2.1. Grey System Theory and Grey Relational Analysis (GRA) Models

Grey system theory is based on the observation and the fact that all-natural and social systems are uncertain systems, which contain various types of uncertainties and noises due to disturbances from internal or external sources, or due to the limitations of human knowledge and cognition. The fundamental characteristics of uncertain systems are the incompleteness in the information of the system or the data available [49]. Incomplete information in the system may involve the elements (parameters), the structure, the boundaries, and the behaviors of a system. Meanwhile, incomplete information of data is demonstrated by the inaccuracies of data and can be categorized into three types according to the original sources, namely, conceptual, level of perspective, and prediction inaccuracies. For instance, the frequently used concepts, such as “large”, “small”, “fat”, “thin”, “good”, “bad”, “young”, and “beautiful”, are inaccurate due to subjectivity in these concepts and lack of comprehensive definition [49].

The fundamental meaning of “grey” is the incompleteness in the information. Since the information quantity and quality of a system form a continuum from a total lack of information to complete information, grey system theory uses “white” to represent completely known information, “black” for unknown information, and “grey” for partially known and partially unknown information [49]. A system of incomplete information is defined as a grey system accordingly [49].

Grey system theory has further developed a series of models to analyze and make use of the information contained in a grey system. Grey Relational Analysis (GRA) (also called Deng’s Grey Incidence Analysis model) is one of the most widely used models of grey system theory, which was initially proposed by Professor Deng Julong [27,49,50,51,52]. GRA utilizes the degree of similarity of geometric curves of available data sequences to determine whether they are closely associated or not. The more similar the curves are, the more relevant the sequences are, and vice versa [27,49]. GRA models include various types of “grey relational degree” numbers (or “degree of grey incidence”). A variation of the models is briefly explained as follows.

Depending on the correlation coefficient of specific points, Deng’s GRA model reflects the inter-influences among the factors analyzed [49,51,53], hence, it has been widely adopted in various fields of research [52,54,55,56,57,58]. According to Liu et al. [49], Deng’s GRA model follows the computing steps as described below.

Step 1, the dependent variable forms the reference sequence x₀ and the independent variables form the comparison sequence x_i (i = 1, 2, 3…, n) [53].

Step 2, for observation time or observation number k (k = 1, 2…, m), grey relational coefficient (also called “grey incidence coefficient, or point relation coefficients), γ_i (k), is calculated according to Equations (1)–(3):

i = 1, 2…, n;

k = 1, 2…, m; (k indicates observation time, or observation number);

ε is called the resolution coefficient, which has a range of value from 0 to 1, and often takes the value of 0.5 even though the rationale behind this assumption is debatable [27,28].

Step 3, based on the results of γ_i (k) from Step 2, grey relational degree (also called Deng’s degree of grey incidence,) β_i(k) is calculated according to Equation (4):

Step 4, rank the grey relational degree β_i(k) according to the numerical values. The higher the correlation degree is, the higher the ranking is [49].

Apart from Deng’s GRA model, there are new variants and developments in GRA, such as Absolute GRA, Relative GRA, the first synthetic GRA (SDGRA), and the second synthetic GRA (SSGRA).

Derived from Deng’s GRA, the Absolute GRA utilizes the specific point grey and analyzes the correlations between the factors [28,49]. Yu et al. [59] and Tung and Lee [60] highlights the deployment of an absolute GRA model for measuring the association/correlation between variables/parameters as:

where

and

where i and k are the same as above.

Further, the Relative GRA utilizes the integral visual angle [49,59]. The original sequence is zeroed, and the mean value is taken as the initial point. Relative GRA would be given by [49]:

where

and

where xi0′(n) is the initial zero of x_i (n), meanwhile i and k are the same as above.

The SDGRA model evolved from Absolute GRA and Relative GRA [49]. It has been further developed to reflect the line of similar degree and the relative to the proximity of the pilot’s rate of change [49]. It is a comprehensive characterization of whether a sequence has a closer connection between several indicators [28].

The SSGRA model has incorporated the advantages of both Deng’s GRA model and the Absolute GRA model [27,28], and “reflects overall closeness between two sequences based on particular points and integral perspectives” [28].

Both SDGRA and SSGRA models can be expressed by the following equation, generally, θ = 0.5 [27,28].

The detailed computing steps to calculate the GRA models discussed above are reported in Javed et al. [27,28] and Liu et al. [49].