift $\kappa$.</figcaption></figure>

<table><caption>TABLE 11. Parameters of the experimental datasets.</caption><thead><tr><th>Prediction [Dataset]</th><th>Label of V</th><th>Size of V</th><th>Label of V&#773;</th><th>Size of V&#773;</th></tr></thead><tbody><tr><td>Age-group [1]</td><td>young</td><td>87</td><td>old</td><td>76</td></tr><tr><td>Fall risk [1]</td><td>high</td><td>66</td><td>low</td><td>97</td></tr><tr><td>Parkinson [1, 8]</td><td>positive</td><td>32<sup>44</sup></td><td>negative</td><td>57<sup>45</sup></td></tr></tbody></table>

the end of the medication, then the amplitude of this impact is the distance $D_\kappa(u_s, u_e)$ calculated in Equation 14.

## C. BINARY PREDICTIONS

The proposed approach can further be used for predicting the age-group, the fall risk and the health status of a subject $u$ using any binary classifier like the MLP [21], the SVM [18], the K Nearest Neighbors (KNN) [28], etc. Suppose that the input statokinesigrams are organized in two classes $V$ and $\bar{V}$ (e.g., young and old, faller and non-faller, healthy and sick). Given a time-shift $\kappa$, the descriptor vectors associated with the statokinesigrams contained in $V$ and $\bar{V}$ are initially computed using Equation 11. The two resulting sets of vectors are taken as input by the selected binary classifier that will predict the class of $u$ as it is described in the Figure 9.

# VI. EXPERIMENTAL RESULTS

The goal of this section is to evaluate the performances of the proposed approach regarding age-group, fall risk and illness prediction through classification experiments realized on open datasets following the principle described in the Figure 9. The C source codes and the experimental data supporting the findings of the current study are available online$^{44,45}$.

## A. EXPERIMENTAL DATASETS

The two open datasets recorded with a FPF [1], [8] initially described in the Table 3 have been selected here as experimental datasets according to the following parameters that are summarized in the Table 11:

1) The dataset proposed by Santos and Duarte [1]$^{II-D}$ was selected for both, age-group and fall risk prediction. For age-group prediction, the class labels were 'young'

<footnote>$^{41}$See the Tables 1 to 8 and the Table 10 of [27]</footnote>

<footnote>$^{44}$https://webperso.etis-lab.fr/sylvain.iloga/index.html</footnote>

<footnote>$^{45}$https://github.com/HmmHBE/HMM-Based-HBE.git</footnote>
<table><caption>**TABLE 12.** Similar items in the recording conditions of the datasets proposed in [1] and [8].</caption><thead><tr><th>Item</th><th>Value</th></tr></thead><tbody><tr><td>FPF references</td><td>OPT400600-1000, AMTI, Watertown, MA, USA</td></tr><tr><td>FPF dimensions</td><td>40cm × 60cm</td></tr><tr><td>Frequency</td><td>100Hz</td></tr><tr><td>Vision</td><td>Eyes open/closed</td></tr><tr><td>Surface</td><td>Firm/Foam</td></tr><tr><td>Feet placement</td><td>20°, Heels kept 10cm apart</td></tr><tr><td>Total time</td><td>720s</td></tr></tbody></table>

and 'old' having respectively 87 and 76 subjects as defined in the dataset. The total score of the Mini-BEST was used to separate the 66 subjects with a 'high' fall risk (score ≤20) from the 97 remaining subjects with a 'low' fall risk (score >20).

2) The 32 subjects suffering from the idiopathic ssec:PublicDatasets's disease of the dataset proposed by de Oliveira et al. [8] II-D were associated with the 57 healthy subjects who declared 'no illness' in the dataset proposed by Santos and Duarte [1] II-D for illness prediction. Therefore the class labels were 'negative' for the 57 healthy subjects and 'positive' for the 32 other subjects.

No experiment has been conducted on the dataset proposed by Audiffren et al. [5] II-D because the statokinesigrams of that dataset were recorded with a WBB which is less accurate and less precise than a FPF. Furthermore, only 50s of data were recorded for each subject in that dataset. This is very low compared to the 720s recorded for each subject in [1] and [8].

The fact that data used in the current work to predict Parkinson's disease were recorded in separate studies [1], [8] may induce undesirable batch effects. Batch effects are systematic, non-biological differences in data caused by variations in the data sources (e.g., instruments, acquisition protocols, etc.), which can lead a classifier to learn source-specific patterns rather than true class-discriminative features. But, as observed in Section 'Data acquisition and processing' of [1] and in Section 'Stabilography' of [8], the statokinegrams of these two datasets were recorded under quite similar conditions. Table 12 enumerates the similar items in the recording conditions of these two datasets. The content of that table supports the conjecture that batch effects may have a limited impact on the classification experiments combining data from these datasets.

