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# Challenges in calibrating multimodal network macroscopic fundamental diagrams: a review and definition of data fusion pipeline

Nandan Maiti, Manon Seppecher, Ludovic Leclercq

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Nandan Maiti, Manon Seppecher, Ludovic Leclercq. Challenges in calibrating multimodal network macroscopic fundamental diagrams: a review and definition of data fusion pipeline. European Transport Research Review, 2025, 17 (1), pp.51. 10.1186/s12544-025-00750-9. http://hal-05395494

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# Challenges in calibrating multimodal network macroscopic fundamental diagrams: a review and definition of data fusion pipeline

Nandan Maiti¹* , Manon Seppecher¹ and Ludovic Leclercq¹

**Abstract**

This study presents a comprehensive evaluation of the real challenges related to the calibration of Network Macroscopic Fundamental Diagrams (NMFDs), with a focus on two key aspects: observability and stability. Observability addresses the estimation of NMFDs using multimodal traffic data, including loop detectors (LDD), floating car data (FCD), and public transport data (PTD), and examines how data quality, spatial coverage, and aggregation intervals influence NMFD characteristics. Stability explores the temporal consistency of NMFDs through longitudinal analysis, investigating their sensitivity to loading and unloading phases, weekday versus weekend dynamics, and recurring demand patterns. Using case studies in Athens and Lyon demonstrates that fusing data sources improves the accuracy of multimodal NMFD estimation, while separating network loading/unloading phases and different days enables reliable NMFD estimation. Building on these insights, we propose a structured NMFD calibration pipeline that integrates bias correction, data fusion, and temporal clustering. This framework supports reproducible, data-driven NMFD estimation and offers practical guidance for urban traffic monitoring and multimodal network management.

**Keywords** Network macroscopic fundamental diagrams, Data fusion, Longitudinal analysis, Network traffic demand, Loop detectors, Floating cars, GTFS

# 1 Introduction

The Network Macroscopic Fundamental Diagrams (NMFDs) represent a pivotal concept in traffic management and urban planning, providing an aggregate view of traffic dynamics over a network. NMFDs characterize the relationship between network-wide traffic variables, such as accumulation, and trip completion rates, enabling a comprehensive understanding of traffic flow and congestion patterns. The production and speed NMFDs [1, 2–6] exhibit unimodal [7–14] or multimodal forms [15–26],

each offering unique insights into different traffic conditions and behaviors. Empirical estimation of NMFDs relies on diverse data sources, including loop detector data (LDD), floating car data (FCD) or GPS data, public transport data (PTD) such as General Transit Feed Specification (GTFS) data, which often requires sophisticated integration of these heterogeneous inputs [12, 23, 27]. The estimation and calibration of NMFDs involve several key steps: observability, homogeneity, scaling, and stability. Observability refers to the estimation of NMFDs using various traffic data sources, often through data fusion techniques to integrate heterogeneous datasets for improved multimodal coverage [23, 28, 29]. Homogeneity involves partitioning the network into smaller, spatially consistent regions to improve estimation accuracy, using

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<footnote>*Correspondence:
Nandan Maiti
nandan.maiti@univ-eiffel.fr
¹University Eiffel, ENTPE, LICIT-ECO7, Bron, Lyon 69500, France</footnote>
criteria such as network topology [30], traffic features [31–33], and control strategies [34]. Scaling addresses the extrapolation of NMFDs from instrumented sub-networks to the full urban network, enabling broader application of limited data [12, 14, 23, 35–37]. Finally, stability focuses on the temporal consistency of NMFDs, with longitudinal clustering used to identify recurrent traffic patterns over time, ensuring more robust and temporally stable calibration [38]. However, empirical estimation of NMFDs poses several challenges, such as variability in data sources and their spatio-temporal coverage, the need for extensive data preprocessing, and the inherent complexity of traffic networks. This paper reviews the challenges of calibration of multimodal NMFD from empirical data sources. While homogeneity (partitioning) and scaling are critical aspects of NMFD estimation, a comprehensive review of this topic is beyond the scope of this paper due to the extensive literature. The first objective is to evaluate observability, existing NMFD calibration methods through heterogeneous data fusion, and to assess the effectiveness of various data source combinations, along with the challenges they present. Next, stability, NMFDs are not universally applicable across all traffic networks over time; rather, they exhibit varied shapes and features depending on the demand patterns of network traffic. This variability underscores their precision in capturing multimodal traffic interactions [6, 38]. The demand profile of a network can be broadly classified into loading and unloading periods, with traffic network dynamics differing significantly between these phases [4,5]. Empirical studies have highlighted the significance of these phases. For instance, [39] found a hysteresis pattern in NMFD by distinguishing the loading and unloading periods within their dataset. Further, [40] introduced a dynamic time-warping-based approach to identify similar NMFD patterns across different days, emphasizing the temporal variations inherent in loading and unloading periods. Therefore, critical questions arise regarding the calibration of NMFDs: How should NMFDs be calibrated to accurately reflect the dynamic nature of traffic networks? When is it essential to perform this calibration to ensure the reliability and accuracy of the NMFDs? How frequently should we update the NMFD parameters to maintain good accuracy? Furthermore, this paper will explore the current challenges related to longitudinal analysis and the identification of loading and unloading periods for representative NMFDs.

## 1.1 Objectives and contribution

The primary objective of this paper is to advance the empirical calibration of NMFDs by addressing key challenges related to observability and stability. Observability concerns the ability to estimate NMFDs using various traffic data sources with differing coverage, resolution,

and aggregation characteristics. Stability focuses on the temporal consistency of NMFDs, including their sensitivity to daily demand patterns and loading-unloading dynamics. Through a structured review, empirical case studies, and a proposed calibration framework, this paper seeks to provide practical guidance and methodological tools for reliable NMFD calibration in multimodal urban networks. The key contributions are as follows:

1. Review: Evaluating the strengths and weaknesses of existing literature on observability and stability in NMFD estimation and calibration using heterogeneous data sources (LDD, FCD, and PTD).

2. Case study: This study uses LDD, FCD, and PTD data from Athens, Greece, and LDD from Lyon, France, to highlight practical challenges and propose new research directions for calibrating unimodal and bimodal NMFDs.

3. NMFDs calibration pipeline: Development of a step-by-step NMFD calibration pipeline, designed to improve the reliability of unimodal and bimodal NMFDs by capturing the dynamic nature of multimodal traffic networks and adapting to available data.

The paper is structured as follows: Sect. 2 reviews relevant literature, identifying gaps in observability and stability in NMFD calibration. Section 3 outlines methodological recommendations. Section 4 presents the empirical case studies. Section 5 introduces the proposed NMFD calibration pipeline. Section 6 concludes and details data availability.

## 2 Existing studies

To structure the literature review and clearly identify existing gaps, we divide the discussion into two key themes: observability and stability.

### 2.1 Observability

Observability refers to the ability to estimate NMFDs using traffic data with adequate spatial and temporal resolution. Common sources, LDD, FCD, and PTD, each have limitations. LDD suffers from uneven spatial distribution, link selection bias, and sensor placement near intersections, leading to errors in flow and speed estimation [12, 41]. Since LDD speed readings are instantaneous, they are less reliable than FCD for network-level speed estimation [41, 42]. However, FCD requires accurate penetration rate estimation and robust fusion methods to integrate with LDD [43]. Both LDD and FCD are suitable for unimodal NMFDs but lack mode specificity [44]. UAVs can capture mode-specific dynamics but are costly and spatially limited [24, 45]. Alternatively, PTD, often sourced from GTFS data or GPS-equipped vehicles,
supports bimodal NMFD estimation by distinguishing public transport from other traffic [23, 25, 46]. While PTD offers reliable bus accumulation, its speed accuracy is limited [24]. Thus, incorporating various combinations of data sources is essential for more accurate unimodal or bimodal NMFD estimations. Applying fusion techniques to different data sets is challenging due to the heterogeneity in spatio-temporal coverage, penetration rates, and availability of data sources. Additionally, missing information from one source can affect the overall estimation process. This manuscript discusses existing fusion methods that combine different data sources, focusing on the most widely available, i.e., LDD, FCD, and PTD, and explores the associated challenges.

