Recently, continuous physiological monitoring integrated with time series analysis and forecasting has drawn biomedical researchers’ attention to prevent respiratory and cardiac complications and death, especially in the postoperative period (ElMoaqet et al., 2016). This is revolving around a vital field when encountering 1–6% (thousands annually) of children being cared for in an intensive care unit (ICU) who will experience a cardiac arrest (Kennedy and Turley, 2011). These arrests happen despite ECG, pulseoximetry, and frequent blood pressure measurements have been continuously monitored. The most active researches provide rigorous predictions based on classification algorithms, which map the input physiological signal(s) to output values to diagnose and label without modeling the underlying physiological data dynamics. On the other hand, the accurate and reliable system of multi-step forecasts is a particular open trial. Furthermore, exploring deep learning algorithms is still necessary for multi-step forecasting models (Masum et al., 2018, Lim and Zohren, 2021). Continuous monitoring should accompany adjustable long-horizon forecasting to achieve a punctual clinical decision making. Accurate forecasting of physiological time series can equip treatment procedures and healthcare professionals to intervene early and prevent adverse clinical events. Time series forecasting has been utilized to design intelligent alarm threshold-based monitoring which is the state-of-the-art technology for hospitalized patients (Masum, 2019). These systems adjustable on high sensitivity alarms forecast an adverse event based on the monitored signal(s) to distinguish anomaly behavior from a normal stream.
However, the desire for intelligent monitoring and benefits is growing recently, these current systems have not been shown to reproducibly improve outcomes in hospitalized patients (Watkinson et al., 2006). New techniques for state detection, such as the fusion of physiological signals from multiple channels, have been developed (Tarassenko et al., 2006). Nevertheless, these methods have not been proven to improve patient outcomes yet. Two common challenges with intelligent monitoring systems are i) the relative rarity of adverse events, especially in the early postoperative period (Watkinson et al., 2006, Watkinson and Tarassenko, 2012); and ii) meeting stationary condition which is the most important assumption for many time series models, and the non-stationary time series should be transformed to stationary ones (Hyndman and Athanasopoulos, 2018, Dickey, 1984).
The first challenge limits the clinical precision of intelligent systems. Additionally, an investigation of patterns of in-hospital deaths shows that late detection of clinical instability results in delayed recognition and reduced successful clinical intervention (Lynn and Curry, 2011). ElMoaqet et. al. in 2016 developed a framework for multi-step ahead prediction models with a performance metric to compensate for and resolve this challenge in intelligent monitoring systems (ElMoaqet et al., 2016). They have proposed a performance metric evaluating near term predictions of critical levels of anomaly in physiological time series. Thereafter, utilizing this metric to build a framework for multi-step ahead prediction models which are capable of forecasting critical levels of anomaly. As for the second challenge, in the analysis of scaling signals, stationary and non-stationary time series determine not only the form of auto-correlations and moments but also the selection of estimators (Kristoufek, 2014). A time series is stationary when the main properties including mean, variance, or auto-correlation structure remain constant all over time. Contrary, if any of these properties vary over time, it is categorized as a non-stationary signal (Hyndman and Athanasopoulos, 2018). Additionally, raw acquired data may require prepossessing and/or decomposition ahead of scaling processes (Jebb et al., 2015).
Either utilizing decomposition models to better understand the underlying dependencies of time series (Hyndman and Athanasopoulos, 2018) or only a normal cleaning of the data, classifying the signal into stationary or non-stationary is necessary for further analysis and forecasting. The theory of scaling processes has affected meaningfully the performance of analyzing techniques as well as forecasting models in several fields of applied science (Pacheco et al., 2012). Aspects of scaling behavior have been proved in finance (Beran, 1992, Beran, 2017), in the analysis of heart rate variability and EEGs as examples of time series in physiology (Cannon et al., 1997, Eke et al., 2000) in mood characterization and other psychological behavioral variables (Delignieres et al., 2006, Jebb et al., 2015), in the modeling of computer network traffic and their lags in Local Area Networks and Wide Area Networks (Lee and Fapojuwo, 2005), and most signals in physiology and neuroscience (Gujral et al., 2020, Xue et al., 2012) among others. Determining stationary or non-stationary conditions is crucial for analysis or estimation purposes as many techniques have been developed for stationary time series forecasting. Once a non-stationary signal encounter to an analysis or modeling technique designed for stationary conditions will result in an ambiguity or drastic performance reduction, respectively. Formal statistical tests for stationary are unit root tests. The most common approach is the augmented Dickey–Fuller test (Said and Dickey, 1984) which tests the null hypothesis that a unit root is present in an auto-regressive (AR) time series model (the null hypothesis i.e. the series is non-stationary). Another widely used test is the Kwiatkowski–Phillips–Schmidt–Shin (KPSS), which determines if a time series is non-stationary because of a unit root or stationary around a mean or linear trend (Kwiatkowski et al., 1992).
