Research on the Branch Road Traffic Flow Estimation and Main Road Traffic Flow Monitoring Optimization Problem

Abstract

Main roads are usually equipped with traffic flow monitoring devices in the road network to record the traffic flow data of the main roads in real time. Three complex scenarios, i.e., Y-junctions, multi-lane merging, and signalized intersections, are considered in this paper by developing a novel modeling system that leverages only historical main-road data to reconstruct branch-road volumes and identify pivotal time points where instantaneous observations enable robust inference of period-aggregate traffic volumes. Four mathematical models (I–IV) are built using the data given in appendix, with performance quantified via error metrics (RMSE, MAE, MAPE) and stability indices (perturbation sensitivity index, structure similarity score). Finally, the significant traffic flow change points are further identified by the PELT algorithm.

1. Introduction

In the road network, main roads are usually equipped with traffic flow monitoring devices, which can record the traffic flow data of the main roads in real time. When multiple branch roads converge into the main road, since some branch roads lack traffic flow monitoring devices, the traffic flow of each branch road needs to be estimated by combining the traffic flow data from the main road with the history trend information of the branch roads. This will provide data and methodological support for issues such as optimizing the timing of traffic signals, alleviating traffic congestion, and planning road resources.

A novel modeling framework that utilizes only historical main-road data is required to reconstruct branch-road traffic volumes and identify critical time points at which instantaneous observations facilitate robust inference of aggregate traffic volumes over specific time periods. And it must be applicable to several common real-world road scenarios, including Y-junctions, multi-lane merging, and signalized intersections, as well as to cases with noisy sensor data.

Least squares methods facilitate both linear and nonlinear economic forecasting, cross-category product gap analysis, and power system transmission line parameter estimation. The nonlinear variant has proven effective for estimating RC thermal models in building energy analysis. For complex optimization challenges, metaheuristic algorithms demonstrate wide applicability. Genetic algorithms optimize flexible power allocation in hybrid energy storage systems, enhance facial expression recognition via reinforcement learning, and improve tuned mass damper designs for wind turbine towers. Hybrid genetic-simulated annealing approaches minimize spare parts inspection costs, optimize composite laminate orientations, enable unsupervised tight sandstone reservoir classification, and refine least squares support vector machine parameters in breakdown diagnosis. Simulated annealing additionally supports tourism management optimization. Robust principal component analysis (RPCA) addresses critical image processing challenges, including single-image glow/hot-pixel removal and low-quality retinal image enhancement, while revealing methodological insights about outlier detection sensitivity to center-point inaccuracies. Change point detection algorithms analyze temporal dynamics across domains, examining both the impact of sliding window selection on detection accuracy and structural shifts in power grid time-series data. The PELT algorithm specifically enables precise microseismic signal identification. Most recently, hybrid models have performed outstandingly in regards to traffic prediction. Li et al. (2025) proposed a spatiotemporal graph model integrating CTM and transformer (the Cell Transformer, abbreviated as CeT) methods. CeT employs discretized lane segments to emulate the cell transmission model, creating a cell space to forecast vehicle counts across all segments based on historical traffic data [1].

2. Materials and Methods

2.1. Scene Overviews

We consider the roads shown in the Figure 1. Firstly, the Y-shaped road in Figure 1a is investigated at the point where the traffic flow of Branch Road 1 and Branch Road 2 converges simultaneously into Main Road 3. Suppose that only the traffic flow monitoring device A1 is installed on Main Road 3, and the traffic flow information of the main road is recorded every 2 min. The time it takes for the vehicle to travel from the branch road to the main road and reach A1 is negligible. The third column in

Table A1. Traffic flow on the main roads considered in this paper (obtained from the 2025 May Day MCM (Mathematical Contest in Modeling) in China).

