Journal of Medical Systems

, 38:107 | Cite as

Time Series Modelling and Forecasting of Emergency Department Overcrowding

  • Farid Kadri
  • Fouzi Harrou
  • Sondès Chaabane
  • Christian Tahon
Systems-Level Quality Improvement
Part of the following topical collections:
  1. Systems-Level Quality Improvement

Abstract

Efficient management of patient flow (demand) in emergency departments (EDs) has become an urgent issue for many hospital administrations. Today, more and more attention is being paid to hospital management systems to optimally manage patient flow and to improve management strategies, efficiency and safety in such establishments. To this end, EDs require significant human and material resources, but unfortunately these are limited. Within such a framework, the ability to accurately forecast demand in emergency departments has considerable implications for hospitals to improve resource allocation and strategic planning. The aim of this study was to develop models for forecasting daily attendances at the hospital emergency department in Lille, France. The study demonstrates how time-series analysis can be used to forecast, at least in the short term, demand for emergency services in a hospital emergency department. The forecasts were based on daily patient attendances at the paediatric emergency department in Lille regional hospital centre, France, from January 2012 to December 2012. An autoregressive integrated moving average (ARIMA) method was applied separately to each of the two GEMSA categories and total patient attendances. Time-series analysis was shown to provide a useful, readily available tool for forecasting emergency department demand.

Keywords

Emergency department Overcrowding Time series ARMA Forecasting 

Introduction

Emergency departments (EDs) are an important component of healthcare systems because they provide immediate and essential medical care for patients, but they are also the most overcrowding component. Unfortunately, the role of the ED as a safety net is now under the threat of overcrowding [1, 2]. The causes of this overcrowding are i) inadequate staffing, hospital bed shortages and inpatient boarding [3, 4], ii) increase in demand (patient flow) for EDs services [5, 6], and iii) influenza season (epidemic period) [7] which often generates a large flows of patients. So, the influx of patients is one of the causes of this overcrowding. The problem of the influx of patients, which is a direct cause of ED overcrowding in several nations [8, 9, 10, 11, 6], affects both private and public health-care systems.

The management of patient flow is a challenge faced by many hospitals, in particular emergency departments. One approach to alleviate and mitigate problems associated with ED overcrowding is to forecast levels of demand for ED care in advance (hours, days) in order to give health-care staff an opportunity to prepare for this demand.

The ability to predict demand in emergency departments is crucial for designing strategies aimed at avoiding overcrowding that may lead to strain situations in these establishments. Time series can be used by managers to make current decisions and plans based on long and/or short-term forecasting.

The main objective of this work was to propose a statistical model based on univariate time series, which was then used to predict daily patient attendances at the paediatric emergency department (PED) in Lille regional hospital centre, France. For this, in addition to the total daily arrivals (expected and unexpected arrivals) at the PED, we considered two significant categories from the GEMSA (Groupes d’Etude Multicentrique des Services d’Accueil, Multicentric Emergency Department Study Group in English) classification: unplanned patients (unexpected arrivals) that return home after PED care, and unplanned patients that are hospitalized after emergency care.

The results yielded by these models will assist hospital managers in their decision-making process to utilize and allocate medical staff better, taking the fluctuant demand on the system and individual zones in the emergency department into consideration. The remainder of this paper is organized as follows: the context and the problem to be resolved in this study are presented in the next two subsections. In second section presents the time-series modelling and how it can be used for forecasting demand in hospital emergency departments. Third section presents the modelling technique used in this work. Fourth section presents a brief description of the data used. The data analysis and validation of the models using real data are presented in Fifth section. Finally, sixth section reviews the main points discussed in this work and concludes the study.

Context of the study

Nowadays, with the growing demand for emergency medical care, the management of hospital emergency departments (EDs) has become increasingly important. For example, according to the reports by the Institute of Medicine of the National Academies (2006) [12], between 1993 and 2003, visits to emergency departments in the United States increased by 26 % while the number of EDs were reduced by 9 % [13]. In France, as well as abroad, much effort has been made over the past few decades to improve emergency department management. However, the number of visits to EDs in France has rapidly increased. Between 1996 and 1999, the annual number of visits to EDs increased by 5.8 %, and increased by 43 % between 1990 and 1998 [14]. In addition, according to the annual public report published by the Medical Emergencies Court of Auditors, the annual number of ED visits doubled from 7 million to 14 million between 1990 and 2004 [15]. Increasing patient demand in EDs is due to many factors, including proximity to these establishments, a need for examinations and to obtain rapid expert medical advice [16].

Modelling and forecasting daily patient volumes provides useful information for hospital emergency departments which may be utilized for allocating resources and planning future expansion. Accurate prediction of patient attendances in an emergency department will help ED management to plan better for the future. For example, organisation of the staff roster and efficient allocation of the resources required to provide a good service in emergency departments. In addition, accurate prediction of daily patient attendances may provide useful information to improve ED efficiency by reducing the number of patients waiting for treatment and health care, as well as increasing the total number of patients treated. It is critical for hospital emergency departments to plan and create strategies to cope with the large number of patients arriving more effectively, and make the best use of the available resources.

