Validating a data-driven framework for vehicular traffic modeling

In IDM, the input parameters are the vehicle's speed v, bumper-to-bumper distance to the leading vehicle (distance headway) s, and the relative speed (Δv). The model outputs the acceleration of a vehicle as:

$$\begin{equation} \frac{\textrm{d} v}{\textrm{d} t} = a\left(1-\left(\frac{v}{v_0}\right)^\delta\right)-a\left(\frac{s^{*}\left(v, \Delta v\right)}{s}\right)^2 \end{equation} \tag{ 1 }$$

where a is the maximum vehicle acceleration, v₀ is the desired velocity, δ is an acceleration exponent, and $s^{*}$ is the desired minimum headway. The first part of the equation describes free flow, in which the acceleration a decreases to zero as the speed approaches v₀. The second part corresponds to the interaction term (braking), where the current distance headway (s) and the desired headway ($s^{*}$) are compared and deceleration is increased as the current headway decreases. The desired minimum headway $s^{*}$ is given by:

$$\begin{equation} s^{*} = s_{0} + \mathrm{max}\left(0,vT+\frac{v\Delta v}{2\sqrt{ab}}\right) \end{equation} \tag{ 2 }$$

where s₀ is the minimum gap allowed, T is the time headway, and b is a positive coefficient defining the rate of deceleration [16]. The ballistic method [53] is used to solve equation (1) and the speed and vehicle positions are determined as:

$$\begin{align*} v \left(t + \Delta t\right) & = v\left(t\right) + \frac{\mathrm{d} v}{\mathrm{d} t} \Delta t \quad\mathrm{and} \\ x \left(t + \Delta t\right) & = x\left(t\right) + v\left(t\right) \Delta t + \frac{1}{2} \frac{\mathrm{d} v}{\mathrm{d} t} \Delta t^2. \end{align*}$$

If the front vehicle is at rest, there is a possibility of calculating both a negative acceleration and velocity in the next time step. This negative velocity is prevented by implementing the following conditional statement [53]:

$$\begin{align} & \textbf{if:} \quad v\left(t\right) + \left(\mathrm{d} v/\mathrm{d} t\right)\Delta t \lt 0 \nonumber \\ & \textbf{then:} \quad v \left(t + \Delta t\right) = 0 \quad\mathrm{and}\quad x \left(t + \Delta t\right) = x\left(t\right) - \frac{1}{2} v^2\left(t\right) / \frac{\mathrm{d} v}{\mathrm{d} t }. \end{align} \tag{ 3 }$$

Each vehicle is simulated identically using typical model parameter values of T = 1.5, a = 0.3, b = 3, δ = 4, $s_{0} = 2$ [16]. The circumference of the circular track is set to L = 314 meters and $v_{0} = 30$ km h⁻¹ based on [54] and given an initial speed of 30 km h⁻¹. Trajectory data for each vehicle are recorded at a 0.1 second interval.

Different from IDM, vehicle acceleration in OVM is dependent on the difference between the vehicle's current speed and optimal speed V(s):

$$\begin{equation} \frac{\textrm{d}v}{\textrm{d}t} = a_{h} \left[ V\left(s\right) - \frac{\textrm{d}x}{\textrm{d}t} \right]. \end{equation} \tag{ 4 }$$

The parameter a_h accounts for heterogeneity among vehicle types and drivers [17], which in our simulation is assumed to be a constant for all vehicles for all time. The optimal velocity (OV) function is a hyperbolic tangent function of distance headway s and is given by:

$$\begin{equation} V\left(s\right) = \alpha \tanh\left[ \beta\left(s-s_{0}\right)\right] + v_{0} \end{equation} \tag{ 5 }$$

The minimum distance headway s₀ along with constants α, β, and v₀ scale the hyperbolic tangent function and determine the response of the OV given the value of distance headway s. Here, as s approaches s₀, the OV reduces to zero to avoid collision. We choose the parameter values as $a_{h} = 1.8$, α = 5.5, β = 0.37, $s_{0} = 9.1$, and $v_{0} = 4.9$ based on empirical evidence [55]. To avoid bias in synthetic data generation, simulation conditions (such as track length, initial conditions, etc) are kept constant to those used with the IDM.