Two main groups of techniques used for correcting batch effects have been designed, and the technical details related to these techniques are available in [29], [30], and [31]. The first group is composed of statistical and preprocessing

<table><caption>**TABLE 13.** Generic confusion matrix of the binary classifier of Figure 9.</caption><thead><tr><th></th><th>V</th><th>V&#773;</th></tr></thead><tbody><tr><th>V</th><td>True(V)</td><td>False(V&#773;)</td></tr><tr><th>V&#773;</th><td>False(V)</td><td>True(V&#773;)</td></tr></tbody></table>

techniques that adjust data distributions before attempting to remove batch-specific mean and variance shifts. They are thus relatively simple to implement, interpretable, and computationally efficient. However, they may underperform when batch effects are complex or nonlinear. Z-score Scaling and Quantile Normalization are important members of this group. The second group consists of machine and deep learning approaches that learn batch-invariant feature representations during training. Unlike statistical and preprocessing techniques, they can handle complex and nonlinear batch effects, while improving generalization. They are mainly limited by their higher computational cost, their need for large datasets, and their lower interpretability. This group includes techniques such as Domain-Adversarial Training and Transfer Learning. In order to correct potential batch effects during the experiment related to prediction of Parkinson's disease in the current work, statistical and preprocessing techniques were preferred due to their simplicity of implementation and their low time cost.

## B. EXPERIMENTAL SETTINGS

Classification experiments were conducted under the system MS Windows 10 Professional edition on a personal computer having 16 GB of main memory and the following processor: Intel(R) Core(TM) i7-8665U CPU @ 1.90GHz 2.11GHz.

The statokinesigrams of the datasets proposed by Santos and Duarte [1] II-D and by de Oliveira et al. [8] II-D that were initially sampled at 100Hz, have been down-sampled with a factor ($\sigma=5$) to obtain the new sampling frequency of 20Hz that was considered during the next steps of the proposed methodology. Each long MC was split to obtain several short MCs of length 20 (the new sampling frequency), except the last one that may be shorter. The Euclidean distance was selected for generating the MCs and the value ($N=10$) was selected to split the interval [0, 100] into ($N+1$) sub-intervals using Equation 3. The initial HMMs were readjusted with ($\varepsilon=1$) as it was the case in [26].<fnref n="46" /> The maximum number of iterations during the HMM training was set to ($\gamma=100$) and the stationary distribution of each HMM was calculated by performing ($r=100$) matrix products following [26].<fnref n="47" />

The proposed approach inherits the main drawback of the stochastic COP variables related to the difficulty to select the appropriate time-shift. For this reason, the time-shifts $\kappa=1, 5, 10$ were initially experimented for each dataset. Then, additional experiments were conducted by

