XI INTERNATIONAL CONFERENCE

“INFORMATION TECHNOLOGY AND IMPLEMENTATION” (IT&I-2024)

Analysis of the accuracy of simulating the human eye movement system based on Volterra models

Vitaliy Pavlenko

Denys Lukashuk

Objective,Subject and Relevance of the Research

• The objective of this research is to evaluate the accuracy of eye movement system (EMS) simulation by examining the errors in calculating multidimensional transient characteristics (MTCs) when using nonlinear dynamic simulation methods based on integro-power series (IPS) and integro-power polynomials (IPP) models.
• The subject of the study includes algorithmic and software tools for extracting EMS dynamic characteristics from eye-tracking data and assessing the accuracy of different simulation methods.
• Research Relevance:

Enhancing the accuracy of modeling the human EMS is crucial for improving research outcomes and EMS diagnostics.

This research can be beneficial in the following fields:

• In medical diagnostic studies (ophthalmology, psychiatry);
• In high-performance sports (athlete training processes);
• In military and civil aviation;
• In biometrics (for authentication).

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Scientific Component of the Project

The developed information technology consists of a sequence of solutions to the following tasks:  

• Identification of the EMS (determining MTCs);
• Construction of diagnostic feature spaces based on EMS identification data;
• Development of a diagnostic model;
• Creation of a psychophysiological state classifier based on the nonlinear dynamic model of the EMS;
• Diagnosis (assessment of abilities and psychophysiological state of the individual).

Stages of implementing model-based classification technology

"Input-Output" Experiments

Positioning of the respondent during EMS studies using the Tobii Pro TX300 eye tracker

EMS responses of the respondent to visual stimuli of varying amplitudes

The research data was based on eye-tracking technology experiments. For the experiment, the respondent's head was fixed in place using a head holder in front of a monitor, and the eye tracker Tobii Pro TX300 was calibrated. Once the calibration was complete, a test was conducted, and the result is displayed on the graph to the right. The y-axis of the graph represents distance, while the x-axis corresponds to time or frame number in the data array.

"Input-Output" Experiments

The test procedure followed this sequence: the respondent’s gaze was initially fixed on a starting point (a red dot). After a set interval, the red dot disappeared, and a stimulus (a blue dot) appeared at one-third of the screen's width (Case A), prompting the respondent to shift their gaze. The red dot then reappeared, and the sequence was repeated for stimuli positioned at two-thirds of the screen’s width (Case B) and at the maximum screen width from the starting point (Case C).

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Test stimulus:

A) – starting position and position of the test stimulus with amplitude а₁=1/3;

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B) – starting position and position of the test stimulus with amplitude а₂=2/3;

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C) – starting position and position of the test stimulus with amplitude а₃=1;

A)

B)

C)

Methods of Identification

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For mathematical modeling of the human EMS, integral nonlinear models are used, which simultaneously take into account the nonlinear and inertial properties of the object of study. In the general case the “input-output” relationship for a nonlinear dynamical system can be represented in terms of the Volterra series as

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If the test signal x(t) = θ(t), where θ(t) is a unit function (Heaviside function), then

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Where ŷ₁(t) = ĥ₁(t), ŷ₂(t) = ĥ₂(t,t), ŷ₃(t) = ĥ₃(t,t,t).

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Structural scheme of the Volterra model

Approximation Method of Identification

This structural scheme provides a clear representation of how the second-order multidimensional transient characteristic is computed using responses to three step signals with amplitudes a₁, a₂, and a₃.

To find the coefficients c , it is necessary to solve a system of linear algebraic equations :

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where

The formula for calculating the second-order MTC:

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Where Δ₂ represents the methodological error resulting from the discarded terms of the IPS of order n+1 and higher.

Compensation Method of Identification

The transient characteristics of the models are computed using the following formulas:

for N=1:

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for N=2:

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Or for 3 signals:

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for N=3

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where a₁, a₂ = 2a₁, a₃ = 3a₁ are the amplitudes of test signals;

Δ_n represents the methodological error resulting from the discarded terms of the IPS of order n+1 and higher.

The computation of transient characteristics can also be visualized through structural schemes.

Structural scheme for N=2 (two signals):

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Structural scheme for N=2 (three signals):

Least Squares Method (LSM)

The scheme represents a second-order model, where block T₂ computes the first- and second-order transient characteristics based on the system of normal equations. The matrix A₂ in block T₂ incorporates all three input step signals (a₁, a₂, a₃) in the form of Heaviside function. The operation of block T₂ and the matrix A₂ are described by the following formulas:

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Accuracy Assessment

To evaluate the accuracy of the developed models for varying amplitudes of the test signals a₁, a₂ and a₃, the metric applied is the normalized root mean square error (NRMSE):

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, j=1, 2, 3;

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where – is the response of the EMS to the test signal in the form of a step function with amplitude a_j, measured at time t_m (where t_M – is the observation time of the EMS responses);

– is the response of the EMS model to the input signal in the form of a step function with amplitude a_j, calculated at time t_m.

Research Results

Let us denote the models of the EMS as follows:

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М1.N/x:<a₁,…, a_x> – model based on the approximation method;

М2.N/x:<a₁,…, a_x> – model based on the LSM;

М3.N/x :<a₁,…, a_x> – model based on the compensation method.

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where:

• N is the model order;
• x is the number of signals used in the model;
• a₁,…, a_x are the amplitudes of the test signals.

Models of N=1 Order

Comparative analysis of the average error values of the EMS models

Transient characteristics of M2.1/2

and M2.1/3 models

EMS and model M2.1/2:a₁, a₂

responses

EMS and model M2.1/2:a₁, a₃

responses

EMS and model M2.1/3

responses

It's important to highlight that both the Volterra series and the Volterra polynomial models yield identical transient characteristics; therefore, results will be shown for only one of these models.

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Models of N=2 Order

Models using the compensation method exhibit greater error compared to analogous models based on the LSM, as demonstrated by both the diagram and the response graphs.

Comparative analysis of the average error values of the EMS models

MTCs of the model M3.2/3

EMS and model М2.2/3 responses

MTCs of the model M2.2/3

EMS and model М3.2/3 responses

Models of N=3 Order

The model M3.3 demonstrates low accuracy, making it unsuitable for reliable simulations of the EMS.

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The models M1.3/3 and M2.3/3 produced identical MTCs. The responses of these models practically coincide with the EMS response for the same input signals. However, the transient characteristics of the third-order models demonstrate instability.

MTCs of the model M3.3

MTCs of the model M2.3

EMS and model M3.3 responses

EMS and model M2.3 responses

Conclusion

For the first time, an error analysis of mathematical models of the EMS in the form of IPS and IPP, derived from eye-tracking data, was conducted using three test step signals of varying amplitudes, employing compensation and approximation identification methods, as well as the LSM. It was established that the most accurate model of the EMS is the quadratic Volterra polynomial, determined based on three EMS responses. It was observed that third-order models exhibited instability in their transient characteristics, which may limit their practical application. Therefore, it is advisable to apply integral quadratic models in diagnostic studies of the human psychophysiological state.