The sources may result in time series being non-stationary accounting for the systematic components as well as the underlying dependencies of the past. It requires a procedure to remove the systematic trend and seasonal effects which are not of interest on the mean level of the series. The most important method is differentiating which converts non-stationary time series into a stationary on the mean of a series. In the simplest case of a linear trend, a series of first differences can effectively “detrend” the original series (Hyndman and Athanasopoulos, 2018). In the case when the time series exhibits a varying trend by itself, then even the first difference may not result in a completely stationary one. Therefore, higher orders of differentiating are applied to make a series stationary. Regularly, in practice, the first or second differences will nearly always make the mean stationary, and it is almost never necessary to go further (Chan and Cryer, 2008). Preventing over differentiating the time series is required to reach the lowest variance of the transformed series (Jebb et al., 2015). Integer order difference data transformations make the series stationary, but the cost is removing all memory from the original series (Lopez de Prado, 2018). Although stationary is a necessary property for inferential purposes, it is a dilemma, especially in the case of time series forecasting that memory preserving is the basis for the predictive models.
The main failure of Auto-Regressive Integrated Moving Average (ARIMA) modeling to times series is imposed by differentiating to achieve stationary memory-erased series (Sutcliffe, 1994). As a complex mathematical remedy, the use of fractional order difference ensures the stationary of the data while preserving as much memory as possible (HOSKING, 1981). It opens up a much wider and more realistic behavior for the trend and seasonal components than traditional integer difference. Nevertheless, fractional differentiating the stochastic process accounts for a burden of computational operation. After Hosking’s paper in the 80s, surprisingly a pause period of using a fractional scheme happened in which the literature on this subject has been scarce i.e. only eight journal articles have been published (Lopez de Prado, 2018).
Once make time series stationary by fractionally differentiate to preserve the memory carries on, the next step is choosing a time series forecasting model. Multi-step forecasting for a long-term horizon is very challenging. In the case of linear statistical models such as ARIMA which is used to predict linear data. However, most real-world applications contain nonlinear data; nonlinear time series forecasting and analysis are yet developing (Kam, 2014). Alternatively, machine learning models are data-driven models which explore patterns within the data to create forecasting models for either linear or nonlinear data. Machine learning has recently been used in healthcare especially in decision support (Frizzell et al., 2017). Particularly, Taieb et al. and Masum et. al. described and compared five different forecast strategies including the Recursive strategy, Direct strategy, Direct-Recursive (DirRec) strategy, Multi-Input Multi-Output (MIMO) strategy, and Direct Multi-Output (DIRMO) strategy (Bontempi et al., 2013, Masum et al., 2019). All strategies utilized the combination of long short-term memory (LSTM), Bidirectional-LSTMs (Bi-LSTM), and Convolutional Neural Networks (CNN). However, they have not used fractional differentiating for preprocessing to make the physiological data stationary. Masum et. al. showed that the forecast model that uses Bi-LSTM with the DIRMO strategy is the more reliable model to forecast heart rate (HR) and blood pressure (BP) time series (Masum et al., 2019). In 2019, Liu and Motani proposed a new approach called generative boosting that includes two parts of the predictive and generative models. Generative boosting utilizes LSTM for both parts leading to a scheme called generative LSTM (GLSTM). The first model consists try to generate synthetic data for the next few time steps, and the second models, try to make long-range predictions based on observed and generated data. Generative boosting mitigates the error propagation in the generative models and reduces the effective prediction horizon in the predictive models. They showed that GLSTM outperforms efficient benchmark models, in such a way that the mean absolute percentage errors (MAPE) of 7.41% and 6.17% were achieved to predict HR and systolic blood pressure (SBP) 20 minutes in advance, respectively (Liu et al., 2019). In 2020, Youssef et. al. proposed a hybrid machine learning algorithm of KNN-LS-SVM instead of LSTM-based models for real-time early warning scores (EWS) estimation and vital signs time-series prediction. They preserved at least one-hour statistical attributes of the different vital signs (i.e., minimum, mean) as input data to forecast statistical attributes one, two, and three hours in advance (Youssef Ali Amer et al., 2020). They achieved the MAPE of predicting a one-hour average heart rate are 4.1, 4.5, and 5% for the next one, two, and three hours respectively for cardiology patients.
This study will present a preprocessing based on fractional-order differentiating followed by deep learning architecture containing directive and iterative steps that utilize U-Net convolutional networks (Ronneberger et al., 2015) and multi-layer Bi-LSTMs. U-Net structure accompanied with skip connections help the entire model transform the raw time series, extract more informative features, and then feed to multi-layer Bi-LSTM for long horizon prediction. The obtained results will be indicated in the result section. Finally, it will end up with a discussion and conclusion.