Pairwise TE and CTE or causation entropy are extensions of the definition of entropy described by Shannon in 1948 [56]. TE was formalized concurrently by Schreiber [57] and Palus et al [58] to assess the information exchange between two variables (X and Y) over time. It is a metric commonly used to detect the coupling strength and direction between time series variables. For example, to detect coupling from ${Y \rightarrow X}$, it quantifies how much information Y can provide about the future state of X using the present states of both variables. With a first-order Markov process assumption, pairwise TE is defined as:

$$\begin{equation} T_{Y \rightarrow X} = \left\langle\log \frac{p\left(x_{n+1} \mid x_n, y_n\right)}{p\left(x_{n+1} \mid x_n\right)}\right\rangle \end{equation} \tag{ 6 }$$

where $\left\langle.\right\rangle$ is the average computed over all the samples, n is the time index, $p(x_{n+1})$ denotes probability, and $p(x_{n+1}|x_{n})$ is the probability of $x_{n+1}$ conditioned on the present state x_n . If there is no influence from Y on X, then $p\left(x_{n+1} \mid x_n, y_n\right) = p\left(x_{n+1} \mid x_n\right)$, and $T_{Y \to X} = 0$. The unit for TE is determined by the base of the logarithm used, i.e. 'nats' for $\log_{e}$, and 'bits' for $\log_{2}$.

When three (or more) variables are involved (X, Y, and Z), pairwise TE may not distinguish the indirect couplings [59]. In such scenarios, CTE can be applied. CTE evaluates the direct influence of Y on X accounting for any indirect influence from Z, and is defined as:

$$\begin{equation} C_{Y \rightarrow X \mid Z} = \left\langle\log \frac{p\left(x_{n+1} \mid x_n, z_n, y_n\right)}{p\left(x_{n+1} \mid x_n, z_n\right)}\right\rangle \end{equation} \tag{ 7 }$$

where $p(x_{n+1} \mid x_n, z_n, y_n)$ is the probability of $x_{n+1}$ conditioned on x_n , y_n and z_n . When Y does not influence X, the values of the numerator and denominator become equivalent, thus $C_{Y \to X\mid Z}$ equals zero.

Both TE and CTE are asymmetric by construction ($T_{X \to Y} \neq T_{Y \to X}$, and $C_{Y \rightarrow X \mid Z} \neq C_{X \rightarrow Y \mid Z}$); allowing for the dominant direction of information flow (coupling direction) to be identified [35]. When examining TE and CTE coupling from finite empirical data, a statistical significance test can be conducted using surrogate data to determine if the resultant value is statistically different than zero [40]. In the present work, the IT measurements and statistical significance tests are performed using the Java Information Dynamics Toolkit for Matlab [60]. The Kraskov estimation method is used for bias correction [61] when estimating the probability density functions.

Here we present an overview of how SINDy identifies governing dynamical systems models from data. SINDy considers a dynamical system of the form:

$$\begin{equation*} \dot{\mathbf{x}}\left(t\right) = \frac{\textrm{d} \mathbf{x}}{\textrm{d}t} = \mathbf{f}\left(\mathbf{x}\left(t\right)\right), \end{equation*}$$

where the vector $\mathbf{x}(t) = [x_1(t); x_2(t); \ldots, x_n(t)] \in \mathbb{R}^{n}$ represents the state of a system at time t and the function $\mathbf{f}(\mathbf{x})$ describes the temporal evolution of the system's state. SINDy identifies the fewest terms that approximate the unknown $\mathbf{f}(\mathbf{x})$ and establishes the model based on a library of candidate basis functions $\boldsymbol{\Theta} (\mathbf{x}) = [\theta_1(\mathbf{x}), \theta_2(\mathbf{x}), \ldots, \theta_p(\mathbf{x})]$:

$$\begin{equation*} f_j = \sum_{k = 1}^{p}{\xi_{jk} \theta_k\left(\mathbf{x}\right)} , \end{equation*}$$

