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

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

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 process values, *X R* [ 3, 3] (Table 1). Linguistic random variable, *Yt* , with the


Table 1. Probability function of random variable *Xt* , fuzzy sets representing linguistic values *L{Y}* of the linguistic random variable *Yt* and probability distribution *P Y*( )

Fuzzy Modelling Stochastic Processes Describing Brownian Motions 303

The set of the linguistic values, *L*(*Y*)={*negative high NH*, *negative low NL*, *positive low PL*,

are represented by respective fuzzy sets. Also, second linguistic random variable, *Yt*<sup>1</sup> , with the name '*Increment at moment t-1*' has been determined with the same set of linguistic values *L*(*Y*). For the tested process, the criterion of independent increments is not fulfilled, thus, the conditional probabilities (probabilities of transitions) 1 (/ ) *PY Y t t* should be found. The empirical joint probability of two linguistic random variables, 1 (, ) *PY Yt t* , has been calculated according to (16) - (19), based on the joint probability of numeric values of pairs, <sup>1</sup> ( ,) *t t p x x* , as well as, the assumed fuzzy events, representing the linguistic values {}{ , , , } *L Y NH NL PH PL <sup>t</sup>* (Table 2). Marginal probability 1 ( ) *P Yt* is presented at the last row of the table. It is not a symmetrical distribution, the highest value of the probability, 0.39251,

Conditional probabilities 1 (/ ) *PY Y t t* , calculated according to (20) – (22) and presented in Table 3, may be treated as the transitions probabilities from fuzzy states {*NH*, *NL*, *PL*, *PH*} at

<sup>1</sup> (, ) *PY Yt t*

PH 0.0178 0.05355 0.10702 0.03572 PL 0.0060 0.06555 0.19024 0.09532 NL 0.0953 0.05955 0.02975 0.06555 NH 0.04765 0.03570 0.0655 0.0298 <sup>1</sup> ( ) *P Yt* 0.16675 0.21435 0.39251 0.22639

Table 2. Joint probability distribution, 1 (, ) *PY Yt t* , of linguistic random variables

<sup>1</sup> (/ ) *PY Y t t*

PH 0.10675 0.24983 0.27266 0.15778 PL 0.03600 0.30581 0.48466 0.42104 NL 0.57150 0.27781 0.07580 0.28955 NH 0.28575 0.16655 0.16688 0.13163 <sup>1</sup> (/ ) *PY Y t t* 1.00000 1.00000 1.00000 1.00000

Table 3. Conditional probability distributions 1 (/ ) *PY Y t t* of linguistic random variables

<sup>1</sup> { } *L Yt* NH NL PL PH

representing empirical set of increments at moments *t* and *t-*1

<sup>1</sup> { } *L Yt* NH NL PL PH

it is a probability that increments take the linguistic value '*Positive Low*'.

moment *t*-1 to the particular fuzzy states at moment *t*.

{ } *L Yt*

{ } *L Yt*

of the process values, the linguistic values

*positive high PH*} has been assumed. In domain

name '*Increment at moment t*' has been assumed, with the set of its linguistic values: *L*(*Y*)={*negative high NH*, *negative low NL*, *zero Z*, *positive low PL*, *positive high PH*}. The linguistic values are represented by respective fuzzy sets. The probability distribution of the linguistic random variable, *P*(*Y*), calculated according to (13) - (15), has been presented in Table 1.

Also, the second linguistic random variable, *Yt*<sup>1</sup> , with the name '*Increment at moment t-1*' has been determined with the same set of linguistic values *L*(*Y*). Increments of the tested process are independent random variables, so conditional probabilities (probabilities of transitions) fulfill the relationship: 1 ( / ) () *PY Y PY tt t* .

The fuzzy knowledge base for the short memory stochastic process consists of the following, five file rules (25 elementary rules) with respective probabilities (according to Table 1.):

R1: 0.014(If *Yt*1 is NH) Then (*Yt* is NH)0.014

Also (*Yt* is NL) 0.220

Also (*Yt* is Z) ) 0.532

Also (*Yt* is PL) 0.220

Also (*Yt* is PH) 0.014;

R2: 0.220 (If *Yt*1 is NL) Then (*Yt* is NH) 0.014


R5: 0.014(If *Yt*1 is PH) Then (*Yt* is NH) 0.014


Also (*Yt* is PH) 0.014.

In the created rule base of the stochastic process, the same probability distributions for random variables, *Xt* and *Xt*<sup>1</sup> , have been assumed. It is result of the simplification, under the assumption of a constant time interval 1 *t* .

