Environmental Data Recovery using Polynomial Regression

for Large-scale Wireless Sensor Networks

Kohei Ohba

, Yoshihiro Yoneda

, Koji Kurihara

, Takashi Suganuma

, Hiroyuki Ito

Noboru Ishihara

, Kunihiko Gotoh

, Koichiro Yamashita

and Kazuya Masu

ICE Cube Center, Tokyo Institute of Technology, Nagatsutacho 4259, Midori-ku, Kanagawa, 226–8503, Japan

Network Systems Laboratory, Fujitsu Laboratories Ltd., Kamikodanaka 4–1–1,

Kawasaki Nakahara-ku, Kanagawa, 211–8588, Japan

Keywords:

Wireless Sensor Networks, Polynomial Regression, Data Recovery, Environment Monitoring.

Abstract:

In the near feature, large-scale wireless sensor networks will play an important role in our lives by monitoring

our environment with large numbers of sensors. However, data loss owing to data collision between the sensor

nodes and electromagnetic noise need to be addressed. As the interval of aggregate data is not ﬁxed, digital

signal processing is not possible and noise degrades the data accuracy. To overcome these problems, we have

researched an environmental data recovery technique using polynomial regression based on the correlations

among environmental data. The reliability of the recovered data is discussed in the time, space and frequency

domains. The relation between the accuracy of the recovered characteristics and the polynomial regression

order is clariﬁed. The effects of noise, data loss and number of sensor nodes are quantiﬁed. Clearly, polynomial

regression offers the advantage of low-pass ﬁltering and enhances the signal-to-noise ratio of the environmental

data. Furthermore, the polynomial regression can recover arbitrary environmental characteristics.

1 INTRODUCTION

Large-scale wireless sensor networks (WSNs) use

wireless sensor nodes to monitor environmental pa-

rameters such as temperature, humidity, pH, light

and air pressure. WSNs have many possible appli-

cations, ranging from structual health monitoring to

ﬁeld monitoring. Thanks to the progress in micro-

electronics based on the integrated circuit technology,

small wireless sensor nodeswith low power consump-

tion have been achieved. However, problems exist

with data loss owing to data collision between the

sensor nodes and electromagnetic noise. As the in-

terval of aggregate data is not ﬁxed in the time and

space domains, digital signal processing using Fourier

or wavelet transforms cannot be applied directly to

the aggregated data. Moreover, noise degrades the

data accuracy. Because the environmental character-

istics have various waveforms, data reliability cannot

evaluate by signal analysis. To overcome these prob-

lems, various techniques, such as data collection tim-

ing (Sivrikaya et al., 2004), redundant system (Ya-

mashita et al., 2014) and data recovery (Doherty et

al., 2000), have been used to increase data reliability.

We apply polynomial regression to environmen-

tal data recovery based on the correlations among

the environmental data. Environmental characteris-

tics are recovered as aggregated data from the sen-

sor nodes using polynomial regression. Thus, data

loss is minimized, and the data can be analysed eas-

ily. Basic sinusoidal environmental variations are as-

sumed to evaluate the data recovery with polynomial

regression. If the sinusoidal characteristics can be

modeled appropriately, arbitrary waveform character-

istics, such as single-shot, periodic and non-periodic

waveforms, can also be modelled. The recovered data

accuracy is evaluated by comparing the recovered and

source characteristics.

We have also proposed a data reliability evalua-

tion ﬂowchart that does not rely on signal analysis

(Yoneda et al., 2014). We also clarify the relation

between the accuracy of the recovered characteristics

and the polynomial regression order, and the effects

of data loss and number of sensor nodes is analysed.

Furthermore, we show that the use of polynomial re-

gression has the advantage of low-pass ﬁltering that

enhances the signal-to-noise ratio (SNR) of the envi-

ronmental characteristics. In addition, we show that

polynomial regression can recover arbitrary environ-

mental characteristics.

In section 2, we introduce the environmental data

recovery technique based on polynomial regression.

