1 Introduction

Machine learning models are ubiquitous nowadays; They are involved in many activities of daily life in which people are aware or unaware of their use. It is essential to know how and why the models provide a particular output and how they can be made more fair and secure, especially in critical applications. Interpretable Machine Learning (IML) strives to construct a bridge between the learned model and human understanding.

There are various ways to achieve model interpretability by applying: i) intrinsic interpretable models, ii) regularization techniques, and iii) posthoc explanation techniques. One trend is to build surrogate models (lower complexity) than the original one and more interpretable to understand the decision process, such as rule-based models. [⁷, ⁹, ⁴⁸].

This work takes advantage of the inherent interpretability that brings Fuzzy Inference Systems (FIS). FIS models are interpretable since they are rule-based models and are described linguistically. The antecedent is represented by fuzzy variables and sets, proposed by Zadeh [⁵⁹]. The inference process of those systems is performed through fuzzy reasoning, which aims to represent human perception and their inference mechanism under uncertainty.

An interesting characteristic of FIS is that their knowledge representation is composed of IF-THEN rules, where their antecedents and consequents are in natural language. For example, a proposition can be “Temperature is hot”, where Temperature is related to an input attribute and hot to the set which partially belong. As was described, this formal notation potentially brings an intrinsic high interpretability degree if their components are well-defined [³²].

The FIS are used in a variety of application domains in ML context, such as medical [¹⁴, ¹⁹, ²⁴, ⁴¹, ⁵⁷, ³⁵], robotics [¹⁵, ¹], decision making [⁶⁰, ¹², ³].

Often this FIS modeling is data-driven, usually by some unsupervised technique (e.g., clustering) to discover their input partition, membership functions parameters, and rule structure. An important issue is that as the number of attributes (input dimension) and fuzzy sets increases, it becomes more susceptible to presenting a combinatorial problem in the rule-finding process. In this approach, we construct the initial rule structure through a decision tree model, which is further transformed to the fuzzy space domain and adapted to a Sugeno-type architecture described in the further sections.

As the input dimension space and feature interaction complexity increases, the resulted decision tree gets deeper; therefore, it becomes pruned to be hardly interpretable. For each leaf node or decision node in the tree, it can extract an IF-THEN decision rule. To tackle the problem of a high number of decision rules is conducted a post-pruning process. The pruning process is carried on by selecting the most relevant elements (antecedents and decision rules).

Granular Computing (GrC) aims to form mean-ingful Information Granules, which represent a collection of objects abstraction and relate them by some similarity in a hierarchical manner [⁵], which allows creating semantically richer structures [⁶⁷]. GrC is inspired by how the human brain works, processing information abstractly at the required level to resolve a given task. The data is organized by some of their characteristics in a hierarchical way to form a granule, a formal representation of this structure with two essential properties, specificity, and coverage.

Specificity is related to the granule representation; the higher specificity is, the less ambiguous it is, and humans can easily understand it. On the other hand, coverage is related to the proportion of individuals designated by the granule.

Intuitively, a well-formed information granule should have higher values in both specificity and coverage. However, these properties are commonly in conflict, as the higher the specificity is, the lower is the coverage; an optimization process usually conducts the granule allocation.

This work proposes a data-driven method to construct a fuzzy rule-based system using the principle of justifiable granularity for selecting the most relevant knowledge base elements, according to the trade-off between specificity and coverage values. This discrimination is conducted over the rules formed by decision tree models, allowing the construction of a variable model complexity useful in IML.

The main contributions of this work are:

In the following sections correspond to: a brief description of the relevant theory, section 2. In section 3 is reviewed the related work. In section 4 is described the hierarchical Information Granules and their generation. In section 5, the neuro-fuzzy model is described. In the section 6 is shown the followed methodology and experiment setup to evaluate the proposal. The results and conclutions are in sections 7 and 8 respectively.

2 Background

The proposed methodology comprehends three main areas: i) Fuzzy systems for establishing natural language interface in the form of fuzzy rules; ii) Decision tree-based models for initial structure rule discovery in the data space domain. iii) Granular computing, aiming at the well-formed Information Granule for rule-based knowledge model representation following the principle of justifiable granularity.

3 Related Work

GrC has been used in Machine Learning problems as a way to define semantic richer data representation to build models with missing information [²⁵], prototype forming for descriptors of facial expressions [⁵⁵]. To establish initial neural network architecture for further optimization [³⁹]. Furthermore, this framework has been used to overcome the limitations of existing Machine Learning models related to data quality [²², ¹⁰, ⁶], interpretability, domain adaptation for regression tasks [²²], and dimensionality reduction [², ¹⁶, ⁵²].