<footnote n="44">⁴⁴Selected from [8]</footnote>

<footnote n="45">⁴⁵Selected from [1]</footnote>

<footnote n="46">⁴⁶See the Equations 14 to 16 of [26]</footnote>

<footnote n="47">⁴⁷See the Section V-A of [26]</footnote>
<table><caption>TABLE 14. Classification results in (%). The bests accuracies and the bests F1 scores are in bold.</caption><thead><tr><th colspan="5">(a) Age-group in [1] using SVM</th></tr><tr><th>κ</th><th>Acc</th><th>F1</th><th>Prec</th><th>Rec</th></tr></thead><tbody><tr><td>1</td><td>76.68</td><td>76.6</td><td>76.7</td><td>76.6</td></tr><tr><td>5</td><td>74.23</td><td>74.1</td><td>74.2</td><td>74.2</td></tr><tr><td>10</td><td>77.91</td><td>77.8</td><td>78.0</td><td>77.9</td></tr><tr><td>{1, 5}</td><td><b>80.36</b></td><td><b>80.4</b></td><td>80.4</td><td>80.4</td></tr><tr><td>{1, 10}</td><td>79.75</td><td>79.7</td><td>79.8</td><td>79.8</td></tr><tr><td>{5, 10}</td><td><b>80.36</b></td><td><b>80.4</b></td><td>80.5</td><td>80.4</td></tr><tr><td>{1, 5, 10}</td><td>78.52</td><td>78.5</td><td>78.5</td><td>78.5</td></tr></tbody></table><table><thead><tr><th colspan="5">(b) Fall risk in [1] using SVM</th></tr><tr><th>κ</th><th>Acc</th><th>F1</th><th>Prec</th><th>Rec</th></tr></thead><tbody><tr><td>1</td><td>76.68</td><td>76.2</td><td>76.6</td><td>76.7</td></tr><tr><td>5</td><td>75.46</td><td>74.4</td><td>75.9</td><td>75.5</td></tr><tr><td>10</td><td>71.16</td><td>70.4</td><td>70.8</td><td>71.2</td></tr><tr><td>{1, 5}</td><td><b>77.91</b></td><td><b>77.3</b></td><td>78.0</td><td>77.9</td></tr><tr><td>{1, 10}</td><td>68.17</td><td>68.5</td><td>68.4</td><td>68.7</td></tr><tr><td>{5, 10}</td><td>71.16</td><td>70.5</td><td>70.8</td><td>71.2</td></tr><tr><td>{1, 5, 10}</td><td>74.23</td><td>73.8</td><td>74.0</td><td>74.2</td></tr></tbody></table><table><thead><tr><th colspan="5">(c) Parkinson in [1, 8] using J48</th></tr><tr><th>κ</th><th>Acc</th><th>F1</th><th>Prec</th><th>Rec</th></tr></thead><tbody><tr><td>1</td><td>98.87</td><td><b>98.9</b></td><td><b>98.9</b></td><td><b>98.9</b></td></tr><tr><td>5</td><td>97.75</td><td>97.8</td><td>97.9</td><td>97.8</td></tr><tr><td>10</td><td>98.87</td><td><b>98.9</b></td><td><b>98.9</b></td><td><b>98.9</b></td></tr><tr><td>{1, 5}</td><td>98.87</td><td><b>98.9</b></td><td><b>98.9</b></td><td><b>98.9</b></td></tr><tr><td>{1, 10}</td><td>98.87</td><td><b>98.9</b></td><td>98.89</td><td>98.89</td></tr><tr><td>{5, 10}</td><td>97.75</td><td>97.8</td><td>97.9</td><td>97.8</td></tr><tr><td>{1, 5, 10}</td><td>98.87</td><td><b>98.9</b></td><td><b>98.9</b></td><td><b>98.9</b></td></tr></tbody></table><table><thead><tr><th colspan="5">(d) Parkinson in [1, 8] using J48 with z-score scaling</th></tr><tr><th>κ</th><th>Acc</th><th>F1</th><th>Prec</th><th>Rec</th></tr></thead><tbody><tr><td>1</td><td>94.38</td><td>94.3</td><td>94.5</td><td>94.4</td></tr><tr><td>5</td><td>95.5</td><td>95.5</td><td>95.5</td><td>95.5</td></tr><tr><td>10</td><td>94.38</td><td>94.3</td><td>94.5</td><td>94.4</td></tr><tr><td>{1, 5}</td><td>97.75</td><td>97.7</td><td>97.8</td><td>97.8</td></tr><tr><td>{1, 10}</td><td>98.87</td><td><b>98.9</b></td><td><b>98.9</b></td><td><b>98.9</b></td></tr><tr><td>{5, 10}</td><td>97.75</td><td>97.7</td><td>97.8</td><td>97.8</td></tr><tr><td>{1, 5, 10}</td><td>97.75</td><td>97.7</td><td>97.8</td><td>97.8</td></tr></tbody></table>

concatenating the final vectors generated by the following subsets of time-shifts: {1, 5}, {1, 10}, {5, 10} and {1, 5, 10}.

Two separate experiments were conducted for the prediction of Parkinson's disease. In the first experiment, the final feature vectors associated with the subjects from [1], [8] were taken as inputs by the selected classifiers without any modification. In the second experiment, these final feature vectors were initially preprocessed using the z-score scaling technique in order to correct potential batch effects before performing the classification. A more detailed analysis of the sensitivity of the proposed approach to its main hyperparameters is available in Section VI-H.

## C. METRICS OF EVALUATION

Consider the binary classifier of Figure 9 that assigns to a subject *u* one of the two classes *V* and $\bar{V}$. Suppose that this classifier experimentally outputs the results summarized in the generic confusion matrix shown in the Table 13.

