H. Bustince and P. Burillo, "Correlation of interval-valued intuitionistic fuzzy sets", Fuzzy Sets and Systems, 1995, Vol. 74, No. 2, pp. 237--244.

An expert system is a computer system that contains and uses expert knowledge. Expert statements that constitute this knowledge are often formulated in terms of natural language that do not have a precise meaning: e.g., the expertize in controlling a car can be described in terms of the rules of the type "if you are driving fast, and an obstacle is very close, break hard" (the italicized words are not formally defined). The problem with describing the meaning of such words to a computer is that, for some values of velocity, the expert is not 100\% sure whether these values mean "fast" or not. So, to describe the meaning of the term $A$ that characterizes some quantity $q$, we must, for each value of $q$, characterize the expert's degree of belief $\mu_A(q)$ that $q$ satisfies the property $A$.

The simplest way to describe uncertainty is to characterize the expert's degree of belief in $A(q)$ by a number $\mu_A(q)$ from the interval $[0,1]$ (e.g., by asking the expert to describe his/her degree of belief $d$ on a scale from 0 to, say, 10, and then assigning $\mu_A(q)=d/10$). The resulting function $\mu_A$ from the set $Q$ of possible value of $q$ to the interval $[0,1]$ is called a fuzzy set.

If we know the degree of belief $\mu_A(q)$ in $A(q)$, then we can estimate the degree of belief in $\neg A(q)$ as $1-\mu_A(q)$.

A fuzzy set is not a perfect description of expert's uncertainty: e.g., it assigns 0.5 both to the case when we have exactly as many arguments in favor of $A(q)$ and in favor of $\neg A(q)$, and to the case when we do not anything about $A(q)$. To distinguish between these two situations, we can ask an expert to estimate two values: his/her degree of confidence $\mu_A(q)$ in $A(q)$, and degree of confidence $\mu_{\neg A}(q)$ in $\neg A(q)$. In the resulting formalism (called intuitionistic fuzzy logic), the degree of belief in $A(q)$ is characterized by an interval $[\mu_A(q),1-\mu_{\neg A}(q)$] (with $\mu_A(q)\le 1-\mu_{\neg A}(q)$).

This description is better but still not perfect because in reality, it is often difficult for an expert to pinpoint his degree of confidence very narrowly. A more realistic description of an expert's degree of confidence is an interval of possible values $[\mu^-_A(q),\mu^+_A(q)]$. If we only consider the degree of belief in $A$, then we get interval-valued fuzzy sets. If we consider interval degrees of confidence for both $A$ and $\neg A$, then we get an interval-valued intuitionistic fuzzy set, in which the degree of belief is described (using an appropriate interval term) by a twin $$[{\bf m}_A(q),1-{\bf m}_{\neg A}(q)],$$ where ${\bf m}_A(q)=[\mu^-_A(q),\mu^+_A(q)]$ and ${\bf m}_{\neg A}(q)=[\mu^-_{\neg A}(q),\mu^+_{\neg A}(q)]$.

An expert can formulate many rules; the more rules, the slower the resulting expert system. Hence, to speed up the expert system, we must delete the rules that are redundant in the sense that they follow from the others. In particular, if we have two rules "if $A$ then ..." and "if $B$ then ..." with different conditions $A$ and $B$, then, we would like to know to what extent the properties $A$ and $B$ are correlated.

If $A$ and $B$ are both described by fuzzy sets, then, as a degree of correlation, we can use the degree to which $A$ and $B$ occur together, i.e., the degree of belief in $A\& B$. If we use the product to describe $\&$, then, for each $q$, we get the formula $I(A,B,q)=\mu_{A\& B}(q)=\mu_A(q)\cdot \mu_B(q)$. So, as a degree of correlation, we can take the average value of $\mu_{A\& B}(q)$ for all $q$: $I(A,B)=|Q|^{-1}\sum_{q\in Q} I(A,B,q)$. The properties $A$ and $B$ are correlated if $A\approx B$, and hence, if $I(A,B)\approx I(A,A)\approx I(B,B)$. So, we can take $K(A,B)=I(A,B)/\sqrt{I(A,A)\cdot I(B,B)}$ as the correlation coefficient that takes values between 0 and 1, and that is equal to 1 iff $A=B$.

If $A$ and $B$ are described by intuitionistic fuzzy sets, then for every $q$, we have an interval $[\mu_A(q),1-\mu_{\neg A}(q)$] of possible values of degree of belief, and therefore, we get an interval of possible values of $\mu_{A\& B}(q)$: $$[\mu_A(q)\cdot \mu_B(q), (1-\mu_{\neg A}(q))\cdot (1-\mu_{\neg B}(q))].$$ As a numerical value $I(A,B,q)$ of correlation, it is natural to take an average value from this interval, i.e., its midpoint $$I(A,B,q)={1\over 2}\cdot(\mu_A(q)\cdot \mu_B(q)+ (1-\mu_{\neg A}(q))\cdot (1-\mu_{\neg B}(q))).$$

For interval-valued intuitionistic fuzzy sets, we have intervals for $\mu_A(q)$ and for $\mu_B(q)$. As a result, the above formula leads to an interval. Again, to get a numerical values, it is natural to take a midpoint of this interval; this idea leads to the formula $$I(A,B,q)={1\over 4}\cdot(\mu^-_A(q)\cdot \mu^-_B(q)+ \mu^+_A(q)\cdot \mu^+_B(q)+$$ $$(1-\mu^-_{\neg A}(q))\cdot (1-\mu^-_{\neg B}(q))+ (1-\mu^+_{\neg A}(q))\cdot (1-\mu^+_{\neg B}(q))).$$

The paper under review introduces the above definitions and describes the properties of thus defined correlation.