Ohba, K., Yoneda, Y., Kurihara, K., Suganuma, T., Ito, H., Ishihara, N., Gotoh, K., Yamashita, K. and Masu, K.

Environmental Data Recovery using Polynomial Regression for Large-scale Wireless Sensor Networks.

DOI: 10.5220/0005636901610168

In Proceedings of the 5th International Confererence on Sensor Networks (SENSORNETS 2016), pages 161-168

ISBN: 978-989-758-169-4

161

Environment characteristic

(Continuously)

Data aggregation

<WSN>

Recover environment characteristic

by polynomial regression

Recovered function

(Continuously)

Aggregate data

(Discretely)

Conventional approach

Proposed approach

Re-sampling

Recovered data

(Discretely)

-Signal analysis-

Frequency analysis

Fourier transform

Wavelet transform

Time series analysis

etc...

Data utilisation

Environment

Figure 1: WSN system using polynomial regression.

In section 3, the reliability of the recovered data is

discussed. The frequency domain characteristics are

evaluated in section 4. In section 5, we conﬁrm the

ability of polynomial regression to recover arbitrary

environmental characteristics and present the conclu-

sions in section 6.

2 ENVIRONMENTAL DATA

RECOVERY USING

POLYNOMIAL REGRESSION

The aggregated data analysis is shown in Fig. 1. If

the interval of the aggregated sampled data is ﬁxed,

the environmentaldata characteristics can be analysed

directly using Fourier or wavelet transforms. How-

ever, when the interval of the data is not ﬁxed, the

data cannot be directly analysed. Therefore, continu-

ous environmental characteristics are recovered from

the aggregated data, and then, the ﬁxed interval data

are resampled from the recovered characteristics.

2.1 Polynomial Regression

There are several ways of expressing the recovered

characteristics, e.g., Fourier series expansion, polyno-

mial regression, interpolation and so on. Polynomial

regression is simple and suitable for expressing con-

tinuous characteristics as it tolerates data loss. How-

ever, polynomial regression is not good at expressing

characteristics with many inﬂection points. If the fre-

quency band is limited, the environmental character-

istics at the limited bandwidth can be expressed us-

ing polynomial expressions. Therefore, polynomial

regression is used in environmental data recovery.

When one-dimensional data are t = [

· ·· t

]

and the environmental data are d = [

,· · · ,d

]

, the

environmental source characteristics function f

(t)

can be recovered and recovered function f

(t) is

(t) =

∑

i=0

(1)

Where a = [

· · · a

]

is the coefﬁcient vector.

The value of a is obtained using at least two multipli-

cation methods. m is the order of polynomial equa-

tion. For two-dimensional data obtained by sensor

nodes arranged in coordinates (x

),· · · , (x

)

and coordinates x = [

· · · x

]

and y =

[

· · · y

]

, the recovered function f

(x,y) is

(x,y) =

∑

j, k≥0

j+k≤m

(2)

Where the coefﬁcient vector a is the column vector of

size

(m+1)(m+2)

× 1.

2.2 Data Reliability Evaluation Flow

2.2.1 Evaluation ﬂow

The reliability evaluation ﬂowchart is shown in Fig.

2. Two-dimensional data are assumed in the evalu-

ation. The environmental source characteristic func-

tion is f

). To consider the effect of noise and

data loss, we deﬁne the sensor node model. When the

noise is expressed as f

), the sampled data with

noise f

) can be expressed as

) = f

) + f

) (3)

To evaluate the effect of data loss, the following func-

tion is added

) =



) (Without data loss)

0 (With data loss)

(4)

SENSORNETS 2016 - 5th International Conference on Sensor Networks

162

(x,y)

)

(x,y)

)

Data loss

0 0.5 1

0.5

Generate

environment

characteristics

Sensor node model

Noise

Sensor nodes

Data

aggregation

Recover

environment

characteristics

by polynomial

regression

Extract error

Analyse

data reliability

evaluation

Environment characteristics

Recovered function

Aggregate data

Error data

Figure 2: Data reliability evaluation ﬂow (based on Yoneda

et al., 2014).

