Extraction of qualitative behavior rules for industrial processes from reduced concept lattice

Abstract

Formal concept analysis (FCA) become an alternative approach to extract and represent knowledge of real world systems. That knowledge can be obtained from implications rules extracted of concept lattices formed by ordered formal concepts. However, in complex systems the number of formal concepts can be large. To deal with this complexity of the FCA, concept reduction techniques can be applied in order to balance the quality of information, and the computational cost for generating and handling the lattice.

In this paper, we develop a novel approach to represent the behavior of physical processes through qualitative rules based on proper implications (minimum representation of the data) extracted from the reduced concept lattice.

As a case study, the cold rolling process was considered. This process characterize by the strong non-linearity among its parameters. The results show that the qualitative behavior of the rolling process is preserved even when the reduction techniques are applied.

The approach can be used to understand the relationship between process parameters through implication rules under different operating conditions of a process. The paper discusses some generic procedures that can be adopted to apply this approach to other industrial processes.

Keywords

Formal concept analysis proper implications cold rolling process

1. Introduction

Formal concept analysis (FCA) [81] is a field of mathematics based on concept lattice theory, which has been proposed as an alternative to represent and extract knowledge of real world systems. FCA considers a data set containing instances, variables and incidence relations among them to represent the systems or physical processes. The extracted knowledge can be obtained from implications ou association rules extracted from concept lattices, in order to describe the qualitative behavior of the process. In this way, it is not necessary a more detailed study about the physical, economics, biological, etc. principles that control the properties of systems [40, 24, 71, 56, 88, 87].

In FCA, data set is represented by a binary relation $I\subseteq G\times M$ , where the elements of $G$ are called objects and the elements of $M$ are called attributes. The triple $(G,M,I)$ is called the formal context. This formal context is represented by a table in which the rows correspond to the objects (process operating conditions) and the columns represent the attributes (process variables) used to evaluate the qualitative behavior of a physical process. From such formal context, formal concepts are obtained, which are pairs $(A,B)$ , where $A\subseteq G$ (the extension), $B\subseteq M$ (the intention) and each object in $A$ (process operating conditions) has all the attributes in $B$ (range of values of the physical variables) and each attribute in $B$ is an attribute common to all objects in $A$ .

The great potential of FCA is provided by the concept lattice, a complete lattice formed by the set of organized formal concepts extracted from the formal context, and an implication rules set $\mathcal{I}$ , extracted from the formal context or lattice concept, expressing the relations between the process variables. Applying different properties to a $\mathcal{I}$ set, we can generate implication sets with certain constraints for specific domains. One of these specific sets is the proper implications set, in which for each implication the premise is minimal and the conclusion is a singleton (unit set) [11]. The proper implications set is useful because it provides a minimum representation of the data, for applications where the goal is to find the minimum set of attributes to achieve specific purposes.

Evidencing from the experience for industrial applications, any new approach, technique or method to extract knowledge from data must seek representations based on implication rules to feed an supervisor system or by a symbolic representation of easy understanding especially by a human operator. FCA gathers these characteristics and allows to change the detail level (granularity) of the behavior implication rules $i\in\mathcal{I}$ generated, which can describe the process in study. As FCA induces potentially high computational complexity, consequently even a small dataset can generate a very large number of formal concepts [5, 38]. Regardless of the cardinality of formal concepts generated in the worst case, all the relationships between concepts are represented in the form of a concept lattice. Although this feature satisfies completeness, but generally results in large number of redundant concepts, thus overloading the lattice structure. Furthermore, the complexity of the relationships between concepts can make the analysis of the final lattice very hard. Determining a concept lattice of an appropriate size and structure in order to express the relevant aspects is one of the most important problems (features) of FCA. To address this problem, concept lattice reduction techniques have been developed based on different characteristics [45, 41, 65, 55, 27, 26, 53, 13, 35, 38, 63, 36]. Such techniques are called concept lattice reduction [21].

In this work, we develop a novel approach to represent the behavior of physical processes through qualitative rules based on proper implications, extracted from the reduced concept lattice. As a case study, we consider the cold rolling process [60], which has a non-linear behavior on several parameters: input thickness, front and back tension, average yield stress and friction coefficient. It should be noted that these relationships are fundamental to the rolling process [88]. Any change on either of them will cause changes on the rolling load, consequently on the outgoing thickness. The FCA is able to capture these changes and to represent them in a simple and easy manner by extracting the knowledge present in the data set in order to understand the behavior of physical process. The results shown that the relationship between certain variables, such as output thickness and rolling load; and between friction coefficient and rolling load are preserved, even when the reduction techniques are applied.

The motivation for this research is to capture knowledge from reduced conceptual lattice, over physical systems through qualitative rules, to provide the understanding of the process by less-experienced operators. Moreover, it provides information about the analyzed process that can be used in the design of on-line control and supervision systems. Thus, this paper discusses, in a generic form, the procedures to be adopted to apply this work to other industrial processes.

The rest of the paper is organized as follows: Section 2 proposes the use of the FCA as an alternative to represent and extract knowledge of physical processes through reduced conceptual lattice. In Section 3, a case study on the cold rolling process is presented followed by Section 4 with the contributions and the conclusions of this work.

2. Related work

There are many techniques for concept lattice reduction, each with different characteristics. Dias and Vieira [21] proposed the classification of techniques for concept lattice reduction in three groups: redundant information removal, simplification, and selection.

The techniques of redundant information removal seek to produce a concept lattice isomorphic to the original. In general, they aim to find a formal context with a minimum number of objects or attributes keeping unchanged the structure of the concept lattice [91, 75, 76, 79, 50, 73, 61, 77, 74, 59, 58, 47, 46, 53, 48, 49]. In general, those techniques operate by reducing attributes, and so they are called attribute reduction techniques.

