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Abstract—We investigated the relative importance and

predictive power of different frequency bands of subcutaneous

glucose signals for the short-term (0-50 min) forecasting of

glucose concentrations in type 1 diabetic patients with data-

driven, autoregressive (AR) models. The study data consisted of

minute-by-minute glucose signals collected from nine deidentified

patients over a five-day period using continuous glucose

monitoring devices. AR models were developed using single and

pairwise combinations of frequency bands of the glucose signal

and compared with a reference model including all bands. The

results suggest that: for open-loop applications, there is no need

to explicitly represent exogenous inputs, such as meals and insulin

intake, in AR models; models based on a single-frequency band,

with periods between 60-120 min or 150-500 min, yield good

predictive power (error <3 mg/dL) for prediction horizons of up

to 25 min; models based on pairs of bands produce predictions

that are indistinguishable from those of the reference model as

long as the 60-120 min period band is included; and AR models

can be developed on signals of short length (~300 min), i.e.,

ignoring long circadian rhythms, without any detriment in

prediction accuracy. Together, these findings provide insights

into efficient development of more effective and parsimonious

data-driven models for short-term prediction of glucose

concentrations in diabetic patients.

Manuscript received November 3, 2009; revised February 12, 2010. This

work was supported in part by the U.S. Army Medical Department, Advanced

Medical Technology Initiative, funded by the Telemedicine and Advanced

Technology Research Center of the U.S. Army Medical Research and

Materiel Command, Fort Detrick, MD, and by the U.S. Air Force Diabetes

Research Program.

Y. Lu is with the Bioinformatics Cell, Telemedicine and Advanced

Technology Research Center, U.S. Army Medical Research and Materiel

Command, Fort Detrick, MD 21702 USA (e-mail: ylu@bioanalysis.org).

A. V. Gribok was with the Bioinformatics Cell, Telemedicine and

Advanced Technology Research Center, U.S. Army Medical Research and

Materiel Command, Fort Detrick, MD 21702, as well as with the Nuclear

Engineering Department, the University of Tennessee, Knoxville, TN 37996

USA (e-mail: agribok@bioanalysis.org).

W. K. Ward is with iSense Corporation and Oregon Health and Sciences

University, Wilsonville, OR 97070 USA (e-mail: kenward503@msn.com).

*J. Reifman is with the Bioinformatics Cell, Telemedicine and Advanced

Technology Research Center, U.S. Army Medical Research and Materiel

Command, Fort Detrick, MD 21702 USA (phone: 301-619-7915; fax: 301-

619-1983; e-mail: jaques.reifman@us.army.mil).

Copyright (c) 2008 IEEE. Personal use of this material is permitted.

However, permission to use this material for any other purposes must be

obtained from the IEEE by sending an email to pubs-permissions@ieee.org.

Index Terms—Autoregressive prediction models, glucose signal

frequency analysis, continuous glucose monitoring, glucose

dynamics, diabetes.

I. INTRODUCTION

ECENT developments in continuous glucose monitoring

(CGM) offer new opportunities and challenges in data

collection and analysis [1], as these CGM devices can sample

subcutaneous glucose concentrations as frequently as every

minute and store the sampled time series for up to several days

for retrospective analysis. On one hand, this abundance of

information opens new opportunities in data analysis and the

understanding of the mechanisms of glucose regulation, while,

on the other hand, it poses additional challenges in terms of

information processing and interpretation. Long and frequently

sampled time-series data collected from CGM devices have

naturally invited the use of techniques that require continual

availability of glucose data, such as Kalman filtering [2] and

techniques that are purely data driven, such as autoregressive

(AR) models [3], for short-term prediction of glucose

concentrations in diabetic patients [4-8]. The obvious

advantages of AR models are their analytical tractability and

their ability to linearly extrapolate future time-series values,

producing a reliable and accurate forecast. However, to yield

accurate predictions, AR models need to be fitted to a

―training‖ signal and the fitting procedure should lead to a

sequence of AR model coefficients that capture the major

frequency components (or bands) in the glucose signal.

