Gaussian Processes for Anomaly Description in Production
Environments
Christian Beecks Kjeld Willy Schmidt
University of Münster and Fraunhofer Institute for University of Münster, Germany
Applied Information Technology FIT, Germany kjeld.schmidt@uni-muenster.de
christian.beecks@uni-muenster.de
Fabian Berns Alexander Grass
University of Münster, Germany Fraunhofer Institute for Applied Information
fabian.berns@uni-muenster.de Technology FIT, Germany
alexander.grass@fit.fraunhofer.de
which are combinations of well-known kernels. By fitting ker-
nel expressions to the corresponding sensor data, we are able
ABSTRACT to decompose the inherent structure of an anomaly and to de-
Concomitant with the rapid spread of cyber-physical systems scribe its individual behavior such as linearity and periodicity
and the advancement of technologies from the Internet of Things, by natural language. For this purpose, we make use of Gaussian
many modern production environments are characterized by vast processes [20] and the Compositional Kernel Search model [11].
amounts of sensor data which are generated throughout differ- We carry out our analysis on the recently proposed IoT dataset
ent stages of production processes. In this paper, we propose a [5], a real-world industry 4.0 dataset, which has been collected
novel method for discovering the inherent structures of anom- within the EU project MONSOON1 . To sum up, we make the
alies arising in IoT sensor data. Our idea consists in modeling and following contributions:
describing anomalies by means of kernel expressions, which are • We propose a machine-learning-based method in order to
combinations of well-known kernels. The results of our empirical model anomalies and to describe their inherent compo-
analysis show that our proposal is suitable for modeling differ- nents.
ently structured anomalies. Moreover, the results indicate that • We enrich the MONSOON IoT dataset with a novel ground
Gaussian processes provide a powerful tool for future algorithmic truth derived from domain experts in order to further
investigations of IoT sensor data. stimulate research of anomaly detection algorithms on
this real-world dataset.
The paper is structured as follows. In Section 2, we outline re-
1 INTRODUCTION lated work. In Section 3, we briefly introduce Gaussian processes
Concomitant with the rapid spread of cyber-physical systems and their application to adapt kernel expressions to sensor data.
and the advancement of technologies from the Internet of Things The preliminary results of our proposed method are reported
(IoT), many modern production environments are characterized and discussed in Section 4, before we conclude our paper with
by vast amounts of sensor data which are generated through- an outlook on future research directions in Section 5.
out different stages of production processes. These sensor data
streams are often considered as valuable information sources 2 RELATED WORK
with a high economic potential and are characterized by high vol- Strongly related to our approach are anomaly detection algo-
ume, velocity and variety. Their data-driven value is indisputable rithms. There is a plethora of these algorithms including Z-Score
for optimizing and fine-tuning industrial production processes. [10], Mahalanobis Distance-Based, Empirical Covariance Estima-
Monitoring sensor data from complex production processes in tion [18] [9], Mahalanobis Distance-Based, Robust Covariance
order to detect outliers or low-performing production behavior Estimation [22] [9], Subspace-based PCA Anomaly Detector [9],
caused by undesired drifts and trends, which we summarize as One-Class SVM [23] [18] [9] [12], Isolation Forest (I-Forest) [16]
anomalies, is a challenging task. Not only due to the massive [18], Gaussian Mixture Model [18] [9] [19], Deep Auto-Encoder
amount of sensor data but also due to different types of anom- [8], Local Outlier Factor [7] [18] [9] [1], Least Squares Anomaly
alies, which are potentially unknown in advance, manual or au- Detector [24], GADPL [14] and k-nearest Neighbour [13] [1] [12].
tomatic inspection systems are frequently supported by anomaly While these algorithms are all possible options for anomaly
detection algorithms. While the last years have witnessed the detection, as shown in different surveys such as [13], [19] and [9],
development of different anomaly detection algorithms, cf. the they are not directly suited for describing the inherent structure
work of Renaudie et al. [21] for a recent performance evaluation of anomalies, which is the major focus of this paper. We choose
in an industrial context, only less effort has been spent to the the means of Gaussian processes for anomaly description due
investigation of the inherent structure of an anomaly. to their capability to not only gather statistical indicators, but
In this paper, we thus propose a novel method to discover the deliver the very characteristics of specific anomalous behavior
inherent structure of an anomaly. Our idea consists in model- from the data [20].
ing and describing anomalies by means of kernel expressions, For describing these characteristics, Lloyd et al. [17] have pro-
posed the Automatic Bayesian Covariance Discovery System
First International Workshop on Data Science for Industry 4.0.
