Probability Theory for Fuzzy Quantum Spaces with Statistical Applications

Kolmogorov probability theory based on set theory belongs to the most important results of mathematics of the 20th century. Naturally, its main advantage is the possibility to use results of the modern measure theory. However, this fact sometimes does not allow larger considerations. In this chapter we want to show this paradox can be eliminated. Of course, we present only some basic ideas. Understanding them enables one to study further results and applications.

Similarly as the Kolmogorov probability theory in the first half of the 20th century, the Zadeh fuzzy set theory played a significant role in the second half of the 20th century. In this chapter we present probability theory on intuitionisic fuzzy sets as well as probability spaces on multivalued logic.

In this chapter we study the existence of a sum of fuzzy observables in a fuzzy quantum space which generalizes the Kolmogorov probability space using the ideas of fuzzy set theory. We also study some properties of the sum of fuzzy observables. To study the above mentioned, we also include the basic notions from the probability theory on fuzzy quantum space in this chapter, i.e. the notion of fuzzy quantum space, a fuzzy observable, an indicator of a fuzzy set, a null fuzzy observable, a Boolean algebra on fuzzy quantum space, fuzzy state etc.

In this chapter we introduce selected limit theorems on fuzzy quantum space, namely Egorov’s theorem, Central limit theorem, Weak and strong law of large numbers, and extreme value theorems for fuzzy quantum space. We also study here the Ergodic theory for fuzzy quantum space and Ergodic theorems and Poincaré recurrence theorems for fuzzy quantum dynamical systems, the Hahn-Jordan decomposition and Lebesgue decomposition for fuzzy quantum space.

In this chapter we compare two methods applied to reduce the dimensionality of data sets. The first method is Principal component analysis, and second method is Factor analysis. We present these methods on data from Atanassov’s intuitionistic fuzzy sets [6]. Earlier we construct an example of applying these methods. The calculations are performed in program R.