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On the Performance of Piecewise Linear

Approximation Techniques in WSNs

Samia Al Fallah

National School of Applied

Sciences, Tangier, Morocco

samia.alfallah@gmail.com

Mounir Arioua

National School of Applied

Sciences, Tetuan, Morocco

m.arioua@ieee.org

Ahmed El Oualkadi

National School of Applied

Sciences, Tangier, Morocco

eloualkadi@gmail.com

Jihane El Asri

National School of Applied

Sciences, Tetuan, Morocco

jihaneelaasri@gmail.com

Abstract—Energy consumption is the major constraint in

the design and the deployment of Wireless Sensor Networks

(WSNs). Since the transmission of data induces high energy

costs in WSN device, many research efforts focus on reducing

the transmission of the raw data by using lossy compression

methods in order to improve energy efﬁciency with an acceptable

data reconstruction tolerance. Thus, an intricate trade-off exists

between energy saving using sampling compression, and the

distortion of reconstructed data samples. In this paper, we

present a survey on Piecewise Linear Approximation methods.

A comparative analysis aims to evaluate the performance of the

selected techniques in terms of energy consumption, compression

ratio and distortion.

Index Terms : WSNs, Lossy compression, PLA, Energy efﬁ-

ciency, Distortion.

I. INTRODUCTION

Wireless Sensor Networks (WSNs) have recently received

a lot of attention due to a wide range of applications such as

animal monitoring, agriculture transforming, health care, IoT,

indoor surveillance and smart buildings [1]. Typical WSN is

expected to consist of a large number of sensors deployed in

regions of interest in order to observe speciﬁc phenomena or

track objects. Energy consumption is one of the most important

factors in WSNs due to the resource constrained transmission

devices [2]. Hence, various approaches have been proposed

to prolong sensor nodes lifetime such as data compression,

due to the fact that the major part of energy is consumed in

data transmission [3] [4] [5] [6]. Some compression algorithms

are designed to support exact reconstruction of the original

data after de-compression (lossless compression) [4]. In other

cases, the reconstructed data is only an approximation of the

original information (lossy compression) [5]. The use of lossy

algorithms may lead to loss of information (distortion), but

generally ensures some additional gains in terms of compres-

sion ratio and most importantly in terms of energy saving. In

this paper, we focus on the performance offered by Piecewise

Linear Approximation (PLA) techniques in terms of energy

saving, compression ratio and reliability of reconstructed data.

With lossy algorithms, the original data is compressed by

eliminating some of the original information in it so that,

at the receiver side, the decompressor can reconstruct the

original signal up to certain accuracy [5]. Depending on the

application, small inaccuracy in the reconstructed data can be

acceptable.

The rest of this paper is organized as follows: Section II

reviews some of research efforts on the performance of lossy

compression methods. PLA techniques are introduced in sec-

tion III, especially LTC, PLAMLiS and Enhanced PLAMLiS.

Section IV provides an overview of some compression metrics

followed by a comparative study of selected techniques in

section V. Finally, the conclusion is given in section VI.

II. RELATED WORKS

Several works have been carried out for what regards

lossy compression schemes [5] [6] [8] [10] [11]. On the one

hand, some of these approaches are based on transforming

the input signal into coefﬁcients in order to facilitate signal

representation [7]. As an example, FFT [8], DCT [9] and

Wavelet transform [10] represent time series into the frequency

domains, but differ in how transformation coefﬁcients are

picked. Speciﬁcally, as mentioned in [11], transformation

methods achieve a good performance in terms of compression

ratio, but unfortunately, incur high energy expenditure due to

their computational cost. On the other hand, adaptive modeling

techniques aim to represent the input signal through linear [12]

