5.1. Implementation
The gene regulatory network controls biological functions by regulating the gene expression levels [
40]. It is meaningful to understand the complex causal relationships within a gene regulatory network in a biological system. Genes are paired into activator and repressor, and this gene pair determines the predicted target gene expression level. In what follows, the proposed PFPN model is used to model the gene regulatory network to predict changes in the expression level of the target gene.
Let be nine genes (nine propositions). The WPFPRs of the gene network are defined as follows:
: IF and and THEN ;
: IF THEN ;
: IF THEN ;
: IF THEN ;
: IF THEN ;
: IF THEN .
Based on the transition principle, the gene regulatory network with nine genes can be modeled by a PFPN, as shown in
Figure 4. According to the gene regulatory network of a biological system, the places in the PFPN with respect to their relative propositions are presented in
Table 1. The places
are called starting places, the places
p7 and
p8 are called intermediate places, and the place
is a terminating place.
To determine the knowledge parameters of local weights, global weights, threshold values, and certainty factors, five experts
are invited to provide their judgements with respect to the six WPFPRs. As it is not easy for the experts to express unprecise knowledge information, a linguistic term set
S is utilized to evaluate the above knowledge parameters:
. All the five linguistic terms can be approximated by PFNs, as outlined in
Table 2.
For the gene regulatory network, the initial picture fuzzy evaluation vectors of local weights, global weights, threshold values, and certainty factors, provided by the five experts, are shown in
Table 3,
Table 4,
Table 5 and
Table 6, respectively. Then, the knowledge parameters of the six WPFPRs are acquired according to the proposed knowledge acquisition approach.
First, the weight of each expert is obtained using Equations (17) and (18). Second, the picture fuzzy evaluation vectors of experts are synthesized into a collective picture fuzzy evaluation vector using Equation (19). Next, the consensus degrees of the experts can be checked according to Equation (20) and improved until they are less than or equal to the consensus degree threshold value
θ (
θ = 0.5). Consequently, the computation results for the four knowledge parameters are displayed in
Table 3,
Table 4,
Table 5 and
Table 6, respectively. Note that the collective picture fuzzy evaluation vectors of local weights and global weights should be defuzzified and normalized, as presented in the last columns of
Table 3 and
Table 4.
For the gene regulatory network, the truth degrees of the starting places are set as follows:
.
With the PFPNs established in
Figure 4, we can obtain:
.
Based on the concurrent inference algorithm of PFPNs, the reasoning process for the considered system is explained below.
- (1)
The enabled place vector is calculated using Equation (25) as follows:
.
- (2)
The token value vector of input places is calculated using Equation (26) as follows:
- (3)
The enabled transition vector is calculated using Equation (27) as follows:
.
- (4)
The output truth degree vector is calculated using Equation (28) as follows:
- (5)
The new marking vector is calculated using Equation (29) as follows:
- (6)
Since , we will continue to next iteration and let .
- (7)
Since , we will continue to next iteration, and .
Since , the reasoning process is over. The final PFNs of all the places are obtained as . The expression level of the target gene is , which means that the degree of positive membership is 0.757, the degree of neutral membership is 0, and the degree of negative membership is 0.125.
5.2. Comparisons and Discussion
To show the effectiveness of the proposed PFPNs, a comparison analysis with the IFPNs [
40] and the conventional FPNs [
34] are made in this part. The expression level of the target gene
by using the IFPNs is
[
40]. For the given gene regulatory network, the knowledge parameters in the FPNs are shown as follows:
,
,
.
Based on the reasoning algorithm of FPNs, the result is obtained as:
.
According to above three FPN models, the ranking results of the intermediate places
,
, and the terminating place
are listed in
Table 7. First, we can find that the ranking results of PFPNs and IFPNs are the same. This can prove the feasibility of the proposed PFPNs, but IFSs are utilized in the IFPN model to handle uncertainty and vagueness in knowledge representation and reasoning. Although IFSs have been successfully applied in various areas, there are situations that cannot be represented by IFSs [
19]. As a generalization of IFSs, the PFSs that consider the degree of positive membership, the degree of neutral membership, and the degree of negative membership are more suitable to describe uncertain information and data. For example, the expression level of the target gene
is
in PFPNs and
in IFPNs. The neutral membership degree is ignored in the IFPNs. Thus, the proposed PFPNs have a wider range of applicability than the IFPNs.
The ranking results derived by the PFPNs and the FPNs are different. The main reason is that the information concerning neutral membership degree and negative membership degree is ignored when the FPN model is used. Thus, the original information will be lost in the knowledge representation and acquisition processes. Furthermore, the global weights are not taken into consideration in the traditional FPNs. This implies a lack of precision in the final reasoning result of FPNs.
In addition, the proposed model considers the conflict and inconsistency among expert evaluations in acquiring knowledge parameters. If conflicts and inconsistencies among experts are ignored, i.e., the five experts are treated equally, then the knowledge parameters are obtained as:
Then the reasoning result of PFPNs is obtained as:
Comparing
with
, it is evident that the obtained expression levels of the target gene
are different. This difference can be explained by the fact that the knowledge parameters acquired without considering the conflict between expert evaluations are inaccurate. For example,
Table 8 shows the global weights derived with and without considering conflict evaluations of the experts (Case 1 and Case 2). In addition, the global weights acquired in the IFPNs [
40] are presented as Case 3. It can be found that the global weights determined in the PFPNs that consider conflict and inconsistency are consistent with the original ones yielded in the IFPNs. Therefore, it is significant to take conflict and inconsistency among experts into account in the knowledge acquisition, and the knowledge parameters obtained in our proposed PFPNs are more reasonable and reliable.
In summary, the PFPNs proposed in this study have the following advantages. First, using PFSs, the FPPNs are more efficient in dealing with the vagueness and imprecision in knowledge representation. Second, via a similarity degree-based expert weighting method, the conflict and inconsistency among expert evaluations can be handled in knowledge acquisition. As a result, the knowledge parameters in PFPNs could be determined accurately based on the opinions of different experts. Third, compared with the reachability tree-based reasoning algorithm in current FPNs, the developed inference algorithm of PFPNs adopts a matrix equation format and can execute knowledge reasoning more efficiently.