A Bayesian Learning Approach for Drone Coverage Network: A Case Study on Cardiac Arrest in Scotland

Tathagata Basu; Edoardo Patelli; Gianluca Filippi; Ben Parsonage; Christy Maddock; Massimiliano Vasile; Marco Fossati; Adam Loyd

cs.LGMar 24, 2026

A Bayesian Learning Approach for Drone Coverage Network: A Case Study on Cardiac Arrest in Scotland

Authors: Tathagata Basu, Edoardo Patelli, Gianluca Filippi, Ben Parsonage, Christy Maddock, Massimiliano Vasile, Marco Fossati, Adam Loyd,

Organizations: UNSW Bengaluru, Bengaluru, India · Civil and Environmental Engineering, University of Strathclyde, Glasgow G1 1XQ, UK · Centre for Maritime Research and Experimentation, 19126 La Spezia, Italy · Mechanical and Aerospace Engineering, University of Strathclyde, Glasgow G1 1XQ, UK ·

Abstract

Drones are becoming popular as a complementary system for Emergency Medical Services (EMS). Although several pilot studies and flight trials have shown the feasibility of drone-assisted Automated External Defibrillator (AED) delivery, running a full-scale operational network remains challenging due to high capital expenditure and environmental uncertainties. In this paper, we formulate a reliability-informed Bayesian learning framework for designing drone-assisted AED delivery networks under environmental and operational uncertainty. We propose our objective function based on the survival probability of Out-off Hospital Cardiac Arrest (OHCA) patients to identify the ideal locations of drone stations. Moreover, we consider the coverage of existing EMS infrastructure to improve the response reliability in remote areas. We illustrate our proposed method using geographically referenced cardiac arrest data from Scotland. The result shows how environmental variability and spatial demand patterns influence optimal drone station placement across urban and rural regions. In addition, we assess the robustness of the network and evaluate its economic viability using a cost-effectiveness analysis based on expected Quality Adjusted Life Year (QALY). The findings suggest that drone-assisted AED delivery is expected to be cost-effective and has the potential to significantly improve the emergency response coverage in rural and urban areas with longer ambulance response times.

Figures & tables

Variable	Estimate	Std. Error	t value	Pr( $>$ $\|$ t $\|$ )
(Intercept)	1.7654	0.0395	44.6802	$<$ 1e-4
$T^{A^{*}}$	0.0290	0.0086	3.3715	0.0008
$I_{b}$	0.0038	0.0025	1.5127	0.1305
$I_{m}$	0.0051	0.0013	3.8861	0.0001
$\chi^{A}$	-0.0019	0.0006	-2.9398	0.0033
$\rho^{w}$	-0.0464	0.0119	-3.9138	0.0001

Table 1: Summary of linear model fit for the ambulance response time

	Take-off time			Landing time			VTOL battery
Variable	Est.	S.E.	$p$ -val	Est.	S.E.	$p$ -val	Est.	S.E.	$p$ -val
(Intercept)	1.4495	0.0269	$<$ 1e-4	3.8494	0.1373	$<$ 1e-4	-0.9255	0.1515	$<$ 1e-4
$\omega_{v}$	0.0518	0.0026	$<$ 1e-4	0.1605	0.0131	$<$ 1e-4	0.1422	0.0144	$<$ 1e-4
$m_{pl}$	0.3421	0.0313	$<$ 1e-4	-0.8646	0.1597	$<$ 1e-4	-0.5602	0.1761	0.0017
$h_{c}$	0.0097	0.0003	$<$ 1e-4	0.0080	0.0016	$<$ 1e-4	0.0064	0.0018	0.0004

Table 2: Summary of linear model fits for take-off time (left section); landing time (middle section); and VTOL battery consumption (right section).

	Cruise time			Cruise battery
Variable	Est.	S.E.	$p$ -val	Est.	S.E.	$p$ -val
(Intercept)	3.6110	0.0353	$<$ 1e-4	-1.7639	0.0364	$<$ 1e-4
$d$	0.0003	$<$ 1e-4	$<$ 1e-4	0.0003	$<$ 1e-4	$<$ 1e-4
$m_{pl}$	-0.0535	0.0401	0.1839	0.0225	0.0414	0.5876
$\Delta_{\chi}$	0.0010	0.0002	$<$ 1e-4	0.0007	0.0002	0.0004
$\Delta_{h}$	-0.0001	0.0001	0.3402	0.0002	0.0002	0.1025

Table 3: Summary of linear model fits for cruise time (left section) and cruise battery consumption (right section).

Figure 1: Aleatory and epistemic uncertainty decomposition for ambulance surrogate, wind surrogates and flight surrogates.