[13], polynomial [14] or autoregressive methods [15]. Hence,

the input time series is collected according to N samples for

each time window transmission. Then the selected compres-

sion method is applied obtaining a set of model parameters

that will be transmitted in the place of the original data. In

the case of linear techniques, PLA represent a time series of

environmental measures with a sequence of line segments up

to a desired approximated accuracy [12] [13]. In the case

of polynomial approaches, the input signal is approximated

through polynomial coefﬁcients, instead of transmitting the

original data samples [14]. In the case of autoregressive

methods, a model of basic coefﬁcients is built using the history

of data samples exploiting the correlation of the signal. As

mentioned in [11], PLA approaches ensure a better energy

cost with a very low computational complexity, contrary to

Polynomial Regression (PR) which induces a high complexity

cost but performs well in terms of accuracy while increasing

the polynomial order. In addition, increasing the length of the

correlation signal increases the length of the autoregressive

model that may lead to high energy consumption [5].

978-1-5386-4609-0/18/$31.00 c

2018 IEEE

In this paper, we perform a comparative study on lossy com-

pression methods, especially PLA approaches. The main goal

is to solve the trade-off existing between energy consumption

for compression, and reliability of reconstructed data at the

receiver side.

III. PLA COMPRESSION METHODS

PLA is a family of linear approximation techniques based

on representing data samples with a sequence of line segments

that preserves original samples within a desired approximation

tolerance. In fact, the objective of PLA methods is to approx-

imate the time series with a sequence of lines (only two end

points for each line) in order to reduce the energy consumption

on data transmission. Since a line segment can be determined

by only two end points, PLA leads to efﬁcient representation of

time series in terms of transmission requirements and memory

[12] [13].

012345

n

0

1

2

3

4

5

x(n)

Fig. 1: Approximation of a time series x(n) by a segment.

At the receiver side, nobservations are approximated

through the vertical projection of the actual samples over the

corresponding line segment (Figure 1). The approximated sig-

nal in what follows is referred to as ˆx(n). The error introduced

is the distance from the actual samples to the segment along

the vertical projection, i.e.|x(n)−ˆx(n)|. Following this simple

idea, several methods have been proposed in the literature.

Lightweight Temporal Compression (LTC) is a lightweight

technique to compress environmental measurements [15]. It

is a simple method to represent a time series by a number

of line segments. Algorithm 1 shows the pseudo code of this

technique.

For both a given time series x(n)and error tolerance ε, the

algorithm ﬁxes the ﬁrst measurement x(1) at the beginning

of a line. The second measurement x(2) is transformed into a

vertical segment whose extremities are x(2)+εand x(2)−ε.

The sensor stores a Highline connecting x(1) and x(2)+ε,

and a Lowline connecting x(1) and x(2)−εas shown in Figure

2(a). With the third measurement x(3), the node tightens these

bounds to ensure that can represent the third measurement

within ε(Figure 2(b)). The process is repeated until a sample

x(s)cannot be accurately represented by any line segments

within the bounds (Figure 2(c)). Once this occurs, the node

transmits a packet containing the ﬁrst endpoint and computes

the midpoint of the upper and lower bounds. Then, the

algorithm starts over looking for a new line segment.

Thereby, LTC algorithm encodes the time series

incrementally, which makes the number of operations

Algorithm 1 LTC Algorithm

Inputs x, ε// Time series, Error tolerance

for i=1 to length(x) do

j=i+1

Highline=Line-function[x(i),x(j)+ε]

Lowline=Line-function[x(i),x(j)-ε]

while j < length(x) do

if Highline below x(j+1)- εor Lowline above x(j+1)+ε

then

Save x(j)

i=j Break

else

if Highline above x(j+1)+εthen

Highline=Line-function[x(i),x(j+1)+ε]

end if

if Lowline below x(j+1)- εthen

Lowline=Line-function[x(i),x(j+1)-ε]

end if

end if

j=j+1

end while

end for

012345

n

0

1

2

3

4

x(n)

(a)

012345

n

0

1

2

3

4

x(n)

(b)

012345

n

0

1

2

3

4

x(n)

(c)

Fig. 2: Steps of the Lightweight Temporal Compression Tech-

nique

(complexity) independent of the correlation of the original

signal [5]. In addition, LTC may be less efﬁcient in terms of

compression ratio when the data values change signiﬁcantly

over time [11].

