Forecasting emergency medical service call arrival rates. (English) Zbl 1223.62161

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

Ann. Appl. Stat. 5, No. 2B, 1379-1406 (2011).

Summary: 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.

MSC:

62P10	Applications of statistics to biology and medical sciences; meta analysis
62H25	Factor analysis and principal components; correspondence analysis
62M20	Inference from stochastic processes and prediction
62M10	Time series, auto-correlation, regression, etc. in statistics (GARCH)
62P99	Applications of statistics
65C60	Computational problems in statistics (MSC2010)

Keywords:

ambulance planning; dynamic factor model; nonhomogeneous Poisson process; integer-valued time series; smoothing splines

Software:

mgcv; gss; gam; R; gamair

References:

[1]	Andrews, B. and Cunningham, S. (1995). LL Bean improves call-center forecasting. Interfaces 25 1-13.
[2]	Andrews, B. and Parsons, H. (1993). Establishing telephone-agent staffing levels through economic optimization. Interfaces 23 14-20.
[3]	Bianchi, L., Jarrett, J. and Choudary Hanumara, R. (1998). Improving forecasting for telemarketing centers by ARIMA modeling with intervention. Internat. J. Forecasting 14 497-504.
[4]	Borchers, D., Buckland, S., Priede, I. and Ahmadi, S. (1997). Improving the precision of the daily egg production method using generalized additive models. Canad. J. Fisheries and Aquatic Sciences 54 2727-2742.
[5]	Brown, L., Gans, N., Mandelbaum, A., Sakov, A., Shen, H., Zeltyn, S. and Zhao, L. (2005). Statistical analysis of a telephone call center: A queueing-science perspective. J. Amer. Statist. Assoc. 100 36-50. · Zbl 1117.62303 · doi:10.1198/016214504000001808
[6]	Channouf, N., L’Ecuyer, P., Ingolfsson, A. and Avramidis, A. (2007). The application of forecasting techniques to modeling emergency medical system calls in Calgary, Alberta. Health Care Management Science 10 25-45.
[7]	Cox, D. R. and Isham, V. (1980). Point Processes . Chapman & Hall, London. · Zbl 0441.60053
[8]	Daskalov, G. (1999). Relating fish recruitment to stock biomass and physical environment in the Black Sea using generalized additive models. Fisheries Research 41 1-23.
[9]	Ferland, R., Latour, A. and Oraichi, D. (2006). Integer-valued GARCH process. J. Time Ser. Anal. 27 923-942. · Zbl 1150.62046 · doi:10.1111/j.1467-9892.2006.00496.x
[10]	Fokianos, K., Rahbek, A. and Tjøstheim, D. (2009). Poisson autoregression. J. Amer. Statist. Assoc. 104 1430-1439. · Zbl 1205.62130 · doi:10.1198/jasa.2009.tm08270
[11]	Goldberg, J. B. (2004). Operations research models for the deployment of emergency services vehicles. EMS Management J. 1 20-39.
[12]	Gross, D. and Harris, C. M. (1998). Fundamentals of Queueing Theory , 3rd ed. Wiley, New York. · Zbl 0949.60002
[13]	Gu, C. (1992). Cross-validating non-Gaussian data. J. Comput. Graph. Statist. 1 169-179.
[14]	Gu, C. (2010). gss: General smoothing splines. R Package Version 1.1-3.
[15]	Harrison, J., Zeevi, A. and Shum, S. (2005). A method for staffing large call centers based on stochastic fluid models. Manufacturing & Service Operations Management 7 20-36.
[16]	Hastie, T. (2009). gam: Generalized additive models. R Package Version 1.01.
[17]	Hastie, T. J. and Tibshirani, R. J. (1990). Generalized Additive Models. Monographs on Statistics and Applied Probability 43 . Chapman & Hall, London. · Zbl 0747.62061
[18]	Hastie, T. and Tibshirani, R. (1993). Varying-coefficient models. J. Roy. Statist. Soc. Ser. B 55 757-796. · Zbl 0796.62060
[19]	Henderson, S. G. (2005). Should we model dependence and nonstationarity, and if so how? In Proceedings of the 37th Conference on Winter Simulation (M. Kuhl, N. Steiger, F. Armstrong and J. Joines, eds.) 120-129. IEEE, Piscataway, NJ.
