Prior Informed Regularization of Recursively Updated Latent-Variables-Based Models with Missing Observations

Abstract

Many data-driven modeling techniques identify locally valid, linear representations of time-varying or nonlinear systems, and thus the model parameters must be adaptively updated as the operating conditions of the system vary, though the model identification typically does not consider prior knowledge. In this work, we propose a new regularized partial least squares (rPLS) algorithm that incorporates prior knowledge in the model identification and can handle missing data in the independent covariates. This latent variable (LV) based modeling technique consists of three steps. First, a LV-based model is developed on the historical time series data. In the second step, the missing observations in the new incomplete data sample are estimated. Finally, the future values of the outputs are predicted as a linear combination of estimated scores and loadings. The model is recursively updated as new data are obtained from the system. The performance of the proposed rPLS and rPLS with exogenous inputs (rPLSX) algorithms are evaluated by modeling variations in glucose concentration (GC) of people with Type 1 diabetes (T1D) in response to meals and physical activities for prediction windows up to one hour, or 12 sampling instances, into the future. The proposed rPLS family of GC prediction models are evaluated with both in-silico and clinical experiment data and compared with the performance of recursive time series and kernel-based models. The root mean squared error (RMSE) with simulated subjects in the multivariable T1D simulator where physical activity effects are incorporated in GC variations are 2.52 and 5.81 mg/dL for 30 and 60 mins ahead predictions (respectively) when information for all meals and physical activities are used, increasing to 2.70 and 6.54 mg/dL (respectively) when meals and activities occurred, but the information is with-held from the modeling algorithms. The RMSE is 10.45 and 14.48 mg/dL for clinical study with prediction horizons of 30 and 60 mins, respectively. The low RMSE values demonstrate the effectiveness of the proposed rPLS approach compared to the conventional recursive modeling algorithms.

1. Introduction

Use of data-driven models has increased in recent decades for system monitoring, fault detection and diagnosis, and advanced control. The accuracy and fidelity of such models become a critical issue when the system has time-varying and nonlinear dynamics. Many computationally efficient data-driven models are linear, and their use in characterizing time-varying or nonlinear systems requires the model parameters be adapted as the operating conditions of the system vary. Some data-driven models such as neural networks necessitate large amounts of data to capture all possible states of the system. Yet, they may also need to be retrained when new types of disturbances occur or the system moves to new operating conditions.

Recursive identification techniques ensure the recently identified model conforms to the current operating conditions of the system. Numerous recursive identification approaches, ranging from recursive least squares to kernel filtering methods [1, 2, 3, 4, 5], are developed and used in adaptive filtering and control applications where decisions are based on the latest estimate of the model. Developments in recursive identification methods, such as parameter update laws and not requiring all data to be stored, have improved the computational and memory requirements of the algorithms for efficient on-line implementation. However, continuously updating the models without considering prior knowledge of the system can make it difficult to tune the complexity and properties of the recursively identified models. Incorporating prior information in recursive identification can encode desirable properties like asymptotic stability and smooth behavior in the recursively identified models. Nevertheless, many recursive identification approaches do not incorporate prior information, a critical omission for nonlinear dynamical systems in biomedical applications where encoded model properties are desired.

Biomedical systems such as the the human body is an example of complex systems with time-varying parameters and nonlinearities. Regulation of this complex system affected by metabolic disorders such as diabetes necessitates the use of model-based controllers that consider the current and future states of the system in making control decisions. Hence, development of dynamic models with accurate prediction capabilities for 30 to 60 minutes (6 to 12 sampling instances) into the future are needed. Furthermore, there is significant variability in the responses of different people with diabetes, which necessitates personalized models. In addition, the metabolic responses of an individual changes from day-to-day and even the personalized model needs to be adjusted frequently to represent the current state of the person whose blood glucose concentration (GC) should be regulated to remain within the desired range.

We have developed a number of model identification approaches leading to recursively updated autoregressive moving average time series models with exogenous inputs (ARMAX) [6, 2, 7] and kernel recursive least squares models (KRLS) [1]. In this work, recursive regularized partial least squares (rPLS) and rPLS models with exogenous inputs (rPLSX) are proposed and their performance is evaluated by developing models for predicting the GC of people with Type 1 diabetes (T1D).

T1D is an autoimmune disorder where β cells in the pancreas are destroyed and thus can no longer produce insulin, characterized by the inability of the body to regulate blood glucose levels [8]. People living with T1D must administer exogenous insulin to enable the transport, storage, and utilization of glucose in their blood stream and further maintain their GC in a safe range (70–180 mg/dL) [6, 1] by injecting insulin several times each day or infusing insulin with an insulin pump. If the blood GC is not tightly regulated, high blood GC (hyperglycemia) can occur and induces chronic complications like retinopathy and kidney failure. On the other hand, dizziness, fainting, or even death can happen due to the insufficient energy supplement for the brain caused by low blood GC (hypoglycemia) [8].

The artificial pancreas (AP) systems can provide automatically adjusted insulin infusion through insulin pumps and reduce the risk of hypoglycemia significantly by using a closed-loop controller that incorporates information from glucose sensors and additional physiological information [6]. The design and formulation of closed-loop control algorithms becomes the critical issue for AP systems and model-based predictive control algorithms are demonstrated to improve the glucose control performance without losing feasibility of implementation on a portable device in in-silico simulations and clinical studies [9, 10]. In model-based control, the future infusion insulin is calculated by minimizing the deviation of GC values predicted by a model from set-points incorporating previous GC data, data of infused insulin, and other available information. Thus, developing a simple but accurate GC prediction model becomes one of the critical issues in the development of model-based closed-loop AP systems.