where the coefficients ξ_jk are typically zero, and entries that are not zero indicate active terms in the dynamics. To find f, time-series measurements of x and their time derivatives $\dot{\mathbf{x}}$ (measured directly or approximated numerically) are sampled at several time steps $t_1, t_2, \ldots, t_m$ and arranged into matrices such that $\mathbf{X} = [\mathbf{x}(1) \; \mathbf{x}(2) \cdots \mathbf{x}(m) ]^T$ and $\dot{\mathbf{X}} = [\dot{\mathbf{x}}(1) \; \dot{\mathbf{x}}(2) \cdots \dot{\mathbf{x}}(m) ]^T$ with $\mathbf{X}, \dot{\mathbf{X}} \in \mathbb{R}^{m \times n}$, where m is the sample size and n is the number of states. The library functions are next evaluated on the data by constructing $\boldsymbol{\Theta} (\mathbf{X}) = [\theta_1(\mathbf{X}) \theta_2(\mathbf{X}) \cdots \theta_p(\mathbf{X})] \in \mathbb{R}^{m \times p}$. Finally, SINDy uses the sparse regression technique to approximately solve:

$$\begin{equation*} \dot{\mathbf{X}} \approx \boldsymbol{\Theta} \left(\mathbf{X}\right) \boldsymbol{\Xi}, \end{equation*}$$

where $\boldsymbol{\Xi} = [\boldsymbol{\xi_1} \boldsymbol{\xi_2} \cdots \boldsymbol{\xi_n}] \in \mathbb{R}^{p \times n}$ is a set of coefficients that determines the active terms in f. An extension of SINDy, referred to as SINDy-PI [46], has been developed for implicit differential equations of the form:

$$\begin{equation*} \mathbf{f}\left(\mathbf{x}, {\dot{\mathbf{x}}}\right) = 0, \end{equation*}$$

and then the sparse model is detected as

$$\begin{equation*} \boldsymbol{\Theta} \left(\mathbf{X}, {\dot{\mathbf{X}}}\right) \boldsymbol{\xi} = 0. \end{equation*}$$

It is also possible to include control input data in the SINDy-PI algorithm.

The equations that describe the two traffic models we utilize for data generation are implicit. Creating a traffic model for a subject vehicle requires incorporating data from adjacent vehicles as control input. Given that SINDy-PI is capable of handling these requirements, we employ it in our study.

We generate synthetic data from two fundamentally different car-following models, IDM and OVM, to validate the proposed data-analytic framework for identifying traffic models. The utilization of synthetic data is motivated by its known ground truth, providing a benchmark for evaluating the performance of the tools employed in this study. We generate data by simulating vehicles on a circular track with a single lane of traffic as illustrated in figure 1. This setup allows for large samples of data to be generated by tracking vehicles within a fixed arena. For IT measures, large datasets are necessary to estimate probability density functions accurately. We simulate vehicles to drive on a single lane which eliminates lateral interactions (influence from adjacent lanes). For each model, we simulate jammed-flow and free-flow traffic conditions by adjusting the vehicle density within a constant track length of 314 meters. For jammed-flow, the number of vehicles (N_v ) is set at 30, while for free-flow, N_v is set to 15. The circular track is simulated by imposing a periodic boundary condition. Periodic boundary conditions are applied to the simulation, treating the last vehicle ($i = N_v$) separately. The position of the vehicle in front of it is considered as the position of the first vehicle (i = 1) plus the track length, achieving the periodic boundary condition. For visualization, vehicle positions are wrapped within the track length.

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Using synthetic data with known ground truth, we evaluate the ability of IT tools to identify coupling direction accurately. For IT analysis, we compute the observables as shown in figure 1. For ith vehicle at a given instant, we compute the distance headway between vehicle i and its immediate front vehicle ($D_{i+1}$), the distance from its immediate rear vehicle ($D_{i-1}$), the distance from i + 2th vehicle ($D_{i+2}$), and the distance the vehicle i travels in the next $\Delta t$ time interval ($D_{\Delta t}$). Subsequently, we employ the IT measures to quantify coupling between these time-series variables to detect the influence of the adjacent vehicles on a subject vehicle. For IT analysis, the simulated vehicle trajectory data was resampled to obtain data points at one second intervals to match human driver reaction time [40]. This resulted in $29\,000$ samples per vehicle after excluding first 1000 samples to eliminate initial transient. The table 1 provides a summary of the variables and samples utilized in IT and SINDy-PI analysis, along with the interpretation of the corresponding analysis.

Table 1. Summary of tools, variables, samples used along with their corresponding interpretation.