[20]	Henderson, S. G. (2009). Operations research tools for addressing current challenges in emergency medical services. In Wiley Encyclopedia of Operations Research and Management Science (J. J. Cochran, ed.). Wiley, New York.
[21]	Kolesar, P. and Green, L. (1998). Insights on service system design from a normal approximation to Erlang’s delay formula. Production and Operations Management 7 282-293.
[22]	Lopes, H. F. and West, M. (2004). Bayesian model assessment in factor analysis. Statist. Sinica 14 41-67. · Zbl 1035.62060
[23]	Matteson, D. S. and Tsay, R. S. (2011). Constrained independent component analysis. Unpublished manuscript.
[24]	McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models , 2nd ed. Chapman & Hall, London. · Zbl 0744.62098
[25]	Opsomer, J., Wang, Y. and Yang, Y. (2001). Nonparametric regression with correlated errors. Statist. Sci. 16 134-153. · Zbl 1059.62537 · doi:10.1214/ss/1009213287
[26]	Pierce, D. A. and Schafer, D. W. (1986). Residuals in generalized linear models. J. Amer. Statist. Assoc. 81 977-986. · Zbl 0644.62076 · doi:10.2307/2289071
[27]	R Development Core Team (2009). R: A Language and Environment for Statistical Computing . R Foundation for Statistical Computing, Vienna, Austria.
[28]	Setzler, H., Saydam, C. and Park, S. (2009). EMS call volume predictions: A comparative study. Comput. Oper. Res. 36 1843-1851.
[29]	Shen, H. and Huang, J. Z. (2008a). Forecasting time series of inhomogeneous Poisson processes with application to call center workforce management. Ann. Appl. Statist. 2 601-623. · Zbl 1400.62350 · doi:10.1214/08-AOAS164
[30]	Shen, H. and Huang, J. (2008b). Interday forecasting and intraday updating of call center arrivals. Manufacturing and Service Operations Management 10 391-410.
[31]	Stock, J. H. and Watson, M. W. (2002). Macroeconomic forecasting using diffusion indexes. J. Bus. Econom. Statist. 20 147-162.
[32]	Swersey, A. (1994). The deployment of police, fire, and emergency medical units. In Handbooks in Operations Research and Management Science 6 151-200. North-Holland, Amsterdam.
[33]	Takane, Y. and Hunter, M. A. (2001). Constrained principal component analysis: A comprehensive theory. Appl. Algebra Engrg. Comm. Comput. 12 391-419. · Zbl 1040.62050 · doi:10.1007/s002000100081
[34]	Tsai, H. and Tsay, R. S. (2010). Constrained factor models. J. Amer. Statist. Assoc. 105 1593-1605. · Zbl 1388.62179 · doi:10.1198/jasa.2010.tm09123
[35]	Tych, W., Pedregal, D., Young, P. and Davies, J. (2002). An unobserved component model for multi-rate forecasting of telephone call demand: The design of a forecasting support system. Internat. J. Forecasting 18 673-695.
[36]	Weinberg, J., Brown, L. D. and Stroud, J. R. (2007). Bayesian forecasting of an inhomogeneous Poisson process with applications to call center data. J. Amer. Statist. Assoc. 102 1185-1198. · Zbl 1333.62297 · doi:10.1198/016214506000001455
[37]	Whitt, W. (2002). Stochastic-Process Limits . Springer, New York. · Zbl 0993.60001 · doi:10.1007/b97479
[38]	Wood, S. N. (2003). Thin plate regression splines. J. R. Stat. Soc. Ser. B Stat. Methodol. 65 95-114. · Zbl 1063.62059 · doi:10.1111/1467-9868.00374
[39]	Wood, S. N. (2006). Generalized Additive Models: An Introduction with R . Chapman & Hall, Boca Raton, FL. · Zbl 1087.62082
[40]	Wood, S. (2008). mgcv: GAMs with GCV/AIC/REML smoothness estimation and GAMMs by PQL. R Package Version 1.6-1.
[41]	Woodard, D. B., Matteson, D. S. and Henderson, S. G. (2010). Stationarity of count-valued and nonlinear time series models. Unpublished manuscript.