The four following metrics of evaluation are adopted in the current work to evaluate the performances of this classifier:

1) The **accuracy** of the classifier which is the proportion of correctly classified subjects, out of the total number of subjects. Equation 15 shows how to compute the classifier accuracy which is generally recommended for balanced classes.

$$Acc = \frac{\text{True}(V) + \text{True}(\bar{V})}{\text{True}(V) + \text{True}(\bar{V}) + \text{False}(V) + \text{False}(\bar{V})} \quad (15)$$

2) The **precision** of the classifier which measures the reliability of its predictions. It is calculated using Equation 16 as the arithmetic mean of the precisions of the two classes. The precision of a class being the proportion of correctly predicted members of that class, out of all the subjects predicted as members of that

class.

$$ \left\{ \begin{array}{l} \text{Prec} = \frac{\text{Prec}(V) + \text{Prec}(\bar{V})}{2} \quad \text{where} \\ \\ \text{Prec}(V) = \frac{\text{True}(V)}{\text{True}(V) + \text{False}(V)} \quad \text{and} \quad (16) \\ \\ \text{Prec}(\bar{V}) = \frac{\text{True}(\bar{V})}{\text{True}(\bar{V}) + \text{False}(\bar{V})} \end{array} \right. $$

3) The **recall** of the classifier which assesses its ability to detect the members of each class. It is calculated using Equation 17 as the arithmetic mean of the recalls of the two classes. The recall of a class being the proportion of members of that class that were correctly identified.

$$ \left\{ \begin{array}{l} \text{Rec} = \frac{\text{Rec}(V) + \text{Rec}(\bar{V})}{2} \quad \text{where} \\ \\ \text{Rec}(V) = \frac{\text{True}(V)}{\text{True}(V) + \text{False}(\bar{V})} \quad \text{and} \quad (17) \\ \\ \text{Rec}(\bar{V}) = \frac{\text{True}(\bar{V})}{\text{True}(\bar{V}) + \text{False}(V)} \end{array} \right. $$

4) The **F1** score of the classifier which balances its precision and its recall to handle class imbalances. It is calculated using Equation 18 as the harmonic mean of the precision and the recall of the classifier.

$$F1 = \frac{2 \times \text{Prec} \times \text{Rec}}{\text{Prec} + \text{Rec}} \quad (18)$$

## D. CLASSIFICATION PERFORMANCES

All the classification experiments were realized with the version 3.9 of the software WEKA [32] through a 10-fold cross-validation (90:10). One of the two following WEKA
<figure><img src="image_10.png" /><figcaption>FIGURE 10. Typical workflow of a technique that assesses the human balance.</figcaption></figure>

classifiers was selected with its default parameters during all the experiments realized in each dataset, the classifier name in WEKA is in brackets: SVM with polynomial kernel (SMO) and Decision Trees (J48) [33]. The vectors generated during each experiment were saved into '.arff' files taken as inputs by WEKA. Table 14 presents the classification results that enable to observe the overall very small gap between the accuracy and the F1 score obtained during each experiment, irrespective of the considered dataset. For age-group prediction in [1], the proposed approach obtained a best accuracy of 80.36% and a best F1 score of 80.4% with the SVM classifier. A best accuracy of 77.91% and a best F1 score of 77.3% were obtained for fall risk prediction in that same dataset [1] with the SVM classifier. For the two experiments related to the prediction of Parkinson's disease in [1] and [8], the application of z-score scaling slightly reduced the performance. Nevertheless, the proposed approach exhibited the same best accuracy of 98.87% and the same best F1 score of 98.9% in the two experiments. This enables to confirm the previously conjectured very limited impact of batch effects. These performances demonstrate that capturing the sequential variations of distances and angles between regularly separated points of the statokinesigrams of a subject is an accurate idea that leads to efficient clinical application.

## E. EXPERIMENTAL TIME COST

In practice, the experimental time cost of the proposed approach is reasonable. Indeed, around 50ms were required to compute each final vector of the Table 10. Beside this, around 7s were required to generate the final vector associated with each subject in [1] and [8].

## F. COMPARISON TO EXISTING WORK

### 1) EXPERIMENTAL COMPARISON

The performances of the proposed approach have been compared to those of the work of Pennone et al. [4] recently published in 2024, the comparison results are presented

<table><caption>TABLE 15. Comparison to existing work on the dataset proposed in [1]. Accuracies are in (%). The best accuracies are in bold.</caption><thead><tr><td>Prediction</td><td>Pennone et al. [4]</td><td>This work</td><td>Accuracy gains</td></tr></thead><tbody><tr><td>Age-group</td><td>65</td><td>80.36</td><td>+15.36</td></tr><tr><td>Fall risk</td><td>74</td><td>77.91</td><td>+3.91</td></tr></tbody></table>

in the Table 15. Indeed, among all the related work surveyed in the current paper that realized experiments in the dataset proposed in [1], it is the only one involving all the 163 subjects and where each prediction simultaneously considers all the test conditions [II-F1].

Table 15 reveals that the proposed approach outperforms [4] for both, age-group and fall risk prediction with accuracy gains of +15.36% and +3.91% respectively. The proposed approach was not compared to the work of de Oliveira et al. [8] because no classification experiments were conducted in that work.

