Adaptive Change Detection for Long-Term Machinery Monitoring Using Incremental Sliding-Window
Adaptive Change Detection for Long-Term Machinery Monitoring Using Incremental Sliding-Window
Teng Wang 0 1 2 3
Guo-Liang Lu 0 1 2 3
Jie Liu 0 1 2 3
Peng Yan 0 1 2 3
0 Key Laboratory of High-Efficiency and Clean Mechanical Manufacture of MOE, School of Mechanical Engineering, Shandong University , Jinan 250061 , China
1 & Guo-Liang Lu
2 Supported by National Natural Science Foundation of China (Grant Nos. 61403232, 61327003) , Shandong Provincial Natural Science Foundation of China (Grant No. ZR2014FQ025), and Young Scholars Program of Shandong University , China, YSPSDU, 2015WLJH30
3 Department of Mechanical and Aerospace Engineering, Carleton University , Ottawa, ON K1S 5B6 , Canada
Detection of structural changes from an operational process is a major goal in machine condition monitoring. Existing methods for this purpose are mainly based on retrospective analysis, resulting in a large detection delay that limits their usages in real applications. This paper presents a new adaptive real-time change detection algorithm, an extension of the recent research by combining with an incremental sliding-window strategy, to handle the multi-change detection in long-term monitoring of machine operations. In particular, in the framework, Hilbert space embedding of distribution is used to map the original data into the Re-producing Kernel Hilbert Space (RKHS) for change detection; then, a new adaptive threshold strategy can be developed when making change decision, in which a global factor (used to control the coarse-to-fine level of detection) is introduced to replace the fixed value of threshold. Through experiments on a range of real testing data which was collected from an experimental rotating machinery system, the excellent detection performances of the algorithm for engineering applications were demonstrated. Compared with state-ofthe-art methods, the proposed algorithm can be more suitable for long-term machinery condition monitoring without any manual re-calibration, thus is promising in modern industries.
Machine monitoring; Change detection; Long-term monitoring; Adaptive threshold
1 Introduction
Detection of structural changes from an operational process
is a major goal in machinery monitoring which enables to
solve many practical problems ranging from early fault
detection, safety protection as well as other process control
problems. Existing works are mainly based on a
retrospective analysis of a data stream composed of numerical
condition monitoring (CM) variables, such as vibration,
sound, power consumption. The basic idea of a standard
retrospective change detection mainly relies on estimating
the logarithm of the likelihood ratio between two
distributions [
1, 2
]. This kind of strategy argues that the
detection of a change can be converted into the detection of the
parameter difference between the two distributions before
and after this change point. As a consequence, the
retrospective change detection aims to estimate this parameter
difference of distributions based upon likelihood ratio
statistics. Change decision can be made by performing a
null hypothesis testing with a threshold. Many effective
tools for this goal such as the cumulative sum metric
(CUSUM) [
3–6
], geometric moving average (GMA) [
7, 8
]
and the generalized likelihood ratio test (GLRT) [
9–12
]
have been widely used. For example, Willersrud et al. [
12
]
developed the GLRT to make efficient downhole drilling
washout detection with the multivariate t-distribution;
Ren˜ones et al. [
6
] used the CUSUM analysis for
multitooth machine tool fault detection. Although these methods
have been experimentally demonstrated the effectiveness
in various fields, due to the requirement of data after
change point, a large detection delay is an essential
limitation of these methods for real applications [
13
]. On the
other hand, real-time change detection aims to detect
changes as soon as possible when a change occurs, this
requirement is crucial in many real-life scenarios such as
security monitoring [
14, 15
], health care [
16, 17
],
automated factory [
18, 19
] as well as machine operation
monitoring studied in this paper. In operation of real-time
change detection, at each time when a datum is input, it
evaluates what extent the input datum is likely to be a
change-point by a certain type of measuring score [20]
which does not need any input data after change time. The
real-time approaches have succeeded in solving many
practical applications (e.g., wind turbine condition
monitoring [
21
], driver vigilance monitoring [
22
]), and thus are
promising [
23
].
The goal of this paper is to further advance this research
line of real-time detection methods. More specifically, our
main contributions in this paper are summarized in two
folds. The first contribution is to apply a martingale-based
framework proposed in our recent article [
24
] to long-term
machine monitoring by combining an incremental
slidingwindow strategy. The basic idea of the original martingale
is to directly learn a statistical regularity from already
observed data, and then detect possible change(s) by
investigating how much each data is deviated from the
regularity using martingale by testing exchangeability.
