**3. Fuzzy knowledge representation of the 'short memory' stochastic process**

#### **3.1 Stochastic process with fuzzy states**

Let *X*(*t*) be a 'short memory' stochastic process, the family of time-dependent random variables, where *X R* , *tT R* and *B* is the Borel σ-field of events. Let *p* be a probability, the normalized measure over the space (,) *B* .

Moreover, assume that according to human experts' suggestions, in the universe of process values, the linguistic random variable has been determined with the set of linguistic values, *L(X)={LXi*}, *i*=1,2,…,*I* e.g. *L(X)={low, middle, high},* according to Zadeh's definition of the linguistic variable (Zadeh, 1975). The meanings of the linguistic values are represented by fuzzy sets *Ai*, *i*=1,2,…,*I* determined on *χ* by their membership functions, ( ) : [0,1] *Ai x* , which are Borel measurable functions, fulfilling the condition

$$\sum\_{i=1}^{l} \mu\_{A\_i}(\mathbf{x}) = \mathbf{1}, \; \forall \mathbf{x} \in \mathcal{X} \; . \tag{12}$$

According to above assumptions, the probability distribution of linguistic values of the process *X*(*t*) can be determined as follows

$$P(X\_t) = \left(P(A\_i) \mid i = 1, 2, \dots, I \quad \right.\tag{13}$$

based on Zadeh's definitions of the probability of fuzzy events (Zadeh, 1968)

$$P(A) = \int\limits\_{\mathcal{X} \subseteq \mathcal{R}''} \mu\_A(\mathbf{x}) dp \,. \tag{14}$$

The following conditions must be fulfilled

$$0 \le P(A\_i) \le 1\text{, } i = 1, 2, \dots, I \quad ; \qquad \sum\_{i=1}^{I} P(A\_i) = 1\text{ .}\tag{15}$$

Let now 1 22 1 *tt tt t t* , , be fixed, so the stochastic process at that moments is represented by two random variables *Xt Xt* ( ), ( ) 1 2 . Assume, that <sup>2</sup> ( ,,) *B p* is a probability space, where 2 2 *R* , *B* is the Borel σ-field of events and *p* is a probability, the normalized measure over <sup>2</sup> ( ,) *B* . The assumptions mean that the probability distribution 1 2 (,) *t t p x x* over the realizations 1 2 , *X X t t* exists.

Let also two linguistic random variables (linguistic random vector) 1 2 (,) *X X t t* be generated in <sup>2</sup> , taking simultaneous linguistic values *LX LX <sup>i</sup> <sup>j</sup>* , *i,j=1,2,…,I*; corresponding collection of fuzzy events , 1,..., *i j <sup>i</sup> <sup>j</sup> <sup>I</sup> A A* is determined on <sup>2</sup> by membership functions

1 2 (,) *Ai j At t x x* , *i,j=1,2,…,I*. Membership functions for joint fuzzy events *Ai Aj* should fulfill

$$\sum\_{i} \sum\_{j} \mu\_{A\_i \times A\_j} (\mathbf{x}\_{t\_1}, \mathbf{x}\_{t\_2}) = \mathbf{1}\_{\prime} \cdot \forall (\mathbf{x}\_{t\_1}, \mathbf{x}\_{t\_2}) \in \mathbb{X}^2 \,. \tag{16}$$

The joint 2D probability distribution of linguistic values (fuzzy states) of the stochastic process *X*(*t*) is determined by the joint probability distribution of the linguistic random vector 1 2 (,) *X X t t*

$$P(X\_{t\_1}, X\_{t\_2}) = \{P(A\_i \times A\_j)\}\_{t\_i, j=1,2,\ldots,l} \tag{17}$$

calculated according to

$$P(A\_i \times A\_j) = \bigcap\_{(\mathbf{x}\_{t\_1}, \mathbf{x}\_{t\_2}) \in \mathcal{X}} \mu\_{A\_i \times A\_j}(\mathbf{x}\_{t\_1}, \mathbf{x}\_{t\_2}) dp \tag{18}$$

and fulfilling

298 Serial and Parallel Robot Manipulators – Kinematics, Dynamics, Control and Optimization

Diffusion coefficient, γ, has been expressed by Einstein as a function of macro- and microscopic parameters of the fluid and particles, respectively. Einstein confirmed statistical

**3. Fuzzy knowledge representation of the 'short memory' stochastic process** 

Let *X*(*t*) be a 'short memory' stochastic process, the family of time-dependent random

