**2. Theoretical foundations of building FGCM**

The basis of building FGCM is the Grey Systems Theory, proposed in 1989 by Deng [10]. Within the framework of this theory, objects and systems with high uncertainty, represented by small samples of incomplete and inaccurate data, are studied. Depending on the character of the available information, the studied systems are divided into three types:


In accordance with the terminology of the grey systems theory, a fuzzy grey cognitive map is a cognitive model of a system in the form of a directed graph defined with use of the following set:

$$\text{FGCM} = \langle \text{C}, F, W \rangle,\tag{1}$$

<sup>⊗</sup> *Xi*ð Þ¼ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> *<sup>f</sup>* <sup>⊗</sup> *Xi*ð Þþ *<sup>k</sup>* <sup>X</sup>*<sup>n</sup>*

BBBBBBB@

accepted in the following form:

c. unipolar sigmoid:

limit cycle (strange attractor).

number ⊗*Wij*.

**23**

can be found in [5, 6, 9].

a. linear function with saturation:

b. bipolar sigmoid (hyperbolic tangent):

(steady state) solution, which is a grey vector lim*<sup>k</sup>*!<sup>∞</sup>

equation (3) ("the fixed point") exists if and only if

0

*DOI: http://dx.doi.org/10.5772/intechopen.89215*

*j* ¼ 1 ð Þ *j* 6¼ *i*

*Cybersecurity Risk Analysis of Industrial Automation Systems on the Basis of Cognitive…*

values of which at each time instant *k* ¼ 0, 1, 2, … belong to some interval *Xi*ð Þ*<sup>k</sup>* , *Xi*ð Þ*<sup>k</sup>* � �; *<sup>f</sup>* is the activation function of the *<sup>i</sup>*-th concept, mapping the argument values into the interval ½ � �1, 1 . The activation function *f*ð Þ∙ , as a rule, is

*f x*ð Þ¼

*f x*ð Þ¼ <sup>1</sup> � *<sup>e</sup>*�*<sup>x</sup>* ð Þ

*f x*ð Þ¼ 1*=* 1 þ *e*

To solve the system of equations (Eq. (3)), it is required to set the initial conditions ⊗ *Xi*ð Þ 0 , which also should be considered as the grey numbers <sup>⊗</sup> *Xi*ð Þ <sup>0</sup> <sup>∈</sup> *Xi*ð Þ <sup>0</sup> , *Xi*ð Þ <sup>0</sup> � �. Most interesting is usually to obtain the equilibrium

To determine the stability of the steady-state solution ⊗ *X*<sup>∗</sup> , one can use the theorem [12], according to which the only equilibrium (steady state) solution of

> *W*<sup>2</sup> *ij* !<sup>1</sup>

where the value of the positive constant *H* depends on the choice of activation function of the concepts: *H* ¼ 1 for function (Eq. (4)); *H* ¼ 2 for function (Eq. (5)); and *H* ¼ 4 for function (Eq. (6)). In the case of negative connection, i.e., for *Wij* <*Wij* <0, we also put in (Eq. (7)) the upper boundary *Wij* of the grey

More detailed information on FGCM construction and their learning algorithms

2

X*n <sup>i</sup>*,*<sup>j</sup>*¼<sup>1</sup>

<sup>1</sup> <sup>þ</sup> *<sup>e</sup>*�*<sup>x</sup>* ð Þ <sup>¼</sup> th *<sup>x</sup>*

2

� �; (5)

½ �¼ <sup>⊗</sup> *Xi*ð Þ*<sup>k</sup>* <sup>⊗</sup> *<sup>X</sup>*<sup>∗</sup> <sup>∈</sup> *<sup>X</sup>*<sup>∗</sup> ,*X*<sup>∗</sup> � � or a

< *H*, (7)

�*<sup>x</sup>* ð Þ*:* (6)

where ⊗ *Xi*ð Þ*k* is the "grey" (interval) variable of the *i*-th concept *Ci* state, the