Moment	Time	Traffic Flow on Main Road 3 in Figure 1a	Traffic Flow on Main Road 5 in Figure 1b	Traffic Flow on Main Road 4 in Figure 1c	Traffic Flow on Main Road 4 in Figure 1c
07:00	0	7.00	32.5	0.8	20.1178
07:02	1	8.50	34.1	2	21.6463
07:04	2	10.00	35.7	3.2	31.0159
07:06	3	11.50	37.3	4.4	40.5070
07:08	4	13.00	38.9	34.6	41.4589
07:10	5	14.50	40.5	37.3	40.3803
07:12	6	16.00	53.1	40	45.0280
07:14	7	17.50	54.7	42.7	23.5751
07:16	8	19.00	56.3	45.4	27.6601
07:18	9	20.50	57.9	20.625	28.5191
07:20	10	22.00	59.5	30.31	22.8609
07:22	11	23.50	61.1	39.395	54.4147
07:24	12	25.00	62.7	47.82	56.3573
07:26	13	26.50	64.3	81.075	54.6330
07:28	14	28.00	51.9	87.35	59.2022
07:30	15	29.50	53.5	92.785	55.9851
07:32	16	31.00	55.1	97.32	37.2741
07:34	17	32.50	56.7	100.895	43.2759
07:36	18	34.00	71.3	81.15	44.3377
07:38	19	35.50	71.9	83.275	46.9259
07:40	20	37.00	72.5	84.26	74.7459
07:42	21	38.50	73.1	84.045	69.1357
07:44	22	40.00	73.7	109.97	74.9930
07:46	23	41.50	74.3	106.375	75.0219
07:48	24	43.00	74.9	101.4	72.3299
07:50	25	44.50	75.5	101.8	52.8119
07:52	26	46.00	64.5	102.2	53.9451
07:54	27	47.50	64.5	79.2	52.3878
07:56	28	49.00	64.5	80.4	55.7438
07:58	29	50.50	64.5	81.6	81.1195
08:00	30	52.00	64.5	82.8	86.6484
08:02	31	51.50	64.5	108.8	81.2547
08:04	32	51.00	64.5	110.8	83.2154
08:06	33	50.50	64.5	112.8	81.3694
08:08	34	50.00	75.5	110.2	53.7771
08:10	35	49.50	75.5	107.6	59.5877
08:12	36	49.00	75.5	76.2	58.5586
08:14	37	48.50	75.5	71.6	54.5368
08:16	38	48.00	75.5	67	80.8018
08:18	39	47.50	75.9	62.4	71.8756
08:20	40	47.00	76.3	76.8	73.5100
08:22	41	46.50	76.7	72.2	68.9525
08:24	42	46.00	63.1	67.6	67.9241
08:26	43	45.50	63.5	63	44.8616
08:28	44	45.00	63.9	63	36.4950
08:30	45	44.50	64.3	44	35.4181
08:32	46	44.00	78.7	44	31.5647
08:34	47	43.50	79.1	44	71.6705
08:36	48	43.00	79.5	44	68.6749
08:38	49	42.50	79.9	82.15	65.5375
08:40	50	42.00	80.3	82.3	58.2082
08:42	51	41.50	80.7	82.45	60.4376
08:44	52	41.00	81.1	82.6	17.2408
08:46	53	40.50	81.5	82.75	12.3009
08:48	54	40.00	70.9	38.9	10.9520
08:50	55	39.50	71.3	38.05	12.9869
08:52	56	39.00	71.7	37.2	36.8780
08:54	57	38.50	72.1	36.35	31.1231
08:56	58	38.00	72.5	52.5	32.2821
08:58	59	37.50	72.9	51.65	28.5905

in Appendix A provides the traffic flow data on Main Road 3 on a certain morning [6:58, 8:58] (7:00 is the first data recording moment, and 8:58 is the last data recording moment, as shown below).