Another signiﬁcant PLA algorithm is Piecewise Linear Ap-

proximation with Minimum number of Line Segment (PLAM-

LiS). This approach represents the time series through a

sequence of line segments [12]. In fact, the goal of this

algorithm (Algorithm 2) is, for both a given time series and

error tolerance ε, to ﬁnd a minimum number of segments to

approximate the time series such that the difference between

any approximation value and its actual value is less than ε.

The endpoints of the line segments must be the points in the

time series.

Algorithm 2 PLAMLiS Algorithm

Inputs x, ε// Time series, Error tolerance

for i=1 to length(x)-1 do

j=i+2

while j < length(x) do

Line=Line-function (x(i),x(j))

for k=i to j do

if Calculate-error(Line, x(k)) < εthen

k=k+1

else

Segment=[x(i),x(j-1)] Break

end if

end for

j=j+1

end while

end for

For each data sample x(i), segments are built associating

x(i)with x(j)(j>i) if the line segment [x(i),x(j)] meets

the error bound ε. Speciﬁcally, the difference between the

approximating value ˆx(k)(i<k<j) and the actual value of

x(k)is computed by Calculate −error function, in order to

verify if the distance |x(n)−ˆx(n)|is not larger than ε. Then

this procedure is iterated for all points of the time series.

After obtaining the set of segments, the algorithm pick the

minimum number of line segments that covers all the points

of the time series [12].

In order to reduce the computational cost of PLAMLiS al-

gorithm, Enhanced PLAMLiS (EPLAMLiS) has been proposed

in the literature [13]. It is based on a recursive segmentation as

shown in the Algorithm 3. The algorithm starts with the ﬁrst

segment [x(1),x(N)]. If this segment approximates all points

within the maximum allowed tolerance ε, the two endpoints

are transmitted and the algorithm ends. Otherwise, the segment

is split in two segments at the point x(i),1<i<N, where

the error is maximum, obtaining two segments [x(1),x(i)]

and [x(i),x(N)]. This procedure is applied for each part of

the line segments until all of the sub line segments meet the

error bound as shown in Figure 3.

Algorithm 3 Enhanced PLAMLiS Algorithm

Inputs x, ε// Time series, Error tolerance

Approximating(x(1), x(N))

Segment=Line-function(x(1),x(N))

for i=2 to N do

if Max-error(Segment, x(i)) > εthen

Approximate(x(1), x(i))

Approximate(x(i), x(N)) Break

end if

end for

012345678

n

0

0.5

1

1.5

2

2.5

3

x(n)

(a)

012345678

n

0

0.5

1

1.5

2

2.5

3

x(n)

(b)

012345678

n

0

0.5

1

1.5

2

2.5

3

x(n)

(c)

012345678

n

0

0.5

1

1.5

2

2.5

3

x(n)

(d)

Fig. 3: Steps of the Enhanced PLAMLiS Compression tech-

nique

EPLAMLiS algorithm aims to be applicable to the sensed

data in sensor networks which have signiﬁcant temporal corre-

lation [11]. Owing to the correlation, the values in data series

are quite similar therefore approximating them by the line

segments will lead to beneﬁts in terms of compression ratio,

and the number of line segments obtained is likely to be small.

IV. COMPRESSION METRICS

Before getting into the comparative study of PLA

techniques, we introduce in the following compression

metrics that assess the overall performance of selected

methods.

Compression ratio is one of the major evaluation parameter

in data compression [17]. It characterizes the compression

effect of the technique, and it is deﬁned as a ratio between

the volume of the compressed and the raw data.

CR =Volume of compressed data

Volume of raw data (1)

In WSN, compression ratio is also considered as one of the

major evaluation parameter. Since it indicates the reduction of

communication energy costs, several researches are focused

on selecting well-performed compression algorithms in view

of lowering energy consumptions in data communication [5]

[6].

