When looking at discharges as time series, autoregressive moving average models are the most popular [30-32]. Exponential smoothing techniques have also been used to forecast monthly [33] and daily patient flow [34].

Jones et al used the classical ARIMA to forecast daily bed occupancy in emergency department of a European hospital [30]. The model which included seasonality terms demonstrated reasonable performance to predict bed occupancy. The authors speculated whether nonlinear forecasting techniques could improve over ARIMA. A recent study confirmed the effectiveness of this forecasting technique in a US hospital setting [35]. ARIMA models were also successfully used to forecast the number of occupied beds during a SARS outbreak in a Singapore hospital [36]. A recent study used patient attendances in a pediatric emergency department to model daily demand using ARIMA [37].

Jones et al [34] compared the ARIMA mode with exponential smoothing and artificial neural networks to forecast daily patient volumes in emergency department. The study revealed no single model to be superior and concluded that seasonal patterns play a major role in daily demand.

Modeling using simulation is typically used to study the behavior of complex systems. An early work in investigated the effects of emergency admissions on daily bed requirements in acute care, using discrete-event stochastic simulation modeling [38]. Sinreich and Marmor [39] proposed a guide for building a simulation tool based on data from emergency departments of 5 Israeli hospitals. Their method analyzed the flow of patients clustered into 8 types along with time elements. The simulation demonstrated that patient processes are better characterized by type of the patients, rather than specific hospitals visited. Yeh and Lin used a simulation model to characterize patient flow through a hospital emergency department and reduced waiting times using a genetic algorithm [40]. A similar experiment was carried out in a geriatric department using a combination of discrete event simulation and queuing model to analyze bed occupancy [19].

Regression models analyze the relationship between the forecasted variable and features in the data. Linear regression that encoded monthly variations was used to forecast patient admissions over a 6-month horizon and outperformed quadratic and autoregressive models [41]. Another study used clustering and Principle Component Analysis PCA to find significant predictors from patient data to model emergency length of stay using linear regression [42]. A nonlinear approach using regression trees was proposed in forecasting patient admissions which demonstrated superior performance over a neural net framework [43].

Barnes et al used 10 predictors to model real-time inpatient length of stay in a 36-bed unit using an RF model [24].

Nonlinear regression is better suited to model the changing dynamics of patient flow. To characterize the outflow of patients from the ward, we resort to regression using RF, kNN, and SVR. In the area of pattern recognition, kNNs [44] are the most effective method that exploits repeated patterns. The kNN algorithm has been successfully applied to forecast to histogram time series in financial data [45]. The nonparametric regression using kNN has been successfully demonstrated for short-term traffic forecasting [46,47] and electricity load forecasting [48,49]. However, kNN regression has not been studied for patient flow.

Another powerful and popular regression technique, SVR, uses kernel functions to map features into a higher dimensional space to perform linear regression. Though this technique has not seen much application in medical forecasting, support vector machines have been successful in financial market prediction, electricity forecasting, business forecasting, and reliability forecasting [50].

Apart from the standard autoregressive methods, we use kNN, RFs, and SVR in forecasting next-day discharges. Because discharge patterns repeat over time, kNN regression can be applied to search for a matching pattern from past discharges. RFs and SVR regression are powerful modelling techniques requiring minimum tuning to effectively handle nonlinearity in the hospital processes.

Recently, RF forecasting was used to predict total patient discharges from a 36 bed unit in an urban hospital [24]. Apart from 4 demographic and 2 timing predictors, this study used 3 clinical predictors for patients: (1) reason for visit: identified by a physician and recorded using International Classification of Diseases: version 9 (ICD-9) diagnosis codes [51], (2) observation status: assigned to patients for monitoring purpose, and (3) pending discharge location. Total number of discharges was estimated from aggregate of individual patient length of stay.

The absence of real-time clinical information in our data makes calculating patient length of stay impossible. Instead, we resort to modelling next-day discharges by observing previous discharge patterns and examining demographics and flow characteristics in the ward.

Our study used retrospective data collected from a recovery ward in Barwon Health, a large public health provider in Victoria, Australia serving about 350,000 residents. Ethics approval was obtained from the Hospital and Research Ethics Committee at Barwon Health (number 12/83) and Deakin University. The total number of available beds depended on the number of staff assigned to the ward. On average, the ward had 36 staffed beds, but fluctuated between 20 and 80 beds with varying patient flow. The physicians in the ward had no teaching responsibilities.

The data for our study came from three tables in the hospital database, as shown in Table 1. Additional real-time data that described patient condition or disease progression were unavailable because diagnosis coding using medical codes is done after discharge. Patient flow was collected for a period of 4 years. Using the admission and discharge times for each patient, we calculated the daily discharges from our ward in study. A total of 12,141 patients were admitted into the ward with a median discharge of 8 patients per day from January 1, 2010, to December 31, 2014. Table 2 summarizes the main characteristics of our data.

