Application of Cloud Model in Qualitative Forecasting for Stock Market Trends

Forecasting stock prices plays an important role in setting a trading strategy or determining the appropriate timing for buying or selling a stock. The use of technical analysis for financial forecasting has been successfully employed by many researchers. The existing qualitative based methods developed based on fuzzy reasoning techniques cannot describe the data comprehensively, which has greatly limited the objectivity of fuzzy time series in uncertain data forecasting. Extended fuzzy sets (e.g., fuzzy probabilistic set) study the fuzziness of the membership grade to a concept. The cloud model, based on probability measure space, automatically produces random membership grades of a concept through a cloud generator. In this paper, a cloud model-based approach was proposed to confirm accurate stock based on Japanese candlestick. By incorporating probability statistics and fuzzy set theories, the cloud model can aid the required transformation between the qualitative concepts and quantitative data. The degree of certainty associated with candlestick patterns can be calculated through repeated assessments by employing the normal cloud model. The hybrid weighting method comprising the fuzzy time series, and Heikin–Ashi candlestick was employed for determining the weights of the indicators in the multi-criteria decision-making process. Fuzzy membership functions are constructed by the cloud model to deal effectively with uncertainty and vagueness of the stock historical data with the aim to predict the next open, high, low, and close prices for the stock. The experimental results prove the feasibility and high forecasting accuracy of the proposed model.

ERGODIC EXTENSIONS OF ENDOMORPHISMS

We examine a class of ergodic transformations on a probability measure space $(X,{\it\mu})$ and show that they extend to representations of ${\mathcal{B}}(L^{2}(X,{\it\mu}))$ that are both implemented by a Cuntz family and ergodic. This class contains several known examples, which are unified in our work. During the analysis of the existence and uniqueness of this Cuntz family, we find several results of independent interest. Most notably, we prove a decomposition of $X$ for $N$-to-one local homeomorphisms that is connected to the orthonormal bases of certain Hilbert modules.

Joint ergodicity along generalized linear functions

A criterion of joint ergodicity of several sequences of transformations of a probability measure space $X$ of the form $T_{i}^{\unicode[STIX]{x1D711}_{i}(n)}$ is given for the case where $T_{i}$ are commuting measure-preserving transformations of $X$ and $\unicode[STIX]{x1D711}_{i}$ are integer-valued generalized linear functions, that is, the functions formed from conventional linear functions by an iterated use of addition, multiplication by constants, and the greatest integer function. We also establish a similar criterion for joint ergodicity of families of transformations depending on a continuous parameter, as well as a condition of joint ergodicity of sequences $T_{i}^{\unicode[STIX]{x1D711}_{i}(n)}$ along primes.

Sharp ergodic theorems for group actions and strong ergodicity

Let $\mu$ be a probability measure on a locally compact group $G$, and suppose $G$ acts measurably on a probability measure space $(X,m)$, preserving the measure $m$. We study ergodic theoretic properties of the action along $\mu$-i.i.d. random walks on $G$. It is shown that under a (necessary) spectral assumption on the $\mu$-averaging operator on $L^2(X,m)$, almost surely the mean and the pointwise (Kakutani's) random ergodic theorems have roughly $n^{-1/2}$ rate of convergence. We also prove a central limit theorem for the pointwise convergence. Under a similar spectral condition on the diagonal $G$-action on $(X\times X,m\times m)$, an almost surely exponential rate of mixing along random walks is obtained.The imposed spectral condition is shown to be connected to a strengthening of the ergodicity property, namely, the uniqueness of $m$-integration as a $G$-invariant mean on $L^\infty(X,m)$. These related conditions, as well as the presented sharp ergodic theorems, never occur for amenable $G$. Nevertheless, we provide many natural examples, among them automorphism actions on tori and actions on Lie groups' homogeneous spaces, for which our results can be applied.

Notions of relative ubiquity for invariant sets of relational structures

Given a finite lexicon L of relational symbols and equality, one may view the collection of all L-structures on the set of natural numbers ω as a space in several different ways. We consider it as: (i) the space of outcomes of certain infinite two-person games; (ii) a compact metric space; and (iii) a probability measure space. For each of these viewpoints, we can give a notion of relative ubiquity, or largeness, for invariant sets of structures on ω. For example, in every sense of relative ubiquity considered here, the set of dense linear orderings on ω is ubiquitous in the set of linear orderings on ω.

Minimal convergence on Lp spaces

Let (X, F, μ) be a probability measure space, p and β real numbers such that 1≤p<+∞ and 0<β<p. For any linear positive operator T satisfying T1, T*1 = 1 we prove the norm and pointwise convergence of the sequence We get then the pointwise and norm convergence in Lp, 0 < β ≥ 1 < p < 2, of the sequence sgn Sif for any positive linear operator on Lp(Ω, A, μ) (μ-σ-finite) verifying ∥(1 − α)I + αS∥p ≤ 1 for a real number 0 < α < 1. In the particular case α = 1, (S is a contraction), β = p−l, this result gives the pointwise and norm convergence of the sequences introduced by Beauzamy and Enflo in 1985 to the asymptotic center of the sequence .

Weakly mixing PET

Suppose that (X, ℬ, μ) is a probability measure space and T is an invertible measure perserving transformation of (X, ℬ, μ). T is called weakly mixing if for any two sets A1A2 ∈ ℬ one has:

Property kα,n on Spaces with Strictly Positive Measure

In this paper we study intersection properties of measurable sets with positive measure in a probability measure space, or equivalently, intersection properties of open subsets on a compact space with a strictly positive measure.The first result in this direction is due to Erdös and it is a negative solution to the problem of calibers on such spaces. In particular, under C.H., Erdös proved that Stone's space of Lebesque measurable sets of [0, 1] modulo null sets, does not have ℵ1-caliber.