Secondly, consider the road shown in Figure 1b. The traffic flow of Branch Road 1 and Branch Road 2 converges onto Main Road 5 simultaneously, and the traffic flow of Branch Road 3 and Branch Road 4 converges onto Main Road 5 simultaneously. Only the traffic flow monitoring device A2 is installed on Main Road 5, and the traffic flow information of the main road is recorded once every 2 min. The fourth column in Table A1 in Appendix A provides the traffic flow data on Main Road 5 during the time period of [6:58, 8:58] on a certain morning. Suppose the time it takes for the vehicle to travel from the intersection of Branch Road 1 and Branch Road 2 to A2 is 2 min, and the travel time of the vehicle from Branch Road 3 and Branch Road 4 to A2 is negligible. According to the historical traffic flow observation records, during the time period of [6:58, 8:58], the traffic flow on Branch Road 1 was stable. The traffic volume of Branch Road 2 increased linearly during the time periods of [6:58, 7:48] and [8:14, 8:58], and remained stable during the time period of (7:48, 8:14). The traffic volume of Branch Road 3 first shows a trend of linear growth followed by stabilization. A periodic pattern is shown as the traffic volume on Branch Road 4. Thirdly, in Figure 1c, the traffic flow of Branch Road 1 and Branch Road 2 simultaneously converges onto Main Road 4. Branch Road 3 is a special traffic control road, and the vehicles on Branch Road 3 are controlled by the traffic signal C when passing through the intersection. The red light time for C is set to 8 min, the green light time to 10 min, and the yellow light time is negligible. Only the traffic flow monitoring device A3 is installed on Main Road 4, recording the traffic flow information of the main road every 2 min. The traffic flow data of Main Road 4 on a certain morning [6:58, 8:58] is provided in the fifth column in Table A1 of Appendix A. Suppose the driving time of the vehicle from the intersection of Branch Road 1 and Branch Road 2 to A3 is 2 min, and the driving time of the vehicle from Branch Road 3 to A3 is negligible.

Finally, in the case of a poor network signal, low visibility, large traffic flow, or too fast traffic speed, the traffic flow monitoring device may produce data errors. Considering the road shown in Figure 1c, suppose that the data recorded by device A3 on a certain day exhibits an error. The observed data within the time period of [6:58, 8:58] are shown in the sixth column in Table A1 of Appendix A. The traffic volume of Branch Road 1 exhibited a trend of “no flow → linear growth → stability → linear decrease to no flow”. The traffic volume of Branch Road 2 increased linearly and decreased linearly during the time periods of [6:58, 7:34] and [8:10, 8:58], respectively, and remained stable during the time period of (7:34, 8:10). The red light duration for signal light C is set to 8 min, the green light duration to 10 min, and the yellow light duration is negligible. When C shows a green light, the traffic flow on Branch Road 3 is either stable or shows a linear changing trend. When C shows a red light, the traffic volume on Branch Road 3 is 0. The time when C shows the green light is unknown.

2.2. Model Hypothesis

We propose the following hypothesis:

Assumption 1.

The traffic flow on the main road is the sum of the traffic flow on each branch road, and the traffic flow on each branch road displays a certain regularity (e.g., the morning and evening peaks; the traffic flow distribution is different during the weekday peak hours), and this regularity can be described by a function.

Assumption 2.

The roads are all one-way lanes, and the blue arrows in Figure 1 represent the direction of traffic flow.

Assumption 3.

The “increase/decrease” trend of traffic flow in scenarios 1–4 refers to the trend of “strictly monotonous increase/strictly monotonous decrease”, “stable” refers to a state in which the traffic flow remains constant at a non-negative fixed value, and the functional relationships describing the variations in traffic flow across all branches are continuous functions.

Assumption 4.

For convenience, let 7:00 be $t=0$ and $t\in [0, 59]$ in the functional relationship.

2.3. Model Analysis and Establishment

2.3.1. Mathematical Model I

According to the historical traffic flow observation records, during the period of [6:58, 8:58], the traffic flow on Branch Road 1 showed a linear growth trend, while that on Branch Road 2 exhibited a trend of linear growth first and then a linear decrease. For the empirical traffic flow data in the third column in Table A1 of Appendix A, we implemented a least squares curve fitting model [2,3,4,5] to reconstruct branch road traffic flows. Python 3.9 was utilized for algorithm implementation. Software implementation details include the least squares method coded in Python with NumPy (1.26.3) for matrix operations and SciPy (1.13.1) for optimization. Specifically, linear regression and piecewise linear models were developed for Branch Road 1 and Branch Road 2. The branch fitting formulas are shown in Equations (1) and (2). The resulting fitted curves are shown in Figure 2 and Figure 3, where RMSE is 0.0000, MAE is 0.000, and MAPE is 0%. Visualization of the main road traffic flow reveals an approximately linear trend. Consequently, the linear model for Branch Roads 1 and 2 achieves an RMSE of 0, indicating negligible overfitting. The fitted values of main road traffic flow align closely with observed data, demonstrating high model fidelity.