### 2) CONCEPTUAL COMPARISON

As it is described in the Figures 10a and 10b, the typical workflow of a study that assesses the human balance is composed of two main steps: the feature extraction step and the classification step. Conceptually, the major difference between the current work and the existing studies resides in the feature extraction step. Indeed, during this step, the existing studies generally process the raw input data by computing various COP variables, in order to derive the final feature vectors. Some existing studies using deep learning models as classifiers skipped the feature extraction step and directly classified the raw data [7]. Conversely, in the current work, none of the existing COP variables is computed during the feature extraction step. Two basic geometric features (angle and distance between regularly separated samples) are rather extracted from the raw input data, then used to construct MCs. These MCs are subsequently used for initializing, then for training one HMM whose metadata
later serve as components of the final feature vectors. This analysis enables to conclude that, the current work conceptually stands out from the existing studies regarding the interpretability of the generated feature vectors because it involves an automatic HMM-based learning process within its feature extraction step. The resulting feature vectors are interpretable because HMMs are robust and interpretable models.

## G. MAIN ASSETS

The technique proposed in the current work:

1) Trains a HMM that automatically learns the overall behavior of the COP during the traversal of its trajectory. No existing technique can do this.

2) Associates a ($N+3$)-dimensional vector $\vec{S}_{i,j,\kappa}$ whose components are easily interpretable with each statokinesigram $S_{i,j}$. Unlike each existing COP variable that aggregates in one single numeric value all the high volume of time-series data contained in a statokinesigram.

3) Generates one unique $n \times (N + 3)$-dimensional vector $\vec{u}_{\kappa}$ associated with each subject $u$ that can be used solely as a relevant descriptor of the subject's balance, irrespective of the test conditions, unlike existing COP variables that must be combined as the components of one descriptor vector.

4) Avoids overfitting during the learning process by performing the down-sampling of the statokinesigrams using a factor $\sigma$ when the original sampling frequency is high.

5) Can be used for various clinical applications including binary predictions (e.g., age-group, fall risk and illness) and the quantification of the impact of a particular medication on a subject's balance.

6) Is highly customizable according to several user-defined parameters including: $\gamma, r, d, \kappa, \sigma$ and $N$.

7) Requires a reasonable time cost. This time cost can be further reduced through parallel implementations as stated in Section VII.

8) Outperforms the existing techniques for age-group and fall risk prediction with accuracy gains of +15.36% and +3.91% respectively, when experimented on the dataset proposed in [1].

9) Exhibits a perfect performances of 100% best accuracy of 98.87% and a best F1 score of 98.9% for the prediction of ssec:PublicDatasets's disease when experimented on the datasets proposed in [1] and [8].

## H. SENSITIVITY TO HYPERPARAMETERS

The proposed approach is highly customizable through six main user-defined hyperparameters (e.g., $\gamma, r, d, \kappa, \sigma, N$) whose variations may affect the experimental performance. It is therefore important to analyze the sensitivity of the proposed approach to these six hyperparameters. This analysis is realized in the next sections and Table 16

summarizes the major insights related to the sensitivity of the proposed approach to its main numeric hyperparameters.

### 1) SENSITIVITY TO $\gamma$

The parameter $\gamma$ is the maximum number of iterations of the Baum-Welch algorithm and its value directly affects both the convergence quality and training time. If $\gamma$ is too small, the algorithm may stop before reaching a local maximum of the likelihood function, resulting in underfitting and poor probability estimations. Conversely, high values of $\gamma$ generally improve the model likelihood and the prediction accuracy up to a point, after which additional iterations yield negligible gains but incur higher computational cost and risk of overfitting. In practice, $\gamma$ should be large enough to allow convergence, but not excessively large to avoid unnecessary computation or fitting noise. A common strategy is to use early stopping when the improvement falls below a predefined threshold. In the current paper, the value ($\gamma$=100) was selected following [26] $^{VI-B}$.

### 2) SENSITIVITY TO $R$

The maximum number $r$ of matrix products that must be performed during the computation of the stationary distribution of a HMM should be chosen such that the difference between $A^r$ and $A^{r-1}$ falls below a small tolerance, indicating that the HMM has reached equilibrium. Choosing $r$ too small leads to an inaccurate stationary distribution which remains influenced by the initial state. Whereas, an excessively large value of $r$ only increases the computational cost without improving the accuracy. In practice, the value of $r$ should be selected adaptively by iterating until convergence within a specified threshold rather than fixing it a priori. In the current paper, the value ($r$=100) was selected following [26] $^{VI-B}$.