\ell(x^{\prime})=-\frac{1}{K}\sum_{k=1}^{K}\left[\sum_{i=1}^{n}e^{k}_{i}\phi\left(P_{i}\left(x^{\prime};A^{k}\right)\right)+\frac{\eta}{n}\sum_{i=1}^{n}\sum_{j=1}^{p}x^{\prime}_{j}\,w^{k}_{ij}A^{k}_{ij}\right]

Algorithm 1 Posterior sampling for drone network design

Figure 2: Prior predictive distribution on total number of active sites for NHS GGC (left) for

\theta_{0}=-1

and for NHS Grampian (right) for

\theta_{0}=-2

Figure 3: Longer sensitivity analysis showing QALY gain against

\beta

for four different combinations.

$\sigma_{\text{mult}}$	Scenario	No. Sites	Baseline Sites	Jaccard	Rank Correlation
$0.5\times$	Lower uncertainty	15	17	0.882	0.950
$1.0\times$	Baseline ( $R^{2}=0.19$ )	17	17	1.000	1.000
$1.5\times$	Higher uncertainty	15	17	0.778	0.965
$2.0\times$	Higher uncertainty	15	17	0.684	0.942

Table 4: Sensitivity of site selection to ambulance response surrogate uncertainty (NHS GGC). The residual standard deviation

\sigma_{\text{amb}}

is multiplied by factors ranging from

0.5\times

2\times

the baseline value to simulate plausible surrogate mis-specification. Jaccard similarity measures the overlap between the selected network and the baseline network (

\sigma_{\text{mult}}=1

); Spearman rank correlation measures the preservation of site-level activation probability ordering. Values close to 1 indicate robust site selection. The baseline

R^{2}\approx 0.19

of the ambulance surrogate is shown for reference.

Method	No. Sites	QALY Gain	Rob $m=1$	Jaccard
Gibbs posterior (proposed)	17	$96.5\pm 8.6$	98.9%	1.00
Deterministic MCLP	17	$95.3\pm 8.7$	99.6%	0.48
Greedy stochastic	17	$92.5\pm 8.8$	99.4%	0.36

Table 5: Benchmark comparison of the proposed Gibbs posterior against a deterministic Maximum Coverage Location Problem (MCLP) solved via greedy forward selection with 2-opt local search, and a greedy stochastic heuristic. All three methods are evaluated at identical budget (

k=17

stations) using the same wind and demand scenarios. QALY gain is reported as mean

\pm

SD across all season-hour scenario combinations. Rob

m=1

: expected coverage when one station fails simultaneously. Jaccard: fraction of sites in common with the Gibbs posterior solution (

=1

by definition for Gibbs).

Figure 4: Summary of our analyses with NHS GGC. The upper-left panel shows number of sites and QALY gain against

\beta

; the upper-right panel shows mean drone response time and number of cases covered in 6 and 8 mins; the middle-left panel shows the number of cases where drone reaches before ambulance; the middle-right panel shows the final network design; the lower-left panel shows the cost-effectiveness of the designs; and the lower-right panel shows the coverage reliability of the designs

	GGC				Grampian
Response	Calls	Mean	$<$ 6 mins	$<$ 8 mins	Calls	Mean	$<$ 6 mins	$<$ 8 mins
Recorded	683	8.43	24%	47%	280	8.60	32%	51%
Drone	610 (616)	4.01	86%	89%	232 (248)	4.33	77%	82%

Table 6: Coverage statistics comparing recorded sas response times and drone-assisted response for NHS GGC and NHS Grampian for the best network design. Mean response times are reported in minutes.

Figure 5: Traceplot of network size and log-posterior for all seasons at the afternoon

Figure 6: Summary of our analyses with NHS Grampian. The upper-left panel shows number of sites and QALY gain against

\beta

Appendix figures & tables24 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 7: Scatterplot and correlation plot for the ambulance response time dataset.

Model	$R^{2}$ Mean	$R^{2}$ SD	RMSE Mean	RMSE SD	MAE Mean	MAE SD
Baseline	-0.000	0.000	0.572	0.014	0.450	0.012
LM	0.188	0.022	0.515	0.014	0.396	0.013
GP1	0.086	0.111	0.546	0.023	0.424	0.021
GP2	0.016	0.090	0.567	0.024	0.436	0.021
GP3	0.080	0.008	0.548	0.011	0.426	0.012
GP4	-0.102	0.024	0.600	0.013	0.463	0.009

Appendix

Table 7: Three-fold cross-validation results for log-transformed ambulance response time models.

Figure 8: Scatterplot and correlation plot for wind models.

