The fuzzy theory is one of the most studied fields of science. This theory has been used in various fields of science by Gu et al. [1], Gong et al. [2], Rasham et al. [3] and Kausar et al. [4]. We know that the works of Moraczewski [5, 6], Feoli et al. [7], Pedersen et al. [8], Srivastava and Shukla [9], Feoli [10], Rapinel et al. [11], Akdeniz Şafak et al. [12], Bingöl et al. [13], Akdeniz Şafak et al. [14], Akýn et al. [15], have examined plant communities in fuzzy environment.
In nature, plants can live individually or in groups. The plant community is a dynamical form and part of vegetation [16]. A plant community can also be defined as the gathering of living plants in mutual and continuous relationship among themselves and with their environment [17]. Plants can be examined and identified both individually and in communities. It is possible to identify plant groups quantitatively and qualitatively. A plant community can be identified with its components, phytosociological structure and its characteristics. Therefore, different researchers have developed different methods in different time periods [18,19,20,21]. As a result, analysis based on the similarities of plant communities show their better recognition, whether it is a newly identified or already existing one.
While conducting plant sociology studies, it can only be carried out by similarity function whether a newly identified union is a new community for the scientific world or is the same as the existing community found in different regions by different researchers. These degrees of similarity with this new study are in fuzzy logic. It could be specified with more sensitive expressions based on membership degrees. Some mathematical applications in the form of similarity indexes given by Sorensen [22] facilitate the understanding of the degree of relationships among the plant species and the similarity among the plant communities. When comparing two or more plant communities in terms of species composition, similarity or difference indexes provide some quantitative evaluation possibilities [23]. The degree of similarity or difference between any two plant populations identified in the field study can be demonstrated by the cluster analysis method [24]. Further summarizing of vegetation data in any study area can only be achieved by ordination [25, 26], and vegetation mapping [27]. Feoli [28] used fuzzy expressions that are partially present in vegetation tables in different biogeographic areas. Pedersen et al. [8] used fuzzy set clustering to examine the floristic and environmental similarity among relevés, whereas they used fuzzy set ordination to relate floristic patterns to environmental variables. We developed the similarity in our own vegetation tables with the help of fuzzy logic. When Zadeh [29] put forward this logic, he did not refer to whether a statement is true or false, but he referred to the degree of belonging.
In our study, the membership degrees in the tables were prepared according to the presence (1) or absence (0) of plant communities determined in the field with the Braun-Blanquet approach which has been calculated in a fuzzy sense beyond {0, 1} logic. In other words, with our work, a classical similarity of plant sociology studies has been extended by fuzzy approach. New outcomes have been obtained in this method. Then, the similarity between fuzzy sets based on the similarity measure was used in order to compare the plant communities with each other. The conclusion that the fuzzy similarity theory can be considered as an extension of the similarity was expressed by Feoli and Orloci [30].
The first use of similarity functions in plant ecology was carried out by Jaccard [31]. Later, it gained significant momentum with the works of Greig-Smith [32], Sokal and Sneath [33], Pielou [34] and Orlóci [35]. The application of the “Similarity Theory (ST)” is now becoming widespread in many disciplines, particularly genetics [36], biomedicine [37], artificial intelligence [38], and forestry sciences [39]. ST has close links with the fuzzy systems theory (FST) as demonstrated by Roberts [40, 41] in plant ecology. For this reason, elaborating the link between ST and FST is the backbone of the study. Our main goal in this study was to compare the membership degrees of 5 communities and 2 sub-communities of forest vegetation, which were determined in the project [42]. That is we clearly demonstrate the similarity degree of them. We achieved this goal by using fuzzy similarity features, not classical logic (Aristotle logic). Another important goal of this study is to understand the nature of plant communities and to arrive at useful classifications of vegetation at different hierarchical levels. With this study, we have made sure that the communities identified in the research field are independent, pure and homogeneous communities. In addition, the fuzzy approach based on similarity and dissimilarity criteria is extremely important in order to create national or international databases of plant communities with correct data. This method is important because repetitive communities will be prevented and correct data flow will be provided. These results help the protection and planning of a country's economy.
