1 Introduction
1.1 Research context
1.2 Motivational objectives
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To offer an up-to-date tutorial of the key developments regarding fuzzy rule-based inference which are tailored to situations where only incomplete domain knowledge is available;
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To provide a systematic comparison between different approaches, so that readers can have an informed choice of what may be the potentially suitable FRI technique(s) to apply given their specific domain problems; and
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To promote the advanced weighted FRI mechanisms to inform the readers about the benefits of using such most recent developed algorithms, which ensure not only the effectiveness of approximate reasoning but also the efficiency (as each time, only two nearest neighbouring rules are required to perform FRI).
1.3 Main contributions
1.4 Paper structure
2 Preliminaries for FRI in fuzzy rule-based systems
3 Fuzzy rule interpolation techniques
3.1 Categorisation of FRI approaches
Methods | Characteristics |
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FRI with only two fuzzy rules | |
FRI with multiple fuzzy rules | |
FRI with fuzzy rules weighted | |
Chen and Lee (2011) | FRI with interval type-2 fuzzy sets (Mendel et al. 2006) |
Chen et al. (2015) | FRI with rough-fuzzy sets |
FRI with adaptivity | |
Hsiao et al. (1998) | Exploiting slopes of fuzzy sets to obtain valid conclusions |
Methods | Characteristics |
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Foundational linear KH FRI based on computation of \(\alpha\)-cut levels | |
Vass et al. (1992) | Extended KH FRI with reduction of invalid conclusions |
Modified \(\alpha\)-cut based method based on coordinate modification | |
Stabilised (general) KH interpolation | |
Modified \(\alpha\)-cut based multidimensional scheme |
Methods | Characteristics |
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Wu et al. (1996) | Using similarity transfers to guarantee valid interpolation |
Adopting generalised concept for interpolation and extrapolation | |
Performing B-spline based interpolation | |
Running FRI with scale and move transformation (T-FRI) | |
Das et al. (2019) | Using fuzzy geometry for linear FRI |
Methods | Characteristics |
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Huang and Shen (2006) | Foundational T-FRI working with two given neighbouring rules involving multiple antecedent variables in various fuzzy membership functions (e.g., complex polygon, Gaussian or bell-shaped) |
Huang and Shen (2008) | Extended T-FRI facilitating both interpolation and extrapolation involving multiple fuzzy rules, with each rule consisting of multiple antecedents |
Backward T-FRI allowing missing antecedent values directly related to the consequent to be interpolated from known antecedents and consequent, supporting backward interpolation and extrapolation involving multiple multi-antecedent fuzzy rules | |
Adaptive T-FRI being capable of restoring system consistency once contradictory results reached during interpolation | |
Chen et al. (2016) | Rough-fuzzy T-FRI allowing representation, handling and utilisation of different levels of uncertainty in knowledge |
Chen and Shen (2017) | Extended T-FRI with interval type-2 fuzzy sets |
Naik et al. (2017b) | Dynamic T-FRI facilitating selection, combination, and generalisation of informative, frequently used interpolated rules for enriching existing rule base while performing interpolation |
Weighted T-FRI allowing attribute weights to be generated given a sparse rule base, being integrated within non-weighted FRI for classification and prediction | |
Chen et al. (2020) | Extended T-FRI being implemented with Takagi Sugeno Kang (TSK) fuzzy models |
Methods | Characteristics |
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Interpolation based on approximation of vague environment of fuzzy rules with application to automatic guided vehicle systems | |
Interpolative method based on graduality | |
Axiomatic approach for interpolation and extrapolation of fuzzy quantities | |
Cartesian based interpolation with each fuzzy set mapped onto a point in high dimensional Cartesian space | |
Group rule interpolation for constructing Adaptive Neuro-Fuzzy Inference System (ANFIS) and an evolutionary computation-supported approach |
3.2 Representative \(\alpha\)-cut based FRI
3.2.1 KH: foundational linear FRI
3.2.1.1 Core principle
The closer a rule’s antecedent \(A^{i}\) (which is a logical aggregation of individual attribute values \(A^{i}_{j}\)) to the observation \(o^{*}\), the closer the rule’s consequent \(B^{i}\) to the outcome \(B^{*}\) that corresponds to \(o^{*}\).
3.2.1.2 Multidimensional implementation
3.2.2 CCL rule interpolation
3.3 Representative intermediate rule based FRI
3.3.1 Representative value of fuzzy set
3.3.2 Scale and move transformation-based FRI (T-FRI)
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Step 1: Closest rules selection
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Step 2: Intermediate fuzzy rule construction
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Step 3: Scale and move factors calculation
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Step 4: Scale and move transformations
3.3.3 Representative modifications to scale and move transformation-based FRI
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Dynamic T-FRI (Naik et al. 2017b)
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Other T-FRI-like approaches
3.3.4 Generalised function-based FRI
3.3.4.1 Generation of interpolated firing rule
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Semantic relation interpolation, which includes any of the semantic revision methods (Ding et al. 1989, 1992; Mukaidono et al. 1990; Shen et al. 1993, 1988), using the semantic revision principle to describe the relation between the antecedent and consequent fuzzy sets within an interpolated intermediate rule.