That framework, however, only works for
at-most-onepoint change detection, thus unsuitable for the cases
containing multiple changes in long-term monitoring
applications. In this paper, we introduce an incremental
slidingwindow strategy for solving this problem.
Recall that, the threshold value for change decision
making is a key factor of detection accuracy. Potential
weakness of the majority of exiting algorithms, e.g., Refs.
[
5–11
], is that they need either human-made instruction/
intervention or an off-line cross-validation to confirm the
value of threshold before operation, and thus make them
largely limited in real applications. Another contribution of
this paper is to develop a new adaptive threshold when
performing change decision making. In particular, we
introduce an alternative factor: a fixed-global parameter
used to control the coarse-to-fine level of detection, instead
of the fixed value of threshold, for change detection (see
Section 3.2 for details). By using this factor, at each step of
change decision, the threshold value can be adaptively
computed from the already observed data.
Besides the methodological extensions of the proposed
method, we also conducted validations on an experimental
setup to investigate effectiveness/priority of the method for
change detection with large datasets. For more details,
please see Section 4.
The rest of the paper is structured as follows. Section 2
presents the outline of martingale framework for machine
monitoring. Section 3 formulates problems addressed in
this paper and provides our proposed methods, followed by
experimental results in Section 4. Section 5 concludes this
paper and shows the future work.
2 Martingale Based Change Detection for Machinery Monitoring
Assuming that we have collected a data stream composed
of numerical CM variables from an operational process of a
machine, i.e., X = {x1,…, xi,…, xn} where
xi, i 2 f1; 2; . . .; ng is the variable value at time i, three
points are provided in the following to support the use of
martingale for change detection:
(1)
(2)
(3)
The changes are detected by testing the null
hypothesis that all n (strangeness) values (which
corresponds to x1; x2; . . .; xn; respectively) are
exchangeable in the index, through the
corresponding exchangeability martingale M1; M2; . . .; Mn,
where Mn is a measurable function of s1; s2; . . .; sn,
satisfying
Mn ¼ EðMnþ1jM1; M2; . . .; MnÞ:
The following Doob’s inequality [
25
] can be used for
rejecting this null hypothesis for a large value of Mn:
Pð9njMn
kÞ
1
k
:
This (exchangeability) martingale is constructed
from a p value, the probability of obtaining a test
statistic at least as extreme as the one that was
actually observed, and the p-value is obtained by a
strangeness value appropriately determined in each
specific application.
On the basis of the above three points, the outline of
performing martingale for change detection is described as
follows (see Ref. [
25
] for more details):
Step 1: The randomized power martingale (RPM) [
26
] is
constructed from the computed s1; s2; . . .; sn by
Mt ¼
i¼1
Ytðep^ie 1Þ; t 2 f1; 2; . . .; ng;
where e 2 ð0; 1Þ and p^is are the value computed from p^
-value functions:
ð1Þ
ð2Þ
ð3Þ
p^iðfs1; . . .; si 1g; siÞ
¼
#fj : sðjÞ [ sðiÞg þ hi#fj : sðjÞ ¼ sðiÞg ;
i
where #f g is a counting function and hi is a random value
from a uniformly distribution of [
0, 1
],
j 2 f1; 2; . . .; i 1g:
Step 2: The following inequality is then tested based on
the Doob’s inequality for any t 2 f1; 2; . . .; ng to test the
hypothesis as below:
H0 : no change :
0\Mt\k ;
HA : change occurs :
Mt
k :
ð4Þ
ð5Þ
That is, if the martingale value Mt is greater than a
predefined threshold k, HA in Eq. (5) is satisfied, i.e., a
change occurs on the time t. Otherwise, the martingale test
satisfying H0 continues to operate as long as 0 \ Mt \ k.
3 Problem Formulation and Proposed Scheme
In Section 2, we have provided the outline of
martingaletest for change detection. Two problems have to be further
considered for long-term machinery monitoring:
(1)
(2)
How to deal with multi-change detection in
longterm monitoring?
Is it possible to adaptively compute the threshold
value when making change decision?
In the following, we will discuss these two problems in
details and provide our proposed schemes.
