A defuzzification-free hierarchical fuzzy system (DF-HFS): Rock mass rating prediction
Introduction
Learning in computer systems may be achieved using two different strategies: learning from data and learning from an expert. Regarding the former, it may be difficult to ensure stability if the data set is of poor quality. This weakness may occur when using insufficient, imbalanced or skewed data sets, an unfavorable sampling strategy or ineffective data pre-processing. In addition, these reasons may be extended because of domain specific circumstances. Therefore, the heuristic approaches with human knowledge are paid considerable attention in the cases in which the data set lacks quality or does not exist. Fuzzy inference systems are robust heuristic solutions that are created to model the fuzzy logic approach [1] by operating the fuzzy rules and fuzzy sets. These systems have been used for handling a large number of challenging classification and regression (or function estimation) problems in several research areas [2], [3], [4], [5], [6], [7], [8].
Although fuzzy inference systems have sufficient opportunities for use, a vital restriction, the curse of dimensionality, may occur when the solution needs to execute a complex system with a large number of input parameters. The complexity in question becomes a serious concern from a structural, logical and cost basis [9]. Note that a single fuzzy inference system is termed as a conventional fuzzy system (CFS) in this paper. An increase in the number of input parameters causes a serious growth in the number of fuzzy rules for a complete rule based system. This situation makes the construction and operation of the inference system difficult and nearly inapplicable due to the computational cost. Conversely, this increase complicates rule creation, as well because establishing a logical association for each linguistic variable becomes challenging for the expert. Even if it is able to be established somehow, it is inevitable that the complexity of fuzzy if–then rules and fuzzy reasoning increases [10]. Under these circumstances, fuzzy system in question suffers from its being uninterpretable. Therefore a divide and conquer approach is applied hierarchically on the complex inference problems in order to handle the curse of dimensionality and allow the system being more interpretable. The systems designed by this approach are named as hierarchical fuzzy systems (HFSs). Briefly, high dimensional CFS is separated into lower dimensional subsystems, and these subsystems are linked in a hierarchical manner [11] in HFSs. As can be seen, HFS is the general name of hierarchic solutions.
In Mamdani style [4] hierarchical fuzzy systems, all of the Mamdani inference steps are performed from fuzzification to defuzzification, and the crisp output is delivered to the following subsystem as its input. To specify the HFS that is customized according to this kind of inference flow, the term of conHFS is used to mean conventional hierarchical fuzzy system in this paper.
Regarding the conHFSs, the only concern that has been considered until quite recently is the effective design of the hierarchic structure. Actually, the usage of different hierarchic structures causes differentiations in the hierarchical system's outputs [12]. Therefore, the accuracy of data transmission in conHFS should also be investigated because it is obvious that the defuzzification steps performed in the inner layers degenerate the fuzziness level of the system while reducing the fuzzy set into a crisp value. This situation may cause two drawbacks: inaccuracy and instability.
The conHFS may be inaccurate because it is not managed to provide the corresponding CFS outputs. In fact, this is a significant point that should not be compromised because of the aforementioned reasons, which encourage building an HFS, and does not aim to diverge the CFS accuracy but the difficulties in the construction and application of CFS. On the other hand, conHFS is instable because it is not robust against the variations in the hierarchic structure. The variation in question may be in the order of the input parameters, the connections of the subsystems, the layers or the overall hierarchic structure (such as incremental, aggregated and cascaded [13]). For each structural variation, processing fuzzy inference may be affected differently. Because the previous information regarding membership values and related fuzzy sets permanently disappears after running the defuzzification step, even if defuzzified results are fuzzified again in the subsequent layer, the inference process of subsequent fuzzy systems tries to handle fuzziness in itself. In other words, subsystems of the hierarchic structures fuzzify and defuzzify the local part of the global problem by using defined rules and associations in it. Therefore, changes applied on the structure of the subsystem or whole hierarchy cause a differentiation in the whole process of the hierarchical system and inevitably the final output.