Tools	Time series variables	Sample size	Interpretation of the analysis
$T_{i+1 \to i}$	$D_{\Delta t}$, $D_{i+1}$	29 000	Detect influence of front vehicle on a subject (i)
$T_{i-1 \to i}$	$D_{\Delta t}$, $D_{i-1}$	29 000	Detect influence of rear vehicle on a subject (i)
$C_{i+1 \to i | i-1}$	$D_{\Delta t}$, $D_{i+1}$, $D_{i-1}$	29 000	Detect front-to-subject influence eliminating indirect influence from rear
$C_{i-1 \to i | i+1}$	$D_{\Delta t}$, $D_{i+1}$, $D_{i-1}$	29 000	Detect rear-to-subject influence eliminating indirect influence from front
$C_{i+1 \to i | i+2}$	$D_{\Delta t}$, $D_{i+1}$, $D_{i+2}$	29 000	Detect front-to-subject influence eliminating indirect influence from lead
$C_{i+2 \to i | i+1}$	$D_{\Delta t}$, $D_{i+1}$, $D_{i+2}$	29 000	Detect lead-to-subject influence eliminating indirect influence from front
SINDy-PI	x1, x2, u1, u2	4000 (train) 2000 (test)	Detect functional relationships

The results of IT analysis are presented in figure 2. Sub-figures in the left column correspond to jammed-flow ($N_v = 30$) while those in the center correspond to free-flow ($N_v = 15$). Results of IT analysis of OVM data is shown in the first three rows, and results of IT analysis of IDM data are shown in the last row.

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3.2.1. IT analysis on OVM data

3.2.2. IT analysis on IDM data

The IT analysis of IDM and OVM data suggests that to create a traffic model for the vehicle i, only variables from the immediate front vehicle (i + 1) need to be included, as the other vehicles have no influence. With this knowledge, we proceed to employ SINDy-PI to investigate its capability for discerning functional relationships among variables, thereby facilitating the completion of model identification. We use data from jammed-flow scenarios where interaction is present in both OVM and IDM models. When applying SINDy-PI, we randomly select a vehicle from the set of 30 vehicles. Notably, we observe that this random selection does not impact the performance of SINDy.

The position and velocity of vehicle i are considered as primary states $\mathbf{x} = [x_{1}, x_{2}]$, position and velocity from vehicle i + 1 are considered as a control input $\mathbf{u} = [u_{1}, u_{2}]$, and velocity and acceleration are represented as the time derivatives of $\dot{\mathbf{x}} = [\textrm{d}x_1, \textrm{d}x_2]$. For both training and testing, we utilize 400 and 200 seconds of data, respectively, with a time resolution of $\Delta t = 0.1$. This accounts for a total of 4000 data points for training and 2000 data points for testing. The testing and training sets remain consistent throughout the evaluations of SINDy-PI. In SINDy-PI, the candidate basis functions included in the library Θ dictate the form of the final model. We assume the final model to be in the form of a polynomial which allows a direct comparison between each coefficient obtained through SINDy-PI and the model itself. Consequently, IDM in equation (1) and OVM in equation (4) are transformed into polynomials (details on these polynomial expansions are provided in appendix). The OVM equation is expanded using a Pade approximation of order [1,2], where this order denotes the power of the polynomials present in the numerator and denominator, respectively, of the series approximation [62, 63].

To evaluate the performance of SINDy-PI, we construct the library of candidate basis functions Θ that include the compulsory terms ($\boldsymbol{\Theta}_C$) derived from the relevant models, as well as we add redundant terms ($\boldsymbol{\Theta}_R$) to create a complete polynomial of a given degree. The compulsory terms can be regarded as the initial hypothesis of the model. The performance of SINDy-PI will be assessed by its ability to accurately retain the compulsory terms, determine their correct coefficients, and reject the redundant terms. When constructing the library, we set the maximum exponent for x to 4 and for u to 2, resulting in terms with highest power of 6 ($x_{1}^4u_{1}^2, x_{1}^4u_{1}u_{2}, \ldots$) and yielding a total of 250 terms

$$\begin{equation*} \boldsymbol{\Theta} = \overbrace{\left[\; \begin{array}{cc}\mid & \mid \\ \boldsymbol{\Theta}_C & \boldsymbol{\Theta}_R \\ \mid & \mid \end{array}\;\right] }^{250~\mathrm{terms}}, \end{equation*}$$

where $\boldsymbol{\Theta}_C$ which is the set of all compulsory terms required to compose either the IDM or OVM and are presented in table 2 and $\boldsymbol{\Theta}_R$ contains the redundant terms.