In general, the models for predicting GC can be divided into two categories [11]: physiological models and empirical (data-driven) models. Physiological models [12, 13, 14, 15] describe the glucose dynamics with equations that follow first principles knowledge, based on the understanding of interactions among GC, consumed carbohydrates, injected insulin, and physical activities. It is hard to develop individual physiological models for each person with limited data and understanding of interactions between glucose, insulin, and other disturbances (i.e., meal, exercises, and stress) and update the parameters in the model in real time [16]. With the development of continuous glucose monitoring (CGM) technology and wearable devices, the GC values and other relevant physiological information such as heart rate, skin temperature, accelerometer data and electrodermal activity can be collected easily [7, 17]. Consequently, data-driven modeling technologies gained prominence and became the preferred techniques in many modeling and control design activities [2, 18] because of their simple and flexible structure and efficiency.

Most current data-driven models that predict future GC levels several time steps into the future are based on recent CGM data, predicted GC values, and manual or automated information about meals and physical activities. They have linear or nonlinear structures, and the recent trends and developments are to make the models adaptive and personalized [10, 19]. The nonlinear models for T1D increased in recent years due to the proliferation of machine learning and deep learning technologies, as well as the improved computational capacity of hardware. Artificial neural networks (ANN) models [20], and recurrent neural network (RNN) models [21, 22, 23] are popular and studied with in-silico and clinical studies. However, large amounts of data capturing the different states of a person are needed to train nonlinear models with large number of model parameters. It may not be feasible to train and personalize such nonlinear models for each individual subject, and sufficient data may not be available from all conditions [24].

Simpler linear representations can readily facilitate the design of computationally-efficient time-varying predictive control algorithms that make use of recursively identified models to track the evolving system to the current operating region. For linear models, the autoregressive with exogenous inputs (ARX) models [2], and recursively updated ARMAX model where the model parameters are updated as new data are received have been reported for building personalized models for use with in-silico simulations and clinical studies [18, 6, 10, 25]. A statistical model [26] has been reported having better performance than ARX model in predicting GC values. In this model, partial least squares (PLS) is performed on the normalized training data set to find the latent structure between the input matrix which is an augmented matrix containing GC values, insulin and meal information, and the output matrix which contains the future GC after the infusion of insulin.

An alternative latent variables based method is to model the key quality features of the system while manipulating the inputs [27, 28, 29]. The quality features of the system after manipulation can be forecasted where the quality features during the manipulation process is known. Thus, in this work, the GC data during the infusion of insulin are used as the output of the model. Furthermore, in order to introduce the prior information into the model, a regularized PLS algorithm is proposed where a linear kernel which contains the prior information is integrated into the PLS algorithm. In previous works, the problem of ill-conditioned covariance matrices involved in the PLS algorithm is addressed through the use of the pseudo-inverse or the incorporation of a regularization term [30]. However, the regularization term is not based on prior knowledge. Moreover, the existing PLS algorithms primarily consider the case where the regressor or covariate variables are known, which does not hold if the independent covariates contain missing data. Motivated by the above considerations, in this work we propose a new PLS algorithm that incorporates regularization from prior knowledge and can handle missing data in the independent covariates. The latent variable (LV) based modeling technique consists of three steps. First, a LV-based model is developed on the historical time series data. In the second step, the score vectors of the new incomplete observation are estimated by using the known data regression method. Finally, the future GC values are predicted as a linear combination of estimated scores and loadings.

The remainder of this paper is organized as follows: In the section 2, the rPLS algorithm is described and the GC prediction model based on PLS and rPLS methods are presented. The proposed GC prediction model is evaluated using both simulation data and clinical data from subjects with T1DM in Section 3. Both models based only on CGM data and multivriable models with CGM and physiological variables data are considered and the results are discussed in Section 3. Finally, some conclusions are given in Section 4.

2. Methods

2.1. PLS model

In statistical models based on LVs, the measured variables are represented by several dependent variables which are, in general, inter-correlated. A general comparison of LV-based methods has been reported [31]. The PLS algorithm [32, 33, 34] is one of the commonly used methods to calculate LVs.

The aim of PLS algorithm is to determine a new set of variables (the LVs) which are linear combinations of the predictor variables, such that the first LV accounts for the largest amount of covariance in the output variables. The second LV explains the most the remaining largest covariance in the output variables after the variance explained by the first LV is removed. The succesvive LVs are determined by following the same approach. More concretely, let X be the normalized predictor matrix with n rows and p columns, and Y be the normalized output matrix with n rows and l columns. The n rows represent observations, p columns correspond to predictor variables, and l columns the output variables. The first step of the PLS algorithm can be expressed as

$$\begin{gathered} \underset{t_1,u_1}{\text{max}}{J}_{PLS}=t_1^T{u}_1 \\ \mathit{\text{subject}}\mathit{\text{to}}:\parallel w_1\parallel =1,\parallel q_1\parallel =1 \end{gathered}$$ (1)

where t₁ and u₁ are the score vectors that are related to the input matrix X and output matrix Y by the weight vectors w₁ and q₁

$$\begin{gathered} t_1=X{w}_1 \\ {u}_1=Y{q}_1 \end{gathered}$$ (2)

The second LVs t₂ and u₂ can be calculated in a similar manner after the deflation of the predictor E₁ and residual of the output F₁

$$p_1=X^T{t}_1/{t_1}^T{t}_1$$ (3)

$$E_1=X-t_1{p}_1^T$$ (4)

$$q_1=Y^T{t}_1/{t_1}^T{t}_1$$ (5)

$$F_1=Y-t_1{q_1}^T$$ (6)

where p₁ is the first loading vector for the predictor variables. By repeating the above iterative procedure, the remaining weight vectors, LV vectors, and loading vectors for predictor and output can be found.

In PLS, the latent structure between input matrix and output is found based on the assumption that there is no accessible prior information about the model. However, it is widely accepted that before building the multivariable regression model, prior information can be helpful to improve the predictive ability of the model[35]. A novel regularized PLS algorithm is proposed in this work, to incorporate the prior information of the model.

