# Time Series Analysis by Fuzzy Logic Methods
## 1 Introduction
Figure 8: (**a**)—record; (**b**)—the relief of the local length of the record; (**c**)—morphological analysis of the record and relief of the local length. Figure 7: (**a**)—record; (**b**)—the relief of the local trend of the record; (**c**)—morphological analysis of the record and relief of the local trend. We compare our work to the most notable accomplishments in morphological analysis. Mathematical morphology, created by [PERSON] and [PERSON] [6; 7; 8; 9], is a mathematical tool for the analysis and processing of geometric structures based on set theory. It employs non-linear morphological filtering using structural elements for image analysis, such as pre-processing, structure enhancement, background selection, and quantitative description through morphological spectra [10; 11; 12]. According to [PERSON], mathematical morphology involves analyzing images under the assumption that they can be described by a particular class of transformations performed on an image, with the image's shape obtained as the maximum invariant of this class [13; 14; 15; 16]. The technical basis is convex analysis in Hilbert spaces, and aims to solve two types of problems: recognition of unknown objects on a known scene and dual search for known objects on an unknown scene. The paper presents a fuzzy logic-based morphology of time series to analyze local properties of the time series, involving two stages: quantitative expression of a property and analysis of its morphology to understand its manifestation in dynamics on the time series. The technical basis is DMA and fuzzy logic [17; 18], and it differs from traditional morphological analysis options that focus on studying nonlocal characteristics, such as trends, seasonality, cyclicality, autocorrelations, and spectrum [19; 20; 21; 22; 23]. Figure 10: (**a**)—record; (**b**)—result of joint monitoring based on the conjunction “min”. Figure 9: (**a**)—record; (**b**)—result of joint monitoring based on the disjunction “max”. Similarly, fuzzy temporal analysis of time series has recently gained popularity, primarily focusing on complex series with many implicit relationships and factors such as those of the financial market and economy [24; 25; 26]. Finally, the authors have a positive experience in examining records for anomalies through straightening of their properties. Elevations on the straightening correspond to anomalies in the records. A natural continuation of research of this kind was a more thorough study of the morphology (geometry) of straightening, not limited to elevations. If the properties behind the straightening are chosen correctly, then such a study is very meaningful, contributing greatly to understanding the morphology of the record, and, through this, about the process behind the record. The morphology of a record is closely related to the nature of the process that it expresses. Morphology allows you to encode, compress a record without losing its significant features, and effectively highlight the most significant features. All of the above were motivations for this work. The work was carried out within the framework of DMA according to its scenarios and its technique. First, the informal logic of the study is addressed and, then, its exact implementation, without which the operation of the algorithms is impossible. The result of the work has a high degree of novelty and differs from traditional variants of morphological analysis of records. In addition, it represents an approach to record analysis in which the researcher has an active position, because the choice of properties and parameters is up to the researcher. In its present form, the presented analysis requires knowledge of the entire record, that is, it has an off-line character and is intended mainly for scientific purposes.