3.3.4.2 Inference with single rule firing
3.3.5 Geometry-based linear FRI (GLFRI)
3.4 Comparison of representative FRI methods over common criteria
No. | Criterion | Interpretation |
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C1 | Validity of conclusion | Interpolated conclusion always leads to a valid fuzzy set |
C2 | Preservation of convexity and normality | If an observation is normal and convex so should the interpolated conclusion also be |
C3 | Compatibility with rule base | For all rules \(r^{i} \in R\) and all \(A^{*} \in {\mathscr {F}}({\mathscr {A}})\): If \(A^{*}=A^{i}\), then \({\mathscr {I}}({\mathscr {A}}^{*})=B^{*}=B^{i}\) |
C4 | Continuity condition | For \(\epsilon > 0\) there exists \(\delta > 0\) s.t. if \(A, A^{*} \in {\mathscr {F}}({\mathscr {A}})\), and \(d(A,A^{*}) \le \delta\) then \(d({\mathscr {I}}(A),{\mathscr {I}}(A^{*})) \le \epsilon\), where \(d(\cdot ,\cdot )\) denotes a certain distance metric |
C5 | Preservation of piece-wise linearity | If fuzzy sets used for interpolation are piece-wise linear, so should interpolated conclusion be |
C6 | Preservation of “in between” | If \(A^{*}\) is in between \(A^{i}\) and \(A^{j}\), then interpolated conclusion \({\mathscr {I}}(A^{*})\) should be in between \({\mathscr {I}}(A^{i})\) and \({\mathscr {I}}(A^{j})\) |
C7 | Use of arbitrary membership function | Mapping \({\mathscr {I}}\) is applicable to any convex and normal form of membership functions |
C8 | Shape invariance | If all fuzzy sets in rule antecedents are of same type of membership function, so should interpolated \({\mathscr {I}}(A^{*})\) be |
C9 | Applicability for multiple rules | Interpolation mapping \({\mathscr {I}}\) can handle fuzzy interpolative reasoning with any number of rules |
C10 | Applicability for multidimensional input | Interpolation mapping \({\mathscr {I}}\) is applicable to any finite number of input variables. |
C11 | Capability of extension to extrapolation | Interpolation mapping \({\mathscr {I}}\) is extrapolatively extensible |
FRI method | C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | C11 |
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KH FRI | \(\times\) | \(\times\) | \(\checkmark\) | \(\times\) | \(\checkmark\) | \(\checkmark\) | \(\times\) | \(\checkmark\) (\(\dagger\)) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) |
(\(\dagger\) only if C1 is satisfied) | |||||||||||
CCL FRI | \(\checkmark\) | \(\checkmark\) | – | – | – | – | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | – |
T-FRI | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) |
Adaptive T-FRI | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) (\(\dagger\)) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\times\) | \(\checkmark\) | \(\times\) |
(\(\dagger\) first-order piecewise linear) | |||||||||||
Backward T-FRI | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) |
CK FRI | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | – |
Dynamic T-FRI | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) |
Generalised function-based FRI | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) (\(\dagger\)) | \(\checkmark\) | \(\checkmark\) (\(\star\)) | – | – | \(\checkmark\) | \(\checkmark\) |
(\(\dagger\) when involving triangular fuzzy sets) | |||||||||||
(\(\star\) for certain combinations of methods in this family) |
4 Weighted FRI methodologies
Short Name | Title | Reference |
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LHTZ2005 | Weighted fuzzy interpolative reasoning method |
Li et al. (2005) |
CC2008 | A new method for multiple fuzzy rules interpolation with weighted antecedent variables |
Chang and Chen (2008) |
CKCP2009 | Weighted fuzzy interpolative reasoning based on weighted increment transformation and weighted ratio transformation techniques |
Chen et al. (2009) |
CC2011a | Weighted fuzzy rule interpolation based on GA-based weight-learning techniques |
Chen and Chang (2011b) |
CC2011b | Weighted fuzzy interpolative reasoning for sparse fuzzy rule-based systems |
Chen and Chang (2011a) |
CLS2013 | Weighted fuzzy interpolative reasoning systems based on interval type-2 fuzzy sets |
Chen et al. (2013b) |
CH2014 | Weighted fuzzy interpolative reasoning based on the slopes of fuzzy sets and particle swarm optimization techniques |
Chen and Hsin (2014) |
DJS2014 | Antecedent selection in fuzzy rule interpolation using feature selection techniques |
Diao et al. (2014) |
CC2016 | Weighted fuzzy interpolative reasoning for sparse fuzzy rule-based systems based on piecewise fuzzy entropies of fuzzy sets |
Chen and Chen (2016) |
CA2018 | Weighted fuzzy interpolated reasoning based on ranking values of polygonal fuzzy sets and new scale and move transformation techniques |
Chen and Adam (2018) |
LSLYS18 | Fuzzy rule based interpolative reasoning supported by attribute ranking |
Li et al. (2018b) |
LSLYS19 | Interpolation with just two nearest neighbouring weighted fuzzy rules |
Li et al. (2020b) |