Emergency department demand forecasting

Rotas, escalation plans and bed plans for next month need a number for each day, and a range around it. This project forecasts daily emergency department attendances 28 days ahead for a fictional acute hospital, compares three models and a simple average, and scores them the way they would be used: twelve monthly forecasts across 2025, each made with only the data that existed on the day.

All the data is synthetic, generated by the project code with a fixed seed. No real hospital, patient or staff data is involved.

In short: an average of exponential smoothing and gradient boosting had the lowest error, a MASE of 0.79 and a mean absolute error 28% below a seasonal naive baseline. Its 80% intervals held on 76.4% of days, a little too narrow. And every model missed Christmas Day by 85 attendances or more.

The planning problem

A rota for the four weeks from 1 December has to be drafted in November. So the useful forecast is the one made at the end of a month for the month that follows, with nothing from that month in hand.

A planner needs three things from it:

1. A daily number for each of the next 28 days, because staffing follows the weekly cycle and the bank holidays, not a monthly total.
2. A range, so the plan can be set against a bad day rather than an average one. An 80% interval is a reasonable default: wide enough to be honest, narrow enough to act on.
3. A warning when the forecast is weak. Christmas, bank holidays and heatwaves are exactly the days when a plan matters most, and they are where models do worst.

What’s in the data

Five years of daily attendances, 1 January 2021 to 31 December 2025, split into ambulance and walk-in arrivals. Every pattern below is planted on purpose and set as a named constant at the top of simulate.py, so the forecasts can be judged against known truth.

Ambulance arrivals are 28.1% of the total, a little higher in winter and lower during heatwaves. The CSV keeps the planted expected value for every day in a planted_expected column, which turns out to be the most useful column in the file: it shows how much of any miss was the model and how much was noise.

