Online Blood Glucose Prediction Using Autoregressive Moving Average Model with Residual Compensation Network Ning Ma and Yuhang Zhao and Shuang Wen and Tao Yang and Ruikun Wu and Rui Tao and Xia Yu and Hongru Li1 Abstract. Blood glucose (BG) prediction plays an important role online software simulator. They provided the foundation for the fur- in daily BG control. Accurate prediction of short-term glucose con- ther development of the mobile prediction. centration can provide early warning for hyperglycemia and hypo- Nevertheless, every prediction algorithm has its own advantages glycemia events. This paper proposed a novel framework that com- and disadvantages. The ARMA model can be constructed easily by bined an online prediction model with a residual compensation net- several steps, but they lack the ability to deal with the nonlinear pat- work. The autoregressive moving average (ARMA) model was used terns [15]. Due to the extremely non-stationary characteristic of the for online blood glucose prediction and the neural network was ap- time series, the single artificial intelligence models sometimes stuck plied for compensation of prediction error. The advantages of this into the local minimum and fail to achieve satisfactory performance. combined framework are: (1) the online ARMA model is efficient With the development of equipment, the generation of data flow is and robust to capture time-varying glucose dynamics, (2) the resid- continuous. Tracking the time-varying characteristic of the system ual compensation network is capable to estimate errors from the on- is crucial. Regarding the non-stationary time series, most scholars line prediction model. The performance of this method was evaluated adopted one online learning method to model the complex system. by the root mean squared error (RMSE) and the mean absolute error The input of the data can adjust the parameters of the model in real- (MAE) in the dataset of OhioT1DM.The results were shown in detail time [12]. The data of blood glucose is non-stationary, aperiodic, and that the mean values of the best RMSE of six patients at 30-min and individuality. Therefore, the use of only one method for BG predic- 60-min horizon were 20.03 and 34.89 respectively, and the best MAE tion may give one-sided results [14]. We need to combine various at 30-min and 60-min horizon were 14.52 and 24.61. Compared with prediction methods to cover the disadvantages. the ARMA model, the combined predictor with a residual compensa- In this paper, we proposed a novel framework that combined an tion network shows better prediction accuracy. Thus, we concluded online prediction model with a residual compensation network. The that the proposed framework was an available approach for online ARMA model was used for online blood glucose prediction and blood glucose level prediction (BGLP). the neural network was applied for compensation of prediction er- ror. The advantages of this combined framework are: (1) the online ARMA model is efficient and robust to capture time-varying glucose 1 INTRODUCTION dynamics, (2) the residual compensation network is capable to esti- mate errors from the online prediction model. The accuracy of this Nowadays, daily BG management is a significant challenge for a pa- method was evaluated by short-term glucose prediction in the data tient with diabetes. Further improvement of glucose control can be set of OhioT1DM. realized through prediction, which allows users to take actions ahead This paper is structured as following five parts: section I presents of time to minimize the occurrence of adverse glycemic events [3]. a brief literature review that discusses related works on short-term Thus, accurate blood glucose prediction plays an important role in glucose prediction technique; section II presents our method for blood glucose control. However, multiple factors influence glucose data preprocessing; section III introduces the principle of the online variability and lead to different responses between individuals under ARMA model and neural network, as well as the overall framework; the same conditions. The prediction of short-term glucose concen- section IV discusses the performance of our method on clinical data, tration has become an urgent problem for researchers. In the past, and section V concludes the paper. various machine learning approaches were proposed to develop data- driven glucose predictive models [22]. John et al. [13] used Recur- rent Neural Networks that trained in an end-to-end fashion to predict 2 DATA PREPROCESSING future blood glucose levels through historical blood glucose data. Jaouher et al. [2] applied an Artificial Neural Networks model to The data used in this paper is provided by the BGLP challenge. predict future blood glucose levels and hypoglycemic events of Type OhioT1DM dataset recorded 8-week CGMs data and corresponding 1 Diabetes Mellitus (T1DM). The results proved that the model was daily events from 6 patients with type 1 diabetes, including numbers accurate, adaptive, and encouraging by clinical implementation. Rey- 540, 544, 552, 567, 584, and 596, respectively. During data collec- mann et al. [19] trained a Support Vector Regression model with an tion and transmission, the errors in calibration or measurements may be produced many missing or outlier data points in