The outcomes obtained in this joint study are given in the results section, together with tables and explanations. The region studied in the project [42] on which this study is based covers Sakarat Mountain and its surroundings; forest, steppe and sub-alpin vegetation types were determined in the research area. Here 5 communities and 2 sub-communities belonging to forest vegetation are evaluated. Sakarat Mountain is located within the provincial borders of Amasya and Tokat. It is also located in the transition zone between the Central Anatolia and Middle Black Sea Region.
The conclusions obtained in the previous project on the Sakarat Mountain are generalized with the above mentioned approach (i.e. fuzzy similarity). The new results are shown in the tables in comparison. Finally, the results achieved and what can be done in the future are explained.
This research paper is organized as follows: Section 2 gives background information about the scheme. Section 3 presents metric 1 with fuzzy similarity of two fuzzy sets. Section 4 introduces Metric 2 with fuzzy similarity of two elements. Section 5 contains some results and discussions about the novelties of this paper. And finally in section 6, the differences between the crisp approximation and fuzzy similarity and the advantages of extending the classical approximation are given.
It is known that in the literature some similarity measures between fuzzy sets and between elements of the sets have been used. The new similarity measures are also used in plant communities and relevés of them. Some advantages of using fuzzyfied similarity measures rather than crisp measures are getting comprehensive knowledge of communities and their elements, making classifications, grouping, checking the accuracy of the results.
The main aim of our joint work is to fuzzify the results obtained in the project [42] of A Phytoecological and Phytosociological Research on the Sakarat Mountain (Amasya). The reason for this is to get some benefits which are explained in the introduction part.
In our work, we have five plant communities and sixty two relevés for extending results obtained in the project [42] mentioned above. Our plant communities are shown as
Let us consider plant communities of {
After the observation on plant communities and relevés, it would be useful to ask the following 2 types of questions in behaviour analysis:
To answer these two fundamental questions in plant sociology we need to give the following two main fuzzy metrics separately for fuzzy similarity with their properties. The reason for this is to establish the new similarity between plant communities (fuzzy sets) and their relevés (fuzzy elements) (See also Lee-Kwang et al. [43]).
Let
Let a fuzzy set
Clearly, it may be symbolized as follows:
Therefore m
We can see some properties of Metric 1 in the following:
The similarity degree is bounded 0 ≤ If The measure is commutative; When the sets
(See also Lee-Kwang et al. [43]).
Consider similarity degrees of any two fuzzy sets in Fuzyy similarity degrees of any two plant communities. = max{{min(0.4,0.6)},{min(0.8,0.3)},{min(0.9,0)},{min(0,0,5)}}, = max{0.4,0.3,0,0} = max{0.4,0.3,0}, = 0.4.
X
0.4
0.6
0
0.8
0.3
0.4
0.9
0
0.8
0
0.5
1
Similarly,
Here our elements are relevés of fuzzy sets (plant communities). The fuzzy similarity measure between two elements 0 ≤ If
(See also Lee-Kwang et al. [43]).
The measure of the fuzzy similarity between two elements satisfies the following properties:
0 ≤ If
Now, we give Table 2 as an example for Metric 2 as follows:
Membership degrees of relevés {
X | ||
---|---|---|
0.2 | 0.5 | |
1 | 1 | |
0 | 0 | |
0.5 | 1 |
By Formula 2, we have
= max{min{0.2,1.0},min{0.5,1.0}} = max{0.2,0.5} = 0.5.
Therefore we have the result of
That is to say, if we need to find the similarity of
is satisfied. These two metrics used here are a new approach for the similarity of phytosociology which we call
In a classical plant sociology works, if the similarity in the compared groups is 50% percent or more, they are considered as “the same community”. If the similarity in the compared groups is less than 50% percent, these groups are evaluated as “the different community”. In classical approaches, degrees of similarity are not mentioned. Whereas, in the fuzzy similarity approach, results for these details are given. This situation is important in the following aspects: it primarily indicates the degree of similarity of the community compared to each other, it also indicates which communities are closer or distant from each other, and that the relevés within the same community have the same or very close membership degrees within themselves is an indicator of homogeneity.