Copyright ©2019 for the individual papers by the papers’ authors. Copying permit- that adapts the Compositional Kernel Search Algorithm [11] by
ted for private and academic purposes. This volume is published and copyrighted
by its editors. 1 www.spire2030.eu/monsoon
Published in the Workshop Proceedings of the EDBT/ICDT 2019 Joint Conference (March 26, 2019, Lisbon, Portugal) on CEUR-WS.org.
Figure 1: An example of the MONSOON IoT dataset with three anomalies.
adding intuitive natural language descriptions of the function Anomaly BIC Kernel Expression
classes described by their models. In [15], these models are ex- 0 -799 C*PER + C*PER + C*PER
panded to discover kernel structures which are able to explain 1 -706 C*SE*PER + C*SE + C
multiple time series at once. 2 -604 C*PER + C*PER + C*PER + C
In this work, we make use of these algorithms in order to 3 -921 C*SE*PER + C*PER + C
describe the inherent structures of anomalies, as shown in the 4 -742 C*PER + C*PER + C*SE + C
following section. 5 -543 C*SE*LIN + C*SE + C*WN + C
6 -630 C*PER + C*SE + C*WN + C
3 GAUSSIAN PROCESSES 7 -1020 C*PER + C*PER + C*PER + C*SE + C
8 -762 C*SE*PER + C*PER + C
In this section, we describe the analysis of anomalies in sensor
9 -1025 C*PER + C*PER + C*SE + C
data via Gaussian processes. To this end, we assume the sensor
10 -424 C*PER + C*SE + C*SE
data to be univariate2 and an anomaly A to be a finite subsequence
11 -849 C*PER + C*PER + C*SE + C
of timestamp-value pairs A = {(ti , vi )}i=i
n with timestamps t ∈ T
i
12 -311 C*SE*PER + C*PER + C
and values vi ∈ R.
13 -860 C*LIN + C*PER + C*PER + C*PER + C
As we do not know in advance the number of values and the
14 -339 C*PER + C*SE + C*SE
distances between individual timestamps, we can also thought
15 -590 C*SE*PER + C*PER + C*SE
of an anomaly A as a mathematical function A : T → R, which
16 -503 C*PER + C*SE + C
assigns every timestamp t ∈ T a real-valued value v(t) ∈ R. By
17 -602 C*SE*PER + C*SE + C*WN + C
considering the individual values v(t) to be random variables
18 -545 C*PER + C*SE + C*SE + C
following a Gaussian distribution, we can formalize the Gaussian
19 -804 C*PER + C*SE + C*WN + C
process as
20 -281 C*PER + C*SE + C*SE
21 -426 C*PER + C*PER + C*SE
v(t) ∼ GP(m(t), k(t, t ′ )),
22 -425 C*SE*PER + C*PER + C*SE
where m(t) = E[v(t)] is the mean function and k(t, t ′ ) = 23 -975 C*SE*PER + C*PER + C
E[(v(t) −m(t)) · (v(t ′ ) −m(t ′ ))] is the covariance function k : T × 24 -1181 C*PER*LIN + C*PER + C*SE
T → R. In other words, a Gaussian process is a stochastic process 25 -880 C*PER*PER + C*PER + C*PER + C
over random variables, where every subset of random variables 26 -455 C*PER + C*PER + C*SE
from the Gaussian process follows a normal distribution. The 27 -542 C*PER + C*SE + C*SE
distribution of the Gaussian process is the joint distribution of all Table 1: Discovered kernel structures and the Bayesian In-
of these random variables and it is thus a probability distribution formation Criterion (BIC) for the encountered 28 anom-
over (the space of) functions in RT . alies.
While the covariance function k defined above is a general
way to model the behavior of data, we aim to describe each
anomaly A by its own covariance function k A . That is, we aim
to learn a covariance function k A , which is then also denoted as
kernel expression in the domain of machine learning, by fitting example, an anomaly A with a highly weighted linear kernel k LIN
combinations of well-known kernels, such as indicates a hidden linearity component while a highly weighted
periodic kernel k PER indicates an inherent periodicity in the
• the constant kernel k C (t, t ′ ) = λ ∈ R, anomaly.