The frequency bands in blood glucose signals reflect the

physiological mechanisms of glucose regulation, with different

mechanisms driving different frequency bands. For example,

in healthy individuals, the pulsatile insulin secreted by the

pancreas is reflected in patterns of blood glucose signal

oscillations with periods between 4 and 15 min [9]. Similarly,

the patterns associated with postprandial glucose regulation of

healthy individuals have predominant periods ranging from 51

to 112 min [10], with larger amplitude oscillations observed

after evening meals and smaller ones after morning meals,

reflecting the circadian rhythmicity of glucose regulation [11].

Recently, Rahaghi and Gough suggested that the spectrum

of oscillations in blood glucose signals of healthy individuals

could be characterized by four major frequency bands [12].

The Importance of Different Frequency Bands

in Predicting Subcutaneous Glucose

Concentration in Type 1 Diabetic Patients

Yinghui Lu, Andrei V. Gribok, W. Kenneth Ward, and Jaques Reifman*

R

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TBME-00930-2009.R1

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They suggest that the lowest frequency band, corresponding to

periods of at least 700 min (Band IV), reflects patterns of

glucose regulation associated with ultradian rhythms of long

periodicity and circadian rhythms. The first mid-frequency

band, spanning oscillations with periods from 150 to 500 min

(Band III), is believed to be primarily associated with the

schedules of exogenous inputs, such as the time of meals and

insulin boluses intake, while the second mid-frequency band,

spanning periods from 60 to 120 min (Band II), is deemed to

reflect insulin secretion in response to continuous enteral

nutrition, constant intravenous glucose infusion, or ingestion

of a meal or insulin boluses [10, 13]. These oscillations are

considered to be intrinsic responses [12] because they can be

interpreted as a step, or an impulse, response of the glucose

regulatory system. The highest frequency band, with periods

between 5 and 15 min (Band I), is assumed to be generated by

rapid pulsatile insulin secretion by pancreatic β cells and is

best observed in fasting, non-diabetic subjects [9]. However,

because β cells are destroyed in type 1 diabetic patients and

insulin is not produced, these high frequencies are absent in

the blood glucose signals of this patient population [14].

In contrast to blood glucose, there is a limited body of work

concerning the frequency analysis of subcutaneous CGM

signals, which, due to time delays and signal attenuation from

blood-to-interstitial transport, may differ from that of the blood

glucose, requiring additional independent studies. By and

large, the existing studies of the spectrum of CGM signals are

limited to Fourier series analysis [15, 16] and are not guided to

educating the development of predictive glucose models. For

example, Breton et al. performed Fourier analysis for

determining the minimal sampling periods of blood and

subcutaneous CGM signals of type 1 diabetic patients and

concluded that CGM signals of periods shorter than 36 min are

nonexistent [15]. This loss of information is attributed to the

lowpass filtering effect caused by the blood-to-interstitial

transport of glucose concentration. In a separate Fourier

analysis study, Miller and Strange [16] suggest that, in type 1

diabetic patients, the amplitude of the second and third

harmonics of CGM signals are correlated with mean HbA1c

values and that, in type 2 diabetic patients, the frequency

content with periods shorter than 72 min is characterized by

white noise.

The efficient development of glucose concentration

predictive models for type 1 diabetic patients, in particular

data-driven AR models, where the model coefficients capture

the frequency content of the signal [4, 5], require a more

detailed analysis of the frequency components of the CGM

glucose signal so that the relative importance and predictive

power of the different frequency bands are properly

characterized. In particular, the answers to the following key,

yet unknown, questions need to be addressed: (1) How to

optimize the glucose signal filtering process so as to eliminate

uninformative signal components while keeping the important

ones? (2) Which frequency components must be present in an

AR model to yield accurate, short-term predictions? (3)

Whether there is a need to explicitly represent exogenous

inputs, such as meals and insulin intake, into the model? and

(4) How much data are needed to develop an accurate AR

model?

In this paper, we attempt to address the above questions by

first associating the frequency content of subcutaneous glucose

signals with those of blood glucose signals and then

investigating the relative importance of the different frequency

bands of the subcutaneous glucose signal in AR modeling.