### 2.1.1 Fusion of LDD and FCD (LDD+FCD)

Although the existing studies [3, 5, 10] started using FCD with LDD to estimate the NMFD with pivotal goal of enriching LDD estimation, [12, 47, 48] adopted FCD implicitly for network speed and LDD for flow estimation. [49] proposed for the first time a fusion algorithm for estimating network average flow ($\hat{q}$) and density ($\hat{k}$) using LDD-derived flow ($\hat{q}_{LDD}$) and density ($\hat{k}_{LDD}$) with FCD-derived flow ($\hat{q}_{FCD}$) and density ($\hat{k}_{FCD}$), considering probe penetration rate ($\rho$) and the number of links with detector to link ratio ($\phi$). The linear combination of $\phi$, and non-calibrated $\rho$ as a weight parameter for variables estimated from LDD and FCD requires sufficient data for calibration (see (1a)).

$$ \hat{q} = \frac{\phi \hat{q}_{LDD} + \sqrt{\rho(1-\phi)} \hat{q}_{FCD}}{\phi + \sqrt{\rho(1-\phi)}} \quad (1a) $$

$$ \hat{k} = \frac{\phi \hat{k}_{LDD} + \sqrt{\rho(1-\phi)} \hat{k}_{FCD}}{\phi + \sqrt{\rho(1-\phi)}} \quad (1b) $$

The underlying assumption in that approach is straightforward: if a loop detector is present at a location, the flow is taken solely from the LDD ($\hat{q}_{LDD}$); if no loop detector is available, the flow is estimated from the FCD ($\hat{q}_{FCD}$). Consequently, when $\rho = 1$, meaning that all vehicles are observed through FCD, there is no need to rely on LDD measurements for that network, and the value of $\phi$ becomes irrelevant. Conversely, when $\phi = 1$, indicating full LDD coverage, Eq.(1a) does not require the use of FCD-derived flow. It could be argued that setting $\phi = 1$ or $\rho = 1$ might aid in estimating the ground truth NMFD. However, the former is linked with the challenges posed by LDD, while the latter could be prohibitively expensive. However, the accuracy of the fusion method should be evaluated against the ground truth NMFD, though estimating this ground truth is impractical. Du et al. [50] proposed a method to estimate $\rho$ using k-means 3-cluster

analysis. This involves clustering probes, aggregating those passing detectors by OD pairs into max, min, and median classes, and then estimating the harmonic mean $\rho$ by matching these clusters. While the systematic approach enhances the accuracy of $\rho$ estimation, it is highly sensitive to outliers, and the assumption of three clusters may impact results in other scenarios. Later, [48] addressed the issue of probe penetration rate by introducing a new method of probe flow estimation as shown in (2). Where $H_i$ is the average number of probes during T observed time on link i with length $l_i$, and the average speed of FCD in link i is $v_i$.

$$ \hat{q}_{FCD} = \frac{\sum max(H_i l_i, v_i T)}{T \sum l_i} \quad (2) $$

The findings revealed that adding more probe vehicles has a minimal impact on estimation accuracy unless detector density is also improved. In simpler terms, this method struggles to provide reliable NMFD when loop detector coverage is sparse. Saffari et al. [43] proposed a method for NMFD estimation using only probe vehicle trajectories, taking into account the spatiotemporal variation of $\rho$. However, this method can only produce accurate results if the last vehicle in all queues at a signal is a probe vehicle, leading to an inherent overestimation of the penetration rate. The study results indicated that with an average $\rho < 0.25$, the method fails to estimate the NMFD accurately. Paipuri et al. [27] compared time-dependent origin-destination (OD) specific penetration and origin-specific penetration and demonstrated the evolution of the average penetration rate over time for both cases. The study showed that time-dependent OD-specific penetration rates identified a prominent hysteresis loop in NMFD compared to the time-average constant penetration rate. Also, they suggested that time aggregation should be done based on a representative homogeneous spatial penetration rate. Lin et al. [51] employed a Back Propagation Neural Network (BPNN) fusion model to integrate LDD and FCD and compared the accuracy with an Adaptive Weighted Averaging (AWA) fusion method. The results indicated that the BPNN fusion model delivers more precise estimations than the AWA fusion model. The main challenge associated with these methods is the estimation of the weighting factor and the initial NMFD parameters for the whole network [52]. To tackle the issue of detector and probe vehicle placement within a network, [37] proposed a principal component analysis (PCA) model designed to pinpoint the critical links in the network. Subsequently, [28] applied a Bayesian data fusion method, combining PCA-reconstructed LDD [37] with upscaled FCD from critical links. While effective, these studies did not specify the optimal number of critical links required. PCA-based models assume
linear relationships, limiting their ability to capture non-linear patterns, while Bayesian models depend heavily on the choice of prior data. Improved results can be achieved by labeling data, e.g., by loading/unloading phases, weekdays vs. weekends, or special events, before applying PCA and Bayesian fusion, enabling more context-aware calibration.

### 2.1.2 Fusion of LDD and PTD (LDD+PTD)

Integrating PTD and LDD data has emerged as a promising approach for investigating interactions between buses and passenger cars sharing the same roadway. This approach leverages a bi-modal NMFD framework to uncover the underlying dynamics of these interactions. An important question remains regarding the potential for PTD data to mitigate biases inherent in LDD data and vice versa. [18] is the first to employ empirical PTD in the formulation of a three-dimensional vehicle Macroscopic Fundamental Diagram (3D-vNMFD) that correlates the accumulation of cars and buses. The public transport (PT) lanes can either be shared with car lanes or dedicated solely to buses. Consequently, the interactions between PT and cars become particularly significant in shared-lane scenarios. Most of the existing studies [18, 20, 22, 23] modeled PT speed as a function of car speed in a homogeneous network, and car speed as a function of linear relationships among PT density ($k_{PT}$) and car density ($k_c$) (as shown in (3a)).

$$ \hat{v}_{PT} = \alpha v_c + \beta \quad (3a) $$

$$ \hat{v}_c = f(k_{PT}, k_c) \quad (3b) $$

However, a significant drawback of this method is that the estimation of PT speed is dependent on its own density. [21] addressed this by proposing a data fusion method to estimate car speed from PTD data without needing $k_c$. Although [21] provided a robust logical derivation of car speed based on PT characteristics, such as the fraction of cars affected by bus operations, the fraction of dwell time experienced by cars, and the impact of the bus network on car traffic, the assumption that PT and cars travel at the same speed in free-flow conditions tends to overestimate PT speed. The car speed can be estimated using cruising speed ($v_{PT}^c$) and free flow speed of PT ($v_{PT}^f$), as shown in (4).

$$ \hat{v}_c = f'(v_{PT}^c, v_{PT}^f) \quad (4) $$

### 2.1.3 Fusion of FCD and PTD (FCD+PTD)

Most studies focus on estimating network traffic variables by combining LDD with either FCD or PTD. FCD accuracy depends on trajectory density and spatial coverage, while PTD offers only approximate car speeds and

is most effective in mixed-traffic lanes. This raises key questions about the relative value of FCD and PTD, and whether integrating both yields more reliable NMFD estimations than relying on a single source. [21] proposed a data fusion algorithm to integrate FCD and PTD to enhance the estimation of average network density for cars and public transport. The algorithm formulates four distinct fusion functions based on the availability of FCD and PTD, incorporating the information about PT stop locations (whether in bus bays or not) within a sub-network at a given time. These scenarios include: (a) all links have FCD, but no PTD is present; (b) no FCD is available, but PTD is available on a few links; (c) some links have both FCD and PTD; and (d) no links have either FCD or PTD. Whereas (a) to (c) used weights from FCD and PTD penetration rates, (d) can rely on a short-term prediction algorithm. [22] utilized PTD and enriched FCD for bimodal NMFD estimation in Shenzhen, China. The study estimated density and flow by using the average traveling speed and vehicle count on a link, which violates Edie's principle [53]. Additionally, the average speed-based method is not applicable in scenarios where FCD is unavailable for every link.

### 2.1.4 Fusion of LDD, FCD, and PTD (LDD+FCD+PTD)

In most cases, macroscopic traffic flow variables are estimated using LDD combined with FCD for unimodal [3, 5, 10, 12, 13, 36, 47–50, 54], and LDD combined with PTD for bimodal NMFD estimation. This integration enhances prediction accuracy. Thus, incomplete or low penetration of FCD and PTD makes estimations biased and inaccurate. Therefore, data fusion methods are necessary to integrate various well-known data sources, such as LDD, FCD, and PTD, to address the limitations inherent in each individual data source. [21] first proposed a Lagrangian method of data fusion for PTD and FCD, combining it with the Eulerian method of LDD for the estimation of 3D-NMFD. Since then, ongoing efforts have aimed to integrate all available data sources to address biases and systematic errors, thereby enhancing the accuracy of the estimations. Later, [23] combined LDD, FCD, and PTD data to work with incomplete datasets for 3D-NMFD estimation. However, they did not propose any advanced fusion methods to address the issue of missing data.

