Learning Causal Graphs in Manufacturing Domains using Structural Equation Models Maximilian Kertel

2025-05-02 0 0 547.47KB 6 页 10玖币

侵权投诉

Learning Causal Graphs in Manufacturing Domains

using Structural Equation Models

Maximilian Kertel

Technology Development Battery Cell

BMW Group

Munich, Germany

maximilian.kertel@bmw.de

Stefan Harmeling

Department of Computer Science

TU Dortmund University

Dortmund, Germany

stefan.harmeling@tu-dortmund.de

Markus Pauly

Department of Statistics

TU Dortmund University

Dortmund, Germany

Research Center Trustworthy

Data Science and Security

UA Ruhr, Germany

pauly@statistik.tu-dortmund.de

Abstract—Many production processes are characterized by

numerous and complex cause-and-effect relationships. Since they

are only partially known they pose a challenge to effective process

control. In this work we present how Structural Equation Models

can be used for deriving cause-and-effect relationships from

the combination of prior knowledge and process data in the

manufacturing domain. Compared to existing applications, we

do not assume linear relationships leading to more informative

results.

Index Terms—Causal Discovery, Bayesian Networks, Industry

4.0

I. INTRODUCTION

To be published in the Proceedings of IEEE AI4I 2022.

Complex manufacturing processes as, e.g. for battery cells

show high scrap rates and thus high production costs and

large environmental footprints. One of the driving factors is

the missing knowledge on the interdependencies between the

process parameters, intermediate product properties and the

quality characteristics [1]. Together we call this the cause-

and-effect relationships (CERs). CERs can be visualized as a

network with the process and product characteristics as nodes

and the CERs as directed edges [1], [2]. It is the goal of our

paper to unify expert knowledge and process data to derive

such a network, which allows the visual identiﬁcation of

•root-causes of erroneous products,

•relevant parameters for process control during successive

production steps and

•important characteristics to predict the quality of the ﬁnal

product.

In complex manufacturing domains, CERs form a linked mesh

of hundreds of involved factors [1]. Typically, CERs are

derived by running Designs of Experiments (DOEs). However,

DOEs can be time-demanding and the production line has to be

stopped in the meantime leading to prohibitively high costs.

Moreover, if there are many potential CERs, the number of

experiments can become infeasible.

At the same time, the Internet of Things (IoT) allows data pro-

cessing and storage along the whole production line, leading

to a vast amount of accessible information. It is thus desirable

to derive the CERs from the existing observational (or non-

experimental) data. For this purpose, Bayesian Networks can

be used to unify expert knowledge and data. From these, CERs

can be derived under the assumption of causal sufﬁciency [3].

This approach is called causal discovery or structure learning.

The most common example in the manufacturing domain [4]–

[6], is the PC algorithm [3]. This algorithm relies on the

assumption of faithfulness and on efﬁcient statistical tests

for conditional independence. In principle the PC algorithm

can be applied with any test for conditional independence.

However, existing nonparametric tests do not scale well [7],

[8]. Most of the applications of the PC algorithm either

discretize the measurements, or researchers approximate the

joint distribution of the variables by a multivariate normal

distribution. For discrete data and normally distributed data

fast tests for conditional independence exist. However, the

former leads to a loss of information, while the latter requires

a linear dependency between the variables to be exact. In case

of manufacturing data this is most likely a misspeciﬁcation

[9]. Simulation studies show, that the performance of the PC

algorithm can be poor in case of non-linearity [10]. This

questions the application of the PC algorithm for large or high-

dimensional manufacturing data.

In recent years, Structural Equation Models (SEM), which can

incorporate arbitrary functional relationships, were increas-

ingly proposed to derive Causal Bayesian Networks. They

replace the assumption on faithfulness by a functional form

of the conditional distributions (see Equation (1)). While the

PC algorithm returns a set of graphs, methods based on SEMs

often derive a single graph. To the best of our knowledge, we

are the ﬁrst to apply SEMs to derive such graphical models

in the manufacturing domain.

The paper is structured as follows. In Section II we present

potential prior knowledge and available data in manufacturing

domains. We continue in Section III by reviewing Bayesian

Networks and SEMs and explain Causal Additive Models

(CAM). In Section IV we present an extension of CAM, called

TCAM, which efﬁciently incorporates prior knowledge. We

apply our method in Section V to process data of the assembly

of battery modules at BMW. We conclude in Section VI.

arXiv:2210.14573v1 [stat.ML] 26 Oct 2022

II. DATA AND CHALLENGES IN COMPLEX

MANUFACTURING DOMAINS

In this section we describe the data sources and propose a

preprocessing of the data. Then, we explain the broad prior

knowledge in manufacturing domains. Finally, we mention

common challenges with production data.

A. Data Sources along the Production Line

The assembly of products consists of production lines,

which again contain several stations, which are passed in a

ﬁxed order and where process steps are carried out. During

those process steps the piece is transformed or it is combined

with other parts in order to achieve a predeﬁned outcome.

All involved parts are assigned to unique identiﬁers. Data of

different types is collected along the production process:

•Process data: the stations take measurements of the

involved parts (e.g. thickness of the piece) and the pa-

rameters of the machine (e.g. weight of applied glue).