clinical data. Al- 1 Northeastern University, China, email: lihongru@ise.neu.edu.cn though, time series models do not consider any physiological factors Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). and only use recent BG data and other inputs that may affect BG lev- for forecasting. In the identification step, data transformation is often els. The missing data will have a significant effect on the accuracy of required to make the time series stationary. Meanwhile, the sliding the models [21]. window technique is more robust to the stochastic changes in the data Online models emphasize the real-time input of data streams, trend and can be applied to smaller datasets [25]. Hence, the sliding hence, the missing data can only be estimated using past data [7]. window technique was added to the ARMA model. Discarding old Our workflow for dealing with missing data problem is as follows. data from the training window can limit the influence of distant past Based on the CGM data, a time grid with a 5-minute sample period trends during model training and can promote the learning of new was derived and the missing data were filled with zeros. Firstly, we trends in the data. made a statistical analysis of the size and number of missing data seg- Differencing was applied to it to remove the trend and stabilize ments through excel software. In both the test set and the training set, the variance because of the trend of the blood glucose data. After there are more than 5% and even 20% missing data. Among them, the that, the sliding window updated the BG data. The method can re- loss of blood glucose between 1-100 is relatively common, which duce the training time of the model because the number of training may be caused by the replacement of CGM in patients. Secondly, sets is always fixed. As far as we know, determining the order of with the statistical results, a backward pushing method or mean value models is a key to the ARMA model. Akaike’s Information Crite- method was implemented for each missing. With the increase of fill- rion (AIC) is widely used to optimize the model parameters in those ing times, the cumulative error will inevitably increase.For the train- models. AIC is an estimation for the likelihood of a model. How- ing set, the missing CGM values were filled with spline and the his- ever, AIC does not have any indication of the absolute quality. The torical average at the same point. When the two values are different, Bayesian Information Criterion (BIC) is a similar criterion for model the weighted method is used to fill. The test set is processed as fol- selection [9]. Then, AIC and BIC were used to select an appropriate lows :(a) the first three positions of the missing segment are filled order in this paper. For the two results, we limit the interval value with extrapolation method;(b) starting from the fourth position of of the global model order, to conduct experiments to find the opti- the missing segment, weight the first-order Taylor series extrapola- mal model parameters. The last step of model building is the diag- tion and average (the historical average at the same point and his- nostic checking of model adequacy. If the model is not adequate, a torical average) to fill;(c) from position 12 of the missing paragraph new tentative model should be identified, which is again followed by uses backward induction. Finally, unbroken data would be obtained the steps of parameter estimation and model verification. The ability for prediction. Although many models with multiple inputs (insulin of the ARMA model in learning small data sets and tracking fast is dose, food intake, etc.) can effectively predict the future BG levels. taken full advantage and can achieve the online update learning. However, the data-collection process of those inputs heavily relies on the subjective inputs provided by the user who wears a CGM device. Since the user may not be professional, the data may be inaccurate 3.2 Residual compensation network and have errors. Due to such limitations, we predicted the future BG 3.2.1 Neural network level only based on the historical BG data. Backpropagation (BP) neural network is a model that can approx- imate various nonlinearities in the data. It is a kind of multi-layer 3 METHODS AND REALIZATION feedforward neural network trained according to the error propaga- In this section, we will introduce the models that are used in the fram- tion algorithm and there are three layers including the input layer, work and explain how the proposed framework works for prediciton. hidden layer, and output layer. In essence, the BP neural network takes the network error square as the objective function and uses the gradient descent method to calculate the minimum value of the ob- 3.1 ARMA model jective functio [6]. Modifying the weight and threshold is the core 3.1.1 ARMA model of the BP neural network. It aims to get the model whose output is consistent with expected results. In this paper, the input layer of the ARMA, which includes the autoregressive (AR) model and moving- neural network is the predictive value of blood glucose, and the out- average (MA) model, is an important method to study the time series put is the prediction error. The structure is shown in Figure 1. [17]. It is widely used in the prediction of finance and wind power [1], [20]. The ARMA could establish linear and nonlinear dynamic models by associating input and output data. And it can be expressed as follows: p q X X yt = βi yt−i + αj t−j + t (1) i−1 j−1 Where p is the order of the autoregressive part, βi is the autoregres- sive parameter, q is the order of the moving average part, αj is the moving average parameter, and t is the error term at time t. In gen- eral, the offline parameter determination uses the Least-squares and the online uses Kalman filter. 