### 2.1.5 Miscellaneous data

As alternatives to traditional data sources, unmanned aerial vehicles (UAVs), mobile probe data, location-based services (LBS), and Google traffic data can be utilized when it is difficult to collect conventional traffic data. UAVs provide high-resolution, class-specific vehicle trajectories that can be used to model bimodal NMFDs, considering not only PTs and cars but also other modes
such as two-wheelers (TWs) and medium/light commer-
cial vehicles, etc [55–58]. The primary advantage of UAV
data over FCD, LBS, or PTD is its comprehensive network
coverage and the high quality of individual trajectories,
enabling the tracking of nearly every vehicle. [26] was the
first to use such data from the Athens pNEUMA project
[45]. Unlike existing studies, [26] proposed a mode-spe-
cific mean speed function in terms of the accumulation of
all modes. This approach introduced alternative explana-
tory variables, such as the accumulation and stop fraction
of each mode, to better capture mode-specific vehicle
dynamics. LBS provides the positioning information of
smartphone devices using either the Global Positioning
System (GPS) or the WiFi Positioning System (WPS),
which is similar to FCD. [59] integrated mobile vehicle
probe data with the well-defined MFD to estimate aver-
age vehicle density in real-time. However, this approach
was not used to estimate the NMFD. The main challenges
with LBS are segmenting individual records into trips
[27] mode detection [60], pre-processing, etc. Later, [27,
61] proposed a trip enrichment scheme that addresses
the challenges of trip OD identification using LBS. [62]
argued that OD matrices could not adequately represent
non-regular users and proposed using total travel dis-
tances based on OD pairs to represent the entire group
of travelers more effectively. [63] used raw mobile data
of 14.4 million unique devices for 21 days to extract trip
information, e.g., trip length, OD pair, etc, to analyze the
performance of the proposed Physics-Informed Machine
Learning with Generalized Bathtub Model (PIML-GBM)
model. However, a robust algorithm is still needed to
estimate mode-specific variables accurately using the
LBS dataset. [64] proposed a country-wide NMFD for the
Netherlands and a location-specific NMFD for Amster-
dam using Google Maps speed data. They introduced
the concept of relative inhomogeneity based on network
density and concluded that network inhomogeneity and
network density are linearly related. Additionally, [64]
identified a unique, well-defined MFD for both loading
and unloading periods, which requires further investiga-
tion at different temporal and spatial aggregation levels.
Existing literature lacks a fusion approach using UAV,
LBS, and Google traffic data for improved network vari-
able estimation. Limited network coverage also restricts
the representativeness of UAV-based data of the overall
network dynamics. Furthermore, previous studies often
focus on short-term analyses, limiting their relevance to
NMFD estimation. Future research should explore fusion
techniques or machine learning (ML) algorithms to
extrapolate local UAV data to the entire network. Table 1
summarizes existing literature on data fusion for NMFD
estimation, focusing on data sizes and aggregation times.
Subsequently, we address the need for optimizing these
parameters to accurately estimate representative NMFDs.

2.1.6 Data aggregation intervals

Most of the existing studies use different temporal aggregation as referred to in Table 1. However, it is not clear how the selection of temporal aggregation intervals plays a role in MFD estimation. It may influence model accuracy, sensitivity, and calibration. In empirical studies, the choice of aggregation time is often dictated by the available data frequency [67, 68]. Short temporal intervals, ranging from 20 seconds to 1 minute, are effective in capturing high variability in traffic dynamics but are susceptible to noise, while longer intervals, such as 5–15 minutes, provide more robust estimates for applications like travel time estimation [67]. However, excessive aggregation risks masking critical transient traffic behaviors, while under-aggregation introduces estimation noise [67, 68, 69] highlighted the impact of temporal granularity on MFD calibration. Their analysis showed that fine-grained intervals, such as 1-minute aggregation, are effective for modeling traffic transitions but introduce challenges in calibration due to heightened variability. Conversely, [69] illustrated that increasing aggregation intervals smooths out fluctuations but reduces modeling accuracy under congested conditions. To address this trade-off, [69] proposed a set-valued FD using quantile regressions to balance transient behavior representation and numerical simplicity. This approach has inspired contemporary studies to adopt confidence intervals for fitted MFD functions, similar to quartile regressions [57, 70].

[71] emphasized that the choice of aggregation inter-
vals is often constrained by practical considerations, such
as sensor placement and computational feasibility. Their
findings underline the importance of aggregated data in
mitigating significant errors in density estimation, espe-
cially when data sparsity is a concern. Similarly, [72]
investigated the effects of temporal aggregation on travel
time reliability estimation using sparse probe vehicle
data. They found that coarser intervals (30–60 minutes)
enhance the explanatory power of variance in regres-
sion models but lose sensitivity to short-term variations.
Medium intervals (15–30 minutes) provide an optimal
balance, reducing noise while preserving critical traffic
details. Conversely, excessively fine intervals (e.g., less
than 10 minutes) risk inaccuracies due to data sparsity.

Therefore, temporal aggregation remains a key consid-
eration in MFD estimation, requiring careful trade-offs
between capturing transient traffic behavior, reducing
noise, and maintaining computational feasibility. The
ongoing evolution of methods, such as confidence inter-
val-based and quartile regression approaches, demon-
strates a concerted effort to refine aggregation practices
for more accurate and reliable traffic modeling.
<table><caption>**Table 1** Overview of NMFDs derived from different data fusion</caption><thead><tr><th rowspan="2">Literature</th><th>Data</th><th>Data duration</th><th>Empirical</th><th>Aggregation</th></tr><tr><th>Fusion</th><th>days (d.)/hours (h.)</th><th>study</th><th>(sec.)</th></tr></thead><tbody><tr><td>Geroliminis and Daganzo [3]</td><td>LDD+FCD</td><td>30 d.</td><td>✓</td><td>300</td></tr><tr><td>Geroliminis and Sun [5]</td><td>LDD+FCD</td><td>30 d.</td><td>✓</td><td>300</td></tr><tr><td>Mahmassani et al. [54]</td><td>LDD+FCD</td><td>6 h.</td><td>✓</td><td>300</td></tr><tr><td>Leclercq et al. [12]</td><td>LDD+FCD</td><td>12 h.</td><td>×</td><td>600</td></tr><tr><td>Saberli et al. [13]</td><td>LDD+FCD</td><td>14 h.</td><td>×</td><td>300</td></tr><tr><td>Tsubota et al. [47]</td><td>LDD+FCD</td><td>30 d.</td><td>✓</td><td>300</td></tr><tr><td>Amb"uhl and Menendez [49]</td><td>LDD+FCD</td><td>4 h.</td><td>×</td><td>300</td></tr><tr><td>Ji et al. [65]</td><td>LDD+FCD</td><td>4 d.</td><td>✓</td><td>300</td></tr><tr><td>Du et al. [50]</td><td>LDD+FCD</td><td>3 h.</td><td>×</td><td>900</td></tr><tr><td>Saeedmanesh and Geroliminis [31]</td><td>LDD+FCD</td><td>4 h.</td><td>✓</td><td>120</td></tr><tr><td>Ambuhl et al. [48]</td><td>LDD+FCD</td><td>4 h.</td><td>×</td><td>300</td></tr><tr><td>Shim et al. [36]</td><td>LDD+FCD</td><td>30 d.</td><td>✓</td><td>900</td></tr><tr><td>Mariotte et al. [14]</td><td>LDD+FCD</td><td>28 d.</td><td>✓</td><td>1080</td></tr><tr><td>Mariotte et al. [66]</td><td>LDD+FCD</td><td>20 h.</td><td>×</td><td>600</td></tr><tr><td>Geroliminis et al. [18]</td><td>LDD+PTD</td><td>5 h.</td><td>×</td><td>300</td></tr><tr><td>Ampountolas et al. [19]</td><td>LDD+PTD</td><td>6 h.</td><td>×</td><td>300</td></tr><tr><td>Loder et al. [20]</td><td>LDD+PTD</td><td>5 d.</td><td>✓</td><td>900</td></tr><tr><td>Dakic and Menendez [21]</td><td>LDD+PTD</td><td>4 h.</td><td>×</td><td>900</td></tr><tr><td>Fu et al. [23]</td><td>LDD+PTD</td><td>3 d.</td><td>✓</td><td>600</td></tr><tr><td>Paipuri and Leclercq [24]</td><td>LDD+PTD</td><td>6 h.</td><td>×</td><td>150</td></tr><tr><td>Johari and Keyvan-Ekbatani [25]</td><td>LDD+PTD</td><td>2.7 h.</td><td>×</td><td>1</td></tr><tr><td>Huang et al. [22]</td><td>FCD+PTD</td><td></td><td>✓</td><td>300</td></tr><tr><td>Dakic and Menendez [21]</td><td>LDD+FCD+PTD</td><td>4 h.</td><td>×</td><td>900</td></tr><tr><td>Fu et al. [23]</td><td>LDD+FCD+PTD</td><td>3 d.</td><td>✓</td><td>600</td></tr><tr><td>Paipuri et al. [27]</td><td>FCD+LBS</td><td>60 d.</td><td>✓</td><td>1200</td></tr><tr><td>Gayah and Dixit [59]</td><td>LBS</td><td>3 h.</td><td>×</td><td>3</td></tr><tr><td>Paipuri et al. [61]</td><td>LBS</td><td>60 d.</td><td>✓</td><td>1200</td></tr><tr><td>Seppecher et al. [62]</td><td>LBS</td><td>60 d.</td><td>✓</td><td>1200</td></tr><tr><td>Ka et al. [63]</td><td>LBS</td><td>30 d.</td><td>✓</td><td>1200</td></tr><tr><td>Knoop et al. [64]</td><td>Google</td><td>210 d.</td><td>✓</td><td>300</td></tr><tr><td>Paipuri et al. [26]</td><td>UAV</td><td>5 d.</td><td>✓</td><td>60</td></tr></tbody><tfoot><tr><td colspan="5">X - simulation data</td></tr></tfoot></table>

