Data Analysis Approach for Incomplete Interval-Valued Intuitionistic Fuzzy Soft Sets

: The model of interval-valued intuitionistic fuzzy soft sets is a novel excellent solution which can manage the uncertainty and fuzziness of data. However, when we apply this model into practical applications, it is an indisputable fact that there are some missing data in many cases for a variety of reasons. For the purpose of handling this problem, this paper presents new data processing approaches for an incomplete interval-valued intuitionistic fuzzy soft set. The missing data will be ignored if percentages of missing degree of membership and nonmember ship in total degree of membership and nonmember ship for both the related parameter and object are below the threshold values; otherwise, it will be filled. The proposed filling method fully considers and employs the characteristics of the interval-valued intuitionistic fuzzy soft set itself. A case is shown in order to display the proposed method. From the results of experiments on all thirty randomly generated datasets, we can discover that the overall accuracy rate is up to 80.1% by our filling method. Finally, we give one real-life application to illustrate our proposed method.


Introduction
I t is true that we are drowning in uncertain and fuzzy data that are ubiquitous in fields such as business management, banking, environmental governance, industrial engineering, evaluation systems and so on. Soft set theory [1,2] is an excellent solution designed for uncertainty. Since soft sets do not have the problem of setting membership functions, it has been extensively use Dina lot of fields as diverse as information system data analysis, decision making [3][4][5][6][7][8][9][10], resource discovery [11], text classification, data mining [12,13], medical diagnosis and so on.
Soft sets combining with other mathematical models and then initiating new, more powerful tools which deal with uncertainty are the main research trends of soft set. There are many integrated types, such as combining the 2-tuple linguistic representation and soft set [14], the belief interval-valued soft set [15], confidence soft sets [16], the linguistic value soft set [17], dual hesitant fuzzy soft sets [18], the Z-soft fuzzy rough set [19], trapezoidal interval type-2 fuzzy soft sets [20,21], bell-shaped fuzzy soft sets [22], possibility neutrosophic soft sets [23], soft-set-based VIKOR approach [24], hesitant linguistic expression soft sets [25], hesitant N-soft sets [26], interval-valued picture fuzzy soft set [27], Q-neutrosophic soft set [28] , totally dependent-neutrosophic soft sets [29], and soft rough sets [30,31] so on. There are some mentionable and notable extended directions besides the above combined models. In the first place, the fuzzy soft set is a combination of fuzzy set and soft set [32]. Object recognition for inexact data from multiple observers is focused [33], providing a decision scheme on account of the model of fuzzy soft sets. A decision making scheme is given by integrating a grey relational analysis and Dempster-Shafer (D-S) theory of evidence and example. In section IV, aiming to verify our method, we generate randomly thirty datasets which are described by IVIFSS. In section V, one real-life application is given to illustrate our contribution. Finally, section VI draws the conclusion for this article.

Preliminaries
Some basic notions about interval-valued intuitionistic fuzzy soft set theory are described in retrospect.

Relevant Definitions
Definition 1. ([58,59]). An interval-valued intuitionistic fuzzy set on a universe X is an object of the form  [0,1]) ,which are defined as the degree of membership and nonmember ship respectively of the element x to the set A, satisfy the following condition: Let U be an initial universe of objects, E be a set of parameters in relation to objects in U, ( ) U ζ be the set of all interval-valued intuitionistic fuzzy sets of U. The definition of the interval-valued intuitionistic fuzzy soft set is given as follows.
φ is a mapping given by The following example is able to illustrate this theory. In this case, only membership functions cannot fully describe the popularity of mobile phones, and then nonmember ship functions should be offered. The membership and the nonmember ship expressions are very individual; it is not easy to determine the accurate value. This theory solves this conflict. For instance, we should observe that mobile phone h1 is least expensive on the membership degree of 0.32 and it is most expensive on the membership degree of 0.41; mobile phone h1 is not least expensive on the nonmember ship degree of 0.39 and it is not most expensive on the nonmember ship degree of 0.52. Table 1 is a complete interval-valued intuitionistic fuzzy soft set. However, it is an indisputable fact that there are some missing data represented by this model in many cases for a variety of reasons. So we propose data analysis method for an incomplete data representation by means of this model as follows.