3.1 Change Detection Using Incremental Sliding
Window
Problem 1. As shown in Eq. (3) and Eq. (5), Mt can be
sequentially processed with a fixed-length L
sliding-window over the given data stream, and all possible change
candidates t 2 f1; 2; . . .; ng are tested. This process
however may be unsuitable for long-term monitoring
applications. A key feature of real machine operations is temporal
variations, i.e., one operation can last for a long time or
only a few seconds. Hence, it is difficult to use a
fixedlength L sliding-window to capture transitions (i.e.,
changes from an operational state to another) in long-term
monitoring. More specifically, a small length of L causes
over-change-detection and a large length of L causes a
large delay. To overcome this problem, we combine the
martingale with an incremental sliding-window strategy
[
27
] to design a real-time change detection algorithm for
Eq. (5).
Proposed scheme: By virtue of incremental
slidingwindow, the length L can be automatically updated
depending on whether a change is detected or not at time t:
If t is no change : ntþ1 ¼ nt; Ltþ1 ¼ Lt þ DL;
If t is a change : ntþ1 ¼ t; Ltþ1 ¼ L1;
ð6Þ
where nt is the starting time when computing the current
martingale and Lt is the length of corresponding sliding
window at time t. The process starts with n1 = 1 and
L1 = 1, and ends with nt þ Lt [ n where n is the length of
a given data stream in off-line applications or ends at an
pre-defined stopping time in on-line applications. Here, it is
worth mentioning that DL is an increasing step to update
the sliding window and was set as 1 in the following
experiments.
3.2 Adaptive Threshold for Change Detection
Problem 2. When making change decision by testing the
null hypothesis shown in Eq. (5), the threshold of k is
essential as it balances the detection precision and recall
(their definitions will be given in Section 4.3). In general,
the value of k is pre-defined empirically or confirmed by a
prior estimation for change detection. It is, however, often
difficult to confirm the optimal value in real-world
applications. To address this problem, unlike existing works that
directly used the original monitored variables for change
detection (e.g., Refs. [
6–12, 25
]), we utilize the Hilbert
space embedding of distribution (HED, also called the
kernel mean or mean map) to map the original data
fxig, i 2 f1; 2; . . .; ng into the Re-producing Kernel Hilbert
Space (RKHS) (see Figure 1). Without going into details,
the idea of using HED for change detection is
straightforward. By this, the probability distribution is now
represented as an element of a RKHS, and the change can be
thus detected by using a well behaved smoothing kernel
function, whose values are small on the data belonging to
the same pattern and large on the data from different
patterns.
Proposed scheme: Inspired by Ref. [
28
], probabilistic
distributions can be embedded in RKHS. The center of the
HED are the mean mapping functions:
lðpxÞ ¼ EðkðfxigÞÞ;
lðfxigÞ ¼ 1t Xt kðxiÞ;
i¼1
where fxig i = 1,2,…,t are assumed to be I.I.D sampled
from the distribution Px. Under mild
conditions, l ðPxÞ(same for l ðfxigÞ) is an element of the
Hilbert space. Mapping l ðPxÞ is attractive because each data
point xi has a one-to-one correspondence with mapping
l ðPxÞ. Thus, we can use the function norm
sðlðPxÞ; kðxtÞÞ(instead of sðPfx1; x2; . . .; xt 1g; xtÞ) used in
Ref. [
1
]) to quantify the strangeness value st for xt. We do
not need to access the actual distributions but rather finite
samples to calculate/estimate the distribution q.
Lemma 1. As long as the rademacher average [
29
],
which measures the ‘‘size’’ of a class of real-valued
functions with respect to a probability distribution, is well
behaved, finite sample yield error converges to zero, thus
they empirically approximate lðPxÞ(see Ref. [
28
] for more
details).
The success of kernel methods largely depends on the
choice of the kernel function k which is chosen according
to the domain knowledge or universal kernels. In this
paper, we employ the widely-used Gaussian radial basis
function (RBF) kernel by
kðxiÞ ¼ exp
1
2r2 jjxi
xjj2 ;
where x and r are the sample mean and standard deviation
of the data stream fx1; x2; . . .; xig: We next construct st to
measure the strangeness of xt to the past data stream up to
time t–1, i.e., fx1; x2; . . .; xi 1g in RKHS, as
st ¼ sðlðpxÞ; kðxtÞÞ ¼ jkðxtÞ
ktc 1j;
where ktc 1 is the kernel center of the data stream, and j j is
distance metric. Here, it is worth mentioning that in real
engineering scenarios, the CM variables are often
composing of multidimensional values measured from multiple
sensors at each time instance, we thus use the Mahalanobis
distance [
30
] to compute the strangeness st, by considering
correlations between variables such that different patterns
in each dimension can be identified and analyzed [
30
],
computed by
st ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðkðxtÞ ktc 1Þ0 X 1 ðkðxtÞ ktc 1Þ
ð10Þ
where P is the covariance matrix.