## 2.2 Stability

The review further examines the temporal stability of NMFDs in homogeneous networks, focusing on determining when NMFDs should be updated to reflect evolving traffic dynamics. We reviewed longitudinal analyses of NMFD characteristics, highlighting methodologies and challenges in identifying network loading and unloading phases. Initially, [6] introduced the concept of loading-unloading NMFDs based on average network occupancy, defining loading and unloading periods through the formation and recovery of hysteresis loops in the NMFDs. Subsequently, [54] advanced the field by proposing dynamic NMFDs, categorizing them based on low, normal, and high demand patterns. Their classification relied on incidental definitions, where low demand was defined as 25% less and high demand as 50% more than normal demand, with normal demand estimated from historical static origin-destination matrices and time-dependent traffic counts. Although this approach

acknowledged the relationship between demand and network loading/unloading periods, it did not provide a justified correlation for using percentage demand variations to define these periods. [73] contributed further by observing different hysteresis loop formations clockwise for production-NMFD and anticlockwise for outflow-NMFD under the same traffic conditions. Their study provided theoretical explanations and simulation results to validate these hysteresis formations. However, the reproducibility of unique NMFDs for identical demand profiles on different days was not addressed. [38] tackled this reproducibility issue by employing a dynamic time-warping-based approach to compare NMFD patterns across different days. They considered the time series of average flow and density, classifying major demand patterns with corresponding NMFDs from one-year data for two cities. Their findings indicated that the saturation flow for unloading NMFDs is slightly lower than for loading. Since previous studies utilized demand profiles
to identify network loading and recovery, this could lead to confusion when categorizing a specific demand as free-flow, as congestion may exhibit the same values. Finally, [34] proposed a novel definition for loading and unloading periods based on network average density and the standard deviation of link densities, an approach also explored by [38]. This method demonstrated distinct hysteresis patterns for both phases of the traffic state, addressing the challenge of similar demand profiles during loading and unloading periods. Based on [34], network loading periods are characterized by an increasing trend in the aggregated network density, and the standard deviation of density across all links, while unloading periods show the reverse trend. However, we argue that the inclusion of the standard deviation term as a necessary condition may be overly restrictive. Since the primary indicator of network loading or unloading is the overall increase or decrease in average network density over time, relying strictly on changes in the standard deviation may lead to the exclusion of valid loading or unloading periods. Therefore, we suggest that while standard deviation may offer additional insight into spatial distribution, it should be considered a supplementary descriptor rather than a defining criterion for identifying loading and unloading phases.

Overall, while existing studies have explored the relationship between demand profiles and network loading/ recovery periods, the identification of these periods and the estimation of NMFDs remain unclear in a network. Future research should aim to refine these definitions and improve the reproducibility of NMFDs across different temporal and spatial contexts for a network. Moreover, a comprehensive longitudinal analysis of daily demand and NMFD patterns is essential for determining consistent patterns in network loading and NMFDs. This extended analysis will facilitate the prediction of NMFDs for specific days of the week.

## 2.3 Summary of the key challenges

The literature review reveals several persisting gaps in current research, particularly in the following areas:

*   **Data Fusion:** Existing methods for integrating heterogeneous data sources like LDD, FCD, and PTD struggle with data quality, missing values, and heuristic calibration. Robust and adaptive fusion frameworks are needed to reliably handle diverse, incomplete, or noisy datasets.

*   **Data Aggregation Time:** The influence of temporal aggregation intervals on NMFD estimation remains unclear. Current studies often select aggregation times arbitrarily, with little analysis of their impact on accuracy and sensitivity. Consequently, systematic guidelines for optimal aggregation are missing.

*   **Longitudinal Analysis:** Inconsistent definitions of loading and unloading periods hinder reproducibility and comparability across studies. Standardized, data-driven longitudinal analysis is needed to reliably capture network dynamics over time.

While NMFDs have been widely studied, a unified calibration framework that systematically addresses key challenges, such as data heterogeneity, limited availability, longitudinal variability, and aggregation effects, is still missing. The following sections offer recommendations to bridge these gaps and demonstrate the proposed NMFD calibration framework's effectiveness through case studies, highlighting its advantages over current methods.

## 3 Methodological recommendations

To enhance the accuracy, robustness, and generalizability of NMFD estimation, this section recommends methodological improvements addressing challenges in data fusion, longitudinal analysis, and data aggregation time. These recommendations are then applied to two empirical datasets in a subsequent case study to demonstrate their practical value in Sect. 4. The following subsections detail these methods and their empirical validation through real-world data analysis.

