March 2024 A nonseparable first-order spatiotemporal intensity for events on linear networks: An application to ambulance interventions
Andrea Gilardi, Riccardo Borgoni, Jorge Mateu
Author Affiliations +
Ann. Appl. Stat. 18(1): 529-554 (March 2024). DOI: 10.1214/23-AOAS1800


The algorithms used for the optimal management of an ambulance fleet require an accurate description of the spatiotemporal evolution of the emergency events. In the last years, several authors have proposed sophisticated statistical approaches to forecast ambulance dispatches, typically modelling the data as a point pattern occurring on a planar region. Nevertheless, ambulance interventions can be more appropriately modelled as a realisation of a point process occurring on a linear network. The constrained spatial domain raises specific challenges and unique methodological problems that cannot be ignored when developing a proper statistical approach. Hence, this paper proposes a spatiotemporal model to analyse ambulance dispatches focusing on the interventions that occurred in the road network of Milan (Italy) from 2015 to 2017. We adopt a nonseparable first-order intensity function with spatial and temporal terms. The temporal dimension is estimated semiparametrically using a Poisson regression model, while the spatial dimension is estimated nonparametrically using a network kernel function. A set of weights is included in the spatial term to capture space-time interactions, inducing nonseparability in the intensity function. A series of tests show that our approach successfully models the ambulance interventions and captures the space-time patterns more accurately than planar or separable point process models.

Funding Statement

J. Mateu is partially supported by grant PID2019-107392RB-I00 from the Spanish Ministry of Science.


Map data copyrighted OpenStreetMap contributors and available from We greatly acknowledge Doct. Piero Brambilla, Doct. Andrea Pagliosa and Doct. Rodolfo Bonora, the regional experts of ambulance interventions working for the local EMS (AREU: Azienda Regionale Emergenza Urgenza). They provided us the ambulance intervention data used in this paper and, more importantly, priceless and continuous assistance to properly understand the complexity of an EMS system and the peculiarities of this problem. We also greatly acknowledge the DEMS Data Science Lab for supporting this work by providing computational resources. This study was carried out within the MOST—Sustainable Mobility National Research Center and received funding from the European Union Next-Generation EU (PIANO NAZIONALE DI RIPRESA E RESILIENZA (PNRR)—MISSIONE 4 COMPONENTE 2, INVESTIMENTO 1.4—D.D. 1033 17/06/2022, CN00000023). This manuscript reflects only the authors’ views and opinions, neither the European Union nor the European Commission can be considered responsible for them. Andrea Gilardi’s present address is Politecnico di Milano, Departimento di Matematica.


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Andrea Gilardi. Riccardo Borgoni. Jorge Mateu. "A nonseparable first-order spatiotemporal intensity for events on linear networks: An application to ambulance interventions." Ann. Appl. Stat. 18 (1) 529 - 554, March 2024.


Received: 1 April 2022; Revised: 1 June 2023; Published: March 2024
First available in Project Euclid: 31 January 2024

Digital Object Identifier: 10.1214/23-AOAS1800

Keywords: Emergency interventions , linear networks , nonparametric methods , point patterns on linear networks , spatial networks , spatiotemporal data

Rights: Copyright © 2024 Institute of Mathematical Statistics


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Vol.18 • No. 1 • March 2024
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