### 3) SENSITIVITY TO $D$

Selecting the appropriate vector distance $d$ in a metric-based algorithm like the proposed approach highly depends on the nature of the data and the objective of the task. When the goal is to preserve geometric relationships in continuous isotropic spaces, the Euclidean distance is typically preferred because it captures straight-line similarity. For data with features that are independent and for which absolute differences are more meaningful, such as sparse or high-dimensional data, the Manhattan distance may be more robust. When the objective is to be less sensitive to outliers, measures like the Minkowski distance or the Cosine similarity can improve performances. The Euclidean distance was therefore selected because the proposed approach automatically learns the sequential variations of geometric features that occur in a statokinesigram.

### 4) SENSITIVITY TO $\kappa$

In stabilometry, choosing the value of the time-shift $\kappa$ is critical for accurately capturing the temporal structure of postural
<table><caption>TABLE 16. Sensitivity of the proposed approach to its main numeric hyperparameters.</caption><thead><tr><th></th><th>Small value</th><th>High value</th><th>How to tune ?</th></tr></thead><tbody><tr><td>γ</td><td>Underfitting &amp; poor probabilities</td><td>Time consuming &amp; overfitting</td><td>Early stopping strategy</td></tr><tr><td>r</td><td>Inaccurate stationary distribution</td><td>Time consuming &amp; no accuracy improvements</td><td>Iterating until convergence</td></tr><tr><td>κ</td><td>Reflects noise or tremor</td><td>Dilution of important transient events</td><td>Empirical or cross-validation</td></tr><tr><td>σ</td><td>Time consuming &amp; unnecessary redundancies</td><td>Misrepresented signal &amp; loss of sway information</td><td>Frequency f/σ between 20Hz and 50Hz</td></tr><tr><td>Ν</td><td>Time consuming &amp; overfitting &amp; noise sensitivity &amp; data sparsity</td><td>Loss of important patterns</td><td>Empirical or cross-validation</td></tr></tbody></table>

control. A small $κ$ focuses on short-term fluctuations and yields information about rapid corrective responses, but may mostly reflect noise or tremor rather than meaningful postural control. In contrast, a large $κ$ captures long-term dependencies and slower postural adjustments. However, it increases the risk of diluting transient but important instability events. Thus, $κ$ should be selected to match the characteristic time scale of postural control, balancing sensitivity to short-term instability with robustness against noise. In practice, the value of $κ$ is most often determined empirically or via cross-validation. It is for this reason that each of the three values ($κ$ = 1, 5, 10) has been first used individually, then combined to maximize the overall accuracy of the human balance assessment.

### 5) SENSITIVITY TO σ

Given an original sampling frequency $f$, choosing the correct down-sampling factor $σ$ in stabilometry is a trade-off between signal fidelity and computational efficiency. A small value of $σ$ preserves the high-frequency components of the COP signal, capturing rapid postural adjustments, but may lead to unnecessary data redundancy and longer processing times. In contrast, a large value of $σ$ reduces data size and noise but risks signal misrepresentation and loss of important sway information if the resulting down-sampled frequency ($f/σ$) falls below twice the highest relevant frequency of postural sway (typically 5 – 10 Hz) as stated by the *Nyquist theorem* [34], [35]. Therefore, $σ$ should be chosen so that ($f/σ$) remains sufficiently high to capture the physiologically meaningful sway dynamics while avoiding oversampling. This often results in effective down-sampled frequencies of about 20 – 50 Hz in related studies. It is for this reason that the value ($σ$ = 5) was selected in the current work to obtain down-sampled frequency of 20 Hz.

### 6) SENSITIVITY TO N

The choice of a suitable value of $N$ used for splitting the interval [0, 100] into ($N$ + 1) slices is a trade-off between resolution and robustness. A high value of $N$ provides finer granularity, allowing the discretization to capture subtle variations in the data, which can improve the expressiveness of the underlying HMM. However, such large values of $N$ may increase sensitivity to noise, augment data sparsity and induce high computational cost, while potentially leading to overfitting. Conversely, a smaller value of $N$ smooths the data, making the representation more robust, but may lead to the loss of important patterns. For all these reasons, we recommend tuning $N$ empirical or through cross-validation. The value ($N$ = 10) was experimented in the current work. It is important to note that unreported experiments realized with the value ($N$ = 20) exhibited slightly lower performances than those obtained with ($N$ = 10), while requiring a more considerable time cost.

## VII. CONCLUSION

The current work tackles the problem of human balance evaluation which has become crucial over the past decades for many clinical applications in stabilometry including age-group, fall risk and illness prediction. Existing techniques used for this purpose include various reliable clinical tests proposed by clinicians (Mini-BEST, FES-I, IPAQ, TMT, etc.) as well as several COP variables derived from a statotokinesigram proposed by researchers (positional, dynamic, stochastic and frequency variables). Unfortunately, these existing COP variables are individually limited by several factors listed in the last column of the Table 4 and none of the existing COP variables is calculated through the automatic learning of the overall behavior of the COP during the traversal of its trajectory.

