# Tailoring Machine Learning for Process Mining

Paolo Ceravolo
Computer Science Department
Università degli Studi di Milano, Italy
<EMAIL_ADDRESS>
Sylvio Barbon Junior
Department of Engineering and Architecture
University of Trieste, Italy
<EMAIL_ADDRESS>
&Ernesto Damiani
Center for Cyber-Physical Systems
Khalifa University, Abu Dhabi, UAE
<EMAIL_ADDRESS>
&Wil van der Aalst
Process and Data Science
RWTH Aachen University
<EMAIL_ADDRESS>

###### Abstract

Machine learning models are routinely integrated into process mining pipelines
to carry out tasks like data transformation, noise reduction, anomaly
detection, classification, and prediction. Often, the design of such models is
based on some _ad-hoc_ assumptions about the corresponding data distributions,
which are not necessarily in accordance with the non-parametric distributions
typically observed with process data. Moreover, the learning procedure they
follow ignores the constraints concurrency imposes to process data. Data
encoding is a key element to smooth the mismatch between these assumptions but
its potential is poorly exploited. In this paper, we argue that a deeper
insight into the issues raised by training machine learning models with
process data is crucial to ground a sound integration of process mining and
machine learning. Our analysis of such issues is aimed at laying the
foundation for a methodology aimed at correctly aligning machine learning with
process mining requirements and stimulating the research to elaborate in this
direction.

_K_ eywords Process Mining $\cdot$ Machine Learning

##  1 Introduction

Process Mining (PM) is a consolidated discipline grounded on data mining and
business process management. The exploitation of traditional PM tasks
(discovery, conformance checking, and enhancement) is today a reality in many
organizations [1, 2]. In the last decade, a wave of new results in artificial
intelligence has triggered the interest of the PM research community in using
supervised or unsupervised Machine Learning (ML) techniques for gaining
insight into business processes and providing advice on how to improve their
inefficiencies.

In today’s practice, ML models are routinely integrated into PM data pipelines
[3] to carry out tasks like data transformation, noise reduction, anomaly
detection, classification, and prediction. For example, ML is playing a key
role in the interface between PM and sensor platforms. Advances in sensing
technologies have made it possible to deploy distributed monitoring platforms
capable of detecting fine-grained events. The granularity gap between these
events and the activities considered by classic PM analysis has often been
bridged using ML models [4, 5] that compute virtual activity logs, a problem
which is also known as log lifting [6]. ML has been proposed as a key
technology to strengthen existing techniques, for example, using trace
clustering to reduce the diversity that a process discovery algorithm must
handle in analyzing an event log [7, 8, 9, 10], to simplify the discovered
models [11, 12, 13], or to support real-time analysis on event streams [14,
15, 16]. ML is adopted to apply predictive models to the executing cases of a
process. This research area, known as predictive process monitoring, exploits
event log data to foresee future events, remaining time, or the outcome of
cases, in support of decision making [17, 18, 19]. Root cause analysis [20]
and data explainability [21] are other tasks that can be applied to event log
data using ML techniques, in order to improve our understanding of a business
process. ML models have also been used in addition to (or in lieu of) classic
linear programming [22] to optimize business processes’ resource consumption
and to provide insights to process re-design [23]. Computational support for
PM tends to converge with the one available for ML models also from the
technology standpoint [24, 25]. This makes their integration seem
straightforward.

In fact, it is not. When PM tasks are mapped to ML tasks, business process-
specific assumptions should drive the construction of training functions and
hyperparameters selection. Some of these assumptions stem from the very nature
of human social systems. For example, it is well known that process variants
are shaped by non-parametric distributions [26]. Quite the contrary, data
normality is beneficial for many ML models, moreover, if the data distribution
is skewed, ML models may be biased toward a particular outcome. In addition,
the ML view on event log data is often oversimplified. The correct encoding of
the procedural nature of event log traces is challenging. Often, the sequence
of executed events is simply captured by a prefix of fixed length. Even more
problematic is encoding concurrency and the interactions constraining the
events in the business process. Encoding event log data into a feature space
compatible with ML algorithms is a critical design choice in other concerns
[27]. It impacts the sample complexity, the data distribution, and the
relevance of the features to put under analysis, for example, to detect
concept drift or to support zero-shot learning [28].

Today, much of the research on integrating ML with PM focuses on developing ML
models to attain high performance in specific business process management
scenarios. Less attention has been paid to designing a general methodology to
select and adapt ML models based on the nature of the PM problem, taking into
account the specific properties of the process data. We argue that, when using
ML models in PM pipelines, it is important to prevent any mismatch between the
assumptions on input data distributions underlying the ML models and the
statistics of the event logs used to feed them [29]. Arbitrarily selecting
algorithms leads to unfair evaluation and sub-optimal solutions. For example,
a given model cannot be compared with another if their implementations
consider different feature spaces [30]. It is also important to make sure that
ML models are exposed to process-specific information, such as the processes’
control-flow constraints. In this paper, we attempt to identify some of the
causes of this mismatch and suggest how to remove them, with the aim of
fostering research on a sound methodology to address the integration between
PM and ML.

We believe that an effort on these aspects must be jointly made by the PM and
Artificial Intelligence research communities. This call to collaboration is
valid in general but particularly in business process management, where data
analysis has to leave the safe harbor of experimental science to sail into the
open sea of decision science. In this paper, we discuss the challenges in a
specific direction, i.e., from PM to ML. More specifically, in Section 2 we
discuss the issues leading to the PM-to-ML mismatch. In Section 3 we introduce
some basic PM notions. In Section 4 we link them to ML principles. Section 5
clarifies the discussion by presenting a couple of samples. Section 6 proposes
research lines for advancing in the direction of a general methodology that
integrates ML models into PM pipelines. Section 7 closes the paper.

