Granular Cognitive Map Reconstruction

Adjusting Granularity Parameters

Wladyslaw Homenda

, Agnieszka Jastrzebska

and Witold Pedrycz

2,3

Faculty of Mathematics and Information Science

Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warsaw, Poland

Systems Research Institute, Polish Academy of Sciences, ul. Newelska 6, 01-447 Warsaw, Poland

Department of Electrical & Computer Engineering

University of Alberta, Edmonton T6R 2G7 AB, Canada

Keywords:

Fuzzy Cognitive Maps, Fuzzy Cognitive Map Reconstruction, Granular Cognitive Maps, Granular Cognitive

Map Reconstruction, Information Granules.

Abstract:

The objective of this paper is to present developed methodology for Granular Cognitive Map reconstruction.

Granular Cognitive Maps model complex imprecise systems. With a proper adjustment of granularity pa-

rameters, a Granular Cognitive Map can represent given system with good balance between generality and

speciﬁcity of the description. The authors present a methodology for Granular Cognitive Map reconstruc-

tion. The proposed approach takes advantage of granular information representation model. The objective of

optimization is to readjust granularity parameters in order to increase coverage of targets by map responses.

In this way we take full advantage of the granular information representation model and produce better, more

accurate map, which maintains exactly the same balance between generality and speciﬁcity. Proposed method-

ology reconstructs Granular Cognitive Map without loosing its speciﬁcity. Presented approach is applied in

a series of experiments that allow evaluating quality of reconstructed maps.

1 INTRODUCTION

Cognitive maps are abstract soft computing mod-

els, which allow describing complex systems ﬂexi-

bly. Cognitive Maps represent knowledge and rela-

tionships within knowledge in a form of a directed

graph. Nodes represent concepts (units or aggre-

gates of information). Edges between the nodes rep-

resent relations between the knowledge gathered in

such map. A very important milestone in research

on cognitive maps is deﬁnition of Fuzzy Cognitive

Maps (FCMs) by B. Kosko in 1986, (Kosko, 1986).

Fuzzy Cognitive Maps combine cognitive maps with

fuzzy sets. They became powerful modeling frame-

work and several practical applications of FCMs have

been proposed, (Papageorgiou and Salmeron, 2013),

(Papakostas et al., 2008).

Later research on imprecise information represen-

tation models has brought further generalizations of

knowledge units, (Zadeh, 1997). Granular Comput-

ing has emerged as an important branch in infor-

mation sciences, (Bargiela and Pedrycz, 2003). In-

formation granules generalize units or aggregates of

knowledge. Modeling capabilities of a model built

on the grounds of granular information are widened.

Granularity elevates existing models, by introducing

a controlled balance between speciﬁcity of knowledge

granules and generality of the described phenomena.

In this paper the authors discuss Granular Cogni-

tive Maps (GCM), a generalization of cognitive maps

based on knowledge granules. In the start point of

our study we place Fuzzy Cognitive Maps, which get

augmented to Granular Cognitive Maps. Most impor-

tantly, we propose a methodology for Granular Cog-

nitive Map reconstruction.

The paper is structured as follows. In Section 2

we present methodology of Granular Cognitive Map

reconstruction. The proposed approach is applied in

Section 3.

2 METHODOLOGY

In this paper authors introduce a methodology for

Granular Cognitive Map reconstruction. We would

like to stress at the beginning that with our procedure

one is able to build a Granular Cognitive Map based

175

Homenda W., Jastrzebska A. and Pedrycz W..

Granular Cognitive Map Reconstruction - Adjusting Granularity Parameters.

DOI: 10.5220/0004869301750184

In Proceedings of the 16th International Conference on Enterprise Information Systems (ICEIS-2014), pages 175-184

ISBN: 978-989-758-028-4

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

only on targets and activations (nomenclature will be

discussed in following paragraphs). In this perspec-

tive, we may also call our methodology a Granular

Cognitive Map construction procedure. We intend to

use the term reconstruction, because in our opinion it

reﬂects better character of the methodology. Granular

Cognitive Maps describe certain systems (sets of re-

lated phenomena). Such system is faithfully modeled

with an ,,ideal” Granular Cognitive Map. Our pro-

cedure aims at reconstruction of this ,,ideal” GCM.