Data table

Week starting	All arrivals	By ambulance
2021-01-04	322	101
2021-01-11	329	103
2021-01-18	332	102
2021-01-25	304	95
2021-02-01	312	94
2021-02-08	308	95
2021-02-15	298	90
2021-02-22	301	92
2021-03-01	278	81
2021-03-08	276	85
2021-03-15	285	85
2021-03-22	276	79
2021-03-29	292	85
2021-04-05	279	77
2021-04-12	267	68
2021-04-19	283	79
2021-04-26	281	77
2021-05-03	270	76
2021-05-10	305	81
2021-05-17	289	78
2021-05-24	288	72
2021-05-31	267	68
2021-06-07	287	71
2021-06-14	284	74
2021-06-21	264	64
2021-06-28	299	76
2021-07-05	294	72
2021-07-12	276	65
2021-07-19	318	75
2021-07-26	294	76
2021-08-02	278	74
2021-08-09	275	67
2021-08-16	277	69
2021-08-23	267	70
2021-08-30	268	66
2021-09-06	281	78
2021-09-13	280	76
2021-09-20	295	82
2021-09-27	303	80
2021-10-04	301	85
2021-10-11	294	84
2021-10-18	287	86
2021-10-25	288	82
2021-11-01	284	81
2021-11-08	299	87
2021-11-15	299	91
2021-11-22	318	97
2021-11-29	305	93
2021-12-06	300	86
2021-12-13	294	87
2021-12-20	289	89
2021-12-27	297	89
2022-01-03	325	105
2022-01-10	356	115
2022-01-17	326	102
2022-01-24	302	94
2022-01-31	344	101
2022-02-07	338	105
2022-02-14	290	95
2022-02-21	319	101
2022-02-28	307	94
2022-03-07	305	91
2022-03-14	297	91
2022-03-21	294	80
2022-03-28	304	88
2022-04-04	297	89
2022-04-11	307	92
2022-04-18	292	80
2022-04-25	304	81
2022-05-02	268	72
2022-05-09	272	73
2022-05-16	289	77
2022-05-23	277	68
2022-05-30	290	76
2022-06-06	294	74
2022-06-13	289	73
2022-06-20	283	77
2022-06-27	285	71
2022-07-04	295	71
2022-07-11	295	76
2022-07-18	295	74
2022-07-25	276	68
2022-08-01	289	76
2022-08-08	296	68
2022-08-15	295	74
2022-08-22	285	73
2022-08-29	298	78
2022-09-05	309	79
2022-09-12	313	85
2022-09-19	305	86
2022-09-26	297	83
2022-10-03	313	84
2022-10-10	297	83
2022-10-17	345	97
2022-10-24	312	88
2022-10-31	321	88
2022-11-07	291	85
2022-11-14	305	88
2022-11-21	297	88
2022-11-28	281	91
2022-12-05	327	103
2022-12-12	321	97
2022-12-19	330	104
2022-12-26	338	106
2023-01-02	362	119
2023-01-09	358	113
2023-01-16	313	99
2023-01-23	336	99
2023-01-30	337	105
2023-02-06	323	102
2023-02-13	324	96
2023-02-20	322	97
2023-02-27	302	95
2023-03-06	345	104
2023-03-13	321	92
2023-03-20	314	95
2023-03-27	330	90
2023-04-03	290	84
2023-04-10	300	86
2023-04-17	312	91
2023-04-24	295	82
2023-05-01	293	74
2023-05-08	277	74
2023-05-15	277	74
2023-05-22	279	76
2023-05-29	269	73
2023-06-05	286	64
2023-06-12	313	80
2023-06-19	294	74
2023-06-26	289	68
2023-07-03	279	66
2023-07-10	308	75
2023-07-17	307	78
2023-07-24	291	70
2023-07-31	295	75
2023-08-07	272	69
2023-08-14	270	74
2023-08-21	284	74
2023-08-28	293	70
2023-09-04	314	85
2023-09-11	286	77
2023-09-18	298	79
2023-09-25	317	81
2023-10-02	313	87
2023-10-09	293	82
2023-10-16	324	95
2023-10-23	306	88
2023-10-30	313	89
2023-11-06	323	88
2023-11-13	324	99
2023-11-20	330	97
2023-11-27	332	96
2023-12-04	315	95
2023-12-11	332	101
2023-12-18	359	112
2023-12-25	312	90
2024-01-01	312	93
2024-01-08	337	108
2024-01-15	319	102
2024-01-22	333	104
2024-01-29	299	88
2024-02-05	327	97
2024-02-12	319	98
2024-02-19	320	98
2024-02-26	335	105
2024-03-04	321	96
2024-03-11	306	85
2024-03-18	314	92
2024-03-25	314	90
2024-04-01	313	90
2024-04-08	310	87
2024-04-15	316	90
2024-04-22	302	82
2024-04-29	315	91
2024-05-06	287	75
2024-05-13	298	80
2024-05-20	277	70
2024-05-27	291	80
2024-06-03	293	69
2024-06-10	288	75
2024-06-17	286	72
2024-06-24	273	68
2024-07-01	290	70
2024-07-08	281	72
2024-07-15	287	67
2024-07-22	272	70
2024-07-29	304	75
2024-08-05	273	71
2024-08-12	296	71
2024-08-19	304	76
2024-08-26	276	70
2024-09-02	304	80
2024-09-09	300	86
2024-09-16	303	77
2024-09-23	309	79
2024-09-30	306	83
2024-10-07	301	86
2024-10-14	298	85
2024-10-21	319	94
2024-10-28	305	90
2024-11-04	327	97
2024-11-11	325	95
2024-11-18	336	103
2024-11-25	322	96
2024-12-02	322	95
2024-12-09	336	105
2024-12-16	349	114
2024-12-23	340	107
2024-12-30	345	108
2025-01-06	328	105
2025-01-13	346	111
2025-01-20	326	105
2025-01-27	321	100
2025-02-03	307	95
2025-02-10	295	89
2025-02-17	321	95
2025-02-24	321	95
2025-03-03	331	93
2025-03-10	320	94
2025-03-17	326	95
2025-03-24	334	94
2025-03-31	331	92
2025-04-07	334	94
2025-04-14	330	93
2025-04-21	320	93
2025-04-28	308	83
2025-05-05	312	85
2025-05-12	324	81
2025-05-19	330	88
2025-05-26	333	85
2025-06-02	315	86
2025-06-09	308	80
2025-06-16	312	73
2025-06-23	291	70
2025-06-30	321	80
2025-07-07	301	72
2025-07-14	315	76
2025-07-21	287	73
2025-07-28	281	76
2025-08-04	294	75
2025-08-11	334	81
2025-08-18	283	75
2025-08-25	310	82
2025-09-01	298	75
2025-09-08	293	79
2025-09-15	292	75
2025-09-22	316	88
2025-09-29	322	87
2025-10-06	334	93
2025-10-13	346	97
2025-10-20	339	98
2025-10-27	344	106
2025-11-03	352	104
2025-11-10	345	99
2025-11-17	327	94
2025-11-24	345	98
2025-12-01	372	109
2025-12-08	346	104
2025-12-15	325	102
2025-12-22	309	100

