Semi-Partial Correlation of Fuzzy Sets

In fuzzy data analysis, a simple correlation coefficient provides us with a good sense of linear relationship between two fuzzy attributes, while a partial correlation coefficient shows the relationship between two fuzzy attributes when the influences of other fuzzy attributes are partialed out from both of the two attributes. But, in some practical applications, we need to use the correlation between two fuzzy attributes when the influences of other fuzzy attributes are removed from only one of the two fuzzy attribute, which is called the semi-partial correlation. The simple and partial correlation analyses on Zadeh’s fuzzy sets have been discussed in previous works [3, 4]. Here, we turn to the analysis of semi-partial correlation between two fuzzy attributes. In this paper, the semi-partial correlation coefficient among fuzzy attributes is analyzed and derived using the membership grades of the fuzzy attributes. Data from previous experiments [3, 4] are used to discuss the semi-partial correlation on fuzzy data set. Key-Words: Fuzzy set, Linear relationship, Simple correlation, Partial correlation, Semi-partial correlation 1 Preliminary When we deal with crisp data, it is very common to find the correlation coefficient between attributes. The correlation coefficients defined on ordinary crisp sets have been well discussed in the conventional statistics [1, 2, 5]. With data volumes get larger so rapidly, many data may be fuzzy [6, 7, 8, 9] but useful, waiting to be explored, and methods to investigate these fuzzy data are certainly needed. In fuzzy data analysis, when researchers are concerned with the way which two fuzzy attributes are related to each other, the correlation analyses defined on fuzzy sets are must. A simple correlation coefficient [3] provides us with a very good sense of linear relationship between two fuzzy attributes. It can show not only the strength of the linear relationship between two fuzzy attributes but also the direction of the relationship. The simple correlation coefficient shows a positive score if two fuzzy attributes are directly related, and shows a negative score if the attributes bear an inverse relationship to each other. It will be close to zero if two attributes have no systematic linear relationship to each other. In real applications, attributes other than the two under consideration are also responsible for the observed relationship, and the effects of these fuzzy attributes may influence the relationship between the observed fuzzy attributes. The question then might be asked, “If we can hold other fuzzy attributes constant, what will be the value of the relationship between our interested fuzzy attributes?” Therefore, the analysis of fuzzy partial correlation [4] is thus developed to show the relationship between two fuzzy attributes when the influences of other fuzzy attributes are partialed out from the observed fuzzy attributes. Now we turn to another important variant of correlation analysis which is called the semi-partial correlation. Such a correlation has different interpretation from the pervious ones we have presented. The coefficient of semi-partial correlation can show the relationship between two fuzzy attributes when the influences of other fuzzy attributes are not removed from both of the two interested fuzzy attributes. A semi-partial correlation is more complicated and has larger variety than a partial correlation can explain, which will be analyzed in this paper. Thus, semi-partial correlation on fuzzy sets is considered here, but first, the simple and partial correlation coefficients on fuzzy sets [3, 4] are reminded in Section 2. Definitions for the semipartial correlation coefficients on fuzzy sets are developed in Section 3. In Section 4, data from previous experiments [3, 4] are used to compute the semi-partial correlation coefficients as we defined. The conclusions are then given in Section 5. 2 Simple and Partial Correlations of Fuzzy Sets Proceedings of the 6th WSEAS International Conference on Applied Informatics and Communications, Elounda, Greece, August 18-20, 2006 (pp208-213)