Towards Gaussian Processes
for Automatic and Interpretable Anomaly Detection in Industry 4.0
Fabian Berns
1
, Markus Lange-Hegermann
2
and Christian Beecks
1
1
Department of Computer Science, University of M
¨
unster, Germany
2
Department of Electrical Engineering and Computer Science,
OWL University of Applied Sciences and Arts, Lemgo, Germany
Keywords:
Anomaly Detection, Gaussian Processes, Explainable Machine Learning, Industry 4.0.
Abstract:
Discerning unexpected from expected data patterns is the key challenge of anomaly detection. Although a
multitude of solutions has been applied to this modern Industry 4.0 problem, it remains an open research issue
to identify the key characteristics subjacent to an anomaly, sc. generate hypothesis as to why they appear. In
recent years, machine learning models have been regarded as universal solution for a wide range of problems.
While most of them suffer from non-self-explanatory representations, Gaussian Processes (GPs) deliver inter-
pretable and robust statistical data models, which are able to cope with unreliable, noisy, or partially missing
data. Thus, we regard them as a suitable solution for detecting and appropriately representing anomalies and
their respective characteristics. In this position paper, we discuss the problem of automatic and interpretable
anomaly detection by means of GPs. That is, we elaborate on why GPs are well suited for anomaly detection
and what the current challenges are when applying these probabilistic models to large-scale production data.
1 INTRODUCTION
Anomaly detection is an important data mining pro-
cess to distinguish expected from unexpected data
patterns. It enables researchers as well as practitioners
to assess the condition and current state of a system
of interest (Chandola et al., 2009). Applications are
manifold, e.g. in medicine (Rajpurkar et al., 2017),
credit card fraud (Sorournejad et al., 2016), or net-
work intrusion detection (Ioannou et al., 2017). Es-
pecially with regards to industrial applications, moni-
toring sensor data from complex processes in order to
detect outliers or low-performing production behav-
ior caused by undesired patterns and trends, which
we summarize as anomalies, is a challenging task
(Beecks et al., 2019). Not only due to the massive
amount of sensor data but also due to different types
of anomalies, manual or automatic inspection systems
are frequently supported by anomaly detection algo-
rithms (Stojanovic et al., 2016). It is often stressed
that anomaly detection should be part of understand-
ing the production process as a whole (Niggemann
and Lohweg, 2015), i.e. that it is not sufficient to de-
tect anomalies but also to generate hypothesis as to
why they appear.
While the last years have witnessed the develop-
ment of different anomaly detection algorithms (cf.
the work of Renaudie et al. (2018) for a recent perfor-
mance evaluation in an industrial context) only less
effort has been spent on the investigation of the in-
herent structure of an anomaly. Utilizing techniques
of machine learning and Artificial Intelligence (AI)
for anomaly detection seems like a natural choice to
not only detect unexpected patterns, but to understand
them. Still, achieving the level of trustworthiness re-
quired for sensitive industrial applications is a non-
trivial task (cf. High-Level Expert Group on Artificial
Intelligence, 2019; Kwon et al., 2019).
With regards to the key requirements for trustwor-
thy AI defined by the High-Level Expert Group on
Artificial Intelligence (2019) of the European Com-
mission, we consider GPs (Rasmussen and Williams,
2006) as an appropriate machine learning model to
fulfill those demands. GPs deliver robust and reliable
statistical data models, which are able to cope with
unreliable, noisy, or partially missing data (Bo
ˇ
skoski
et al., 2012). Since these models are composed from
simple components defined by explainable hyperpa-
rameters, they are interpretable by nature and deliver
the capabilities to trace their predictions back to those
Berns, F., Lange-Hegermann, M. and Beecks, C.
Towards Gaussian Processes for Automatic and Interpretable Anomaly Detection in Industry 4.0.
DOI: 10.5220/0010130300870092
In Proceedings of the International Conference on Innovative Intelligent Industrial Production and Logistics (IN4PL 2020), pages 87-92
ISBN: 978-989-758-476-3
Copyright
c
 2020 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
87