### 3.1 Data fusion

This section investigates data fusion techniques to estimate unimodal and bimodal NMFDs using different combinations of data sources. Table 2 shows how network macroscopic variables (flow, density, speed) can be estimated by fusing different data sources. The diagonal of Table 2 shows the variables using a unique data source, and the other elements represent the additional benefit of data fusion. For example, in the first row of the first column, we estimate the average network flow ($\hat{q}_{\text{LDD}}$), and the network local speed ($\hat{v}_{\text{LDD}}$) of an equipped network where the detectors were installed only with LDD. Similarly, in the second row of the second column, only FCD are used to access the average FCD speed ($\bar{v}_{\text{FCD}}$) by calculating total travel time ($TTT_{\text{FCD}}$), total travel distance ($TTD_{\text{FCD}}$) with total network length ($L$) and the aggregation time ($\Delta t$). The ($\bar{v}_{\text{FCD}}$), which can be treated as the network average speed ($\hat{v}$) based on the permissible penetration rate ($\rho_{\text{FCD}}$). The FCD penetration can be estimated using the accumulation ($N$) from LDD, and the FCD count ($n_{\text{FCD}}$). Public transport variables, given in the third row in the third column, similarly to FCD, PT flow ($\bar{q}_{\text{PT}}$), density ($\bar{k}_{\text{PT}}$), and the average speed in the network ($\bar{v}_{\text{PT}}$) can be estimated using total travel time ($TTT_{\text{PT}}$), total travel distance ($TTD_{\text{PT}}$) in the network. One could estimate the speed of each individual mode in the Eulerian framework by assuming that the speed
<table><caption>**Table 2** Input data requirements and fusion rules for NMFD calibration. Each data source provides direct measurements and derived variables. Fusion steps combine sources to estimate multimodal network states</caption><thead><tr><th>Data Source</th><th>Directly Available</th><th>Derived Variables</th><th>Used in Fusion</th></tr></thead><tbody><tr><td>LDD</td><td>Counts, average speeds</td><td>q&#770;<sub>LDD</sub> = &sum;<sub>i</sub> q&#770;<sub>i,LDD</sub>l<sub>i</sub>,<br/>v&#770;<sub>LDD</sub> = &sum;<sub>i</sub> v&#770;<sub>i,LDD</sub>/n, k&#770;<sub>LDD</sub> = q&#770;<sub>LDD</sub>/v&#770;<sub>LDD</sub></td><td>Basis for network flow and<br/>accumulation</td></tr><tr><td>FCD</td><td>Vehicle trajectories or average<br/>link speeds</td><td>k&#770;<sub>FCD</sub> = TTT<sub>FCD</sub>/L&Delta;t, q&#770;<sub>FCD</sub> = TTT<sub>FCD</sub>/L&Delta;t, v&#770;<sub>FCD</sub> = q&#770;<sub>FCD</sub>/k&#770;<sub>FCD</sub></td><td>Speed reference, probe<br/>penetration estimation</td></tr><tr><td>PTD (GTFS)</td><td>Continuous vehicle locations<br/>(bus trajectories)</td><td>k&#770;<sub>PT</sub> = TTT<sub>PT</sub>/L&Delta;t, q&#770;<sub>PT</sub> = TTT<sub>PT</sub>/L&Delta;t, v&#770;<sub>PT</sub> = q&#770;<sub>PT</sub>/k&#770;<sub>PT</sub></td><td>Multimodal extension, car/<br/>PT separation</td></tr><tr><td>Fusion Step</td><td>Inputs</td><td>Outputs</td><td>Application</td></tr><tr><td>LDD+FCD</td><td>Counts + trajectories</td><td>q&#770; = q&#770;<sub>FCD</sub>/&rho;<sub>FCD</sub>; &rho;<sub>FCD</sub> = n<sub>FCD</sub>/N or or q&#770; = q&#770;<sub>LDD</sub>,<br/>v&#770; = v&#770;<sub>FCD</sub>/k&#770; = q&#770;/v&#770;</td><td>NMFD calibration</td></tr><tr><td>LDD+PTD</td><td>Counts + PT trajectories</td><td>v&#770;<sub>car</sub> = q&#770;<sub>LDD</sub>-q&#770;<sub>PT</sub>/q&#770;<sub>LDD</sub>-q&#770;<sub>PT</sub>/v&#770;<sub>LDD</sub>, q&#770;<sub>car</sub> = q&#770;<sub>LDD</sub>-q&#770;<sub>PT</sub>, q&#770;<sub>PT</sub>, k&#770;<sub>PT</sub>, v&#770;<sub>PT</sub></td><td>Multimodal NMFD<br/>estimation</td></tr><tr><td>FCD+PTD</td><td>PT &amp; FCD Trajectories</td><td>q&#770; = q&#770;<sub>FCD</sub>/&rho;<sub>FCD</sub>, v&#770; = v&#770;<sub>FCD</sub>, q&#770;<sub>PT</sub>, k&#770;<sub>PT</sub>, v&#770;<sub>PT</sub></td><td>NMFD calibration (if &rho;<sub>FCD</sub><br/>available), PT MFD</td></tr><tr><td>LDD+FCD+PTD</td><td>Counts + PT &amp; FCD Trajectories</td><td>v&#770; = v&#770;<sub>FCD</sub>, q&#770; = q&#770;<sub>LDD</sub>, k&#770; = q&#770;/v&#770;, q&#770;<sub>car</sub>, v&#770;<sub>car</sub>, v&#770;<sub>PT</sub>, q&#770;<sub>PT</sub>, k&#770;<sub>PT</sub></td><td>Full multimodal NMFD cali-<br/>bration pipeline (see Fig. 12)</td></tr></tbody></table>

values in the detector represent the average speed of all modes passing through the detector within a specific time interval. Thus, for LDD and PTD fusion in the first row with the first and third columns, we can estimate the car's speed (assuming cars are only in non-PT mode) by providing flow and speed from PTD and LDD. We can derive the car speed in Eq. (5) from LDD using PTD information by assuming only cars and PT crossed the detector.

$$ 
\begin{aligned}
\hat{v}_{\text{LDD}} &= \frac{\hat{q}_{\text{LDD}}}{\frac{\hat{q}_{\text{cars}}}{\hat{v}_{\text{cars}}} + \frac{\hat{q}_{\text{PT}}}{\hat{v}_{\text{PT}}}}, & \hat{q}_{\text{cars}} &= \hat{q}_{\text{LDD}} - \hat{q}_{\text{PT}} \\
\hat{v}_{\text{cars}} &= \frac{\hat{q}_{\text{cars}}}{\frac{\hat{q}_{\text{LDD}}}{\hat{v}_{\text{LDD}}} - \frac{\hat{q}_{\text{PT}}}{\hat{v}_{\text{PT}}}}
\end{aligned}
\qquad (5) $$

While LDD count is widely recognized for accumulation and flow estimation, it is generally not recommended for speed estimation in a region because it captures the localized speed of the vehicles [12, 23, 41, 42] analyzed the LDD of 28 cities and found that speeds from the LDD are biased and directly dependent on the position with respect to the intersection. Thus, FCD can be used as a reliable source for space-mean-speed estimation at a link or region level. [12] proposed a novel analytical approach to correct the LDD speed using the network flow parameters, such as wave speed, jam density, saturation flow, and free-flow speed with the signal cycle, which can be difficult to estimate for an unfamiliar network. Recently, [41] proposed a data-driven ML method to correct LDD local speeds to spatial mean speeds for NMFD estimation using sample FCD. The detailed model design and validation are presented in [41]; here, we focus on its integration into NMFD calibration. Once we estimate the spatial mean speed from LDD, we can accurately estimate

network traffic variables without requiring expensive FCD. Therefore, we must correct the LDD bias prior to data fusion for multimodal and unimodal network variable estimation, as detailed in Fig. 1.

The existing study on NMFD uses data exclusively from LDD, an equipped network, which is a subset of the whole network. However, the objective of NMFD is to capture the overall network dynamics, which are not adequately represented when relying solely on data from a network subset. Consequently, future research should focus on scaling NMFDs from the equipped network subset to the entire network. [14] proposed a scaling factor based on the ratio of total network length to the equipped network length. However, validating this scaling factor is crucial. Later, [35] proposed a link-hierarchical scaling approach for sparse sensor networks, but validation with synthetic or simulated networks is needed to assess its behavior under higher congestion and diverse network structures. In this context, refining LDD correction methods to estimate an NMFD for the entire network presents a promising avenue for future research.

## 3.2 Aggregation periods

Table 1 presents various data fusion methods employed to estimate NMFDs, highlighting the significant variation in data duration (ranging from 2 hours to 365 days) and aggregation times (from 1 second to 18 minutes). This variability underscores the need for an empirical study to determine the optimal aggregation time and data size required to estimate a representative NMFD accurately. To identify the optimal aggregation period for MFD estimation, start by defining the objective: capturing macroscopic traffic variables (flow, density, speed) while balancing representativeness and smoothness. Representativeness indicates whether the aggregated data
<figure><img src="image_5.png" /><figcaption>Fig. 1 A systematic method for correcting ldd and estimating macroscopic traffic variables using heterogeneous data fusion</figcaption></figure>

captures key patterns, such as peak hours and congestion, while smoothness refers to noise reduction without overly smoothing crucial variations. An in-depth analysis is essential to understand how aggregation time affects hysteresis in NMFDs, as a longer aggregation time can significantly influence the averages of loading and unloading variables. Analyze the available data for temporal variability, periodic patterns, and noise levels. To test the aggregation (Δt) intervals (e.g., 5, 10, 15, 30, 60 minutes) by calculating aggregated flow, density, and speed for each aggregation period as $\bar{q}_{\Delta t}$, $\bar{k}_{\Delta t}$, and $\bar{v}_{\Delta t}$. Evaluate these intervals based on their ability to represent key traffic patterns, reduce noise, and align with statistical properties and ground truth. Use visual inspection, variance analysis, and autocorrelation to compare intervals. Then, consider the application context, shorter intervals for real-time monitoring, and longer intervals for long-term planning. Finally, select the optimal period that best balances accuracy, smoothness, and computational efficiency. A brief overview of the research direction on aggregation time estimation is presented in Fig. 2. The optimal interval $\Delta t^*$ minimizes the total error $E(\Delta t)$, defined as a weighted sum of noise error (variance of aggregated data) and detail error (deviation from original data). By testing intervals (e.g., 5, 10, 15 minutes, etc) and selecting the one with the lowest $E(\Delta t)$, we ensure the aggregated data accurately represents traffic patterns while reducing noise. Alternatively, varying aggregation periods based on network loading and unloading could be considered.