In this joint study, the fuzzy similarity analysis has been applied on five communities and two sub-communities belonging only to forest vegetation among three different vegetation types (forest, steppe and subalpine vegetation) identified in the project [42] titled “A Phytoecological and Phytosociological Research on the Sakarat Mountain (Amasya)”. The main part of this joined work is given in this section. Here, we obtained two kinds of fuzzy similarity tables of which similarity of fuzzy sets and similarity of fuzzy elements in the sets in our approach.
In this study, fuzzy similarity measures were used. The fuzzy similarity methods used in our study are more general and comprehensive than classical phytosociological methods. The method, earlier used, in the mentioned project is quite classical. The relations of five communities and two sub-communities of forest vegetation with each other were analyzed by fuzzy similarity methods. Numerical analysis results are given in Table 3 below. As can be seen from this table,
How the values in Table 3 are calculated is explained in detail with an example in Metric 1.
Fuzzy similarity of plant communities (FSPC) and their numerical values.
FSPC | Numerical values | FSPC | Numerical values |
---|---|---|---|
0.50 | 0.00 | ||
0.25 | 0.00 | ||
0.16 | 0.00 | ||
0.25 | 0.16 | ||
0.25 | 0.16 | ||
0.00 |
If we used the crisp similarity in this table we would have only 1 value and the rest of them would be 0. The numerical values in Table 3 are the degrees to which the plant taxa seen in plant communities are common (the same). For the example, the result of
Descriptions of the relevés (Table 4-19: 4-A1.1, 4-A1.2, 4-A1.3, 4-A1.4, 4-A2.1, 4-A2.2, 4-A2.3, 4-A2.4; Table 4-B1.1, 4-B1.2, 4-B1.3, 4-B1.4, 4-B2.1, 4-B2.2, 4-B2.3, 4-B2.4):
The relevés in Table 4–19 (Table 4-A1, Table 4-A2, Table 4-B1 and Table 4-B2) include five communities. We can express them as follows:
The distinct groupings in the parts where certain relevés are located show that the relevés forming the communities are regularly separated. It clearly shows that the similarities of the grouped relevés are in balance within themselves and that they are clearly different from other relevés. In addition, the fact that the relevés within the same community have the same or very close membership degrees indicates homogeneity.
How the values in Tables 4-A1, Tables 4-A2, Tables 4-B1 and Tables 4-B2 are calculated is explained in detail with an example in Metric 2.
The reason for dividing Table 4 to several parts is due to the fact it is too large. Here, each table given below is quarter part of the main table of fuzzy similarity between elements.
In Tables 4-A1, it shows the similarity degrees of
- | 1.00 | 1.00 | 0.75 | 1.00 | 0.75 | 1.00 | 1.00 | 1.00 | 1.00 | 0.50 | 0.50 | 0.50 | 0.50 | 0.5 | |
1.00 | - | 1.00 | 0.75 | 1.00 | 0.75 | 1.00 | 1.00 | 1.00 | 1.00 | 0.50 | 0.50 | 0.50 | 0.50 | 0.5 | |
1.00 | 1.00 | - | 0.75 | 1.00 | 0.75 | 1.00 | 1.00 | 1.00 | 1.00 | 0.50 | 0.50 | 0.50 | 0.50 | 0.5 | |
0.75 | 0.75 | 0.75 | - | 0.75 | 0.75 | 0.75 | 0.75 | 0.75 | 0.75 | 0.50 | 0.50 | 0.50 | 0.50 | 0.5 | |
1.00 | 1.00 | 1.00 | 0.75 | - | 0.75 | 1.00 | 1.00 | 1.00 | 1.00 | 0.50 | 0.50 | 0.50 | 0.50 | 0.5 | |
0.75 | 0.75 | 0.75 | 0.75 | 0.75 | - | 0.75 | 0.75 | 0.75 | 0.75 | 0.50 | 0.50 | 0.50 | 0.50 | 0.5 | |
1.00 | 1.00 | 1.00 | 0.75 | 1.00 | 0.75 | - | 1.00 | 1.00 | 1.00 | 0.50 | 0.50 | 0.50 | 0.50 | 0.5 | |
1.00 | 1.00 | 1.00 | 0.75 | 1.00 | 0.75 | 1.00 | - | 1.00 | 1.00 | 0.50 | 0.50 | 0.50 | 0.50 | 0.5 | |
1.00 | 1.00 | 1.00 | 0.75 | 1.00 | 0.75 | 1.00 | 1.00 | - | 1.00 | 0.50 | 0.50 | 0.50 | 0.50 | 0.5 | |
1.00 | 1.00 | 1.00 | 0.75 | 1.00 | 0.75 | 1.00 | 1.00 | 1.00 | - | 0.50 | 0.50 | 0.50 | 0.50 | 0.5 | |
0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | - | 0.85 | 1.00 | 0.85 | 0.5 | |
0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.85 | - | 0.85 | 0.50 | 0.5 | |
0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 1.00 | 0.85 | - | 0.85 | 0.5 | |
0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.85 | 0.50 | 0.85 | - | 0.5 | |
0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | - |
0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.00 | |
0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.00 | |
0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.00 | |
0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.00 | |
0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.00 | |
0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.00 | |
0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.00 | |
0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.00 | |
0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.00 | |
0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.00 | |
0.71 | 1.00 | 0.85 | 0.85 | 0.57 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.00 | |