Since we used RBF as the kernel function as given in
Eq. (8), an isolated data point can be certified if st a r,
ð8Þ
ð9Þ
where a is a fixed-global factor controlling the confidence
level of detection and r is the standard deviation computed
from existed data (that is, an adaptive threshold).
Based on this fact, kernelized change decision can be
made by re-writing Eq. (5) as follows,
If 0\Mt\a K rt 1 : no change,
If Mt
a K rt 1 : change occurs,
ð11Þ
where K is a projection coefficient of data from RKHS to
martingale space and rt 1 can be computed adaptively
from the data stream up to time t 1. In real
implementation, the employed Gaussian function is often
standardized as a normal Gaussian function (i.e., l = 0 and r = 1).
Figure 2 gives typical confidence levels corresponding to
different a in Gaussian distribution. Thus K can be fixed as
K 2:17 by an off-line estimation. The estimation is made
as follows: (a) given a set of data streams containing
changes, we first make a definition of detection accuracy as
q = N/P where N is the number of correctly detected
changes and P is the number of total detected changes, and
set up a threshold value k to guarantee a perfect accuracy,
i.e., q = 100%; (b) then, decrease the value of k gradually
to make sure q not decrease; (c) once q decreases, K is
computed by
k
K ¼ 5 r
since five times of r can guarantee all changes can be
detected.
4 Experimental Verification
In this section, we aim to demonstrate the effectiveness and
priority of the proposed adaptive change-detection
algorithm for long-term machine monitoring. Thus, Experiment
I and Experiment II are conducted to answer the following
two questions:
(1)
(2)
Will the proposed incremental sliding-window be
more suitable than the fixed-length sliding-window
for long-term machine monitoring?
Can the adaptive detection algorithm be effective for
change detection and how does it perform with large
datasets?
4.1 Experimental Setup
4.2 Experiment I: Performance of Incremental
Sliding-Window
In our method, we propose to use incremental
slidingwindow, instead of length-fixed sliding window, for
longterm machine monitoring. To evaluate the effectiveness/
priority of this strategy for change detection, a set of testing
threshold values from 1 to 20 at an interval of 1 were
checked. Particularly, Figure 5 shows three detection cases
of using incremental sliding-window for the testing data
stream (shown in Figure 4). It is apparent that the
performance differs greatly with different values of k, and the
best performance has been achieved when k = 7.
Specifically, the smaller value of brings higher false alarms but
larger value takes a lager detection delay. Here, it is
demonstrated again the important roles of threshold value k
for change detection. In the following, since the value of
k ¼ 7 has been shown the success of change detection for
the testing data, we compare this performance with that of
using a fixed-length L sliding window detection. From
Figure 6 that shows the results of three cases of
L = {3000, 5000, 7000}, we can observe that the small
length of L = 3000 causes over-detection (i.e., more false
alarms) and the large length of L = 7000 cause a large
detection delay; it tends to achieve a good detection
performance when L = 5000, while one notes that it is often
difficult to fix the length of L of the sliding-window
considering the great temporal variations of collected CM
signal from real machine operations.
In overall, taking the results shown in Figure 5 and
Figure 6 together, the followings were interestingly found:
•
•
The length-fixed sliding-window martingale requires
more parameters, i.e., k and L, for performing change
detection, which requires a more complicated prior
estimation of them before usage;
By the incremental sliding-window martingale, only
one parameter: k is required, which inspires an
extension of change detection by adaptive threshold.
Both of them inspire the adaptive threshold given in
Section 3.2, which will be evaluated in the following.
4.3 Experiment II: Performance of Adaptive
Threshold
In this section, we will evaluate the proposed adaptive
threshold for machine monitoring. Here, it is noted that
since in the Section 4.2, we have demonstrated the priority
of using incremental sliding-window for long-term
machine monitoring, in this section, we only test the
performance of adaptive threshold with incremental
slidingwindow.
Figure 7 shows the results where all changes in the
testing CM data (the same data shown in Figure 4) have
been successfully detected without any false alarms when
setting a as 3.0 but when k decreases, more false alarms are
brought out. In addition, it is also observed that on the basis
of considering the projection coefficient K from RKHS to
martingale space and the global fixed confidence level a,
the threshold value can be computed/adjusted adaptively
according to the standard deviation computed from the past
data at each step of change decision making as mentioned
in Eq. (11), in other words, the threshold value is not fixed
in the whole process, that is different from many existing
works [
7–11, 25
]. All of them are consistent with the
analysis previously made in Section 3.2. Moreover, to
verify the effectiveness of the adaptive threshold with large
Precision is the probability that a detection is actually
correct, i.e., a true change. Recall is the probability that the
detection recognizes a true change.