#### **4.2 Exemplary fuzzy models constructed based on realizations of stochastic processes**

#### **4.2.1 Fuzzy model constructed based on data of a floating particle**

In the object literature the problem of the fulfilling the Wiener process assumptions by empirical data is often raised, e.g. the expected values of empirical increments are non-zero or increments do not fulfill the criterion of probabilistic independence. These facts have been also observed based on data representing increments of one coordinate, *ttt* <sup>1</sup> *xxx* , *x R* [ 3.3,3.3] , describing the behavior of the particle floating in some liquid. It was assumed, that data stand for the realization of a certain stochastic process *Y*(*t*). Also, the linguistic random variable*Yt* has been determined, with the name '*Increment at moment t*'.

name '*Increment at moment t*' has been assumed, with the set of its linguistic values: *L*(*Y*)={*negative high NH*, *negative low NL*, *zero Z*, *positive low PL*, *positive high PH*}. The linguistic values are represented by respective fuzzy sets. The probability distribution of the linguistic random variable, *P*(*Y*), calculated according to (13) - (15), has been presented in

Also, the second linguistic random variable, *Yt*<sup>1</sup> , with the name '*Increment at moment t-1*' has been determined with the same set of linguistic values *L*(*Y*). Increments of the tested process are independent random variables, so conditional probabilities (probabilities of

The fuzzy knowledge base for the short memory stochastic process consists of the following, five file rules (25 elementary rules) with respective probabilities (according to Table 1.):

R1: 0.014(If *Yt*1 is NH) Then (*Yt* is NH)0.014

R2: 0.220 (If *Yt*1 is NL) Then (*Yt* is NH) 0.014

R5: 0.014(If *Yt*1 is PH) Then (*Yt* is NH) 0.014


In the created rule base of the stochastic process, the same probability distributions for random variables, *Xt* and *Xt*<sup>1</sup> , have been assumed. It is result of the simplification,

In the object literature the problem of the fulfilling the Wiener process assumptions by empirical data is often raised, e.g. the expected values of empirical increments are non-zero or increments do not fulfill the criterion of probabilistic independence. These facts have been also observed based on data representing increments of one coordinate, *ttt* <sup>1</sup> *xxx* ,

assumed, that data stand for the realization of a certain stochastic process *Y*(*t*). Also, the linguistic random variable*Yt* has been determined, with the name '*Increment at moment t*'.

, describing the behavior of the particle floating in some liquid. It was

**4.2 Exemplary fuzzy models constructed based on realizations of stochastic** 

**4.2.1 Fuzzy model constructed based on data of a floating particle** 

Also (*Yt* is NL) 0.220

Also (*Yt* is Z) ) 0.532

Also (*Yt* is PL) 0.220

Also (*Yt* is PH) 0.014;


Also (*Yt* is PH) 0.014.

transitions) fulfill the relationship: 1 ( / ) () *PY Y PY tt t* .

under the assumption of a constant time interval 1 *t* .

Table 1.

**processes** 

*x R* [ 3.3,3.3]

The set of the linguistic values, *L*(*Y*)={*negative high NH*, *negative low NL*, *positive low PL*, *positive high PH*} has been assumed. In domain of the process values, the linguistic values are represented by respective fuzzy sets. Also, second linguistic random variable, *Yt*<sup>1</sup> , with the name '*Increment at moment t-1*' has been determined with the same set of linguistic values *L*(*Y*). For the tested process, the criterion of independent increments is not fulfilled, thus, the conditional probabilities (probabilities of transitions) 1 (/ ) *PY Y t t* should be found.

The empirical joint probability of two linguistic random variables, 1 (, ) *PY Yt t* , has been calculated according to (16) - (19), based on the joint probability of numeric values of pairs, <sup>1</sup> ( ,) *t t p x x* , as well as, the assumed fuzzy events, representing the linguistic values {}{ , , , } *L Y NH NL PH PL <sup>t</sup>* (Table 2). Marginal probability 1 ( ) *P Yt* is presented at the last row of the table. It is not a symmetrical distribution, the highest value of the probability, 0.39251, it is a probability that increments take the linguistic value '*Positive Low*'.

Conditional probabilities 1 (/ ) *PY Y t t* , calculated according to (20) – (22) and presented in Table 3, may be treated as the transitions probabilities from fuzzy states {*NH*, *NL*, *PL*, *PH*} at moment *t*-1 to the particular fuzzy states at moment *t*.