The optimal aggregation interval $\Delta t^*$ minimizes the trade-off between noise reduction and loss of temporal

detail. This can be formulated as an optimization problem as (6):

$$ 
\begin{aligned}
E(\Delta t) &= \alpha E_{\text{noise}}(\Delta t) + (1 - \alpha) E_{\text{detail}}(\Delta t) \\
E_{\text{noise}}(\Delta t) &= \text{Var}(\bar{q}_{\Delta t}, \bar{k}_{\Delta t}, \bar{v}_{\Delta t}) \\
E_{\text{detail}}(\Delta t) &= \frac{1}{N} \sum_{i=1}^{N} |X_i - X_{\Delta t, i}|
\end{aligned}
\quad (6) $$

Here, $\alpha$ is a weight factor to balance noise reduction and detail retention. $\text{Var}(\cdot)$ is the variance of the aggregated traffic variables. $X_i$ is the original available data, and $X_{\Delta t, i}$ is the aggregated data at interval $\Delta t$ for total $N$ data points. The optimal aggregation time ($\Delta t^*$) is the value of $\Delta t$ that minimizes the total error $E(\Delta t)$ as shown in (7):

$$ \Delta t^* = \operatorname*{argmin}_{\Delta t} E(\Delta t) \quad (7) $$

This approach provides a systematic method for determining the optimal aggregation period for macroscopic traffic analysis.

### 3.3 Stability

A comprehensive longitudinal analysis of day-to-day traffic patterns over an extended period is essential to understanding the variability in demand and NMFD patterns within a network. Comparing NMFDs for working days and non-working days could significantly enhance perimeter control strategies and overall network traffic management. Additionally, there is a notable gap in the literature regarding the analysis of NMFDs during
<figure><img src="image_6.png" /><figcaption>**Fig. 2** A systematic method for determining the optimal aggregation period for macroscopic traffic analysis</figcaption></figure>

special events and disruptions, such as floods, closures of one transport mode, road closures, or large gatherings like football matches and political events. Understanding NMFD parameters under these conditions is crucial for effective traffic management during disruptions. Furthermore, the development of a straightforward methodology to identify potential clusters of network traffic demand and their corresponding NMFDs is highly anticipated. Additionally, conducting day-to-day comparisons of NMFDs within clusters (intra-cluster) and between different clusters (inter-cluster) will provide valuable insights into daily variations in NMFDs. Network loading and unloading demand patterns significantly influence

the shape and parameters of the NMFDs. Most studies have focused on examining the direction and size of the hysteresis loops in an NMFD based on daily demand patterns [6, 23, 73,74]. Consequently, there is a need for more empirical data to validate the stability and repeatability of the NMFDs loading and unloading regimes. Additionally, a standardized method for identifying stable loading and unloading curves in NMFDs is highly recommended to facilitate the clear distinction between these curves in the temporal domain. Figure 3 presents a methodological flowchart for identifying network variables related to loading-unloading, weekdays-weekends, and special periods by fusing corrected-LDD and FCD
<figure><img src="image_7.png" /><figcaption>**Fig. 3** A systematic method of longitudinal analysis for determining different calibration periods for estimating NMFDs</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Fig. 4** Study location with loop detectors, public transport stops, and links information</figcaption></figure>

data. We propose analyzing spatiotemporal network demand patterns using dynamic time warping [38] or k-means clustering [55].

## 4 Case studies

We present empirical analyses of both bimodal and uni-modal NMFDs, based on the data fusion summarized in Table 2. In the bimodal analysis, we considered only cars and PTs. To investigate the loading and unloading demand patterns and their corresponding NMFD shape, we utilized a substantial dataset spanning six months of LDD data, as well as four days of FCD and PT data from downtown Athens, Greece. Figure 4(a) presents a map view of the city, and the link-level details can be found in Fig. 4(b). The total network coverage area is approximately (7.3 × 4.8)km². Data available includes

LDD and FCD from September 2022 to February 2023. We extracted PTD from an open-source repository, the GTFS data of Athens Transportation [75] from November 1st to November 4th, 2022. The LDD in the network provides critical data, including total vehicle counts and average speeds. A total of 142 detectors were installed across the network. The FCD average speed was estimated on each link by utilizing the mean speed of these probe vehicles, with data collected at five-minute intervals. The PTD data was given in a GTFS format, which includes the trip ID, stop information, route information, PT line ID, and the distance and travel time between each stop. The LDD and PT stop locations are shown in Fig. 4(b), where mostly LDD are installed on the major links, but PT is distributed over the network randomly.
## 4.1 NMFD estimation using data fusion

This section presents the unimodal and bimodal NMFDs derived from different data fusions, as illustrated in Table 2. We defined the total network length (L) as the sum of the lengths of all links equipped with detectors. The available aggregation times (Δt) for LDD, FCD, and PTD are 60 min, 5 min, and 1 sec, respectively.

Figure 5 illustrates and compares the unimodal and multimodal NMFDs derived from different data fusion methods. Figure 5(a) demonstrates the relationship between speed and total density, where unimodal speed is derived from LDD and FCD, while mode-specific speed is derived from PT and cars. It is evident that the unimodal LDD consistently overestimates speed compared to LDD+FCD due to positional bias in the LDD data, as

most LDD detectors are situated downstream of intersections, capturing higher speeds than the average link speed. In contrast, FCD provides an average link speed. Furthermore, car speeds in the free-flow regime are significantly higher than the average link speed because cars travel faster than slower vehicles in free-flow conditions; however, this difference diminishes at congestion. Figure 5(b) shows the flow-density relationship, with unimodal flow calculated from LDD and mode-specific flow from PTD and cars. The sum of car flow and PT flow represents the network average flow. The car data indicate significantly higher flow rates than PT. Figures 5(c) and 5(d) further detail mode-specific speed and flow against mode-specific density for PT and cars, respectively. Figure 5(c) reveals class-specific NMFDs, highlighting

<figure><img src="image_9.png" /><figcaption>**Fig. 5** Multi-modal NMFDs (a) Unimodal speed-NMFDs were estimated from LDD and FCD, while bimodal speeds were derived from ptd and cars compared to the total density in the network derived from LDD+FCD+PTD (b) unimodal flow-NMFDs were derived from ldd, while bimodal flows were derived from ptd and cars compared to the total density in the network derived from LDD+FCD+PTD (c) mode-specific speed vs mode-specific density for pt and cars from PTD+LDD (d) mode-specific flow vs mode-specific density for pt and cars from PTD+LDD</figcaption></figure>
a clear difference between the free-flow speeds and maximum densities of cars and public transport. The car speed-NMFD appears as a scaled version of the PT speed-NMFD. Figure 5(d) shows that the PT NMFD only observed a small saturation flow of around 180 buses/ hour/lane with a maximum bus density of 16 buses/km/ lane, compared to cars, which achieves around 1100 cars/ hour/lane and 40 cars/km/lane. In the Athens network, PT vehicles represent approximately 7% of the total accumulation. While this is smaller than the car share, PT vehicles operate at lower and more stable speeds, with dwell times and priority signals that can influence traffic dynamics, especially in shared corridors. Including PT in the NMFD ensures a complete multimodal representation and aligns with applications in cities where PT has a larger modal share. Therefore, we demonstrated the bimodal and unimodal NMFDs of an equipped network of LDD by incorporating FCD and PTD.

## 4.2 Data aggregation

To empirically assess the impact of data aggregation intervals on NMFD parameters, we employed the Lyon LDD dataset from [41], which provides network-wide occupancy and vehicle counts. Using this dataset, we estimated flow-based NMFDs by plotting average network flow against occupancy for various time aggregation intervals (Δt = {6, 12, 18, 24, 30, 36, 48, 54, 60} minutes). Figure 7 illustrates the resulting NMFDs across these intervals. We first examine key NMFD parameters, such as free-flow speed, critical occupancy, and maximum flow, then analyze the hysteresis behavior observed during network loading and unloading phases. Figure 6 presents the NMFDs separated by loading and unloading periods for each aggregation level. While NMFDs within the same loading or unloading phase generally align well

across aggregation intervals, a deviation is noticeable in the loading phase for larger intervals (Δt = {48, 54, 60}), as highlighted by the dotted oval in Fig. 6. In this region, the NMFDs exhibit a flatter slope compared to smaller aggregation intervals (Δt ≤ 30), likely due to the loss of data granularity caused by larger averaging windows. Whereas, NMFDs during unloading periods align well across different time aggregations, exhibiting minimal fluctuations even at smaller aggregations.