##  2 The Issues Landscape

An important problem underlying our discussion is how to take into account
process data specificity in ML model selection and (hyper-) parameter tuning.
Of course, processing event logs poses all the usual challenges of data pre-
processing and preparation. We will not discuss standard data pre-processing
techniques such as outlier removal [31, 32], noise filtering [33, 34], and
missing entries recovery [35] as they can be tackled by current statistical
techniques. Rather, we will focus on issues that are specific to process data,
including their statistical distribution and event concurrency. Indeed,
careless assumptions on the encoding of input data may result in biased models
with reduced generalization capability.

###  2.1 Data Distribution

When choosing an ML model for a PM task, it is tempting to assume that the
process data fed to the model will follow a normal distribution. Indeed, data
normality is beneficial for many types of ML models. Models like Gaussian,
naive Bayes, logistic and linear regression explicitly rely on the assumption
that the data distribution is bi-variate or multivariate normal. Many
phenomena of interest for business process analysis, such as the duration of
some activities, are known to follow normal or log-normal distributions
111See, for instance, the “lunch break” duration distributions at
https://www.statista.com/statistics/995991/distribution-of-lunch-breaks-by-
length-in-europe/. For other PM data, however, assuming normality is not
always advisable. For example, process variants are specific activity
sequences that occur through a process from start to end. Variants’ occurrence
in an activity log is typically following a non-parametric trend that complies
with the Pareto principle [26]. A normal distribution cannot always be assured
also for the pairwise dependency relationship between activities, a key
statistical information exploited by process discovery algorithms [36].
Indeed, in this case, the normality assumption has been verified for some
event logs, including some popular benchmarks we will discuss in Section 5
(the “Road traffic fines” [37] and “Receipt phase of an environmental permit
application process” [38]). However, the normality of dependencies in less
regular, “spaghetti” like, processes is not observed, as in the “BPI Challenge
2015 Municipality 1”[39]. There are reasons to believe that dependencies in
loosely specified logs may follow some power-law trend as well, and require
careful parameter fitting in statistical analysis. Imbalanced data sets or
non-stationary environments may also cause serious difficulties. For example,
if the training data is skewed towards a particular class or outcome, the
model may be more likely to predict that class or outcome even when it is not
the most likely one. Independent component analysis [40] provides ways to
reveal Gaussianity and non-Gaussianity. Of course, non-normal distributions
can be transformed to normal ones using Box-Cox transformations [41], and
unbalanced data sets can be balanced [42, 43] but, as we shall see, such data
transformations should be applied with caution, as they have consequences on
the performance of the models.

In any case, PM data regarding distributions of variants cannot be expected to
always follow a Gaussian behavior, demanding estimation techniques to sample
from the sequential process underlying log generation. _Markov Chain Monte
Carlo_(MCMC) techniques are sometimes used for sampling from an unknown
probability distribution (for instance, the distribution of variants) by using
data to construct a Markov chain whose equilibrium distribution approximates
the unknown one. MCMC techniques can be combined with Kalman filtering[44] to
control uncertainty. Of course, an explicit estimate of the data distribution
may not even be necessary. Some ML models work well also in the case of non-
normally distributed data. Simple yet effective ML models like decision trees
and random forests do not assume any normality and work reasonably well on raw
event data. Also, linear regression is statistically effective if the model
errors are Gaussian, an assumption less stringent for process data than the
normality of the entire data set. Kernel methods, e.g., Gaussian processes and
support vector machines, provide flexible models that are practical to work
with but require proper hyperparameter variables to fit the data.

###  2.2 Concurrency

Another key attention point is concurrency. How to use ML to predict the
behavior of highly concurrent systems and processes is still an open problem,
and the research done in the AI community has only scratched its surface (see
[45] for a recent review). Most ML approaches view event logs as merely
sequential data [46], rather than sequential manifestations of a concurrent
system. This may lead to under-sampling the log space and to insufficient
training to handle apparently out-of-order event sequences [47]. To address
this issue, it is important to provide ML models with control-flow information
about the iterative or concurrent execution of tasks as additional context
alongside event logs. One approach that has been explored is the use of Bi-
directional Long-Short Term Memory (BiLSTM) architectures. Thapa et al. [48]
leveraged BiLSTM to detect concurrent human activities in a smart home
environment. Additionally, Thapa et al. [49] adapted the LSTM algorithm into a
synchronous algorithm called sync-LSTM, enabling the model to handle multiple
parallel input sequences and generate multiple synchronized output sequences.
The field of predicting the behavior of highly concurrent systems using ML is
rapidly evolving, as indicated by the recent survey conducted by Neu et al.
[50]. Researchers are actively exploring new techniques and methodologies to
improve the understanding and prediction of concurrency in various domains.

###  2.3 Non-stationary Behaviour

Even when the process data distributions can be fitted precisely, running
processes, especially the ones involving resources that learn and age like
people and equipment, change over time. This gives rise to non-stationary
behavior. This problem is a critical one since ML models’ learning capacity
decreases under non-stationary conditions [16]. Concept drift detection
techniques are therefore required. In traditional data mining applications,
concept drift is identified when, at two separate points in time, a concept,
i.e., the relation between a data instance and its associated class, changes
[51]. In PM, many aspects of drift should be carefully monitored, including
the appropriateness of the event trace with respect to the model, the
dependency relationship between activities, and the interdependence between
the activities and the available resources or cycle time. Each aspect should
be appropriately encoded and monitored using statistical analysis [52].