2.2. rPLS model

The rPLS algorithm uses the PLS method for finding the latent structure by maximizing the covariance between X and Y. The objective of rPLS method is

$$\begin{gathered} \underset{t,u}{\text{max}}{J}_{\mathit{\text{rPLS}}}=t^{T}u-{\lambda }_w{w}^{T}Rw \\ \text{subject}\text{to}:\parallel w\parallel =1,\parallel q\parallel =1 \end{gathered}$$ (7)

where R is the regularization matrix, and it is designed as a first-order stable spline kernel [36], also called the Tuned/Correlated (TC) kernel. given by

$$\begin{gathered} R_{ij}(\eta )=\lambda {\alpha }^{\mathit{\text{max}}(i,j)} \\ \eta =[\lambda ,\alpha ],\lambda \ge 0,0\le \alpha <1 \end{gathered}$$ (8)

where λ and α are the two hyperparameters of the TC kernel encoding the prior information. The kernel penalizes the LV weight coefficients over time, causing the coefficients to decay exponentially, which encodes the exponential stability of the system. Moreover, the weighting coefficients are highly correlated to coefficients near to it. A smooth model is desired, which is achieved by the diagonals closer to the main diagonal having higher correlations [37, 38] as shown in Figure 1. The TC kernel ensures that appropriate time-dependencies on the LV weights are represented in the identified model. The hyperparameters can be determined through cross-validation. For multivariate regression model, the TC kernel can be extended as

$$R\left({\eta }_1,\dots ,{\eta }_k\right)=\left[\begin{array}{cccc}R\left({\eta }_1\right)& 0& \dots & 0\\ 0& R\left({\eta }_2\right)& \dots & 0\\ ⋮& ⋮& \ddots & 0\\ 0& 0& \dots & R\left({\eta }_k\right)\end{array}\right]$$ (9)

where k is the number of different variates in the input data.

Figure 1:

TC kernel matrix (λ = 1, α = 0.99)

The rPLS algorithm is summarized in Algorithm 1. After performing the rPLS algorithm, the input matrix X and output matrix Y can be expressed as

$$\begin{gathered} X=T{P}^T+E \\ Y=T{Q}^T+F \end{gathered}$$ (10)

where T = [t₁, t₂, …, t_a] = [τ₁, τ₂, …, τ_n]^T, t_i is the ith latent variable and τ_j is the score vector of the jth observation, and a is the number of LVs. P = [p₁, p₂, …, p_a] and Q = [q₁, q₂, …, q_a] are the loading matrix for the input matrix X and output matrix Y . E and F are the residual matrix of X and Y, respectively.

2.3. Regression with known data and missing information

For a given n × p predictor matrix X and a n × l output matrix Y with each row corresponding to an observation and each column representing a variable, an rPLS model can be developed and expressed as equation (10), where the score matrix T can be considered as either a collection of column vectors t_i (ith latent variable) or a collection of row vectors τ_j (scores of the jth observation). For a new observation z not used in developing the rPLS model, based on the assumption that it belongs to the same population as the n observations in X, its score vector τ can be calculated as

$$\tau =R^{T}z$$ (18)

where R = W(P^TW)⁻¹.

Consider a situation where only some of the variables in the new observation z are observed. The variables can be rearranged in such a way that the object z can be written as a concatenation of a shorter measured variables vector and an unobserved variables vector. Assuming that the first r variables are measured, without loss of generality, the object vector can be partitioned as

$$z=\left[\begin{array}{l}{z}^{\ast }\hfill \\ z^{\#}\hfill \end{array}\right]$$

where z^# denotes the observed variables and z* represents the missing measurements. The data matrix X can be partitioned into two submatrices under the same assumption

$$X=\left[\begin{array}{ll}{X}^{\ast }\hfill & X^{\#}\hfill \end{array}\right]$$ (19)

where X* and X^# are submatrices with the r observed variables and the remaining (p − r) unmeasured variables, respectively (Figure 2).

Figure 2:

Illustration of observed and unobserved variables data partitioning

The objective of the regression method with known data is to estimate the scores of a new individual from the training data set X [29] where the same variables to be missing in each row of the input matrix X. Thus,

$$T=X^{\ast }B+U$$ (20)

where B is the coefficient matrix that can be calculated by using the least squares method B = (X*^TX*)⁻¹ X*⁻¹T.

Therefore, the score vector τ of the new observation z can be calculated from

$$\widehat{\tau }=z^{\ast }B$$ (21)

Then, the output of the model is given by

$$\widehat{y}=Q*\widehat{\tau }$$ (22)

3. Glucose prediction using rPLS model with exogenous inputs

Blood glucose is the major energy source in the human body. It is converted from absorbed carbohydrates in meals and is transferred to liver, muscle, and adipose cells when appropriate amount of insulin is present in the blood-stream. In order to apply the LV-based technique to model the glucose dynamics and forecast future GC values, plasma insulin concentration (PIC) must be known. PIC can be calculated by the PIC estimator [39, 10] and used as part of the exogenous input variables. This provides more accurate information on the amount of insulin available in a subject compared to the generic insulinon-board values and pre-announced meal information. Furthermore, physical activity is one of the factors that influences GC dynamics and the performance of prediction models can be improved significantly when physical activity information is utilized in the modeling process [25, 7, 17]. The presence, duration and intensity of physical activity can be determined from energy expenditure. Energy expenditure is estimated [40] from physiological signals and is taken into account as one of the exogenous inputs while building LV-based GC prediction models. The GC values reported by the CGM is the main variable used for managing the glucose dynamics with feedback control. Once the GC values during the infusion of insulin is modeled, the GC values after insulin infusion can be forecasted easily. Thus, the input matrix for n observations is arranged as

$$X=\left[\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_n\end{array}\right]=\left[\begin{array}{cccc}{u}_{GC,1}& u_{PIC,1}& u_{\mathit{\text{Meal}},1}& u_{EE,1}\\ u_{GC,2}& u_{PIC,2}& u_{\mathit{\text{Meal}},2}& u_{EE,2}\\ ⋮& \ddots & \ddots & ⋮\\ u_{GC,n}& u_{PIC,n}& u_{\mathit{\text{Meal}},n}& u_{EE,n}\end{array}\right]$$