Model	$R^{2}$ Mean	$R^{2}$ SD	RMSE Mean	RMSE SD	MAE Mean	MAE SD
Baseline	-0.055	0.048	0.451	0.044	0.365	0.035
LM1	0.154	0.136	0.404	0.062	0.325	0.040
GP1	0.856	0.118	0.157	0.049	0.127	0.045
GP2	0.275	0.160	0.371	0.043	0.309	0.045
GP3	0.883	0.043	0.147	0.019	0.118	0.018
GP4	0.715	0.078	0.232	0.030	0.189	0.031

Appendix

Table 8: Ten-fold cross-validation results for log wind speed models.

Model	$R^{2}$ Mean	$R^{2}$ SD	RMSE Mean	RMSE SD	MAE Mean	MAE SD
Baseline	-0.153	0.349	0.127	0.021	0.100	0.015
LM1	0.282	0.247	0.099	0.014	0.077	0.011
GP1	-0.055	0.244	0.121	0.017	0.096	0.011
GP2	-3.347	2.744	0.227	0.052	0.177	0.033
GP3	-0.069	0.240	0.122	0.017	0.097	0.012
GP4	-5.175	4.268	0.269	0.079	0.211	0.059

Appendix

Table 9: Ten-fold cross-validation results for log wind direction models.

Model	$R^{2}$ Mean	$R^{2}$ SD	RMSE Mean	RMSE SD	MAE Mean	MAE SD
Baseline	-0.144	0.321	24.056	3.383	19.274	3.013
LM1	0.290	0.241	18.792	2.512	14.717	2.304
GP1	-0.133	0.351	23.867	3.271	19.153	3.267
GP2	-1.700	0.854	36.473	4.268	30.022	3.278
GP3	-0.117	0.353	23.723	3.488	19.083	3.355
GP4	-2.229	1.296	39.438	6.233	31.300	4.798

Appendix

Table 10: Ten-fold cross-validation results for wind direction models.

Model	$R^{2}$ Mean	$R^{2}$ SD	RMSE Mean	RMSE SD
LM1	0.28	0.22	3.60	1.33
GP6	-1.60	3.64	5.76	2.80

Appendix

Table 11: Ten-fold cross-validation results for sine component of the wind through circular treatment.

Model	$R^{2}$ Mean	$R^{2}$ SD	RMSE Mean	RMSE SD
LM1	0.17	0.15	3.84	0.67
GP6	0.57	0.09	2.78	0.51

Appendix

Table 12: Ten-fold cross-validation results for cosine component of the wind through circular treatment.

Figure 9: Scatterplot and correlation plot for VTOL models.

Model	$R^{2}$ Mean	$R^{2}$ SD	RMSE Mean	RMSE SD	MAE Mean	MAE SD
Baseline	-0.045	0.041	0.179	0.021	0.149	0.018
LM1	0.859	0.130	0.061	0.030	0.042	0.011
GP1	0.819	0.184	0.066	0.040	0.047	0.027
GP2	0.819	0.131	0.072	0.029	0.045	0.012
GP3	0.866	0.130	0.060	0.032	0.041	0.016
GP4	0.818	0.137	0.072	0.031	0.047	0.010

Appendix

Table 13: Ten-fold cross-validation results for take-off time surrogate models.

Model	$R^{2}$ Mean	$R^{2}$ SD	RMSE Mean	RMSE SD	MAE Mean	MAE SD
Baseline	-0.065	0.049	0.458	0.088	0.368	0.065
LM1	0.468	0.242	0.324	0.129	0.228	0.068
GP1	0.129	0.290	0.412	0.124	0.301	0.070
GP2	0.455	0.255	0.327	0.134	0.233	0.077
GP3	0.224	0.288	0.385	0.117	0.275	0.064
GP4	0.416	0.250	0.339	0.130	0.240	0.068

Appendix

Table 14: Ten-fold cross-validation results for landing time surrogate models.

Model	$R^{2}$ Mean	$R^{2}$ SD	RMSE Mean	RMSE SD	MAE Mean	MAE SD
Baseline	-0.032	0.028	0.456	0.069	0.367	0.060
LM1	0.288	0.135	0.376	0.059	0.291	0.044
GP1	-0.037	0.108	0.456	0.066	0.359	0.053
GP2	0.294	0.138	0.374	0.059	0.290	0.045
GP3	0.006	0.281	0.441	0.073	0.344	0.054
GP4	0.281	0.139	0.377	0.058	0.291	0.041

Appendix

Table 15: Ten-fold cross-validation results for VTOL battery surrogate models.

Figure 10: Scatterplot and correlation plot for cruise models.

Model	$R^{2}$ Mean	$R^{2}$ SD	RMSE Mean	RMSE SD	MAE Mean	MAE SD
Baseline	-0.046	0.072	0.379	0.039	0.313	0.033
LM1	0.943	0.025	0.086	0.015	0.064	0.010
GP1	-0.046	0.072	0.379	0.039	0.313	0.033
GP2	-1.850	1.251	0.604	0.102	0.502	0.100
GP3	-0.046	0.072	0.379	0.039	0.313	0.033
GP4	-1.875	1.302	0.606	0.105	0.501	0.102

Appendix

Table 16: Ten-fold cross-validation results for cruise time surrogate models.