• the linear kernel k LIN (t, t ′ ) = (t − l) · (t ′ − l), The resulting kernel expressions are reported and discussed
|t −t ′ | 2
• the squared exponential kernel k SE (t, t ′ ) = exp − 2l 2 , in the next section.
′
2 sin2 t −t
• or the periodic kernel k PER (t, t ′ ) = exp l2
2
.
4 PRELIMINARY RESULTS
In order to individually fit a kernel expression to each anomaly
based on the aforementioned kernels, we use the compositional In this section, we report and discuss the results of our pre-
kernel model, as utilized for instance in [17]. This allows us to liminary performance evaluation. For this purpose, we use the
decompose an anomaly into individual components, which can be recently introduced MONSOON IoT dataset [5] which comprises
ranked by their contribution towards explaining the data. As an 357,383 data records in total. This dataset is based on a real pro-
duction line of coffee capsules and the attribute under observation
2 It is noteworthy that this approach also applies to multivariate data. is the plastification time, that is the time which is needed to melt
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(plastify) the plastic melt for the actual injection molding cycle.
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More information about this process can be found in [3].
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An overview of this attribute value, i.e. the pastification time,
26
as a function of the cycle number is shown in Figure 1. As can
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be seen in the figure, while the normal plastification time is at
25
approximately 4.2 seconds, it drops down to less then 3 seconds
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in case of an anomaly. Supported by domain experts, we figured
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out 28 anomalies in total in this dataset, of which three are shown
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in the above figure.
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23
25
In the first series of experiments, we computed the best fit-
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ting kernel expressions by means of the ABCD algorithm. The
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results are shown in Table 1 for each anomaly. Together with the
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kernel expression of the corresponding anomaly, we also show
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21
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the Bayesian Information Criterion (BIC) value which models
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the trade-off between model accuracy and size. As can be seen in
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the table, all anomalies are well described by their corresponding
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kernel expression (lower BIC values indicate better fit and vice
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versa). Surprisingly many kernel expressions do not show a lin-
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ear component k LIN , although some anomalies clearly show this
18
linear tendency. We figure out that this is due to overfitting of
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the kernel expression in the ABCD algorithm. We aim to address 17
this issue in future research.
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16
In the second series of experiments, we evaluated how suitable
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a kernel expression of a certain anomaly fits to other anomalies.
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15
The results in form of the corresponding BIC values are summa-
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rized in Table 2. As can be seen in this table, kernel expressions
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of a certain anomaly do in general not fit to other anomalies.
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One reason for this behavior is the high degree of idiosyncrasy
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of the anomalies. Another reason might be the overfitting issue
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mentioned above.
To sum up, we have investigated the potential of describing
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anomalies in IoT sensor data by means of kernel expressions.
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Our preliminary results indicate that our proposal is well suited
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for this purpose. As one major challenge, we figure out that the
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problem of overfitting needs to be addressed in future research.
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9
5 CONCLUSIONS AND FUTURE WORK
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8
In this paper, we have addressed the problem of discovering the
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inherent structures of anomalies arising in IoT sensor data. To this
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end, we have proposed to model and describe anomalies by means
7
of kernel expressions, which are combinations of well-known
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kernels. The results of our empirical analysis show that our pro-
40
6
posal is suitable for modeling differently structured anomalies.
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Moreover, the results indicate that Gaussian processes provide a
5
powerful tool for future algorithmic investigations of IoT sensor
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data.
4
In future work, we aim to address the problem of overfitting
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by modifying the grammar used within the ABCD algorithm for
3
computing the kernel expressions. In addition, we aim to further
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develop our proposal in order to not only describe anomalies
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2
but also detect anomalies (which is not the focus of the current
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paper). For this purpose, we aim to measure similarity in IoT
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sensor data by incorporating Gaussian processes into adaptive
0
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distance-based similarity models, such as the Signature Matching
Distance [6], and query processing algorithms [2, 4].
Kernel
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ACKNOWLEDGMENTS Table 2: Evaluation of the BIC for every kernel expression
The project underlying this paper has received funding from against every anomaly.
the European Union’s Horizon 2020 research and innovation
program under grant agreement No 723650 (MONSOON). This
paper reflects only the authors’ views and the commission is not
responsible for any use that may be made of the information it The MIT Press.
contains. [21] David Renaudie, Maria A. Zuluaga, and Rodrigo Acuna-Agost. 2018. Bench-
marking Anomaly Detection Algorithms in an Industrial Context: Dealing
with Scarce Labels and Multiple Positive Types. In IEEE International Confer-
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