Problem to be solved

The anticipation and/or management of patient influx is one of the most crucial problems in emergency departments (EDs) throughout the world [17]. To deal with this influx of patients, emergency departments require significant human and material resources, as well as a high degree of coordination among human and material elements [18, 19]. Unfortunately, these resources are limited. The consequences of this influx of patients has resulted in problems of emergency department overcrowding [11, 2]. ED overcrowding affects i) patients: very lengthy waits, patients leave the ED without being treated, violence of angry patients against staff, reduced access to emergency medical services and increase in patient mortality [20, 21], and ii) the quality of treatment and prognosis by medical staff who are often overloaded thus leading to a decrease in physician job satisfaction [22, 23]. Consequently, ED managers must control the problems related to the care load flow (demand) to improve the organization and the management of resources and the internal restructuring reflected by resource pooling, including technical platforms.

Time series and emergency department admissions

Due to the importance of forecasting the number of patient arrivals in the hospital system to maintain performance and to help enhance the management of hospital establishments, several forecasting techniques have been developed. These techniques can be broadly divided into two categories: qualitative and quantitative methods. Qualitative-based forecasting methods predict the future, usually using opinion and management judgment of experts in specified fields [24]. Quantitative methods [25], on the other hand, rely on mathematical models. These approaches are based on the analysis of historical data and assume that past data patterns can be used to forecast future data points. Techniques in this category are mostly based on time-series methods [26]. The advantage of time-series techniques is their simplicity and effectiveness, and they are more attractive for practical applications. Many time-series forecasting techniques are referenced in the bibliography, and they can be broadly categorized into two main classes: univariate and multivariate techniques. Univariate techniques involve the analysis of a single variable while multivariate analysis examines two or multiple variables simultaneously.

In the field of health care systems, increasing attention has been accorded to time-series models to predict patient arrivals in hospital establishments [27, 28, 29, 30]. The literature review shows that time-series analysis has been largely applied in the hospital sector to forecast patient arrivals, length of stay, and for projecting the utilization of inpatient days. In this regard, some of the research efforts made by previous researchers deserve mentioning [31] presented Holt-Winters exponential-smoothing and autoregressive integrated moving average models for forecasting monthly discharges and differences in occupancy in several hospitals [32] used three statistical methods (moving averages and seasonal decomposition methods) to predict the number of ED visits at any time of the week to a university hospital in New Mexico. They found that simple models can be used to describe the number of ED visits for each hour of the week [33] developed a statistical model for an emergency department at a hospital in Israel based on 3 years of daily time-series data. They used a regression model with a linear trend, and 3 types of seasonal factor representing the effects of the day of the week, the month of the year, and the type of day (holiday, half working day, or full working day) [34] employed two univariate time-series analysis methods (Box–Jenkins method and ad hoc approach) to model and forecast monthly patient volumes between 1986 and 1996 at King Faisal University family and community medicine primary health care clinic, Al-Khobar, Saudi Arabia [35] used Box–Jenkins models to forecast the number of daily emergency admissions and the number of beds occupied by emergency admissions on a daily basis at Bromley Hospitals NHS Trust in the United Kingdom [36] studied ED patterns in a hospital in Tenerife, Spain. Their analysis was based on a time series of the number of ED presentations every hour over a 6-year period (1997 to 2002) [37] used autoregressive (AR) and Welch spectral estimation methods to analyze the electroencephalogram (EEG) signals. The parameters of autoregressive (AR) method were estimated by using Yule–Walker. The results demonstrate superior performance of the covariance methods over Yule–walker AR and Welch methods [38] used ARIMA models to forecast ward occupancy, due to SARS infection, for up to 3 days in advance. The authors used seasonal decomposition methods to obtain detailed estimates of the effects of the hour of day, the day of week, and the week of the year on attendances [39] reviewed the past 25 years of research into time-series forecasting covering the period 1982–2005. The authors divided the time-series models into linear and nonlinear models. They discussed the strengths and weaknesses of each method [40] used two statistical methods: i) exponential smoothing and ii) Box–Jenkins Method, to forecast the number of patients present each month from 2000 to 2005 at the ED of a regional hospital in Victoria, Australia [41] applied an Adaptive Auto Regressive-Moving Average (A-ARMA) to analyze the electromyographic (EMG) signals. The author found that A-ARMA present good models with few parameters and their ability to, with small p and q, represent a very rich set of stationary time series in a parsimonious way [28] evaluated the use of four statistical methods (Box–Jenkins, time-series regression, exponential smoothing, and artificial neural network models) to predict daily ED patient volumes over 27 months (from 2005 to 2007) in three different hospital EDs in Utah and southern Idaho (USA). The authors compared four models with a benchmark multiple linear regression model already available in the emergency medicine literature. They claim that regression-based models incorporating calendar variables account for site-specific special-day effects, and allow for residual autocorrelation providing a more appropriate, informative, and consistently accurate approach for forecasting daily ED patient volumes [42] studied the temporal relationships between the demands for key resources in the emergency department (ED) and the inpatient hospital in 2006, and developed multivariate forecasting models. The authors compared the ability of the models they developed (multivariate models) to provide out-of-sample forecasts of ED census and the demands for diagnostic resources with a univariate benchmark model [43] used a time-series method to predict daily emergency department attendances. The authors applied an autoregressive integrated moving average (ARIMA) method separately to three acuity categories and total patient attendances at the ED of an acute care regional general hospital from July 2005 to March 2008. They concluded that time-series analysis provides a useful, readily available tool for predicting emergency department workload that can be used for staff roster and resource planning [44] presented a hybrid system (called Online Multi-Agent Monitoring System) to solve the patient’s critical state monitoring problems in real time. The medical monitoring system combines multi-agent approaches with time-series models (ARIMA).

