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Forecasting emergency medical service call arrival rates

David S. Matteson, Mathew W. McLean, Dawn B. Woodard, Shane G. Henderson

arXiv:1107.4919v1stat.AP

TL;DR

Hourly EMS call forecasting must handle low count volumes and estimates that affect staffing costs and response times. The paper combines integer-valued time-series models with dynamic latent factors and calendar covariates; it reports substantially improved forecast performance, with simulation results providing a practical measure of that performance.

  • Problem

    Rudimentary EMS arrival estimates can be noisy, and low hourly counts make continuous or Gaussian modeling problematic for forecasting ambulance demand.

    Source: §1 · p. 2

  • Method

    An integer-valued time-series model combines dynamic latent factors, constrained calendar effects, smoothing splines, and residual-dependence modeling for hourly call rates.

    Source: §1 · pp. 2, 4

  • Results

    The method substantially improves forecast performance relative to simpler methods, and the data-based simulation distinguishes their effectiveness in line with out-of-sample RMSE analysis.

    Source: §5 · p. 27

  • Takeaways & Limitations

    A data-based simulation can provide a practical measure of forecast performance, and relative operating cost can be extended to customized cost functions and applications.

    Source: §5 · pp. 27–28

    Smoothing-parameter selection may be unreliable when autocorrelation is ignored, particularly with too few days relative to smooth factors and loadings or with long-range dependence.

    Source: §3.3 · p. 10

Abstract

from arXiv · show

We introduce a new method for forecasting emergency call arrival rates that combines integer-valued time series models with a dynamic latent factor structure. Covariate information is captured via simple constraints on the factor loadings. We directly model the count-valued arrivals per hour, rather than using an artificial assumption of normality. This is crucial for the emergency medical service context, in which the volume of calls may be very low. Smoothing splines are used in estimating the factor levels and loadings to improve long-term forecasts. We impose time series structure at the hourly level, rather than at the daily level, capturing the fine-scale dependence in addition to the long-term structure. Our analysis considers all emergency priority calls received by Toronto EMS between January 2007 and December 2008 for which an ambulance was dispatched. Empirical results demonstrate significantly reduced error in forecasting call arrival volume. To quantify the impact of reduced forecast errors, we design a queueing model simulation that approximates the dynamics of an ambulance system. The results show better performance as the forecasting method improves. This notion of quantifying the operational impact of improved statistical procedures may be of independent interest.

1. Introduction.

EMS arrival-rate forecasts support ambulance planning, but hourly demand is often estimated with rudimentary methods and Gaussian models can be inaccurate when counts are low. The paper proposes count-valued hourly forecasting that combines dynamic factors, calendar covariates, smoothing splines, and serial-dependence models, reporting substantially improved forecast performance.

1. Introduction.

  • Noisy arrival-rate estimates can contribute to overstaffing and higher costs or understaffing and slower response times.The paper notes that common practice may average only four corresponding hourly observations; a formal time-series approach can account for week-to-week differences and neighboring hours.Source: §1 · p. 2
  • The proposed integer-valued dynamic factor model, with smoothing splines and time-series models for residual dependence, significantly reduces hourly call-arrival forecast error.Day-of-week and week-of-year effects enter through constraints on factor loadings; splines improve long-horizon forecasts, while the time-series component supports adaptive short-term forecasts.Source: §1 · p. 2
  • Prior call-center forecasting often treated integer observations as continuous, which is problematic at low arrival rates; Gaussian EMS models also perform poorly for typical low hourly counts.The paper also notes that Gaussian modeling conflicts with the Poisson assumption used in operations-research staffing methods.Source: §1 · pp. 2–3
  • The model assumes Poisson hourly volumes and a random intensity forecastable from past observations, partitioned into stationary and nonstationary components.Unlike a Cox process, the random intensity may depend on previous observations as well as its own history.Source: §1 · p. 4
  • The factor structure reduces model parameters, captures intraday patterns, and incorporates day-of-week and week-of-year effects through loading constraints.Smoothing splines impose smooth evolution in factor levels and loadings; stationary integer-valued time-series models capture remaining serial dependence.Source: §1 · p. 4

2. Notation and data description.

The analysis models repeated hourly call counts over 24-hour days as observations from a nonhomogeneous counting process with latent intensity. It uses Toronto EMS priority-call data from 2007–2008 and accounts for calendar effects and remaining serial dependence.

2. Notation and data description.

  • Hourly call counts are modeled as a nonhomogeneous counting process with latent nonnegative intensity, with counts conditionally Poisson given that intensity.The analysis treats an arrival as a call for which one or more ambulances were deployed.Source: §2 · pp. 5–6
  • Across a typical 24-hour day, call arrivals increase quickly in late morning, peak in early afternoon, then decline slowly to a trough between 5 and 6 a.m.Source: §2 · p. 5
  • The model conditions on past observed counts and deterministic covariates, including day-of-week and week-of-year information.The paper defines intensity as the conditional expectation given the covariates and prior observations, while noting that it may also depend on unobserved random variables.Source: §2 · p. 5
  • The conditional intensity inflation rate models remaining serial dependence after covariates, potentially reflecting sporadic events such as inclement weather or unusual traffic.The authors argue that explicitly modeling this dependence may improve short-term forecast accuracy.Source: §2 · p. 6
  • The data cover emergency-priority calls received by Toronto EMS from January 2007 through December 2008 when an ambulance was dispatched.Scheduled patient transfers are excluded; for multiple calls associated with one event, only the first arrival time is included, and calls lacking a reported location were removed.Source: §2 · p. 6
  • Seven days with recording gaps of at least two consecutive hours and New Year’s Eve and Day were excluded from the analysis.The New Year dates had intraday patterns fundamentally different from the rest of the year; removing the seven gap-affected days had negligible impact on results and simplified comparisons and simulation.Source: §2 · pp. 6–7

3. Modeling.

The model combines a dynamic latent factor representation of nonstationary intraday call arrivals with integer-valued time-series models for remaining serial dependence. Calendar covariates enter through constrained loadings, while smoothing splines support smooth factor and loading estimates.