Based on the four major frequency bands suggested by

Rahaghi and Gough [12], we applied subband AR modeling

[17] to CGM signals of type 1 diabetic patients and

determined the predictive power of the different frequency

bands and their dependencies on prediction horizon. We found

that, provided enough training data were available, the AR

models captured all the frequency information present in the

subcutaneous glucose concentration signal, obviating the need

to explicitly represent exogenous inputs into the model, such

as meals and insulin, for open-loop applications. We also

found that the frequency band associated with the intrinsic

response of glucose regulation was indispensable for obtaining

accurate predictions up to 50 min ahead, although the energy

content of this frequency band in the glucose signal is low

(~1.5% of the total signal’s energy). Finally, we concluded that

a training signal as short as 300 min, i.e., one that excludes

low circadian rhythm frequencies, was capable of producing

accurate predictions, potentially shortening data collection

time and expediting model development. Together, these

findings provide insights into the development of more

effective and parsimonious AR models for short-term

predictions of subcutaneous glucose concentrations in diabetic

patients.

II. METHODS

A. Study Population

In this paper, we analyzed the temporal dynamics and

frequency content of subcutaneous glucose time-series data of

nine deidentified type 1 diabetic patients. Subcutaneous

glucose measurements were collected on a minute-by-minute

basis for each of the nine subjects for approximately 5 days

with the iSense CGM system [4, 6]. Subjects were confined to

the investigational site for the whole duration of the study and

limited to mild physical activity. Subjects were included if

they were between 18 and 70 yr of age, had been diagnosed

with type 1 diabetes and treated with insulin for at least 12 mo,

had a body mass index of <35.0 kg/m2, and had glycated

hemoglobin (HbA1c) of >6.1%. Subjects were excluded if

they had acute and severe illness apart from diabetes, had a

clinically significant abnormal electrocardiogram, hematology,

or biochemistry screening test, or had any disease requiring the

use of anticoagulants. In addition, subjects were excluded if

they were pregnant or lactating.

The subjects were provided three meals per day at 9 a.m., 1

p.m., and 7 p.m. (plus a mid-afternoon snack at 4 p.m.) and

continued their normal insulin therapy, which was provided

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TBME-00930-2009.R1

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either by an external continuous insulin pump or by multiple

daily subcutaneous injections. In addition, each subject

received a bolus of regular or ultra rapid insulin immediately

before each meal (excluding the snack) either by subcutaneous

injection or via the subcutaneous catheter of the insulin pump.

Figure 1a shows the raw CGM signal for one patient (subject

#6) in our study collected over 4,000 min (66.7 h), where the

time points of food and insulin intake are illustrated by vertical

lines. The figure illustrates the typical daily variations in

glucose concentration, including a drastic increase in

concentration between 6 and 9 a.m. although no food or other

nutrients were taken during the night. This is due to a circadian

rhythm known as the ―dawn phenomenon,‖ which is explained

by an increase in insulin resistance caused by certain

hormones, and occurs in both diabetic and non-diabetic

individuals alike [18, 19].

B. Frequency Analysis

To study the relative importance and predictive power of the

four major frequency bands suggested by Rahaghi and Gough

[12] for subcutaneous CGM signals, we developed four

bandpass filters, where each filter only passed glucose signals

in one of the four period bands (Band I: 5-15 min, Band II: 60-

120 min, Band III: 150-500 min, and Band IV: ≥700 min) of a

raw CGM time-series signal. Figure 1b shows the

corresponding four bandpass-filtered signals extracted from

the raw time-series signal in Fig. 1a. For example, through

bandpass filtering of Band IV, only patterns associated with

circadian rhythms, such as the 24-h dawn phenomenon, and

ultradian rhythms with periodicities ≥700 min were extracted

from the raw signal. However, because the Band I frequencies

in Fig. 1b are expected to consist mainly of measurement noise

devoid of any significant physiological information [15, 16],

we performed limited analysis for this high-frequency band.