In [80], the authors classify the attributes as absolutely necessary, relatively necessary or absolutely unnecessary. An attribute is considered absolutely necessary if it is present in all minimal sets. It is considered relatively necessary if it is present in at least one minimal set. And finally, if an attribute is not in any minimal set, is considered unnecessary.

In [53], the author shows that the classification of attributes proposed by [80] is independent of: concept lattices, property-oriented concept lattices and object-oriented concept lattices (property-oriented concept lattice and object-oriented concept lattices are originated from rough set theory [57]). As a consequence, only one of those three types of lattices needs to be considered for obtaining the minimal set of attributes.

The techniques of simplification seek an abstraction of the concept lattice, i.e. a high-level overview that preserves only the essential aspects [28, 17, 7, 4, 66, 67, 41, 30, 40]. The technique is one that, from a formal context or a concept lattice, abstracts non-essential differences between concepts, objects or attributes.

Clustering techniques can be applied to sets of objects, attributes or formal concepts. In [17] clustering of objects is performed in order to reduce the dimensionality of concept lattices.

The techniques of selection act by selecting formal concepts, objects or attributes based on a relevance criterion [42, 43, 64, 2, 63, 70, 8, 9, 12, 10, 62, 68, 5, 90]. Some authors propose to use all user background knowledge to create constraints on attributes.

The method proposed by [5] assigns a weight to each attribute in order to express its relevance, and then selects formal concepts considered relevant.

In this work, in order to extract qualitative behavior from reduced concept lattice, we consider the selection techniques.

3. Formal concept analysis

In the FCA, the data obtained from the operation of a physical process are presented as a formal context, a triple $(G,M,I)$ , where $G$ is a set of elements called objects (process operating conditions), $M$ is a set of elements called attributes (variables of the physical process) and $I\subseteq G\times M$ is called an incidence relation. If $(g,m)\in I$ , one says that “the object $g$ has the attribute $m$ ”. Table 1 shows an example of a formal context.

Table 1
Example of formal context

Obj/Att	a	b	c	d	e	f
1	x	x				x
2	x
3	x	x	x	x		x
4		x		x	x	x
5	x

Given a set of objects $A\subseteq G$ from a formal context $(G,M,I)$ , the set of attributes which is common to all those objects is termed $A^{\prime}$ . Similarly, for a set $B\subseteq M$ , $B^{\prime}$ is the set of objects that have all the attributes from $B$ . i.e., $A^{\prime}=\{m\in M|\forall g\in A:(g,m)\in I\}$ and $B^{\prime}=\{g\in G|\forall m\in B:(g,m)\linebreak\in I\}$ .1

The notation $x^{\prime}$ will be used as abbreviation $\{x\}^{\prime}$ , whether $x$ is an object or an attribute.

By using such derivation operators, the notion of formal concept is defined as a pair

(A,B)\in\mathcal{P}(G)\times\mathcal{P}(M)

such that

A^{\prime}=B

and

B^{\prime}=A

, where

A

is called the extent and

B

the intent of the concept. For example, from the formal context of Table 1, it can be seen that the pair

(\{3,4\},\{\mbox{b},\mbox{d},\mbox{f}\})

is a formal concept with extent

\{3,4\}

and intent

\{\mbox{b},\mbox{d},\mbox{f}\}

The set of formal concepts can be ordered by the partial order $\preceq$ such that for any two formal concepts $(A_{1},B_{1})$ and $(A_{2},B_{2})$ , $(A_{1},B_{1})\preceq(A_{2},B_{2})$ if and only if $A_{1}\subseteq A_{2}$ (equivalently, $B_{2}\subseteq B_{1}$ ). The set of concepts ordered by $\preceq$ constitutes a complete lattice [19], is called lattice. The concept lattice obtained from a formal context $(G,M,I)$ is denoted by $\mathcal{B}(G,M,I)$ . Figure 1 presents the line diagram [29] of the concept lattice originated from the formal context of Table 1.2

All line diagrams in this paper were drawn using the Conexp software [84].

Each node in the line diagram represents a formal concept. The objects are shown inside white boxes drawn below some concept nodes and the attributes inside gray boxes drawn above some concept nodes. The boxes are distributed in such a way that the extent of a concept is obtained by collecting all objects from the concept node to the lattice infimum, and its intent is obtained by visiting all attributes from the concept node to the lattice supremum.

Figure 1.

Concept lattice originated from the formal context.

3.1 Implication rules

The implication rules are a specialty of the notion of association rules, because, in general, the first can be seen as a special case of the second, with 100% confidence. Differently of the association rules, the implication rules support inference axioms, although the associations are more flexible than implications because they permit discovery of approximate dependencies.

In order to describe the qualitative behavior of the process, in this work, we are interested in implications that we are certain that will occur (confidence 100%). With association rules, we do not have certain that given the premise the conclusion will happen. This makes it difficult to understand the behavior of the physical process.

On the other hand, it is known that, to find the association rules, for example, applying the apriori algorithm, the generation of the set of frequent itemsets is computationally expensive. Thus, FCA seeks for implications by means of the formal concepts, which are pairs of objects and attributes. In this case, the size of the search space could reduces and the computation may be more efficient [15].

Given a formal context $(G,M,I)$ or a concept lattice $\mathcal{B}(G,M,I)$ , one can extract exact implication rules (hereafter named as implications). Some definitions are given as follows [29]:

.

Given a formal context $(G,M,I)$ , An implication is an expression $P\rightarrow Q$ , where $P,Q\subseteq M$ .

An implication $P\rightarrow Q$ , extracted from a formal context, or respective concept lattice, have to be such that $P^{\prime}\subseteq Q^{\prime}$ . In other words: every object which has the attributes of $P$ also has the attributes of $Q$ . Note that, in $P\rightarrow Q$ , $P$ is the left hand side (lhs) and $Q$ is the right hand side (rhs).

.