In this paper, a new strategy, which aims to provide conservation of fuzziness in the data while transmitting it between layers of the hierarchic structures, is proposed and is called the defuzzification-free hierarchical fuzzy inference system (DF-HFS). In DF-HFS, the redundantly repeated and misleading defuzzification steps are removed from inner layers, and the fuzzy result of the aggregation step is directly transferred to the higher layer. Thus, the fuzzy information is propagated from the first layer to the topmost layer precisely.
Experiments are performed on the logical XOR and rock mass rating (RMR). A logical problem is selected as a subject because it is very simple to create those generally accepted and indisputably accurate rules without considering the number of input parameters. More importantly, it is possible to create the hierarchical system rules, which are same as the corresponding CFS. In fact, this is the most critical point for the measurement of data loss between layers. Therefore, the logical XOR problem is a distinguished case in this study. On the other hand, the RMR calculation is selected as a second subject because the conventional RMR calculation does not address the uncertainty of the RMR parameters. In fact, the conventional RMR measurement is applied by using sharp class boundaries and fixed rating scales, even though this certainty cannot be reached in the fieldworks [14]. Therefore, the RMR problem is worth solving by a rule based system [15], [16]. Additionally, calculating RMR by using the CFS solution is highly inapplicable because of the large number of input parameters and fuzzy rules. Thus, the hierarchical approach becomes inevitable for the RMR prediction.
To make a comparison and provide interpretations, the hierarchical classifying-type fuzzy system (HCTFS) proposed by [17] is also implemented, and the details for HCTFS are given in Section 2. As a result, DF-HFS, conHFS and HCTFS are implemented in logical XOR and RMR experiments for the quantitative comparison. For the comparisons, the CFS output is used as the reference point. The obtained results reveal that among the other hierarchical inference flows, DF-HFS provides the closest outputs to the corresponding CFS. Moreover, once the hierarchical system rules are created identically with the related CFS rules, the DF-HFS provides identical outputs with CFS unlike the other hierarchical solutions. In these circumstances, it enables the implementation of more flexible and robust HFSs against the structural variations as well.
Section snippets
Literature summary
Hierarchical fuzzy systems address the issue that solving a complex problem with a large number of input parameters by a single fuzzy inference system has some logical, structural and cost basis restrictions. One of the pioneer studies for hierarchical fuzzy systems is proposed by Raju and Zhou [11] to overcome these restrictions. Afterwards, the computational and structural advantages of HFSs are discussed [9], [17], [18], [19], [20], and approximation capability of differently modelled HFSs
Method of DF-HFS
In conHFS built with Mamdani style inference [4], the crisp output of a subsystem is transferred to the higher layer subsystem as an input. Although this strategy solves the CFSs' curse of the dimensionality problem by reducing the inference cost and allowing simple rule creation, the redundantly repeated defuzzification steps cause the data to be degenerated because each defuzzification generalizes the fuzzy information by reducing it into a single value [9], [13]. Therefore, an amount of loss
RMR data set
The rock mass rating (RMR) developed by Z.T. Bieniawski is a strategy which is utilized in classification of rock masses that is universally used for tunnels, mines, slopes and foundations to provide insight during the planning of the construction [15]. Conventionally, there are two steps for the RMR calculation as follows: basic RMR calculation and adjusted RMR calculation. The basic RMR calculation is provided by using five parameters, which are the uniaxial compressive strength of rock
Experiments and results
In the experiments, four methods, which are implemented by utilizing the FuzzyNet [42] library, are listed below with reminder explanations:
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conHFS: it is the conventional implementation of HFS, where the defuzzification step is implemented in each subsystem.
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HCTFS: it is proposed in [17] as an AHFS-based hierarchic structure. Although the defuzzification step is omitted in the inner layers, the structure of the hierarchy is fixed. The defuzzification method of HCTFS is center-average.
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DF-HFS: it
Conclusion
When the use of only one fuzzy system becomes inapplicable due to the increase in the number of input parameters and indirectly the number of fuzzy rules, the curse of the dimensionality problem occurs. Due to hierarchical fuzzy systems, this problem may be solved as a structural, logical and cost basis. However, the conventional HFS (conHFS) repeats the misleading defuzzification steps redundantly in the inner layers. This detail causes degeneration in the information during data transmission
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