^{1}

^{*}

^{2}

^{*}

^{2}

^{*}

In this paper, a new method for Principal Component Analysis in intuitionistic fuzzy situations has been proposed. This approach is based on cross entropy as an information index. This new method is a useful method for data reduction for situations in which data are not exact. The inexactness in the situations assumed here is due to fuzziness and missing data information, so that we have two functions (membership and non-membership). Thus, method proposed here is suitable for Atanasov’s Intuitionistic Fuzzy Sets (A-IFSs) in which we have an uncertainty due to a mixture of fuzziness and missing data information. For the demonstration of the application of the method, we have used an example and have presented a conclusion.

This study can be very significant for statistical analyzes, since in many situations of data collection not only the data are recorded inexactly, but also they might be incomplete due to data missing. In other words, since this study applies to intuistionistic fuzzy situation, which is a generalized form of the fuzzy situation, it will have significant implications for all fuzzy situations. Intuitionistic fuzzy set [

Principal Component Analysis (PCA) is a favorite statistical method for classification and data reduction. This method is very useful in other sciences too. Szmidt and Kacprzyk introduced a PCA based on Pearson correlation for intuitionistic fuzzy sets [

The paper is divided into 5 main sections. In Section 2, we provide an overview of some of the basic concepts of intuitionistic fuzzy sets which are required for the present study. In Section 3, the definition of cross entropy and its characteristics will be provided. In Section 4, our novel method will be explained and demonstrated through Szmidt and Kacprzyk’s example [

In this section, we will present those aspects of intuitionistic fuzzy sets that are necessary for our discussions. As a generalized form of fuzzy sets, intuitionistic fuzzy sets have two functions: membership function and non- membership function, so that their sum is less that 1, which means that intuitionistic fuzzy sets, unlike fuzzy sets, keeps room for hesitancy such as unanswered items in a questionnaire in social sciences. As shown here, we will use such blank spaces in questionnaires as information in our classification.

An intuitionistic fuzzy set A in reference set X is given by:

where

with the condition that

and

Denote the degree of membership and non-membership of X to A, respectively.

Obviously, each fuzzy set

For each intuitionistic fuzzy set in X, we will call

the intuitionistic index of x in A. It is the hesitancy degree of x to A. It is obvious that

For each fuzzy set

As shown above the hesitancy of the index for the intuitionistic fuzzy sets is meaningful whereas it is null for the fuzzy sets. It is for this reason that it is argued that intuitionistic fuzzy sets is more information-sensitive that fuzzy sets.

In this section, we discus cross entropy for two probability distributions and in the second part we present a complete form of cross entropy for intuitionistic fuzzy sets. Information indices, such as cross entropy and mutual information, are not only information measures but also correlation measures, since they contain the characteristic of distance. The reason we preferred cross entropy over mutual information is that the latter is based on probability functions. For the same reason, mutual information has been used, by other researches, for random variables. Therefore in the next subsection we will focus on cross entropy for probability functions and, then, we will discuss it in regard to fuzzy and intuitionistic fuzzy sets.

In information theory, the cross entropy between two probability distributions measures the average number of bits needed to identity an event from a set of possibilities if a coding is used based on a given probability distribution q rather than the “true” distribution p. Cross entropy is a measure of the divergence of two probability plans based on entropy structure.

The cross entropy for two distributions p and q over the same probability space is defined as follows:

where H(p) is the entropy of p which is defined as

with the possible values of X being

For discrete probability distributions p and q the k-l divergence of q from p is defined as

Regarding the similarities that exist between fuzzy membership function and probability function, Shang and Jiang defined the cross entropy for two fuzzy sets in the same reference set [

Let A and B be two FSs defined on X. Then

is called fuzzy cross entropy, where n is the cardinality of the finite reference set X.

10 years after Shang and Jiang offered the first definition of fuzzy cross entropy; Valachos and Sergiadis modified and generalized the concept of fuzzy cross entropy for intuitionistic fuzzy situations.

Let A and B be two IFSs defined on X. Then

is called intuitionistic fuzzy cross entropy, where n is the cardinality of the finite reference set X.

That Formula is the degree of discrimination and divergence of A from B. However, it is not symmetric [

For two IFSs A and B,

is called a symmetric discrimination information measure for IFSs. It is obviously seen that the three conditions below is satisfied. It should be noted that the definition of discrimination information measure for IFSs is due to the third condition, i.e. the condition of being symmetrical.

After the text edit has been completed, the paper is ready for the template. Duplicate the template file by using the Save As command, and use the naming convention prescribed by your journal for the name of your paper. In this newly created file, highlight all of the contents and import your prepared text file. You are now ready to style your paper.