Data Analysis Approaches for Incomplete Interval-Valued Intuitionistic Fuzzy Soft Sets
First of all, we give some new related definitions. According to these definitions, we propose the data analysis approaches for incomplete data representation by means of this model. An example is given to illustrate it.

be missing degree of membership and nonmember ship of elements
, respectively. n is the number of objects.
are the number of the missing lower degrees of membership and nonmember ship of are the number of the missing upper degrees of membership and nonmember ship of an ,respectively. m is the number of parameters.

be the missing degree of membership and nonmember ship of elements
as lower and upper membership degree and nonmember ship degree for parameter a ε respectively, where they are formulated as where 1 q is the number of existing membership degrees of the objects.
) as the lower and upper membership degree and nonmember ship degrees for the object b h , where it is formulated as where 2 q is the number of existing membership degrees of the parameters.

Data Analysis Approaches for Incomplete Interval-Valued Intuitionistic Fuzzy Soft Sets
Based on the above definitions, we give our algorithm as follows: , the remainder data which belong to the same row with the missing data are reliable; otherwise, this missing data should be ignored. (d) When the missing value is one of membership degree or nonmember ship degree, for , we fill the missing data by the following equations: a a a a a (16) (f) Finally, we can get a complete interval-valued intuitionistic fuzzy soft set.   Table 2, which has missing values denoted by "*".

One Example for the Proposed Approaches
We apply our proposed method to convert the incomplete interval-valued intuitionistic fuzzy soft set into a complete one.
Step 1: Input the incomplete interval-valued intuitionistic fuzzy soft set ) , ( E ϑ and the parameter set E.
Step2: Find Step3: Compute  Step5: Because there are eight groups of missing data which involve both membership degree and nonmember ship degree, we calculate ) by Equation (16)  ; Finally, we convert this incomplete IVIFSS into one complete IVIFSS which is shown in Table 3.

Experimental Results
In this part, we use specific examples to prove the effectiveness and accuracy of our data filling method. Firstly, we define the deviation rate as follows: where i p is the deviation rate, i q is the actual data value, and i q is the filled data value.
Therefore, the accuracy rate of our filling algorithm is: The average accuracy rate is: where n is the number of data we have filled in.
Aiming to verify our method, we generate randomly thirty datasets which are described by IVIFSS. The first ten datasets involve ten objects and five parameters. The next ten datasets have 50 objects and ten parameters. The last ten datasets have 100 objects and 15 parameters. For every dataset, we set the number of the test missing data as four groups, among which two groups miss the related lower and upper membership degrees and nonmember ship degrees, the other two groups miss one of membership degree and nonmember ship degree. For example, there is one dataset which has ten objects and five parameters displayed in Table 4. The test data are randomly chosen from the initial data set. We randomly choose [ ] as the test missing data.
Next, we use our method to fill in the missing data and get the following results. .
In order to evaluate the accuracy of the predicted data for the missing data, we compute the accuracy rate for all of missing data, respectively. As a result, we obtain that the average accuracy is 95.1%. We repeat this process 10 times on this dataset, in which the missing data is randomly chosen. Finally, we get the average accuracy for this dataset as 93.2%. This above verifying process is made on the thirty datasets. We find that the lowest accuracy rate is 58.8% and the highest one is up to 96.1% on these datasets. In summary, the overall accuracy rate on all of thirty datasets is 80.1%.

One Real-Life Application
In this section, we apply our proposed data analysis methods for incomplete interval-valued intuitionistic fuzzy soft set into one real-life application as follows.
One university held a competition to create the university website for students, aiming to inspire the students' positive regard for the university. The competition committee will reward the best design. Five evaluation indexes are used, such as ''distinct purpose'', ''effective communication'', "good navigation'', ''excellent layouts'' and ''acceptable load time''. The feelings of different evaluators about the seven designed websites from the above five aspects are fuzzy and unclear. So we use the model of the interval-valued intuitionistic fuzzy soft set to express this fuzziness. Let  Table   5. However, because of some reasons, there are some missing data which are not recorded. We have to apply our proposed methods to complete this data set. The process is shown as follows: as missing degrees of membership and nonmember ship of ; the remainder data which belong to the same row with the missing data are reliable.
Step 4: There is one group of missing data which involves one of membership degree or Step5: There are four groups of missing data which involve both membership degree and nonmember ship degree, we calculate

Conclusions
The model of the interval-valued intuitionistic fuzzy soft set has been widely used since it was proposed. In actual data processing, we have to face up to missing data, which leads to unsuccessful and improper applications based on the model of the interval-valued intuitionistic fuzzy soft set. This paper focuses on data analysis methods for an incomplete interval-valued intuitionistic fuzzy soft set. The related filling idea fully considers and employs the characteristics of this model itself. The experimental results verify that the overall accuracy rate on all of thirty randomly generated datasets is up to 80.1% by our filling method. One real-life application illustrates our contribution.

Conflicts of Interest:
The authors declare no conflict of interest.