Second premise speaking for the term reconstruction

derives from the validation scheme, which we use to

assess quality of our methodology.

2.1 From a Fuzzy Cognitive Map to a

Granular Cognitive Map

Let us start the discussion with a drawing that visual-

izes a cognitive map. Figure 1 shows a cognitive map

Figure 1: A cognitive map.

consisting of n = 3 nodes. Nodes are connected with

directed edges that represent relationships. In Fuzzy

Cognitive Maps such relationship may take a value of

a real number from the [−1, 1] interval. Connections

are gathered in a weights matrix, which is n×n in size

and is denoted as W . Information relevant to FCM

states is available in activations. Activations, denoted

as X , are in n × N matrix, where N is the number of

observations. Activations are forces that are relevant

for n nodes in N observations. Activations take values

of real numbers from [0, 1] interval.

Activations together with weights matrix allow to

compute map responses in N observations, according

to a general formula:

Y = f (W ∗ X) (1)

where ∗ is an operation performed on matrices W and

X, which produces a matrix W ∗ X of size nxN, and

f is a mapping applied to individually to elements of

W ∗ X. Matrix product is an example of such opera-

tion and it is utilized in this study.

Let us denote i-th row, j-th column and an element

in i-th row and j-th column of a matrix A as A

i·

, A

· j

and A

i j

, respectively. In order to compute map’s re-

sponse to k-th activations (response in k-th iteration),

we apply a formula:

·k

= f tras(W · X

·k

) (2)

and, more speciﬁcally, i-th node response in k-th iter-

ation is computed by multiplying:

= f trans(W

i·

· X

·k

) (3)

where f trans is a nonlinear non-decreasing transfor-

mation function. f trans : R → [0, 1]. In this paper we

use sigmoid function:

f sig(z) =

1 +exp(−τz)

, τ > 0 (4)

We chose the τ parameter equal to 2.5 based on ex-

periments.

Map responses are collated with ,,ideal”, real val-

ues, which are called targets (T GT ). The closer map

responses are to targets, the better the map. Quality

of a Fuzzy Cognitive Map is assessed by calculating

error statistics regarding discrepancies between Y and

T GT .

In our previous research we have proposed

a methodology for Fuzzy Cognitive Map reconstruc-

tion. The proposed procedure is based on gradient-

based error minimization. Outcome of FCM recon-

struction is a weights matrix that describes connec-

tions within a map. Weights are adjusted so that dif-

ferences between Y and T GT are the smallest. Let

us assume that we have such reconstructed FCM.

Thereby, we have optimized weights matrix, denoted

as W

f in

, activations X and targets T GT .

Augmentation to Granular Cognitive Map may

occur at two levels: weights and responses. In our ex-

periments we focus on the ﬁrst approach, where aug-

mentation to knowledge granules concerns weights.

Research on methodological details on Granular Cog-

nitive Maps in general and on augmentation from

FCM to GCM is in (Pedrycz and Homenda, 2012).

Augmented, granular weights, denoted as W

ﬁn

together with activations X produce granular outputs:

Y = f (W

ﬁn

?X) (5)

where ? is a speciﬁc operator, applicable to chosen

representation model of knowledge granules.

In Granular Cognitive Maps information (or ag-

gregates of information) are represented with knowl-

edge granules. There are several knowledge descrip-

tion models that may be used to deﬁne granules, in-

cluding:

• intervals,

• triangular fuzzy numbers,

• parabolic fuzzy numbers,

• others.

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In this study we use intervals for granules representa-

tion.

In the process of Granular Cognitive Maps con-

struction and exploration following elements are rele-

vant: activations (X), granular weights (W

ﬁn

), granu-

lar map responses (Y), additional parameters of gran-

ularity and targets (TGT ).

Elevation to granular information representation

model requires adjustment of additional parameters.

Speciﬁcs of granularity parameters depend on chosen

granules representation scheme. In the case of inter-

vals additional parameters, relevant for GCM, are:

• size of knowledge granule ε,

• symmetry parameter γ.