$$f_{11}\left(t\right)=0.5t+7, 0\le t<60$$

(1)

$$f_{12}(t)=\left\{\begin{array}{c}t, 0\le t<30\\ -t+30, 30\le t<60\end{array}\right.$$

(2)

2.3.2. Mathematical Model II

According to the historical traffic flow observation records, during the time period of [6:58, 8:58], the traffic flow on Branch Road 1 was stable. The traffic volume of Branch Road 2 increases linearly during the time periods of [6:58, 7:48] and [8:14, 8:58], and remains stable during the time period of (7:48, 8:14). The traffic volume of Branch Road 3 shows a trend of linear growth first, followed by stabilization. The traffic volume on Branch Road 4 shows a periodic pattern.

For the dataset in the fourth column in Table A1 of Appendix A, we initialize model parameters using iterative algorithm outputs as estimates. These parameters were refined via nonlinear least squares optimization [5], minimizing the residual sum of squares through iterative refinement. Following problem specifications, distinct functional forms were implemented for each branch: a constant model for Branch 1, piecewise linear models for Branches 2 and 3, and a Fourier series model (to capture periodic variations) for Branch 4. The Levenberg–Marquardt algorithm [6] is used to fit the function model, where the regularization parameter is set 0.01, since the approximate effect is good. For Branch 4, we implemented a second-order Fourier spectral decomposition [7,8] containing fundamental and second harmonic terms, balancing accuracy with computational efficiency. Software implementation details are seen in

Table A2. The error indexes and model stability indexes for each method.

Model	II	III	IV
Method	LS-FSM	SA-NLS	NLS	GA–SA	RPCA
Software implementation details	NumPy pandas SciPy scikit-learn (1.6.1)	NumPy pandas SciPy	NumPy pandas SciPy scikit-learn	NumPy pandas SciPy scikit-learn	NumPy pandas scikit-learn CVXPY (1.6.6)
RMSE	3.6526	11.9175	10.7004	3.6733	5.8200
MAE	3.0606	9.4187	8.0558	2.8329	3.6315
MAPE	5.0600%	32.2188%	20.6800%	7.4200%	10.5100%
PSI	0.0861	0.4094	0.1284	0.0184	0.0200
SSS	99.68%	99.76%	99.44%	99.89%	97.96%
RMSE without time delay compensation	3.8152	12.4331	10.9357	4.6824	7.6431

. The branch fitting formulas are shown in Equations (3)–(6).

$$f_{21}(t)=16.5204$$

(3)

$$f_{22}(t)=\left\{\begin{array}{c}0.4t+12.7760, 0\le t<24\\ 22.3760, 24\le t<36\\ 0.3903t+7.9344, 36\le t<60\end{array}\right.$$

(4)

$$f_{23}(t)=\left\{\begin{array}{c}0.4t+11.5610, 0\le t<32\\ 24.1610, 32\le t<60\end{array}\right.$$

(5)

$$\begin{array}{ll}{f}_{24}(t)=& 7.4599-2.9964\cos ({\displaystyle \frac{2\pi }{27.7}}t)-3.1779\sin ({\displaystyle \frac{2\pi }{27.7}}t)\\ & -5.1064\cos ({\displaystyle \frac{4\pi }{27.7}}t)-6.3322\sin ({\displaystyle \frac{4\pi }{27.7}}t)\end{array}$$

(6)

We obtain the fitted results and shown in Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7, where RMSE is 3.6526, MAE is 3.0606, MAPE is 86.10%, PSI is 0.0861 under a 3% perturbation of the parameter, SSS is 99.68% with time delay compensation, and RMSE is 3.8152 without time delay compensation. Incorporating a 2 min time delay compensation significantly enhances model accuracy, as the resulting temporal patterns of branch road traffic flows demonstrably align with empirical traffic behavior. Residual analysis via the fast Fourier transform (FFT) method revealed no significant spectral peaks exceeding the statistical threshold (49.6105), confirming that the second-order expansion sufficiently captured inherent periodicity without higher-order terms.

Remark 1. 