components and the according training data (Lloyd
et al., 2014; Duvenaud et al., 2013).
In this position paper, we discuss the problem
of automatic and interpretable anomaly detection by
means of GPs. That is, we elaborate on why GPs are
well suited for anomaly detection and what the cur-
rent challenges are when applying these probabilistic
models to large-scale production data. To this end,
our contributions are two-fold:
• We introduce GPs as an interpretable model for
anomaly detection.
• We outline challenges that have to be addressed in
order to scale GPs to large-scale production data
This position paper is structured as follows. We
outline related work in Section 2. We introduce GPs
in Section 3, while the concept and rationale for
anomaly detection by means of these statistical data
models and the remaining challenges in that field are
explained in detail in Section 4. We conclude our pa-
per with an outlook on future work in Section 5.
2 RELATED WORK
In the era of Industry 4.0, the field of anomaly de-
tection has become crucially important. As a re-
sult, there is a plethora of classical anomaly detec-
tion algorithms that have been proposed in recent
years such as Z-Score (Domingues et al., 2016), Ma-
halanobis Distance-Based, Empirical Covariance Es-
timation (Pedregosa et al., 2011; Chandola et al.,
2009), Robust Covariance Estimation (Rousseeuw,
1984; Chandola et al., 2009), Subspace-based PCA
Anomaly Detector (Chandola et al., 2009), One-Class
SVM (Sch
¨
olkopf et al., 2001; Pedregosa et al., 2011;
Chandola et al., 2009; Eskin et al., 2002), Isolation
Forest (I-Forest) (Liu et al., 2008; Pedregosa et al.,
2011), Gaussian Mixture Model (Pedregosa et al.,
2011; Chandola et al., 2009; Phua et al., 2010), Deep
Auto-Encoder (Candel et al., 2018; Gong et al., 2019),
Local Outlier Factor (Breunig et al., 2000; Pedregosa
et al., 2011; Chandola et al., 2009; Auslander et al.,
2011), Self-Organizing Maps (Von Birgelen et al.,
2018), Least Squares Anomaly Detector (Tavallaee
et al., 2010), GADPL (Graß et al., 2019), Automata
(Voden
ˇ
carevi
´
c et al., 2011), and k-Nearest Neighbor
(Goldstein and Uchida, 2016; Auslander et al., 2011;
Eskin et al., 2002).
Current approaches (Chalapathy and Chawla,
2019; Zenati et al., 2018; An and Cho, 2015; Zhang
and Chen, 2019; Sabokrou et al., 2018; Suh et al.,
2016; Berkhahn et al., 2019; Li et al., 2019; Kawachi
et al., 2018; Guo et al., 2018; Wang et al., 2019; Dias
et al., 2020) frequently make use of generative models
for anomaly detection, e.g. Variational Autoencoders
(Kingma and Welling, 2014), Generative Adversarial
Networks (Goodfellow et al., 2014), GP Latent Vari-
able Models (Damianou et al., 2016), or Normaliz-
ing Flows (Rezende and Mohamed, 2015), in partic-
ular for sequence data (Bowman et al., 2016). These
models can be trained automatically for the usage of
anomaly detection (M
¨
uller et al., 2020).
While these algorithms are all possible approaches
for anomaly detection, as shown in different surveys
(Goldstein and Uchida, 2016; Phua et al., 2010; Chan-
dola et al., 2009), they are not directly suited for de-
scribing the inherent structure of anomalies, which is
the major focus of this position paper. We choose
GPs (Rasmussen and Williams, 2006) for anomaly
description due to their capability to not only gather
statistical indicators, but deliver the very characteris-
tics of specific anomalous behavior from the data.
For automatically describing the underlying data
characteristics, Lloyd et al. (2014) have proposed the
Automatic Bayesian Covariance Discovery System
that adapts the Compositional Kernel Search Algo-
rithm (Duvenaud et al., 2013) by adding intuitive nat-
ural language descriptions of the function classes de-
scribed by their models. Hwang et al. (2016) fur-
ther expand on those concepts by expanding these
models to discover kernel structures which are able
to explain multiple time series at once. Recently,
the 3CS (Berns et al., 2020) and LARGe algorithms
(Berns and Beecks, 2020) have shown to outperform
the aforementioned approaches in terms of efficiency.
As these methods are all based on GPs, we give a
short introduction of GPs in the following section.
3 GAUSSIAN PROCESSES
A Gaussian Process (GP) (Rasmussen and Williams,
2006) is a stochastic process over random variables
{ f (x) | x ∈ X }, indexed by a set X , where every fi-
nite subset of random variables follows a multivariate
normal distribution. The distribution of a GP is the
joint distribution of all of these random variables and
it is thus a probability distribution over the space of
functions { f : X → R}. A GP is formalized as
f (·) ∼ GP

m(·), k(·, ·)

, (1)
where the mean function m : X → R and the covari-
ance function k : X ×X → R are defined ∀x, x
0
∈ X via
m(x) = E [ f (x)] (2)
k(x, x
0
) = E

( f (x) − m(x)) ·

f (x
0
) − m(x
0
)

(3)
IN4PL 2020 - International Conference on Innovative Intelligent Industrial Production and Logistics
88