Construction of a GCM requires a conceptual settle-

ment of a conﬂict between speciﬁcity and generality.

Generality of knowledge granule is directly linked to

the ε parameter - size of the granule. In the case of

interval-based model it is the length of the interval.

The bigger given granule, the more general concept it

can describe. Speciﬁcity criterion is at variance with

generality. Speciﬁcity is translated to precision. The

more speciﬁc knowledge granules, the less fuzzy the

description of the corresponding phenomena or rela-

tions. In the process of a Granular Cognitive Map re-

construction we manipulate with the length of inter-

vals with the aim to achieve a compromise between

generality and speciﬁcity.

The γ parameter determines symmetry of knowl-

edge granules. Center of a knowledge granule is

where feature evaluation is at its peak. γ = 0.5 means

that the granule is symmetrical. Following formulas

allow to elevate represented knowledge from fuzzy to

granular model based on intervals:

−

= a

− ε∗ γ ∗|range

| (6)

= a

+ ε∗ γ ∗|range

| (7)

where a

−

and a

are lower and upper limits of the in-

terval that represents the knowledge granule a. Aug-

mentation from fuzzy to granular model can occur

at various levels. In this study we elevate weights

towards granular weights. As a result, the maximal

length of the interval is 2.

In the case of Granular Cognitive Map quality

assessment we are interested in coverage. Con-

cept of coverage is informally illustrated in Figure 2.

Granular map responses in Figure 2 are denoted as

Figure 2: Coverage of a target by a granular map response.

−

, y

]. These are upper and lower limits of the in-

terval, which represents the granule. In the Figure 2

target is not covered by the map response. We aim

to construct such a map that produces outputs, which

cover as many targets as possible.

A Granular Cognitive Map gives granular map re-

sponses that should cover real, observed values - tar-

gets. There are many possible deﬁnitions of coverage.

We would like to discuss two most intuitive kinds of

coverage for an interval-based GCM:

• weak coverage,

• strict coverage.

Weak coverage account all targets that fall into map

responses. Weak coverage of an i-th node in k-th ob-

servation is calculated as:

covweak

(

1, if T GT

∈ [y

−

, y

]

0, otherwise

(8)

Weak coverage for all targets is calculated as follows:

covweak =

∑

k=1

∑

i=1

covweak

N ∗ n

(9)

Weak coverage is averaged by dividing by the number

of observations and the number of nodes. It informs,

to what extent the Granular Cognitive Map covered a

single data point.

In contrast, strict coverage is increased only if all

nodes within k-th observation were covered by map

responses.

covstrict

(

1, if T GT

∈ [y

−

, y

]

0, otherwise

(10)

Strict coverage requires that the whole column of tar-

gets is covered by map responses. Strict coverage of

all targets is calculated as follows:

covstrict =

∑

k=1

covstrict

(11)

Strict coverage punishes columns that were covered

incompletely. It is coverage criterion much harder to

satisfy.

2.2 Methodology for Granular

Cognitive Map Reconstruction

The developed procedure for Granular Cognitive Map

reconstruction aims at ﬁnding such a map that max-

imizes coverage of targets by map responses with

given constraints regarding granularity parameters. In

this section we discuss the methodology of the GCM

reconstruction. Moreover, we present a procedure for

validation.

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177

Figure 3: Granular Cognitive Map reconstruction procedure.

Let us recall that information granularity entails

certain restrictions. The conﬂict between granule gen-

erality and speciﬁcity on one hand limits the model,

but on the other hand it balances precision with ver-

satility. In a Granular Cognitive Map based on in-

tervals as granules representation scheme, general-

ity/speciﬁcity is controlled with lengths of intervals.

Figure 3 summarizes the methodology of a GCM

reconstruction and quality assessment.

The algorithm in Figure 3 in its full state recon-

structs a Granular Cognitive Map and assesses its

quality based on 3 datasets:

• train not distorted,

• train distorted,

• test.

Dataset denoted as not distorted train is an ideal, per-

fect, dataset, which is never available in real-life in-

formation processing. It is only available for experi-

ments on artiﬁcially generated data. In this paper we

introduce the developed methodology of GCM recon-

struction. Therefore, we use the perfect train dataset

for model quality assessment purposes.