0.71 | 0.50 | 0.50 | 0.50 | 0.50 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.00 | |
0.71 | 1.00 | 0.85 | 0.85 | 0.57 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.00 | |
0.71 | 0.50 | 0.50 | 0.50 | 0.50 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.00 | |
0.43 | 0.50 | 0.50 | 0.50 | 0.50 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.00 |
0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.71 | 0.71 | 0.71 | 0.71 | 0.43 | |
0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 1.00 | 0.85 | 1.00 | 0.85 | 0.50 | |
0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.85 | 0.50 | 0.85 | 0.50 | 0.50 | |
0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.85 | 0.50 | 0.85 | 0.50 | 0.50 | |
0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.57 | 0.50 | 0.57 | 0.50 | 0.50 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
- | 0.71 | 0.71 | 0.71 | 0.57 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.00 | |
0.71 | - | 0.85 | 0.85 | 0.57 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.00 | |
0.71 | 0.50 | - | 0.50 | 0.50 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.00 | |
0.71 | 0.50 | 0.50 | - | 0.50 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.00 | |
0.57 | 0.57 | 0.50 | 0.50 | - | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.00 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | - | 0.85 | 0.85 | 0.71 | 0.85 | 0.57 | 0.57 | 0.85 | 0.28 | 0.57 | 0.00 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.85 | - | 0.85 | 0.71 | 0.85 | 0.57 | 0.57 | 0.85 | 0.28 | 0.57 | 0.00 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.85 | 0.85 | - | 0.71 | 0.85 | 0.57 | 0.57 | 0.85 | 0.28 | 0.57 | 0.00 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.71 | 0.71 | 0.71 | - | 0.71 | 0.57 | 0.57 | 0.85 | 0.28 | 0.57 | 0.00 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.85 | 0.85 | 0.85 | 0.71 | - | 0.57 | 0.57 | 0.85 | 0.28 | 0.57 | 0.00 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.57 | 0.57 | 0.57 | 0.57 | 0.57 | - | 0.57 | 0.57 | 0.28 | 0.57 | 0.00 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.57 | 0.57 | 0.57 | 0.57 | 0.57 | 0.57 | - | 0.57 | 0.28 | 0.57 | 0.00 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.85 | 0.85 | 0.85 | 0.85 | 0.85 | 0.57 | 0.57 | - | 0.28 | 0.57 | 0.00 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.28 | 0.28 | 0.28 | 0.28 | 0.28 | 0.28 | 0.28 | 0.28 | - | 0.57 | 0.00 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.57 | 0.57 | 0.57 | 0.57 | 0.57 | 0.57 | 0.57 | 0.57 | 0.57 | - | 0.00 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | - |
Tables 4-A2 show the similarity degrees of the
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.66 | 0.66 | 0.66 | 0.66 | 0.66 | 0.66 | 0.66 | 0.66 | 0.66 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 |
The following Tables 4-B1 are for
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.66 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.66 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.66 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.66 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.66 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.66 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.66 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.66 | |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.16 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.16 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.16 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.16 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.16 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.16 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.16 |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.16 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.16 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.16 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.16 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.16 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.16 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.16 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.16 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.16 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.16 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.16 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.16 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.16 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.16 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.16 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.16 |
The following Tables 4-B2 are for
- | 0.83 | 0.66 | 0.83 | 1.00 | 0.82 | 0.66 | 1.00 | 0.83 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | |
0.83 | - | 0.66 | 0.83 | 0.83 | 0.83 | 0.83 | 0.66 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | |
0.66 | 0.66 | - | 0.66 | 0.66 | 0.66 | 0.66 | 0.66 | 0.66 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | |
0.83 | 0.83 | 0.66 | - | 0.83 | 0.83 | 0.83 | 0.66 | 0.83 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | |
1.00 | 0.83 | 0.66 | 0.83 | - | 0.82 | 0.66 | 1.00 | 0.83 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | |
0.82 | 0.83 | 0.66 | 0.83 | 0.83 | - | 0.83 | 0.66 | 0.83 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | |
0.66 | 0.66 | 0.66 | 0.66 | 0.66 | 0.66 | - | 0.66 | 0.66 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | |
1.00 | 0.83 | 0.66 | 0.83 | 1.00 | 0.82 | 0.66 | - | 0.83 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | |
0.83 | 0.83 | 0.66 | 0.83 | 0.83 | 0.83 | 0.83 | 0.66 | - | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | |
0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | - | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.25 | - | 0.25 | 0.25 | 0.25 | 0.25 | |
0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.25 | 0.25 | - | 0.25 | 0.25 | 0.25 | |
0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.25 | 0.25 | 0.25 | - | 0.25 | 0.25 | |
0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.25 | 0.25 | 0.25 | 0.25 | - | 0.25 | |
0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | - |
0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | |
0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | |
0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | |
0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | |
0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | |
0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | |
0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | |
0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | |
0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 |
0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 |
- | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | - | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | - | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | - | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | - | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | - | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | - | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | - | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | - | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | - | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | - | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | - | 0.25 | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | - | 0.25 | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | - | 0.25 | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | - | 0.25 | |
0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | - |
So far, until the analysis in the sense of fuzzy logic, in the classical methods used in new syntaxon researches, if the similarity features to the known syntaxa are more than fifty percent, it is stated that the examined species are similar to the previous ones; otherwise, if the common features were less than fifty percent, it was decided that the examined species were a new species. In the examinations made using fuzzy logic, the similarity degrees are given exactly (e.g. 57 % instead of 1 i.e. the similarity degree belongs to the closed interval [50,100 ] or 49 % instead of 0 i.e. the similarity degree belongs to [0, 50)), that is, in detail, more than the classical methods. This study has a superiority over classical methods as it gives more detailed information than previous studies, in other words it gives us more precise information on which group is closer (not a new syntaxon) and which is farther away (it is a new syntaxon). This study brings a different perspective to new researches in plant sociology.
According to the authors of this paper, there are no conflicts of interest to report regarding the article that is being presented.
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for profit sectors.
M.Ü.B.-Conceptualization, Methodology, Software. S.A.Ş.-Formal Analysis, Writing-Review Editing, Validation, Writing-Original Draft. Ö.A.-Supervision. All authors read and approved the final submitted version of this manuscript.
The data used in this study are taken from the project [42] of “A phytoecological and phytosociological research on the Sakarat mountain (Amasya)” supported by The Scientific and Technological Research Council of Türkiye (TÜBİTAK), (Project number: TÜBİTAK TOVAG-HD 105O018). We thank the institution for its support. We would also like to extend our deepest gratitude to Prof. Dr. Haci Mehmet Baskonus for his support.
All data that support the findings of this study are included within the article.
The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.