Fuzzy Modelling Stochastic Processes Describing Brownian Motions

Wiener process, as a special mathematical model of Brownian motions, has been investigated and modelling in many probabilistic examples. In the topic literature it is easy to find many procedures of numeric probabilistic simulations of the Wiener process. Fuzzy modelling does not give us more accurate models than probabilistic modelling. Fuzzy knowledge-based modelling allows to determine linguistic description of non-precise relationships between variables and to derive the reasoning procedure from non-crisp facts. More over, using the notions of probabilities of fuzzy events, it is possible to determine a frequency of a conclusion as well as its expected value. Wiener process and a random walk are very often used for modelling phenomena in physics, engineering and economy. In the area of robot control theory these processes can represent some time-varying parameters of the environments where the object of control operates. Fuzzy models of these processes can constitute a part of a fuzzy model of a tested complex system. In paragraph 2. of this chapter, the mathematical descriptions of Brownian motions has been reminded, according to the theory of probability and stochastic processes. Some basics of fuzzy modelling has been presented in paragraph 3., to show the method of creating the knowledge base and rules of reasoning. Attention is focused on identification techniques for building empirical probabilities of fuzzy events from input-output data. Exemplary calculations of knowledge bases for real stochastic processes, as well as, some remarks on future works have been presented in paragraphs 4 and 5.


Introduction
Wiener process, as a special mathematical model of Brownian motions, has been investigated and modelling in many probabilistic examples. In the topic literature it is easy to find many procedures of numeric probabilistic simulations of the Wiener process. Fuzzy modelling does not give us more accurate models than probabilistic modelling. Fuzzy knowledge-based modelling allows to determine linguistic description of non-precise relationships between variables and to derive the reasoning procedure from non-crisp facts. More over, using the notions of probabilities of fuzzy events, it is possible to determine a frequency of a conclusion as well as its expected value. Wiener process and a random walk are very often used for modelling phenomena in physics, engineering and economy. In the area of robot control theory these processes can represent some time-varying parameters of the environments where the object of control operates. Fuzzy models of these processes can constitute a part of a fuzzy model of a tested complex system. In paragraph 2. of this chapter, the mathematical descriptions of Brownian motions has been reminded, according to the theory of probability and stochastic processes. Some basics of fuzzy modelling has been presented in paragraph 3., to show the method of creating the knowledge base and rules of reasoning. Attention is focused on identification techniques for building empirical probabilities of fuzzy events from input-output data. Exemplary calculations of knowledge bases for real stochastic processes, as well as, some remarks on future works have been presented in paragraphs 4 and 5.
particles are frequent and irregular. It is usually assumed by mathematicians, that the displacements of the particle 12 , ,..., ,... n ZZ Z at particular time steps, are independent, identically distributed random variables. The stochastic process { , 1, 2,..., .,...} i Zi n  is named random walk. In macroscopic scale, if the time between two observations of the particle, t   , is larger than the time between successive crashes, then the increment of the particle positions, t XX   , is a sum of many small displacements, Wiener process is also known as the Brownian motion process (Fisz, 1967;van Kampen, 1990;Kushner, 1983;Sobczyk, 1991). The increments, t XX   , 0 t     , are normal distributed random variables with the expected value and variance as follows: are also normal distributed with parameters: and with a non-zero covariance matrix.
Xt   is the standard Wiener process. Probability, that a particle occurs in some interval [a, b], at the moment t, is given by the relationship is a conditional probability density function, and 11 (, ) f tx is given by formula (1)  Xt t   , i=1,…,n, is the scalar stochastic Wiener process and particular scalar stochastic processes are independent. As an example of the 3D stochastic Wiener process we can show three coordinates of the Brownian particle trajectory. The Wiener process is also the special diffusion stochastic process, fulfilling the Fokker-Planck diffusion equation where the solution is given by the normal probability density function, and the diffusion coefficient is equal to 2 /2   (van Kampen, 1990). In macroscopic scale, in physics and in industrial practice, the probability value f(x)dx that scalar variable X assumes its value from the interval [x, x+dx] is equivalent to the quotient n/N (concentration of particles), where n defines the power of subset of particles, whose feature X determines the value over the interval [x, x+dx], and N is the population size. This idea is consistent with Einstein's experiments who considered collective motion of Brownian particles. He assumed that the density (concentration) of Brownian particles (,) xt  at point x at time t, met the following diffusion equation: where γ is a diffusion coefficient. The solution has the known exponential form. From the analytical form of the solution the second central moment of the displacement is expressed as www.intechopen.com 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 character of the diffusion law (cited by van Kampen, 1990).

Stochastic process with fuzzy states
Let X(t) be a 'short memory' stochastic process, the family of time-dependent random variables, where XR    , 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)={LX i }, 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 A i , i=1,2,…,I determined on χ by their membership functions, () : which are Borel measurable functions, fulfilling the condition According to above assumptions, the probability distribution of linguistic values of the process X(t) can be determined as follows based on Zadeh's definitions of the probability of fuzzy events (Zadeh, 1968 The following conditions must be fulfilled The following relationships should be fulfilled for the conditional distributions of fuzzy states (probability of the transitions) 21 1,...,

Rule based fuzzy model
The proposed model of the stochastic process, formulated into fuzzy categories, for two moments 12 ,, tt 21 tt  , is a collection of file rules, in the following form (Walaszek-Babiszewska, 2008, 2011: or as a collection of the elementary rules in the form where the weights w i , w j/i , w ij represent probabilities of fuzzy states, determined by (13) - (15), (20) - (22) and (17)

Reasoning procedures
Considering reasoning procedure, we assume that some non-crisp (vague) observed value of the stochastic process at moment t 1 is known and equal to 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 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 & Filev, 1994;Hellendoorn & Driankov, 1997). Weights of rules, representing the probability of a fuzzy event in antecedent (w i ), as well as, the conditional probability of a fuzzy event at the consequence part (w j/i ), can be used to determine probabilistic characteristics of the conclusion. It is worth to note, that fuzzy www.intechopen.com conclusions (27) and (28) computed as the aggregated outputs of all active i-th file rules, can be determined by the following formula (Walaszek-Babiszewska, 2011) where membership functions, / ' ji A  , 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 () t PX of the output linguistic random variable.

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,    (15), has been presented in Table 1. Also, the second linguistic random variable, 1 t Y  , 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: 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.): In the created rule base of the stochastic process, the same probability distributions for random variables, t X  and 1 t X   , have been assumed. It is result of the simplification, under the assumption of a constant time interval 1 t   .

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,  The fuzzy knowledge base of the behavior of some particle, determined by changes of its coordinate t Y , consists of four file rules (20 elementary rules) with respective probabilities, as follows: Probabilities (weights) at the consequent stand for transitions probabilities.

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) 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.  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

Conclusion and future works
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. The robotics is an important part of modern engineering and is related to a group of branches such as electric & electronics, computer, mathematics and mechanism design. The interest in robotics has been steadily increasing during the last decades. This concern has directly impacted the development of the novel theoretical research areas and products. This new book provides information about fundamental topics of serial and parallel manipulators such as kinematics & dynamics modeling, optimization, control algorithms and design strategies. I would like to thank all authors who have contributed the book chapters with their valuable novel ideas and current developments.