The perfect data is in real life never available. Sys-

tem modeling methodologies take into account two

kinds of distortions: random and systematic. We as-

sume existence of perfect data, but in fact we oper-

ate on distorted dataset. Therefore, the map is trained

with respect to the randomly distorted train dataset.

Random distortions are introduced prior to the ex-

ecution of the GCM reconstruction algorithm. Distor-

tions are added to targets as random values from the

normal distribution with standard deviation equal to

0.4. In consequence, T GT

contains signiﬁcant num-

ber of 0s and 1s, which cannot be model outputs with

sigmoid function as in Formula 4. Due to asymptotic

properties of sigmoid function we cannot expect the

map to reach targets, which are equal to 0 or 1.

Test dataset is used to assess quality of the map.

At the start point in our algorithm we get pre-

trained FCM with weights matrix W

f in

. We have also

activations X and distorted targets T GT

. We ele-

vate weights matrix W

f in

to granular weights matrix,

denoted with boldface font: W

ﬁn

using initial val-

ues of granularity parameters ε and γ. Granularity of

information is propagated, activations with granular

weights produce granular map responses Y

ini

At this point we have a Granular Cognitive Map

that is not yet optimized. The optimization aims

at coverage maximization and it can be performed

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through adjustment of several parameters of the Gran-

ular Cognitive Map:

• weights W

ﬁn

• ε - single value for all weights or ε matrix with

adjusted value for each weight,

• γ - single value for all weight or γ matrix with ﬁt-

ted value for each weight.

We may adjust simultaneously or successively one of

the above elements or more. In Figure 3 the step of

optimization is in the blue box. In this article we focus

on adjustment of the granularity parameters: ε and γ.

We do not interfere with the weights matrix. Instead,

we try do explore to the greatest extent the beneﬁts

of chosen granular knowledge granules representation

model - intervals.

Coverage maximization task is computationally

challenging. The optimization procedure has to in-

dependently adjust multiple parameters and the max-

imization criteria (see Formulas 11 and 9) are discon-

tinuous. Therefore, we have applied particle swarm

optimization method. PSO (introduced in (Kennedy

and Eberhart, 1995) and (Shi and Eberhart, 1998))

does not require that the optimization problem be dif-

ferentiable. It can search within a very large space

of candidate solutions. The drawback of choosing a

metaheuristics is that we do not have any guarantee

that the optimal solution will be found.

In literature there is a discussion on practical as-

pects of optimization in Fuzzy Cognitive Maps learn-

ing and exploration, for example: (Papakostas et al.,

2012), (Stach et al., 2005) and (Stach et al., 2004).

The topic of Granular Cognitive Maps and optimiza-

tion has not been yet researched and documented.

Optimized Granular Cognitive Map gives new,

granular responses denoted as Y.

The quality of the reconstructed Granular Cog-

nitive Map is assessed on the three aforementioned

datasets. We calculate coverage statistics with respect

to all three datasets before and after the optimization:

• before optimization: coverage of T GT

ini(tial)

, coverage of T GT by Y

ini

and coverage

of T GT

T (est)

by Y

T(est)ini(tial)

• after: coverage of T GT

by Y, coverage of T GT

by Y and coverage of T GT

by Y

3 EXPERIMENTS

In this section authors apply the proposed methodol-

ogy in a series of experiments. Different approaches

to Granular Cognitive Map reconstruction were tested

and compared for the same map (n = 8, N = 24, the

same X and T GT

datasets). We reconstruct the GCM

by adjustment of granularity parameters: ε and γ for

the interval-based representation of knowledge gran-

ules. Optimization procedure maximizes weak cov-

erage deﬁned in Formula 9. In this paper we adjust

matrix of ε and/or matrix of γ. Matrices contain sepa-

rate parameters for each weight.

Please note that ε

i j

∈ [0, 2], i, j = 1, .. . , n. 2 is

maximal length of the interval for granular weights,

for example in the case when granule center is in 0.

ε deﬁnes knowledge granule size. γ

i j

are symmetry

parameters and γ

i j

∈ [0, 1], i, j = 1, . . ., n. For γ = 0.5

the granule is symmetrical and granule center is in the

middle of the interval.