A time series decomposition of our data revealed strong seasonal variations and high nonlinearity in daily discharge patterns. There was a defined weekly pattern–discharge from ward peaked on Fridays and dropped significantly on weekends (see Figure 2). This seasonal nature is in tune with previous studies [9,32]. Aggregating the daily discharges into a monthly time series revealed defined monthly patterns (see Figure 3). The data displayed no significant trend. In addition, the daily discharge pattern was found to be highly nonlinear. Our forecasting methods must be able to handle such data dynamics.

We describe the following diverse methods that are applicable to forecasting under complex data dynamics: (1) ARIMA, (2) autoregressive moving, (3) forecasting using kNN discharge patterns, (4) RF, and (5) SVR. Autoregressive methods model the temporal linear correlation between nearby data points in the time series. Nearest patterns lift this linearity assumption and assumes that short periods form repeated patterns. Finally, RF and SVR look for a nonlinear functional relationship between the future outcomes and descriptors in the past.

We extracted all data from the database tables (as in Table 1) for our ward in study. Patient flow was analyzed for a period of 5 years. We formatted our data as a matrix where each row corresponds to a day and each column represents a feature (descriptor). Two main groups of features were identified: (1) ward level and (2) patient level. Our feature creation process resulted in 20 ward-level and 88 patient-level predictors, as listed in Table 3. The ward-level descriptor: trend of next-day discharge was calculated by fitting a locally weighted polynomial regression [63] from past discharges. An example of this regression fitting is shown in Figure 12.

Our training and testing sets are separated by time. This strategy reflects the common practice of training the model using data in the past and applying it on future data. Training data consisted of 1460 days from January 1, 2010, to December 31, 2013. Testing data consisted of 365 days in the year 2014. The characteristics of the training and validation cohort are shown in Table 4. Most stays were short, around 65% of patients stayed for less than 5 days.

In this section, we describe the results of comparing our different forecasting methods. The model parameters for kNN forecast, RF, and SVR models were tuned to minimize forecast errors.

For kNN regression, the optimum value of pattern length: d and number of nearest neighbours: k, was obtained by analyzing forecast RMSE for values d ϵ (1,100) and k ϵ(5,1000). Minimum RMSE of 3.77 was obtained at d=70 and k=125.

The SVR parameters C (penalty cost) and ϵ (amount of allowed error) were determined by choosing the best value from a grid search, that minimized the model RMSE. Similarly, the optimum number of variables in building each node of the RF was chosen by examining its effect on minimizing the out-of-bag estimate.

We compared the naive forecasting methods with our proposed 5 models using MFE, MAE, RMSE, and sMAPE. The results are summarized in Table 5, whereas Figure 13 compares the distribution of actual discharges with different model forecasts.

The naive forecasts are unable to capture all variations in the data and resulted in the maximum error when compared with other models.

The variations in seasonality and trend are better captured in ARIMA and ARMAX models. The time series consisting of past 3-month discharges were used to generate the next-day discharge forecast. The ARMAX model also included the day of week and ward occupancy as exogenous variables, which resulted in better forecast performance over ARIMA.

Interestingly, kNN was more successful than ARIMA and ARMAX in capturing the variations in discharge, demonstrating about 3% improvement in MAE, when compared with ARMAX. However, the kNN model tends to under forecast (MFE = 1.09), possibly because of resorting to median values for forecast. In comparison, RF and SVR forecast models demonstrated better performance. This can be expected because they are derived from all the 108 features. However, RF demonstrated a relative improvement of 3.3 % in MAE over SVR model (see Table 5). When looking at forecast errors for each day of week, RF model confirmed better performance, as shown in Figure 14.

The process of SVR with RBF kernel maps all data into a higher dimensional space. Hence, the original features responsible for forecast cannot be recovered, and the model acts as a black box. Alternatively, RF algorithm returns an estimate of importance for each variable for regression. Examining the features with high importance could give us a better understanding of the discharge process.

The features in random forecast model were ranked on importance scores. The top 10 significant features are described as follows. The day of week for the forecast proved to be the most important feature. Other features were number of patients in the ward during the day of forecast, the trend of discharges measured using locally weighted polynomial regression, number of discharges in past 14th day, number of discharges in past 21st day, number of patients who had visited only one previous ward, the number of males in the ward, number of patients labelled as: “public standard,” and current month of forecast.

Improved patient flow and efficient bed management is key to counter escalating service and economic pressures in hospitals. Predicting next-day discharges is crucial but has been seldom studied for general wards. When compared with emergency and acute care wards, predicting next-day discharges from a general ward is more challenging because of the nonavailability of real-time clinical information. The daily discharge pattern is seasonal and irregular. This could be attributed to management of hospital processes such as ward rounds, inpatient tests, and medication. The nonlinear nature of these processes contributes to unpredictable length of stay even in patients with similar diagnosis.