Synthetic data. Source: projects/ed-demand-forecasting. Weeks with an outage day average the days that were recorded.

Approach

Backtest design

Twenty-four forecast origins, one at each month end from 31 December 2023 to 30 November 2025. Each forecast covers the first 28 days of the following month. The twelve origins that forecast 2024 are for calibrating the prediction intervals and for design choices. The twelve that forecast 2025 are the ones scored: 335 days per model, because the outage on 11 March has no actual to score against.

Every model receives the history up to its origin and nothing else. Outage days in that history are filled with the same weekday one week earlier, which only looks backwards. The rolling-origin backtesting article explains why this beats a single train/test split.

Models

1. Seasonal naive. Repeat the last observed week. It is the baseline any other model has to beat.
2. Exponential smoothing (ETS). Holt-Winters from statsmodels, with a damped additive trend and multiplicative weekly seasonality, refitted at every origin. It knows nothing about bank holidays or the time of year.
3. Gradient boosting (GBM). scikit-learn’s HistGradientBoostingRegressor as a direct multi-horizon model: one regressor, with the horizon as a feature. Every (origin, horizon) pair inside the history becomes a training row, 29,526 of them at the first 2025 origin and 38,878 at the last.
4. Average of ETS and GBM. The mean of the two point forecasts.

The GBM predicts the log of attendances relative to the 28-day mean at the origin, so the trees never have to extrapolate the trend. Its features are the horizon, the day of week, flags for bank holidays, the working day after one, and Christmas Day, and four ratios to that 28-day level. Every lagged value is chosen so that it existed at the origin:

def direct_features(y, calendar, origin, horizon):
    cumsum = np.concatenate(([0.0], np.cumsum(y)))
    target = origin + horizon
    level = _window_mean(cumsum, origin, LEVEL_DAYS)
    # The four most recent observed days with the same weekday as the target.
    back = target - WEEK * np.ceil(horizon / WEEK).astype(int)
    same_weekday = (y[back] + y[back - 7] + y[back - 14] + y[back - 21]) / 4
    last_week = _window_mean(cumsum, origin, WEEK)
    last_year_week = _window_mean(cumsum, target - YEAR + 3, WEEK)  # centred on target - 364
    last_year_month = _window_mean(cumsum, target - YEAR + 14, LEVEL_DAYS)
    ...

For a target ten days ahead, “same weekday” means 14 days before the target, not 7, because 7 days before the target is still in the future. At forecast time y stops at the origin, so a feature that reached past it would raise an IndexError instead of leaking quietly.

An earlier version also had day of year as a feature. It did worse on the 2024 origins: with three or four winters of history, the trees learnt the timing of past respiratory waves, and that timing moves every year. The feature set above is the one that scored best on the 2024 origins. To be straight about it: I had also seen the earlier version’s 2025 scores before making that change, so the 2025 results aren’t a perfectly clean hold-out for the gradient boosting model. The choice holds up on 2024 alone, which is why I kept it.

