**Definition 7.**

$$\underline{B}X\_{\beta} = \left\{ i \in I \, \middle| \, \frac{|\, [i]\_{IND(B)} \cap X|}{|[i]\_{IND(B)}|} \ge \beta \right\} \tag{7}$$

$$\mathbb{Z}X\_{\beta} = \left\{ i \in I \, \middle| \, \frac{| [i]\_{IND(B)} \cap X |}{| [i]\_{IND(B)} |} > 1 - \beta \right\} \tag{8}$$

*where set B is a subset of the attribute set A; set X is a subset of the universe I; β is the variable precision, which ranges from 0.5 to 1; BX<sup>β</sup> is the lower approximation; and BX<sup>β</sup> is the upper approximation*.

Compared with the rough set, VPRS extends the range of the upper and lower approximation, thus restricting the sensitivity of the rough set model to noisy data.

The rough set uses the indiscernibility relation to classify equivalence classes, but it is unsuitable for numerical data, especially in the case of the application of big data and high accuracy. One way to overcome this difficulty is to use the similarity relation, so we can extend the rough set based on the similarity relation.

This extension essentially involves modifications of the two concepts of the rough set, the indiscernibility relation and equivalence class. The classical indiscernibility relation is more suitable for those descriptive attributes, while elements of an attribute set that satisfy the indiscernibility relation are divided into an equivalence class. However, when dealing with numeric data, the effect of this method will be considerably reduced. The rough set, based on the similarity relation extends the indiscernibility relation into a similarity relation, and the equivalence class classified by the indiscernibility relation is replaced by a similarity relation.

#### *2.2. Rough Set and Functional Dependence*

With the establishment of the rough set model with variable precision based on the similarity relation, the decision rules between approximately equivalence classes divided by the conditional attribute set and the approximate equivalence classes divided by decisive attribute set can be worked out. However, this is not enough to explain the correlation between the conditional attribute set and the decisive attribute set, and so, functional dependence is introduced. Although the rough set and functional dependence are two different fields, many concepts of functional dependence can be explained from the perspective of the rough set.

**Definition 8.** *Functional dependence, the complete dependence between universe I and attribute A, can be expressed as C* → *d , where C* ⊆ *A, d* ∈ *A*.

**Definition 9.** *Partial dependence, the partial dependence between universe I and attribute A, can be expressed as C* → *pd , where C* ⊆ *A, d* ∈ *A*.

**Inference 1.** *Complete dependence of attribute set, any attribute* ∈ *D, D* ⊆ *A, C* → *d works. Accordingly, there is functional dependence between C and D, and this relationship can be expressed as C* → *D, where D is an attribute set*.

**Inference 2.** *Partial dependence of attribute set, any attribute* ∈ *D, D* ⊆ *A, C* → *pd works. Accordingly, there is partial functional dependence between C and D, and this relationship can be expressed as C* → *pD, where D is an attribute set*.

We now explain complete dependence and partial dependence from the perspective of the rough set. For a decision-making system based on the rough set, λ = 1 means that the decisive attribute set completely depends on the conditional attribute set, that is, there is complete dependence between the decisive attribute set and conditional attribute set. 0 < λ < 1 means that there are some factors that affect the decisive attribute set in addition to the conditional attribute set, that is, the decisive attribute set partially depends on the conditional attribute set, so the following inferences are introduced.

**Inference 3.** *Complete dependence in the rough set, in a decision-making system DS* = (*U*, *C* ∪ *D*)*, only occurs when p* λ = 1*, and the complete dependence C* → *D comes into effect*.

**Inference 4.** *Partial dependence in the rough set, in a decision-making system DS* = (*U*, *C* ∪ *D*)*,* 0 < λ < 1 *indicates that there is partial dependence C* → *pD between C and D, and the degree of partial dependence p equals* λ.

After calculating the accuracy λ of the model, according to inference 3 and inference 4, if λ = 1, the decisive attribute set completely depends on the conditional attribute set. If 0 < λ < 1, the decisive attribute set partially depends on the conditional attribute set. In other words, there is a certain correlation between the conditional attribute set and the decisive attribute set, and λ can be used as a parameter to measure the degree of correlation.

This section proposes the rough set model with variable precision, based on the similarity relation and some inferences related to functional dependence. This method will not only dig out the correlation between the conditional attribute set and the decisive attribute set, but also use accuracy λ to measure the degree of correlation.