If $X$ is a set of attributes, then $X$ respects an implication $P\rightarrow Q$ iff $P\not\subseteq X$ or $Q\subseteq X$ .

.

An implication $P\rightarrow Q$ holds in a set $\{X_{1},\ldots,X_{n}\}\subseteq M$ iff each $X_{i}$ respects $P\rightarrow Q$ ; and $P\rightarrow Q$ is an implication of the context $(G,M,I)$ iff it holds in its set of object intents (an object intent is the set of its attributes).

.

An implication $P\rightarrow Q$ follows from a set of implications $\mathcal{I}$ , iff for every set of attributes $X$ if $X$ respects $\mathcal{I}$ , then it respects $P\rightarrow Q$ .

.

A set of implications $\mathcal{I}$ is said to be complete in $(G,M,I)$ iff every implication of $(G,\linebreak M,I)$ follows from $\mathcal{I}$ .

.

A set of implications $\mathcal{I}$ is said to be redundant iff it contains an implication $P\rightarrow Q$ that follows from $\mathcal{I}\backslash\{P\rightarrow Q\}$ .

.

An implication $P\rightarrow Q$ is considered superfluous iff $P\cap Q\not=\emptyset$ .

From the concept lattice of Fig. 1, one can extract the following implication set $\mathcal{I}=\{d\rightarrow b;c\rightarrow b;d\rightarrow f;c\rightarrow d;f% \rightarrow b;c\rightarrow f;f\rightarrow d;e\rightarrow b;a,d\rightarrow c;e% \rightarrow d;a,f\rightarrow c;e\rightarrow f;c\rightarrow a;ae\rightarrow c\}$ .

For physical process will be convenient that each implication represents a minimum condition of an operating condition, which would allow an easy understanding of the qualitative behavior of the process. For this, we will require that the complete set of implications $\mathcal{I}$ of a formal context $(G,M,I)$ has the following characteristics, to be used as representative of the process:

•
the right hand side of each implication is unitary: if $P\rightarrow m\in\mathcal{I}$ , then $m\in M$ ;
•
superfluous implications are not allowed: if $P\rightarrow m\in\mathcal{I}$ , then $m\notin P$ ;
•
specializations are not allowed, i.e. left hand sides are minimal: if $P\rightarrow m\in\mathcal{I}$ , then there is no $Q\rightarrow m\in\mathcal{I}$ such that $Q\subset P$ .

A complete set of implications in $(G,M,I)$ with such properties is the so called set of proper implications [72] or unary implication system (UIS) [11].

.

Let $\mathcal{J}$ be the complete closed set of implications of a formal context $(G,M,I)$ . Then the set of proper implications $\mathcal{I}$ for $(G,M,I)$ is defined as:

$\displaystyle\{P\rightarrow m\in\mathcal{J}|P\subseteq M\text{ and }m\in M% \setminus P\text{ and }$ $\displaystyle\forall Z\subset P:Z\rightarrow m\notin\mathcal{J}\}.$

However, in some cases we must extract such implications from a subset of formal concepts, which forms a given section of a concept lattice, and that section might not form a complete concept lattice [64].3
³
The sets of objects or attributes of the considered concepts are not a closure systems.

This is the case for techniques of the selection of concepts that do not have access to the whole original concept lattice [21]. Thus it might not be possible to obtain a complete (with respect to some formal context) set of proper implications. In order to deal with this problem, we propose to extract only implications with some representative support value. Dias and Vieira [22] have discussed on how to extract proper implications with support greater than zero ( $P\rightarrow m\in\mathcal{J}|P\neq\oslash$ ) from a subset of formal concepts.

Considering a formal concept $C\in\mathcal{B}(G,M,I)$ , the set of implications to be obtained will share the following properties with the original set of proper implications [72]:

.

It will not have specializations of its implications.

.

The right hand side of its implications will be unitary.

.

It will be consistent with the original derivation operators when applied to only the elements of the concept.

.

It will have only implications with support greater than zero.4
⁴
Implications with support equal to zero are not inconsistent, but possible to logically exist. It should be noted that, in applications related to physical processes, the most appropriate proper implications are those with support greater than zero.

.

The set of implications must reflect the lattice formed by the actual set of chosen concepts.

Propositions 1–4 are guaranteed by the proper use of minimum generators [11] as lhs (left hand side), and property 5 is guaranteed by ensuring those properties globally (topological ordering given by the cover relation of the lattice, smallest intentions first). The order in which the set of concepts is explored is important to ensure that the lhs of an implication is as small as possible.

Recently, despite the computational cost, minimal generators have gained prominence in FCA. Its importance is due, mainly, to the fact that they favor the principle of minimum description length (MDL), i.e. the best hypothesis for a given set of data is the one that leads to the best compression of the data [31].
3.2 Techniques for reduction of concept lattice

In order to deal with the complexity of a concept lattice, techniques for concept lattice reduction [21] seek a balance between quality of information, computational cost to generate and maintain the lattice and easiness of analysis [5, 53, 63, 47, 43, 70, 90, 49, 52, 16, 83, 6].

In this subsection we present a brief review of reduction techniques. Among the various reduction techniques discussed and classified by Dias and Vieira [21], we selected the techniques stability [43] and minimal generators [5] belonging to the selection class. The stability technique is the main technique of reduction observed in the literature and presents satisfactory results [43, 38, 64]. On the other hand, the selection technique, based on minimum generators, has gained prominence in the FCA. Minimum generators represents, without loss, the knowledge present in the formal concepts. A brief description of the techniques is explained in the following subsections.

.

A technique for concept lattice reduction is one that aims to reduce the concept lattice complexity, both in terms of magnitude and inter-relationships, while maintaining relevant information.