We also developed multi-bandpass filters that only passed

the frequencies associated with each of the pairwise

combinations of three of the four bands, II, III, and IV. For

example, for filters that passed the pairwise combination of

bands II+III, we used a multi-bandpass filter that only passed

the frequencies in each of these two bands, eliminating all

other frequencies, including those in the 120-150 min gap

between these bands. To obtain a reference glucose

concentration signal and corresponding model against which

all other subband signals and models could be compared with,

we used a lowpass filter with a cut-off frequency of 1/3,600 Hz

(equivalent to a period of 60 min), which removed the high-

frequency content of the CGM signal up to the lower bound of

Band II. This cut-off frequency was selected for two reasons.

First, as discussed above, Fourier series analysis from different

studies of diabetic patients provides substantial evidence that

CGM signals with periods shorter than 36 min are nonexistent

[15] and that periods shorter than 72 min are characterized by

white noise [16]. Second, our group has shown that, to obtain

consistent AR coefficients and robust models from CGM

signals, it is necessary to remove frequencies with periods

shorter than ~90 min [4]. The resulting reference signal is

illustrated in Fig. 1a.

Finally, to analyze the overall frequency content of glucose

concentration signals of type 1 diabetic patients, we used the

Welch’s method with a Hamming window of 50% overlap to

compute the power spectral density (PSD) of the raw CGM

signals [20].

C. Autoregressive (AR) Modeling

An AR model is a type of linear model where a future signal

yn+1, at discrete time n+1, is represented by a linear

combination of previous signal observations yn–i, i = 0, 1, 2,…,

m–1, plus white noise εn+1,

1

m

by

,

1

0

1

n

i

inin

y

(1)

where m denotes the order of the model, i.e., the number of

previous observations used to represent yn+1, and bi represents

fixed model coefficients. The coefficients bi describe the

temporal correlations between each of the previous signals yn–i,

i = 0, 1, 2,…, m–1, and the next one yn+1, and capture the

frequency content of the underlying signal [3]. Therefore, we

may use the set of coefficients bi and previous signals yn–i to

make one-step-ahead predictions for yn+1, i.e.,

m

y

,

ˆ

1

0

1

i

inin

yb

(2)

where ŷn+1 denotes the predicted value for yn+1. Equation (2)

can also be used to make k-step-ahead predictions, with k = 2,

3,…, by iteratively substituting the k–1 predicted values for the

corresponding k–1 yet unobserved signals. For example, we

may make 2-step-ahead predictions ŷn+2 by substituting the

0

200

400

Glucose (mg/dL)

Raw Signal

Reference Signal (Periods >1h)

0 12 2436 48 60

Relative Magnitude (mg/dL)

Time (h)

(b)

Night time

No food or insulin

Night time

No food or insulin

9 a.m. 1 p.m. 4 p.m. 7 p.m.

9 a.m. 1 p.m. 4 p.m. 7 p.m. 9 a.m. 1 p.m. 4 p.m. 7 p.m.

Band III (150 - 500 min)

Band II (60 - 120 min)

Band I (5 - 15 min)

Band IV (> 700 min)

(a)

Fig. 1. Subcutaneous glucose concentration of a typical type 1 diabetic

patient. (a) Raw and reference (i.e., filtered) glucose signals. (b) Relative

magnitude of the corresponding bandpass-filtered time series of the original

raw signal representing the four major frequency bands.

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TBME-00930-2009.R1

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predicted value ŷn+1 for its unobserved signal yn+1 on the right

side of Eq. (2).

An AR model can also be represented in the frequency

domain. By performing a Z-transform of Eq. (1), we can

convert its discrete-time representation into a frequency-

domain representation,

)(

zH

,

1

1

1

0

) 1(

m

i

i

izb

(3)

where H(z) denotes the corresponding transfer function of the

AR model. Because the coefficients bi in an AR model capture

the frequency content of the glucose signal, the PSD P(ω) of

this all-pole transfer function H(z),

)(

m

,

1

1

2

1

0

) 1(

i

ij

ieb

P

(4)

where ω denotes the radian frequency, can be used to

approximate the spectrum of the underlying glucose signal

[21].