In [³¹] is discussed the importance of adopting the GrC paradigm in rule-based systems as a way to improve interpretability. The principle of justifiable granularity has been used to discover robust information clusters in the context of data-driven system modeling [⁶⁸].

There are various works in the context of GrC which support the hypothesis of more robust generation rule-based systems [⁴⁴, ⁵⁴], rule reduction using complex fuzzy measures with GrC [⁴⁷]. Also, the hierarchical representation of granule modeling has been used to solve hierarchical classification problems [²³], for building interpretable models in data stream learning environments [²⁷].

The use of GrC in the context of Machine Learning comprehends the discovery of the Information Granules in the data space domain and forms them by some formal description, e.g., intervals, fuzzy sets, rough sets, hyperboxes.

Some recent approaches in GrC adopt cognitive science perspectives and fuzzy logic to support intelligent decision-making [¹⁸, ⁵⁶]. GrC has promoted the adoption of fuzzy logic for data abstraction to be capable of processing various data types in classification tasks using granular decision trees [²⁹, ³⁰]. A top-rated operator in deep learning for extracting relevant features, the convolution, had been adapted to operate with fuzzy sets with a granular perspective for classification problems [¹¹]. In [⁶⁶] is proposed a polynomial-feature granulation method based on long short-term memory network for oxygen supply network prediction.

In this work, a decision-tree model discovers the rule base before characterizing its elements in a granular paradigm. Given an a priori defined granules collection, this work selects the best ones to create higher-level Information Granules. It then performs the fuzzification to develop a new Sugeno-type fuzzy rule base with learning capability due to their analogous neural network representation.

4 Granules Construction

Different levels of abstractions define the granule construction process. The level of these granules is denoted by the subindex Ai. In the first level of granularity, A0 corresponds to the original data space, so that a zero-level granule is equivalent to a given instance of the dataset A0≈x(i).

A level 1 Information Granule (A1) is characterized as the tuple (m,r) that correspond to a range [m−r,m+r], notice that in this particular scenario, it is defined with a symmetric proportion from the median (could be further extended). A level 2 Information Granule (A2) is described as an implication relation; at this abstraction level, the interaction between different domain Information Granules is considered. A level 3 Information Granule in this work is defined as a decision set.

The notation to denote a granulation process is through eq. 3:

where G is the mapping process to form a higher Information Granule given a set of lower-level granules, A′i⊆Ai denote a set that belongs to the i-level granular space, and Ai+1 is a formed higher level granule. For instance, the first abstraction level starts from the crisp data space, such that G(X)=G(A0)=A1. For notation purposes Gl(A) denotes the process of perform l-level abstraction processes for a given granule A.

The notation to denote a degranulation process is through eq. 4:

where G−1 is the mapping process to form a lower Information Granule given a granule, Ai denote the i-level granular space, and Ai−1 is a formed lower level granule.

It is essential to notice that the raw representation of a granule is a set that is formed by granules of lower levels. Intuitively, a granule should be represented by a model that requires low information.

For a level 1 Information Granule is defined the following metrics to measure the coverage (eq. 5) and specificity (eq. 6).

Coverage:

Specificity:

where A1 is an Information Granule, xk∈X, m, and r characterize a range (level 1 granule); m is the median of the range, and r the distance to the rage limits in a symmetric way. card(⋅) stands for the cardinality of a given set:

where A′1⊆A1, that in their raw representation is a set that belongs to the A1 space, namely a set of ranges (denoted by a tuple (m,r)) which conforms an IF-THEN rule.

The operator ∧ represent the conjuntion operation. ρ:A1×X is a logical function that forms the proposition “x belongs to the granule A1(i)”. Y is the target domain:

In the degranulation process of a level 2 granule, the raw representation of the operation is a set of level 1 granules, and those granules form the antecedent part in the rule structure.

For a level 2 Information Granule are defined the following metrics to measure the coverage (eq. 9) and specificity (eq. 10).

Coverage:

Specificity:

where A2 is an Information Granule formed by a implication relationship ∧iρ(A1(i);x)→Y, N is the number of instances that correspond to the dataset; and, dim(X(i)) is the number of considered features in the dataset. card(⋅) stands for the cardinality of a given set.