Another important metric adopted in apprising compression

techniques is energy consumption for compression. It can be

deﬁned as the energy needed to accomplish the compression

task. For each type of compression algorithms, we calculate

the number of operations accounting additions, subtractions,

multiplications, divisions and comparisons. Depending on the

type of the micro-controller used in the study, we map the

corresponding number of clock cycles and subsequently we

calculate the energy consumed for processing of each algo-

rithm.

Total energy consumption is the sum of the energies for

compression and transmission. For assessing the performance

of compression algorithms in WSN, a proper criterion is

needed, which focuses on the energy efﬁciency of each al-

gorithm. Energy Saving Beneﬁt (ESB) exposes the energy

saving introduced by compression algorithms [17]. The ESB

expression is formulated as:

η=Euncomp−Ecomp

Euncomp

∗100 (2)

Where Euncomp is the total energy cost without compres-

sion, and Ecomp is the total energy cost with compression.

The energy consumption without compression is expressed

as follows:

Euncomp =Ptran ∗L∗Ttran (3)

Where Ptran is the transmit power, Lis the volume of raw

data and Ttran is the time overhead on transmitting one byte.

However, the energy consumption with compression, it can

be formulated as:

Ecomp =PMCU ∗L∗TMCU +Ptran ∗L∗Ttran ∗CR (4)

Where PMCU is the computation power of the compression

algorithm and TMCU is time overhead on compressing one

byte. By using equation (3) and (4), the relation of ηbecomes:

η=1−CR −PMCU∗TMCU

Ptrans∗Ttr ans

∗100 (5)

Thus, the evaluation criterion includes almost all the main

metrics to evaluate compression, and provides important infor-

mation on whether data compression can bring energy saving

or not.

Otherwise, in the case of lossy compression, and at the

receiver side, the reconstructed data is only an approximation

of the original information. Hence, this loss of information

can be measured by a distortion parameter deﬁned as follows:

D=1

N

n

i=1

|ˆx(i)−x(i)|(6)

Where x(i)is an element of a given time series and ˆx(i)

represents its compression version.

The prescribed signal representation accuracy depends nec-

essarily on WSN applications. Hence, selected data compres-

sion methods exploit signal correlation in order to minimize

energy expenditure and ensure a high reliability of recon-

structed data [11].

V. T ESTS AND RESULTS

This section provides a comparison of the performance

of each type of compression algorithms described in the

previous section. We have selected in this study TI MSP430

micro-controller, using 16 bits ﬂoating point package for the

calculations [18]. The TI MSP430 is powered by a current of

I= 330 μA, a voltage of V= 2.2 V and a clock rate of C=1

MHz. Hence, the energy consumed per a clock cycle is given

by:

E0=V∗C∗I=0.726μJ

Table I exposes the CPU cycles needed for each type of

calculation. Hence, energy for compression is computed by

recording the number of clock cycles needed for each type of

operations while executing a data compression algorithm.

TABLE I: CPU cycles needed for processing

Operation Clock cycle

Addition X+Y 184

Subtraction X-Y 177

Multiplication X*Y 395

Division X /Y 405

Comparison X<=>Y 37

For this analysis, we have selected the TI CC2420 RF

transceiver [19] which follows IEEE 802.15.4 standard [20].

Energy cost for the transmission of one bit can be deﬁned as

follows:

ET=U∗I

D=0.23μJ

Where I=17.4 mA is the current consumption for transmis-

sion at a voltage of U= 3.3 V for an effective data rate of

D=250 kbps.

In this study, we have considered a time series of N=24

temperature samples collected during one day (one sample per

hour) in Tetuan city as shown in Figure 4.

