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Computers and Chemical Engineering 120 (2019) 46–53
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Computers and Chemical Engineering
journal homepage: www.elsevier.com/locate/compchemeng
Determining the number of segments for piece-wise linear
representation of discrete-time signals
R
Jiandong Wang
a , ∗
, Yan Yu
b
, Kuang Chen
b
a
College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao, Shandong, China
b
College of Engineering, Peking University, Beijing, China
a r t i c l e i n f o
Article history:
Received 26 April 2018
Revised 6 August 2018
Accepted 25 August 2018
Available online 22 September 2018
Keywords:
Data segmentation
Piecewise linear representation
Linear regression
Confidence intervals
a b s t r a c t
Piece-wise linear representation (PLR) separates a discrete-time continuous-valued signal into a few short
data segments, each of which is represented by a straight line. One prerequisite parameter in the PLR is
the number of data segments. This paper proposes a new method to determine the number of data
segments. The method is based on a key observation on two balancing effects of the number of data seg-
ments on the confidence intervals of line approximations. The confidence intervals define parallelogram-
shaped spaces, and an index is formulated as the weighted percentage of these spaces occupied by data
points in data segments. The number of data segments is determined as the one achieving the maximum
value of the formulated index. Numerical and industrial examples are provided to illustrate the effective-
ness of the proposed method.
©2018 Elsevier Ltd. All rights reserved.
1. Introduction
Modern industrial plants are commonly rich in data and poor
in information. Computerized technologies are desperately needed
to help operators and engineers extracting useful information from
data in a quick and reliable manner. Piece-wise linear representa-
tion (PLR) separates a long continuous-valued data sequence by a
few short data segments, each of which is represented by straight
lines. The PLR is one of the most frequently used techniques for in-
formation extraction ( Cheung and Stephanopoulos, 1990; Fu, 2011;
Janusz and Venkatasubramanian, 1991; Keogh et al., 2004; Zhou
et al., 2017 ). It has been applied to process monitoring, plant con-
trol, fault detection and diagnosis, e.g., monitoring state transitions
in chemical plants ( Sundarraman and Srinivasan, 2003 ), monitor-
ing operations of oil sand primary separations ( Sammaknejad et al.,
2014 ), generating alarms for abnormal conditions of patients in in-
tensive care units ( Charbonnier and Gentil, 2007 ), soft-sensing for
waster water treatment plant control ( Thurlimann et al., 2018 ).
One basic formulation of data segmentation is to find the best
approximation of a data sequence using K segments ( Keogh et al.,
2004 ). The number K of segments is a prerequisite parameter.
R
This work was partially supported by the National Natural Science Foundation of
China under Grant no. 61433001, and by the Research Fund for the Taishan Scholar
Project of Shandong Province of China.
∗
Corresponding author.
E-mail addresses: jiandong@sdust.edu.cn (J. Wang), yuyan_coe@pku.edu.cn (Y.
Yu), chenkuang@pku.edu.cn (K. Chen).
Vasko and Toivonen (2002) found the number of segments by
comparing the relative reduction of approximation errors with the
counterpart from permutated data. Zaliapin et al. (2004) chose
the number of segments as the one achieving the maximum of
a determination function of approximation errors. Salvador and
Chan (2005) proposed a so-called ‘L’ method to determine the
number of segments at the knee point of a function composed
by approximation errors. Frenzen et al. (2010) derived a relation
between the number of segments and the desired approximation
error for an arithmetic function. Zhou et al. (2017) sought the in-
flection point with the largest tuning angle among three adjacent
approximation errors.
This paper proposes a new method to determine the number of
data segments for the PLR of discrete-time continuous-valued sig-
nals. A PLR method uses straight lines as basis functions to approx-
imate data points in each data segment. The number of data seg-
ments has two balancing effects on the widths of confidence inter-
vals of line approximations: If there are more data segments, then
approximation errors usually are reduced, so that confidence in-
tervals of data approximations tend to be narrower; however, con-
fidence intervals would be wider if data segments enclose fewer
data points. In order to measure the balancing effects, an index
is formulated as the weighted percentage of the parallelogram-
shaped spaces defined by confidence intervals, to be occupied by
data points in data segments. Owing to the balancing effects, it
is a reasonable choice to determine the number of data segments
as the one achieving the maximum value of the formulated index.
Such a number is associated with confidence intervals that incor-
https://doi.org/10.1016/j.compchemeng.2018.08.034
0098-1354/© 2018 Elsevier Ltd. All rights reserved.