LASPATED: A Library for the Analysis of Spatio-Temporal Discrete Data

Vincent Guigues; Anton Kleywegt; Giovanni Amorim; André Krauss; Victor Hugo Nascimento

doi:10.5334/jors.544

LASPATED: A Library for the Analysis of Spatio-Temporal Discrete Data

Journal of Open Research Software

Volume 14 (2026): Issue 1

By: Vincent Guigues , Anton Kleywegt , Giovanni Amorim, André Krauss and Victor Hugo Nascimento

Open Access

|Jul 2026

Figures & Tables

Space discretization of a region containing the city of Rio de Janeiro into $10 \times 10 = 100$ rectangles, 76 of which have nonempty intersection with the region and are shown in the figure.

Space discretization of a region containing the city of Rio de Janeiro into hexagons using the Uber library H3 with scale parameter equal to 7.

Space discretization of a region containing the city of Rio de Janeiro into 160 administrative districts.

Aggregated estimates of the intensities for 3 estimators: (i) Empirical: the empirical intensities, (ii) Rectangular $10 \times 10$ : Model M1 with the $10 \times 10$ rectangular space discretization, and (iii) Covariates, rectangular $10 \times 10$ : Model M2 with the $10 \times 10$ rectangular space discretization. Top left: emergencies of all priorities, top right: high-priority emergencies, bottom left: intermediate priority emergencies, bottom right: low-priority emergencies.

Aggregated estimates of the intensities for 3 estimators: (i) Empirical: the empirical intensities, (ii) Hexagonal 7: Model M1 with hexagonal discretization with scale parameter 7, and (iii) Covariates, hexagonal 7: Model M2 with hexagonal discretization with scale parameter 7. Top left: emergencies of all priorities, top right: high-priority emergencies, bottom left: intermediate priority emergencies, bottom right: low-priority emergencies.

Aggregated estimates of the intensities for 3 estimators: (i) Empirical: the empirical intensities, (ii) District: Model M1 with the space discretization by districts, (iii) Covariates, district: Model M2 with the space discretization by districts. Top left: emergencies of all priorities, top right: high-priority emergencies, bottom left: intermediate priority emergencies, bottom right: low-priority emergencies.

Space discretization of a region containing the city of Rio de Janeiro into $100 \times 100 = 10000$ rectangles, 4916 of which have nonempty intersection with the region and are shown in the figure.

Space discretization of a region containing the city of Rio de Janeiro into hexagons using the Uber library H3 with scale parameter equal to 8.

Discretization of a region containing the city of Rio de Janeiro into 34 subregions, given by the Voronoi diagram of ambulance stations in Rio de Janeiro.

Table 2

Mean relative error $e_{*}$ . In column 1, m/n denotes $N_{i, t} = m$ for non-holiday time periods t, and $N_{i, t} = n$ for holiday time periods t.

SAMPLE SIZE N_i,t	HOLIDAY	COV	NO COV 1	NO COV 2	NO COV 3	EMP
52	0	0.006	0.012	0.048	0.048	0.048
520	0	0.002	0.004	0.015	0.015	0.015
780	0	0.001	0.003	0.012	0.012	0.012
51/1	1	0.067	0.063	0.138	0.138	0.154
510/10	1	0.027	0.050	0.065	0.065	0.049
765/15	1	0.018	0.038	0.054	0.054	0.040

Test region $S = {[0, 10]}^{2}$ . The intensity function $λ$ is different in the blue and red subregions, but the user does not know about these subregions. For estimation purposes, the user discretizes $S$ into $10 \times 10 = 100$ square zones.

Intensities as a function of location, for time intervals that start at odd times and even times. **(a)** Time intervals that start at odd times. **(b)** Time intervals that start at even times.

Mean relative estimation errors $\frac{1}{| I | | T |} \sum_{i \in I} \sum_{t \in T} | \frac{λ_{i, t} - {\hat{λ}}_{i, t}^{w}}{λ_{i, t}} |$ as a function of the penalty parameter w, for four sample sizes and six estimators (including the empirical estimator at $w = 0$ ).

Table 1

The column headings denote the estimators as follows: Reg 1: estimator with partition $G_{4}$ of time intervals and with neighbor-based spatial regularization; Reg 2: estimator with partition $G_{4}$ of time intervals and without spatial regularization; Reg 3: estimator with partition $G_{2}$ of time intervals and with neighbor-based spatial regularization; Reg 4: estimator with partition $G_{2}$ of time intervals and without spatial regularization; CV: cross validation; Emp: empirical estimator. For each sample size and for each estimator Reg 1, Reg 2, Reg 3, Reg 4, each cell of the table gives two numbers separated by a slash (/): the first number is the minimum mean relative error over different values of penalty parameter w, and the second number is the value of w that attains the minimum. For the cross validation estimator, the first number is the mean relative error using the penalty parameter $w^{*}$ , and the second number is the value of $w^{*}$ . For the empirical estimator, the first number is the mean relative error using the penalty parameter $w = 0$ .

SAMPLE SIZE N_i,t	REG 1	REG 2	REG 3	REG 4	CV	EMP
1	0.47/1.2	0.67/2.0	0.34/0.6	0.51/1.4	-	1.539/0
10	0.31/0.06	0.30/0.2	0.26/0.06	0.17/0.2	0.42/0.2	0.543/0
50	0.18/0.008	0.12/0.04	0.15/0.004	0.08/0.036	0.37/0.04	0.256/0
500	0.08/0.0	0.03/0.004	0.08/0.0	0.02/0.0028	0.26/0.002	0.07687/0

Comparison of true, empirical, and regularized estimates of the Poisson process intensities. True intensities as shown in Figure 12.

Mean relative errors $\frac{1}{| I | | T |} \sum_{t \in T} \sum_{i \in I} \frac{| \int_{R_{i}} λ (x, y, t) d (x, y) - {\hat{λ}}_{t, i}^{w} |}{\int_{R_{i}} λ (x, y, t) d (x, y)}$ as a function of the penalty parameter w. Intensities $λ (x, y, t)$ are as shown in Figure 15.

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DOI: https://doi.org/10.5334/jors.544 | Journal eISSN: 2049-9647

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Language: English

Page range: 51 - 51

Submitted on: Nov 15, 2024

Accepted on: Jun 24, 2026

Published on: Jul 8, 2026

Published by: Ubiquity Press

In partnership with: Paradigm Publishing Services

Keywords:

Spatio-temporal Poisson process,

intensity estimation,

arrival rate estimation,

nonparametric rate estimation,

fitting functions of space and time,

spatio-temporal discretization,

statistical software

© 2026 Vincent Guigues, Anton Kleywegt, Giovanni Amorim, André Krauss, Victor Hugo Nascimento, published by Ubiquity Press
This work is licensed under the Creative Commons Attribution 4.0 License.

Volume 14 (2026): Issue 1

LASPATED: A Library for the Analysis of Spatio-Temporal Discrete Data

Figures & Tables

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Table 2

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Table 1

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Figure 15

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