Integral nonlinear models are used to simulate the human eye movement system (EMS) while accounting for its nonlinear dynamics and inertial properties. Multidimensional transient characteristics (MTCs) of the EMS were identified based on experimental input-output data obtained from eye-tracking responses to visual test stimuli. These transient characteristics include first-order and diagonal cross-sections up to the second and third orders of MTCs. The study aimed to evaluate the accuracy of EMS simulation models by analyzing the calculation errors of transient characteristics using nonlinear dynamic identification methods based on Volterra integro-power series (IPS) and integro-power polynomial (IPP). Computational methods, including the least squares method (LSM), approximation, and compensation, were used to derive the models. Models developed using the LSM and approximation methods produced consistent transient characteristics when the same test signals were applied, highlighting the convergence of the Volterra series within the identified region. The findings showed that increasing the number of test signals enhanced the accuracy of the EMS models. Quadratic models were identified as the most reliable, providing a balance between precision and computational efficiency. Cubic models closely matched EMS responses but exhibited instability in their transient characteristics, making them less practical for EMS application. The compensation method, while computationally less demanding, proved unsuitable for tasks requiring high accuracy due to significant errors in the resulting models. Quadratic IPP models developed with LSM based on three response datasets are recommended for future studies, as they provide a stable and precise framework for modeling EMS dynamics and exploring psychophysiological state assessment.
Eye-tracking technology [
Despite the advancements in eye-tracking systems, there is a growing need for new mathematical models to accurately simulate the human eye movement system (EMS). Additionally, specialized equipment is necessary to support experimental research involving these systems. Eyetracking technology relies on sophisticated devices, known as eye trackers, to precisely determine the coordinates of eye movements.
In medical applications and experimental psychology, the ability to simulate the EMS is vital for
effective control, monitoring, and diagnostics. Developing a comprehensive mathematical model of
the EMS, which takes into account individual human variability, is essential for creating advanced
treatment methodologies [
This paper presents the findings on the accuracy of simulating the dynamic characteristics of the human EMS. These characteristics were derived from experimental "input-output" data, capturing responses to visual stimuli using cutting-edge eye-tracking technology (a simulation task). The study focuses on the nonlinear and inertial properties of the EMS, which are crucial for developing accurate and reliable models.
To simulate the human eye movement system, integral nonlinear models [
The aim of this research is to evaluate the accuracy of EMS simulation by examining the errors in calculating MTCs when using nonlinear dynamic simulation methods based on IPS and IPP models. This study also addresses the development of algorithmic and software tools for extracting EMS dynamic characteristics from eye-tracking data and assessing the accuracy of different simulation methods.
In this study, the approximation method [
Approximation Identification Method. The approximation identification method for NDS (method
of linear combinations of responses) in the time domain is grounded in isolating the n-th partial
component (PC) of the NDS response by constructing linear combinations of responses to test
signals with different amplitudes. This approach is an adaptation of methods originally based on
the Volterra series. It is proved in [
Assertion 1. Let test signals a1 (t), a2x(t) aNx(t) be sequentially applied to the input of the NDS, where N is the degree; a1, a2 aN are different real numbers, non-zero, satisfying the condition |aj| 1 for j=1,2,...,N; x(t) is an arbitrary function. Then, the linear combination of the system's responses to these inputs equals the n-th PC of the response to the input signal x(t) with an accuracy up to the discarded terms of the Integro-Power Series of order N+1 and higher: where yn [x(t)] = yn (t);
N c j y[a j x(t)] = yn[x(t)] + , j=1 (1) if j are real coefficients such that where
A N c = b , (2) A N = aa112 aa222 aa NN2 , c = cc12 , b = bb12 , a1N a2N aNN cN bN here bl = 1 when l = n; bl = 0 when l n, l N}.
The system (2) always has a solution, and it is unique since its determinant differs from the Vandermonde determinant only by the factor a1, a2 aN. Thus, for any real numbers aj, which are non-zero and pairwise distinct, it is possible to find numbers cj such that the linear combination (1) of the NDS responses equals the n-th term of the Integro-Power Series with an accuracy up to the discarded terms of the series. By satisfying the conditions for forming the system of linear algebraic equations (2), we obtain the relation (1).
When test signals in the form of step functions (Heaviside functions t)) with amplitudes a1, a2 aN are applied to the input of the system being identified, we obtain estimates of the diagonal cross-sections of the NDS multidimensional transient characteristics:
N hˆn (t,...,t) = yˆn (t) = c(jn) y(a jθ(t)) = c1(n) y(t | a1 ) + c2(n) y(t | a2 ) + ... + cN(n) y(t | aN ), n =1, N, (3) j=1 where y(t|aj) = y(aj t)) are the NDS responses to the test signal with amplitude aj .