Moreover, assume that according to human experts' suggestions, in the universe of process values, the linguistic random variable has been determined with the set of linguistic values, *L(X)={LXi*}, *i*=1,2,…,*I* e.g. *L(X)={low, middle, high},* according to Zadeh's definition of the linguistic variable (Zadeh, 1975). The meanings of the linguistic values are represented by fuzzy sets *Ai*, *i*=1,2,…,*I* determined on *χ* by their membership functions, ( ) : [0,1] *Ai*

> ( ) 1, *i*

*x x*

According to above assumptions, the probability distribution of linguistic values of the

( ) () *n A R P A x dp* 

Let now 1 22 1 *tt tt t t* , , be fixed, so the stochastic process at that moments is

Let also two linguistic random variables (linguistic random vector) 1 2 (,) *X X t t* be generated

*A A* is determined on <sup>2</sup>

, taking simultaneous linguistic values *LX LX <sup>i</sup> <sup>j</sup>* , *i,j=1,2,…,I*; corresponding collection

, *tT R* and *B* is the Borel σ-field of events. Let *p* be a

*x* ,

*B* .

1

*B* . The assumptions mean that the probability distribution

*i*

*R* , *B* is the Borel σ-field of events and *p* is a probability, the normalized

*I*

. (12)

*PX PA i I* ( ) ( , 1,2,..., *t i* , (13)

( )1

*i*

*P A* 

. (14)

. (15)

*B p* is a probability

by membership functions

character of the diffusion law (cited by van Kampen, 1990).

probability, the normalized measure over the space (,)

which are Borel measurable functions, fulfilling the condition

1

based on Zadeh's definitions of the probability of fuzzy events (Zadeh, 1968)

0 ( )1 *P Ai* , *i*=1,2,…,*I* ;

represented by two random variables *Xt Xt* ( ), ( ) 1 2 . Assume, that <sup>2</sup> ( ,,)

*i* 

*I A*

**3.1 Stochastic process with fuzzy states** 

process *X*(*t*) can be determined as follows

The following conditions must be fulfilled

of fuzzy events , 1,..., *i j <sup>i</sup> <sup>j</sup> <sup>I</sup>*

1 2 (,) *t t p x x* over the realizations 1 2 , *X X t t* exists.

space, where 2 2 

in <sup>2</sup> 

measure over <sup>2</sup> ( ,)

variables, where *X R*

$$0 \le P(A\_i \times A\_j) \le 1, \text{ } \forall i, j = 1, \dots, I \quad \text{and} \quad \sum\_{i=1}^{I} \sum\_{j=1}^{I} P(A\_i \times A\_j) = 1 \tag{19}$$

(Walaszek-Babiszewska, 2008, 2011). From the joint probability distribution (17), the conditional probability distribution of the fuzzy transition

$$P[(X\_{t\_2} = A\_j) \mid (X\_{t\_1} = A\_i)] \text{ , } j \equiv 1, 2, \dots, l; \text{ } i \text{-const} \tag{20}$$

can be determined according to

$$\begin{aligned} \left\{ \begin{array}{l} P[\{X\_{t\_2} \neq \{X\_{t\_1} = A\_i\}\} \ \}\_{j=1,\ldots,l} = \\ = \frac{\{P(X\_{t\_2} = A\_j, X\_{t\_1} = A\_i) \ \} \ \}\_{j=1,\ldots,l} .\end{array} . \end{aligned} \tag{21}$$

The following relationships should be fulfilled for the conditional distributions of fuzzy states (probability of the transitions)

$$\sum\_{j=1,\dots,I} P[(X\_{t\_2} = A\_j \;/\ (X\_{t\_1} = A\_i))] = 1; \; i = \text{const.} \tag{22}$$

#### **3.2 Rule based fuzzy model**

The proposed model of the stochastic process, formulated into fuzzy categories, for two moments 1 2 *t t*, , 2 1 *t t* , is a collection of file rules, in the following form (Walaszek-Babiszewska, 2008, 2011):

$$\forall A\_i \in L(X) \text{ \textquotedblleft } i = 1, \dots, I$$

$$R^{(i)} \colon w\_i[\![If(X\_{t\_1} \text{ is } A\_i) \text{)}] \to \text{Then} (X\_{t\_2} \text{ is } A\_1) w\_{1/i}$$

$$\text{Also} (X\_{t\_2} \text{ is } A\_j) w\_{j/i} \tag{23}$$

$$\text{Also} (X\_{t\_2} \text{ is } A\_j) w\_{1/i}$$

Fuzzy Modelling Stochastic Processes Describing Brownian Motions 301

2 1 {( ( )) /[ ']} *E X is A X is A A t jt*

computed as the aggregated outputs of all active *i*-*th* file rules, can be determined by the

<sup>2</sup> <sup>2</sup> / ' / 2 () () ( ) *j i <sup>i</sup> t it i <sup>A</sup> <sup>j</sup> iA t*