Figure 7 (right) quantifies the hysteresis loop size, the range of occupancy over which the divergence between loading and unloading branches occurs. We find that as Δt increases, the hysteresis loop becomes smaller, indicating that higher aggregation smooths out transient dynamics and blends the distinct traffic states of loading and unloading into a single mean NMFD. This effect is especially evident when comparing lower aggregation intervals (Δt = {6, 12, 18}), which show a pronounced hysteresis loop, to higher ones (Δt = {48, 54, 60}), where the loop nearly vanishes.

As discussed in Section 3.2, larger aggregation intervals improve smoothness but obscure temporal traffic dynamics. Although Figs. 6 and 7 show that NMFD shapes for different aggregation intervals may appear broadly similar, the loop-size analysis confirms that smaller Δt values better capture the early separation between loading and unloading phases. This distinction is less evident visually but becomes clear when the hysteresis size metric is applied. Using the optimisation framework in Eqs. (6)-(7), we find that Δt = 12 min provides the best trade-off between noise reduction and preservation of hysteresis dynamics, consistent with the more pronounced loops observed for smaller aggregation intervals in Fig. 7.

<figure><img src="image_10.png" /><figcaption>**Fig. 6** Comparing loading and unloading periods NMFD separately for data aggregation periods (Δt = {6, 12, 18, 24, 30, 36, 48, 54, 60} minutes)</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>**Fig. 7** Flow-occupancy NMFDs, visualized as light scatter plots with lines connecting data points, are aggregated over periods of $\Delta t = \{6, 12, 18, 24, 30, 36, 48, 54, 60\}$ minutes. Loading (forward arrows) and unloading (backward arrows) periods are differentiated to highlight NMFD hysteresis loops. The right figure shows the size of the hysteresis with aggregation times</figcaption></figure>

## 4.3 Stability analysis

Next, we will discuss the stability of NMFDs by evaluating the loading and unloading periods of time-series demand patterns using six months of daily demand data. We used the definition of loading and unloading periods, proposed by [34], excluding the $sd_i(t)$ terms, as shown in Eq. (8).

$$ period = \begin{cases} \text{loading: } \tilde{k}(t) \geq \tilde{k}(t-1) \\ \text{unloading: } \tilde{k}(t) \leq \tilde{k}(t-1) \end{cases} \quad (8) $$

Equation (8) presents a straightforward criterion for differentiating between loading and unloading phases. However, short-term fluctuations in network accumulation can lead to brief, misleading sign changes. To mitigate these noise effects, we first smooth the density series using a 3-sample moving median, where the smoothed density, $\tilde{k}(t) = \frac{1}{w}\tilde{k}(t+\tau)$, using a centered window of length $w = 2m+1$ with a recommended window size, $m=1$. Additionally, we require that the sign of the change remains consistent for at least two consecutive aggregation intervals before switching phases. These steps, smoothing and ensuring persistence, help prevent misclassifications caused by isolated fluctuations. Figure 8 illustrates the demand profile from September 2022 to February 2023. It is evident from Fig. 8 that the loading

periods consist of morning and evening peaks, although the exact timing of the loading and unloading periods varies based on the daily demand profile. Therefore, it is essential to identify similar demand patterns during loading and unloading periods to accurately estimate the associated NMFDs. [38] proposed a dynamic-time wrapping-based approach to identify similar patterns of demand, but it became computationally extensive with an increased number of days, yielding a distance matrix of $n \times n$, where $n$ is the number of days. Therefore, we reevaluate the efficiency of simple classification algorithms based on the Euclidean distance classification in our study. We choose the K-means algorithm [76] to identify the demand clusters.

Based on the analysis, the number of clusters is $k=4$ for the $n=180$ days data (using Elbow, Silhouette Method). Figure 9 (a) represents Cluster 1 days with a distinct peak in the morning (around 8–10 AM) and another peak in the evening (around 5–7 PM), indicating typical rush hour patterns. Cluster 4 (Fig. 9 (d)) is similar to Cluster 1 but with less pronounced peaks and lesser overall density, indicating relatively fewer days. Figure 9 (b) shows a more spread-out pattern with less pronounced peaks, possibly indicating steady traffic throughout the day with occasional spikes, whereas Figure 9 (c) demonstrates the same patterns with higher unloading demand.
<figure><img src="image_12.png" /><figcaption>**Fig. 8** Identifying loading and unloading periods in demand patterns using Eq. (8) for six months of demand data</figcaption></figure>

The comparative analysis of flow-density (Fig. 9 (e-h)) and speed-density (Fig. 9 (i-l)) plots across the demand clusters shows that although free flow speeds across the clusters are similar, the observed densities differ. Further analysis revealed that Clusters 1 and 4 represent weekdays and Clusters 2 and 3 represent weekends. Between weekdays and weekends, clusters 1 and 3 demonstrated higher demand during the transition of loading and unloading periods. So, using K-means for longitudinal demand analysis provides a better understanding of similar or repeatable NMFDs within a network.

The other key questions we aim to address are: How does the shape of NMFDs differ during loading and unloading periods across different clusters? How does the shape of NMFDs change for each individual day within a cluster? Is the free-flow portion of NMFDs consistent across all clusters? Lastly, how can we easily identify loading and unloading NMFDs within a network?

To address the first two questions, we analyzed data from November 2022. Figure 10(a) displays the demand profile and the identified clusters. Figures 10(b) and 10(c) reveal that in the critical region, loading traffic states are consistently above unloading states across all clusters, whereas in free-flow conditions, all clusters exhibit a unique trend. This indicates that NMFDs in a region maintain nearly equal relationships during free flow across all days. Examining the day-to-day variations, Figures 10(d) to 10(g) show that the average NMFDs for weekdays follow similar patterns, with overlapping 95%

confidence intervals (CI) indicated by the shaded areas. However, weekends deviate from weekday patterns, featuring a narrower critical density region and an earlier decline in flow for both loading and unloading periods.

When most studies typically start by identifying network loading and unloading periods, then proceed to estimate the NMFDs. In contrast, we investigate the reverse approach by addressing the third question: identifying the loading and unloading curves directly from the NMFDs. This reverse approach allows for the direct estimation of the loading and unloading curve from NMFDs without the need to verify loading-unloading periods. We applied the K-means algorithm to cluster the congested regime into two classes: one representing the upper bound of the NMFDs and the other representing the lower bound. Figure 11 shows flow-density plots for September and October 2022, with two distinct classes indicated by cross and dot markers. Comparing the traffic states from each class by labeling the time series revealed that the upper-bound traffic state class occurs during morning and evening peaks (7 a.m. to 10 a.m. and 2 p.m. to 6 p.m.), while the lower-bound class corresponds to off-peak times. These findings align with the previous results shown in Fig. 10. Therefore, it is possible to estimate the loading and unloading regime in NMFDs by applying the K-means classification directly to the NMFDs, without prior analysis of the network demand profile.
<figure><img src="image_13.png" /><figcaption>**Fig. 9 (a-d)** Demand clusters based on K-means clustering (e-h) Average flow-density plots for the demand clusters 1 to 4, respectively (i-l). Average speed-density plots for the demand clusters 1 to 4, respectively</figcaption></figure>