###  2.4 Zero-shot Learning

A related topic is using ML models to identify solutions never observed during
training, the so-called zero-shot learning [53]. There are several zero-shot
learning approaches, but a commonality is that unstructured auxiliary
information is encoded during the training process instead of using explicit
labels. The training process aims to learn to connect new input elements to
encodings that have the greatest similarity in terms of auxiliary information.
In this way, the system can propose an outcome never observed during the
training stage. Zero-shot is relevant in PM when the availability of labeled
process data is limited, as the process may be recently developed, unused, or
its outcomes inaccessible. In these situations, relying on historical
observations to guide learning tasks is insufficient or erroneous. This
scarcity has drawn the attention of the PM community to _contrastive learning_
, a manner of unsupervised learning that learns representations by contrasting
positive and negative pairs. Graph-related contrastive learning methods apply
this notion to all types of graph data. Some popular unsupervised
representation learning methods imply the idea of contrastive learning. For
instance, DeepWalk [54] and node2vec [55] generate Markov chains of nodes
based on random walking on graphs, forcing the neighboring nodes of a graph to
have similar representations. More recent proposals such as DGI [56] and
InfoGraph [57] combine contrastive learning with ordinary supervised training
to maximize the mutual information of node and graph levels.

Much work is also being done on generative engines for logs based on
likelihood-based models, like auto-encoders and Generative Adversarial
Networks (GANs)[58]. However, ordinary GANs show some limitations when applied
to generate “clean” process data, where low confidence variants are due to
failures of the monitoring context rather than to adversarial constructions
[59]. In addition, the GANs’ _objective function_ , i.e. the difference
between the generated and the original distribution of traces in the event
log, is not always suitable for evaluating the quality of the generated
process variants, and even less for comparing different generators.
Performance measures should be used instead, and the trained algorithms should
be able to provide an answer with different information details, for example,
predicting a performance result knowing or not the availability of resources
currently in use.

###  2.5 Data Encoding

Supervised ML algorithms are trained on collections of examples, each encoded
as a vector in a multidimensional feature space. An appropriate encoding
method can reduce the sample complexity and reduce the space or time
complexity of the model [27]. In PM, even more, than selecting individual
features, it is important to capture the interconnections between the
different process dimensions. The event logs analyzed in PM contain
information from several complementary dimensions, such as event data,
executing traces, resource consumption, and cycle time. Each event can be
described as a multidimensional object, but its value for the process
execution lies in the interdependence with the other events composing the
process case instance, the resources available in the system, and the temporal
limits constraining the case, which in turn depend on the other cases
executed, executing, or to be executed in the system. Therefore, capturing the
constraints due to the alternative, optional or mandatory dependency between
events is crucial in PM. Encoding methods should also identify features
subject to concept drift. Extracting insights from this type of functional
data is not straightforward; covariance control[60] is needed to take into
account the hidden relationships between the different dimensions.

Despite all this, little effort has been spent by the PM community to study
the impact of encoding methods on the performance of PM pipelines. Only a few
comparative studies are available [61, 27, 62, 63]. Basic techniques, such as
the one-hot encoding scheme [64] or frequency-based encoding [9], are often
adopted. For numerical attributes, general statistics have been used, such as
average, maximum, minimum, and sum [18]. The $k$-gram encoding schema [8] is
also quite popular. Each activity in the trace is represented as the sequence
of $k$ activities executed to reach it. As an alternative, arrays encoding
traces as the frequency of their activities at each position have been
proposed [65]. These encoding techniques can incorporate some control-flow
information, but cannot fully account for concurrency. To better capture
dependency between activities, techniques borrowed from other domains have
been proposed, including text mining [66, 67] and graph embedding [68, 69].
Graph embedding methods emerged from the necessity of representing graphs as
low-dimensional vectors to be exploited by downstream ML models. These methods
rely themselves on ML models (usually, supervised learners) to compute highly
informative but low-dimensional vectors of fixed length [70]. When applied to
event logs encoding, such methods outperform the others, at the cost of higher
time complexity and loss of transparency, as the resulting vectors are
organized in a latent space losing any reference to the event log attributes
or their statistical properties [71]. In any case, the representation of the
control-flow is purely sequential and concurrency is not captured by these
methods too. Recently, emerging attention on techniques for encoding control-
flow information into a feature space is observed, for example by representing
the degree of parallelism or optionality of activities [72, 73]. Another trend
is aimed at constructing multi-perspective views of traces, representing the
data-flow and control-flow into the same encoding [74, 75]. However, the
application of these methods is still limited.

Generally speaking, the encoding procedures used to map PM data to ML models
are not documented enough in the PM literature. Sometimes, the feature space
selected is not explicitly presented, the steps followed to encode data are
not well specified, or the adopted code is not shared. _Ablation studies_ ,
removing parts of the data representation and studying the removal’s impact on
performance, are still the exception rather than the norm. We argue that the
formalization of the encoding procedure allows explaining this key design
choice to be justified by the specific analytical goals and the assumptions
applying to the algorithms considered. We will propose such a formalization in
Section 4.

##  3 Basic Notions in PM

To make this paper self-contained, in this section we recall some of the basic
concepts of PM. An event log is a collection of events generated in a temporal
sequence and stored as tuples, i.e., recorded values from a set of attributes.
Events are aggregated by case, i.e., the end-to-end execution of a business
process. For the sake of classification, all cases following the same trace,
i.e., performing the same sequence of business process activities, can be
considered equal as they belong to the same process variant.