(27) or (28), depending on the type of input data and the interpretation of a fuzzy model. Also, it is possible to determine probability of the fuzzy conclusion, taking into account a

**4. Creating fuzzy models of stochastic processes - Exemplary calculations** 

First example show the fuzzy representation of the simplest form of the considered above stochastic processes, the one-dimensional time-discrete stochastic process of the increments, *XXX ttt*<sup>1</sup> . The increments, at given *t*, are normal distributed random variables, so it is useful to use the standard normal probability distribution function, over the domain of the

> *x* ( ) *PL*

linguistic random variable 2 *Xt* . The fuzzy expected value of the following prediction,

'

marginal probability distribution 2 ( ) *P Xt* of the output linguistic random variable.

*<sup>A</sup> <sup>i</sup> i j*

 *x w x ww x* . (30)

, of the conclusions from elementary rules are given by

(Table 1). Linguistic random variable, *Yt* , with the

0.014

events *P*(*Y*)

*P*(*NL*)= 0.220

*P*(*PL*)= 0.220

*P*(*Z*)= 0.532

Fuzzy sets (events) Probability of fuzzy

 *x* ( ) *PH x*

*L X*( ) , of linguistic values of the

, (29)

conclusions (27) and (28) represent some functions,

following formula (Walaszek-Babiszewska, 2011)

where membership functions, /' *<sup>A</sup> j i*

process values, *X R* [ 3, 3]

[-2.5, -2) 0.01654 0.5 0.5

[-2, -1.5) 0.044057 0 1

*NH* ( ) *x*

[-1.5, -1) 0.091848 1 0 [-1, -0.5) 0.149882 0.5 0.5

[-0.5, 0) 0.191463 0 1

[0, 0.5) 0.191463 1 0 [0.5, 1) 0.149882 0.5 0.5

[1, 1.5) 0.091848 0 1

[1.5, 2) 0.044057 1 0

[2, 2.5) 0.01654 0.5 0.5 P(PH)= [2.5, 3] 0.00486 0 1 0.014

*L{Y}* of the linguistic random variable *Yt* and probability distribution *P Y*( )

Table 1. Probability function of random variable *Xt* , fuzzy sets representing linguistic values

*x ab* [,) *p*(*x*)

**4.1 Fuzzy model of the stochastic time-discrete increments** 

 *NL*( ) *x* ( ) *<sup>Z</sup>* 

[-3, -2.5) 0.00486 1 0 *P*(*NH*)=

or as a collection of the elementary rules in the form

$$
\forall A\_i \in L(X) \; , \; \forall A\_j \in L(X) \; , \; i, j = 1, 2, \dots, l
$$

$$
R^{(i,j)} \colon w\_{ij} [\![If \left(X\_{t\_1} \text{ is } A\_i \right) \text{ Then } \left(X\_{t\_2} \text{ is } A\_j \right)] \tag{24}
$$

where the weights *wi*, *wj/i*, *wij* represent probabilities of fuzzy states, determined by (13) – (15), (20) – (22) and (17) - (19), respectively. The weights stand for the frequency of the occurrence of fuzzy events in particular parts of rules and show the probabilistic structure of the linguistic values of the linguistic random vector 1 2 , *X X t t* . The weights do not change logic values of the conditional sentences.

#### **3.3 Reasoning procedures**

Considering reasoning procedure, we assume that some non-crisp (vague) observed value of the stochastic process at moment *t1* is known and equal to 1 ' *X A <sup>t</sup>* , , or some crisp value 1 1 *X x t t* of the stochastic process at moment *t1* is given. Then, the level of activation of the elementary rule (24) is determined according to one of the following formulas

$$\tau\_i = \max\_{\mathbf{x}} \min \{ \mu\_{A\_i}(\mathbf{x}), \mu\_{A^i}(\mathbf{x}) \}\,. \tag{25}$$

$$\tau\_i = \mu\_{A\_i}(\mathbf{x}\_{t\_i}^\*) \text{ , } i = 1, \dots, I \text{ , } \tag{26}$$

repectively (Yager & Filev, 1994; Hellendoorn & Driankov, 1997). The conclusion according to the generalized Mamdani-Assilian's type interpretation of fuzzy models has the following form

$$\mu\_{A\_{j/l}^{+}}\left(\mathbf{x}\_{t\_2}\right) = T\left(\pi\_{i'}\mu\_{A\_{j}}\left(\mathbf{x}\_{t\_2}\right)\right), \; j = 1, \ldots, l; \; i = \text{const}; \tag{27}$$

thus the conclusion derived based on logic type interpretation of fuzzy models is as follows

$$\mu\_{A\_{ij\_{l\bar{l}}}}(\mathbf{x}\_{t\_2}) = \mathbf{l}(\tau\_i, \mu\_{A\_{\bar{l}\_j}}(\mathbf{x}\_{t\_2})), \ j = 1, \ldots, l; \ i = const.} \tag{28}$$

where *T* denotes a *t*-norm and I means the implication operator. Aggregation of the conclusions from particular rules is usually computed by using any s-norm operator (Yager & Filev, 1994; Hellendoorn & Driankov, 1997).