Where f

) represents the sampled data consid-

ering the effect of noise and data loss. The amount of

data in f

) decreases compared with the number

of f

). The error of the recovered data is deﬁned

(x,y) = f

(x,y) − f

(x,y) (5)

The data accuracy that are sampled at ﬁxed intervals

using the above continuous functions are compared

with the accuracy of the evaluated data. The root-

mean-square error (RMSE) at each comparison point

is deﬁned as data reliability. RMSE is given

RMSE =

mean( f

(x,y)

) × 100(%) (6)

In the evaluation, we carry out 1000 iterations to min-

imize the effect of noise variation and data loss.

2.2.2 Parameter Setting

To evaluate the data recovery reliability, the following

conditions are considered.

0 1 2 3 4 5

−1.5

−1

−0.5

0.5

1.5

t : Partition region size

Data

(a) one-dimensional case

Partition region size

(b) two-dimensional case

Figure 3: Examples of the recovery function.

Sinusoidal Environmental Characteristics. The

correlations among the actual environmental data are

complex. However, to determine a generalized index,

it is preferable to use simple data characteristics. In

this study, a sinusoidal wave is assumed as the ba-

sic environmental characteristic because any arbitrary

characteristic can be expressed as a linear combina-

tion of a sinusoidal wave. The following equations

are the sinusoidal functions used for one- and two-

dimensional data.

(t) =

sin



2π

+ θ



(7)

(x,y) =

sin

2π

(x−x

)

+(y−y

)

+θ

(8)

Where (x

) is showing the position of the wave

generation source, A

is the peak-to-peak amplitude,

L is the wavelength and θ is the phase.

Observation Region. The observation region is the

region where the polynomial regression is applied.

The observation region is partitioned and then

polynomial regression is applied to each partition.

Each data recovery function is joined to express

the characteristics of the observation region. The

partitions of the region are determined by the cycle

Environmental Data Recovery using Polynomial Regression for Large-scale Wireless Sensor Networks

163

0.5

0.25

1.5

2.5

0.5 1 1.5 2

RMSE (%)

Partition region Size

5th order

6th order

7th order

8th order

9th order

(a) one-dimensional case

0.5

0.25

1.5

2.5

0.5

1.5

RMSE (%)

Partition region Size

5th order

6th order

7th order

8th order

9th order

(b) two-dimensional case

Figure 4: Precision of polynomial regression.

(wavelength) of the highest frequency of the environ-

mental characteristics.

Number of Sensor Nodes. The sensor nodes are

arranged at equal intervals in the observation region,

including the upper boundary. In the case of two-

dimensional structures, the sensor nodes are set on a

grid. The density of the sensor nodes is represented

by the number of sensor nodes N in the observation

region. When the analysis is in the time domain, the

one-dimensional coordinate axis is evaluated with

respect to the time axis. In this case, the number of

sensor nodes in the observation region represents the

number of sampled data.

Noise. Electromagnetic noise generated at the sensor

interface of an ampliﬁer and an analogue-to-digital

converter and electromagnetic noise in the environ-

ment mainly contribute to data noise. The SNR is

deﬁned by following equation.

SNR = 10log

var( f

)

var( f

)

(9)

White noise (Gaussian noise) is added in the reliabil-

ity evaluation.

0.001

0.01

0.1

1 10 100 1000

RMSE (%)

Number of Sensor Nodes N

order = 9

30 dB

40 dB

50 dB

noise-free

×4

×1/2

(a) 1-dimensional case

0.001

0.01

0.1

10 100 1000 10000

RMSE(%)

Number of Sensor Nodes N

order = 9

30 dB

40 dB

50 dB

noise-free

×4

×1/2

(b) 2-dimensional case

Figure 5: Required number of sensor nodes.

Data Loss. Data loss occurs because of data colli-

sions or intermittent failures in the wireless commu-

nication. To simulate the effect of data loss, we use a

pseudorandom data generation technique.

