Comparing some plant communities in a region of Türkiye via fuzzy similarity

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 AS₁(Pino sylvestri-Fagetum orientali ass.nova), AS₂, (Acero campestri-Pinetum sylvestri ass.nova), AS₃, (Querco pubescenti-Pinetum sylvestri ass.nova), AS₄ (Carpino orientali-Quercetum boissieri ass.nova), AS₅ (Rhododendro lutei-Fagetum orientali ass.nova; AS_5a (carpino orientali-quercetosum macrantherae subass.nova); AS_5b (vaccinio arctostaphylli-loniceretosum orientali subass.nova) and relevés as r₁,r₂,r₃,⋯,r₆₂ := r₁(1)r₆₂. A member is involved in communities with the membership degrees as shown in Table 3.

Let X = {AS₁, AS₂, AS₃, AS₄, AS₅} be the set of plant communities we deal with.

Let a fuzzy set AS_i in the universal set X is defined by the membership function AS_i(r_k), for r_k ∈ AS_i ⊂ X;k = 1(1)62. Here AS_i;i = 1(1)5 be any plant community taken from X. Then we can give the following equality called as a fuzzy similarity (FS) of AS_i and AS_j. (1) FS(ASi,ASj)=max{min{μASi(rk),μASj(rk)}}; i,j=1(1)5; k=1(1)62, rk∈ASi⊂X,=max{min[μASi(r1),μASj(r1)],min[μASi(r2),μASj(r2)],⋯,min[μASi(r62),μASi(r62)]}. \begin{array}{*{20}{c}}{FS(A{S_i},A{S_j}) = \max \{ \min \{ {\mu _{A{S_i}}}({r_k}),{\mu _{A{S_j}}}({r_k})\} \} ; i,j = 1(1)5; k = 1(1)62, {r_k} \in A{S_i} \subset X,}\\{ = \max \{ \min [{\mu _{A{S_i}}}({r_1}),{\mu _{A{S_j}}}({r_1})],\min [{\mu _{A{S_i}}}({r_2}),{\mu _{A{S_j}}}({r_2})], \cdots ,\min [{\mu _{A{S_i}}}({r_{62}}),{\mu _{A{S_i}}}({r_{62}})]\} .}\end{array} Namely, min[μAS_i(r₁),μAS_j(r₁)] = m₁(i, j); min[μAS_i(r₂),μAS_j(r₂)] = m₂(i, j);⋯;min[μAS_i(r₆₂),μAS_j(r₆₂)] = m₆₂(i, j);i, j = 1(1)5.

Clearly, it may be symbolized as follows:

FS(AS₁,AS₂) = max{ m₁(1,2), m₂(1,2),⋯, m₆₂(1,2)},FS(AS₁,AS₃) = max{ m₁(1,3), m₂(1,3),⋯, m₆₂(1,3)},

Therefore m_k(i, j) = min[μ_ASi(k),μ_ASj(k)],i, j = 1(1)5;k = 1(1)62.

We can see some properties of Metric 1 in the following: 1)

FS(AS_i,AS_j) is the maximum membership degree in the intersection FS(AS_i ∩ AS_j).

(See also Lee-Kwang et al. [43]).

Here our elements are relevés of fuzzy sets (plant communities). The fuzzy similarity measure between two elements r_k,r_l ∈ AS_i ⊂ X in fuzzy sets, i = 1(1)5;k,l = 1(1)62 is defined as follows: FSe(rk,rl)=max{min{μASi(rk),μASi(rl)};i=1(1)5;k,l=1(1)62, =max{mi(k,l)}, =max{m1(k,l),m2(k,l),m3(k,l),m4(k,l),m5(k,l)}, mi(k,l)={mi(1,2),mi(1,3),⋯,mi(1,62),mi(2,3),mi(2,4),⋯,mi(2,62),mi(3,4)mi(3,5),⋯, mi(3,62),⋯,mi(61,62)},i=1(1)5. \begin{array}{*{20}{l}}{F{S_e}({r_k},{r_l}) = \max \{ \min \{ \mu A{S_i}({r_k}),\mu A{S_i}({r_l})\} ;i = 1(1)5;k,l = 1(1)62,}\\{\;\;\;\,\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \max \{ {m_i}(k,l)\} ,}\\{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\, = \max \{ {m_1}(k,l),{m_2}(k,l),{m_3}(k,l),{m_4}(k,l),{m_5}(k,l)\} ,}\\{\;\;\;\;\;{m_i}(k,l) = \{ {m_i}(1,2),{m_i}(1,3), \cdots ,{m_i}(1,62),{m_i}(2,3),{m_i}(2,4), \cdots ,{m_i}(2,62),{m_i}(3,4){m_i}(3,5), \cdots ,}\\{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{m_i}(3,62), \cdots ,{m_i}(61,62)\} ,i = 1(1)5.}\end{array} The fuzzy similarity measure between two elements r_k,r_m ∈ AS_i ⊂ X in fuzzy sets, i = 1(1)5;k,m = 1(1)62 is defined as follows: (2) FSe(rk,rm)=max{min{μASi(rk),μASi(rm)}};i=1(1)5;k,m=1(1)62. F{S_e}({r_k},{r_m}) = \max \{ \min \{ {\mu _{A{S_i}}}({r_k}),{\mu _{A{S_i}}}({r_m})\} \} ;i = 1(1)5;k,m = 1(1)62. Metric 2 satisfies the following three properties: (1)

0 ≤ FS_e(r_k,r_m) ≤ 1.

The measure of the fuzzy similarity between two elements satisfies the following properties: (1)

0 ≤ FS_e(r_k,r_l) ≤ 1.

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, FS(AS₁,AS₂) was calculated as 0.5. That is, the fuzzy similarity of community AS₁ to community AS₂ is 0.5. Here, FS(AS₁,AS₂) expression is the degree of similarity of AS₁ and AS₂.

How the values in Table 3 are calculated is explained in detail with an example in Metric 1.

Tables 4-A2 show the similarity degrees of the r_i,r_j relevés. In the case of i = 1(1)31; j = 32(1)62. Here, relevés with a degree of similarity of 0.25 seem to be dominant in number (For the comparison to crisp case, see the explanation for Table 4-A1). Moreover, Tables 4-A2 is constructed by FS_e(r_i,rj), Fuzzy similarity of elements (relevés) (fuzzy similarity between r_i,r_j;i = 1(1)31; j = 32(1)62).

The following Tables 4-B1 are for FS_e(r_i,r_j), Fuzzy similarity of elements (relevés) (fuzzy similarity between r_i,r_j;i = 32(1)62; j = 1(1)31). Tables 4-B1 show the similarity degrees of the r_i,r_j relevés. In the case of i = 32(1)62; j = 1(1)31. Here, it is seen that the relevés with a similarity degree of 0.25 are dominant in number (For the comparison to crisp case, see also the explanation for Tables 4-A1).

The following Tables 4-B2 are for FS_e(r_i,r_j), Fuzzy similarity of elements (relevés) (fuzzy similarity between r_i,r_j;i, j = 32(1)62). Tables 4-B2 show the similarity degrees of the r_i, r_j relevés. In the case of i, j = 32(1)62. Here, it is seen that the relevés with a similarity degree of 0.25 are dominant in number (For the comparison to crisp case, see also the explanation for Tables 4-A1). In addition, in these Tables (4–19), we can see also that the relevés taken during the field studies were carried out correctly.

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.