There are also Autoregressive Moving Average models with eXternal inputs (ARMAX) and their variants (ARX, ARIMAX, SARIMAX…), which take into account the explanatory variables [26, 45]. The explanatory variables may represent meteorological measurements (temperature, wind direction…), or epidemics. Finally, multiple time-series models, which are natural extensions of the univariate ARMA models in the sense that a vector of dependent variables replaces the dependent variable, can be found in the literature. These models include vector autoregression (VAR), and its extensions (VMA, VARMA…), which is one of the most successful for multivariate time-series modelling [46, 47]. These models allow measurements from several time series to be processed simultaneously.

Time-series methods are interesting tools for predicting emergency department demand (number of patient arrivals). They are very useful as an initial significant analysis of patient flow because of their ease of use, implementation and interpretation within the framework of modelling and forecasting of emergency department overcrowding caused by the influx of patients.

In the next section, we present the method used in this study to analyze and forecast demand in the paediatric emergency department in Lille hospital centre, France.

Univariate time-series forecasting

ED patient arrivals are defined as a temporal pattern; the number of patient arrivals in the ED varies considerably according to the hour of day, the day of week, the week of month and the month of year. To deal with the large influx of patients, we need a modelling approach that can give decision-makers a reliable estimate of future patient arrivals based on current known levels.

A time series is a sequence of measurements indexed over time or a set of chronologically ordered observations, and time-series forecasting is the practice of using past and present values of one or more time series to predict future values of the time series. Generally, time-series analysis has two main objectives: one is to identify the nature of the phenomenon represented by the sequence of observations, the other is to forecast (or predict) future values of the time-series variable. Time-series forecasting methods assume that historical data is a good indicator of future demand.

In this paper, the main time series of interest is daily attendances in an emergency department. We wished to explore the practical problem of how well ED demand can be forecast, at least in the short term i.e., one day, using only this time-series data. Hence, in this study we focused on time-series analysis methods. The use of observations available at time t of the number of patient arrivals at the emergency department from a time series to forecast its value at some future time t + l can provide a basis for a variety of applications such as: customer demand, medication inventory control, economic and business planning, and general control of health-care systems.

The results presented in the literature show that ARMA models [48, 49] provide a very accessible time-series analytical tool both in terms of methodological constraints and the level of mathematical models used that are less complex linear stochastic equations. This analysis tool is mainly used for predicting future values and identifying the structure of the time series. For this purpose, in this study we used ARMA models to analyze and forecast daily patient arrivals at the paediatric emergency department in Lille regional hospital centre, France.

Univariate time series: ARIMA modelling

In practice, most phenomena present a time dimension that is usually dissimulated but should be taken into account. This is the case of patient flow data for which the patient flow \( {Y}_t \) at time t depends on previous values (\( {Y}_{t-1},{Y}_{t-2},\dots \)). The three univariate time-series models widely applied to model patient volume in emergency departments are autoregressive processes (AR), moving average processes (MA), and autoregressive moving average processes (ARMA). Adding non-stationary models to the mix leads to the integrated ARMA or autoregressive integrated moving average (ARIMA) model popularized in the work by [48]. ARMA models are very comprehensive linear models and can represent many types of linear relationships, such as autoregressive (AR), moving average (MA), and mixed AR and MA time-series structures.

The AR model is intuitively appealing because it describes how an observation directly depends upon one or more previous measurements plus white noise. For time-series observations\( {\; Y}_t \), the AR model of order p, which is also written as AR(p), is defined by:
$$ {Y}_t={\displaystyle {\sum}_{i=1}^p{a}_i{Y}_{t- i}+{\varepsilon}_t} $$
(1)
where \( {a}_j \) are non-seasonal AR parameters, and \( {\varepsilon}_t \) is zero-mean Gaussian noise \( {\varepsilon}_t\sim N\Big(0,{\sigma}^2\Big) \).
MA models describe how an observation depends upon the current white noise term as well as one or more previous errors. An MA model of order q, also denoted by MA(q), is defined as:
$$ {Y}_t={\varepsilon}_t+{\displaystyle {\sum}_{j=1}^q{b}_j{\varepsilon}_{t- j}} $$
(2)
where \( {b}_j \) are non-seasonal MA parameters. The ARMA model was first presented by Box and Jenkins [48]. ARMA models are more sophisticated stochastic models that combine elements of moving average methods and regression methods. One advantage of an ARMA process is that it involves fewer parameters than an MA or AR process alone [50]. The mixed AR and MA or ARMA model of order (p, q), also denoted autoregressive moving average, ARMA(p,q) is written as:
$$ {Y}_t={\displaystyle {\sum}_{i=1}^p{a}_i{Y}_{t- i}}+{\displaystyle {\sum}_{j=1}^q{b}_j{\varepsilon}_{t- j}+{\varepsilon}_t} $$
(3)
where \( {Y}_t \)is the variable to be predicted using previous samples of the time series, \( {\;\varepsilon}_t \)is a sequence of i.i.d. (independent and identically distributed) terms which have zero mean. The model parameters \( {a}_i \) (auto-regressive part) establish a linear relationship between the value predicted by the model at time t and the past values of the time series. The model parameters \( {b}_j \) (moving average part) establish a linear relationship between the value predicted by the model at time k and a Gaussian distribution of i.i.d. samples [51].
In time series analysis, the lag operator or backshift operator (B) defined as \( {\mathrm{B}}^{\mathrm{k}}{\mathrm{Y}}_{\mathrm{t}}={\mathrm{Y}}_{\mathrm{t}-\mathrm{k}}, \)operates on an element of a time series to produce the previous element. Using backshift operator B, we can define the Eq. (3) as follow:
$$ \begin{array}{l}\varnothing (B){Y}_t=\theta (B){\varepsilon}_t,\hfill \\ {}\varnothing (B)=\left(1-{a}_1 B-{a}_2{B}^2-\dots -{a}_p{B}^p\right)\hfill \\ {}\theta (B)=\left(1-{b}_q B-{b}_q{B}^2-\dots - {b}_q{B}^q\right)\hfill \end{array} $$
(4)