3 DATA RELIABILITY OF

PERIODIC CHARACTERISTICS

The reliability of recovered data that are re-sampled

from the recovered function is evaluated by com-

paring with the environmental source characteristics

function. The data reliability is evaluated using the

conditions described in the previous section. Fig-

ure 3 shows the sinusoidal signal that is assumed as

the environmental characteristics of one- and two-

dimensional conditions.

The SNR at each sensor node is set at 40 dB.

Therefore, the reference position of the RMSE is de-

termined at 1.0%. When the number of sensor nodes

is increased, the reference position of the RMSE is

0.5%. Without sensor node noise, the reference posi-

tion of the RMSE is 0.25%.

SENSORNETS 2016 - 5th International Conference on Sensor Networks

164

1 10 100 1000

Precision of Each

Sensor Node (SNR) [dB]

Number of Sensor Nodes N

order = 9

1.0%

0.50%

0.25%

×4

6dB

(a) 1-dimensional case

10 100 1000 10000

Precision of Each

Sensor Node (SNR) [dB]

Number of Sensor Nodes N

order = 9

1.0%

0.50%

0.25%

×4

6dB

(b) 2-dimensional case

Figure 6: Sensor accuracy (based on Yoneda et al., 2014).

3.1 Application Range of the

Polynomial Regression

Firstly, the relation between the partition region cycle

and RMSE was analysed by changing the order

of the polynomial without the sensor node noise.

The number of sensor nodes is ten for the one-cycle

partition region in the one-dimensional case and 10

× 10 for the one-cycle partition region in the two-

dimensional case. We also examined the ﬁve-cycle

partition region and monitored the maximum error.

Results for the one- and two-dimensional sinusoidal

signals (Fig. 3) are shown in Fig. 4. For 0.25% error

and one-cycle partition region, the order of the poly-

nomial should be higher than seven for one- and two-

dimensional signals.

3.2 Effect of Sensor Node Number

The number of sensor nodes is thought to strongly af-

fect the data evaluation reliability. The relation be-

tween the number of sensor nodes and RMSE was

analysed for SNR of 50, 40 and 30 dB at each sen-

sor node. And ninth-order polynomial was used

in the analysis. The results for the one- and two-

dimensional cases are shown in Fig. 5. Obviously,

the errors reduced with the number of sensor nodes.

The increasing number of sensor nodes reduced the

RMSE owing to noise. Figure 6 shows the results

for the required SNR at each sensor node when the

RMSE is 0.25%, 0.5% and 1%. The precision of each

sensor node is improved by increasing the number of

sensor nodes.

There are two ways to improve the data evalua-

tion reliability. The ﬁrst is to increase the number of

sensor nodes and the second is to use high SNR. If

the number of sensor nodes is increased four times,

the RMSE decreases 50%. If the SNR of each sensor

node is improved by 6 dB, the RMSE decreases by

50%.

3.3 Effect of Data Loss

The relation between data loss rate and the RMSE

was analysed. A ninth-order polynomial and 40-dB

SNR at each sensor node was assumed. The number

of sensor nodes was selected to satisfy the RMSE of

0.5% and 0.25%. In the one-dimensional case, 36 and

149 nodes were selected for the analysis. In the two-

dimensional cases, 225 and 841 were selected.

The results for the one- and two-dimensional

cases are shown in Fig.7. The RMSE increases with

data loss rate, of course. However, by increasing

the number of sensor nodes, the RMSE decreases.

The number of sensor nodes satisﬁes the RMSE of

0.25% adequately, whereas the data loss rate is 60%

for RMSE of 0.5% in the one-dimensional case and

65% in the two-dimensional case. These results sug-

gest that a redundant system can enhance the data re-

liability.

4 RELIABILITY IN THE

FREQUENCY DOMAIN

The reliability of the recovered data using polynomial

regression was also evaluated in the frequency do-

main (Ohba et al., 2015). The fast Fourier transform

(FFT) was applied to the recovered data.