Weights of rules, representing the probability of a fuzzy event in antecedent (*wi*), as well as, the conditional probability of a fuzzy event at the consequence part (*wj/i*), can be used to determine probabilistic characteristics of the conclusion. It is worth to note, that fuzzy

1 2

: [ ( )] ( )

*R w If X isA Then X is A w*

1 1/

(23)

/

/

(,): [ ( ) ( )] *i j R w If X is A Then X is A ij ti t <sup>j</sup>* (24)

( )

*i*

or as a collection of the elementary rules in the form

logic values of the conditional sentences.

**3.3 Reasoning procedures** 

1 1 *X x t t*

form

2

*Also X is A w*

( )

*t j ji*

*t J Ji*

*i ti t i*

2

( ) *A LX <sup>i</sup>* , ( ) *Aj L X* , *i*,*j*=1,2,…,*I*

where the weights *wi*, *wj/i*, *wij* represent probabilities of fuzzy states, determined by (13) – (15), (20) – (22) and (17) - (19), respectively. The weights stand for the frequency of the occurrence of fuzzy events in particular parts of rules and show the probabilistic structure of the linguistic values of the linguistic random vector 1 2 , *X X t t* . The weights do not change

Considering reasoning procedure, we assume that some non-crisp (vague) observed value of the stochastic process at moment *t1* is known and equal to 1 ' *X A <sup>t</sup>* , , or some crisp value

elementary rule (24) is determined according to one of the following formulas

'

'

& Filev, 1994; Hellendoorn & Driankov, 1997).

 

 2 2 / ( ) ( , ( )) *j i <sup>j</sup> A A t it*

 2 2 / ( ) ( , ( )) *j i <sup>j</sup> A A tit*

 *x x* 

 *xT x* 

of the stochastic process at moment *t1* is given. Then, the level of activation of the

maxmin[ ( ), ( )] '*<sup>i</sup> i AA <sup>x</sup>*

 

*x x* , (25)

*x* , *i*=1,…,*I* , (26)

, *j*=1,…,*I*; *i*=const; (27)

, *j*=1,…,*I*; *i*=c*onst*, (28)

<sup>1</sup> ( ) *<sup>i</sup> i At*

repectively (Yager & Filev, 1994; Hellendoorn & Driankov, 1997). The conclusion according to the generalized Mamdani-Assilian's type interpretation of fuzzy models has the following

thus the conclusion derived based on logic type interpretation of fuzzy models is as follows

where *T* denotes a *t*-norm and I means the implication operator. Aggregation of the conclusions from particular rules is usually computed by using any s-norm operator (Yager

Weights of rules, representing the probability of a fuzzy event in antecedent (*wi*), as well as, the conditional probability of a fuzzy event at the consequence part (*wj/i*), can be used to determine probabilistic characteristics of the conclusion. It is worth to note, that fuzzy

*Also X is A w*

( )

1 2

conclusions (27) and (28) represent some functions, *L X*( ) , of linguistic values of the linguistic random variable 2 *Xt* . The fuzzy expected value of the following prediction,

$$E\{\left(X\_{t\_2}\text{ is}\varphi(A\_j)\right)/\left[X\_{t\_1}\text{ is}\ A'\right]\}=\bar{A}\_{\prime}\tag{29}$$

computed as the aggregated outputs of all active *i*-*th* file rules, can be determined by the following formula (Walaszek-Babiszewska, 2011)

$$
\mu\_{\stackrel{\frown}{A}}(\mathbf{x}\_{t\_2}) = \sum\_{i} w\_i \mu\_{A\_i^{\vee}}(\mathbf{x}\_{t\_2}) = \sum\_{i} w\_i \sum\_{j} w\_{j/i} \mu\_{A\_{j/i}^{\vee}}(\mathbf{x}\_{t\_2}) \,. \tag{30}
$$

where membership functions, /' *<sup>A</sup> j i* , of the conclusions from elementary rules are given by

(27) or (28), depending on the type of input data and the interpretation of a fuzzy model. Also, it is possible to determine probability of the fuzzy conclusion, taking into account a marginal probability distribution 2 ( ) *P Xt* of the output linguistic random variable.