To select a justifiable Information Granule, it is necessary a measure their formation quality. Due to the contradictory behavior of coverage and specificity can form a Pareto front to select the best candidates. Notice that the Pareto front is formed by the product and can be computed to any granule at any abstraction level (eq. 11):

There is a parameter that serves to prioritize one of the terms [³⁸], γi. If 0≥gammai<1 there is pondered more the coverage, on the contrary, if γi>1 then the specificity gets more relevant in the calculus. For each abstraction level, a different value γi can be defined.

4.1 Hierarchical Quality Measurement

The quality construction of a given of H level granule is measure with the proposed metric (eq. 12), which perform successive degranulation process and multiply their Pareto front values to the lower level granules:

V(Ah)={Qlh(Ah)×V(A′h−1)|∀A′h−1∈G−1(Ah)}. (12)

The optimization problem is shown in equation 13, which aims to find the more appropriate level 1 granules. The aptitude function is the hierarchical measure (eq. 12), their restrictions are: 1) The solution set must be the minimum cardinality (showed as the function card(⋅)) 2) The aggregation of the values vi (hierarchical Pareto) should be equal or less the to regularization parameter α, and 3) The number of elements in the solution set must be equal or less than the regularization parameter ξ. These two regularization parameters allow finding the best level 1 granules smaller set with at least a cumulative value of α and does not have more than ξ elements:

AH*=arg⁡max⁡A′1⊆…⊆AHV(G2(A′1))subject to:1)   min⁡card({A′1}),2)   ∑i=1card({G−1(AH)})V(G(A′1))i≤α,3)   card({A′1})≤ξ. (13)

The resulted Information Granule collection reconstructs the decision set, formally by G2(A*). This process can be treated as a post-pruning technique, the bias of the model increases while its variance decreases. The initial structure for building the knowledge base of the FIS to be optimized is formed by the decision set. Figure 1 shows the general block diagram of the proposed methodology for data-driven fuzzy rule base construction. The details of the neuro-fuzzy model characteristics are in section 5.

Fig. 1 Block diagram of the proposed methodology for building a fuzzy rule-based system from decision set created by decision-tree model using the principle of justifiable granularity 

5 Sugeno-Type Neuro-Fuzzy Model

The Sugeno fuzzy systems allow their construction in a systematic form to generating fuzzy rules in a data-driven manner, they were proposed by Takagi, Sugeno, and Kang [⁴⁵]. These systems are also composed of IF-THEN rules, but only their antecedent belongs to the fuzzy space. The consequent part is a crisp function that maps the input space.

As in the Mandani type fuzzy systems, the rules might fire all at once to get a crisp output value, can compute a simple weighted average of function outputs, which is less time-consuming than defuzzification in Mamdani type fuzzy systems. The knowledge base can be described as follows:

where xi is a fuzzy variable (which models the feature space), their fuzzy values are represented by the terms low, high, etc. y is the output space described by the function σj∈J(n)(x;w), where J represents the output classes, x are the input crisp values, and ware the function’s coefficients.

Once the knowledge base is designed, some optimization methods can adjust their membership function parameters to fit the data better. This optimization process comprehends the membership function and consequent crisp function parameters. In this approach, to maintain the interpretability characteristic, sigmoid and linear functions are selected.

An analogous neuro-fuzzy architecture carries out the optimization of fuzzy model parameters. This architecture comprises five layers: input, fuzzification, inference, implication, and defuzzification layer. The connections between the layers are not fully connected to maintain coherent antecedent relationships in the fuzzy rules. In the fuzzification layer are only connected the membership functions belonging to the input domain. In the implication layer, each crisp function is only related to the rule’s output (a given class). Figure 2 shows a visual representation of a Sugeno-type neuro-fuzzy model.

Fig. 2 The Sugeno Neuro-fuzzy representation is composed of a non-fully connected 5-layer artificial neural network. All the computations correspond to the involved operations in a Sugeno-type fuzzy inference system for q inputs, n rules, and crisp consequent functions. Each rule is only linked to one class j∈J target space 

The parameters to optimize the model belong to the antecedent part of the fuzzy rule and the (trainable) parameters in the crip consequent function.

The neural architecture is shown in figure 2. The first layer is non-fully connected among neurons that represent the fuzzification process. Wich maps a crisp input to the fuzzy space:

where Brj∈Vk is a fuzzy set that belongs to the fuzzy variable Vk, the domain of each fuzzy variable is shared by its corresponding attribute domain in the dataset. Only the membership functions directly related to the attribute are evaluated by the input value, which results in a semi-connected layer.