In relation to ARMA models, ARIMA models are extended to include differencing. ARIMA processing has been shown to be the most successful approach to model a broad variety of time series [29, 48, 49]. In many cases it is a very useful method to fit time-series data. However, a lot of phenomena tend to vary from one season to another, thus the seasonal fluctuations may cause problems when fitting an ARIMA model. The main feature of seasonal data is that there are high correlations among observations from a certain time span (day, week, month or quarter). When a time series exhibits seasonality, it is useful to try to exploit the correlation between the data at successive periods of time. SARIMA models will allow us to do that. If the data series contain seasonal fluctuations, the SARIMA model can be used to measure the seasonal effect or eliminate seasonality.

Identification of ARIMA models

The main steps required to obtain the orders (p, q) and parameters (\( {\mathrm{a}}_{\mathrm{i}} \), \( {\mathrm{b}}_{\mathrm{j}} \)) that appear in Eq. 3 are recalled below [48, 49] (see Fig. 1):
  • Time Series stationarity: Identification of the order of differentiation

    The identification of the structure of a time series starts with the observation of its stationarity. The non-stationarity of a time series can be seen on the graph of the series (the increase or decrease in the trend) and through patterns of the autocorrelation function (ACF) of the series. To make a stationary series, Box proposed applying a differentiation term, i.e. replacing the original series with the series of differences of adjacent points.

  • Model estimation: identification of AR and MA orders

    The order of both the AR and the MA parts must be estimated. Autocorrelation and partial autocorrelation computation is used to obtain these orders. The order of the autoregressive term present in the final ARIMA model is equal to p, which corresponds to the number of significant peaks in the Partial autocorrelation function (PACF). The order of the moving average term in the final ARIMA model is equal to q, which is the number of significant peaks in the Autocorrelation function (ACF).

    The estimated autoregressive parameters AR(p) and moving average MA(q) may be established using a non-linear estimation method of maximum likelihood (maximum likelihood estimator). This calculation is performed according to the orders of the ARIMA parameters defined in the model identification phase.

  • Selection and validation of the ARIMA models

    The most suitable models chosen should offer adequate predictions. Generally, in order to evaluate models, the data used are split into two groups: i) the training group, used to build the ARIMA model, and ii) the validation group, to evaluate the ARIMA model. The post-verification of the model is achieved using two tests: i) testing of the significance of the parameters and ii) validation of the white noise residue hypothesis. Model validation refers to various statistical specification tests to check the validity of the model.

    To evaluate the predictive abilities of the models, several measures of a model’s ability to fit data have been developed. The following can be cited: percentage variability in regression analyses (R2), Root Mean Square Error (RMSE), Mean Absolute Percentage Error (MAPE) [49], Absolute Deviation (MAD), Mean Square Error (MSE) [52], Bayesian Information Criterion (BIC) [53].

    Residual analysis is the most important step in model validation. Several statistical tests (Jarque-Bera, Lilliefors test…) and plots of the residuals (histogram, Henry’s line, probability–probability plot or PP plot, and the normal Quantile-Quantile plot or QQ plot…) can be used to examine the goodness of fit to the historical data of the model selected. The model residuals are described as good if they have various properties: normality, homoscedasticity, and independence.

Fig. 1

Time-series modelling using ARIMA (Box and Jenkins) procedure

Data analysis

In this section we describe the data sources and the data on which we based our analysis and the forecasting models that we used.

Description of the data used

Lille Regional Hospital Centre (CHRU) serves four million inhabitants in Nord-Pas-de-Calais, a region characterized by one of the largest population densities in France (7 % of the French population). The paediatric emergency department (PED) in Lille regional hospital centre (CHRU) is open 24 h a day and receives 23 900 patients a year on average. Besides its internal capacity, the PED shares many resources, such as administrative patient registration, clinical laboratory, scanner, magnetic resonance imaging (MRI), X-rays and blood bank, with other hospital departments.

This retrospective study was conducted utilizing a dataset extracted from the database of Lille regional hospital centre paediatric emergency department (PED). The data used in this paper are the time series of daily patient attendances at Lille regional hospital centre PED, from January 2012 to December 2012.

The purpose of this paper is to describe how statistical forecasting methods were used to make short-term (i.e. one day) predictions for daily ED attendances at Lille regional hospital centre. Before any attempt to model and forecast three times series (G2, G4 of GEMSA classification and total daily attendances, described in the next section), it was critical to conduct preliminary descriptive analyses of the data, paying particular attention to the identification of important features such as seasonal patterns, cyclical variations, trends, outliers, and any other noteworthy fluctuations in the series.