######  Definition 1 (Event, Attribute)

Let $\Sigma$ be the event universe, i.e., the set of all possible event
identifiers; $\Sigma^{*}$ denotes the set of all finite sequences over
$\Sigma$. Events have various attributes, such as timestamp, activity,
resource, associated cost, and others. Let $\mathcal{AN}$ be the set of
attribute names. For any event $\textit{e}\in\Sigma$ and attribute
$\textsc{a}\in\mathcal{AN}$, the function $\\#_{\textsc{a}}(\textit{e})$
returns the value of the attribute a for event e.

The set of possible values of each attribute is restricted to a domain. For
example, $\\#_{\textsc{activity}}:\Sigma\rightarrow\mathcal{A}$, where
$\mathcal{A}$ is the set of the legal activities of a business process, e.g.
$\mathcal{A}=\\{a,b,c,d,e\\}$. If e does not contain the attribute value for
some $\textsc{a}\in\mathcal{AN}$, then $\\#_{\textsc{a}}(\textit{e})=\bot$. It
follows that an event can also be viewed as a tuple of attribute-value pairs
$e=(\mathcal{A}_{1},...,\mathcal{A}_{m})$, where $m$ is the cardinality of
$\mathcal{AN}$.

######  Definition 2 (Sequence, Sub-sequence)

In a sequence of events $\sigma\in\Sigma^{\ast}$, each event appears only once
and time is non-decreasing, i.e., for
$1\leq\textit{i}\leq\textit{j}\leq|\sigma|:\\#_{\textsc{timestamp}}(e_{i})\leq\\#_{\textsc{timestamp}}(e_{j})$.
Thus $\langle e_{1},e_{2},e_{3}\rangle$ denotes three subsequent events. A
sequence can also be denoted as a function generating the corresponding event
for each position in the sequence: $\sigma(i\rightarrow n)\mapsto\langle
e_{i},...,e_{n}\rangle$, with $e_{n}$ the last event of a sequence. In this
way, we can define a sub-sequence as a sequence $\sigma(i\rightarrow j)$ where
$0\leq i<j<n$.

######  Definition 3 (Case, Event Log)

Let $\mathcal{C}$ be the case universe, that is, the set of all possible
identifiers of a business case execution. $\mathcal{C}$ is the domain of an
attribute $\\#_{\textsc{case}}\in\mathcal{AN}$. We denote a case
$c\in\mathcal{C}$ as $\langle e_{1},e_{2},e_{3}\rangle_{c}$, meaning that all
events are in a sequence and share the same case. For a case $\langle
e_{1},e_{2},e_{3}\rangle_{c}$ we have $\\#_{\textsc{case}}$($e_{1}$) =
$\\#_{\textsc{case}}$($e_{2}$) = $\\#_{\textsc{case}}$($e_{3}$) = $c$. An
event log $L$ is a set of cases $L\subseteq\Sigma^{*}$ where each event
appears only once in the log, i.e., for any two different cases, the
intersection of their events is empty. When the case identifier is not used as
a grouping attribute, an event log $\hat{L}$ can be simply viewed as a set of
events, thus $\hat{L}\subseteq\Sigma$.

######  Definition 4 (Variant, Event Log)

The cases $c_{1}$ and $c_{2}$ follow the same variant if $\langle
e_{1},e_{2},e_{3}\rangle_{c_{1}}$ and $\langle
e_{4},e_{5},e_{6}\rangle_{c_{2}}$ have the same sequence of activities, e.g.
$\\#_{\textsc{activity}}$($e_{1}$) = $\\#_{\textsc{activity}}$($e_{4}$) = $a$,
$\\#_{\textsc{activity}}$($e_{2}$) = $\\#_{\textsc{activity}}$($e_{5}$) = $b$,
$\\#_{\textsc{activity}}$($e_{3}$) = $\\#_{\textsc{activity}}$($e_{6}$) = $a$.
We call this sequence a trace. This implies an event log can also be viewed as
a multi-set of traces. We denote an event log as a multi-set by writing
$\overline{L}=[\langle a,b,c\rangle^{3},\langle a,b,a\rangle^{11},\langle
a,c,b,a\rangle^{20}]$. The superscript number of a trace details the number of
cases following this variant. For example, $\langle a,b,a\rangle^{11}$ means
we have a variant with $11$ cases following the trace $\langle a,b,a\rangle$.

##  4 A Formalisation of PM Data Encoding

Despite the variety of encoding methods discussed in Section 2, we argue that
available approaches fail to capture key process-level information such as the
interplay between cases, or between activity execution and availability of
resources. Most of the encoding methods in use today focus on the control-
flow, according to an inter-case view. Methods focusing on the intra-case view
have been proposed but are rarely applied [76]. Similarly, proposals for
encoding the data-flow [77] are available in the literature, but never adopted
in comparative studies or surveys. Another recent trend is stressing the need
of capturing constraints connected to concurrency [72, 73]. In this section,
we discuss in detail how PM data is encoded to suit ML models’ training
procedures. For the sake of space, we limit our discussion to supervised
learning, probably the most widely applied ML approach. Generally speaking,
supervised techniques train models to compute functions
$f:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d^{\prime}}$ where the input is a
$d$-dimensional vector $\mathbf{x}$ and the output is a
$d^{\prime}$-dimensional vector $\mathbf{y}$. Each dimension is a measurable
piece of data, a.k.a feature or attribute. For popular ML tasks, the output is
mono-dimensional. In regression, the output is a real-valued scalar value,
while in classification, the output is a natural number indexing a “class”.
However, nothing prevents having multidimensional vectors in output. In
structured learning, input and output may be a structure like a block matrix,
divided into sub-matrices to represent algebraic entities such as graphs,
tensors, etc. The training process to approximate $f$ requires a set of
examples
$\\{(\mathbf{x}_{1},\mathbf{y}_{1}),...,(\mathbf{x}_{n},\mathbf{y}_{n})\\}$
where inputs and outputs are paired. We can then define this training set as
an example matrix
$\mathbf{X}:=[\mathbf{x}_{1},...,\mathbf{x}_{n}]^{\top}\in\mathbb{R}^{n\times
d}$ and a label matrix
$\mathbf{Y}:=[\mathbf{y}_{1},...,\mathbf{y}_{n}]^{\top}\in\mathbb{R}^{n\times
d^{\prime}}$, given by the number $n$ of vectors and the number $d$ of
dimensions in the vector space.