where

$$\begin{gathered} u_{GC,i}=[y(i),y(i+1),\dots ,y(i+L-1)] \\ {u}_{PIC,i}=\left[u_{PIC}(i),u_{PIC}(i+1),\dots ,u_{PIC}(i+L-1)\right] \\ {u}_{\mathit{\text{Meal}},i}=\left[u_{\mathit{\text{Meal}}}(i),u_{\mathit{\text{Meal}}}(i+1),\dots ,u_{\mathit{\text{Meal}}}(i+L-1)\right] \\ {u}_{EE,i}=\left[u_{EE}(i),u_{EE}(i+1),\dots ,u_{EE}(i+L-1)\right] \end{gathered}$$

and y(i), u_PIC(i), u_Meal(i), and u_EE(i) are the ith GC measurement, estimated PIC, estimated meal absorption rate, and energy expenditure, respectively. L is the past window length. For a prediction horizon PH, the corresponding output matrix can be constructed as

$$Y=\left[\begin{array}{c}{y}_1\\ y_2\\ ⋮\\ y_n\end{array}\right]$$

where y_i = [y(i + L − PH), y(i + L − PH + 1), …, y(i + L − 1)], the input and output vector are shown in Figure 3. It is clear that the variables in the output matrix Y are part of the input matrix X, thus, for a new observation, the variables that have the same index as variables in the output vector are not observable, and the input matrix X can be rearranged as shown in equation (19).

Figure 3:

Illustration of the arrangement of input and output matrix

Furthermore, the recent GC dynamics usually have a larger effect on future GC estimates. Thus, to design the TC kernel, the kernel parameters corresponding to GC variables are set to smaller values than the hyperparameters that correspond to other variables (i.e., PIC, meal absorption rate, and energy expenditure). By doing so, the GC values can have a larger contribution to the LVs that describe the major variance in the estimates of future GC values. Furthermore, the parameter values in the regularization matrix decrease along the diagonal for each corresponding variable since the latest GC, PIC, meal absorption rate, and energy expenditure values have larger effects on future GC values.

At each sampling time, when a new GC value is reported by the CGM, PIC value and meal absorption rate can be estimated by the PIC estimator [39] providing the preceding insulin information. Accordingly, two new observation vectors can be generated: the complete observation vector x_new that contains 4L variables in the case where PIC, estimation of meal effect, and energy expenditure are used as exogenous inputs and an incomplete observation vector $x_{\mathit{\text{new}}}^{\ast }$ that is composed of (4L − PH) values where the future information such as meal, exercise, and PIC are assumed to be known from a model in the model-based AP control system. The complete observation vector x_new which contains new information about the glucose dynamics is used to update the training dataset X and the incomplete observation vector $x_{\mathit{\text{new}}}^{\ast }$ is the input of the developed LV-based model to predict future GC values.

For online glucose prediction using LV-based modeling technique with exogenous inputs, the following steps are iterated for every new sample after initialization by using historical data for one day.

1. Estimate u_PIC and u_Meal using PIC estimator.

2. Generate the complete observation vector x_new and incomplete observation vector $x_{\mathit{\text{new}}}^{\ast }$ using the measured GC values and estimated PIC, meal absorption rate, and energy expenditure.

3. If the number of observations in the training dataset X is smaller than N, add the new complete observation x_new to the dataset X, otherwise, replace the earliest observation in X with the new observed x_new.

4. Normalize the training predictor matrix X and the corresponding output matrix Y.

5. Develop the rPLS model on the predictor X and its corresponding response Y as stated in (10).

6. Estimate score vector $\widehat{\tau }$ using the incomplete observation vector x_new∗ as stated in (21).

7. Predict the future GC values with prediction horizons up to PH using (22).

8. Process the next data point.

4. Results and Discussion

The performance of the proposed LV-based glucose prediction models developed by PLS and rPLS algorithms were evaluated with both in-silico and clinical subject data along with the GC prediction models based on recursively updated ARMAX models and kernel recursive least squares (KRLS) model. In ARMAX models, the model parameters are updated using recursive least squares (RLS) algorithm. The model predictions were compared with GC values measured by CGM sensor with a sampling time of 5 minutes. To develop the LV-based GC prediction model, the data from the first day are collected as the initial training dataset. The past window length (also referred to as time lag) L is set to be 36 samples (3 hours). As new GC data becomes available, the new observations are augmented with L of 36 and 24 samples to generate the so-called complete observation and incomplete observation. The training data set can be updated with the new complete observation with the longer L, and the incomplete observation is treated as the new input of the model to predict the following one hour ahead GC values.

For the given N testing data points where y defines the measured GC data and $\widehat{y}$ represents the GC prediction values, the model accuracy can be quantitatively analyzed by calculating the metrics:

1. Mean absolute relative deviation (MARD):

   $$\mathit{\text{MARD}}=\frac{1}{N}{\sum }_{i=1}^N\left|\frac{y(i)-\widehat{y}(i)}{y(i)}\right|\times 100\%$$ (23)

   which indicates how close the prediction results and the actual data match.
2. Root mean squared error (RMSE):

   $$\mathit{\text{RMSE}}=\sqrt{\frac{1}{N}{\sum }_{i=1}^N{(y(i)-\widehat{y}(i))}^2}$$ (24)

   where the deviation of predictions from data is indicated.

3. Clark error grid analysis [41] has been shown to be a reliable method for evaluating the accuracy of CGM sensor data and it has been used to assess the performance of a GC prediction model by plotting GC data versus the predicted values. In the following results, the percentages of the model predictions that lie in each “clinically accurate” regions are reported (Zone A indicating most accurate).