4.1 FFT Analysis

The one-dimensional sinusoidal environmental char-

acteristics are assumed to be the same as in the pre-

vious sections. Recovered data at ﬁxed intervals are

obtained by sampling the data recovered by polyno-

mial regression. FFT is applied to the recovered data.

The signal-to-noise and distortion ratio (SNDR) and

spurious-free dynamic range (SFDR) were evaluated.

Environmental Data Recovery using Polynomial Regression for Large-scale Wireless Sensor Networks

165

0.1

0 20 40 60 80 100

RMSE (%)

Data Loss Rate (%)

N = 36

N = 149

0.5

60%

0.25

(a) 1-dimensional case

0.1

0.5

0 20 40 60 80 100

RMSE (%)

Data Loss Rate (%)

N = 15x15

N = 29x29

65%

0.25

(b) 2-dimensional case

Figure 7: Data loss robustness (based on Yoneda et al.,

2014).

The SFDR is used to discuss the effect of harmonic

distortion.

When the fundamental frequency is f

, the num-

ber of FFT points is FFT

POINT

and the sampling fre-

quency is F

, f

FFT

POINT

(10)

Therefore, the input signal frequency f

and wave-

length of the input signal per division λ

= m f

(11)

(12)

Where D

is the number of divisions and m is an inte-

ger number.

4.2 Data Reliability in the FFT Analysis

4.2.1 Effect of Input Signal Cycle (Wavelength)

The relation between signal cycle (wavelength) in the

polynomial regression and the evaluation indices of

gain, SNDR and SFDR was analysed using FFT. In

the analysis, a ninth-order polynomial, 40-dB SNR at

each sensor node, 32 divisions dividing FFT points

into partition region and 1024 of FFT points are as-

sumed. The results are shown in Fig. 8. This is

showing the relation between input frequency cycle

(wavelength) for polynomial regression and the eval-

uation indexes. Decreases of 1 dB are tolerated by the

SNDR and SFDR and for wavelength with the max-

imum partition of 1.6 cycles. Above 1.6 cycles, the

partition region signals are ﬁltered out. The gain is

ﬂat up to the three-cycle partition region. Thus, the

polynomial regression acts as a low-pass ﬁlter. This

means that the SNDR and SFDR improvebecause the

polynomial regression limits the bandwidth of the en-

vironmental signals.

4.2.2 Effect of the Number of Sensor Node

The number of sensor nodes per partition region is

evaluated. The results are shown in Fig. 9. The FFT

results for the source environmental signals were 40-

dB SNDR and 59-dB SFDR. For 13 sensor nodes, the

SNDR is the same as the result of the source envi-

ronmental signals. For 25 sensor nodes, the SNDR is

the same as the result of source environmentalsignals.

Higher SNDR and SFDR are possible by increasing

the number of sensor nodes. By increasing the num-

ber of sensor nodes four times, both SNDR and SFDR

improved by 6 dB.

4.2.3 Frequency Spectrum

The frequency spectrum is evaluated by FFT. The re-

sults are shown in Fig. 10. Based on the results of

Figs. 8 and 9, the 1.6-cycle (wavelength) input sig-

nal region and 25 sensor nodes per partition region

were assumed. Compared with the spectrum of the

source environmental signal, the noise level of the

high-frequency region is ﬁltered out. Table 1 sum-

marizes the FFT results and Table 2 summarizes the

conditions of the FFT analysis. By limiting the ob-

servation region in the polynomial regression, both

SNDR and SFDR are improved.

Table 1: FFT analysis result.

SNDR[dB] SFDR[dBc]

(A) FFT 40.0 59.0

(B) FFT

using recovered data

45.2 59.9

(=B-A) Difference +5.2 +0.9

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166

-12

-9

-6

-3

100

0 1 2 3 4

SNDR [dB] SFDR [dBc]

Gain [dB]

1.6

SNDR [dB]

SFDR [dBc]

λin

Figure 8: SNDR, SFDR and gain vs. input wavelength

(based on Ohba et al., 2015).