In their original format, PM log entries do not belong to a vector space. This
is because the events in an event log are grouped by case and this grouping is
essential to keep a connection with business process execution.

Our goal here is to formalize the procedure to encode the cases into vectors
in a way that can be used as a template to describe the specific encoding
chosen for a PM application. Our starting point is $\hat{L}\subseteq\Sigma$, a
log view as a set of event identifiers. This representation can be mapped into
a vector space $\mathbf{X}$ by applying a suitable _transformation function_
grouping event by case and returning vectors of size equal to or less than the
event size.

######  Definition 5 (Encoding function)

Given an event log $\hat{L}\subseteq\Sigma$, an _encoding function_
$\Gamma:\Sigma\rightarrow\mathbf{X}^{n\times d}$ represents $\hat{L}$ in the
vector space $\mathbf{X}$. The encoding function $\Gamma$ is valid if it
defines a transformation where two elements of $\Sigma$, $e_{i}$ and $e_{j}$
are aggregated on the same element $\mathbf{x}\in\mathbb{R}^{d}$ if
$\\#_{\textsc{case}}$($e_{i}$) = $\\#_{\textsc{case}}$($e_{j}$), with
$n\leq|\mathcal{C}|$, i.e. the vectors in $\mathbf{X}$ are a subset of the
cases in $\mathcal{C}$.

We propose a canonical representation of $\Gamma$ as a composition of a
_filtering function_ $\pi$, a _dimensioning function_ $\rho$, a _grouping
function_ $\eta$, and a _valuation function_ $\nu$, i.e.,
$\Gamma=\nu\circ\eta\circ\rho\circ\pi$. One or more of these components can
implement the identity function with null effects.

In particular, $\pi:\Sigma\rightarrow\Sigma_{\alpha}$ imposes a condition on
the events’ attributes or the attributes’ values, $\forall
e\in\hat{L}\land\textsc{a}\in\mathcal{AN}:P(\\#_{\textsc{a}}(e))$, where $P$
is a predicate, thus $|\Sigma_{\alpha}|\leq|\Sigma|$. For example, filtering
the events by their timestamp $\forall e\in\hat{L}:$ YYYY-MM-DD
$\geq\\#_{\textsc{timestamp}}(e)\leq$ YYYY-MM-DD. The function
$\rho:\Sigma_{\alpha}\rightarrow D$ defines the dimensions of the vector
space, creating new dimensions based on a range of values in the original
dimensions or, less commonly, grouping multiple dimensions into a single one.
Often, the set $D$ is the union of multiple attribute domains, i.e.
$D=\mathcal{A}_{k=1}\cup\mathcal{A}_{k=2}\cup\cdots\mathcal{A}_{k=l}$. The
function $\eta:\Sigma_{\alpha}\rightarrow\mathbf{X}^{n\times d}_{\alpha}$,
with $d=|D|$, assigns to $\mathbf{X}_{\alpha}$ the values of the attributes in
$e$ and groups events by case so that
$\forall\mathbf{x}\forall\textsc{a}_{k}:\mathbf{x}_{i,j}=\\#_{\textsc{a}_{k}}(\textit{e})\iff\\#_{\textsc{a}_{k}}(\textit{e})=D_{j}\land\\#_{\textsc{case}}(e)=c_{i}$.
The number of elements in the vector space equals the number of cases to
include in the example matrix, thus $n\leq|\mathcal{C}|$. Because the sets
$\Sigma_{\alpha}$ and $D$ can be view as columnar matrices
$M_{\Sigma_{\alpha}}^{n\times 1}$ and $M_{D}^{d\times 1}$, the size of
$\mathbf{X}_{\alpha}$ is equal to $M_{\Sigma_{\alpha}}\times M_{D}^{\top}$,
i.e. the set of events we selected with $\pi$ is multiplied by the dimensions
we identified with $\rho$. It is worth mentioning that, when grouping is
applied, each vector component becomes an array of attribute values rather
than a single value. The function $\nu$ aims at transforming these arrays of
attribute values into real-valued scalar values. We define
$\nu:\mathbf{X}^{n\times d}_{\alpha}\rightarrow\mathbf{X}^{n\times d}$ to
clarify the components of the two matrices are valuated differently.

