Rough Sets

References:

• Z. Pawlak, "Rough Sets: Theoretical Aspects of Reasoning about Data," in Theory and Decision Library: Series D, vol. 9, W. Leinfellner and G. Eberlein eds, Kluwer Academic Publishers, 1991.
• Xiaohua Hu and Nick Cercone, "Learning in Relational Databases: A Rough Set Approach," in Computational Intelligence, v.11, no.2 (1995), pp. 323-338.

Terminology from Rough Set Theory

Information Systems

Formally, an information system is a triple S = (U, A, V) where U = {x₁,...,x_n} is a (finite) set of objects (called the universe), and A is a set of attributes for which each object of the universe has a value (from the domain set V). The set of attributes is partitioned into the condition attributes (C), and the decision attributes (D). In the context of machine learning, there is typically only one decision attribute, which takes a value of true/false to indicate positive examples and negative examples.

Indiscernibility

A pair of objects x_i, x_j are indiscernible over a set of attributes B iff they have the same value for every attribute of B. Obviously, objects that are indiscernible over B will also be indiscernible over any subset of B. However, objects that were discernible over B may become indiscernible over a superset of B. In the context of machine learning, where we want to create simple rules, we will be interested in eliminating "redundant" attributes, which have little or no effect on the discernibility of objects.

Approximation Spaces

Indiscernibility over a set of attributes B defines an equivalence relation on the universe of objects. The lower approximation of a target set X (denoted B_*(X)) is the union of all the equivalence classes that are fully contained in X. The upper approximation of X (denoted B^*(X)) is the union of all equivalence classes that intersect X. Clearly, B_*(X) ⊆ X ⊆ B^*(X).

Consider the information system presented in the following table:

In this case, the objects of the universe are people (Sarah, Dana, Alex, Annie, Emily, Pete, John, Katie). Suppose we wish to describe the set X of people who did get a sunburn (X = {Sarah, Annie, Emily}). When using the set of attributes B = C = {Hair, Height, Weight, Lotion}, we find that each equivalence class has exactly one element, so we can define X precisely as B_*(X) = X = B^*(X) = [Sarah] ∪ [Annie] ∪ [Emily]. If, however, we consider only the attribute Lotion, then there are only two equivalence classes:

• [Sarah] = {Sarah, Annie, Emily, Pete, John}
• [Dana] = {Dana, Alex, Katie}

Reduct of Attributes

In order to simplify the learned rules, we will be interested in eliminating condition attributes. First, we will look for absolute reducts, which are subsets B of the set C of condition attributes such that B preserves the indiscernibility equivalence classes of C. In the "sunburn" example above, we observe that taking B = {Height, Weight, Lotion} (i.e. eliminating Hair) leaves every element of the universe distinct. Other absolute reducts include the following:

• {Hair, Height, Lotion}
• {Hair, Height, Weight}

Consider, however, that for our purposes, absolute reducts are too strict. That is, we do not care if we make formerly discernible objects to become indiscernible, as long as they have the same value for the decision attribute(s). For example, eliminating the attribute Height makes Pete and John indiscernible, but since neither of them got a sunburn, we have not lost any ability to predict a sunburn. Thus, B = {Hair, Weight, Lotion} is a relative reduct of C with respect to D. Notice that Height is in every absolute reduct, but it need not be in a relative reduct. Next, we will discuss these concepts formally.

Positive Regions and Dependency

• Let S = (U, A, V) be an information system, with A = C ∪ D and B ⊆ C.
• Let IND(B) denote the set of equivalence classes of U with respect to B.
• Let IND(D) denote the set of equivalence classes of U with respect to D.

Define the positive region of B in IND(D) as

POS_B(D) = ∪{ B_*(X) : X ∈ IND(D) }.

That is, POS_B(D) includes all objects that can be sorted into classes of IND(D) without error, based solely on the classification information in IND(B).

Furthermore, we say that the set of attributes D depends in degree k on the subset R of C if

k(R, D) = card(POS_R(D)) ÷ card(POS_C(D))

Clearly, k(R, D) ≤ k(C, D) ≤ 1. Also, the following are equivalent to each other:

• k(R, D) = k(C, D).
• R is a reduct of C with respect to D.
• POS_R(D) = POS_C(D).

Computing the Best Reduct

In general, the problem of finding all reducts is NP-hard. Therefore, we will consider a greedy algorithm for finding the "best" reduct.

Now, it is known that there is a (possibly empty) subset of attributes, called the core, which is common to all reducts. Furthermore, it can be shown that CO = CORE(C, D) = {a ∈ C : POS_C(D) ≠ POS_C-{a}(D)}. So, we test every attribute to see if it belongs to the core, then we pass the core to the following algorithm:

1. REDU = CO.
2. AR = C - REDU.
3. While k(REDU, D) < k(C, D) - margin, do the following:
   1. Find attribute a ∈ AR such that k(REDU ∪ {a}, D) is maximized.
   2. Set REDU = REDU ∪ {a}; AR = AR - {a}.
4. Output REDU.

Example

Consider the "sunburn" data above. First, let us compute the core. Every object is discernible, so POS_C(D) = U.

So, CORE(C, D) = CO = {Hair}, and POS_CO(D) = {Alex, Emily, Pete, John}, and k(CO, D) = 4/8 = 0.5. Now we proceed with the algorithm.

The algorithm terminates after one iteration, with the answer REDU = {Hair, Lotion}. The reduced table is as follows:

Learning Rules

So, finally, we are ready to read the rules out of the table:

• (Hair ≠ Brown ∧ Lotion = No) ⇒ Sunburn = Yes
• (Hair = Blonde ∧ Lotion = Yes) ∨ (Hair = Brown) ⇒ Sunburn = No