20

22

24

26

28

30

Temperature °C

2PM

8AM

2AM8PM

3PM

Fig. 4: Collected temperature samples of Tetuan City

The analysis aims to evaluate the performance of lossy

compression methods presented in section II, in terms of

compression effectiveness and energy saving. For each type

of compression, we have changed the compression ratio by

tuning the error tolerance.

A. Compression Ratio vs Energy for compression

The performance in terms of energy for compression as a

function of the compression ratio is studied.

0 50 100 150 200 250

Energy for compression (μJ)

0.2

0.4

0.6

0.8

1

Compression Ratio

EPLAMLiS

PLAMLiS

LTC

Fig. 5: Energy for compression vs Compression Ratio.

Figure 5 shows the energy for compression as a function

of the compression ratio for each compression method. For

increasing values of the error tolerance ε, the compression ratio

becomes systematically small for all schemes, but the energy

consumed for compression differs. In fact, PLAMLiS algo-

rithm require a large amount of energy, contrary to EPLAMLiS

and LTC which requires a small energy expenditure.

The energy for compression is strongly related to the com-

plexity of the algorithm. LTC encodes the time series sample

by sample incrementally regardless of the error tolerance

value. Thus, the number of operations depends weakly on

the compression ratio. EPLAMLiS has to work fewer for

increasing values of εdue to the fact that the number of opera-

tions (divide and reiterate) becomes smaller, and consequently

the energy consumption is reduced. For PLAMLiS case, for

each point of the time series, the algorithm ﬁnds the longest

segment that meets the error tolerance. For high values of ε,

these segments become longer. For this reason, the algorithm

becomes more complex when the error bound is increased, as

a result the energy for compression increments.

B. Compression Ratio vs Total Energy

Total energy consumption presents the sum of computa-

tional and transmission energy. Figure 6 shows the inﬂuence

of the compression ratio on the energy saving. In fact, the

three curves have almost the same shape as Figure 5, with a

little difference in the slope. Thus, this difference is due to

the transmission energy that decreases when the compression

ratio becomes smaller. As a result, only LTC and EPLAM-

LiS can achieve some energy saving, contrary to PLAMLiS

that requires more energy expenditure. Hence, compared to

EPLAMLiS, LTC presents the most signiﬁcant energy saving.

50 100 150 200 250 300

Total Energy (μJ)

0.2

0.4

0.6

0.8

1

Compression Ratio

EPLAMLiS

PLAMLiS

LTC

Fig. 6: Total Energy Consumption vs Compression Ratio.

C. Compression Ratio vs Distortion

The distortion is the major parameter in measuring the

reliability of the compression method. Figure 7 shows the

variation of the distortion as a function of the compression

ratio.

For all compression methods, the distortion increases with

increasing values of ε, which makes it inversely proportional

to the compression ratio. Compared to LTC, EPLAMLiS has

the lowest distortion for a given compression ratio, due to the

fact that the endpoints of all line segments must be points in

the data series, which is not the case concerning LTC. For this

reason, EPLAMLiS shows a high level of signal representation

accuracy.

VI. CONCLUSION

In this paper, we have compared the performance of PLA

techniques in terms of compression ratio, energy saving and

0 0.1 0.2 0.3 0.4 0.5 0.6

Distortion (C°)

0.2

0.4

0.6

0.8

1

Compression Ratio

EPLAMLiS

PLAMLiS

LTC

Fig. 7: Distortion vs Compression Ratio.

accuracy of reconstructed data for wireless sensor network

devices. The obtained results revealed that there is a trade-

off between the energy saving and the reliability of recon-

structed data. LTC is a lightweight compression method that

incurs the smallest energy expenditure. However, compared to

EPLAMLiS algorithm, LTC represents a drawback in terms

of distortion of reconstructed data samples. For EPLAMLiS

case, the signal is decompressed with high level of accuracy

but at the cost of some energy expenditure. Future works aim

to propose a combined algorithm which will be based on the

compelling features of LTC and EPLAMLiS algorithms in

order to optimize the trade-off between energy cost and data

accuracy.

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