For example, the basic one-hot encoding schema corresponds to a null $\pi$, a
$\rho$ with $D=\bigcup_{k=1}^{l}\mathcal{A}_{k}$, an $\eta$ for grouping the
events of a same case, and a $\nu:\mathbf{X}^{n\times
d}_{\alpha}\rightarrow\\{0,1\\}^{n\times d}$, returning $\mathbf{x}_{i,j}=1$
if at least a value $\\#_{\textsc{a}_{k}}(\textit{e})=D_{j}$ is observed for
the case $\\#_{\textsc{case}}(e)=c_{i}$, and $0$ if not. The popular _activity
profile_ schema [7] encodes an event log into a vector of activity values by
simply counting all events of a case that include that activity. The encoding
function maps the events in $\hat{L}$ into $\mathbf{X}$ by executing the four
canonical transformations as follows. First, it verifies to consider only
events associated with activity values $\forall
e\in\Sigma:\\#_{\textsc{activity}}(\textit{e})\neq\bot$. Then it defines the
dimensions of $\mathbf{X}$ with $\rho$ so that $D=\mathcal{A}$, where
$\mathcal{A}$ is the set of legal business process activities. Third, it
aggregates the data by case with $\eta$. Finally, it performs the evaluation
with $\nu$, assigning the count of the components in $\mathbf{x}_{i,j}$ for
each case $c_{i}$. For instance, the log $\overline{L}=[\langle
a,b,c\rangle^{3},\langle a,b,a\rangle^{11},\langle a,c,b,a\rangle^{20}]$, is
transformed in the first matrix in 1 with $\pi$, in the second matrix with
$\rho$, in the third matrix in with $\eta$, to finally get the fourth matrix
in 1 with $\nu$.

$\begin{bmatrix}e_{1}\\\ e_{2}\\\ e_{3}\\\ e_{4}\\\ e_{5}\\\
...\end{bmatrix}\begin{bmatrix}a\\\ b\\\ c\\\
\end{bmatrix}\begin{bmatrix}a&b&c\\\ a&b&c\\\ a&b&c\\\ [a,a]&b&\bot\\\
[a,a]&b&\bot\\\ ...\end{bmatrix}\begin{bmatrix}1&1&1\\\ 1&1&1\\\ 1&1&1\\\
2&1&0\\\ 2&1&0\\\ ...\end{bmatrix}$ (1)

We believe that if the PM community would get used to clarifying the
definition of the following functions when defining an encoding procedure, the
literature will benefit in terms of the comparability of the results. For
example, a data-fow approach will require clarifying the contribution of the
different dimensions in encoding cases. An intra-case approach will require
modifying the $\eta$ function to encode multiple cases into a single vector.

Cases | Number of Variants | Coverage of Cases
---|---|---
56482 | 1 | 37,6%
102853 | 2 | 68,4%
132758 | 4 | 88,3%
142926 | 7 | 95,0%
148887 | 17 | 99,0%
150270 | 131 | 99,9%
150370 | 231 | 100,0%
Table 1: Managing Road Traffic Fines Event Log

((a)) Unbalanced event log

((b)) Balanced event log

Figure 1: Two decision trees generated from our sample event log. In 1(a) the
data in input conforms to the case distribution observed in the event log. As
a consequence, the most frequent variants take the lion’s share and the
numeric feature amount decides multiple split points. In 1(b) data is balanced
oversampling those variants with low occurrence. The split points in the tree
use categorical features only. Decision tree is an example of an algorithm
significantly affected by uncritical training using the case distribution of
event logs.

##  5 Illustrative Examples

We will now use two examples to illustrate the concepts introduced above.

The first example refers to the real-live event log of road traffic fines
[37]. The events captured in the event log include creating a fine notice,
recording the penalty amount, verifying if the payment is received,
registering an appeal to the prefecture, and others. The reader interested in
more details is referred to [78]. As illustrated in Table 1, the occurrence of
trace variants follows a Pareto distribution with only $4$ variants covering
more than 88% of the recorded cases and with $100$ variants that have a single
occurrence. The most occurring variant is $\langle
Create~{}Fine,~{}Send~{}Fine,~{}Insert~{}Fine~{}Notification,~{}Add~{}Penalty,$
$~{}Send~{}for~{}Credit~{}Collection\rangle^{56482}$, the second is $\langle
Create~{}Fine,~{}Payment\rangle^{46371}$, the third is $\langle
Create~{}Fine,~{}Send~{}Fine\rangle^{20385}$, and so on.

Let us now try to develop predictive analytics on this event log. For example,
we could ask ourselves why certain cases exhibit a duration that is
significantly longer than others. To study the problem, we are interested in
searching for patterns correlated to long duration. Using encoding, we can
represent the cases in the event log as vectors composed of categorical data,
such as the executed activities, and of numerical data such as the number of
penalties and the trace duration222The methods used for encoding the event log
in a vector space are available in the PM4PY library
https://pm4py.fit.fraunhofer.de/documentation#decision-trees. A decision tree
can then be used to highlight the factors influencing case duration. We
express it as a simple binary problem: being below or above a threshold of 200
days. Figure 1 illustrates the results we obtain. Figure 1(a) presents a
decision tree conforming to the case distribution observed in the event log.
The entire set of cases in $L$ is encoded in $\mathcal{X}$. As a consequence,
the most frequent variants take the lion’s share of the examples used to train
the decision tree. Figure 1(b) presents the decision tree obtained by
balancing the case distribution among variants, oversampling those variants
with low occurrence. This is, for example, achieved by creating $\mathcal{X}$
taking an equal number of occurrences to the traces in $L$.

Because the split points of the tree are chosen to best separate examples into
two groups with minimum mixing, the cases with low occurrence tend to be
ignored. Indeed, the tree in Figure 1(a) relies on the numeric feature amount
to decide on multiple split points. On the contrary, the tree in Figure 1(b)
defines the split points using categorical features only. This is due to the
fact that the variants not associated with a penalty amount were quite rare,
and by increasing their representation for balancing the data set we prevented
the algorithm to use the penalty amount as a discrimination feature.