4.1. In silico study with UVa/Padova T1D simulator

The simulated subject data were generated with a sampling time of 5-min using the FDA-accepted University of Virginia/Padova T1D simulator [42]. The simulations included 6 days with meal information generated from Table 1 and Table 2, where the mean values and standard deviation (STD) of time, duration, and amount of carbohydrate of each meal are defined, respectively. The meal characteristics vary daily based on the parameters in Tables 1 and 2.

Table 1:

Means of variables in the scenario for simulation study

Events	Meal				Exercise
	B	L	D	S	TM	Bk
ST	08:00	12:30	18:30	21:00	10:30	16:30
Dur	20	30	30	10	30	30
Quantity	60	75	80	40	5	60

B: breakfast, L: lunch, D: dinner, S: snack, TM: treadmill, Bk: bike, ST: start time, Dur: duration(minutes) Quantity: Carbohydrates (g) for meal, speed (mph) for treadmill, power (W) for bike

Table 2:

Standard deviation and maximum variation of variables in the scenario for simulation study

Events	Standard deviation			Maximum variation
	ST	Dur	Quality	ST	Dur	Quality
Meal	15	5	20	120	10	25
TM	15	5	0.5	120	20	2
Bk	15	5	10	120	20	25

ST: start time, Dur: duration (minute), TM: treadmill, Bk: bike

Quantity: Carbohydrates (g) for meal, speed (mph) for treadmill, power (W) for bike

To investigate the effects of exogenous inputs (i.e., PIC and meal absorption rate), the LV-based GC prediction model with and without exogenous inputs were compared. For the case where exogenous inputs were not used, the LV-based GC prediction model are labeled as rPLS and PLS model, respectively. For the case where exogenous inputs containing estimation of PIC and meal absorption rate were used, the LV-based model are labeled as rPLSX and PLSX model, respectively. L was set to 36 and the prediction horizon PH was set to 12, such that the future GC values during the next one hour can be predicted. The number of LVs is 10. While 3 or 4 LVs seem enough for the glucose-only models (Figure 4), 10 LVs are appropriate for the multivariable case that includes physiological variables discussed in the next sections. For ease of comparison, 10 LVs are used for both cases.The hyperparameters for TC kernel are η = [λ, α] = [1, 0.1] and [η₁, η₂, η₃] = [λ₁, α₁, λ₂, α₂, λ₃, α₃] = [145, 0.1, 145, 0.99, 145, 0.99] for LV-based model without and with exogenous inputs, respectively.

Figure 4:

Variance explained by each LV in LV models

The MARDs and RMSEs are summarized in Table 3. When exogenous inputs are not used in modeling, the prediction accuracy of PLS and rPLS method are almost the same where the average RMSE is about 17.75 mg/dL and the average MARD is 5.64% for 6-steps-ahead prediction. The average RMSE is about 37.27 mg/dL and the average MARD is 13.96% for 12-steps-ahead prediction. This is because as the number of LVs increases, both rPLS and PLS methods explain 100% variance in the output matrix (Figure 4). The first LV in rPLS model explained more variance in the output data which indicates that the major LV can explain more variations in the output matrix by using the regularization term and it is possible to develop an LV model with fewer LVs by using rPLS compared to PLS algorithm.

Table 3:

MARD (%) and RMSE (mg/dL) results (Mean/STD) for glucose prediction of in-silico subjects using UVa/Padova T1DM Simulator

Model	PH= 30 min		PH= 60 min
	MARD	RMSE	MARD	RMSE
PLS	5.64/2.86	17.75/7.42	13.96/6.21	37.27/13.30
rPLS	5.64/2.86	17.74/7.41	13.96/6.92	37.26/13.30
PLSX	1.88/1.29	5.37/2.86	4.22/3.12	11.04/6.20
rPLSX	1.52/1.24	4.37/2.66	3.65/2.90	9.78/5.90

When exogenous inputs were used in the modeling process, the average RMSE for 6-steps-ahead prediction reduced to 5.37 mg/dL and 4.37 mg/dL for PLSX and rPLSX model, respectively. Similarly, the average MARD for 6-steps-ahead prediction reduced to 1.88% for PLSX model and 1.52% for rPLSX model. For 12-steps-ahead prediction, the average RMSE and MARD are 11.04 mg/dL and 4.22% for PLSX model, and 9.78 mg/dL and 3.65 % for rPLSX model. The average RMSE and MARD indicate that the prediction accuracy of the LV-based GC prediction model can be improved significantly by using exogenous inputs.

The RMSE and MARD of both LV-based models for different PH (Figure 5) indicate that when exogenous inputs are used, the increases in RMSE and MARD are smaller as the PH is increasing. This shows the potential to develop long-term GC prediction models based on LV models with exogenous inputs.

Figure 5:

Changes in RMSE and MARD with PH for subjects in UVa/Padova T1DM simulator

The percentages of GC predictions in different accuracy regions of the Clarke Grid (Table 4) indicate that the percentage of GC predicted by PLSX and rPLSX models in zone A (most accurate) for 6-steps-ahead prediction is 99.67% and 99.81% compared to 94.46% by LV-based models without exogenous inputs. For 12-steps-ahead prediction, the percentage of predictions in zone A increased 19.76% for PLSX (from 77.59% for PLS to 97.35% for PLSX) and 20.43% for rPLSX model.

Table 4:

Clark error grid results (%, Mean/STD) for glucose prediction of in-silico subjects using UVa/Padova T1DM simulator

Model	PH= 30 min					PH= 60 min
	zone A	zone B	zone C	zone D	zone E	zone A	zone B	zone C	zone D	zone E
PLS	94.46/5.39	4.51/3.70	0.01/0.02	1.00/1.77	0.02/0.04	77.59/15.02	18.87/11.53	0.02/0.09	3.51/4.18	0.01/0.03
rPLS	94.46/5.39	4.51/3.69	0.01/0.02	1.01/1.78	0.02/0.04	77.59/15.07	18.87/11.57	0.02/0.09	3.51/4.18	0.01/0.03
PLSX	99.67/0.64	0.26/0.50	0/0	0.07/0.20	0/0	97.35/4.37	2.31/3.84	0/0.03	0.34/0.60	0/0
rPLSX	99.81/0.57	0.15/0.44	0/0	0.04/0.15	0/0	98.02/3.60	1.69/3.07	0.01/0.03	0.28/0.58	0/0

Figure 6 compares 6-steps-ahead predictions based on PLSX and rPLSX models. The LV-based models with exogenous inputs capture the glucose dynamics, and the GC values predicted by rPLSX model are more accurate around the peaks and valleys (nadirs).