Prediction intervals

The intervals are empirical, not model-based. For each origin, I take the log residuals, log(actual ÷ forecast), from the previous 12 origins, split them by model and horizon week, and use the 10th and 90th percentiles as multiplicative bounds. The first 2025 interval therefore uses only 2024 errors, and no interval ever sees an actual from its own forecast window.

past = out[out["origin"].isin(origins[i - INTERVAL_WINDOW : i]) & (out["date"] <= origin)]
quantiles = past.groupby(["model", "week"])["log_residual"].quantile(list(INTERVAL))

Prediction intervals for planners covers this method and quantile regression in more depth.

Results

Errors are in attendances per day. Bias is the mean of forecast minus actual, so a negative number means under-forecasting. MASE is scaled by the in-sample error of a one-week seasonal naive at each origin; the forecast accuracy metrics article explains that choice.

Data table

Model	MASE
Seasonal naive	1.09
Exponential smoothing	0.81
Gradient boosting	0.79
Average of ETS and GBM	0.79

Synthetic data. Source: projects/ed-demand-forecasting.

The seasonal naive scores above 1 because its denominator is a one-week-ahead naive forecast, and forecasting up to four weeks ahead is harder. The other three beat it by a wide margin. The average edges the GBM, 0.787 against 0.791, but that gap is far smaller than the month-to-month swings, and the GBM won six of the twelve months on its own. I’d call them tied.

Accuracy by horizon

Data table

Days ahead	Seasonal naive	Exponential smoothing	Gradient boosting	Average of ETS and GBM
1	29.2	25.4	19.8	22.6
2	47.5	31.6	30.1	30.8
3	27.3	19.2	27.8	23.5
4	37.1	24.0	24.3	24.1
5	30.7	28.6	32.1	28.2
6	18.9	15.3	16.7	15.9
7	44.3	31.9	29.9	30.8
8	23.8	27.6	22.5	25.0
9	37.5	16.9	15.4	16.2
10	20.8	15.8	13.9	13.0
11	37.9	27.1	26.3	26.1
12	27.7	23.3	25.9	24.4
13	36.4	32.3	30.8	30.5
14	39.0	24.6	23.3	23.5
15	29.3	25.6	25.6	25.5
16	41.7	21.2	24.8	23.0
17	35.6	28.6	26.4	27.5
18	33.7	22.6	16.1	19.3
19	35.3	25.8	29.3	27.1
20	30.6	26.0	24.3	25.1
21	38.3	20.8	19.0	18.7
22	32.9	25.6	25.6	24.9
23	32.0	18.9	17.2	17.9
24	38.1	26.9	24.4	25.7
25	42.4	32.3	32.7	31.0
26	28.9	30.0	26.1	26.6
27	30.2	27.9	25.5	26.3
28	31.8	22.4	22.5	22.3

Synthetic data. Source: projects/ed-demand-forecasting. Each point averages at most 12 days, so expect wobble.

Error barely grows across the 28 days. The average’s MAE by horizon week runs 25.1, 22.6, 23.7 and 24.9. The level of demand moves slowly, and most of the remaining error is day-to-day noise that no model can predict at any horizon.

That noise has a size. A forecast that knew the planted expected value for every day, with no error at all, would still have an MAE of 20.0 against the recorded counts. The best model reaches 24.1. Measured against the planted expected value instead of the noisy actual, the GBM’s own error is 13.0 and the average’s 13.1, against 14.7 for ETS and 25.7 for the seasonal naive.

Intervals

Data table

Model	Coverage
Seasonal naive	74.3%
Exponential smoothing	75.5%
Gradient boosting	75.8%
Average of ETS and GBM	76.4%

Synthetic data. Source: projects/ed-demand-forecasting. Intervals come from past backtest residuals by horizon week.

Every model’s 80% interval held on 74% to 76% of days, so the intervals are a few points too narrow. The 2024 residuals that calibrated them came from a calmer year: on the 2024 origins the average’s MAE was 21.9, against 24.1 in 2025, and 2025 had two months where every model was wrong in the same direction (below). For a planner the practical reading is that the interval missed on 23.6% of days rather than the 20% it promised. Widening it, or refreshing it after a bad month, would be the fix.