The current work attempts to attenuate these limitations through the proposal of a new HMM-based stochastic technique that trains HMMs for automatically learning the sequential variations of distances and angles that occur in the statotokinesigrams of a given subject $u$, irrespective of the test conditions. Meta-data derived from these HMMs are gathered for deriving one unique COP variable $\vec{u}$ which is a $n \times (N+3)$-dimensional vector whose components accurately describe the balance of $u$. The vector $\vec{u}$ can be used for predicting the health status of $u$ as well as for evaluating the impact of a particular medication on his balance.

Classification experiments conducted on two open datasets [1], [8] with the vectors generated by the proposed approach showed bests accuracies of 80.36% for age-group prediction in [1], of 77.91% for fall risk prediction in [1] and of 98.87% for the prediction of Parkison's disease in [1] and [8]. This performance is +15.36% and +3.91% better than the technique recently proposed in 2024 by Pennone et al. [4] for age and fall risk prediction. The following perspectives can be explored:
1) Instead of using Equation 3 for splitting the interval [0,100] into (N+1) slices during the transformation of a statokinesigram into a MC, future work can analyze the impact of using continuous observation HMMs [23], [36] on the performances of the proposed approach. Indeed, unlike standard (or discrete) HMMs that can only observe discrete symbols, continuous observation HMMs offer the interesting possibility of observing continuous data as symbols by using Gaussian Mixture Models [37] to model the emission probability distribution.

2) Another alternative and potentially effective strategy involves the use of data-adaptive partitioning techniques like quantile-based discretization and k-means clustering to organize the angle-distance pairs into (N+1) clusters. Such an approach is expected to mitigate data sparsity and facilitate a more reliable estimation of the HMM parameters in future work, particularly in the context of short-duration trials or data recorded with WBBs.

3) In the present study, the final feature vector is obtained by averaging (with Equation 10) and concatenating (with Equation 11) HMM-based metadata across recording conditions. Future work may investigate alternative pooling strategies, such as attention-weighted mechanisms, condition-conditioned embeddings, or Fisher score representations, which could provide more discriminative and informative subject-level descriptors.

4) Future research may focus on the clinical interpretation of HMM-derived metadata by linking specific state occupancies and transition patterns to established balance constructs (e.g., sway velocity bursts, mediolateral dominance). Such mappings would enhance the practical relevance and clinical interpretability of the model outputs for practitioners.

5) Future work can also rely on the models and vectors obtained during the experiments realized in the current work for illness prediction to develop a software that will be used by clinicians for the prediction of the ssec:PublicDatasets's disease. To achieve this goal, the clinician will only need to provide the statokinesigrams of the subject recorded under similar experimental settings as those of [1] and [8].

6) In the same way, Equation 14 can serve as basis for the implementation of another software in future work that will gradually evaluate the impact of a particular medication on the balance of a subject during a specific period.

7) In addition to the FPFs, the performances of the proposed approach can be evaluated for statokinesigrams recorded by other innovative devices as radars and event cameras that have never yet been experimented in stabilometry.

8) The mean vector computation step of the proposed methodology can be replaced in future work by the

technique proposed in [38], where the authors generate one descriptor vector after training HMMs for learning the content of a finite set of vectors according to user-defined properties.

9) The time cost of the proposed approach can be reduced in future work if the mean vector generation blocks of the Figure 4 are executed in parallel for each of the *n* test conditions. More precisely, *n* processors {$p_1, p_2, ..., p_n$} can be used for this purpose, each processor $p_i$ ($1 \le i \le n$) being responsible of the computation of the vector $\vec{S}_{i,\kappa}$.

10) The time cost can be further reduced if a parallel version of the Baum-Welch algorithm is executed. This can be implemented using Message Passing Interface (MPI) following [39] or using a Field-Programmable Gate Array (FPGA) chip following [40].

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

**A. J. TSAGMO DENKENG** received the master's degree with thesis in computer science from the Faculty of Sciences, University of Yaoundé I, Cameroon, under the supervision of Thomas Djotio, in 2022. He is currently pursuing the Ph.D. degree in computer science. His Ph.D. studies are supported by CY Cergy Paris Université, France, the LÉONIE CHAPTAL Foundation, France, and the University of Yaoundé I, Cameroon. Since May 2025, he has been a Non-Permanent Member

of the ETIS-UMR 8051 Research Laboratory, CY Cergy Paris Université. His researches focus on the use of machine learning techniques for the analysis of temporal data from various sources (radars and force plateforms).