The inference layer is also non-fully connected, fires at a certain strength value in the range [0,1]. The implication operation calculates a t-norm (∗˜) as a product:

The implication layer performs a normalization operation that conforms a step to the weighted average to the output:

After the normalization process, for each rule that has a crisp function consequent related to an

output class j∈J, a ∗˜ as the product is calculated:

where σ corresponds to sigmoid function that transform the input space, σ(xi;wl)=11+e−wlxi. After the sigmoid transformation, all values are computed by the softmax function, which maps proportionally for each class, values in the range [0,1] and ∑j=1Jhj=1, it commonly interprets those values as a probability output:

The output value dimension corresponds to the number of classes in the problem domain. Getting an actual label as a result value could be computed just as the argument position with the higher probability value:

This neuro-fuzzy model adjusts its parameters to fit a given target better. To measure the error of the prediction is used the Cross-Entropy loss function (LossCE):

where N is the number of instances on the batch; y is the target value in the form of one-hot-encoding, and hj is the predicted probability of belonging to the class j. In the loss function are defined regularization l1 and l2 norms to constrain the weight values and prevent overfitting. These regularizers can help to improve the interpretability by only relevant considering relevant features.

The trainable parameters on the neuro-fuzzy model are the design parameters of the fuzzy sets and the set of crisp consequent function parameters; in this setup, the membership functions are defined by Gaussian functions, then the trainable parameters are the mean and σ values, where σ>0.

A gradient descent optimization method is applied to find the best parameters. The learning rule is shown in equation 21:

where θ are the parameter vector values; E(θ) is the gradient of the error value of the model with the parameters θ; η is the learning rate value in 0<η<1.

The hyper-parameters of the model are:

6 Experimentation

Ten publicly available datasets^fn are used in order to evaluate the proposed model under different domain applications. All datasets correspond to classification tasks; their characteristics are shown in the table 1.

The methodology consists of primary three steps:

7 Results

The proposed model was evaluated by 5-fold cross-validation, in 10 publicly available datasets for classification, with 6 different values for the hyper-parameter α which serve up to select the compression level by selecting the most relevant elements (according to eq. 12). The obtained results for overall Sugeno-type neuro-fuzzy model for classification (showed in table 2) were f1-scores of 0.77, 0.81, 0.81, 0.82, 0.83, 0.82 when α-values were set to 0.2, 0.4, 0.6, 0.8, 0.95, 1 respectively, with the maximimun rules parameter (ξ) set to 50.

Considering all different values of α, in average the model reduction (in terms of elements) was 73.51% with σ=17.23. From a rule percentage reduction perspective, the average compression was 57.53% with σ=17.34, and relative error concerning the Simple Decision Tree model of −1.4% with σ=0.021, and 5.55% with σ=0.022 respect to Random Forest model (in this experiment the number of weak learners was set to 100).

Figure 3 shows the performance comparison between the decision tree-based models and the proposed one at diverse α values.

Fig. 3 Performance comparison between the decision tree-based models (Decision Tree and Random Forest) and the proposed granular model (Fuzzy-GrM) at diverse σ values. As lower the α value, the higher compression the model is 

At the global level, considering all analyzed datasets, the minimum elements compression percentage given the following values of the prunning value α were: 81.25% with α=0.2, 58.33% with α=0.4, 50% with α=0.6, 23.07% with α=0.8, 81.25% with α=0.2, 11.11% with α=0.95, 0% with α=1. At rule compression percentage were obtained: 57.14% with α=0.2, 33.33% with α=0.4, 25% with α=0.6, and no rule compression at all for higher values for α.

In dataset results tables (10-7), there are comparisons of the f1-score of the proposed model (F-GrM), Decision Tree (DT), and Random Forest (RF). Figure 5 shows as higher compression is applied to the model, the higher the variance is. Figure 4 shows the overall performance comparison of different hyperparameters.

Fig. 4 Overall performance comparison with diverse values of alpha, applied in the ten different classification datasets 

Fig. 5 The relationship between performance and model compression percentage in terms of antecedents 

A paired sample t-test over the mean f1-score values was used to formally validate the results in all experiments with a confident value of 95%. Table 12 shows the comparison between neuro-fuzzy (with different compression rates) and decision tree models.

All mean f1-score values for the proposed model and random forest have a significant difference (RF had substantially better general performance than the proposed model).