Table 2. We present the true model coefficients obtained from the polynomial expansion of IDM and OVM alongside sample SINDy results for comparison. The SINDy coefficients presented correspond to library of $\boldsymbol{\Theta}_3$ for IDM and library of $\boldsymbol{\Theta}_6$ for OVM. Rows displaying 'n/a' indicate the absence of the corresponding term in the library of the respective model.

Model coefficients and coefficients estimated by SINDy-PI
Library	IDM		OVM
θi	Model	SINDy-PI	Model	SINDy-PI
1	3.6000	3.4265	26.4497	26.4497
x1	2.4000	2.4000	10.9867	10.9867
x2	−1.8000	−1.8533	−25.8060	−25.8060
$x_1 x_2$	n/a	n/a	−4.4848	−4.4848
$x_1^2$	0.3000	0.3089	1.2075	1.2075
$x_2^2$	−1.3074	−1.3462	n/a	n/a
$x_1^2 x_2$	n/a	n/a	−0.2464	−0.2464
$x_2^3$	−0.4743	−0.4884	n/a	n/a
$x_2^4$	−0.0843	−0.0868	n/a	n/a
$x_1 x_2^4$	−0.0005	−0.0005	n/a	n/a
$x_1^2 x_2^4$	−0.0001	−0.0001	n/a	n/a
u1	−2.4000	−2.4000	−10.9867	−10.9867
$u_1 x_1$	−0.6000	−0.6178	−2.4149	−2.4149
$u_1 x_2$	n/a	n/a	4.4848	4.4848
$u_2 x_2$	0.6324	0.6512	n/a	n/a
$u_1 x_1 x_2$	n/a	n/a	0.4928	0.4928
$u_2 x_2^2$	0.4743	0.4884	n/a	n/a
$u_2 x_2^3$	0.1666	0.1716	n/a	n/a
$u_1 x_2^4$	0.0005	0.0005	n/a	n/a
$u_1 x_1 x_2^4$	0.0001	0.0001	n/a	n/a
$u_1^2$	0.3000	0.3089	1.2075	1.2075
$u_1^2 x_2$	n/a	n/a	−0.2464	−0.2464
$u_2^2 x_2^2$	−0.0833	−0.0858	n/a	n/a
$u_1^2 x_2^4$	−0.0001	−0.0001	n/a	n/a
dx2	16.0000	15.5400	14.3367	14.3367
$dx_2 x_1$	8.0000	8.0000	2.4916	2.4916
$dx_2 x_1^2$	1.0000	1.0296	0.1369	0.1369
$dx_2 u_1$	−8.0000	−8.0000	−2.4916	−2.4916
$dx_2 u_1 x_1$	−2.0000	−2.0592	−0.2738	−0.2738
$dx_2 u_1^2$	1.0000	1.0296	0.1369	0.1369

3.3.1. SINDy-PI analysis of IDM and OVM

We systematically evaluate the performance of SIDNy-PI starting with using only the compulsory terms in the library for each model, denoted as $\boldsymbol{\Theta}_0 = \boldsymbol{\Theta}_C$. Next, we incrementally introduce redundant terms by selecting from $\boldsymbol{\Theta}_R$ up to a specified polynomial degree N:

$$\begin{align*} \boldsymbol{\Theta}_N = \left[\boldsymbol{\Theta}_C \; \boldsymbol{\Theta}_R^{N}\right] \mathrm{, for }\,\, N = \left\{0,1,2,\dots,6\right\}~\mathrm{such \,\,that}\!:\\ \begin{cases} \boldsymbol{\Theta}_0 = \boldsymbol{\Theta}_C\\ \boldsymbol{\Theta}_1 = \left[\boldsymbol{\Theta}_C \; \boldsymbol{\Theta}_R^{1}\right],~\mathrm{terms\,\, up\,\, to \,\,degree}\,\, 1 \\ \boldsymbol{\Theta}_2 = \left[\boldsymbol{\Theta}_C \; \boldsymbol{\Theta}_R^{2}\right],~\mathrm{terms\,\, up\,\, to\,\, degree}\,\, 2 \\ \vdots\\ \boldsymbol{\Theta}_6 = \left[\boldsymbol{\Theta}_C \; \boldsymbol{\Theta}_R^{6}\right],~\mathrm{terms\,\, up\,\, to\,\, degree}\,\, 6 \end{cases} \end{align*}$$