Results presented in this section allow to review

the inﬂuence on γ parameter with restricted ε on the

coverage. We use common plotting scheme in each

subsection. The most important aspect of this section

is that as a result of optimization, we substantially

increase coverage and maintain the same generality

of the model.

In Figures in the following section, in each data

point the total speciﬁcity of the map before optimiza-

tion is the same as after the optimization. Thanks to

the readjustment methodology we increase coverage

and retain the same balance between speciﬁcity and

generality. Such improvement is performed only by a

manipulation with granularity parameters.

The particular GCM reconstruction methodolo-

gies applied and presented in this section are based

on adjustment of:

• ε,

• γ,

• ε and γ successively,

• ε and γ simultaneously.

Optimization was performed with PSO in R with

default parameters. The number of iterations was set

to 4000. Duration of experiments presented in this

section was varying. A single experiment course for

parallel optimization of 64 variables for 10 values of

γ on a standard PC took approximately 15 hours.

The character of the aforementioned datasets is

varied. It was already highlighted that the distorted

train dataset - the one that is used for GCM train-

ing contains 0s and 1s. The model, due to asymptotic

properties of the sigmoid function, cannot cover these

values. It will be easy to spot that for the distorted

train dataset coverage statistics are generally low. Not

distorted train dataset is the ,,ideal” dataset, which de-

scribes perfect map responses. Test dataset contains

separate values that are not related to training data in

any way. Test dataset and ,,ideal” train dataset are

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179

used to assess map quality. The higher the coverage,

the better GCM responses cover targets.

3.1 GCM Reconstruction Through

Adjustment of ε

First possible scenario of optimization is adjustment

of the ε matrix. In this scenario other parameters re-

main without any modiﬁcation, we focus on the ba-

sic granularity parameter: knowledge granule size.

The optimization concerns n

values of ε, separate for

each weight.

The optimization procedure was performed 11

times for varying values of the symmetry parameter

γ, starting from 0 to 1 by 0.1. As a result we were

able to plot results in 3D perspective. The plots are

cut and rotated to illustrate outputs in the most conve-

nient way. In each plot coverage is collated with γ and

ε. Values of γ can be read directly from γ axis. Values

of ε are illustrated in less straightforward way.

In the case, when we adjust ε in 3D plots one can

see coverage versus restricted ε. This restrictions con-

cern two aspects:

• upper limit of individual ε values: ε

i j

6 (2 ∗ x)

• sum of all ε:

∑

i=1

∑

j=1

i j

6 (n

∗ x)

where x is on the ε axis. Introducing named re-

strictions allows to manipulate with balance between

speciﬁcity and generality. With the proposed proce-

dure we beneﬁt from the granular information repre-

sentation model to the greatest extent, by maintaining

ﬂexibility of the phenomena description. The sum of

all ε describes generality of the whole map, while re-

strictions on individual ε control speciﬁcity of a single

information granule.

Figure 4 illustrates weak and strict coverage be-

fore and after adjustment of the ε matrix. As one may

expect, the highest values of coverage are for lenient

limits on the ε. The stronger we limit the generality

criterion, the lower coverage we get.

The poorest results are for distorted train dataset.

This is because T GT

has a lot of 1s and 0s, which

cannot be covered by the model.

Second important observation is that highest cov-

erage for the training datasets is for symmetrical gran-

ules, around γ = 0.5. For the test dataset optimal value

of γ is slightly smaller.

The methodology based on ε matrix adjustment

produces moderately good results. One can see that

the optimization produced generally better models,

but not in each case. For the ,,ideal” train dataset

(ﬁrst column) we have increased the coverage in al-

most each case. Similarly, strict coverage was im-

proved. The worst results are for strict coverage on

distorted dataset.

3.2 GCM Reconstruction Through

Adjustment of γ

In this approach we reconstruct the Granular Cogni-

tive Map by adjustment of the alternative granularity

parameter: γ. In consequence, we allow asymmetrical

granular weights W

f in

. Sizes of the intervals, which

represent knowledge granules remain the same. We

look for the optimal center of the granule and move

proportionally left and right limits of the intervals.