10 100 1000

SNDR [dB]

SFDR [dBc]

6dB

×4

SNDR = 40.0 [dB]

SFDR = 59.0 [dBc]

×4

SNDR [dB] SFDR [dBc]

Number of Sensor Nodes

6dB

Figure 9: SNDR and SFDR vs. number of sensor

nodes(based on K. Ohba, 2015).

-100

-80

-60

-40

-20

1 10

100

1000

λin = 31/32 (A)

in = 31/32 (B)

using recovered data

m (=fin/f0)

Power spectrum [dB]

in = 3λ

in = 1

Figure 10: Frequency spectrum with and without recovered

data.

Table 2: FFT analysis conditions.

Parameters Conditions

Noise 40[dB]

Polynomial regression order 9

FFT points 1024

Division 32

5 ARBITRARY

CHARACTERISTICS

RECOVERY BY POLYNOMIAL

REGRESSION

In sections 3 and 4, it was clariﬁed that polynomial

regression can recover the sinusoidal environmental

characteristics. Polynomial regression for arbitrary

characteristics is also validated by selecting the or-

der of the polynomial equation for each partition re-

gion. Scale-space ﬁltering (SSF) (Witkin, 1984) and

Akaikefs information criterion (AIC) (Akaike, 1974)

were used to select the partition region and the order

of the polynomial regression based on aggregate data.

The SSF detects extreme values by the convolution

of the Gaussian function. The partition region is ob-

tained as the region between the extreme points. The

AIC is a statistical measure that estimate the quality

of the environmental source characteristics from ag-

gregate data, including noise and data loss. The order

of the polynomial regression for the partition region is

obtained by the SSF. By detecting the extreme values

by SSF and estimating the quality of source charac-

teristics using polynomial regression between the ap-

propriately selected extreme points by AIC, arbitrary

characteristics can be recovered (Haze, et al., 2012).

Figure 11 shows extreme values of arbitrary envi-

ronmental characteristics with 40-dB SNR detected

by SSF. The observation region is divided into par-

tition regions using the extreme values and the order

of polynomial regression is thus optimized. The re-

gions divided by the criterion of extreme values are

the partition regions, and the order of the polynomial

regression is set at each partition region. Figure 12

shows the recovered data from arbitrary characteris-

tics with 40-dB SNR using the SSF, AIC and poly-

nomial regression. The RMSE is under 0.1%; thus,

the polynomial regression can obviously recover the

arbitrary environmental characteristics.

0 0.5 1 1.5 2 2.5 3 3.5 4

−0.5

0.5

1.5

2.5

Data

0.5

Criterion of Extreme value

Environment characteristics: f

Extreme value

Figure 11: Extreme values of arbitrary characteristics.

Environmental Data Recovery using Polynomial Regression for Large-scale Wireless Sensor Networks

167

0 0.5 1 1.5 2 2.5 3 3.5 4

−0.5

0.5

1.5

2.5

Data

RMSE = 0.0029538 (0.41747%)

−0.03

−0.02

−0.01

0.01

0.02

0.03

Error

Environment characteristics: f

Recovered function: f

Error data: f

Figure 12: Data recovery function based on arbitrary char-

acteristic.

6 CONCLUSIONS

We used polynomial regression for environmental

data recovery from large-scale WSNs.

(1) A data reliability evaluation procedure for WSN

was proposed.

(2) The recovered data reliability depends on the or-

der of the polynomial regression, the number of sen-

sor nodes and the effect of noise and data loss were

quantiﬁed.

(3) From FFT analysis, it is seen that polynomial re-

gression act as a low-pass ﬁlter. Data recovery using

polynomial regression enhances the SNDR or SFDR

in the WSNs system.

(4) Polynomial regression can recover arbitrary envi-

ronmental characteristics and can be used with SSF

and AIC.

In conclusion, environmental data recovery tech-

nique using polynomial regression can be applied to

large-scale wireless sensor networks.

ACKNOWLEDGEMENTS

Part of this work was a joint research project of Tokyo

Institute of Technology and Fujitsu Laboratories Lim-

ited. We would like to express our thanks to all who

supported this project.

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