3.2.1 Reduction technique based on the stability index

This technique proposes the stability index, which aims to measure how much the intention of a concept depends on its extension [43]. The stability index $\alpha$ of the formal concept $(A,B)$ is such that, $\alpha(A,B)=\frac{|\{C\subseteq A|C^{\prime}=B\}|}{2^{|A|}}$ [38]. This index can be used to define a threshold from which all formal concepts with lower values are removed. The idea is to select the most stable formal concepts, i.e., those that best describe the process under analysis. So, the proper implications are extracted based on the selected formal concepts.

In this paper, the stability index is applied in the selection of formal concepts that represent physical processes. Therefore, the algorithm proposed in [64] was used.

3.2.2 Reduction technique based on minimum generators

Formal concepts can be represented in a compact manner through the minimal generators [29]. Let $(A,B)$ be a formal concept obtained from the context $(G,M,I)$ . A generator $(A,B)$ is a subset $D\subseteq M$ , such that $D^{\prime}=A$ . The set of all generators of $(A,B)$ is $\textit{ger}(A,B)=\{D\subseteq M|D^{\prime}=A\}$ . The set of the minimal generators of $(A,B)$ is $\textit{mger}(A,B)=\{D\in\textit{ger}(A,B)|\nexists m\in D:(D\setminus\{m\})^{% \prime}=A\}$ .

The selection of formal concepts from minimum generators was proposed in [5]. For each attribute of the formal context a weight is assigned in order to describe its relevance ( $m_{1}\succ m_{2}$ ) – which describes that the attribute $m_{1}$ is more relevant than the attribute $m_{2}$ . This relevance of the attributes is transported to the relevance of the formal concepts, through the analysis of the generators and minimum generators of each formal concept. A formal concept is considered relevant if it has at least one minimum generator whose sum of the attributes’ weight divided by the number of attributes is greater than a minimum relevance defined by the user. The most relevant concepts are then selected.

Minimum generators allows the selection of formal concepts that best represent the physical process under analysis.

3.3 Extraction and evaluation of qualitative behavior of physical processes using proper implications

For the understanding of physical processes, the representation of qualitative behavior through implications may be appropriate because it is easy to understand for less experienced operators. The set of implications to be extracted corresponds to proper implications. In this paper, an implication $P\rightarrow m\in\mathcal{I}$ (Definition 8) expresses that a conjunction of ranges of values for a given subset of variables (attributes represented by $P$ ) implies a range of values for a given variable ( $m$ ).

The proper implications must be evaluated in relation to their expressiveness. For this, suppose $r$ is a record of a validation database $\mathcal{V}$ . Given an implication $P\rightarrow m\in I$ and assuming that $\mathcal{M}(r)$ is the set of mapped attributes for values of a record $r\in\mathcal{V}$ , we have the following possibilities:

(a)
if $P\not\subseteq\mathcal{M}(r)$ , then $r$ is not represented by $P\rightarrow m$ ;
(b)
if $P\subseteq\mathcal{M}(r)$ and $m\in\mathcal{M}(r)$ , then $r$ is represented by $P\rightarrow m$ .

The cases $a$ , $b$ permit to derive the following metric for analysis of the knowledge expressed in the set of proper implications $\mathcal{I}$ :

$\displaystyle T_{r}=100\times\frac{nr}{|\mathcal{V}|}$ (1)

where $T_{r}$ is the representativity rate; $nr=|\{P\subseteq\mathcal{M}(r)\text{ and }m\in\mathcal{M}(r)|P\rightarrow m\in% \mathcal{I}\text{ and }r\in\mathcal{V}\}|$ is the number of records represented and $|\mathcal{V}|$ the number of records used for analysis.
4. Case study: Cold rolling process

The cold rolling process has been highly automated and reached a high standard industrial level. In this process, the objectives are always larger productivity and better quality of the final product in terms of thickness and shape. The literature shows several non-conventional strategies to reach these objectives, as neural networks [87, 44, 54, 14], fuzzy logic [34] and genetic programming [69]. The only condition for the new strategies is to represent correctly the behavior of the process in a quantitative and qualitative way for different operating conditions so that they can be used in the design of on-line control and supervision systems.

The mathematical model of the cold rolling process can be obtained through the study of variables involved in the process, as Alexander’s mathematical model [1] considered one of the most complete. However, the model is known by requesting numerical solution with significant computational effort. A more natural alternative to understand this kind of process is through symbolic representation, expressing the cause-effect relation among the several parameters through qualitative rules [18].

Into classical theories there are several models to calculate the rolling load required for the deformation process. These models, Eq. (2), are non-linear functions of several parameters. It is possible to observe that any change in the input thickness $h_{i}(mm)$ , in the output thickness $h_{o}(mm)$ , in the back tension $t_{b}(N/mm^{2})$ , in the front tension $t_{f}(N/mm^{2})$ , in the yield stress $\bar{y}(N/mm^{2})$ and/or friction coefficient $\mu$ , will cause changes on the rolling load $L$ ( $N/mm$ ) and, consequently, on the outgoing thickness.

$\displaystyle L=f(h_{i},h_{o},\mu,t_{b},t_{f},\bar{y},E,R)$ (2)

where $E(N/mm^{2})$ is the Young’s modulus of the strip material and $R(mm)$ is the roll radius, considered constant parameters.

During the rolling process, the cylinders are compressed against the strip by a force transmitted by the back rolls. Since the rolling mill is not perfectly rigid, the output thickness can be expressed by the elastic equation of the rolling mill, Eq. (3).

$\displaystyle h_{o}=g+\frac{L_{w}}{D}$ (3)

where $g(mm)$ is considered the roll-gap or “gap”, $W(mm)$ the width of the material being rolled, $D(N/mm)$ the mill modulus and $L_{w}(N/mm)$ rolling load per unit width $(L_{w}=L_{w}\times W)$ .

In order to illustrate the behavior of a rolling mill, Fig. 2 shows variations in the nominal operational condition due to the variation in several parameters [25]. Curve (I) corresponds to an increment in the back and/or front tensions of stress. Curve (II) represents a decrement in the input thickness; curve (III) corresponds to the operational condition; curve (IV) represents an increment in the output thickness and curve (V) shows possible increment in the friction coefficient and in the average yield stress or a decrement in the back and/or front tensions.

Figure 2.

Operational condition of a rolling mill.

4.1 Data for analysis of the rolling process

A synthetic database representing the rolling process was provided by the Data Science Research Group (DSRgroup).5

⁵
http://www.icei.pucminas.br/projects/dsrgroup/.

The data has been produced from a nominal operating condition point searching all space of the free variables in a regular manner using Alexander’s mathematical model [1]. The nominal operating condition for the rolling process was defined as:

•

$h_{i}=5.0$ $(mm)$

•

$h_{o}=3.6$ $(mm)$

•

$\mu=0.12$

•

$t_{b}=4.325$ $(N/mm^{2})$

•

$t_{f}=89.22$ $(N/mm^{2})$

•

$y=256.325+468.187\epsilon^{0.4275}$

•

$\bar{y}=460.106$ $(N/mm^{2})$

•

$L=875310$ $(N/mm)$

Data for the material and the rolling mill were defined as:

•

$E=200054$ $(N/mm^{2})$

•

$v=0.330$ Poisson’s ratio of the strip

•

$W=500.0$ $(mm)$

•

$R=292.1$ $(mm)$

•

$D=4903.300$ $(N/mm)$

•

$g=1.846$ $(mm)$

•

$S_{i}=250.556$ $(N/mm^{2})$

•

$S_{i}=S_{o}=534.900$ $(N/mm^{2})$

The reference database has ten million records, a high number that makes it difficult to apply the FCA. In [89], statistical techniques was proposed for the construction of representative datasets, maintaining the generalization capacity of the model constructed from the data. Using these approaches, a new equation $n=(z/e)^{2}\times(f\times(1-f))$ is given, where $n$ is the sample size, $z$ is the confidence level, $e$ is the error around the average and $f$ is the population proportion. By setting $z$ at 90% (1,645), $f$ at 0.5 and $e$ at 2%, we have $n$ $\cong$ 1691 records. So, it was selected 10 disjoint samples of 1691 records using the following procedure:6

⁶

This total of samples is sufficient to report the results with 90% confidence [32].

For each variable were selected records that contained the maximum, minimum and near average values;

The register representing the nominal operating condition was chose;

The remaining elements were selected randomly in order to reach the total of 1691 records.

In this paper, the objective is to represent, through the FCA, all the sample space of the data and to demonstrate the potentialities of the FCA to represent the process qualitatively. The Table 2 displays the maximum and minimum values of each variable for the total set of selected data.

Table 2

Maximum and minimum values for each variable of the rolling process

	$h i$	$h o$	$\mu$	$t_{f}$	$t_{b}$	$\bar{y}$	$L$
	$(mm)$	$(mm)$		$(N/mm^{2})$	$(N/mm^{2})$	$(N/mm^{2})$	$(N/mm)$
Minimum	4.6	3.492	0.096	6.3686	0.3087	40.8793	10207.049
Maximum	5.4	3.708	0.144	11.8274	0.5733	50.3415	28722.925

Following the previous procedures 10 samples were selected. Among the selected samples, half was used for modeling through the FCA (formal context, concept lattice and extraction of proper implications) and the other half used for validation; i.e., analysis of the representativeness of the implications.

4.2 Discretization of the rolling process variables

The values of the variables of the rolling process are real numbers, as can be seen in Table 2. However, the FCA works with discrete values. In this case, some alternatives are the use of pattern structures [37], FCA fuzzy [3] or discretization of the continuous values [82]. Among these alternatives, it was chosen to use a discretization process proposed in [88]. An equidistant discretization was performed on the data, applying a total of 15 cut-off conditions in each variable, which resulted in 105 attributes (7 variables divided into 15 intervals each). Table 3 shows the discretization applied in the rolling load $(L)$ , friction coefficient $(\mu)$ and output thickness $h_{o}$ considering the minimum and maximum values described in the Table 2. For example, the [10207.049, 11441.440] range was mapped to the $L=0$ attribute. The lower bound is always closed and the upper bound is always open, except the last, that is, the one mapped in the $L=14$ attribute. A similar procedure was used for the other variables.

Table 3
Discretization applied in the rolling load $(L)$ , friction coefficient $(\mu)$ and output thickness $h_{o}$

	Rolling load		Friction coefficient		Output thickness
	$L$ $(N/mm)$		$\mu$		$h_{o}$ $(mm)$
Interval	Inf	Sup	Inf	Sup	Inf	Sup
0	10207.0	11441.4	0.096	0.0992	3.492	3.5064
1	11441.4	12675.8	0.0992	0.1024	3.5064	3.5208
2	12675.8	13910.2	0.1024	0.1056	3.5208	3.5352
3	13910.2	15144.6	0.1056	0.1088	3.5352	3.5496
4	15144.6	16379.0	0.1088	0.112	3.5496	3.564
5	16379.0	17613.4	0.112	0.1152	3.564	3.5784
6	17613.4	18847.8	0.1152	0.1184	3.5784	3.5928
7	18847.8	20082.2	0.1184	0.1216	3.5928	3.6072
8	20082.2	21316.6	0.1216	0.1248	3.6072	3.6216
9	21316.6	22551.0	0.1248	0.128	3.6216	3.636
10	22551.0	23785.4	0.128	0.1312	3.636	3.6504
11	23785.4	25019.7	0.1312	0.1344	3.6504	3.6648
12	25019.7	26254.1	0.1344	0.1376	3.6648	3.6792
13	26254.1	27488.5	0.1376	0.1408	3.6792	3.6936
14	27488.5	28722.9	0.1408	0.144	3.6936	3.708

Applying the discretization process in the 5 samples selected to construct the models, a set of formal contexts $(G,M,I)$ was produced, in which:

•

$G$ represents the sample records;

•

$M$ represents the intervals of the variables; i.e., $M=\{m_{1},m_{2},\ldots,m_{n}\}$ correspond to the variables $h_{i},h_{o},\mu,t_{b},t_{f},\bar{y}$ and $L$ discretized, where $E$ and $R$ have been set (see Section 4.1); and

•

$I$ represents the existence of a record $g(g\in G)$ , which has a variable in the range $m(m\in M)$ .

where $|G|=$ 1961, $|M|=$ 105 and $I$ has a density of approximately 7% in each formal context built.

An aspect that needs to be highlighted is the one related to the number of intervals (discretization). The number of intervals for each parameter has an influence in the number and quality of the rules obtained. So, the number of intervals can be considered as a regulatory mechanism of the detail level of the rules obtained. However, this regulation is highly dependent on the quality of the data. On the other hand, it is important to observe that the discretization interval (to build the formal context) should not be inferior to the one of the sampling interval considered to collect data of a physics process. This is due to the fact of the extracted rules in this interval cannot be significant.

4.3 Qualitative behavior of the process through proper implications

The formal contexts built from the data samples were named contexts #1 through #5, respectively. An example of implication obtained from the context #1 is $h_{o}=0\rightarrow L=14$ , which expresses the fact that low output thickness ( $h_{o}=0$ ) results in a high rolling load value ( $L=14$ ). More specifically, if $h_{o}$ is in the [3.492, 3.5064] range, then $L$ is in the [27488.533, 28722.925] range. This is consistent with the physical behavior of the process and with the database records generated by the Alexander’s model (Table 2).7

⁷
The algorithm proposed in [22] was used to extract proper implications with support greater than zero through the concept lattice. This algorithm is part of the EF-Concept Analysis package [20] in evolution and available at http://www. icei.pucminas.br/projetos/dsrgroup/.

The Table 4 represents the number of formal concepts, their proper implications, and the representation rate $T_{r}$ obtained for each context. For example, for the formal context #1, more than 87,000 proper implications presented a representative rate $T_{r}$ of 99.29%. In other words, less than 1% of the records used in the validation process were not represented by the set of proper implications. The behavior is similar for the other formal contexts. The results presented show that the proper implications describe a significant part of the behavior of the rolling process. These outcomes are the baseline for comparing the application of FCA with and without the reduction in the conceptual lattice.

Table 4

Summary of the results related to proper implications for the original validation contexts

	Context #1	Context #2	Context #3	Context #4	Context #5
# Concepts	21027	21051	21054	20872	20999
# Implications	87650	86186	86005	86360	87050
$T_{r}$ (%)	99.29	98.52	98.81	98.81	98.58

On the other hand, the rolling process involves non-linear relations that make the analysis difficult. Fortunately, some process behaviors are known as, for example, the relationship between variables:

•

Rolling load $L$ and output thickness $h_{o}$ ;

•

rolling load $L$ and friction coefficient $\mu$ .

Rolling load and output thickness have inverse behavior, i.e., the smaller the output thickness, the greater the rolling load. On the other hand, the higher the friction, the greater the rolling load required in the production process. These relationships are considered fundamental in the rolling process [88].

These behaviors can be observed if we consider all the proper implications $P\rightarrow m$ , whose premise $P$ has at least one attribute representing the variables output thickness ( $h_{o}$ ) and/or friction coefficient ( $\mu$ ) and whose consequent $m$ is an attribute referring to the variable rolling load ( $L$ ). Consider the following proper implications obtained from the formal context #1:

$i$ : $h_{o}=1$ , $t_{f}=9$ , $\bar{y}=3\rightarrow L=14$ ; $j$ : $h_{o}=5$ , $\mu=13\rightarrow L=13$ ; and $k$ : $h_{o}=1$ , $t_{b}=2$ , $h_{i}=13\rightarrow L=14$ .

In this case, 2 implications, $i$ and $k$ , contribute with occurrences of both $h_{o}=1$ (lower output thickness) and $L=14$ (greater rolling load), and an implication, $j$ which contributes with $\mu=13$ (higher friction coefficient) and $L=13$ (greater rolling load). In [88] the author demonstrates that given a determined value of input thickness $h_{i}$ the rolling load is more sensitive to the output thickness $h_{o}$ and friction coefficient $\mu$ .

The Fig. 3a–e presents, from an easy-to-view temperature graphic, the behavior of the rolling load variables $L$ and output thickness $h_{o}$ based on the number of proper implications obtained for the formal contexts #1 to #5, respectively. The Fig. 3f presents the average for the five formal contexts. The color tone at each position represents the number of proper implications. Light tones represent little or no implication and dark tones represent a large number of proper implications.

For example, the position ( $h_{o}=0$ , $L=14$ ) in the Fig. 3a indicates a large number of proper implications (dark color), indicating that the operating conditions represented by the data (generated using the Alexander’s model considered in Section 4.1) contribute to higher values of the rolling load. On the other hand, the position ( $h_{o}=0$ , $L=0$ ) has few proper implications (light color), indicating few operating conditions such as, combinations of parameter values, which lead to lower values of rolling load. A similar behavior of the rolling process, with respect to the values of the two considered variables, can be can be observed in all the figures.

Figure 3.

Mapping the relationship between rolling load ( $L$ ) and output thickness ( $h_{o}$ ) through proper implications.

Similarly, for the output thickness and rolling load, the Fig. 4a–f analyze the relationship between rolling load $L$ and friction coefficient $\mu$ .

Figure 4.

Mapping the relationship between rolling load ( $L$ ) and friction coefficient ( $\mu$ ) through proper implications.

Figure 5.

Reductions in the number of formal concepts, quantitative of proper implications and representativeness caused by the stability index for the context #1.

In the next section the same procedure will be applied, after the usage of stability reduction techniques [43] and minimum generators [5]. In this part, the knowledge obtained through the FCA prior to the application of reduction technique will be considered a baseline.

4.3.1 Application of reduction techniques and results analysis

Initially, the selection of concepts is applied through the criterion of the stability index. Figure 5 presents the reductions in the number of formal concepts, the number of proper implications and the representativeness as a function of the stability index for the context #1. First of all, note that different threshold (or cut points) values can be chosen and the optimal values are, obviously, dependent on the data. Here, the threshold were indicated by domain expert. Notice that, different threshold values can be chosen and the optimal values are dependent on the data. In this work, the threshold was indicated by the domain expert, which seeks the threshold more adequate to obtain more representative rules that satisfy the previous knowledge about of the process being analyzed. Using a stability index of 0.1 no formal concept is removed. Between 0.13 and 0.24 the number of formal concepts is reduced to 20918 (approximately 1%). From 0.25 to 0.37 the number of formal concepts is reduced to 11417 (45.7%). The selection of the stability index is dependent on the intended application. A stability index equal to 0.37 was chosen based on observed reductions. Values less than 0.37 show a reduction in the number of formal concepts very low, and greater values provide a reduction of more than 50%. This value will be used in all formal contexts.

In Fig. 5, it can also be observed the number of proper implications and the index of representativeness. Note that certain reductions increase the number of proper implications. When a concept that generates its proper implications is removed, its superior neighbours can generate more specific and more signifi- cant implications. Consequently, the number of proper implications may increase [22]. This increment is the result of the application of the algorithm proposed in [22] to extract proper implications from a subset of formal concepts that do not form a conceptual lattice [64]. Regarding the representativeness ( $T_{r}$ ), note that the values follow the same behavior on the number of implications, showing that the implications resulting from the reduced lattices are consistent. In particular, for a stability index equal to 0.37, the set of implications obtained is able to perfectly represent the physical process under analysis.

The Fig. 6a and b present the relationships between rolling load and output thickness and rolling load and friction coefficient for the proper implications of the reduced formal contexts considering the average values. Comparing to the ratios prior to the application of the reduction technique, shown in Figs 3 and 4, it is noted that the behavior of the process was preserved for the considered operating conditions.

Figure 6.

Mapping the average relations between parameters of the rolling process through proper implications obtained after the reduction through the stability technique which consider average values.

Next, the concept selection is applied through the analysis of minimum generators. Initially, it is necessary to determine the relevance of attributes; i.e., the relevance of the variables of the rolling process and, consequently, of the attributes $M$ . In the rolling process, Zárate and Bittencout [86] proposed the following relative relevance for the variables: $\mu\succ h_{o}\succ h_{i}\succ\bar{y}\succ t_{b}\succ t_{f}$ . In this paper, $\mu\succ h_{o}$ means that $\mu$ is more relevant (greater influence on the rolling load) than $h_{o}$ . Based on this relative relevance, the following weights are assigned to the attributes corresponding to the variables (for $i=0,\ldots,14$ , i.e. the weights are identical for the same variable): 0.7 for the attribute $L=i$ , 0.6 for $\mu=i$ , 0.5 for $h_{o}=i$ , 0.4 for $h_{i}=i$ , 0.3 for $\bar{y}=i$ , 0.2 for $t_{b}=i$ and 0.1 for $t_{f}=i$ . For example, every attribute related to the rolling load variable ( $L$ ) has the weight 0.7 on the proposed scale. The relevance, as well as the set of proposed weights, was used in another application, and satisfactory results were observed [23].

Figure 7 presents the reductions of the number of formal concepts, the number of proper implications and representativeness caused by the minimum relevance for the reduction technique based on the mini-mum generators for the context #1 using the mentioned set of weights. Up to a minimum relevance of approximately 0.1 no reduction is achieved. Using between 0.11 and 0.6 reductions range from 1% to 95%. It is observed that, as with the stability technique (Fig. 5), no large reductions were found when there was a small variation in the minimum relevance. Among the reduction configurations evaluated, a minimum relevance of 0.4 was chosen. This value results in a reduction of 37.69%, 37.73%, 37.66%, 37.39% and 37.89% in the number of formal concepts for contexts #1 through #5, respectively.

Figure 7.

Reductions in the number of formal concepts, quantitative of proper implications and representativeness in function of the minimum relevance for the reduction technique based on minimum generators for the context #1.

A 37% reduction in the number of formal concepts resulted in an average of 88,381 proper implications and an average representativeness rate of 98%. Obviously, the mentioned values are lower than those obtained through the FCA without applying the reduction. However, they are better than the results presented by the stability index for a similar reduction in the number of formal concepts.

The Fig. 8a and b present the relationships between rolling load and output thickness and between rolling load and friction coefficient through proper implications obtained after reduction by the minimum generator technique considering the average values. Even before an average reduction of 37% in the number of formal concepts, the previously analyzed behavior of the rolling process was preserved.

Figure 8.

Mapping the average relations between parameters of the rolling process through proper implications obtained after reduction by the technique of minimum generators considering the average values.

We propose to use proper implications extracted from reduced concept lattice to represent the behavior of the process being studied in a symbolic and qualitative form. In proper implications set, the right hand side of each implication is unitary, superfluous implications are not allowed, specializations are not allowed, i.e. left hand sides are minimal. This set is appropriate because each implication represents an operating condition of the physical process. In other words, each implication expresses that a minimal conjunction of ranges of values for certain variables imply in a range of values for a particular variable. In the case study, the cold rolling process was considered. The relation between the variables output thickness and rolling load; and between friction coefficient and rolling load were analyzed through the extracted implications from original and reduced concept lattice. It is known that output thickness and rolling load are inversely proportional. On the other hand, friction coefficient and rolling load are proportional. The results demonstrated that even if using an average reduction of 40% in the number of formal concepts, reduction techniques based on minimum generators and stability are able to select the formal concepts that best represent the process. The proper implications generated by the conceptual reduced lattice by such techniques represented, satisfactorily, the analyzed part of the rolling process.

It is important to note that the FCA approach extracts relations among the parameters involved in the process, independently of antecedent and consequent. This means that FCA tries to show the qualitative relations learned from the data. Note also that FCA seeks an implication base from which other implication rules can be deduced.

5. Conclusions and future research

In this work, formal concept analysis (FCA) has been applied to extract and represent knowledge of physical process. The FCA can be used to understand the qualitative behavior processes without the need of complex mathematical models. From a set of collected data, it is possible to apply the FCA and use the proper implications to bear an intelligent system for on-line control and supervision.

We propose use proper implications extracted from reduced concept lattice to represent the behavior of the process being studied in a symbolic and qualitative form. In proper implications set, the right hand side of each implication is unitary, superfluous implications are not allowed and specializations are not allowed; i.e., left hand sides are minimal. This set is appropriate because each implication represents a condition of operation of the physical process. In other words, each implication expresses that a minimal conjunction of ranges of values for certain variables imply in a range of values for a particular variable.

As case study, the cold rolling process was considered. How the relation between the variables output thickness and rolling load; and between friction coefficient and rolling load are previously knowledge of domain expert, it was possible to validate the extracted implications from original and reduced concept lattice. It is important to note that the previous knowledge is essential to validate our approach. We highlight that this knowledge is based on physical principles, which govern of the process, which can be obtained by means of modeling or simulation with incomplete knowledge as proposed by Qualitative Reasoning theory [39].

The results shown that even if using an average reduction of 40% in the number of formal concepts, reduction techniques based on minimum generators and stability are able to select the formal concepts that best represent the process.

The FCA produces qualitative behavior rules where the dependent parameters of the process can or cannot be the consequent variables of the rules extracted. This can be useful in industrial processes for automation, control and design of supervisory systems, where it is important to consider rules that have the dependent parameters as consequent. This can help the operator to identify which parameters can change the operating condition, such as the controllable variables for a control system.

It is important to note that the FCA approach extracts relations among the parameters involved in the process, independently of antecedent and consequent. This means that the FCA tries to show the qualitative relations learned from the data. Notice also that the FCA looks for a implications base from which other implication rules can be deduced. the FCA produces qualitative behavior rules where the dependent parameters of the process can or cannot be the consequent variables of the rules extracted. This can be useful in industrial processes for automation, control and design of supervisory systems, where it is important to consider rules that have the dependent parameters as consequent. This can help to identify which parameters can change the operational condition, such as the controllable variables for a control system.

As future works we intend to apply the FCA in physical process with different characteristics. Finally, some FCA extension has recently been applied for complex data based on rough set [51, 85, 33, 78]. It is important to highlight the work of [78] which proposes a formalism for the stages of preprocessing and data mining, essential stages of a KDD process. The proposal is an extension of the Pawlak’s Information System (rough set theory) which establishes that knowledge is an ability to classify objects, which intrinsically the techniques, methods and algorithms of data mining seek. In this work we look for the objects (operating conditions) that can characterize a behavior of the physical process under study. We pretend to use these extensions in order to extract knowledge from the industrial process.

Footnotes

Acknowledgments

We would also acknowledge the financial support received from the Foundation for Research Support of Minas Gerais state, FAPEMIG; the National Council for Scientific and Technological Development, CNPq; Coordination for the Improvement of Higher Education Personnel, CAPES.

Nomenclature

$\mu$ : Friction coefficient

$\bar{y}$ : Average yielded tensile stress ( $N/mm^{2}$ )

$g$ : Gap ( $m m$ )

$h_{i}$ : Strip input thickness ( $m m$ )

$h_{o}$ : Strip output thickness ( $m m$ )

$M$ : Stiffness Rolling Mill modulus ( $N/mm$ )

$L$ : Rolling load ( $N/mm$ )

$L_{w}$ : Rolling load x Strip width ( $N$ )

$T q$ : Rolling Torque ( $N-mm/mm$ )

$R$ : Roll radius ( $m m$ )

$t_{f}$ : Front tension stress ( $N/mm^{2}$ )

$t_{b}$ : Back tension stress ( $N/mm^{2}$ )

$W$ : Strip width ( $m m$ )

$E$ : Young modulus of the strip material

$S_{i}$ : Yield stress in plane-strain in the entry

$S_{0}$ : Yield stress in plane-strain in the exit

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Extraction of qualitative behavior rules for industrial processes from reduced concept lattice

Abstract

Keywords

1. Introduction

2. Related work

3. Formal concept analysis

Table 1 Example of formal context

.

.

.

.

.

.

.

.

.

.

.

.

.

.

3.2.1 Reduction technique based on the stability index

3.2.2 Reduction technique based on minimum generators

3.3 Extraction and evaluation of qualitative behavior of physical processes using proper implications

5 http://www.icei.pucminas.br/projects/dsrgroup/.

Table 3 Discretization applied in the rolling load ( L ) , friction coefficient ( μ ) and output thickness h o

7 The algorithm proposed in [22] was used to extract proper implications with support greater than zero through the concept lattice. This algorithm is part of the EF-Concept Analysis package [20] in evolution and available at http://www. icei.pucminas.br/projetos/dsrgroup/.

Footnotes

Acknowledgments

Nomenclature

References

Table 1
Example of formal context

⁵
http://www.icei.pucminas.br/projects/dsrgroup/.

Table 3
Discretization applied in the rolling load $(L)$ , friction coefficient $(\mu)$ and output thickness $h_{o}$

⁷
The algorithm proposed in [22] was used to extract proper implications with support greater than zero through the concept lattice. This algorithm is part of the EF-Concept Analysis package [20] in evolution and available at http://www. icei.pucminas.br/projetos/dsrgroup/.