⊗*Wji*⊗ *Xj*ð Þ*k*

*x*, ifj j *x* ≤ 1; sgn *<sup>x</sup>*, ifj j *<sup>x</sup>* <sup>&</sup>gt;1, (

1

CCCCCCCA

,ð Þ *i* ¼ 1, 2, …, *n* , (3)

(4)

where *C* ¼ f g *Ci* is the set of concepts (vertices of the graph), ð Þ *i* ¼ 1, 2, …, *n* ; *F* ¼ *Fij* � � is the set of connections between concepts (arcs of the graph); and *<sup>W</sup>* <sup>¼</sup> *Wij* � � is the set of the relationships between the concepts determining the weights of these connections, ð Þ *i*, *j* ∈ Ω. Here, Ω ¼ *i*1, *j* 1 � �, *i*2, *j* 2 � �, …, *iL*, *j L* � � � � is the set of the pairs of adjacent (interconnected) vertices indices, *L*≤*n n*ð Þ � 1 .

In contrast to the traditional FCM representation, the weights of FGCM connections are set with the use of "grey" (interval) numbers ⊗*Wij*, defined as

$$\otimes W\_{\vec{\eta}} \in \left[ \underline{W\_{\vec{\eta}}}, \overline{W\_{\vec{\eta}}} \right], \text{where } \underline{W\_{\vec{\eta}}} < \overline{W\_{\vec{\eta}}}, \left[ \underline{W\_{\vec{\eta}}}, \overline{W\_{\vec{\eta}}} \right] \in [-1, 1], \tag{2}$$

where *Wij* and *Wij* are, respectively, the lower and the upper boundaries of the grey number. So, the weight of connection between *i*-th and *j*-th concepts (*Ci* ! *Cj*) can take any value within the given range of change *Wij*,*Wij* h i∈½ � �1, 1 . In the particular case, when *Wij* <sup>¼</sup> *Wij*, we get <sup>⊗</sup>*Wij* <sup>∈</sup> *Wij*,*Wij* h i—<sup>a</sup> "white" (crisp, usual) number.

It is assumed that the change of the concepts state in time is described by equations

*Cybersecurity Risk Analysis of Industrial Automation Systems on the Basis of Cognitive… DOI: http://dx.doi.org/10.5772/intechopen.89215*

$$\otimes X\_i(k+1) = f\left(\otimes X\_i(k) + \sum\_{\substack{j=1 \\ j \neq i}}^n \otimes W\_{ji} \otimes X\_j(k)\right), (i = 1, 2, \dots, n), \tag{3}$$

where ⊗ *Xi*ð Þ*k* is the "grey" (interval) variable of the *i*-th concept *Ci* state, the values of which at each time instant *k* ¼ 0, 1, 2, … belong to some interval *Xi*ð Þ*<sup>k</sup>* , *Xi*ð Þ*<sup>k</sup>* � �; *<sup>f</sup>* is the activation function of the *<sup>i</sup>*-th concept, mapping the argument values into the interval ½ � �1, 1 . The activation function *f*ð Þ∙ , as a rule, is accepted in the following form:

a. linear function with saturation:

information on the state of several similar concepts of the original FCM, thus aggregating the corresponding concepts. In our case, on the contrary, the original

corresponds to the number of basic subsystems of the system under study, and the decomposition of FGCM implies a representation of each concept of the original FGCM in the form of independent (local) FGCM, describing the behavior of

The basis of building FGCM is the Grey Systems Theory, proposed in 1989 by Deng [10]. Within the framework of this theory, objects and systems with high uncertainty, represented by small samples of incomplete and inaccurate data, are studied. Depending on the character of the available information, the studied

• "white" systems (the internal structure and the properties of the system are

• "black" systems (the internal structure and the properties of the system are

In accordance with the terminology of the grey systems theory, a fuzzy grey cognitive map is a cognitive model of a system in the form of a directed graph

where *C* ¼ f g *Ci* is the set of concepts (vertices of the graph), ð Þ *i* ¼ 1, 2, …, *n* ; *F* ¼

In contrast to the traditional FCM representation, the weights of FGCM connec-

where *Wij* and *Wij* are, respectively, the lower and the upper boundaries of the grey number. So, the weight of connection between *i*-th and *j*-th concepts (*Ci* ! *Cj*)