Figure 6:

Comparison of measured and 30-min-ahead predicted GC values (CGM) for in-silico subject 19

4.2. In-silico study with multivariable T1DM simulator - Physical activities asdisturbance

To further evaluate the prediction performance of the LV-based models with exogenous inputs, the multivariable glucose-insulin-physiological variables simulator (mGIPsim) developed in our group [43] is used to generate in-silico subject data. In this simulator, a physical activity compartment model is added to compute the effects of physical activities on GC dynamics. It describes the effects of exercise on glucose utilization and provides physiological data as outputs in addition to GC values. For the 20 subjects in mGIPsim, appropriate doses of insulin calculated by a model predictive controller (MPC) were given for a two-weeks-long simulation. The meal information and exercise scenarios were the same for the 20 in-silico subjects. In this study, two different scenarios were defined: regular daily routine (Case I) and the irregular daily routine (Case II).

Case I:

The subjects are assumed to take regular daily activities, a daily scenario includes four meals (breakfast, lunch, dinner, and night snack), and one exercise (treadmill running or biking). The mean values of start times, durations, and quantities (carbohydrate for meal, speed for treadmill, and power for bike) of all events are summarized in Table 1. The standard deviation and maximum variation of the parameters in each event are described in Table 2. It is assumed that the variables that define the daily activities follow Gaussian distribution. Thus, given the standard deviations of each variable, random variations which are limited by the maximum variation in Table 2 can be introduced into each event to generate more realistic scenarios for different days of the simulation.

Case II:

The daily scenario was first generated in the same manner as case I, though the night snack of the first Friday and the breakfasts of the two Saturdays are deleted. Furthermore, the in-silico subjects do exercises only on Mondays, Wednesdays, Saturdays and Sundays rather than every day, and the duration and intensity of the physical activity are assumed to be longer and higher on weekends than week days.

After generating a training dataset using data collected from the first day, the GC prediction model based on LV techniques can be updated online and can forecast the future GC values up to 60-min where the number of LVs is 10 and the kernel parameters are [η₁, η₂, η₃, η₄] = [λ₁, α₁, λ₂, α₂, λ₃, α₃, λ₄, α₄] = [145, 0.1, 145, 0.99, 145, 0.99, 145, 0.99].

For Case I, the MARDs and RMSEs are summarized in Table 5. The RMSEs of the proposed LV-based models are 4.40 mg/dL and 2.52 mg/dL for 6-steps-ahead prediction by using PLSX and rPLSX models, respectively. The MARDs of the PLSX and rPLSX models are 1.31% and 0.6% with PH=30 min. For 12-steps-ahead (60 min) prediction, the RMSE and MARD of the PLSX model are 7.29 mg/dL and 2.23%, and for the rPLSX model 5.81 mg/dL and 1.50%, respectively. Both the MARDs and RMSEs in Table 5 indicate that GC values predicted by the proposed LV-based models are more accurate than those predicted by ARMAX model and KRLS model. Thus the proposed LV-based models have better performance in capturing glucose dynamics while the subject perform their regular daily routines. Furthermore, the rPLSX model gives more accurate predictions than the PLSX model.

Table 5:

MARD (%) and RMSE (mg/dL) results (Mean/STD) for glucose prediction of 20 in silico subjects

Model	PH= 30 min				PH= 60 min
	Case I		Case II		Case I		Case II
	MARD	RMSE	MARD	RMSE	MARD	RMSE	MARD	RMSE
ARMAX	1.50/0.42	8.25/3.50	1.70/0.68	10.78/8.29	6.05/1.92	25.00/6.45	6.11/1.48	25.38/6.74
KRLS	3.38/0.96	11.26/1.96	3.60/0.96	11.72/1.93	9.15/2.29	29.31/3.88	9.92/2.52	30.26/3.78
PLSX	1.31/0.29	4.40/0.59	1.48/0.33	4.87/0.75	2.23/0.64	7.29/1.22	2.72/0.84	8.51/1.49
rPLSX	0.60/0.15	2.52/0.56	0.65/0.16	2.70/0.70	1.50/0.41	5.81/1.24	1.73/0.46	6.54/1.58

The trends of RMSE and MARD with PH (Figure 7) indicate an increase as the PH becomes larger. For 5- and 10-min-ahead prediction, ARMAX model gives more accurate predictions. LV-based models have smaller prediction errors for the larger PH, and rPLSX model can forecast future GC values more accurately.

Figure 7:

Changes in RMSE and MARD with PH for Case I simulations

Table 6 summarizes the percentage of data distributed in different regions of the Clarke grid with PH of 30-min where 100.00%, 98.59%, and 99.05% of the GC estimates lie in region A predicted by LV-based, ARMAX, and KRLS models, respectively. The distribution of estimates with PH of 60-min are summarized in Table 6 as well, where 99.86%, 99.83%, 94.95%, and 89.08% of the estimates lie in region A for the PLSX, rPLSX, ARMAX, and KRLS models, respectively. For short-term predictions (30-min-ahead), all the models can give reliable “clinical accuracy” with over 99% of predictions in region A. The proposed LV-based models give more accurate prediction than the ARMAX and KRLS models. For longer-term predictions (60-min-ahead), the accuracy of ARMAX and KRLS model predictions decreases, while the proposed LV-based models can still predict GC values with high “clinical accuracy”.