Perturbation Sensitivity Index:

$$\mathbf{P}\mathbf{S}\mathbf{I}={\displaystyle \frac{1}{\mathbf{n}}}{\sum }_{\mathit{t}}^{\mathit{n}}\left|{\overline{\mathit{Q}}}_4^{\left(\mathit{\delta }\right)}(\mathit{t})-{\overline{\mathit{Q}}}_4(\mathit{t})\right|$$

(7)

We applied ±3% perturbations to the parameters of the branch road flow estimation function, recomputed the prediction error for main road traffic flow, and assessed the stability of the model’s output. Results indicate that the perturbation sensitivity index remains within the range of ±2 units across all tested perturbations, demonstrating robust model performance.

Structure Similarity Score:

$$\mathbf{S}\mathbf{S}\mathbf{S}={\displaystyle \frac{\mathbf{1}}{\mathbf{3}}}{\sum }_{\mathit{i}\mathbf{=}\mathbf{1}}^{\mathbf{3}}{\displaystyle \frac{\mathbf{1}}{\mathbf{59}}}{\sum }_{\mathit{t}\mathbf{=}\mathbf{1}}^{\mathbf{59}}[\mathbf{s}\mathbf{i}\mathbf{g}\mathbf{n}({\overline{\mathit{Q}}}_{\mathit{i}}(\mathit{t}))=\mathbf{s}\mathbf{i}\mathbf{g}\mathbf{n}({\overline{\mathit{Q}}}_{\mathit{i}}^{\left(\mathit{\delta }\right)}(\mathit{t}))]$$

(8)

The sign sequences (reflecting monotonicity properties) of branch functions are compared before and after perturbation, with structural similarity quantified. Experiments demonstrate that a structure similarity score exceeding 95% indicates that the model robustly preserves essential physical relationships in branch flow dynamics under data disturbances.

2.3.3. Mathematical Model III

According to the historical traffic flow observation records, during the time period of [6:58, 8:58], the traffic flow on Branch Road 1 exhibited a trend of “no flow→ increase → decrease → stability → decrease to no flow”. The traffic volume of Branch Road 2 increased linearly and decreased linearly during the time periods of [6:58, 8:10] and [8:34, 8:58], respectively, and remained stable during the time period of (8:10, 8:34). When C shows a green light, the traffic flow on Branch Road 3 either remains stable or shows a linear changing trend, and the first green light begins to light up at 7:06. When C shows a red light, the traffic volume on Branch Road 3 is 0. Based on the fact that hybrid models have recently performed outstandingly in regards to traffic prediction [8,9], this provides us with many ideas. However, these methods rely on multi-source sensors or other circumstances and cannot be fully applied to scenarios where there are no monitoring devices on branch roads.

For the dataset in the fifth column in Table A1 of Appendix A, we implemented constrained least squares estimation [10] to determine parameter vectors by minimizing the residual sum of the squares. Piecewise linear models are subsequently established for Branches 1, 2, and 3, per problem requirements. When initial parameter adjustments failed to satisfy constraints for traffic flow on Branch 1, we integrated simulated annealing with nonlinear least squares (initial temperature: 1; step size: 0.5, adjusted every 50 iterations) to refine the model. Software implementation details are seen in Table A2. The branch fitting formulas are shown in Equations (9)–(11).

$$f_{31}(t)=\left\{\begin{array}{l}0,0\le t<8\hfill \\ 3.3t-26.4,8\le t<16\hfill \\ -0.5t+34.4,16\le t<26\hfill \\ 21.658,26\le t<35\hfill \\ -5t+196.658,35\le t<39\hfill \\ 0,39\le t<60\hfill \end{array}\right.$$

(9)

$$f_{32}(t)=\left\{\begin{array}{l}3.909t+0.8,0\le t<16\hfill \\ 63.346,16\le t<33\hfill \\ -0.884t+92.518,33\le t<60\hfill \end{array}\right.$$

(10)

$$f_{33}(t)=\left\{\begin{array}{l}15.218 ,\text{the green light}\hfill \\ 0 ,\text{the red light}\hfill \end{array}\right.$$

(11)

This hybrid approach fits the functional relationship well, with results presented in Figure 8, Figure 9 and Figure 10. RMSE is 11.9175, MAE is 9.4187, MAPE is 32.2188%, PSI is 0.4094, SSS is 99.76% under a 3% perturbation of the parameter with time delay compensation, and RMSE is 12.4331 without time delay compensation. The modeling error is primarily attributable to traffic flow discontinuities induced by signal control. Incorporating the 18 min signal cycle significantly enhances model accuracy, while the inferred branch road flow patterns align with empirical traffic characteristics.