We evaluate SINDy-PI corresponding to each library $\boldsymbol{\Theta}_N$ and compare these results to the true model coefficients of the IDM and OVM. The performance of SINDy-PI is quantified in terms of sensitivity, specificity, and accuracy. Terms identified by SINDy-PI are labeled as true positive (TP) if they are present in the true model and false positive (FP) if they are not.

As shown in table 2, the total number of positives (active terms) for the IDM and OVM are $\mathit{P} = 25$ and $\mathit{P} = 18$ respectively, and the number of negatives, $\mathit{N}$, depends on the library used for testing. Sensitivity, specificity, and accuracy are then measured as follows:

$$\begin{align*} \mathrm{sensitivity} & = \frac{\mathrm{TP}}{\mathit{P}} \\ \mathrm{specificity} & = \frac{\mathit{N - FP}}{\mathit{N}} \\ \mathrm{accuracy} & = \frac{\mathrm{TP} + \mathrm{\textit{N} - FP}}{\mathit{P + N}}. \end{align*}$$

Tables 3 and 4 present the results in percentage. Additionally, we measure error computed by comparing the coefficient of each term identified by SINDy-PI with the true coefficient of the respective model. We calculate the error for each FP term ($\mathrm{FP}_{e}$) by computing the absolute difference between the estimated coefficient returned by SINDy-PI and the true value, which is zero since those terms are absent in the model. The error for each TP term ($\mathrm{TP}_{e}$) is determined as the ratio of the absolute difference between the estimated coefficient and true coefficient, divided by the true coefficient. Tables 3 and 4 shows the maximum values for both types of errors across all six evaluated libraries.

Table 3. presents the sensitivity, specificity, accuracy, and maximum error for the coefficients identified by SINDy-PI for the IDM evaluated at each size of library $\boldsymbol{\Theta}_N$. There are 25 terms required to represent the IDM that SINDy-PI identified in each evaluation for a sensitivity of 100$\%$. Additional terms identified by SINDy-PI are considered false positives.

Results from SINDy-PI for IDM
Evaluated Library $\boldsymbol{\Theta}_N$	$\boldsymbol{\Theta}_0$	$\boldsymbol{\Theta}_1$	$\boldsymbol{\Theta}_2$	$\boldsymbol{\Theta}_3$	$\boldsymbol{\Theta}_4$	$\boldsymbol{\Theta}_5$	$\boldsymbol{\Theta}_6$
No. of terms in $\boldsymbol{\Theta}_N$	38	38	48	72	104	142	180
Sensitivity ($\%$):	100	100	100	100	100	100	100
Specificity ($\%$):	76.92	76.92	86.96	100	92.41	64.10	70.97
Accuracy ($\%$):	92.11	92.11	93.75	100	94.23	70.42	75.00
Max ($\mathrm{TP}_{e}$) ($\%$):	4.904	7.753	7.556	7.511	4.820	4.820	4.820
Max ($\mathrm{FP}_{e}$):	$3\times 10^{-15}$	$2\times 10^{-13}$	$3\times 10^{-15}$	n/a	$1\times 10^{-14}$	$4\times 10^{-10}$	$9\times 10^{-11}$

Table 4. presents the sensitivity, specificity, accuracy, and maximum error for the coefficients identified by SINDy-PI for the OVM evaluated at each size of library $\boldsymbol{\Theta}_N$. There are 18 terms required for the OVM that SINDy-PI identified in each evaluation for a sensitivity of 100$\%$. Additional terms identified by SINDy-PI are considered false positives.