In this section we solve an example presented in Szmidt and Kacprzyk’s paper through ours new method [

Szmidt and Kacprzyk solved it with correlation matrices [

We recall the cross entropy to solve this problem. We begin by computing the 1-D. The results are given in

The eigenvectors for eigenvalue are given in Tables 3-5.

As we saw, based on the four attributes that were four Atanassov’s intuitionistic fuzzy sets, we managed to obtain the symmetrical form of their cross entropy that formed a matrix showing the divergence of the four A-IFSs. Then we used 1-enteries of the matrix, because we were interested in correlation not divergence. Based on the matrix we obtained the eigenvectors and eigenvalues, with the eigenvectors forming the components of the matrix. And, based on the eigenvalues we obtained the amount of the variation of the data and through that we sorted the components. As you see in

No. | Attributes | Class | |||
---|---|---|---|---|---|

Outlook | Humidity | Windy | Temperature | ||

1 | (0, 0.33, 0.67) | (0, 0.33, 0.67) | (0.2, 0, 0.8) | (0, 0.33, 0.67) | N |

2 | (0, 0.33, 0.67) | (0, 0.33, 0.67) | (0, 0.33, 0.67) | (0, 0.33, 0.67) | N |

3 | (1, 0, 0) | (0, 0.33, 0.67) | (0.2, 0, 0.8) | (0, 0.33, 0.67) | P |

4 | (0.2, 0.11,0.69) | (0, 0.33, 0.67) | (0.2, 0, 0.8) | (0, 0, 1) | P |

5 | (0.2, 0.11,0.69) | (0.6, 0, 0.4) | (0.2, 0, 0.8) | (0.4, 0.11, 0.49) | P |

6 | (0.2, 0.11,0.69) | (0.6, 0, 0.4) | (0, 0.33, 0.67) | (0.4, 0.11, 0.49) | N |

7 | (1, 0, 0) | (0.6, 0, 0.4) | (0, 0.33, 0.67) | (0.4, 0.11, 0.49) | P |

8 | (0, 0.33, 0.67) | (0, 0.33, 0.67) | (0.2, 0, 0.8) | (0, 0, 1) | N |

9 | (0, 0.33, 0.67) | (0.6, 0, 0.4) | (0.2, 0, 0.8) | (0.4, 0.11, 0.49) | P |

10 | (0.2, 0.11,0.69) | (0.6, 0, 0.4) | (0.2, 0, 0.8) | (0, 0, 1) | P |

11 | (0, 0.33, 0.67) | (0.6, 0, 0.4) | (0, 0.33, 0.67) | (0, 0, 1) | P |

12 | (1, 0, 0) | (0, 0.33, 0.67) | (0, 0.33, 0.67) | (0, 0, 1) | P |

13 | (1, 0, 0) | (0.6, 0, 0.4) | (0.2, 0, 0.8) | (0, 0.33, 0.67) | P |

14 | (0.2, 0.11,0.69) | (0, 0.33, 0.67) | (0, 0.33, 0.67) | (0, 0, 1) | N |

Temperature | Windy | Humidity | Outlook | Attribute |
---|---|---|---|---|

0.9351 | 0.9482 | 1.0001 | 1 | Outlook |

0.9557 | 0.9610 | 1.0000 | 1.0001 | Humidity |

1.0073 | 1.0000 | 0.9610 | 0.9482 | Windy |

1.000 | 1.0073 | 0.9557 | 0.9351 | Temperature |

Temperature | Windy | Humidity | Outlook | Attribute |
---|---|---|---|---|

0 | 0 | 0 | 0.0091 | Eig. (1-D) |

0 | 0 | 0.0002 | 0 | |

0 | 0.1054 | 0 | 0 | |

3.9028 | 0 | 0 | 0 |

Vec (1) | Vec (2) | Vec (3) | Vec (4) |
---|---|---|---|

0.2831 | 0.5834 | 0.5763 | 0.4974 |

0.2679 | 0.7092 | 0.4166 | 0.5017 |

0.6560 | 0.3371 | 0.4520 | 0.5016 |

0.6462 | 0.2073 | 0.5385 | 0.4993 |

Components | eig. | %eig. | %eig. cumulative |
---|---|---|---|

C1 | 3.9028 | 97.1474 | 97.1474 |

C2 | 0.1054 | 2.6235 | 99.7709 |

C3 | 0.009 | 0.2240 | 99.9950 |

C4 | 0.0002 | 0.0049 | 100 |

we have replaced cross entropy as an information index for Pearson correlation, and we have tried to improve the method.