•End-of-Line (EoL) tests take additional quality measure-

ments of the intermediate or ﬁnal products.

•Station information: at some production steps the pieces

are spread out to identical stations, such that parts can be

processed in parallel and every piece is assigned to one

of the stations.

•Bill of Material (BoM): the BoM contains the information

which pieces were merged together and on which position

they have been worked in.

•Supplier data: suppliers transmit data on provided goods.

The preprocessing of the data, which is depicted in Figure 1,

consists of the following steps:

1) Collect the data for every intermediate product.

2) Iteratively merge the data of all subcomponents of a ﬁnal

product.

Measurements of identical subcomponents, which are placed

in the same position, can be found in the same column.

Eventually, the ﬁnal tabular data set contains all measurements

that can be associated with a ﬁnal product.

B. Prior Knowledge

As the stations are passed in a ﬁxed order, we know that

CERs across different stations can only act forward in time.

Additonally, in many manufacturing organizations, tools as the

Failure Mode and Effect Analysis (FMEA) [11] are imple-

mented to extract expert knowledge on CERs in the production

process and to provide the information in a structured form.

C. Challenges of Data Analysis in Manufacturing

Often, similar information is recorded multiple times along

the production line, leading to multicollinearity [4]. Also,

sensors might deliver non-informative data by recording im-

plausible values. Industrial data is also reported to be drifting

over time. However, even in shorter time intervals, data of

a series production contains thousands of observations. This

distinguishes the manufacturing domain from other applica-

tions of causal discovery as medicine, genetics or the social

sciences.

III. STRUCTURE LEARNING OF GRAPHICAL MODELS

A. Some Preliminaries on Graphical Models

Let G= (V,E)be a directed acyclic graph (DAG) [12,

Chapter 6] with nodes V= (V1, . . . , Vp)and edges E. The

node Viis called a parent of Vjif the edge Vi→Vjis in

E. We denote the set of all parents of Vjas pa(Vj). A tuple

of nodes (Vj1, . . . , Vj`), such that Vjkis a parent of Vjk+1 for

all k= 1,...,(`−1), is called a directed path. Nodes that

can be reached from Xjthrough a directed path are called the

descendants of Xj.

In the following we denote random vectors with bold letters

as Zand random variables as Z. Let X= (X1, . . . , Xp)be a

random vector representing the data generating process. For a

graph Gwith nodes X1, . . . , Xp, we call (X, G)a Bayesian

network if the local Markov property holds, i.e.

Xi⊥Xj|pa(Xi)

for any Xjthat is not a descendant of Xiin G. Here, X⊥Y|Z

denotes the conditional independence of Xand Ygiven Z. In

that case, we can deduce additional conditional independencies

for Xfrom the graph Gusing the concept of d-separation [12].

For a Bayesian Network (X, G), it then holds that Xi⊥Xj|S

if Xiand Xjare d-separated by Sin G. On the other hand,

if there is a graph G, such that Xi⊥Xj|Simplies that Xi

and Xjare d-separated given Sin G, then Xis called faithful

with respect to G. As multiple graphs can contain the same

d-separations, this graph Gis in general not unique.

To promote the intuition, assume that Xhas a joint density f.

Then Xi⊥Xj|Scan be characterized by

f(xi|Xj=xj,S=s) = f(xi|S=s),

where f(xi|Z=z)denotes the conditional density function

of Xigiven Z=z. Thus, if we already know S, then Xjdoes

not provide additional information on Xi. Assume that we are

interested which variable in {Xj,XS}causes the variable Xi

to be out of the speciﬁcation limits. Then we know, that the

root causes can be found within S.

B. Graph Learning with Structural Equation Models

While the PC algorithm is the classic approach for deriving

a Causal Bayesian Network, recent research focused on identi-

fying it using acyclic SEMs [10], [13]–[15]. They assume that

there exists a permutation Π0(1, . . . , p) = π0(1), . . . , π0(p)

and functions {f`, ` = 1, . . . , p}, such that

X`=f`(X`1, . . . , X`v, ε`), ` = 1, . . . , p, (1)

where π0(`k)< π0(`)for all k= 1, . . . , v and ε1, . . . , εpare

i.i.d. noise terms. As the estimation of f`in Equation (1) is

difﬁcult in high dimensions, one typically restricts the function

class and the distribution of the noise terms. In this work, we

assume that the functions follow the additive form

f`(X`1, . . . , X`v, ε`) = c`+X

k:π0(k)<π0(`)

fk,`(Xk) + ε`,(2)

where ε`∼ N (0, σ`)and c`∈R. To ensure the uniqueness

of the fk,` and without loss of generality, we set E(X`)=0

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

LearningCausalGraphsinManufacturingDomainsusingStructuralEquationModelsMaximilianKertelTechnologyDevelopmentBatteryCellBMWGroupMunich,Germanymaximilian.kertel@bmw.deStefanHarmelingDepartmentofComputerScienceTUDortmundUniversityDortmund,Germanystefan.harmeling@tu-dortmund.deMarkusPaulyDepartmentofSta...

展开>> 收起<<

Learning Causal Graphs in Manufacturing Domains using Structural Equation Models Maximilian Kertel.pdf

共6页,预览2页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Learning Causal Graphs in Manufacturing Domains using Structural Equation Models Maximilian Kertel

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: