OPTIMIZATION IN ENGINE DESIGN VIA FORMAL CONCEPT ANALYSIS USING NEGATIVE ATTRIBUTES

Jan Zizka et al. (Eds) : CCSIT, SIPP, AISC, CMCA, SEAS, CSITEC, DaKM, PDCTA, NeCoM - 2016
pp. 159–172, 2016. © CS & IT-CSCP 2016 DOI : 10.5121/csit.2016.60115
OPTIMIZATION IN ENGINE DESIGN VIA
FORMAL CONCEPT ANALYSIS USING
NEGATIVE ATTRIBUTES
Rodríguez-Jiménez, J. M, Cordero, P, Enciso, M and Mora, A
Universidad de Málaga, Andalucía Tech,
Boulevar Louis Pasteur SN, Málaga, Spain
jmrodriguez@ctima.uma.es
ABSTRACT
There is an exhaustive study around the area of engine design that covers different methods that
try to reduce costs of production and to optimize the performance of these engines.
Mathematical methods based in statistics, self-organized maps and neural networks reach the
best results in these designs but there exists the problem that configuration of these methods is
not an easy work due the high number of parameters that have to be measured.
In this work we extend an algorithm for computing implications between attributes with positive
and negative values for obtaining the mixed concepts lattice and also we propose a theoretical
method based in these results for engine simulators adjusting specific and different elements for
obtaining optimal engine configurations.
KEYWORDS
Nowledge Discovery, Formal Concept Analysis, Negative Attributes, Attribute Implications
,Engine Design
1. INTRODUCTION
An engine, or motor, is a machine designed to convert one form of energy into mechanical
energy. There are several kinds of engines that are used in different situations. The most popular
are heat engines, including internal combustion engines and external combustion engines, that
burn fuel to create heat, which then creates a force. Some of these engines create electricity that is
used in electric motors that convert electrical energy into mechanical motion. Engines are not
only artificial machines because in nature we can find some examples in biological systems with
the form of molecular motors, like myosin in muscles, that use chemical energy to create forces
and eventually motion in animals and plants.
The performance of an engine measure different kind of properties like engine speed, torque,
power, efficiency or sound levels. Focusing in have the best result in one of them could do that
other properties have a poor result so engineers have to take into account all of these properties in
general.

160 Computer Science & Information Technology (CS & IT)
The cost of engine design is high so engineers have to use simulators to configure models. In the
design of engines, simulators are an excellent tool to save cost of production because in them
engineers can adjust different parameters and observe the global result. Complex engines could
have thousands of parameters that have to be adjusted for obtaining optimal results in different
ways. The problem is: what parameter they have to adjust? How these parameters have to be
modified?
In general usage, design of experiments (DOE) or experimental design is the design of any
information-gathering exercises where variation is present. Engineers could manage this process
under full control, where statistics are usually used, or not, using artificial intelligence methods.
Formal planned experimentation is often used in evaluating physical objects, structures,
components and materials. Design of experiments is thus a discipline that has very broad
application across all the natural and social sciences and, of course, engineering.
Datasets are necessary to collect all the information in experiments results. There are several
kinds of datasets but in engine design all data is measured in numerical values. These datasets can
be reduced, without lose of information, to Formal Concept Analysis compatible datasets where a
binary relation determines if an object has a property or attribute. In this area we can use different
methods and properties that have been developed and can not be used in the original dataset.
The mining of negative attributes from datasets has been studied in the last decade to obtain
additional and useful information. There exists an exhaustive study around the notion of negative
association rules between sets of attributes. However, in Formal Concept Analysis, the needed
theory for the management of negative attributes is in an incipient stage. We proposed in a
previous work an algorithm, based on the NextClosure algorithm [1], that allows obtaining mixed
implications. The proposed algorithm returns a feasible and complete mixed implicational system
by performing a reduced number of requests to the formal context.
Knowledge discovering is nowadays a well established discipline focussed on the development of
tools and techniques to reveal useful information hidden in data sets. Its main goal is to detect
patterns to improve decision making and is approached using pattern recognition, clustering,
association and classification. Part of these patterns is expressed as implications (or association
rules) which allow us to address information using a formal notation and to manage them
syntactically by using logic.
In Formal Concept Analysis we have 3 related items such that we can manage them to obtain
knowledge: Implications, Concepts and Context.
Figure 1. Items in Formal Concept Analysis

Computer Science & Information Technology (CS & IT) 161
Contexts are data sets with a binary relation I, between a set of objects G and their attributes M.
Implications are formulas in the form A→B where A and B are subsets of a certain set (universe)
of attributes M. Both subsets of attributes are considered to be conjunctive cubes, i.e. A=a1 ∧ …
∧ an and B= b1 ∧ … ∧ bm.
Concepts are closed sets that have a pair formed by a subset of objects and a subset of attributes
that are related with two mappings that are called derivation operators. We can calculate the
closure of a subset of attributes with respect to a set of implications and establish the concept
associated.
Formal Concept Analysis have a extended set of applications in real life that allows to manage
information to extract knowledge from a context, a set of implications or a set of concepts: Social
Networks, Marketing, Recommendation Systems,...
We are going to focus in the existence of logics developed to specify and manage sets of
implications. The pioneer of these logics was the one introduced by W.W. Armstrong [2], which
was proven to be sound and complete.
In this work we use implications in the area of formal concept analysis [3] and assume the
common interpretation in this environment: given a formal context K over a set of attributes ,
the implication A→B asserts that any object which have all the A attributes, also has all the B
attributes. Although we focus on knowledge discovery techniques to extract implications in
formal concept analysis, this problem is similar to the extraction of functional dependencies or
association rules from an arbitrary data set.
One of the former researchers who points out the importance of this problem in datasets was H.
Mannila [4] and it was also studied by other researches from the database areas like S. Navathe
[5]. In this work we will address the mining of implications with negation. The extended
implications allow us to relate items which conflict with each other. While classical implications
express that {``cyclist with short and sharp accelerations are great sprinters"}, implications with
negations allow us to express that {``cyclist with short and sharp accelerations are not great
climbers"}.
Since implication formulas are built using a binary connective which relates two conjunctive
clauses, negation is only considered at the attribute level, i.e. the negation A is considered the
conjunctive clause of its negated attributes: naa ∧∧Κ1 . We want to increase the
expressiveness to exploit all the properties that Formal Concept Analysis had, but not result in
much higher costs.
Notice that extended implications cannot be considered as the negation of a classical implication.
We do not want to express that a certain implication does not holds but the evidence of the
absence of a certain attribute because this is a solved question.
Applications of mixed attributes are usual in data mining but not in Formal Concept Analysis.
There are some usual approaches, for example detecting errors [6], which provide benefits to
knowledge discovery.

2. PRELIMINARIES
In this section, the basic notions related with Formal Concept Analysis (FCA) [7] and attribute
implications are briefly presented. See [3] for a more detailed explanation.
A formal context is a triple K=<G,M,I> where G and M are finite non-empty sets and I⊆GxM
is a binary relation. The elements in G are named objects, the elements in M attributes and
<g,m>∈I means that the object g has the attribute m. From this triple, two mappings ↑:2G
→ 2M
and ↓:2M
→2G
, named derivation operators, are defined as follows: for any A⊆G and B⊆M,
A↑
= {m∈M| for each g∈A: <g, m>∈I}
B↓
= {g∈G| for each m∈B: <g, m>∈I}
A↑
is the subset of all attributes shared by all the objects in A and B↓
is the subset of all objects
that have the attributes in B. The pair (↑,↓) constitutes a Galois connection between 2G
and 2M
and, therefore, both compositions are closure operators.
A pair of subsets <A, B> with A⊆G and B⊆M such A↑
=B and B↓
=A is named a formal
concept. A is named the extent and B the intent of the concept. These extents and intents coincide
with closed sets wrt the closure operators because A↑↓
=A and B↓↑
=B. Thus, the set of all the
formal concepts have a lattice structure, named concept lattice}, with the relation
<A1, B1> ≤ <A2, B2> if and only if A1 ⊆A2 (or equivalently, B2 ⊆B1)
The concept lattice can be characterized in terms of attribute implications. An attribute
implication is an expression A→B where A, B⊆M and it holds in a formal context if A↓
⊆ B↓
.
That is, any object that has all the attributes in A has also all the attributes in B. It is well known
that the sets of attribute implications that are satisfied by a context satisfy the Armstrong's
Axioms:
[Ref] Reflexivity: If B⊆A then ├ A→B.
[Augm] Augmentation: A→B ├ A∪ C→B∪ C.
[Trans] Transitivity: A→B, B→C ├ A→ C.
A set of implications B is an implicational system for K if: (1) any implication from B holds in K
and (2) any implication that K satisfies follows (can be inferred) by using Armstrong's Axioms
from B.
3. OUR APPROACH FOR EXTENDING FORMAL CONCEPT ANALYSIS
WITH NEGATIVES ATTRIBUTES
From now on, the set of all the attributes is denoted by M and its elements by the letter m
possibly with sub indexes. The elements in MM ∪ are going to be denoted by the first letters in
the alphabet: a, b, c … So, the symbols a, b, c … could represent positive or negative attributes.
Capital letters A, B, C … denote subsets of MM ∪ . If A ⊆ MM ∪ , then A denotes the set of
the opposite of attributes in A. That is, A = { a | a∈A} where a =a. Moreover, for A⊆ MM ∪
, the following sets are defined:

Pos(A)= {m∈M| m∈A}
Neg(A)= {m∈M| m ∈A}
Tot(A)= Pos(A)∪Neg(A)
and, therefore, Pos(A), Neg(A), Tot(A)⊆M.
The traditional derivation operators defined in Formal Concept Analysis are modified in [8] to
consider the new framework.
Definition 1: Mixed Concept-forming Operators. Let K=<G, M, I> be a formal context. We
define the operators ⇑:2G
→2 MM ∪ and ⇓:2 MM ∪ →2G
as follows: for A⊆G and B⊆
MM ∪ ,
A⇑
= {m∈M|<g, m>∈ I for all g∈A}∪{ m ∈ M | <g, m>∉I for all g∈A}
B⇓
= {g∈G| <g, m>∈ I for all m∈B}∩{g∈G|<g, m>∉I for all m ∈B}
We prove in [9] that the adaptation of the derivation operators form a Galois connection.
Theorem 1: Let K=<G, M, I> be a formal context. The pair of derivation operators (⇑,⇓)
introduced in Definition 1 is a Galois Connection.
Example 1: Chemical analysis of oil in motors allows detecting problems inside them.
Table 1. Presence of chemical residuum in analysis of oil.
Fe Cu Pb Al Si
o1 0 1 1 1 0
o2 0 1 0 1 0
o3 1 0 0 1 0
o4 0 0 0 0 1
{o1,o2}↑
={Cu, Al}≠{ Fe ,Cu, Al, Si }={o1,o2}⇑
Also, we adapt the definitions of formal concept and implication to allow the use of negative
attributes inside them.
Definition 2: Let K=<G, M, I> be a formal context. A mixed formal concept in K is a pair of
subsets <A,B> with A⊆G and B⊆ MM ∪ such A⇑
=B and B⇓
=A.
Definition 3: Let K=<G, M, I> be a formal context and let A,B⊆ MM ∪ , the context K
satisfies a mixed attribute implication A→B, if A⇓
⊆ B⇓
.
There are special cases of mixed attributes subsets that have important properties for the proposed
algorithm and reduce the complexity of it. Missaoui [10] works with intents that have empty
support, i.e. have any m∈M such that mm in the intent, so these subsets of attributes doesn't
appear in the real life. We want to discard them because they are not useful for our purposes.
Definition 4: Let K=<G, M, I> be a formal context and a set A⊆ MM ∪ is named consistent
set if Pos(A)∩Neg(A)=∅. The set of all consistent sets is denoted Ctts.

Definition 5: Let K=<G, M, I> be a formal context and a set A⊆ MM ∪ is said to be full
consistent set if A∈Ctts and Tot(A)=M.
We are going to extend the logic that propose Armstrong's Axioms with a sound and complete
system proposed by Missaoui with a global framework and not for particular problems.
Therefore, the following axioms are added to the Armstrong's axioms: for all a, b∈ MM ∪ and
A⊆ MM ∪ ,
[Cont] Contradiction: ├ a a → MM .
[Rft] Reflection: Aa→ b ├ Ab → a .
The algorithm for calculating the implicational system that we propose in [8] uses the axiomatic
system and, considering the full consistent sets, the dataset have to be checked 3|M|
-2|M|
times in
the worst case.

4. ALGORITHM FOR MIXED CONCEPTS AND MIXED ATTRIBUTE
IMPLICATIONS
If we want to calculate both, mixed implicational system and set of mixed concepts, we can
extend algorithm 2 [8]. We know that we only have to modify a few lines because it works with
closures.
To adapt the algorithm, we have to pay attention to the condition While Y< M because it has to be
changed to While Y≠M for detecting the concepts, so the time of calculating arise. Because we
only have to add full consistent sets, that are the minimal concepts that have the intents of the
objects, we don't need to change this condition, and only add these intents.
If we choose the first option we have to do 2|M|
operations that always be bigger or equal than |G|.
So, the modifications are in lines 12-13, that take the closed sets that we ignore in Algorithm 1,
and lines 15-16 that add to the sets of intents of concepts the full consistent sets.
With this algorithm we extract all possible implications and concepts from the context. In the
worst case, we have to check 3|M|
-2|M|
+|G| times the dataset that not depend of different amount of
objects. The dataset could be mixed-clarified and the number of possible objects has a maximum
in 2|M|
.
5. OPTIMIZING ENGINE DESIGN
In engine simulators, each component has specific measures that determine how these
components work. Width of a simple pipe could increase or decrease the amount of fuel that the
engine receive, so all this measures are important in the development of each engine. Each
measure could be considered as an attribute of the engine and also, the relation among them
determine the global efficiency. All these measures are collected in a dataset that show how the
engine works but it is difficult to decide specifically which of them we can change and how we
have to change it because we don't know how these changes affect engine performance. Taguchi
[11] refers that development engineers have to apply parameter design methods to make the basic
functions approach the ideal functions under real conditions. These design activities should be

conducted by research and development departments before the product is finally created.
Taguchi's work is based in statistical methods that actually are replaced with neural networks
processes or Multi-objective Covariance Matrix Adaptation Evolutionary Strategy methods [12,
13]
The method that we propose tries to focus on these specific parts of the engine that could increase
the performance and change these values for obtaining the best result. Researchers could observe
that is similar to a self-organized map [14], with the difference that use properties of mixed
concept lattices and we only have to change a reduced number of attributes, reducing the cost of
the procedure.
Our starting point is different experimental configurations of the engine that experts consider as
initial results. Let G be the set of different collected configurations (objects) and M the set of
elements that could be measured adding a value called objective function (attributes). Value of
the objective function depends on different values of attributes, for example number of
revolutions, but could not be associated with a particular equation. However, in the next example,
for demonstration purposes, we have to consider a polynomial function.
Example 2: Let consider domain of possible values [0, 10] for all attributes and the objective
function is a2
+(b-5)+(c-5)-(d-5)-e2
, so objective domain is [-115,115]. The attributes with higher
weights are a and e, so changing these values we can modify the objective function in different
ways.
Let's consider that experts determine that the optimal value for the objective is 100 and we want
to know what are the preferred size for different parts of the engine for obtaining the nearest
value to the proposed optimal.
Table 2: Dataset with engine configurations
T Part a Part b Part c Part d Part e Objective
c1 3 1 7 2 3 1
c2 4 3 2 3 4 -3
c3 6 5 4 4 2 32
c4 6 0 3 6 5 3
c5 2 7 7 3 1 9
Let T be a dataset with size |G|*|M| that considers configurations in rows and measures of each
element of the engine in columns. Elements in T, (tx,y with x∈G, y∈M) are numbers that
represents each different value for the attributes but not necessary represents a formal context.
We have to point to the objective function and determine the selected value for this function that
experts consider as optimal. Different values for objective function could be near from the
optimal value but we need the optimal or, in cases where it is not possible, the nearest one.
A specific number k, with 1<k<|G|, have to be selected that determine the size of the control
fixed group. This group Gk is compounded by k configurations which have the nearest objective
values with respect to the optimal value. For each attribute m∈M, we can fix an interval
Inm=[lm,um] being lm=inf(tx,m) and um=sup(tx,m) with x∈Gk.

Now we can adapt T to a compatible dataset T1
in Formal Concept Analysis in this way. Let
define the values of the adapted dataset as t1
x,y=1 if tx,y∈Iny=[ly,uy]; 0 otherwise
This adaptation represents all the values that are in the fixed intervals with a binary relation. It is
obvious that, for all g∈Gk, m∈M, t1
g,m=1 so we have to focus in the absence of attributes in the
sense of Formal Concept Analysis. The absence of attributes means that these values are out of
the fixed intervals (fixed by Gk) and points to possible candidates for replacing their values.
Example 3: Fixing k=2 and optimal value for objective 100 we have that Gk={c3,c5} and
Ina=[2,6], Inb=[5,7], Inc=[4,7], Ind=[3,4], Ine=[1,2]
Table 3: Adapted formal context to T
T1
Part a Part b Part c Part d Part e Objective
c1 1 0 1 0 0 1
c2 1 0 0 1 0 -3
c3 1 1 1 1 1 32
c4 1 0 0 0 0 3
c5 1 1 1 1 1 9
The problem that we have to solve is what tx,y∈T we have to change and what values could have
tx,y.
Using Algorithm 3 with the context <G-Gk, M, T1
>, we can obtain the mixed concepts lattice.
The top element is the mixed concept <G-Gk, (G-Gk)⇑
>. Since the lattice has an order, the upper
concept that has the less number of negative attributes (not zero) determines the selected set of
attributes to change. We are going to call this subset of attributes N.
For avoiding local fix points for the objective function, we have to consider 4 different values for
each attribute that we are going to change in T: original value, average between minimum
possible value for attribute y and ly, average between maximum possible value for attribute y and
uy, and average for the interval Iny, for each y∈N, i.e. for each configuration, we have to consider
4|N|
combinations.
If changing attributes in N we do not obtain better objective values, we have to follow in order
the mixed concept lattice and get the next mixed concept for obtaining the new N. If proposed
changes do not offer a better objective value for all the objects that are not in Gk, we have to
finish our search.
This method have to be repeated recursively while distance between objective value and optimal
value in Gk are higher than distance between objective value and optimal value in G-Gk after
checking all possible combinations.
Example 4: Obtaining mixed concept lattice for G-Gk, the top element of the lattice point to the
concept with attributes abe . Parameter a is positive so it is not necessary to be changed.
Focusing in negative attributes, we have to change parameters b, e in c1, c2 and c4, resulting Table
4.

Figure 2. Mixed concept lattice for T1
restricted to c1, c2 and c4
Table 4. Adapted formal context for T
T Part a Part b Part c Part d Part e Pre. Obj. New Obj.
c1 3 8.5 7 2 0.5 1 17.25
c2 4 8.5 2 3 0.5 -3 18.25
c3 6 5 4 4 2 32
c4 6 8.5 3 6 0.5 3 36.25
c5 2 7 7 3 1 9
We can observe that all the objectives values have changed and, in this case, c1, c2 and c4 have an
objective value higher than c5 that was in Gk in the first step, and value of c4 is higher than c3 so
the distance to the optimal value decreases. Values for b increases and values for e decreases.
In the next step, Gk=c3,c4 and, recursively, we change in second step groups a, c, d; third step a, c,
d; fourth step a, c, d; fifth step a, b, d, e and we find that has the same objective value that optimal
selected by experts.
In 2 steps we have to check 3*(42
)+9*(43
)+44
=880 combinations that are less than all possible
combinations for each configuration, for example, if we only take into account positive integers,
there are 105
=100000 combinations. If we consider decimals our method is not affected but the
possible combinations grow.
In this example, if we consider minimum value for y instead of average between minimum
possible value for y and ly and maximum value for y instead of average between maximum
possible value for y and uy, for each y∈N, we only have to do 112 combinations to reach the
proposed optimal value, but this example have a particular solution using interval extremes, that
is not the general case.
This method could be included in simulators as an artificial intelligence tool for adjusting
parameters. Being an automatic tool, engineers do not need to examine and calibrate the
simulator trying to detect what parameter has to change and what value is needed. Configuration
for optimal value is not always reached due to parameters restrictions and the proposed original
configurations, but a close value could be obtained.
As other methods, detecting local fix points is a problem that can be solved with a different set of
initial configurations as a start point.

6. EXPERIMENTS
Due to legal problems in the use of confidential data, we can not check the method proposed in
the previous section in real simulators, so we have to do simulations with different equations.
These equations have been chosen for its simplicity of calculation and complexity and diversity
of results according to the parameters entered.
6.1.Experiment 1
Table 5. Results from Experiment 1
k |G| Objective Init. Dist. Final Dist. Iter. % Reduc.
2 5 930 577.73 0.29 4.1 0.05
2 10 930 418.70 0.75 4.7 0.18
2 20 930 197.20 1.11 4.2 0.56
3 5 930 796.80 0.68 4.1 0.09
3 10 930 224.65 0.30 5.6 0.14
3 20 930 117.62 0.41 3.6 0.35
2 5 27030 14403.09 0.05 11.0 0.00
2 10 27030 14339.90 0.59 18.0 0.00
2 20 27030 10332.63 0.78 19.3 0.01
3 5 27030 15115.54 0.10 11.0 0.00
3 10 27030 13309.24 0.44 14.0 0.00
3 20 27030 13819.18 0.34 21.5 0.00
2 5 27900 19507.26 0.12 19.8 0.00
2 10 27900 15730.50 0.43 31.1 0.00
2 20 27900 14026.64 0.96 46.7 0.01
3 5 27900 16117.79 0.05 15.5 0.00
3 10 27900 17525.48 0.81 34.7 0.01
3 20 27900 11296.91 0.76 38.1 0.01