To investigate the relative importance and predictive power

of each of the three Bands (II, III, and IV) associated with the

different dynamics of glucose regulation, we applied the

method of regularized least squares to fit the coefficients bi

and obtain AR models of order 30 (m = 30), as proposed in

[4]. Regularization yields smoothly varying AR-model

coefficients bi, a requirement for obtaining stable and accurate

models with clinically acceptable time lags [4]. For all

calculations, we used consecutive 2,000 min (or 2,000 data

points) of the glucose signal of a subject to fit the AR model.

For each subject, we developed separate AR models for each

of the three bands as well as for the three pairwise

combinations of bands. For comparison purpose, we also

developed (reference) AR models for each subject using the

reference signal consisting of all three bands. Moreover, to

further investigate the glucose dynamics within the different

frequency bands and their combinations and compare them to

the spectrum estimated from the raw CGM signal using the

Welch’s method, we calculated the PSD using Eq. (4) for each

of the AR models.

The predictive power was quantified by root mean squared

error (RMSE) deviations, defined as the square root of the

mean of the squared differences between the predicted value ŷn

and the observed value yn. The predictive performance of the

models were evaluated for each subject using their

corresponding testing data between 2,000 and 4,000 min, and

each model was evaluated for different prediction horizons,

ranging from 1 to 50 min.

III. RESULTS

Figure 2 shows the PSDs estimated using the raw CGM

signals for each of the nine patients in the study, where the

signal energy was plotted as a function of the signal period

instead of frequency to facilitate physiological interpretation.

The majority of the signal’s energy fell within the two longest

period ranges, Bands III and IV. In contrast, Band II contained

a relatively small amount of the total energy (~1.5%), while

the energy in Band I was only ~0.6% of the total, which

supports the conclusion that, as in blood glucose signals [14],

CGM signals of type 1 diabetic patients lack high-frequency

pulsatile insulin secretion patterns.

Figure 2 also illustrates the differences in the signal

spectrum profile for each of the frequency bands. For example,

Band IV was characterized by two distinct peaks at ~12 and

~24 h associated with well-established ultradian and circadian

periods [11]. Similarly, Band III was characterized by five

distinct peaks at approximately 3.0, 3.5, 5.0, 6.0, and 8.0 h.

The periods at 3.0 and 6.0 h exactly coincided with meal and

insulin schedules at 1 p.m., 4 p.m., and 7 p.m., whereas the

periods at 3.5 and 5.0 h were likely related to the 4.0 h time

interval between meal and insulin schedules at 9 a.m. and 1

p.m. We speculate that the 8.0 h period is not associated with

meal and insulin schedules, but rather with ultradian

oscillations associated with sleep [13]. Finally, analysis of the

spectrum profile for Band II indicated a number of peaks,

including two predominant ones with periods at ~80 and ~105

min (results not shown in the scale used in Fig. 2). This finding

is corroborated by the work of Simon et al. [10], who

identified the predominant period of the intrinsic regulatory

response in blood glucose to vary from 51 to 112 min.

Figure 3 shows the mean PSDs (averaged over the nine

subjects) for Bands II, III, and IV and their pairwise

combinations estimated from the corresponding AR-model

coefficients using Eq. (4). The results show that, as expected,

each AR model only captured the frequency content of the

corresponding frequency band(s) of the CGM signals.

Fig. 2. Power spectral densities (PSD) of the raw glucose time series of nine

patients and their averaged spectrum estimated using the Welch’s method.

The four gray areas correspond to the four modeled frequency bands.

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Comparison with Fig. 2 indicated that the PSDs estimated by

the subband AR models were, in general, less resolved than the

ones obtained from the raw signal (i.e., Fig. 2). For instance,

Fig. 2 shows five salient periods in Band III while Fig. 3b

depicts only two of them, at ~3.0 and ~6.0 h. This is mainly

attributed to the constraint, or regularization, imposed on the

fitting of the AR model coefficients bi [4].