Where the best models still fail

Because the data is synthetic, each miss can be split into the model’s own error (forecast minus the planted expected value) and noise (actual minus expected). The chart shows only the model’s part.

Data table

Day type (days)	Seasonal naive	Exponential smoothing	Gradient boosting	Average of ETS and GBM
Ordinary days (313)	24.7	13.7	12.4	12.3
Bank holidays (8)	59.3	46.4	22.2	34.0
Working day after a bank holiday (5)	26.4	21.5	12.2	16.5
Planted heatwave days (9)	28.4	14.9	27.1	20.9

Synthetic data. Source: projects/ed-demand-forecasting. Heatwaves are planted in the data but unknown to every model.

Christmas Day. Actual 238, planted expected 241.9. The forecasts were 323 (GBM), 329 (average), 335 (ETS) and 341 (seasonal naive). The GBM has a Christmas flag, but it can’t use it: min_samples_leaf is 200, and the training window holds three Christmas Days, 84 rows across the 28 horizons. The trees can’t give Christmas a leaf of its own.

Data table

Date	Actual	Forecast (Average of ETS and GBM)	Forecast (Average of ETS and GBM) (low)	Forecast (Average of ETS and GBM) (high)
2025-11-10	407
2025-11-11	400
2025-11-12	328
2025-11-13	308
2025-11-14	362
2025-11-15	313
2025-11-16	296
2025-11-17	362
2025-11-18	372
2025-11-19	338
2025-11-20	311
2025-11-21	293
2025-11-22	304
2025-11-23	310
2025-11-24	417
2025-11-25	319
2025-11-26	349
2025-11-27	341
2025-11-28	372
2025-11-29	300
2025-11-30	317
2025-12-01	449	386	333	432
2025-12-02	379	353	304	394
2025-12-03	317	338	292	378
2025-12-04	375	341	294	381
2025-12-05	403	338	291	377
2025-12-06	338	319	275	357
2025-12-07	343	317	274	355
2025-12-08	411	380	344	431
2025-12-09	360	354	320	401
2025-12-10	325	338	305	383
2025-12-11	345	335	303	380
2025-12-12	352	336	304	381
2025-12-13	301	318	287	360
2025-12-14	326	318	288	361
2025-12-15	438	376	334	424
2025-12-16	330	356	316	401
2025-12-17	317	338	300	381
2025-12-18	326	334	297	377
2025-12-19	269	335	298	378
2025-12-20	271	318	282	358
2025-12-21	325	320	284	361
2025-12-22	355	388	347	440
2025-12-23	343	354	317	401
2025-12-24	317	338	303	383
2025-12-25	238	329	295	373
2025-12-26	298	329	295	373
2025-12-27	284	319	286	362
2025-12-28	325	317	284	359

Synthetic data. Source: projects/ed-demand-forecasting. Gaps in the actual line are data-feed outage days.

Bank holidays. Here the flags earn their place. The GBM’s own error on the eight 2025 bank holidays was 22.2, against 46.4 for ETS, which forecasts a bank holiday as if it were an ordinary weekday. Averaging the two dilutes the gain to 34.0, so the “best” model is worse than the GBM alone on exactly the days a planner worries about.

Heatwaves. No model knows about heat. On 12 to 14 August the actuals were 382, 368 and 362, against the average’s 305, 290 and 288. ETS’s smaller heatwave error in the chart is luck, not insight: its forecasts sat higher than the GBM’s on all nine heatwave days, for reasons that have nothing to do with heat.