Analysis of the data used

Descriptive account of arrivals

Figure 2 provides the actual total number of patient arrivals per month from January 2012 to December 2012. In general, the flow of patients varied between winter and/or epidemic periods (November – March) and normal periods (April – October). It appears that the traffic was relatively light from July to September, as shown in this figure.
Fig. 2

Actual number of arrivals per month from January 2012 to December 2012

Figure 3 presents the daily arrivals at the paediatric emergency department over the entire period (from January 2012 to December 2012). This figure shows that the daily arrivals do not tend to increase or decrease over the duration of the data set. So, on average over the entire period of year (from January to December 2012), it appears that there is no trend in this time series. The number of patients arriving at the PED varies considerably according to the day of the week, as shown by the height of individual spikes in Fig. 3. This could be explained by:
Fig. 3

Daily paediatric emergency department arrivals from January to December 2012

  • More daily arrivals on certain days of the week: more arrivals were observed on Sundays and Mondays than the rest of the week, as shown in Fig. 4. According to this Figure, with the cumulative distribution function, CDF = 0.5, a difference of 11 patients per day of the week was recorded, with arrival numbers of 59 patients registered on Wednesday and 71 patients on Monday. Also, in 80 % of cases (CDF = 0.8), a difference of 15 patients was recorded between Wednesday and Monday.
    Fig. 4

    Distribution function per day of the week of the number of patients received at the PED for the period of 2012

  • Special event days that may lead to abnormally high PED arrivals, like holidays, sporting events and festivals in the region.

Classification of patients in emergency departments

Two main schemes exist by which such patients may be categorized: the Multicentric Emergency Department Study Group (Groupes d'Etude Multicentrique des Services d’Accueil, GEMSA), and the clinical classification of emergency patients called CCMU (Classification Clinique des maladies des Urgences) [54]. This paper only focuses on the GEMSA classification, which was developed by the Commission of Emergency Medicine of the French Resuscitation Society (Commission de Médecine d’Urgence de la Société de Réanimation de langue Française). The GEMSA classification identifies six groups of patients according to the outcome on leaving the ED, as summarized in Table 1. Each group is associated with a different care load. The classification criteria were established according to the input and output mode of the patients, and the programming or not of the care activity. The GEMSA classification can provide useful information about the patient’s arrival, which may be planned or unplanned, and/or significant and prolonged support. In addition, GEMSA classification traces the organization of the care activities and the patient’s pathway within the emergency department. This classification could be used to predict resource consumption in an ED. Groups G4 and G6 are characterized by an intense workload (heavy workload) for the medical and nursing staff and the utilization of significant radiological as well as biological means.
Table 1

Patient category and admission mode according to the GEMSA classification

GEMSA classification

Admission mode

Description

GEMSA 1 (G1)

Unplanned

Patient dead on arrival or died before any resuscitation

GEMSA 2 (G2)

Unplanned

Patient not convened, unexpected arrival, returned home after emergency care

GEMSA 3 (G3)

Planned

Patient convened, expected arrival, returned home after emergency care

GEMSA 4 (G4)

Unplanned

Patient not convened, unexpected arrival, hospitalized after emergency care

GEMSA 5 (G5)

Planned

Patient convened, expected arrival, hospitalized after emergency care

GEMSA 6 (G6)

Unplanned

Patient requiring immediate or prolonged care (intensive care)

Figure 5 shows the distribution of visits according to the GEMSA classification recorded from January to December 2012. As it can be seen in Fig. 4, over 80 % of patients who arrived in the PED were unplanned, G2; these patients usually returned home after treatment at the PED. G4, unexpected arrivals, accounted for over 10.7 %; most of these patients were hospitalized after treatment in the PED.
Fig. 5

Distribution of visits to the PED according to the GEMSA classification recorded from database 2012

Figures 6 and 7 present the daily arrivals at the paediatric emergency department from January 2012 to December 2012, for the two patient categories, i.e. G2 and G4, respectively. According to Figs. 6 and 7, it can be observed that the time series do not exhibit a significant trend. It can be noted that it is important to predict the total number of arrivals at the PED and the number of arrivals of patients in these two categories (G2 and G4), in order to avoid overcrowding, to organize resources better and to reduce the need for care services downstream of the PED.
Fig. 6

Daily G2 arrivals at the paediatric emergency department from January 2012 to December 2012

Fig. 7

Daily G4 arrivals at the paediatric emergency department from January 2012 to December 2012

The catalogue of descriptive statistics for the two categories (G2 and G4) is presented in Table 2. According to this table, the maximum daily patient arrivals at the PED were 82 and 19 for G2 and G4 respectively. Also, the interquartile range (difference between the first and the third quartile) was equal to 15 and 4 for the categories G2 and G4 respectively. In the case of category G4, the number of patients arriving at the PED varied between 5 and 9 in case of the first quartile and the third quartile respectively (see Table 2).
Table 2

Descriptive statistics of the two GEMSA categories (G2 and G4 arrivals)

Statistics

Category G2

Category G4

Mean

53

8

Median

52

7

Standard deviation

10.88

3.10

Maximum

82

19

Minimum

22

1

First quartile

45

5

Third quartile

60

9

The challenge in this study was to determine and validate forecasting models for daily patient attendances at the paediatric emergency department (PED) in Lille regional hospital centre, France. For this, we considered the two patient categories (G2 and G4) presented above, and total daily attendances at the PED. In the next section, we present the time-series forecasting method used in this study.

Experimental results

According to the GEMSA classification admission mode (Table 1), we considered two main types of patient arrivals: G 2 and G 4 unplanned admission modes, and total arrivals at the PED.