Table 2. Joint probability distribution, 1 (, ) *PY Yt t* , of linguistic random variables representing empirical set of increments at moments *t* and *t-*1


Table 3. Conditional probability distributions 1 (/ ) *PY Y t t* of linguistic random variables

The fuzzy knowledge base of the behavior of some particle, determined by changes of its coordinate *Yt* , consists of four file rules (20 elementary rules) with respective probabilities, as follows:

R1: 0.16675 (If *Yt*1 is NH) Then (*Yt* is NH) 0.28575

Also (*Yt* is NL) 0.57150

Fuzzy Modelling Stochastic Processes Describing Brownian Motions 305

important to recognize a probabilistic character of the changes, especially the changes determined in linguistic categories, like: *Positive Big, Positive Small, Zero, Negative Small, Negative Big.* To determine characteristic of the process with fuzzy states, first, we have

 *DX(n)=X(n)-X(n-1)* (33) and a joint probability distribution *p(DX(n),DX(n-1*)) of non-fuzzy values of the process. The range of the increments values, a real number interval [-4.8, 4.8], has been divided into 14 disjoint intervals and the frequency of the occurrence of measurements in particular intervals has been determined. The disjoint intervals have been used for the description of membership functions of particular linguistic values of the set

The empirical joint probability distribution of the linguistic random variables, *P(L{DX(n-1)}, L{DX(n)})*, has been calculated and presented in Table 4. In the last row, the marginal probability values of one linguistic random variable, are presented. It is almost symmetrical distribution, with the highest value of the probability, 0.6114, for the linguistic value of increments equal to '*Zero'*. Conditional probability distributions for particular linguistic values of the variable *DX*(*n*) have been also calculated and they represent weights of particular consequent parts of the rule-base fuzzy model (34). The model of the knowledge

R1: 0.6114 IF (*DX(n-1*) IS *Z)* THEN (*DX(n)* IS *Z*) 0.6450

R2: 0.2123 IF (*DX(n-1)* IS *NS*) THEN (*DX(n*) IS *Z*) 0.6739

ALSO (*DX(n)* IS *NS*) 0.1945

ALSO (*DX(n)* IS *PS*) 0.1462

ALSO (*DX(n)* IS *PB*) 0.0113

ALSO (*DX(n)* IS *NB*) 0.0030

ALSO (*DX(n)* IS *PS*) 0.2032

ALSO (*DX(n)* IS *NS*) 0.1114

ALSO (*DX(n)* IS *PB*) 0.0111

ALSO (*DX(n)* IS *NB*) 0.0004

ALSO (*DX(n)* IS *Z*) 0.4181

ALSO (*DX(n)* IS *NB*) 0.0791

ALSO (*DX(n)* IS *PS*) 0.0714

R3: 0.1474 IF (*DX(n-1*) IS *PS*) THEN (*DX(n)* IS *NS*) 0.4258 (34)

calculated the increments

*L*{*DX*(*n*)}={*NB,NS,Z,PS,PB*}.

base consists of the following five file rules with weights:

Also (*Yt* is PL) 0.036

Also (*Yt* is PH) 0.10675;

R2: 0.21435 (If *Yt*1 is NL) Then (*Yt* is NH) 0.16655


R4: 0.22639(If *Yt*1 is PH) Then (*Yt* is NH) 0.13163


Also (*Yt* is PH) 0.15778.

Probabilities (weights) at the consequent stand for transitions probabilities.

#### **4.2.2 Fuzzy model of the stochastic increments observed in some technological situation**

In a certain technological situation some parameter of a non-homogeneous grain material was measured at discrete moments (Figure 1). It is assumed that observed values *X*(*n*), *n*=1,…,400 represent realization of a certain stochastic process whose variance is high and changes are very quick. For human experts, engineers of the technological process, it is very

Fig. 1. Realization of the stochastic process *X*(*n*)

The fuzzy knowledge base of the behavior of some particle, determined by changes of its coordinate *Yt* , consists of four file rules (20 elementary rules) with respective probabilities,

R1: 0.16675 (If *Yt*1 is NH) Then (*Yt* is NH) 0.28575

R2: 0.21435 (If *Yt*1 is NL) Then (*Yt* is NH) 0.16655


R4: 0.22639(If *Yt*1 is PH) Then (*Yt* is NH) 0.13163


In a certain technological situation some parameter of a non-homogeneous grain material was measured at discrete moments (Figure 1). It is assumed that observed values *X*(*n*), *n*=1,…,400 represent realization of a certain stochastic process whose variance is high and changes are very quick. For human experts, engineers of the technological process, it is very

Probabilities (weights) at the consequent stand for transitions probabilities.

