# Hybrid Learning for General Type-2 TSK Fuzzy Logic Systems
## 1 Introduction  
Learning techniques in soft computing exist for the purpose of adjusting models so they can accurately represent data in some domain. Although there are various approaches to these learning techniques, we can categorize learning techniques into two groups: hybrid and non-hybrid learning techniques. A non-hybrid learning technique is composed of a single algorithmic process which achieves the learning of a model, whereas a hybrid learning technique is composed of a sequence of two or more algorithms where in each step a portion of the final model is achieved. Some examples of non-hybrid techniques are a learning algorithm for multiplicative neuron model artificial neural networks [1], an optimized second-order stochastic learning algorithm for neural network training using bounded stochastic diagonal Levenberg-Marquardt [2], design of interval type-2 fuzzy logic systems by utilizing the theory of an extreme learning machine [3], the well-known backpropagation technique for artificial neural networks [4], etc. Yet by combining some of these direct approaches with each other or with other techniques, their performance can greatly improve, such that some steps can compensate performance loss or simply focus on optimizing a portion of the model. Examples of such hybrid models are the scoring criterion for hybrid learning of two-component Bayesian multinets [5], hybrid learning particle swarm optimization with genetic disturbance intended to combat the problem of premature convergence observed in many particle swarm optimization variants [6], a hybrid MonteCarlo algorithm used to train Bayesian neural networks [7], a learning method for constructing compact fuzzy models [8], etc. When dealing with raw data where models must be created, transforming such data into more manageable and meaningful information granules can greatly improve how the model performs as well as reducing the computational load of the model. An information granule is a representation of some similar information which can be used to model a portion of some domain knowledge. By forming multiple information granules, these can represent the totality of the information from which data is available; therefore, forming a granular model. Granular computing [9; 10] is the paradigm to which these concepts belong. Information granules which intrinsically support uncertainty can be represented by general type-2 fuzzy sets (GT2 FSs), and in turn these GT2 FSs can be inferred by a general type-2 fuzzy logic system (GT2 FLS) [11]. Although, when dealing with type-2 fuzzy logic systems, they are either in the form of interval type-2 FSs [12] or general type-2 FSs, interval type-2 FSs being a simplification of general type-2 FSs. In simple terms, uncertainty in a GT2 FS is represented by a 3D volume, while uncertainty in an IT2 FS is represented by a 2D area. It is not until recent years that research interest has gained momentum for GT2 FLSs; examples of such published research are fuzzy clustering based on a simulated annealing meta-heuristic algorithm [13], similarity measures \(\alpha\)-plane representation [14], hierarchical collapsing method for direct defuzzification [15], a multi-central fuzzy clustering approach for pattern recognitions [16], etc. Apart from Mamdani FLSs which represent consequents with membership functions, there also exists the representation by linear functions. These FLSs are named Takagi-Sugeno-Kang fuzzy logic systems (TSK FLSs). Examples of TSK FLS usage are evolving crane systems [17], fuzzy dynamic output feedback control [18], analysis of the dengue risk [19], predicting the complex changes of offshore beach topographies under high waves [20], clustering [21], etc. As published research with GT2 FLS is still very limited, most of it uses Mamdani FLSs, and so far only two published journal papers using a GT2 TSK FLS exist, for controlling a mobile robot [22], and data-driven modeling via a type-2 TSK neural network [23].