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

**A. MOHAMMAD MOURAD** received the master's degree with thesis from the Faculty of Sciences, University of Maroua, Cameroon, in 2023, under the supervision of Sylvain Iloga. He was a Software Engineer with the National Advanced School of Engineering, University of Maroua, in 2020. He has been a Teacher in computer science with the Higher Teachers' Training College, University of Maroua, since 2021, where he carried out a teaching internship of one year,

in 2024. From June to July 2025, he carried out a Research Internship with the ETIS-UMR 8051 Laboratory, CY Cergy Paris Université, in France. His researches mainly concern data modeling using hidden Markov models.

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

**SYLVAIN ILOGA** received the Ph.D. degree in computer science from the University of Yaoundé I, Cameroon, in January 2018. Since January 2010, he has been a Teacher with the Department of Computer Science, Higher Teachers' Training College of Maroua, Cameroon. From September 2017 to August 2019, he completed a Research and Teaching Internship with the IUT of Cergy-Pontoise, Neuville University, France. In January 2019, he was qualified as an Associate Professor

with Section 27 (computer science), France. From October 2022 to July 2024, he was a Teacher with the ESIEE-IT CCI Paris Île-de-France, France. In November 2023, he was subsequently promoted to the rank of an Associate Professor in Cameroon. Beside this, he is an Associate Member of the Research Laboratory ETIS-UMR 8051, CY Cergy Paris Université, ENSEA, CNRS F-95000 in Cergy, France. His researches mainly focus on data modeling through machine learning using hidden Markov models, with the final goal to design their implementation into reconfigurable architectures.
<figure><img src="image_14.png" /></figure>

**R. M. MBALLA MBA** received the master's degree in public health from the Catholic University of Central Africa, Cameroon, in 2015, the Ph.D. degree in general medicine from the Faculty of Medicine and Biomedical Sciences, University of Yaoundé I, Cameroon, in 2018, and the University Diploma degrees in geriatrics and in gerontology capacity from the Sorbonne University of Paris, France, in 2020 and 2022, respectively. After several years of clinical experience as a General Practitioner in Cameroon, since 2023, she has been a Geriatrician with the Central Hospital of Yaoundé, Cameroon. Beside this, she is a ICOPE Trainer in the VIHeillir Project in collaboration with French National Agency for Research on AIDS and Viral Hepatitis in Cameroon. Her ongoing researches mainly focus on frailty and sarcopenia.

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

**T. DJOTIO NDIÉ** directs the WiCoNET Research Laboratory and the NetSAp Team with the National Advanced School of Engineering (ENSPY), University of Yaoundé I, Cameroon. His Flagship Project, WiABox, delivers low-cost wireless infrastructure that closes the digital divide in rural communities across the Global South. His research sits at the crossroads of network security, wireless networking for development, and network management, and extends into machine-learning-driven precision agriculture, software-defined networking (SDN), network-traffic engineering, disintermediation, and digital-transformation tools. Adopting a holistic, multidisciplinary lens, he designs connectivity solutions that simultaneously advance rural livelihoods, health services, quality education, and sustainable food systems. By aligning his work with sustainable development goals 2, 3, 4, 9, and 17—zero hunger, good health and well-being, quality education, industry innovation and infrastructure, and global partnerships, he consistently turns cutting-edge networking research into real-world impact for underserved communities.

<figure><img src="image_16.png" /></figure>

**HAJER BAAZAOU** received the Ph.D. degree in computer science (business intelligence for spatial data), in 2005. She received her accreditation to supervise research (H.D.R.) in (knowledge-based systems: ontology building and user profile management), in 2014. She is currently a Professor of computer science with the Engineering Graduate School CY-Tech of CY University Paris, a member of the ETIS Laboratory, the MIDI team. Her main research results have been published in more than 20 journal articles, book chapters and more than 60 indexed conferences and workshops. She has led and participated in several international and national projects, as well as journal and conference program committees. Her research focuses on data integration, semantic technologies and information systems, with applications mainly in the medical and environmental fields.

<figure><img src="image_17.png" /></figure>

**OLIVIER ROMAIN** received the Engineering degree in electronics from ENS Cachan, the master's degree in electronics from Louis Pasteur University, and the Ph.D. degree in electronics from Pierre and Marie Curie University, Paris. From 2012 and 2019, he was the Head of the Department of Architecture, ETIS-UMR 8051 Laboratory. He was the Director of the Laboratory, from 2020 to 2023. In addition, he is a University Professor in electrical engineering with CY Cergy Paris Université. His research focuses mainly on systems on chips for diffusion and on biomedical applications.

<figure><img src="image_18.png" /></figure>