3. Modeling.

  • The model combines integer-valued time-series models with dynamic latent factors to represent nonstationary intraday arrivals and remaining serial dependence.The factor structure is parsimonious; the time-series component captures dependence not explained by the factors.Source: §3 · p. 7

3.1. Dynamic latent factor model.

  • A small set of day-varying loadings combines shared intraday factors, reducing the intensity matrix’s dimensionality while preserving a common daily pattern.The log-scale factor representation ensures positive intensity estimates; the factors are orthogonal and constant across days.Source: §3.1 · pp. 7–8

3.2. Factor modeling with covariates via constraints.

  • Calendar constraints express loadings as L = HB, incorporating day-of-week effects and smoothly varying week-of-year effects.Rows of H can also encode groupings such as weekdays, weekends, or special days.Source: §3.2 · p. 8

3.3. Factor model estimation via smoothing splines.

  • Generalized additive models estimate smooth intraday factors and loading functions, with performance iteration offering computational efficiency but no general convergence guarantee.Automatic smoothing-parameter selection may break down when residual autocorrelation is ignored, especially with too few days or long-range dependence.Source: §3.3 · pp. 9–10

3.4. Adaptive forecasting with time series models.

  • Residual dependence persists across many lags, so the model adds a conditional intensity-inflation recursion; under constant baseline intensity, its linear form is an IntGARCH(1,1).The fitted recursion’s standardized residual autocorrelation appears adequately removed.Source: §3.4 · p. 11
  • Nonlinear alternatives include a threshold model that adapts more outside its central residual range and deterministic day-part regimes that can introduce jumps and have no stationarity.The deterministic regimes divide periods at 10 a.m. and 4 p.m.Source: §3.4 · p. 12

3.5. Estimation algorithm.

  • Estimation alternates factor-model updates, then fits the conditional intensity process by conditional maximum likelihood using the estimated baseline intensity.The recursion’s estimates depend on its initialization; contiguous observation blocks are reinitialized at ηt = 1.Source: §3.5 · pp. 12–14

4. Empirical analysis.

The empirical analysis evaluates factor fit, forecast performance, and operational implications using year-swapped Toronto EMS data. Results favor a four-factor constrained and smoothed model with residual time-series adjustment and show lower simulated costs.

4.1. Interpreting the fitted model.

  • With K=4, factors capture the mean diurnal pattern, weekday/weekend differences, and Friday/Saturday evening increases; constrained loadings vary smoothly through the year.The estimated intensity preserves each weekday's relative shape while changing smoothly over the weeks of the year.Source: §4.1 · pp. 15, 17, Figure 2 · p. 16

4.2. Goodness of fit and model selection.

  • Graphical tapering in goodness-of-fit improvement consistently selects K=4; additional factors can capture noise, and this selection approach does not fully account for uncertainty.The study also considers out-of-sample performance when assessing factor number.Source: §4.2 · pp. 18–19, Figure 4 · p. 19

4.3. Out-of-sample forecast performance.

  • Adding constraints, smoothing splines, and an IntGARCH residual adjustment successively improves forecast RMSE; nonlinear alternatives offer only slight gains over IntGARCH.With one training year, factor models begin to overfit at K=5; regime switching has the lowest RMSE for both test sets, although nonlinear models' out-of-sample performance is similar to IntGARCH.Source: §4.3 · pp. 19, 21, 22, Table 1 · p. 20

4.4. Queueing model simulation to approximate ambulance operations.

  • The queueing simulation uses an M/M/s system with constant service rates and approximates the EMS goal of reaching a target proportion of calls quickly by targeting immediate service.Staffing levels can change instantaneously at each period boundary in the simulation, unlike real practice.Source: §4.4 · p. 23
  • Across forecasting methods, mean hourly cost is lowest for the factor model with IntGARCH, followed by the factor model alone and simple prediction.The queueing model simplifies ambulance operations by omitting the spatial component.Source: §4.4 · pp. 23, 26, Figure 7 · p. 26

5. Conclusions.

The proposed forecasting method captures nonstationary call arrivals and improves forecast performance. A queueing simulation provides a practical measure of forecast performance, and the authors identify several directions for future work.

5. Conclusions.

  • The factor model captures nonstationary call-arrival behavior, while the CIIR process adaptively forecasts deviations from the diurnal pattern.The analysis also reports substantial forecast-performance improvements over simpler methods and assesses successive refinements, including covariates and smoothing splines.Source: §5 · p. 27
  • The data-based simulation distinguishes forecasting methods, and its results coincide with out-of-sample RMSE analysis.The authors describe it as a practical measure of forecast performance.Source: §5 · p. 27
  • Relative operating cost measures forecast performance and can be extended to customized cost functions and varied applications.Source: §5 · p. 28
  • Future work will examine additional covariates, other nonlinear time-series models, Bayesian methods for count-valued observations, and spatial or spatial-temporal arrival analysis.Source: §5 · p. 28

SUPPLEMENTARY MATERIAL

Supplementary materials provide additional tables and code for model estimation, RMSE calculations, and queueing-model simulation.

SUPPLEMENTARY MATERIAL

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