Data table

Date	Actual	Forecast (Average of ETS and GBM)	Forecast (Average of ETS and GBM) (low)	Forecast (Average of ETS and GBM) (high)
2025-07-11	323
2025-07-12	284
2025-07-13	334
2025-07-14	358
2025-07-15	302
2025-07-16	321
2025-07-17	315
2025-07-18	289
2025-07-19	324
2025-07-20	296
2025-07-21	323
2025-07-22	305
2025-07-23	282
2025-07-24	285
2025-07-25	314
2025-07-26	255
2025-07-27	248
2025-07-28	267
2025-07-29	307
2025-07-30	290
2025-07-31	286
2025-08-01	284	287	247	321
2025-08-02	264	270	232	302
2025-08-03	272	268	231	300
2025-08-04	354	325	280	364
2025-08-05	320	303	261	339
2025-08-06	300	290	250	325
2025-08-07	289	289	249	324
2025-08-08	275	288	258	314
2025-08-09	292	272	244	297
2025-08-10	227	270	242	295
2025-08-11	380	326	292	356
2025-08-12	382	305	273	333
2025-08-13	368	290	260	316
2025-08-14	362	288	258	314
2025-08-15	263	288	256	325
2025-08-16	269	272	242	308
2025-08-17	312	270	240	305
2025-08-18	292	317	282	358
2025-08-19	274	306	272	345
2025-08-20	281	291	258	329
2025-08-21	299	291	258	328
2025-08-22	301	288	260	317
2025-08-23	266	276	250	304
2025-08-24	268	275	248	303
2025-08-25	335	316	286	348
2025-08-26	432	329	298	362
2025-08-27	271	293	265	323
2025-08-28	265	292	264	322

Synthetic data. Source: projects/ed-demand-forecasting. Gaps in the actual line are data-feed outage days.

Noise that looks like a miss. On 26 August, the day after the summer bank holiday, 432 people attended. The planted expected value was 349, and the GBM forecast 351. With real data this would look like an 81-attendance model failure, and someone would be tempted to fix the model. It was noise.

Level shifts. February was over-forecast by 23.5 to 27.0 a day by every model. The planted winter wave faded after January, the planted expected level fell from 334.7 to 314.7, and the models carried January’s level forward. October was under-forecast by 27.4 to 30.3 a day by ETS, the GBM and the average, because the planted expected level jumped from 304.9 in September to 334.1. The seasonal naive, which just copies the last week of September, was least wrong in October, the only month it won.

Limits

• The seasonal shape is planted, not observed. I put the winter peak into attendances. NHS England’s monthly A&E report for June 2016 notes that A&E attendances usually peak in the summer months, while emergency admissions generally peak in winter. For a real hospital, the seasonal shape has to come from its own data.
• No COVID-era disruption. Real 2021 data would need special handling, or excluding from training.
• Twelve origins is a small test. 335 scored days per model is enough to separate the seasonal naive from the rest, not enough to separate the GBM from the average.
• Thin interval calibration. Each horizon week’s interval rests on about 84 past residuals, 12 origins by 7 days.
• One daily series. Rotas need arrivals by hour, and bed plans need admissions, not attendances.

What I’d do next

• Use each model where it’s strong. Take the GBM on flagged days and the average elsewhere, and test that rule on the 2024 origins before believing it.
• Handle Christmas explicitly. A multiplicative adjustment estimated from past Christmases, or a smaller minimum leaf size, checked on the 2024 origins so the rest of the year doesn’t pay for it.
• Add a weather forecast. Temperature for heatwaves, using forecast temperatures available at the origin, not observed ones, to avoid leaking the future.
• Adaptive intervals. Widen the band after a month where coverage slips, or fit quantile GBMs with loss="quantile" and compare their pinball loss with the empirical intervals.
• More origins. Weekly origins would give about four times as many forecasts at each horizon, and a firmer ranking.
• Turn it into a bed plan. Forecast admissions instead of attendances and feed them to the bed occupancy simulation as its arrival rate.
• Keep the parts coherent. Separate forecasts for ambulance and walk-in arrivals won’t add up to the total; hierarchical forecast reconciliation makes them agree.

Run it yourself

uv run python projects/ed-demand-forecasting/run.py

It regenerates the data, runs all 24 origins, rewrites every chart and download on this page, and prints the tables quoted here, in about 20 to 30 seconds. A rerun gives byte-identical files. The full code is in the download below, with a README that explains each file.