Table 6:

Clark error grid results (%, Mean/STD) for GC predictions of 20 in silico subjects in Case I

Model	PH= 30 min					PH= 60 min
	zone A	zone B	zone C	zone D	zone E	zone A	zone B	zone C	zone D	zone E
ARMAX	99.71/0.36	0.19/0.17	0.04/0.10	0.01/0.03	0.05/0.12	94.95/3.08	4.17/2.76	0.32/0.49	0.46/0.38	0.10/0.09
KRLS	99.05/1.02	0.94/1.02	0/0	0.01/0.01	0/0.02	89.08/5.55	9.69/4.76	0.12/0.23	1.08/0.71	0.03/0.04
PLSX	100.00/0.01	0/0.01	0/0	0/0	0/0	99.86/0.14	0.14/0.14	0/0	0/0	0/0
rPLSX	100.00/0.01	0/0.01	0/0	0/0	0/0	99.83/0.19	0.17/0.19	0/0	0/0	0/0

Figure 8 shows the comparison of measured and predicted GC values for PH of 30-min and 60-min, respectively. For 30-min-ahead prediction, the predictions of LV-based models overlap with the data, while the predictions by the ARMAX and KRLS model oscillate in the valleys and peaks. Furthermore, the rPLSX model gives more accurate prediction around the peaks. For 60-min-ahead prediction, the predicted GC values by ARMAX and KRLS models have larger deviations from the data, while the predictions by LV-based models can still match the data.

Figure 8:

Comparison of measured and predicted GC values using LV-based, ARMAX, and KRLS methods for in-silico subject 8 in Case I

For case II, the MADRs and RMSEs are presented in Table 5. The MARDs of GC predicted by LV-based models are 1.48% and 0.65% for PLSX and rPLSX with PH of 30-min, respectively. For 60-min-ahead prediction, the MARDs are 2.72% and 1.73% for PLSX and rPLSX models, respectively. The MARDs of ARMAX and KRLS models are 1.70% and 3.60% with PH of 30-min, and are 6.11% and 9.92% for 60-min ahead prediction. The RMSEs for PH of 30-min are 4.87mg/dL, 2.70 mg/dL, 10.78 mg/dL, and 11.72 mg/dL for PLSX, rPLSX, ARMAX, and KRLS models, respectively. And the RMSEs for PH of 60-min are 8.51 mg/dL, 6.54 mg/dL, 25.38 mg/dL, and 30.26 mg/dL for LV models based on PLSX and rPLSX, ARMAX model, and KRLS model, respectively. The MARDs and RMSEs indicate that the proposed PLSX models can give more reliable predictions for both PHs of 30-min and 60-min, and both are smaller for rPLSX compared to PLSX.

Figure 9 shows the relationship between the RMSE and MARD with PH for different algorithms. When PH increases, future disturbances (such as exercise and meal) will change the glucose dynamics, which is hard for the model to capture. Thus, the RMSE and MARD increase with longer PH, as expected. ARMAX and KRLS models have smaller prediction errors for 1- and 2-steps-ahead prediction, however, as the PH increases, their prediction errors become larger. For PLSX and rPLSX models, the MARD and RMSE are smaller for long-term prediction, and the prediction error of rPLSX model is smaller.

Figure 9:

Changes in RMSE and MARD with PH for simulation study: Case II

GC predictions in different regions of Clarke error grid are reported in Table 7 where 99.57% and 99.90% of GCs predicted by LV-based models are in zone A for 30-min-ahead prediction. The ARMAX and KRLS models have good predictions as well, their predictions in zone A are 99.39% and 98.56%, respectively. For 60-min-ahead prediction, GC predictions by ARMAX and KRLS mode are in zone A are reduced (94.61% and 87.44%), while the prediction accuracy of LV-based models in zone A remain high (98.97% and 99.36%). The rPLSX model can predict the GC values accurately with over 99% of predictions in zone A.

Table 7:

Clark error grid results (%, Mean/STD) for GC predictions of 20 in silico subjects in Case II

Model	PH= 30 min					PH= 60 min
	zone A	zone B	zone C	zone D	zone E	zone A	zone B	zone C	zone D	zone E
ARMAX	99.39/0.78	0.29/0.25	0.09/0.16	0.04/0.08	0.18/0.49	94.61/2.33	4.44/2.14	0.33/0.41	0.40/0.33	0.21/0.35
KRLS	98.56/1.24	1.39/1.20	0/0.01	0.04/0.06	0/0	87.44/5.57	11.32/4.76	0.24/0.28	0.98/0.79	0.03/0.04
PLSX	99.57/0.31	0.33/0.23	0/0	0.10/0.17	0/0	98.97/0.57	0.90/0.43	0/0	0.13/0.25	0/0
rPLSX	99.90/0.09	0.08/0.08	0/0	0.02/0.03	0/0	99.36/0.41	0.56/0.35	0/0.01	0.08/0.14	0/0

The comparison of measured GC values and predictions (Figure 10) for 30-min and 60-min PH indicate that all the models can predict the glucose dynamics with high accuracy for PH of 30-min and the deviations of predictions from data are small. For 60-min-ahead prediction, the prediction performance of both the ARMAX model and KRLS model becomes degraded and delay between data and predictions can be observed. However, GC values predicted by the proposed LV-based models are still accurate, and the rPLSX model can predict the GCs around the peaks more accurately with no delays in predictions with respect to data.

Figure 10:

Comparison of measured and predicted GC values using LV-based, ARMAX, and KRLS methods for in-silico subject 3 in Case II

An issue facing the ARMAX and KRLS methods is the significant collinearity in the data, especially in biomedical applications where the states are not directly measurable and the readily obtained measurements that are often highly correlated with the response of the predicted outputs. This challenges many conventional recursive modeling approaches. The PLS and rPLSX methods are able to handle this collinearity by identifying relations in the latent variables, and thus overcoming the drawbacks of the conventional recursive identification methods.