Model	$R^{2}$ Mean	$R^{2}$ SD	RMSE Mean	RMSE SD	MAE Mean	MAE SD
Baseline	-0.043	0.065	0.377	0.040	0.313	0.034
LM1	0.941	0.024	0.088	0.016	0.066	0.009
GP1	-0.043	0.065	0.377	0.040	0.313	0.034
GP2	0.277	0.268	0.307	0.049	0.250	0.046
GP3	-0.043	0.065	0.377	0.040	0.313	0.034
GP4	0.273	0.274	0.307	0.049	0.251	0.046

Appendix

Table 17: Ten-fold cross-validation results for cruise battery surrogate models.

Figure 11: Joint vs Independent coverage probability.

	Location parameter of logistic transformation ( $a$ )
Test	Value	No. Sites	Cases	QALY	Cost per QALY
1	1	13	550	93	4612
2	2	14	553	93	4727
3	3	15	555	94	4783
4	4	15	556	94	4843
5	5	15	556	94	4880

Appendix

Table 18: Median estimates with respect to different hyper parameters presented in the model. Total 450 combinations are considered for the sensitivity analysis involving GGC area and for temperature parameter

\beta=15

is fixed. For the sake of computation 8 equidistant hours of the day along with all seasons are chosen with 500 iterations for MCMC run. Estimates are obtained by discarding first 100 simulations.

Figure 12: Posterior activation probability across different season hour combination for NHS GGC.

Figure 13: Posterior activation probability across different season hour combination for NHS Grampian.

$\beta$	Season	Mean	Median	Mean	Median	Mean	Median	(frac.)	(mean)
		ESS log-posterior		ESS network size		Acceptance rate		Collapsed	Moves
5	autumn	71.29	70.00	77.69	77.69	0.33	0.33	0.00	1321.46
	spring	69.20	68.57	74.09	72.32	0.33	0.33	0.00	1319.46
	summer	67.67	68.13	75.10	73.61	0.33	0.33	0.00	1325.25
	winter	68.64	65.89	74.36	71.95	0.33	0.33	0.00	1313.88
7	autumn	72.00	68.48	77.96	77.01	0.33	0.34	0.00	1334.21

Appendix

Table 19: MCMC convergence diagnostics for NHS GGC across all temperature parameter values

\beta

and seasons.

$\beta$	Season	Mean	Median	Mean	Median	Mean	Median	(frac.)	(mean)
		ESS log-posterior		ESS network size		Acceptance rate		Collapsed	Moves
5	autumn	10.47	9.80	10.91	9.13	0.10	0.10	0.00	409.08
	spring	10.16	9.45	10.41	9.49	0.10	0.10	0.00	410.04
	summer	10.76	10.76	11.27	10.46	0.10	0.10	0.00	415.13
	winter	11.25	10.86	11.60	11.00	0.10	0.10	0.00	406.92
7	autumn	9.35	8.94	9.39	8.99	0.10	0.10	0.00	413.13

Appendix

Table 20: MCMC convergence diagnostics for NHS Grampian across all temperature parameter values

\beta

and seasons.

Figure 14: Traceplot of network size and log-posterior for all seasons at the afternoon for NHS Grampian.

Quantity	$\bar{Q}$	$\mathrm{VaR}_{0.05}$	$\mathrm{CVaR}_{0.05}$	$\mathrm{CVaR}/\bar{Q}$
Final design (aleatory uncertainty)
Cost-effective design $x^{\star}$ ( $8$ sites)	$78.58$	$65.55$	$62.59$	$0.80$
Under design uncertainty (posterior of $x$ )
Posterior-predictive $\mathrm{CVaR}_{0.05}$	—	—	$22.44$	—
Posterior mean $\mathrm{CVaR}_{0.05}$ ( $95\%$ CrI)	—	—	$47.78$ $[12.87,\,65.58]$	—

Appendix

Table 21: Tail-weighted performance of the network design for NHS GGC and

\alpha=0.05

). The upper panel reports tail statistics under aleatory / wind uncertainty for the final cost-effective final design and the lower panel propagates the Gibbs posterior over designs through the tail functional.

Figure 15: Tail risk against network size across the design posterior for NHS GGC and

\alpha=0.05

. Each grey point is one posterior design draw and the black line represents the posterior-weighted mean

\mathrm{CVaR}_{0.05}

for each network size. The final cost-effective design (

8

sites) is marked with red triangle.

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