The size of M was fixed in 9 attributes which have their domains in [0,10] that allows decimal
values. The objective function for this experiment is ∑=
3
1i
tx,i+( ∑=
6
4i
tx,i)2
+( ∑=
9
7i
tx,i)3
for each
configuration x∈G. This function has values in [0,27930] and has 3 groups where we can
interexchange values with the same result, i.e. if we change values in tx,4 and tx,5 objective
function does not change.
We are going to use different values for k and |G| for the objectives 930, 27030 and 27900 that
are the different configurations obtained decreasing one of the three groups to their lower values
and the other groups to the upper values. 10 experiments were done with different random initial
values for each fixed value for k, |G| and objective and the average results are presented in Table
5. Initial distance is the difference in absolute distance from the best original objective value to
the optimal value, final distance is the absolute distance from the best objective value to the
optimal value after applying the method, iterations are the number of recursive executions of the
method and reduction is the proportion between initial distance and final distance.
In this experiment we can see that, for the objective 930, there are multiple combinations of
values that reach this objective and our method has difficult to locate one of them specifically due
to the several fix points. For objectives 27030 and 27900, the number of possible combinations is
lower, so our method can focus in particular combinations with a relevant percentage of reduction
from the original distance to the optimal value. Is an interesting observation that low size of |G|
has the nearest solutions to the optimal value.
6.2.Experiment 2
The size of M was fixed in 9 attributes which have their domains in [0,10] that allows decimal
values. The objective function for this experiment is ∑=
9
1i
(tx,i-i)2
for each configuration x∈G.
This function has values in [0,485] and has a unique combination of values that reach the
objective value 0. The difficulty to reach the optimal value in 0 is that each attribute have to get a
value that is not in the extremes of its domain.
We are going to use different values of k and |G| simulating 10 experiments with different
random initial values for each fixed value for k and |G|. The average results are presented in
Table 6.
In this experiment, final distance is, more or less, similar in the average value of each experiment.
We have a specific case where k=3 and |G|=5 where the solution is the nearest one to the optimal
value. Since initial distance is lower when |G| increase, the percentage of reduction grows, but we
can see that there is not a special relation between k and |G| in this experiment for reaching the
nearest solution to the optimal value.

Table 6. Results from Experiment 2
k |G| Init. Dist. Final Dist. Iter. % Reduc.
2 5 72.87 2.68 5.8 3.66
2 10 52.87 2.67 7.5 5.06
2 20 54.30 2.66 8.5 5.33
2 30 44.40 2.70 9.8 6.09
3 5 93.97 0.73 8.0 0.77
3 10 66.30 2.56 8.9 3.86
3 20 52.78 2.36 8.1 4.48
3 30 45.14 2.11 9.2 4.69
4 5 89.72 2.03 8.7 2.27
4 10 64.41 2.56 7.8 3.97
4 20 47.16 2.29 9.3 4.86
4 30 47.08 2.46 10.9 5.21
7. CONCLUSIONS AND FUTURE WORK
In this paper, a new algorithm for obtaining mixed concept lattices from a context is introduced.
This algorithm is an important tool that let us to consider how attributes in the formal context are
related among them, focusing our attention to relevant information.
The algorithm also produces an implicational system that we do not use in this work but is a
complementary tool that gives us specific information about relations between attributes and was
the essential idea to design the method that we propose.
This method for adjusting parameters in simulators is not a new idea, but we want to extract the
knowledge hidden in datasets using formal concept analysis and, in particular, using the
knowledge that negative attributes provide.
Experiments let us to check that this method could be an interesting tool for simulators with a
relevant reduction in production and development costs.
In a future work, we want to develop new algorithms to reduce the time of calculating the mixed
concepts lattice adapting different algorithms from Formal Concept Analysis.

Values of k and |G| have to be studied with different situations, for example changing
|M|, checking what the best initial values for these parameters are.
ACKNOWLEDGEMENTS
Supported by grant TIN2011-28084 and TIN2014-59471 of the Science and Innovation Ministry
of Spain, co-funded by the European Regional Development Fund (ERDF).
We want to thanks Dr. Juan Cabrera and Dr. Juan Castillo for the approach to solving the
problem discussed in this article.
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