Identification of NDS using the Least Squares Method. The method of NDS identification based on the Volterra polynomial model in the time domain relies on approximating the NDS response y(t) to an arbitrary deterministic signal x(t) in the form of an integro-power polynomial of N-th order (N the order of the approximation model):
N N t t n yN (t) = yˆn (t) = ... wn (t − τ1,..., t − τn ) x(τi )dτi .
n=1 n=1 0 n times0 i=1
Valid assertion [
Assertion 2. Let test signals a1x(t), a2x(t) aLx(t) be sequentially applied to the input of the NDS; a1, a2 aL are different real numbers satisfying the condition 0<aj1 for j=1,2,...,L; x(t) is an arbitrary deterministic signal, then
N N t t n N ~yN (a j x(t)) = yˆ n (a j x(t)) = a nj 0 n t.i.m.es0 wn (t − τ1 ,...,t − τ n ) x(τi )dτi = a nj yˆ n (t) n=1 n=1 i=1 n=1 for j, j = 1, L.
The partial components in the approximation model yˆn (t) are found using the least squares method. This allows obtaining estimates for them, where the sum of squares of deviations of the NDS responses being identified, y[a j x(t)] from the model responses ~yN [a j x(t)] , is minimal, thus ensuring the minimum mean square criterion
L L N 2 J N = (y(a j x(t)) − ~yN (a j x(t)))2 = y(t | a j ) − a nj yˆ n (t) → min .
j=1 j=1 n=1 (4) (5) (6)
AAyˆ = A y , where
A = aa12 aa1222 aa12NN , y = yy((tt || aa12)), yˆ = yyˆˆ12((tt)) .
aL aL2 aLN y(t | aL ) yˆ N (t)
If the identified system is supplied with test signals in the form of step functions with
amplitudes a1, a2 aL, we obtain estimates of the transient characteristics hˆ1(N) (t) and the diagonal
cross-sections of the transient characteristics of the eye movement system hˆ(N) (t, t) , hˆ(N) (t, t, t)
2 3
hˆ(N ) (t,..., t) [
N or
Responses of the investigated EMS models are generally calculated based on expressions:
~y j (t | a j ) = a j yˆ1(t) + a 2j yˆ 2 (t) + ... + a Nj yˆ N (t), j = 1, L ,
~y(t | a j ) = a j hˆ1(N) (t) + a 2j hˆ2(N) (t, t) + ... + a Nj hˆ N(N) (t,..., t), j = 1, L . (9)
Compensation Identification Method. The formalism of the method for determining the
intersections of n-th order transient characteristics of nonlinear dynamic systems is based on the
following assertion [
Assertion 3. Let the test inputs be the sum of n step signals xk (t) = ak θ(t − τ k ) (k=1,2,...,n), shifted in time by 1, ..., n. Then, for an NDS with a single input and a single output, the estimate of the intersection of the n-th order transient characteristic is hˆn (t − τ1 ,..., t − τ n ) = n! n ak −1 1 (−1) n+kn=1δk y(t | δ1 ,..., δ n ) .
k =1 δ1,...,δn =0 where y(t | δ1,..., δn ) is the NDS response at time t when subjected to a multi-step input signal with amplitudes ak, obtained as a result of processing experimental data based on (11). If δ k = 1 , the test input contains a step signal shifted by k; otherwise, if δk = 0 , it does not contain it.
In certain cases, we have: for n=1
Minimizing criterion (6) boils down to solving a system of normal equations Gauss, which in vector-matrix form can be expressed as: for n=2 or for n=3 hˆ1(t) = y(t |a1) ;
a1 hˆ2 (t,t) = 21a12 y(t |a2 ) − 2 y(t | a1) ; hˆ2 (t,t) =
1 2a1a2
y(t |a3) − y(t | a1) − y(t |a2 ) ; 1 hˆ3 (t, t, t) = 6a13 y(t |a3 ) − 3y(t | a2 ) + 3y(t | a1 ). where a1, a2 = 2a1, a3 = 3a1 are the amplitudes of test signals. (7) (8) (10) (11) (12) (13) (14)
The responses of the second and third-order models are calculated accordingly using the expressions:
~y(t | a j ) = a jhˆ1(t) + a 2jhˆ2 (t,t) . ~y(t | a j ) = a jhˆ1(t) + a 2jhˆ2 (t,t) + a3jhˆ3(t,t,t) , j = 1, L . (15) (16)
The EMS's responses to the test step signals, defined as x(t) = aj t) with amplitudes aj (j=1, 2, 3):
a1=1/3, a2=2/3, a3=1 were analyzed. These responses formed the basis in the construction of
Volterra models [
For EMS simulation in work [
The software tools were developed using the Python programming environment.
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): M (y(tm | a j ) − ~y(tm | a j ))2 εaj = m=0 M y2 (tm | a j ) , j=1, 2, 3; (17)
m=0
where y(tm | a j ) and ~y(tm | a j ) are the responses of the EMS and the model of the EMS to the test
signal in the form of a step function with amplitude aj, measured/ computed at the time instant tm
(tm is the observation time of the EMS responses); j=1,2,3.
y(t | a1 ) or y(t | a2 ) or y(t | a3) , as shown in Fig. 3. Fig. 4 shows transient characteristics graphs of
the M2.1/2 models, which were calculated based on two responses: y(t | a1 ) and y(t | a2 ) , or
y(t | a1 ) and y(t | a3 ) , or y(t | a2 ) and y(t | a3 ) and the transient characteristic graph of the model M2.1/3.