Typically, for open wards, a floor manager uses previous experience to foresee the number of available beds. In this paper, we attempt to model total number of next-day discharges using 5 methods. We have compared the forecasting performance using MAE, RMSE, and sMAPE. Our predictors are extracted from commonly available data in the hospital database. Although the kNN method is simple to implement, requiring no special expertise, software packages for other models are available for all common platforms. These models can be implemented by the analytics staff in hospital IT department and can be easily integrated into existing health information systems.

In our experiments, forecast based on RF model outperformed all other models. Forecasting error rate is 31.9% (as measured by sMAPE) which is in the same ballpark as the recent work of [24], though we had no real-time clinical information. An RF model makes minimum assumptions about the underlying data. Hence, it is the most flexible, and at the same time, comes with great overfitting control. Similarly, SVR also demonstrated superior performance, compared with the autoregressive and kNN models. The RBF kernel maps the features into a higher dimensional space during the regression process. Hence, the physical meaning of the features is lost, making it difficult to interpret the model. Finally, RFs and SVR are able to handle more features. This extra information in the form of patient demographics and past admission and discharge statistics contributed to improve the predictive performance when compared with other models.

The kNN regression also performed well as it assumes only the locality in the data. But it is not adaptive, and thus less flexible in capturing complex patterns. The kNN regression assumes similar patterns in past discharges extrapolate to similar future discharge, which is not true for daily discharges from ward. ARMAX model outperformed the traditional ARIMA forecasts since it incorporated seasonal information as external regressors. As expected, a naive forecast of using the median of past discharges performed worst.

We noticed a weekly pattern (Figure 2) and monthly pattern (Figure 3) in discharges from the ward. Other studies have also confirmed that discharges peak on Friday and drop during weekends [5,9,10]. This “weekend effect” could be attributed to shortages in staffing or reduced availability of services like sophisticated tests and procedures [10,69]. This suggests discharges are heavily influenced by administrative reasons and staffing.

Feature importance score from an RF model helps in identifying the features contributing to the regression process. The day of forecast proved to be one of the most important features in the RF model. Other important features included trend based on nonlinear regression of past weekdays, number of discharges in the past days, ward occupancy in previous day, number of males in the ward, and number of general patients in ward.

When looking at for each day of the week, the RF and SVR model consistently outperformed other models. Sundays and Thursdays proved to be the easiest to predict for all models (Figure 14). This can be expected since these days had the least variation in our data. Fridays proved to be the most difficult to forecast. Retraining the RF model by omitting “day of the week” increased the forecast error by 1.39% (as measured by sMAPE).

Patient length of stay is inherently variable, partly due to the complex nonlinear structure of medical care [8]. The number of discharges from a ward is strongly related to the length of stay of the current patients in the ward. Hence, the variability in ward-level discharges is compounded by the variability in individual patient length of stay. In our study, the daily discharge pattern from ward shows great variation for each day of week. Apart from patient level details, we believe that a knowledge of hospital policies is also required to capture such nonlinearity.

In our study, we were able to validate that the weekend patterns affect discharges from a general ward. The RF model was able to give a reasonable estimate of number of next-day discharges from the ward. Clinical staff can use this information as an aid to decisions regarding staffing and resource utilization. This foresight can also aid discharge planning such as communication and patient transfer between wards or between hospitals.

An estimate of number of free beds can also help reduce emergency department (ED) boarding time and improve patient flow [12,23]. ED boarding time is the time spent by a patient in emergency care when a bed is not available in the ward. ED boarding time severely reduces the hospital efficiency. High bed occupancy in ward directly contributes to ED overcrowding [70]. In our data, 42.81% of patients were admitted from the emergency care. An estimate of daily forecasts can be helpful in deciding the number of beds in wards to ease patient flow.

We acknowledge the following limitations in our study. First, we focused only on a single ward. However, it was a ward with different patient types, and hence the results could be an indication for all general wards. Second, we did not use patient clinical data to model discharges. This was because clinical diagnosis data were available only for 42.81% of patients who came from emergency. In a general ward, clinical coding is not done in real time. However, we believe that incorporating clinical information to model patient length of stay could improve forecasting performance. Third, we did not compare our forecasts with clinicians/managing nurses. Finally, our study is retrospective. However, we have selected prediction period separated from development period. This has eliminated possible leakage and optimism.

This study set out to model patient outflow from an open ward with no real-time clinical information. We have demonstrated that using patient-level and ward-level features in modelling forecasts outperforms the traditional autoregressive methods. Our proposed models are built from commonly available data and hence could be easily extended to other wards. By supplementing patient-level clinical information when available, we believe that the forecasting accuracy of our models can be further improved.