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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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