The first step in any time-series analysis is to plot the observations against time to obtain simple descriptive measures of the main properties of the series. Time-series plots provide a preliminary understanding of the time behaviour of the series. The graph should highlight important features of the series, such as trends and seasonality. The data series of daily attendances in emergency departments of GEMSA patient categories (i.e. G2, G4, and total patient attendances), collected from January 2012 to December 2012, were firstly used to develop a descriptive model. It is important to verify beforehand the periodicity (or seasonality) of the time series that needs to be modelled. To this end, the autocorrelation function analysis (ACF), also called correlogram, is usually used to determine the periodicity in the time series analyzed. It is well known that the distance between extremum points in the autocorrelation functions gives the period of the time series. Plots of these times series and the corresponding auto-correlation functions (ACF) are shown in Figs. 8 and 9, respectively.
Fig. 8

Daily attendances at the paediatric emergency department of G2, G4 and total arrivals from January 2012 to December 2012

Fig. 9

ACF of the three arrival types (G2, G4, and Total arrivals), Jan 2012-Dec 2012

From the time-series plot depicted in Fig. 8, it is apparent that the series does not have a long-term trend or seasonality. The stationarity of these three series was confirmed by the Phillips-Perron Test, i.e. p-value of the test = 0.01, which is lower than 5 % [55]. This led us to study stationary series with ARMA(p,q), (i.e. ARIMA(p,0,q)), According to Fig. 9, no apparent periodicity can be observed in the ACF. The similarity between the autocorrelation functions of the daily attendances time series can also be seen. The ACF also decreases exponentially to 0 as the lag increases. The presence of a significant short-term dependence (short-memory) can also be observed in all the time series. The short-memory property, or short-term dependence, describes the low-order correlation structure of a series. It is well known that the ARMA processes, also termed short memory, developed by Box and Jenkins, model short-term correlations in a time series.

These data, which are shown in Fig. 8, were scaled to be zero mean with a unit variance, and then used to develop a model using the following equation:
$$ \left({\widehat{Y}}_i=\frac{Y_i-\mu {Y}_i}{\sigma {Y}_i}\right) $$
(5)
These operations were reversed (multiplied by st.dev,\( {\;\sigma Y}_i \), and mean added, \( {\mu Y}_i \), for each time series) after the forecasting procedures so that the predictions correspond to the original series. The three transformed time series (G2, G4 and total) are depicted in Fig. 10.
Fig. 10

Auto-scaled daily attendances of the three time series

Proposed ARMA models

The best-fit model for G4 was ARMA(1,1), which is a non-seasonal stationary auto-regressive moving average model. The best-fit model for G2 and total attendances were ARMA (2, 1). The estimated parameters with their sampling standard deviations are shown in Table 3.
Table 3

Parameters of the three models

Parameters

G2, ARMA (2,1)

G4, ARMA (1,1)

Total, ARMA (2,1)

a1

0.36 ± 0.013

0.24 ± 0.016

0.22 ± 0.070

a2

0.07 ± 0.012

0.01 ± 0.069

b1

0.4 ± 0.030

0.235 ± 0.031

0.23 ± 0.030

The mathematical equations of the ARMA models for daily patient arrivals at the PED (G2, G4 and Total, respectively) are as follows:
$$ \left(1-0.36 B-0.07{B}^2\right){Y}_t^{G2}=\mu +\left(1-0.4 B\right){\varepsilon}_t $$
(6)
$$ \left(1-0.24 B\right){Y}_t^{G4}=\mu +\left(1-0.235 B\right){\varepsilon}_t $$
(7)
$$ \left(1-0.22 B-0.01{B}^2\right){Y}_t^{t otal}=\mu +\left(1-0.23 B\right){\varepsilon}_t $$
(8)
Figure 11 shows the observed and predicted time series for G2, G4 and total attendances (obtained by the selected models ARMA(2,1) ARMA(1,1) and ARMA(2,1) respectively). According to Fig. 11, the observed data for the three time series are well-adjusted by the three models selected (Total, G2 and G4). Furthermore, to illustrate the quality of the ARMA-based models selected, one common and simple approach is to regress predicted versus observed values (or vice versa). Ideally, on a plot of observed versus predicted values, the points should be scattered around a diagonal straight line (\( Y=\widehat{Y} \)). This plot can be used directly to present a goodness of fit as vertical deviations from the ‘perfect’ line indicate any biases present (either overall, or in certain sections of the data).
Fig. 11

Observed and predicted daily attendances at the emergency department per patient category, Jan 2012 - Dec 2012

The scatter plots of observed versus predicted attendances for the three best-fit models selected, and the regression line are shown in Fig. 12. It shows that the points are distributed along the regression line, i.e. for the three studied times series, the slope of the regression line between observed and predicted values is not significantly different from 1 and the y-intercept is not significantly different from 0. Therefore, the models were successful in accounting for most of the significant autocorrelations present in the data, and there is no indication of a curvature or other anomalies. This type of graph presents very similar information to a residuals versus predicted plot, widely used in statistical diagnostics [50]. According to Fig. 12, it can be seen that the scatter plots of observed and predicted daily attendances, or patient arrivals, at the paediatric emergency department (PED) indicate a reasonable performance of the selected models.
Fig. 12

Scatter plot of daily attendances at the emergency department per patient category, observed versus predicted, Jan 2012-Dec 2012