The results: improvement in coverage after γ ad-

justment are illustrated in Figure 5. Plots in Figure 5

are not directly comparable with plots in the previous

subsection, because optimization procedure allowed

to modify the γ matrix.

This approach to GCM reconstruction gave satis-

fying results. Coverage has improved in each case.

Noteworthy is very good coverage on not distorted

train dataset and test dataset. Manipulation with sym-

metry parameters resulted high coverage for rigorous

speciﬁcity criterion. The initial values of γ (on the

horizontal axis) were reset. Chosen optimization al-

gorithm (PSO) is well suited for our approach. Weak

coverage for the same value of ε on the ideal train

dataset are similar for each initial value of γ. The pro-

cedure is stable and it produces comparable results.

From the theoretical point of view, adjustment of

γ in the procedure of Granular Cognitive Map recon-

struction plays secondary role. Symmetry parameter

does not play any role in the most crucial problem

of granularity: balance between generality and speci-

ﬁcity. Nevertheless, it was shown, that adjustment of γ

improves coverage. Therefore, methodology of Gran-

ular Cognitive Map reconstruction should take into

account both granularity parameters and take full ad-

vantage of granular knowledge representation model.

Based on this premises, in the following subsec-

tions we present GCM reconstruction methodologies,

that adjust both ε and γ matrices.

3.3 GCM Reconstruction Through

Successive Adjustment of ε and γ

In order to fully beneﬁt from the assumed granular

knowledge representation model, we propose an ap-

proach to Granular Cognitive Map reconstruction that

is based on adjustment of both granularity parameters:

granules size ε and symmetry γ. Figure 6 illustrates

differences in coverage prior and after the optimiza-

tion of weak coverage.

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Figure 4: Weak (top row) and strict (bottom row) coverage on not distorted train dataset (ﬁrst column), distorted train dataset

(second column) and test dataset (third column) before and after adjustment of the ε matrix.

In the GCM reconstruction scheme discussed in

this subsection ﬁrstly we adjust the size of knowledge

granules with respect to restrictions on the sum of all ε

(

∑

i=1

∑

j=1

i j

6 (n

∗ x)) and restrictions on the indi-

vidual value of each ε

i j

(ε

i j

6 (2 ∗ x)). Subsequently,

values of γ are adjusted. In other words, in the ﬁrst

step we maximize coverage and maintain the balance

between generality and speciﬁcity. Next, we tune the

map by adjustment of granules symmetry.

Optimization procedure produced more accurate

GCMs. Coverage on each dataset was better, than if

we adjust only ε matrix. In contrast, adjustment of

only γ parameters gave similar results. For the largest

values of granule size, weak coverage on the ideal

dataset gets saturated to 1. Most importantly, cov-

erage on the test dataset has improved. Presented ap-

proach gives satisfying results.

3.4 GCM Reconstruction Through

Simultaneous Adjustment of ε and γ

The last approach to Granular Cognitive Map recon-

struction, which we discuss in this article is simulta-

neous adjustment of both granularity parameters: ε

and γ. Please note that ε values are limited by re-

strictions on the sum and on the individual value (as

mentioned in the previous subsections). Figure 7 il-

lustrates improvements in coverage provided by the

GCM before and after the reconstruction procedure.

The proposed procedure produces better ﬁtted

Granular Cognitive Maps that do not loose prior bal-

ance between generality and speciﬁcity.

The GCM reconstruction strategy based on si-

multaneous adjustment of ε and γ is very successful.

Weak and strict coverage has improved in each case.

On the ,,ideal” training dataset weak coverage reaches

1 even for strong restrictions on the size of knowledge

granules. Weak coverage on the test dataset has im-

proved as well. Strict coverage is much harder to ob-

tain, since it requires all nodes in given observation to

GranularCognitiveMapReconstruction-AdjustingGranularityParameters

181

Figure 5: Weak (top row) and strict (bottom row) coverage on not distorted train dataset (ﬁrst column), distorted train dataset

(second column) and test dataset (third column) before and after adjustment of the γ matrix.

be covered by map response. Nevertheless, strict cov-

erage is also satisfactory. The only dataset that has

still rather poor coverage is distorted training dataset.