3.1.2 Online model The ARMA includes three iterative steps including model identifica- Figure 1. The basic structure of BP neural network. tion, parameter estimation, and diagnostic checking. Stationarity is a necessary condition in building an ARMA model which is useful In Figure 1, n is the number of nodes in the hidden layer, p and q are the number of nodes in the input layer and output layer respec- tively. The number of hidden layers can be determined according to the empirical formula: √ n= p+q+a (2) Where a is the adjustment constant between 1 and 10. The number of input layers is determined by correlation analysis. Then, the best number of hidden layers is determined by the experiment to follow equation (2). It is generally believed that increasing the number of hidden layers can reduce the network error and improve the accuracy, but also complicate the network, thus increasing the network training time and the tendency of overfitting. 3.2.2 Framework of residual compensation Both the ARMA model and BP neural network have achieved suc- cesses in their own linear or nonlinear domains. Neither of them is suitable for all circumstances. The statistical methods have their lin- ear limitations, which means that they cannot simulate the real-time series with nonlinear mode well [5]. On the other hand, a single BP Figure 2. Flow diagram of online ARMA model with residual compensa- neural network is not enough to capture the time patterns contained tion network. in highly complex time series. In the training process of the BP neu- ral network, there may be problems of the model following error and tion analysis, the range of input variables may be different from 6 uncertainty, resulting in the generation of overfitting or underfitting patients. model [11]. A hybrid methodology can be a good strategy for practi- Step 4. As an important supplement to model prediction, a com- cal use [24] , [4]. It combines different models to capture different as- mon three-layer neural network is applied to predict the residual. The pects of the underlying patterns. Ji et al. [10] used the ARMA model neural network predicts the errors in the future based on a series of to predict linear components of the time series, and the TDNN model errors in the past and can overcome the influence of various uncer- to predict nonlinear components. Results showed that the model had tainties changes on system stability. the advantages of both two methods and the prediction accuracy of Step 5. Analysis of blood glucose predictions and residual time the model was improved. However, only the optimal combinations of series in statistically. The output display value range of the CGM is different models can obtain the best hybrid models, the framework of [40,400], and the error is basically within the range of [-50,50]. For the hybrid models becomes very important. this reason, some rules are employed to correct discrete data points In this paper, we proposed a novel framework that combined an appropriately in the research. online ARMA model with a residual compensation network (RCN- Step 6. Combine the results of the two-step prediction and get the ARMA) to predict BG. The blood glucose data which belongs to final After the five steps, we have got the prediction results of the BG chaotic time series contains linear and nonlinear components [8]. and the error. Combine the results of the two-step prediction by the Due to the randomness and volatility of BG, the ARMA model in- direct sum method and get the final BG prediction. evitably produces large errors in the prediction of nonlinear non- stationary time-series data, which has a certain tendency and period- 4 RESULTS AND DISCUSSION icity [23]. The BP neural network has good data error tolerance, but it is insufficient for linear prediction. Since the ARMA model cannot 4.1 Evaluating indicator capture the nonlinear structure of the BG data. The residuals of the linear model will contain information about the nonlinearity. The BP For model evaluation, general and commonly used evaluation meth- neural network is valid for satisfying the prediction effect of most ods are sensitivity, specific, root mean square error (RMSE), and non-linear properties. Hence, the BP neural network was applied to mean absolute error (MAE) [18]. In this paper, two widely used eval- predict residuals. The framework aimed to reduce the uncertainty of uation indexes were applied to compare the prediction capacity. The model selection and improve the model forecasting performance by error indexes define as below: dealing with both linear and nonlinear patterns in time series. The v u N flow diagram of the RCN-ARMA is shown in Figure 2. u1 X RM SE = t (ŷi − yi )2 (3) The specific prediction process of RCN-ARMA is as follows: N i=1 Step 1. The sliding window updates the input for the ARMA N model. AIC