2.3.4. Mathematical Model IV

According to the historical traffic flow observation records, in Figure 1c, the traffic volume of Branch Road 2 increased linearly and decreased linearly during the time periods of [6:58, 7:34] and [8:10, 8:58], respectively, and remained stable during the time period of (7:34, 8:10). The red light duration for signal light C is set to 8 min, the green light duration to 10 min, and the yellow light duration is negligible. When C shows a green light, the traffic flow on Branch Road 3 is either stable or shows a linear changing trend. When C shows a red light, the traffic volume on Branch Road 3 is 0. The time when C shows the green light is unknown.

We employ three parameter optimization methods: constrained least squares, a hybrid genetic algorithm–simulated annealing (GA–SA) approach [11,12,13,14,15,16,17,18], and robust principal component analysis (RPCA) [19,20,21]. The hybrid GA–SA model was ultimately selected to characterize branch road traffic flows using the dataset in the sixth column in Table A1 of Appendix A. This hybrid algorithm enhances the classical simulated annealing framework by integrating the population-based search mechanism of genetic algorithms. The resulting optimizer combines probabilistic state transitions with global population search capabilities. In traffic monitoring applications, main road sensors may generate sparse yet high-amplitude errors due to network delays or equipment failures, resulting in sudden data drops. While the GA–SA hybrid fits functional traffic models, RPCA effectively isolates such anomalies while preserving the underlying low-rank traffic matrix structure.

In the GA–SA model, we set the population size to 15, the uniformly sampled mutation rate from 0.5 to 1.0, the crossover rate at 0.7, the initial temperature to 5230, and the restart temperature ratio to $2\times {10}^{-5}$, after adjusting the parameters. In RPCA model, we set the regularization coefficient to ${\displaystyle \frac{\mathbf{1}}{\sqrt{\mathbf{10}}}}$. Software implementation details can be seen in Table A2 The branch fitting formulas are shown in Equations (12)–(14). The GA–SA model displays the best effect, and the result is shown in Figure 11, Figure 12 and Figure 13. Meanwhile the green light first commenced at 7:06 (t = 3). In the GA–SA model, RMSE is 13.6733, MAE is 2.8329, MAPE is 7.42%, PSI is 0.0184, SSS is 99.89% under a 3% perturbation of the parameter with time delay compensation, and RMSE is 4.6824 without time delay compensation. The error indexes and model stability indexes for NLS and RPCA are shown in Table A2 of Appendix B.

$$f_{41}(t)=\left\{\begin{array}{l}0,0\le t<13\hfill \\ 1.21t-15.972,13\le t<27\hfill \\ 16.97,27\le t<48\hfill \\ -4.2425t+220.61,48\le t<52\hfill \\ 0,52\le t<60\hfill \end{array}\right.$$

(12)

$$f_{42}(t)=\left\{\begin{array}{l}1.4482t+14.18,0\le t<17\hfill \\ 38.68,17\le t<35\hfill \\ -1.47t+90.13,35\le t<60\hfill \end{array}\right.$$

(13)

$$f_{43}(t)=\left\{\begin{array}{l}25.25,\text{the green light}\hfill \\ 0 ,\text{the red light}\hfill \end{array}\right.$$

(14)

3. Results

According to the expression of Mathematical Model I to Mathematical Model IV, we obtain the traffic flow values on each branch road at moment 7:30 and 8:30, respectively (see

Table 1. Numerical values of branch traffic flow in Mathematical Model I.

Moment	Branch Road 1	Branch Road 2
7:30	14.5	15
8:30	29.5	7.5

,

Table 2. Numerical values of branch traffic flow in Mathematical Model II.