This is no surprise though. Distorted dataset contains

0s and 1s, which cannot be covered by map response.

3.5 Comparison of Presented

Approaches to GCM Reconstruction

In this subsection we summarize and evaluate applied

approaches to Granular Cognitive Map reconstruc-

tion. Table 1 compares mean weak coverage on the

three discussed datasets.

Table 1: Comparison of proposed approaches to Granular

Cognitive Map reconstruction.

mean of weak coverage

approach train ND train D test

adjustment of ε 0.775 0.413 0.658

adjustment of γ 0.892 0.466 0.776

adjust. of ε then γ 0.894 0.456 0.761

adjust. of ε and γ 0.917 0.461 0.790

Table 1 illustrates differences in mean weak cover-

age for the 4 different approaches to Granular Cogni-

tive Map reconstruction discussed in this paper. Val-

ues in Table 1 are means of weak coverage plotted

in Figures 4, 5, 6 and 7. With this statistics we may

quantitatively compare applied methodologies.

The most successful strategy of Granular Cogni-

tive Map reconstruction adjusts ε and γ simultane-

ously. In the last row in Table 1, mean weak cover-

ages are the highest. The poorest results are when we

adjust only the ε matrix. Quality of Granular Cog-

nitive Maps reconstructed by adjustment of γ matrix

and by successive adjustment of ε and the γ is similar.

Coverage is the worst for the distorted train

dataset. This is because of speciﬁcs of this dataset

(it contains 0s and 1s). We have chosen such train-

ing dataset on purpose. As a result, the reconstructed

GCMs are not overﬁtted. The developed methodol-

ogy for experiments with GCM reconstruction was

properly constructed, as it produced high coverages

for the ,,ideal” dataset and for the test dataset.

The proposed approach was focused on Granular

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Figure 6: Weak (top row) and strict (bottom row) coverage on not distorted train dataset (ﬁrst column), distorted train dataset

(second column) and test dataset (third column) before and after successive adjustment of ε and then γ.

Cognitive Map reconstruction with given restrictions

on the generality/speciﬁcity balance criterion. The

aim of the discussed strategies of optimization was to

produce better-ﬁtted Granular Cognitive Maps that do

not loose prior balance between generality and speci-

ﬁcity. Such requirements were implemented through

explicit limitations on values of individual ε and on

sum of all ε. As outcome, reconstructed maps cover

more targets, but maintain required balance that de-

termines their precision.

4 CONCLUSIONS

In the article authors have proposed a general method-

ology for Granular Cognitive Map reconstruction.

Presented approach has been concertized with 4 dis-

tinct GCM optimization schemes. We have discussed

and compared Granular Cognitive Map reconstruc-

tion procedures that adjust granularity parameters.

Theoretical assumptions of the proposed methodol-

ogy focus on most important indicators of granularity:

balance between generality and speciﬁcity.

The paper is supported by a series of experiments,

which illustrate results of the proposed Granular Cog-

nitive Map reconstruction procedures. We showed

that proper optimization schemes allow increasing

coverage and maintaining the same generality of a

whole map. GCM reconstruction strategy that pro-

duces the most accurate model - a map that covers the

greatest number of targets is simultaneous adjustment

of both granularity parameters: ε and γ. Discussed

approaches of ε and/or γ optimization take full beneﬁt

of granular knowledge representation model.

The objective of the research discussed in this

article is to propose appropriate methodologies for

Granular Cognitive Maps training and exploration.

In future research authors plan to investigate other

knowledge granules representation models, most im-

portantly fuzzy numbers and bipolar knowledge rep-

resentation schemes.

GranularCognitiveMapReconstruction-AdjustingGranularityParameters

183

Figure 7: Weak (top row) and strict (bottom row) coverage on not distorted train dataset (ﬁrst column), distorted train dataset

(second column) and test dataset (third column) before and after simultaneous adjustment of ε and γ.

ACKNOWLEDGEMENTS

The research is partially supported by the National

Science Center, grant No 2011/01/B/ST6/06478, de-

cision no DEC-2011/01/B/ST6/06478.

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