For the M1.1 identification method does not allow to calculate the transient characteristics based
on two or three responses [
For N=2, based on the two responses y(t|a1) and y(t|a2), or y(t|a1) and y(t|a3), or y(t|a2) and y(t|a3), the corresponding multidimensional transient characteristics 1(t|aj,ak) and 2(t,t|aj,ak), j, k = 1,2,3; j k were obtained. In the models M1.2/2 and M2.2/2, identical MTCs 1(t|aj,ak) and 2(t,t|aj,ak) were obtained for the same experimental data.
Fig. 5 displays the primary-order transient characteristics graphs, while Fig. 6 shows the diagonal cross-sections of the second-order MTCs for the EMS models M1.2/2 and M2.2/2. These intersections were computed based on the responses: y(t|a1) and y(t|a2), or y(t|a1) and y(t|a3), or y(t|a2) and y(t|a3).
The transient characteristics of the EMS model M3.2/2 are illustrated in Fig. 7. The corresponding responses for the same model are given in Fig.8.
The graphs of the multidimensional transient characteristics 1(t) and 2(t,t), which were determined based on the three responses y(t|a1), y(t|a2), y(t|a3) for the EMS model M2.2/3, are shown in Fig. 9. The corresponding graphs comparing the M2.2/3 responses with the EMS responses to identical test signals are presented in Fig. 10. Analogous results were obtained for the M3.2/3 and are shown in Fig. 11 and Fig. 12.
For N=3, the graph of the primary-order transient characteristic, along with the graphs of the diagonal cross-sections of the second and third-order multidimensional transient characteristics for the EMS model M3.3, are displayed in Fig.13. The responses of the EMS model M3.3 are shown in Fig.14. Analogous results for the EMS models M1.3 and M2.3 are shown in Fig. 15 and Fig.16.
Models M2.2/2:a1, a2 M2.2/2:a1, a3 M2.2/2:a2,a3 M2.2/3:a1, a2, a3
Models M3.2/2: a1, a2 M3.2/3:a1,a2,a3 M3.3/3: a1,a2,a3
For N=3, based on the three responses y(t|a1), y(t|a2), y(t|a3) the transient characteristics 1(t), 2(t,t), 3(t,t,t) were obtained. For the models M1.3/3 and M2.3/3, identical MTCs were obtained, and the responses of the models practically coincide with the responses of the EMS for the same input signals. At the same time, Figs. 13 and 15 demonstrate that the transient characteristics of the third-order models exhibit instability.
On Fig. 17, a comparative analysis diagram of errors based on the percentage NRMSE criterion constructed using identification software tools for EMS models: M2.1/1, M2.1/2, M2.1/3, is presented. On Fig. 18, the same analysis is provided for models M2.2/2, M3.2/2, M2.2/3, M3.2/3 and M3.3 (based on average values). The EMS models M1.3 and M2.3 are not shown in the diagram because they have negligible deviations from the EMS responses.
Python-implemented tools for nonlinear dynamic identification were applied to develop mathematical models of the human eye movement system (EMS) based on Volterra integro-power series (IPS) and integro-power polynomials (IPP). This study aimed to evaluate the accuracy of the developed models in the form of first-order transient characteristics and multidimensional transient characteristics of the second and third orders, derived from eye-tracking data under the influence of visual test stimuli (step signals of varying amplitudes). To construct models based on empirical data, computational identification methods were employed, including the compensation method, approximation method, and least squares method (LSM). The identification errors of EMS models were assessed using normalized root mean square error (NRMSE), reflecting deviations between model responses and EMS responses to identical test signals.
Models obtained using the approximation and LSM methods on the same test signals exhibited identical transient characteristics, as these characteristics coincided within the convergence region of the Volterra series. It was found that the accuracy of models, represented by transient characteristics, improved with an increasing number of test signals. For the linear model, the average error decreased from 11.4% with one test signal to 7.6% with three test signals when LSM was applied. For the quadratic model, an average error of 9.1% was observed with two test signals, which decreased to 4.5% when three signals were used. For the cubic model, using three test signals resulted in nearly identical responses between the model and the EMS. However, it was established that third-order models exhibited instability in their transient characteristics, limiting their practical applicability.
The compensation method required fewer computational resources but produced models with significant errors, rendering them unsuitable for diagnostic studies. The best result for the quadratic model, achieved using three test signals, yielded an average error of 8.4%.
This study, for the first time, analyzed the errors of EMS mathematical models in the form of IPS and IPP derived from eye-tracking data using three test step signals of varying amplitudes. The analysis employed compensation, approximation identification methods, and the LSM. It was determined that the most accurate EMS model is the quadratic Volterra polynomial, identified based on three EMS responses. Therefore, integral quadratic models are recommended for diagnostic studies of the human psychophysiological state.
The authors have not employed any Generative AI tools. 442. doi:10.1007/978-3-031