((a))

((b))

Figure 2: (a) The Heuristic Miner Algorithm [79] was used to discover a model
from the “Road Traffic Fines” [37] event log. The discovered model specifies
alternative routes that can be followed to complete the process. In
particular, executing Payment or Send Fine implies the following alternative
paths. (b) The Heuristic Miner Algorithm [79] is used to discover a model from
the “Artificial Patient Treatment” [80] event log. The discovered model
specifies that Blood test, X-ray scan, and Physical test are executed in
parallel. Any order can be followed in executing these activities.

It is important to note that, in general, we cannot say if proactive balancing
is better than using data as they are, and even which is the correct balancing
factor to be applied. The strategy to be preferred strongly depends on our
goal. If we want to analyse an event log in order to identify procedures that
can be automated and learn the decision rule to be used, our interest is in
the frequent behaviour. The real distribution of the event log, or even a
distribution pruned from rare examples [26], must address the learning
procedure we adopt. If our goal is anomaly detection [81] or root cause
analysis [82] rare examples have to be represented.

Our next example is related to the need of capturing concurrency (Section 2).
While cases included in an event log are described as sequences of activities,
the behaviour they describe should be interpreted differently based on the
model that generated them. To capture control-flow behaviour, one needs to
encode the dependency relationships in event logs. By executing the Heuristic
Miner algorithm [79] on the “Road Traffic Fines” [37] event log, we observe
alternative paths can be followed to complete the process. If a case includes
the execution of the Payment activity, it will not include Send Fine and the
following activities. The same algorithm applied to the “Artificial Patient
Treatment” [80] will reveal the concurrent execution of the Blood test, X-ray
scan, and Physical test activities. All these activities are required to
complete the diagnostic stage, except for X-ray scan, which may be skipped,
but the order of execution is not relevant. Thanks to process models, PM
techniques do consider concurrency. Two sequences $\langle a,b,c\rangle$ and
$\langle a,c,b\rangle$ can have the same conformance to the model if the model
describes $b$ and $c$ as concurrent activities, while the conformance value
will be different if $b$ and $c$ are in sequence or relate to alternative
paths. Unfortunately, most ML models view event logs merely as sequential
data. When cases get encoded into a vector space, the inference the ML model
can produce is based on the distance in this space. The distance between
$\langle a,b,c\rangle$, and $\langle a,c,b\rangle$ is accounted in the same
way in the vector space, and we cannot differentiate between the sequences
based on the reference process model. This limitation impairs capturing
concurrent behaviour that is not detected by simply matching the two
sequences. In terms of our example, an ML procedure could effectively predict
the lead time of a case knowing that the Payment activity was executed.
Training an ML algorithm to predict the conformance to the diagnostic protocol
of a delivered treatment is more complex, and will require a higher amount of
training data, as the ML model needs to incorporate examples on the
equivalence of the different orders of execution of the Blood test, X-ray
scan, and Physical test activities. Encoding this equivalence in vector space
spaces, for example, defining suitable pictograms to feed a CNN, is still an
open challenge.

##  6 Toward an Integrated Methodology

Guided by the above considerations about encoding, we will now outline the
strategy to be used to properly integrate PM and ML. In the previous sections,
we argued that when PM tasks are mapped to ML tasks, PM-specific assumptions
should drive the construction of training functions and hyper-parameters
selection. Simple ML classification and regression algorithms model the data
by a single Gaussian grounded on mean and co-variance. On the other hand,
kernel methods like Gaussian Processes and Support Vector Machines, have
opened the possibility of flexible models that are practical to work with, but
require non-trivial hyper-parameter tuning to fit behavioural data[83].

Figure 3 provides a synoptic view of mapping PM tasks to ML ones.

Figure 3: From task to Task, an overview of PM and ML relationship

As an example of non-trivial mapping, let us consider the non-linear
relationship between data samples and the expected outcomes addressed by
robust ML algorithms with adjusted hyper-parameters. At this point, linear
projections as PCA are not effective as t-SNE visualisation [84] to obtain
insights from the data. Other challenges with moderate difficulty are related
to label availability and imbalanced scenarios [81]. In this case, semi-
supervised ML techniques and generative models can tackle the label issue, as
well as sampling or synthesising methods are the second ones. Problems related
to data quality, in which the difficulty is to build an approximation to have
a proper data distribution accentuated, can be solved by enlarging the
training data and by a proper tuning of the ML algorithm. Alternatively, the
training process can be enriched using generative models [85]. To handle the
difficulties outlined in Section 1, when using non-pictorial traces
representation “process-friendly” GANs can be considered, like Sequence GANs
(SGANs), in which the adversarial samples are designed from discrete
sequences, like events. The application of GANs is not limited to data
augmentation, as it can be used also for improving data quality for process
model generalisation [85]. Preliminary results are available on using GAN-
generated data to improve predictive tasks (e.g., lead time of incomplete
cases) under an adversarial framework [86].

Coming from non-stationary process behaviour, sampling methods are a promising
way to reduce the impact of non-stationary distributions of event log data
[87]. After bringing the data to at least a near-stationary behaviour, the
business process can naturally change its pattern over time, leading to a
burdensome problem called concept drift [88, 89, 52, 16]. In dealing with this
problem, a significant part of the PM community has focused on detecting and
managing its onset. Regardless of the success of these attempts, we still
consider this problem an open issue, since the event data stream is modelled
as a complete trace stream, known from the start to the end activity. In
reality, the drift onset occurs at an arbitrary position of the event stream,
well before the endpoint is reached and the rest of the trace is known. Some
researchers are addressing this information deficiency by using statistical
adaptations based on the Hoeffding Bounds [90]. In principle, it is possible
to rely on statistical assumptions about the confidence interval of the data
to make a decision on the drift onset. In other words, it is possible to
create ML models and perform predictions supported by an approximated
conjecture about the future, obtained from the available event log data. The
use of “stateful” ML models with memory, in particular, Deep Learners based on
the LSTM architecture, could enable handling drifts. However, this kind of
challenge demands experienced ML practitioners and a robust computational
structure.

