# Time Series Analysis by Fuzzy Logic Methods
## 1 Introduction  
Until recently, when working with a record (time series), a researcher could rely only on statistical methods of spectral-temporal analysis. Recently, the situation has changed. New methods for analyzing a record have appeared, associated with the development of artificial intelligence, and these methods allow researchers to take more active positions in relation to records, so that, in particular, they can express experience and knowledge at the formal level. The subject of this work is one of these methods. Informally, the method is used when a researcher is interested in some property of a record of a local character. More precisely, he or she is interested in the dynamics of manifestation of a property in the record, because the investigator concerned believes that having this knowledge gives him or her an opportunity to better understand the nature of the process behind the record. Thus, two things must be strictly understood: what a local property (hereinafter referred to as property) of a record is and what its dynamics are. All this is conducted on the basis of discrete mathematical analysis (DMA), which is a new approach to data analysis that focuses on the researcher and occupies an intermediate position between hard mathematical methods and soft analytical ones [1; 2; 3; 4; 5]. DMA in data analysis uses scenarios of classical mathematics, in which the fundamental foundations of the scenarios are replaced by fuzzy models of their discrete counterparts. The solution of the problem within the DMA framework consists of two parts. The first part is informal, and consists of examining the logic of the researcher, introducing the necessary concepts, and explaining the scheme and principles of the solution. The second part has a formal character, wherein, with the help of the DMA apparatus, all concepts receive a strict definition within the framework of fuzzy mathematics (FM) and fuzzy logic (FL), and schemes and principles become algorithms. Guided by this, we turn to the concepts of Dynamics and Property. **Dynamics**: The informal first part of the dynamics concept in solving a problem is as follows. With a given relief and questions of a local morphological nature, we send the researcher on a journey through the relief and ask the researcher to answer the followingquestion: \"To what extent, being on the relief at a given moment in time, does he or she feel himself or herself to be on the relief's surface, slope, peak, depression, and so on?\"  
**Dynamics** The formal second part of the dynamics concept in solving a problem involves formalizing the researcher's answers by means of fuzzy logic. The result is a family of morphological measures that express the local morphological structure of the relief (background, trend, peak, depression, and so on). **Property** The informal first part of the property concept in solving the problem involves the researcher sliding over the record, estimating by positive numbers the degree of manifestation of the property of interest to him or her in the same small fragments of the record. These scores are assigned by the researcher to the fragment centers. So, from the original record, the researcher arrives at a non-negative function, which is called straightening the record, with respect to the property of interest to the researcher. **Property** The formal second part of the property concept in solving the problem is the third part of the article. The researcher researching a record can be interested in any property. Nevertheless, the reality, in regard to the stability of such interest, is that there is a circle of basic straightening that most researchers want to deal with. Behind each of the straightening is a fundamental mathematical concept. The paper presents the following rectifications: energy (dispersion, continuity), irregularity (frequency, length), and spread (Cauchy fundamentalism, continuity).