Given a finite dataset D = {X , Y } with input X =
{x
i
| x
i
∈ X ∧ 1 ≤ i ≤ n} representing the underlying
index values, such as timestamps or locations, and tar-
get Y = {y
i
| y
i
= f (x
i
), x
i
∈ X } representing the actual
data values, such as sensor values or other complex
measurements, a GP can be used to statistically repre-
sent the dataset D by optimizing the hyperparameters
θ of both mean and covariance function. This opti-
mization is frequently carried out by maximizing the
log marginal likelihood L (Rasmussen and Williams,
2006; Kim and Teh, 2018) of the GP:
L (m, k, θ | D) = −
1
2
·

(y − µ)
T
Σ
−1
(y − µ)+
log | Σ | + nlog(2π)] (4)
As can be seen in Equation 4, the marginalization
of a GP for a given dataset D of n records results in
a mean vector µ ∈ R
n
, and a covariance matrix Σ ∈
R
n×n
which are defined as y[i] = f (x
i
), µ[i] = m(x
i
),
and Σ[i, j] = k(x
i
, x
j
) for 1 ≤ i, j ≤ n, respectively.
How GPs are utilized in the context of anomaly
detection and how the resulting challenges can be
solved are described in the following section.
4 GPs FOR ANOMALY
DETECTION
GP models are robust Bayesian models that intrinsi-
cally prevent overfitting. This sturdiness is a clear
advantage of GP models in dynamic and noisy pro-
duction environments with always changing circum-
stances like evolving processes or sensors modifica-
tions. GP models yield smooth models even when
only a few data points are given, which is particu-
lar important in production, where the batch size is
steadily decreasing.
In addition to recognizing an anomaly, it is impor-
tant to understand and interpret its underlying struc-
ture. Only this makes it possible to rectify the un-
derlying problem. Being interpretable is immanent in
GPs, due to their rigid mathematical structure. Their
covariance functions can induce combinations of var-
ious physically motivated behaviors like trends, pe-
riodicity, differential equations (Lange-Hegermann,
2018, 2020), and change points, and each of these
covariance functions comes with hyperparameters,
which are often interpretable physical constants like
periods. The model prediction can even be disas-
sembled into parts stemming from the individual in-
terpretable parts of the covariance function. All this
physically interpretable structure can be learned from
data, and additionally domain knowledge can be pre-
encoded into the models by experts.
What makes the application of GPs difficult to
large-scale, streaming production data is the super-
cubic computation time complexity of optimizing
GPs. Though efficient solutions exist (Snelson and
Ghahramani, 2007), they do not provide a fast solu-
tion for automatic and interpretable anomaly detec-
tion. For this reason, we identified the following ma-
jor challenges.
4.1 Efficient Algorithms for Large-scale
GP Models
The first challenge is concerned with the develop-
ment of efficient retrieval algorithms for GP models.
This includes the central question of how to efficiently
search and determine suitable covariance functions
for anomaly detection. Instead of applying a greedy
search through the space of possible covariance func-
tions, as done by state-of-the-art algorithms (Duve-
naud et al., 2013; Lloyd et al., 2014; Steinruecken
et al., 2019), one could learn the impact of individual
hyperparameters in order to specifically derive new
covariance functions. In addition to the development
of such intelligent covariance search heuristics, an-
other challenge is to process and infer GP models for
multivariate, event-based sensor data in (near) real
time. This demands for resource-efficient streaming
GP retrieval algorithms that scale to state of the art big
data processing frameworks such as Apache Hadoop,
Spark, and Storm.
4.2 Model Selection for GPs in Anomaly
Detection
Besides the development of efficient algorithms, the
second major challenge lies in the development of
suitable GP model selection approaches. While
prominent model quality estimators, such as the like-
lihood function, tend to prefer complex models con-
taining many hyperparameters, Laplace approxima-
tions enable to capture the model evidence of GP
models more properly. The particular challenge is
thus to couple Laplace approximations with unsuper-
vised GP Latent Variable Models (Damianou et al.,
2016) in order to enable conclusions on the underly-
ing physical processes and individual sensor dimen-
sions.
To sum up, the development of real-time GP
model retrieval and selection algorithms for multivari-
ate data streams that are open-source and are based
on open industry standards are necessary for detect-
ing, analyzing, and understanding anomalies in an
domain-agnostic, automatic, and interpretable man-
ner.
Towards Gaussian Processes for Automatic and Interpretable Anomaly Detection in Industry 4.0
89

5 CONCLUSION
In this paper, we have argued for the application of
GPs for automatic and interpretable anomaly detec-
tion. GPs are robust bayesian machine learning tools,
which enable inference for noisy as well as unreliable
data. GP models yield smooth models even when only
a few data points are given, which is particular useful
in industrial scenarios, where creating those records
is potentially expensive.
Along with the advantages, several challenges
have to be met. In particular, concepts and algorithms
are needed to facilitate efficient GP model retrieval
and selection on large-scale streaming data. We aim
to address these challenges in our future work and to
elaborate our developments in various application do-
mains.
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