###  6.1 Hyper-parameter Tuning

Once a class of ML models has been chosen, hyper-parameter tuning must be
performed to instantiate the ML model that delivers the desired accuracy (and
possibly some required non-functional properties, like explainability).
Searching the model space by trial and error can be burdensome. Automated
Machine Learning (AutoML) is a reasonable alternative to tackle these problems
grounded on sharing previous knowledge for similar tasks. AutoML can help to
handle the classification problems called Zero-shot [91, 92] or Cold-start
[93], for which little context information (and even the complete list of
classes) may not be available at the start of the training, by taking
advantage of meta-features and information on similar models, akin to how
human experts start an old-fashioned search for desirable models driven by
their experience on related tasks [94]. Some PM research works based on AutoML
discuss how to find a suitable PM pipeline by recommending steps. For example,
[95] proposed a solution to suggest the encoding method, since the higher
number of methods might lead to a tricky selection. Furthermore, there are
encoding methods able to fit particular data. It is remarkable that
traditional process mining tasks can be leveraged when matched with
intelligent decision support approaches.

###  6.2 Final Recommendations

In this final section, we present a set of recommendations that aim to be
valuable for both PM practitioners and researchers.

####  6.2.1 RECOMMENDATION 1: Choose data representation carefully

When working with PM data structures, it is crucial to carefully translate
them into a metric feature space that can be manipulated by ML algorithms.
Additionally, it is important to preserve context information, such as
control-flow constraints, which are essential for process analysis. The choice
of encoding techniques should align with problem-specific goals and
constraints.

####  6.2.2 RECOMMENDATION 2: Fit the data distributions

PM often deals with non-Gaussian, non-stationary distributions. To achieve
optimal performance in production, it is advisable to estimate the data
distribution instead of relying on the best Gaussian mix approximation.
Building training sets interactively poses a significant challenge in PM.
Leveraging ML approaches such as AutoML and Active Learning can help reduce
the manual burden and improve the process.

####  6.2.3 RECOMMENDATION 3: Do not assume the availability of a labelled
training set

In business process environments, obtaining pre-existing labelled training
sets for PM tasks is uncommon. Constructing a training set by correctly
sampling the data space is essential, particularly due to the high diversity
of process execution conditions in PM tasks.

####  6.2.4 RECOMMENDATION 4: Consider zero-shot learning

During the training of ML models, the complete set of possible outcomes (co-
domain of $f$) may only be partially known. For instance, in process
optimisation, the cost of certain sequences may not be available at the time
of training the regression model. It is essential to assess the completeness
of the available information when formulating the problem statement to ensure
the quality of model inference.

####  6.2.5 RECOMMENDATION 5: Ensure minimum ML quality at an early stage via
constraints

As the estimation of data distribution converges over time, an extended
convergence period is unacceptable as it results in a high model error during
training. It is possible to impose control flow constraints on ML models when
they are known in advance based on domain requirements and regulations.

####  6.2.6 RECOMMENDATION 6: Incorporate domain knowledge

Domain knowledge plays a critical role in effective PM. Integrating domain-
specific information and constraints into ML models can significantly enhance
their performance and interpretability. It is important to actively involve
domain experts in the feature engineering and model validation processes.

####  6.2.7 RECOMMENDATION 7: Evaluate model interpretability

PM tasks often require interpretable models to gain insights into process
behaviour and make informed decisions. It is essential to evaluate the
interpretability of ML models and choose algorithms that provide transparent
explanations of their predictions. This becomes particularly crucial when
dealing with critical processes or compliance and regulatory requirements.

####  6.2.8 RECOMMENDATION 8: Continuously monitor and update ML models

Process environments are dynamic, and changes over time can impact the
performance of ML models. Establishing a framework for monitoring and
evaluation allows the assessment of models’ performance and facilitates their
timely updates as needed. Continuous learning and retraining of models ensure
their accuracy and relevance in evolving process scenarios.

####  6.2.9 RECOMMENDATION 9: Share knowledge and best practices

Promote knowledge sharing and collaboration within the PM community. Encourage
the dissemination of successful case studies, research findings, and best
practices to foster learning and advancement in the field. Engage in
conferences, workshops, and online forums to connect with fellow practitioners
and researchers and stay updated with the latest developments in PM.

By following these recommendations, PM practitioners and researchers can
improve the effectiveness and efficiency of process mining applications,
enabling better process understanding, optimisation, and decision-making.

##  7 Conclusions

The growing use of ML methods in PM necessitates a robust and comprehensive
methodology for integrating these algorithmic techniques. This paper aimed to
address the challenges associated with the ML/PM mapping and identify the
fundamental principles for establishing a methodological foundation in this
field. Through the analysis conducted in this study, we have provided a set of
recommendations that can guide practitioners and researchers in effectively
applying ML to PM tasks. These recommendations encompass various aspects of
the PM process, from data representation to model evaluation and monitoring.
By following these recommendations, PM practitioners and researchers can
enhance the effectiveness and efficiency of their ML-driven process mining
applications. It is important to acknowledge that the field of ML in PM is
constantly evolving, and new challenges and opportunities will continue to
arise. As such, ongoing research and collaboration among practitioners and
researchers are crucial to refine and expand upon the proposed
recommendations. By embracing a methodological foundation that integrates ML
techniques in PM, we can unlock the full potential of process mining and
leverage the power of data-driven insights to drive process understanding,
optimisation, and decision-making in various domains and industries.

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