Moment	Branch Road 1	Branch Road 2	Branch Road 3	Branch Road 4
7:30	16.5204	18.776	17.561	3.58
8:30	16.5204	25.4979	24.161	5.54

,

Table 3. Numerical values of branch traffic flow in Mathematical Model III.

Moment	Branch Road 1	Branch Road 2	Branch Road 3
7:30	23.1	59.435	15.218
8:30	0	52.738	0

and

Table 4. Numerical values of branch traffic flow in Mathematical Model IV.

Moment	Branch Road 1	Branch Road 2	Branch Road 3
7:30	1.68	36.6185	24.67
8:30	16.34	39.61	0

).

Furthermore, to determine function parameters, sufficient data points surrounding these key time points are required. Therefore, change point detection [22,23] is employed to identify optimal segmentation intervals. Specifically, the PELT algorithm [24] is applied to the main road traffic flow time series to detect significant change points.

Let the main road traffic sequence be $y_t$ and find the set of change points ${\rm T}=\left\{{{\rm T}}_1,{{\rm T}}_2,\dots ,{{\rm T}}_k\right\}$, such that the residuals of the segmented linear model are minimized, as follows:

$$\mathrm{m}\mathrm{i}\mathrm{n}{\sum }_{k=0}^K\mathcal{F}\left(y_{{{\rm T}}_k:{{\rm T}}_{k+1}}\right)+\beta K,$$

(15)

where $\mathcal{F}$ is the segmented cost function (e.g., mean square error), and $\beta$ is the penalty coefficient, which controls the number of change points. Figure 14 presents the results for Model II, identifying change points at times t = 0, 4, 19, 25, 37, 40 and observation times 07:00, 07:08, 07:38, 07:50, 08:14, 08:20. The penalty coefficient is set to 3, resulting in the lowest MAE of 3.38. For Mathematical Model III, which is presented in Figure 15, the detected change points occur at t = 3, 8, 12, 17, 21, 26, 30, 35, 39, 44, 48, 53, 57, with observations recorded at 07:06, 07:16, 07:24, 07:34, 07:42, 07:52, 08:00, 08:10, 08:18, 08:28, 08:36, 08:46, 08:54. The penalty coefficient is set to 8, resulting in the lowest MAE of 8.29. Model IV, which is presented in Figure 16, yields change points at t = 3, 8, 12, 17, 19, 21, 26, 30, 35, 39, 44, 48, 49, 53, 57 and observations at 07:06, 07:16, 07:24, 07:34, 07:38, 07:42, 07:52, 08:00, 08:10, 08:18, 08:28, 08:36, 08:38, 08:46, 08:54. The penalty coefficient is set to 3, resulting in the lowest MAE of 10.27. Software implementation details include the PELT algorithm coded in Python with NumPy, ruptures (1.1.9) for signal segmentation and pandas (2.3.0) for data processing.

4. Discussion

Collectively, Models I-IV underpin the development of comprehensive traffic monitoring systems. Crucially, the robust optimization framework derived from Model IV enables practical implementation within smart city infrastructures by

(1)

Rapid identification of incident locations;

(2)

Dynamic generation of optimal rerouting strategies.

5. Conclusions

This paper has four main limitations:

(1)

The strict monotonicity of traffic flow assumed in Assumption 3 may not hold in actual road conditions;

(2)

The fixed signal timing in Model III/IV reduces adaptability to adaptive signal systems;

(3)

The computational complexity of the GA–SA hybrid limits real-time deployment (>2 s/iteration);

(4)

The validation datasets lack extreme weather conditions. Future work will integrate reinforcement learning for dynamic signal optimization.

Nevertheless, The model proposed in this study exhibits high adaptability, featuring multi-class configurations tailored for diverse road structures and complex constraints. Algorithmically, this work introduces three key innovations:

(1)

Periodic fluctuation modeling through the Fourier spectral method.

(2)

Constrained optimization for enforcing continuity in piecewise functions.

(3)

Hybrid global optimization combining genetic algorithms with simulated annealing.

These advancements collectively enhance the model’s explanatory capability and generalization performance. Furthermore, the variable change-point detection algorithm quantitatively identifies critical monitoring moments, providing a cost-efficient framework for practical traffic data collection.