4.3. Modeling with clinical study data

To further evaluate the performance of the proposed LV-based GC prediction models, models were developed with clinical data for PHs up to one hour. Eleven datasets with each contains about 2 days in duration of data collected from nine subjects with T1DM [44] are used. The study was conducted in adolescents with T1DM who attended the Yale Children’s Diabetes Clinic (New Haven, CT), where GC was reported by CGM every 5 minutes. Table 8 summarizes the demographic information on all nine subjects.

Table 8:

Demographic information of clinical subjects (Mean/Standard Deviation)

Age(year)	18.89/1.62
Body weight (kg)	60.56/9.58
Height (cm)	168.22/9.19
BMI (kg/m2)	21.39/3.17
Total daily insulin dose (U)(basal)	54.58(24.71)/20.77(11.87)
Duration of time with diabetes (month)	72.44/44.88
HbA1c (%)	7.36/0.61

The LV-based models are updated online after a one-day initialization period and the future GC values up to one hour are forecasted. 10 LVs which explain about 98% of the variance in the output matrix for both PLSX and rPLSX method are used and the parameters of the TC kernel are [η₁, η₂, η₃, η₄] = [λ₁, α₁, λ₂, α₂, λ₃, α₃, λ₄, α₄] = [145, 0.1, 145, 0.99, 145, 0.99, 145, 0.99].

The MARD and RMSE values (Table 9) for all the clinical datasets indicate that the LV-based GC prediction models have higher prediction accuracy and capture the glucose dynamics accurately. The MARDs of LV-based models are 5.92% and 5.68% for PH of 30-min, and are 10.65% and 10.45% for PH of 60-min, respectively, and their RMSEs are 10.65 mg/dL and 10.45 mg/dL for PH of 30-min, and 15.50 mg/dL and 14.48 mg/dL for PH of 60-min, respectively. The ARMAX and KRLS models have MARD and RMSE values that are twice as large (MARD: 11.23% and 13.57% for PH of 30-min, and 17.06% and 25.81% for PH of 60-min, and RMSE: 21.97 mg/dL, 23.25 mg/dL, 32.62 mg/dL, and 43.22 mg/dL, respectively).

Table 9:

MARD (%) and RMSE (mg/dL) results (Mean/STD) for GC prediction of clinical Subjects

Model	PH= 30 min		PH= 60 min
	MARD	RMSE	MARD	RMSE
ARMAX	11.85/5.32	21.97/7.54	17.06/9.30	32.62/16.66
KRLS	13.57/2.93	23.25/5.44	25.81/4.89	43.22/9.71
PLSX	5.92/2.29	10.65/4.67	8.61/4.28	15.50/7.38
rPLSX	5.68/2.80	10.45/4.97	7.84/3.89	14.48/6.68

The changes in RMSE and MARD with PH (Figure 11) show their increase with longer PH, as expected. The flatter slopes of RMSE and MARD curves as a function of PH for the LV-based GC prediction models compared to the slopes of ARMAX and KRLS models match the results with simulated data and indicate the potential for using the regularized PLS models (rPLSX) to forecast GC values for longer prediction horizons.

Figure 11:

Changes in RMSE and MARD with PH for clinical study data

Table 10 summarizes the Clarke error grid results where 95.28%, 81.80%, and 78.61% GCs predicted by the proposed LV-based models, ARMAX model and KRLS model lie in zone A for PH of 30-min. For the PH of 60-min, only 73.40% of predictions with ARMAX model lie in zone A and only 50.73% predictions with KRLS model lie in zone A. Over 90% of predictions are in zone A for the LV-based models, 91.45% with PLSX and 92.08% with rPLSX model.

Table 10:

Clark error grid results (%, Mean/STD) for GC prediction of clinical subjects

Model	PH= 30 min					PH= 60 min
	zone A	zone B	zone C	zone D	zone E	zone A	zone B	zone C	zone D	zone E
ARMAX	81.80/13.93	17.03/13.07	0.22/0.53	0.95/1.16	0/0	73.40/19.22	23.41/17.31	1.00/1.47	1.77/1.81	0.43/1.09
KRLS	78.61/7.68	20.11/6.82	0.21/0.58	1.08/1.23	0/0	50.73/9.89	46.02/8.24	0.90/1.41	1.94/1.95	0.41/1.15
PLSX	95.28/5.70	4.18/4.73	0.03/0.09	0.51/1.12	0/0	91.45/8.90	7.82/8.06	0.06/0.19	0.67/1.26	0/0
rPLSX	95.28/6.00	4.15/4.90	0.06/0.19	0.52/1.23	0/0	92.08/7.47	7.19/6.50	0.12/0.29	0.61/1.23	0/0

The comparison of predicted GC values and the measured GC data from CGM sensors (Figure 12) illustrate that LV-based models, in particular rPLSX, are accurate in forecasting GC values for PH of 30-min and match the peaks and valleys without distortions. For PH of 60-min, the LV-based GC prediction models still give reliable predictions of future GCs, but in a few valleys the predicted GC is slightly higher than the actual data at the nadirs (Figure 12 near sample 150), implying that some of hypoglycemic events might be missed if the GC values are near the hypoglycemia threshold.

Figure 12:

Comparison of measured and predicted GC values for a clinical subject

5. Conclusions

Glucose concentration prediction models based on LV techniques provide accurate predictions for 30- and 60-minute prediction windows both with simulated data and clinical data. LV-based models with exogenous inputs are more accurate with the regularized (rPLSX) method than conventional PLS. The LVs generated from rPLS algorithm utilizes the prior information of the model and explain more variations of the outputs. A comparison of the performance of these recursive LV-based models with ARMAX models (model parameters are updated recursively using RLS algorithm) and KRLS model yields smaller MARD and RMSE values for LV-based models for 30-min and 60-min prediction horizons. This superior performance was observed both when the subjects follow the same daily activities, or when the daily events of subjects vary from day to day. For the retrospective clinical data, the LV-based models provided significant improvement in prediction accuracy compared with the ARMAX and KRLS model.