In addition, we also use a single performance indicator
F1 defined as
F1 ¼
2
Recall Precision
Recall þ Precision
:
Apparently, F1 is a harmonic mean between precision
and recall, and a high value of F1 ensures reasonably a
high balance between precision and recall.
Figure 8 shows the detection performances for different
values of a 2 {0.92, 1.84, 2.30, 2.50, 2.75, 3.00, 3.22, 3.69,
4.15, 4.61, 5.00}. Specifically, it can be found in
Figure 8(a) that with an increasing value of a, the detection
precision increases and achieves the best performance
when a [ 3. On the other hand, for the recall shown in
Figure 8(b), our proposed method can always obtain a
perfect performance, that is 100% which means the all true
changes can be successfully detected for tested values of a.
These results can be more clearly observed in Figure 8(c)
where F1 can achieve the best performance when a [ 3.
All of these results are not surprising because three times of
r guarantees approximate 99.8% (as shown in Figure 2)
data points have been contained in a Gaussian distribution.
Considering a smaller a brings a smaller detection delay as
shown in Figure 7, it is thus recommended a ¼ 3 when
using the proposed method in real applications.
5 Conclusions
In this paper, we have extended our recent work [
25
] to
long-term machine monitoring where two schemes are
proposed: 1) using the incremental sliding-window to solve
the problem of multi-change detection; and 2) developing
an adaptive threshold when making change decision.
Experimental results on an experimental setup
demonstrated great successes of the proposed method in
multichange detection in long-term monitoring. With this, it can
be concluded that the improved algorithm is feasible for a
new generation of long-term machine monitoring systems.
In view of this, further work will be done to continue
verifying the capability of the improved algorithm for
detecting a wider range of changes when operating a
machine to make it ready for commercial exploitation.
In addition, considering that the detection delay is one of
essential aspects to be considered when design a detection
method, another future work is to extract informative
features to represent the raw collected data for modeling in
order to further decrease the delay of our method when
detecting changes
Open Access This article is distributed under the terms of the
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Teng Wang, received his bachelor degree from Hefei University of
Technology in 2016, and now is a master candidate at Shandong
University. His research interest includes machine monitoring,
prognostics and health management.
Guo-Liang Lu, received the bachelor degree and master degree from
Shandong University, Jinan, China, in 2006 and 2009, respectively.
He received his PhD degree from the Graduate School of Information
Science and Technology of Hokkaido University, Sapporo, Japan, in
March 2013. He is currently an associate professor in Shandong
University. His research interests mainly include signal processing,
machine monitoring, machine vision and visual servo control.
Jie Liu, is currently an Associate Professor in the Dept. of
Mechanical & Aerospace Engineering at Carleton University, Ottawa,
Canada. He obtained his bachelor degree. in Electronics and Precision
Engineering from Tianjin University (China) in 1998, his master
degree in Control Engineering from Lakehead University (Canada) in
2005, and his PhD degree in Mechanical Engineering from the
University of Waterloo (Canada) in 2008. Before joining Carleton, he
worked as an Postdoctoral Fellow in the Dept. of Mechanical
Engineering at UC Berkeley for one and half years. Dr. Liu has been
engaged in interdisciplinary research in the areas of prognostics and
health management, intelligent mechatronic systems, and battery
management systems for more than thirteen years. His research
results have been disseminated through over 35 journal publications
and 20 conference papers. He is a Steering Committee Member of
Annual IEEE PHM Conferences, an Associate Editor of IEEE
Transactions on Reliability, an IEEE Senior Member, and a registered
Professional Engineer in Ontario, Canada.
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Peng Yan, received the bachelor degree and master degree from Southeast University in 1997 and 1999 respectively, and the PhD degree from the Ohio State University, Columbus, OH, in 2003 , all in Electrical Engineering. From 2004 to 2005, he worked as postdoctorate researcher at the Ohio State University . From 2005 to 2011 , he held various industry positions including a Senior Staff Engineer at Seagate Technology at Shakopee MN and a Staff Scientist at United Technology Research Center (UTRC) at Easthartford CT . Currently he is a full professor at the school of Mechanical Engineering of Shandong University, China. His research interests include robust control and control of high precision mechatronics .