Model selection and validation

The models were selected and validated according to the reviewing criteria used for assessing model performance (“Identification of ARIMA Models” section). In this paper the models selected were evaluated by calculating the performance metrics RMSE and \( {R}^2 \). The quality of fit of the ARMA models selected was first evaluated in terms of the RMSE criterion, which represents the mean deviation of the predicted values with respect to the observed ones. RMSE is defined as:
$$ RMSE=\sqrt{\frac{{\displaystyle \sum }{\left(\widehat{Y}- Y\right)}^2}{n}} $$
(9)
where \( Y \) are the measured values, \( \widehat{Y} \) are the corresponding predicted values and n is the number of samples. This statistic is easier to interpret since it has the same units as the values plotted on the vertical axis. Low RMSE values indicate good model performance, that is, a good match between predicted and measured values. The RMSE value depends on the mean value of the variable, which hinders comparisons between models for predicting variables that have different mean values. Lower RMSE values (RMSE< 1) are indicative of a model that represents the observed values better.
We also evaluated the goodness of fit using the coefficient of determination, or \( {\mathrm{R}}^2 \), which corresponds to the percentage of variability explained by the model. A higher R2 indicates better model accuracy. The latter is defined as:
$$ {R}^2=1-\frac{SSR}{SSY} $$
(10)
where SSR is the sum of squared residuals, also known as the sum of squared errors of prediction. It is a measure of the discrepancy between the data observed and an estimation model (predicted values):
$$ SSR={\displaystyle {\sum}_{i=1}^N{\left({Y}_i-{\widehat{Y}}_i\right)}^2} $$
(11)
Here, \( {Y}_i \)are the observed values, and \( \widehat{Y_i} \) are the predicted values obtained from the selected model. A small SSR indicates a close fit of the model to the data. If RSS is equal to zero the model is perfect, i.e. for all N samples the observed values coincide with the predicted values. SSY is the sum of the squared differences between the observed values and the average observed data:
$$ SSY={\displaystyle {\sum}_{i=1}^N{\left({Y}_i-\overline{Y}\right)}^2} $$
(12)
where \( {Y}_i \)are the observed values, and \( \overline{Y} \) is the mean of the observed data. SSY is assumed to be a theoretical reference model where for each experimental response (observed data) a constant value is calculated as the average experimental response. The coefficient of determination \( {R}^2 \) is also defined as the square of the correlation between the response values and the predicted response values. This statistic measures how successful the fit is in explaining the variation of the data. A value of \( {R}^2=1 \), upper bound and desired value, denotes that the model fits the data perfectly. The degree of fit declines the further the statistics are from one. For example, a value of \( {R}^2=0.99 \) means that 99 % of the total sum of squares in the training set is explained by the model, and that only 1 % is in the residuals.
As shown in Table 4, the high \( {R}^2 \) and the low RMSE of the best-fit models show that all three models closely represented the observed time series.
Table 4

Statistical validation measures applied to data from Fig. 8

Parameters

G2, ARMA (2,1)

G4, ARMA (1,1)

Total, ARMA (2,1)

R2

0.99

0.99

0.99

RMSE

0.781

0.015

0.141

Residual analysis

The diagnostic verification of model adequacy is the last task in ARMA model building. Firstly, it is necessary to check whether the residual distribution follows a Gaussian distribution. To this end, the residual normality hypothesis was verified in this study by examining Henry’s line and the histogram. The latter is the simplest graphical tool that allows us to visually check the normality of the residuals. It can be recalled that for the line of Henry, the normality of the residual distribution can be recognized by the quality of the alignment of the points. If residuals are distributed according to a normal distribution then the points should be aligned.

When conducting a residual analysis, a scatter plot of the residuals is one of the most commonly used plots to detect nonlinearity and outliers. Generally, if the points in a residual plot are randomly dispersed around zero a linear regression model is appropriate for the data, if not a non-linear model is more appropriate. Figure 13 shows the residuals of the three ARMA models selected.
Fig. 13

Residuals of the three ARMA models selected: G2, G4, and Total time series

Figure 13 shows a random pattern, indicating a good fit for the three ARMA models selected. The verification of the normality of the residuals using Henry’s line and the histogram for G2 time series data is illustrated in Fig. 14a and b, respectively. Figure 15a and b illustrate the verification of the normality of the residuals of the G4 time series data. Finally, the verification of the Normality of the Residuals using Henry’s line and the histogram for the Total time series is illustrated in Fig. 16a and b. Figures 14, 15 and 16 show that the assumption of a normal distribution for the residuals appears to be reasonable.
Fig. 14

Gaussian distribution test a Henry’s line, b histogram, Category G2

Fig. 15

Gaussian distribution test a Henry’s line, b histogram, Category G4

Fig. 16

Gaussian distribution test a Henry’s line, b histogram, Total

We then checked the independence of the residuals (more specifically the absence of autocorrelation). The residuals were assumed to be non-autocorrelated. To determine whether residuals are non-autocorrelated, the ACF of the residuals is examined. If the assumption is satisfied, the ACF of the residuals should be large for any non-zero lag. If the ACF is significantly different from zero, this implies that there is dependence between observations. The ACF should have no significant spikes at early lags, or the residual error might not be random. Furthermore, the ACF of the residuals drawn for the selected models in Fig. 17 indicate that the residuals were not significantly different from a white noise series. According to Fig. 17, the residuals are approximately uncorrelated. As the residuals are normally distributed and uncorrelated, it can be deduced that the model fits the data well.
Fig. 17

ACF of residual errors a G2 b G4 c Total

Forecasting

Forecasting is a technique for predicting the future. In hospital management systems, forecasting is still needed because having partial knowledge from forecasting is better than having no knowledge. Thus, the better the management is able to estimate the future; the better it should be able to prepare for it.