# 5 NMFD calibration pipeline

The literature review highlights the need for a well-defined and systematic approach for calibrating NMFDs. We propose a calibration pipeline for the NMFD, where calibration refers to the estimation of the parameters of the macroscopic flow-density relationship (e.g., free-flow speed, critical density, capacity, branch asymmetry) so that the model reproduces the fused multimodal traffic states observed in the network. The pipeline ingests heterogeneous traffic data, fuses them into a consistent multimodal state time series, and then fits the NMFD model by minimizing the difference between observed and modelled flows during both loading and unloading phases. Key elements in this process include bias correction, data fusion, and longitudinal analysis. We refer to this structured approach as the NMFD calibration pipeline, as it presents an overall methodology depending on the available data sources, as illustrated in Fig. 12. Historically, we developed an NMFD calibration package in a Python module that includes all steps [77]. This pipeline begins with the systematic errors and bias correction of

the available data. In our study, we correct the potential biases in LDD speed estimation. The correction process of [41] refines the foundational traffic variables network flow, network density, and network speed, making them more representative of actual conditions across multiple modes. Following this correction, multisource data fusion is performed by integrating corrected LDD, FCD, and PTD observations. The NMFD calibration pipeline requires three main data sources. (a) LDD must provide both vehicle counts and average speeds, with spatial coverage on at least 3–5% of network links to ensure representativeness and reliable flow estimates [35]. (b) FCD is required at least on LDD-equipped links to derive accurate space-mean speeds and vehicle counts. But, if not, all LDD links having FCD, we can estimate a correction factor to estimate the spatial average speed from LDD speed using sparse LDD and FCD as per [41]. Broader FCD coverage allows direct estimation of link speeds from probes, bypassing the correction step. (c) PTD continuous vehicle trajectory data (e.g., GPS traces) is preferable for constructing PT flow, but static timetable data may
<figure><img src="image_14.png" /><figcaption>**Fig. 10** (a) Demand clusters for Nov 2022 (b,c) NMFDs corresponding to the demand clusters with labeling loading and unloading periods (d,e,f,g) Day-wise average NMFDs with the 95% ci for loading and unloading</figcaption></figure>

be sufficient if it reasonably reflects actual operations. In our case study, GTFS real-time vehicle location feeds were used instead of static schedules, as they capture dwell times, delays, and actual speeds, which are essential for accurate multimodal NMFD estimation. This fusion enables the estimation of both unimodal and bimodal network-level traffic variables, allowing for a nuanced understanding of mode interactions within the transport system. The next key phase involves a longitudinal analysis to identify and cluster traffic patterns over time. This process includes labeling NMFDs based on varying traffic conditions such as network loading versus unloading periods, weekday versus weekend dynamics, and special event scenarios. These temporal distinctions provide critical insights into the shifting behavior of network traffic states. Ultimately, this analysis supports the derivation of unimodal and bimodal NMFD relationships for each identified cluster and labeling period, as suggested in the flowchart under the NMFD Relationships section.

## 5.1 NMFD calibration package

The source code for the automatic calibration of NMFDs is available on GitHub at [77]. The package provides a comprehensive pipeline for multimodal MFD calibration using heterogeneous traffic data. The core modules of the repository include:

* `preprocess_LDDCorrection.py`: This module adjusts Local LDD speeds to better reflect average link-level speeds. To train the machine learning model, a few days of FCD from the same network are required. Once trained, the model can be applied to the remaining LDD to correct speed estimates across the network.

* `data_fusion.py`: This module applies data fusion techniques to integrate traffic data from LDD, FCD, and PTD derived from GTFS feeds, for the estimation of unimodal and bimodal NMFD variables. When only LDD and FCD are available, the output corresponds to a unimodal NMFD. However, if PTD is available alongside either LDD or FCD, the module can estimate bi-modal NMFDs, distinguishing between car and public transport modes, as detailed in Table 2.

* `MFD_longitudinal_analysis.py`: This module performs a longitudinal analysis to detect variations in NMFD shapes and cluster traffic patterns over time. It distinguishes between loading and unloading phases based on the definition in Equation 8, and further differentiates between weekdays and weekends. As a result, the module produces four distinct sets of NMFDs: loading-weekdays, loading-weekends, unloading-weekdays, and unloading-weekends.
<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /><figcaption>**Fig. 11** Loading unloading regimes identification in NMFDs using k-mean algorithm</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>**Fig. 12** A step-by-step methodological pipeline for NMFD estimation and calibration</figcaption></figure>
The repository is structured to facilitate reproducibility and extension. Detailed instructions for installation and usage can be found in the repository's documentation, README.md.

# 6 Conclusion

This study investigates the observability and stability of multimodal NMFDs through empirical estimation and calibration using diverse, multimodal traffic data. Calibrated NMFDs are used in real-time traffic state estimation and control, short-term forecasting, multimodal performance monitoring, and long-term planning. Each application implies different quality requirements: for example, control and estimation demand robustness near the free-flow and critical regimes, while planning requires accurate representation of capacity and hysteresis patterns. In this study, we assess calibration quality through (i) preservation of hysteresis loops (loop-size metric, Sect. 3.2, Sect. 4.2) and (ii) temporal stability of NMFD parameters (Sect. 3.3, Sect. 4.3). Although we did not validate goodness-of-fit between observed and modelled flows here, this remains a desirable direction for calibrating NMFDs. Through an extensive review of existing methods and a detailed case study using real-world data from Athens and Lyon, we shed light on the practical challenges and limitations of current approaches, particularly those related to data heterogeneity, bias, data aggregations, and temporal variability. The 1-hour aggregation in Fig. 8 corresponds to the native resolution of the Athens LDD dataset, which also contains the FCD and PT data used in this stability analysis. In contrast, the Lyon dataset has a 6-minute LDD resolution but lacks FCD and PT data, and was therefore only used in the aggregation-interval sensitivity analysis (Figs. 6 and 7). Our findings highlight the limitations of relying solely on LDD, which often overestimates speeds for unimodal NMFD due to sensor placement near downstream intersections, and underscore the importance of integrating FCD and PTD to obtain a more balanced and accurate view of multimodal network dynamics. Bimodal NMFDs based on data fusion reveal that free-flow car speeds are significantly higher than average link speeds, with this disparity diminishing under congested conditions. Public transport speeds remain consistently lower across all regimes.

The stability analysis revealed that traffic states evolve dynamically across both modes and temporal contexts, such as weekday versus weekend and loading versus unloading conditions, underscoring the non-static nature of NMFDs under varying demand patterns. Our longitudinal evaluation showed that loading states consistently lie above unloading states within the critical density region, while free-flow regimes exhibit a distinct relationship across clusters. Weekday NMFDs displayed stable

patterns with overlapping confidence intervals, whereas weekend NMFDs featured a narrower critical density range and an earlier onset of flow reduction. Temporal clustering further distinguished peak-hour upper-bound states from off-peak lower-bound states, demonstrating its effectiveness for refined NMFD characterization. Additionally, we found that simple clustering methods, such as k-means, can reliably identify recurring demand patterns and their corresponding NMFDs, offering a practical and computationally efficient tool for temporal traffic pattern analysis. In addition to addressing data integration challenges, this study identifies a critical research gap in the selection of appropriate data aggregation intervals for NMFD estimation. We propose a structured framework for determining suitable aggregation times, supported by empirical evidence from the case studies. While adaptive smoothing techniques may reduce short-term variability, our results show that sensitivity to aggregation intervals remains essential for preserving key NMFD dynamics such as hysteresis loops. A key contribution of this work is the development and application of an integrated NMFD calibration pipeline, implemented as an open-source Python package to ensure reproducibility. This framework systematically incorporates data correction, multimodal data fusion, and longitudinal clustering into a cohesive methodology. Applied in real-world settings, it enables the estimation of calibrated, interpretable NMFDs that accurately reflect the complex behavior of urban multimodal networks.

Overall, the proposed pipeline contributes a systematic framework for multimodal NMFD calibration by integrating heterogeneous data sources. This provides a basis for more consistent and reproducible calibration practices, while highlighting directions for future improvements and applications.

## Acknowledgements

The authors would like to thank Prof. Eleni I. Vlahogianni and her team from NTUA Athens for their support in providing traffic data.

## Author contributions

NM: Writing - original draft, Writing - review & editing, Software, Methodology, Data curation, Investigation, Formal analysis, Conceptualization. MS: Writing - review & editing, Supervision. LL: Writing - review & editing, Methodology, Investigation, Visualization, Conceptualization, Supervision.

## Funding

This article has been published open access with support of the European Conference of Transport Research Institutes (ECTRI). This research has received funding from the European Union's Horizon Europe research and innovation program under Grant Agreement no. 101103808 (ACUMEN).

## Data availability

This study uses two primary datasets from Athens, Greece, and Lyon, France. The Athens case study employs LDD, FCD, and PTD. PTD is publicly available in GTFS format via the Athens transportation data catalog: https://catalog.hcapdata.gr/dataset/dromologia-osy. LDD can be obtained upon request from OASA: https://www.oasa.gr/. FCD (probe speed data) was provided by HERE Technologies (https://www.here.com/) under a consortium agreement through the ACUMEN project (EU Horizon Europe, Grant No. 101103808). The
Lyon dataset includes LDD and is openly accessible via the CEREMA AVATAR platform: https:// avatar.cerema.fr/cartographie.

# Declarations

## Competing interest

No conflict of interest exists in the submission of this manuscript, and the manuscript is approved by all authors for publication.

Received: 27 May 2025 / Accepted: 15 October 2025

Published online: 29 October 2025

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