Fig. 1. Realization of the stochastic process *X*(*n*)

**4.2.2 Fuzzy model of the stochastic increments observed in some technological** 

Also (*Yt* is NL) 0.57150

Also (*Yt* is PL) 0.036

Also (*Yt* is PH) 0.10675;

Also (*Yt* is PH) 0.15778.

as follows:

**situation** 

important to recognize a probabilistic character of the changes, especially the changes determined in linguistic categories, like: *Positive Big, Positive Small, Zero, Negative Small, Negative Big.* To determine characteristic of the process with fuzzy states, first, we have calculated the increments

$$DX(n) \equiv X(n) \cdot X(n-1) \tag{33}$$

and a joint probability distribution *p(DX(n),DX(n-1*)) of non-fuzzy values of the process. The range of the increments values, a real number interval [-4.8, 4.8], has been divided into 14 disjoint intervals and the frequency of the occurrence of measurements in particular intervals has been determined. The disjoint intervals have been used for the description of membership functions of particular linguistic values of the set *L*{*DX*(*n*)}={*NB,NS,Z,PS,PB*}.

The empirical joint probability distribution of the linguistic random variables, *P(L{DX(n-1)}, L{DX(n)})*, has been calculated and presented in Table 4. In the last row, the marginal probability values of one linguistic random variable, are presented. It is almost symmetrical distribution, with the highest value of the probability, 0.6114, for the linguistic value of increments equal to '*Zero'*. Conditional probability distributions for particular linguistic values of the variable *DX*(*n*) have been also calculated and they represent weights of particular consequent parts of the rule-base fuzzy model (34). The model of the knowledge base consists of the following five file rules with weights:


Fuzzy Modelling Stochastic Processes Describing Brownian Motions 307

Fig. 2. Realization of the stochastic processes of increments *DX*(*n*), *DX*(*n-1*) and the

Fisz, M. (1967). *Probability and Statistics Theory* (in Polish), PWN, Warsaw, Poland

Springer, ISBN 3-540-62721-9, Berlin, Germany

In this chapter the new approach to fuzzy modelling has been presented. Knowledge base in the form of weighted fuzzy rules represents in the same time the probability distribution of the fuzzy events occurring in the statements. Considered examples show the creating a few simple models of stochastic increments processes. In the future, in modelling the Wiener process, the time dependent probability of the increments should be taken into account.

Hellendoorn, H. & Driankov, D. (Eds.), (1977). *Fuzzy Model Identification; Selected Approaches*,

Kushner, H. (1983). *Introduction to Stochastic Control*, PWN (Polish edition), ISBN 83-01-

Sobczyk K. (1996). *Stochastic Differential Equations*, WNT (Polish edition), ISBN 83-204-1971-9,

Van Kampen, N.G. (1990). *Stochastic Processes in Physics and Chemistry*, PWN (Polish edition),

Walaszek-Babiszewska, A. (2008). Probability Measures of Fuzzy Events and Linguistic

Walaszek-Babiszewska, A. (2011). *Fuzzy Modelling in Stochastic Environment; Theory,* 

213, ISBN 978-960-6766-41-1, Cambridge, UK, February 20-22, 2008

Fuzzy Modelling – Forms Expressing Randomness and Imprecision, In: *Advances on Artificial Intelligence, Knowledge Engineering and Data Bases*, L.A. Zadeh, J. Kacprzyk et al. (Eds.), *Proceedings of the 7th WSEAS International Conference AIKED'08* pp.207-

*knowledge bases, examples,* LAP Lambert Academic Publishing, ISBN 978-3-8454-

predicted mean value

**6. References** 

**5. Conclusion and future works** 

02212-4, Warsaw, Poland

ISBN 83-01-09713-2, Warsaw, Poland

1022-7, Saarbrucken, Germany

Warsaw, Poland

 ALSO (*DX(n)* IS *PB*) 0.0056 R4: 0.0184 IF (*DX(n-1*) IS *NB*) THEN (*DX(n)* IS *Z*) 0.6044 ALSO (*DX(n)* IS *PS*) 0.2363 ALSO (*DX(n)* IS *NS)* 0.1263 ALSO (*DX(n)* IS *PB*) 0.0330 R5: 0.0105 IF (*DX(n-1)* IS *PB*) THEN (*DX(n)* IS *NB*) 0.4762 ALSO (*DX(n)* IS *NS*) 0.4286

ALSO (*DX(n)* IS *Z*) 0.0952.


Table 4. Joint empirical probability distribution of two linguistic random variables representing increments

To determine the predicted value *DX(n)=b\*,* for given value (crisp or fuzzy) *DX(n-1)=a\**, the reasoning procedure, described in 3.3 is used, e.g. for *DX(n-1)=1.55,* predicted value is approximated as equal to *DX(n)=0.30538.* This value depends on many parameters of the fuzzy model and the reasoning procedure. It is very useful to create the computing system with many options of changing the reasoning parameters. In Fig. 2 the predicted, mean values of the increments has been underlined by thick line.

Fig. 2. Realization of the stochastic processes of increments *DX*(*n*), *DX*(*n-1*) and the predicted mean value