Figure 4 shows the predictive performance of the subband

AR models, where RMSEs, averaged over the nine subjects,

were plotted as a function of prediction horizon for the

reference model as well as for the single-band models (Fig. 4a)

and the pairwise-band models (Fig. 4b). As expected, Fig. 4a

shows that the reference model, which captured the full

dynamic range of the glucose signal, yielded the smallest

RMSEs across the 0-50 min prediction horizons. For

prediction horizons of <25 min, the AR models obtained with

Band II or Band III frequencies had essentially the same

predictive performance as the model obtained with the

reference signal. This result is instructive because it suggests

that, for short prediction horizons, middle-frequency

dynamics, resulting from intrinsic oscillations or schedules of

exogenous inputs, are sufficient to produce accurate models

and that low-frequency dynamics, associated with circadian

rhythms, may not be necessary. For longer prediction horizons

(25-50 min), Band III models outperformed Band II models

because the former contained more low-frequency content

required for predicting longer horizons. This was evident by

the performance of Band IV model, which had the worst

predictive performance for short horizons (<40 min) while

outperforming Band II and III models for prediction horizons

of >45 min. We also confirmed that Band I signals were not

informative, as simulations showed that models based solely

on this frequency band consistently underperformed the other

models (results not shown).

Figure 4b compares the performance of the pairwise-band

models. In stark contrast with the single-band models, the

results indicate that when we considered frequencies from

Band II, with either Band III or Band IV, the resulting

pairwise models were as predictive over the 0-50-min

prediction horizon as those obtained with the reference model.

These results suggest that although a frequency band may not

have sufficient predictive power by itself, in combination with

other bands, the resulting models could be very accurate.

Importantly, they also indicate that the energy content of a

signal alone was not necessarily an appropriate metric for

indicating its predictive power (as discussed above, Band II

accounts for only ~1.5% of the total energy of the glucose

signal). Conversely, models constructed using frequencies

from Bands III+IV, which account for ~93.0% of the signal’s

total energy, showed inferior performance when compared

with the other pairwise models and some of the single-band

models.

These results suggest that the prediction accuracy of AR

models is highly dependent on which frequencies of the CGM

signal are captured by the AR model coefficients bi and weakly

dependent on the resolution of the captured frequencies. For

example, as discussed above, the AR models for Band III only

captured two (Fig. 3b) of the five salient periods in this

frequency band (Fig. 2). Nevertheless, for a 25-min prediction

horizon, the Band III AR models yielded an average RMSE of

<2 mg/dL (Fig. 4a).

To provide further insights into the role of the different

frequency bands in AR-model predictions, we analyzed 35-

min-ahead predictions for a typical patient (subject #6) for

models based on the six possible combinations of single and

pairwise frequency bands, as illustrated in Fig. 5.

Figure 5a shows that while the Band II model was capable

of yielding smooth predictions for the high-frequency

oscillations, such as those around 3,000 min, it either

underpredicted or overpredicted the low-frequency trends,

systematically generating a prediction bias. Conversely, Fig.

012345

0.5

1

1.5

2

x 10

(a)

14

PSD (magnitude 2/ h)

Band II

Period (h)

02468 10

2

4

6

x 10

(b)

18

Band III

Period (h)

10 15202530

1

2

3

x 10

(c)

8

Period (h)

Band IV

0 10

Period (h)

20 30

2

4

6

8

10

12x 10

15

Band III+IV

0 10

Period (h)

2030

0.5

1

1.5

2

2.5

x 10

(e)

15

Band II+IV

0246810

2

4

6

8

10

12x 10

14

Period (h)

PSD (magnitude 2/ h)

Band II+III

(f)(d)

Fig. 3. Mean power spectral densities (PSD) of single-frequency bands (a-c)

and pairwise-frequency bands (d-f) estimated through the AR-model

coefficients.

010 20 304050

0

10

20

30

40

50

RMSE (mg/dL)

0 10203040 50

0

5

10

15

20

RMSE (mg/dL)

Prediction Horizon (min)

Reference Model

Band IV

Band III

Band II

300 min training data

Reference Model

Band II+IV

Band II+III

Band III+IV

(b)

(a)

Fig. 4. Average root mean squared error (RMSE) of the autoregressive

model predictions as a function of prediction horizon for the reference model

based on all frequency bands, and for models based on: (a) single-frequency

bands and (b) pairwise-frequency bands.

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