Once a model has been fitted to the data, future values of the time series can forecast. In this sub-section, the fitted ARMA models from “Description of the Data Used” section are examined for their predictive capability. The models were compared with the actual data values, which were not included in the model fitting procedure. The fitted series and forecasts of total, G2 and G4 patient attendances time series are shown in Figs. 18, 19 and 20, respectively.
Fig. 18

ARMA (2,1) fitted total series (01 Nov–22 Dec 2012) and forecasted (22 Dec to 28 Dec 2012)

Fig. 19

ARMA (2,1) fitted G2 series (01 Nov–22 Dec 2012) and forecasted (22 Dec–28-Dec 2012)

Fig. 20

ARMA (1,1) fitted G4 series (01 Nov–22 Dec 2012) and forecasted (22 Dec–28 Dec 2012)

Table 5 presents the comparison between the observed data and the data predicted by the three models obtained by fitting January to December 2012. The standard deviation of errors in the forecasting period for the three ARMA models, G2, G4 and Total are given in Table 6.
Table 5

Comparison between the observed data and the predicted data obtained by fitting January to December 2012

Horizon H

Total

Forecast

Total error

G2

Forecast

G2 error

G4

Forecast

G4 error

1

76

73.5237

2.4763

66

62.8051

3.1949

7

8.8255

1.8255

2

71

69.2891

1.7109

65

62.4135

2.5865

6

8.3917

2.3917

3

69

67.9322

1.0678

60

62.0918

2.0918

7

8.5370

1.5370

4

65

66.4108

1.4108

57

61.7788

4.7788

5

8.6298

3.6298

5

73

66.4114

6.5886

62

61.4768

0.5232

8

8.6511

0.6511

6

66

65.6456

0.3544

56

61.1853

5.1853

8

8.6524

0.6524

7

71

65.9898

5.0102

56

60.9041

4.9041

5

8.6535

3.6535

Table 6

Standard deviations of forecast errors for 22 Dec to 28 Dec 2012

Series

Standard deviations of forecast errors

Total

2.28

G2

1.73

G4

1.78

According to Table 5, it can be noted that the mean forecast error for total arrivals is 2.66 (3 patients), with a forecasting error rate of 3.79 %. The maximum error rate is 9.03 %, recorded for horizon H = 5. The mean error for the second category, G2, is 3.32, with a forecasting error rate of 5.5 % (4 patients). The maximum error rate recorded for this category was 9.26 %. Also, the mean forecast error for the category G4 is 2.05 (2 patients), with a forecasting error rate of 31 %; this is due to the divergence of some actual data values from the average. The maximum error rate was 72 % (4 patients) recorded for horizon H = 7. According to Table 5, the ARMA (1,1) model selected for category G4, predicts the mean value of daily patient arrivals, as shown in Table 2. The mean number of arrivals per day at the PED in the case of category G4 was 8 patients, so 50 % of arrival numbers were between 6 and 9 patients per day. This model remains reasonable as it predicts a number of patients greater than or equal to the real mean number of patient arrivals.

According to Table 6, the maximum standard deviation of error recorded for total patient arrivals was less than 3. The maximum standard deviation of error for G2 and G4 was less than 2. The results indicate that the three models proposed for total, G2 and G4 arrivals provide an acceptable description of daily patient admissions to the paediatric emergency department (PED) in Lille hospital.

Conclusion

Daily patient arrivals at the paediatric emergency department (PED) in Lille hospital, France, were studied using univariate times-series analysis. Forecasting, a useful tool in hospital management systems, was used in this paper to study paediatric emergency department attendances. Accurate forecasting of patient attendances will of course facilitate timely planning of staff deployment and allocation of resources within such health care establishments.

Firstly, this paper reports the development of univariate ARMA models to describe daily attendances in a paediatric emergency department (PED). Based on the data analysed from January to December 2012, the best ARMA models for the three arrival categories (G2, G4 and total) were non-seasonal stationary auto-regressive moving average models. The results indicate that the models proposed provide an acceptable description of PED admissions. Secondly, the models developed were used to forecast the number of daily arrivals at the PED. The results in the case studied indicate that the forecasting performance of the models proposed is acceptable. The approach proposed and lessons learned from this study may assist other regional hospitals and their emergency departments in carrying out their own analyses to aid planning. The results have also shown the suitability of our approach in predicting the number of patient arrivals at the PED in Lille. The time series is essentially linear and therefore ARMA modelling offered robust predictions in many cases.

However, this study raises several questions about related series. The forecasting of ED visits on a finer time scale, such as hourly, would be very interesting. Accurate forecasting of these series would facilitate planning of nursing rosters and allocation of staff within the department, and could potentially assist in bed occupancy prediction. It may be possible to find a better model using other fitting techniques, which would lead to more accurate forecasts. The scope can be expanded to multivariate forecasting models. We could also forecast daily attendances at the PED using other explanatory variables such as meteorological measurements (temperature…), and epidemic events. To this end, the Autoregressive Moving Average model with eXternal inputs (ARMAX) could be used.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Farid Kadri
    • 1
    • 2
  • Fouzi Harrou
    • 3
  • Sondès Chaabane
    • 1
    • 2
  • Christian Tahon
    • 1
    • 2
  1. 1.Univ. Lille Nord de FranceLilleFrance
  2. 2.UVHC, TEMPO Lab., “Production, Services, Information” TeamValenciennesFrance
  3. 3.Chemical Engineering ProgramTexas A&M University at QatarDohaQatar

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