It is assumed that the change of the concepts state in time is described by

, where *Wij* <*Wij*, *Wij*,*Wij*

1 � �, *i*2, *j*

� � is the set of connections between concepts (arcs of the graph); and *<sup>W</sup>* <sup>¼</sup>

the pairs of adjacent (interconnected) vertices indices, *L*≤*n n*ð Þ � 1 .

tions are set with the use of "grey" (interval) numbers ⊗*Wij*, defined as

� � is the set of the relationships between the concepts determining the weights

FGCM ¼ 〈*C*, *F*,*W*〉, (1)

2 � �, …, *iL*, *j*

h i

h i

h i

*L* � � � � is the set of

∈½ � �1, 1 , (2)

∈½ � �1, 1 . In the

—a "white" (crisp,

• "grey" systems (partial information about the system is known); and

FGCM has a small dimension, the number of forming its basic concepts

**2. Theoretical foundations of building FGCM**

systems are divided into three types:

completely known);

completely unknown).

defined with use of the following set:

of these connections, ð Þ *i*, *j* ∈ Ω. Here, Ω ¼ *i*1, *j*

⊗*Wij* ∈ *Wij*,*Wij*

h i

can take any value within the given range of change *Wij*,*Wij*

particular case, when *Wij* ¼ *Wij*, we get ⊗*Wij* ∈ *Wij*,*Wij*

this concept.

*Digital Forensic Science*

*Fij*

*Wij*

usual) number.

equations

**22**

$$f(\mathbf{x}) = \begin{cases} \quad \text{x, if } |\mathbf{x}| \le \mathbf{1}; \\ \quad \text{sgn } \boldsymbol{\varkappa}, \text{if } |\boldsymbol{\varkappa}| > \mathbf{1}, \end{cases} \tag{4}$$

b. bipolar sigmoid (hyperbolic tangent):

$$f(\mathbf{x}) = \frac{(\mathbf{1} - e^{-\mathbf{x}})}{(\mathbf{1} + e^{-\mathbf{x}})} = \mathbf{th}\left(\frac{\mathbf{x}}{2}\right);\tag{5}$$

c. unipolar sigmoid:

$$f(\mathbf{x}) = \mathbf{1}/(\mathbf{1} + \mathbf{e}^{-\mathbf{x}}).\tag{6}$$

To solve the system of equations (Eq. (3)), it is required to set the initial conditions ⊗ *Xi*ð Þ 0 , which also should be considered as the grey numbers <sup>⊗</sup> *Xi*ð Þ <sup>0</sup> <sup>∈</sup> *Xi*ð Þ <sup>0</sup> , *Xi*ð Þ <sup>0</sup> � �. Most interesting is usually to obtain the equilibrium (steady state) solution, which is a grey vector lim*<sup>k</sup>*!<sup>∞</sup> ½ �¼ <sup>⊗</sup> *Xi*ð Þ*<sup>k</sup>* <sup>⊗</sup> *<sup>X</sup>*<sup>∗</sup> <sup>∈</sup> *<sup>X</sup>*<sup>∗</sup> ,*X*<sup>∗</sup> � � or a limit cycle (strange attractor).

To determine the stability of the steady-state solution ⊗ *X*<sup>∗</sup> , one can use the theorem [12], according to which the only equilibrium (steady state) solution of equation (3) ("the fixed point") exists if and only if

$$\left(\sum\_{i,j=1}^{n} \overline{W}\_{\vec{\eta}}^{2}\right)^{\frac{1}{2}} < H,\tag{7}$$

where the value of the positive constant *H* depends on the choice of activation function of the concepts: *H* ¼ 1 for function (Eq. (4)); *H* ¼ 2 for function (Eq. (5)); and *H* ¼ 4 for function (Eq. (6)). In the case of negative connection, i.e., for *Wij* <*Wij* <0, we also put in (Eq. (7)) the upper boundary *Wij* of the grey number ⊗*Wij*.

More detailed information on FGCM construction and their learning algorithms can be found in [5, 6, 9].