Results from SINDy-PI for OVM
Evaluated Library $\boldsymbol{\Theta}_N$	$\boldsymbol{\Theta}_0$	$\boldsymbol{\Theta}_1$	$\boldsymbol{\Theta}_2$	$\boldsymbol{\Theta}_3$	$\boldsymbol{\Theta}_4$	$\boldsymbol{\Theta}_5$	$\boldsymbol{\Theta}_6$
No. of terms in $\boldsymbol{\Theta}_N$	24	26	34	56	94	136	180
Sensitivity ($\%$):	100	100	100	100	100	100	100
Specificity ($\%$):	100	100	100	100	100	100	100
Accuracy ($\%$):	100	100	100	100	100	100	100
Max ($\mathrm{TP}_{e}$) ($\%$):	0.008	0.007	0.008	0.008	0.012	0.004	0.008
Max ($\mathrm{FP}_{e}$):	n/a	n/a	n/a	n/a	n/a	n/a	n/a

From table 3, we observe that SINDy-PI accurately identified all TP terms for IDM data across all libraries, achieving a sensitivity of 100%. Notably, the library $\boldsymbol{\Theta}_5$ led to the highest number of FP, resulting in a specificity of 64.1% and an overall accuracy of 70.42%. No FP terms are identified when evaluated with $\boldsymbol{\Theta}_3$, resulting in specificity and accuracy of 100%. The estimated coefficients for the TP terms fall within a range of 8% of the actual coefficients. The coefficients for the FP terms are on the order of $1\times 10^{-10}$ or lower, markedly smaller than any coefficients found in the actual set. Consequently, when the model is not known in advance, these can be confidently disregarded. When applied to the OVM, SINDy-PI achieves a perfect accuracy of 100% across all evaluated libraries, as shown in table 4. Furthermore, the error in the TP coefficients is below 0.01%.

These results indicate that SINDy-PI accurately identified the TP terms and their corresponding coefficients for the respective model. The reduced specificity observed in the IDM results is attributed to the ballistic method used in generating data. If the front vehicle is at rest, the ballistic update method to solve IDM (section 2.1) introduces discontinuities. SINDy-PI accommodates these discontinuities by incorporating additional terms. Moreover, we perform additional tests using varying sampling rates with CTE and SINDy-PI. The supplementary analysis produced comparable results, affirming the robustness of our findings.

3.3.2. SINDy-PI analysis of IDM and OVM with noise

Next, we evaluate the robustness of SINDy-PI's performance in the presence of measurement noise, which is common with real-world data, including traffic systems [64]. We add Gaussian noise $\mathcal{N}(0, \sigma_i^2)$ of ten increasing magnitudes where σ_i ranges from $1\times 10^{-10}$ to $1\times 10^{-1}$. Additionally, at each σ_i and library size $\boldsymbol{\Theta}_{N}$, we generate ten independent datasets for the evaluation of SINDY-PI to account for stochastic noise. A model returned by SINDy-PI is considered successful if it meets three criteria: $\mathrm{sensitivity} = 100\%$, $\mathrm{TP}_{e} \unicode{x2A7D} 10\%$, and $\mathrm{FP}_{e} \unicode{x2A7D} 1$. The summary of these results from ten evaluations for a given σ_i and library size $\boldsymbol{\Theta}_{N}$ are presented in figure 3.

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Looking at the results with the IDM data in figure 3(a), we see that the accuracy of SINDy-PI falls sharply with added noise of $\sigma \gtrsim 1\times 10^{-4}$ for smaller libraries of $\boldsymbol{\Theta}_{0}$ to $\boldsymbol{\Theta}_{2}$ which increases to $\sigma \gtrsim 1\times 10^{-6}$ at the largest library of $\boldsymbol{\Theta}_{5}$. A similar trend is observed for OVM data shown in figure 3(b), where SINDy-PI performance deteriorates with both increased noise levels and an increased library size. As the noise level increases, performance declines because the added noise magnitude is comparable to the smallest coefficient present in each model, as outlined in table 2. When the library size is large, this effect is more pronounced at smaller magnitudes of σ. These results are consistent with previous observations on SINDy's performance with noisy data [46, 65].

The simulation data used to assess SINDy's performance is generated using the hyperbolic tangent representation, consistent with the true OV model as visualized in figure B1. We utilize the Pade approximation of the OVM model as a basis for comparison with the SINDy-PI results, which utilizes a polynomial basis library.

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