It is important to note that the fuzzy granule model has considerably fewer rule elements even in configurations with no significant difference in the mean f1-scores with respect to the DT model.

According to the validation, there is a significant difference between models where alpha is 0.2, 0.95, and 1; in the first value, which the model compresses the most the rules (around 82%), the initial decision tree model result is higher by 0.02. However, in alpha values 0.95 and 1, which the model comprises the lowest the rules (around 41% and 39%, respectively), the proposed model is higher by 0.03 and 0.02, respectively. There is no significant difference between model results in intermediate alpha values, although the decision sets were compressed between 71% and 49%.

8 Conclusion and Future Work

This work addressed the problem of input space partition to create the base rule structure for a FIS by selecting the meaningful elements from a decision tree model given a compression rate value (controlled by the hyperparameter α).

The motivation for choosing a subset of the elements in rule-based systems is to maintain their structure as simple as possible, directly impacting their interpretability. Fewer antecedents lead to understanding the phenomena with less effort; also, the number of rules affects too. In order to build a knowledge base with the greatest interpretability, it is necessary to create it with fewer antecedents and rules.

The antecedents were characterized as type-one granules by ranges, while the implications were represented as type-two granules through relationships between granules of the lower level. The abstraction of decision sets is made by hierarchical structure between the different levels in the granules. The transition from one level to another is defined through the granulation and degranulation operations.

The methodology for extracting the relevant elements was based on the principle of justifiable granularity, which has a higher value of specificity and coverage. In this approach, the optimization method selects all level granules simultaneously through the hierarchy established.

The reconstruction of the selected granules creates a new decision set with fewer elements than the original one due to the optimization restrictions: i) minimize the number of level-one granules; ii) the cumulative Pareto front values must be equal or less than a regularizer hyper-parameter α; iii) a hyperparameter ξ restricts the number of level-one granules to consider in the selection.

The presented work defined a hierarchical measurement that helps to consider the best lower-level information granules better suited to form high-quality higher-level Information Granules. This well-formed Information Granules search is defined as an optimization problem with restrictions. The objective is to find the smallest set of best lower-level granules that maximize the hierarchical quality measurement composed by the Pareto front at different levels of specificity and coverage. As the number of elements decreases in the rule-based system, the bias increases; to increase the model variance, the resulting pruned decision set is converted to a Sugeno-type neuro-fuzzy model for classification tasks that is further optimized. The proposed Sugeno-type neuro-fuzzy model for classification has the following characteristics:

The results support that using a FIS with the proposed method for rule selection can compress the contained information in a classical decision set without significantly compromising the performance model. The different compression rate values (α) impact the model performance.

As shown in the results, a higher compression rate (lower α value, e.g., 0.2) degrades the model performance significantly; however, this small decision set in a complex domain might be helpful to have a more interpretable model to understand phenomena better.

An attractive characteristic is that intermediate compressing rate values (e.g., between 0.4 and 0.8) achieve a considerable reduction in the decision set without significant difference performance.

In problems where the performance is crucial, higher α values are recommended, which less compress the decision set but still might be considerable (e.g., at least a mean of 39% in this experimental setup). The rule compression achieved by the proposed method is relevant in the context of IML due that simplifies the decision model by i) reducing the number of elements (antecedents and rules), decreases as well the complexity of the systems, which improves the interpretability; ii) fuzzy logic brings an interface in natural language to the human and promotes a better understanding of the model; iii) the compression of the model can be controlled by α hyper-parameter and be set to the most convenient value for a specific application domain (see figure 3 compare the behavior of α in different domain applications).

This proposal opens the opportunity to further research in decision-set-based Information Granules for smaller and more interpretable models; this methodology can be used with different models/methods that generate decision sets. An extension of the current work is considered to design a higher-level information granule, specificity and coverage metric to select the most relevant decision set source elements for ensemble methods.

Another possible extension of this work is to use type-2 fuzzy logic, which better manages decision processes under uncertainty and achieves better performance with a smaller knowledge base in terms of rules. The overlapping of rule partition with different consequent could characterize uncertainty in higher-level fuzzy sets.

The proposed Hierarchical Decision Granules Optimization method can be adapted to any rule-based system by defining specificity and coverage metrics for each granule level. It can be interesting to incorporate different information frameworks such as probabilistic and rough sets to enhance the intrinsic semantic meaning in the Information Granule, therefore generate richer explanations taking advantage of the natural language interface that brings the fuzzy logic.