and BIC are used to confirm the order of ARMA. Then 1 X M AE = |ŷi − yi | (4) predict the blood glucose by the online ARMA model. N i=1 Step 2. Compared to the predicted value with the raw data, the residual time series, which is used to the compensation network, can Where: ŷi represents the predicted value, yi represents the real value be constructed. and N represents the size of the data set. Two rules were applied in Step 3. The correlation analysis of the predicted values and resid- the evaluation: 1) as long as the corresponding timestamp had the raw ual time series is carried out to determine the input of the residual data, the RMSE and MAE indexes of the test set would be calculated; compensation network [16]. According to the results of the correla- 2) If there was a null value in the input model data, it meant that the data was given insufficiently, and the value at this time would not be recorded. We only use the first and the code used during the experiment is available on Github. In this paper, the predictions of the model were recorded from the thirteenth point of the test set. And the results were recorded as two decimal places rounded. 4.2 Results In this section, the results and analysis of the proposed framework are presented. The online AR, BP, and ARMA models were used for 30-min ahead predictions. The mean values of the RMSE and MAE for six patients are shown in Table 1. Then the RCN-ARMA was used to 30-min and 60-min ahead predictions. The experiments Figure 4. Forecasting results of patient 567 for 30-min ahead predictions. were conducted on patients with different inputs by establishing an online ARMA model and a residual compensation network (RCN- ARMA). The optimal value of the sliding window was selected by one of the main reasons affecting the prediction effect of the model; the experimental method and keeps the same in two networks. Due to (b) the addition of the error compensation model improves the hys- the heterogeneity of the patients themselves, the selected parameters teresis of the predicted value of the model; (c) the mixed prediction had some differences. The 30-min ahead predictions of ARMA and results show sharp fluctuations and a certain amount of peak data that RCN-ARMA for 540, 567 patients are graphically shown in Figure are negative effects of adding compensation. 3 and Figure 4. Table 2 shows the RMSE and MAE of the different contributors for 30-min and 60-min ahead predictions. Based on the results in table2, mean RMSE and MAE of 30-min and 60-min ahead Table 2. RMSE and MAE of the RCN-ARMA model for 6 patients (PH=30 predictions respectively with the online ARMA and RCN-ARMA and 60 minutes). are shown in Table 3. The above tables contain the results of three ID 540 544 552 567 584 596 cases, and the reliability of the conclusions is enhanced through a 30-RMSE 22.19 17.66 17.40 21.12 23.88 17.93 comparison of multiple cases. 30-MAE 16.29 13.27 12.95 14.94 16.99 12.68 60-RMSE 40.03 31.873 30.06 38.42 38.71 30.27 Table 1. Mean values of the RMSE and MAE for different models (predic- 60-MAE 30.32 24.25 22.88 29.58 29.03 22.39 tion horizon (PH) =30 minutes.) Method AR BP ARMA RMSE 21.80 33.45 21.44 MAE 15.93 24.54 15.17 Table 3. Mean values of the RMSE and MAE for ARMA and RCN-ARMA model. PH=30 minutes PH=60 minutes Method RMSE MAE RMSE MAE ARMA 21.44 15.17 38.78 28.42 RCN-ARMA 20.03 14.52 34.89 26.41 Drop value 1.41 0.65 3.89 2.01 As can be seen from Table 2 and Table 3: (a) for different patients, the model prediction effect is different and reflects the specificity of blood glucose data; (b) prediction ability of the model got worse with the increase of the prediction step. This is a major issue that needs to be addressed urgently; (c) through the correlation analysis of predicted value residuals, it implies that a significant correlation Figure 3. Forecasting results of patient 540 for 30-min ahead predictions. relationship exists for the multi-step ahead forecast error series of ARMA. Thus, it is very useful for the error forecast models to select Table 1 shows that different models have different prediction ef- effective input variables in this multi-step ahead forecasting model; fects on blood glucose prediction. The ARMA model is better than (d) compared with the online ARMA model, the evaluating indicator the other two models in prediction. The reason is that the online of RCN-ARMA all decreased, especially for 60-min ahead predic- ARMA model has an advantage in tracking real-time changes of tions; (e) from the drop value, the change of two different step size data.And the AR model which does not contain the moving average evaluation indexes gradually increases. The overall effect decreases model (MA) is a special form of ARMA model. There is a big differ- with the increase of prediction step size for both models. The im- ence in MAE between the two. Therefore, we choose online ARMA provements of the proposed combined framework compared with a as the base model. Figure 3 and Figure 4 clearly illustrated that (a) certain individual model increase with increasing prediction steps for the predicted value of ARMA has obvious lag on the whole, which is the continuous multi-step ahead forecasting. 4.3 Discussion [4] P.S.G. de Mattos Neto, G.D.C. Cavalcanti, and F. 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