# Time Series Analysis by Fuzzy Logic Methods
## 1 Introduction
The set is open to replenishment, which is dictated by practice, and provides greater flexibility to the analysis of a record within the framework of DMA. Let us summarize the intermediate result. The researcher chooses a property, builds its rectification on the record and studies the dynamics of the manifestation of the property on the record through the morphology of its rectification. Morphology is a collection of fuzzy structures in time associated with the morphological properties of straightening. In the simplest case, this is a set of fuzzy structures constructed in a work that are responsible for the simplest basic morphological properties of straightening. Further, NL makes it possible to express not only the complex property of one straightening, but also the properties related to several instances of straightening at the same time. In other words, it becomes possible to study a record based on the totality of their properties, in the form of their manifestations, on the record. The purpose of the work is to create a morphological analysis of a record within the framework of DMA using NM and NL. The structure of the work is based on two parts. The first part involves modeling in the form of fuzzy answer structures, analyzing the researcher's record in regard to 16 questions concerning its local morphological structure (background, trend, top, bottom, and so on). The second part implements the systemic point of view on the record, connecting a system of rectifications with it, that quantitatively express the manifestation of one or another local property of the record. Applying the first part with the second gives the main result. Let us formulate it. The result of the work is a procedure for morphological analysis of a record based on a system of its rectifications, carried out using a family of fuzzy structures responsible for the simplest basic morphological properties of the rectification  
Let us look at some examples. **Example 1**: _The more interesting (fundamental) the property behind the rectification, the more interesting its dynamic monitoring (dynamic analysis). This is exactly what the property is in this example. This is the local dispersion of a record, closely related to its continuity, and, therefore, to the regularity of the process behind it. Figure 1a shows the original record in black. The relief of its local dispersion is shown in Figure 1b. Four properties were chosen for its dynamic analysis: the surface of the relief, the boundary of the hill on it, its slope and top._  
_The fuzzy dynamic expressions are constructed according to the algorithms of the article, which serve as the basis for monitoring. The result is a morphological analysis of the variance, that is, its dynamic coding by a property that is most strongly fuzzy at the moment._Figure 1c shows the result of such coding: 0 (illac)--background, 1 (green)--border, 2 (blue)--slope, 3 (red)--peak. The colors help you understand what the encoding looks like on and through the rectifier of the original recording. The coloring of the stochastic corridor around the original record in Figure 1a shows its different stages and is useful for understanding what is happening on the record. **Example 2**. : _Here, the property is the local length of the record, as well as the dispersion, which are of fundamental importance, and depend on many factors. The most important factors are the local differential and spectral characteristics of the record. In this case, the basis for monitoring the straightening relief is its seven morphological properties, which are the following: the background and three groups of dual characteristics, left/right foot, left/right slope, top/bottom._  
_Fuzzy dynamic expressions are built for the above, acting as the basis for monitoring the \"length\" straightening on the original record. The background, left foot, right slope and top are shown in the third drawing in Figures 2-5. The first two are the same (original notation and \"length\" straightening).